Mathematics of Complexity and Dynamical Systems

Mathematics of Complexity and Dynamical Systems

This book consists of selections from the Encyclopedia of Complexity and Systems Science edited by Robert A. Meyers, published by Springer New York in 2009.

Robert A. Meyers (Ed.)

Mathematics of Complexity and Dynamical Systems With 489 Figures and 25 Tables

123

ROBERT A. MEYERS, Ph. D. Editor-in-Chief RAMTECH LIMITED 122 Escalle Lane Larkspur, CA 94939 USA [email protected]

Library of Congress Control Number: 2011939016

ISBN: 978-1-4614-1806-1 This publication is available also as: Print publication under ISBN: 978-1-4614-1805-4 and Print and electronic bundle under ISBN 978-1-4614-1807-8 © 2012 SpringerScience+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. This book consists of selections from the Encyclopedia of Complexity and Systems Science edited by Robert A. Meyers, published by Springer New York in 2009. springer.com Printed on acid free paper

Preface

Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. They are therefore adaptive as they evolve and may contain self-driving feedback loops. Thus, complex systems are much more than a sum of their parts. Complex systems are often characterized as having extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The conclusion is that a reductionist (bottom-up) approach is often an incomplete description of a phenomenon. This recognition, that the collective behavior of the whole system cannot be simply inferred from the understanding of the behavior of the individual components, has led to many new concepts and sophisticated mathematical and modeling tools for application to many scientific, engineering, and societal issues that can be adequately described only in terms of complexity and complex systems. This compendium, entitled, Mathematics of Complexity and Dynamical Systems, furnishes mathematical solutions to problems, inherent in chaotic complex dynamical systems that are both descriptive and predictive in the same way that calculus is descriptive of continuously-varying processes. There are 113 articles in this encyclopedic treatment which have been organized into 7 sections each headed by a recognized expert in the field and supported by peer reviewers, and also a section of three articles selected by the editor. The sections are:        

Ergodic Theory Fractals and Multifractals Non-Linear Ordinary Differential Equations and Dynamical Systems Non-Linear Partial Differential Equations Perturbation Theory Solitons Systems and Control Theory Three EiC Selections: Catastrophe Theory; Infinite Dimensional Controllability; Philosophy of Science, Mathematical Models In

Descriptions of each of the sections is presented below, together with example articles of exceptional relevance in defining the mathematics of chaotic and dynamical systems.  Ergodic Theory lies at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Problems, techniques, and results are related to many other areas of mathematics, and ergodic theory has had applications both within mathematics and to numerous other branches of science. Ergodic theory has particularly strong overlap with other branches of dynamical systems (e.g. see articles on  Chaos and Ergodic Theory,  Entropy in Ergodic Theory,  Ergodic Theory: Recurrence and  Kolmogorov–Arnold–Moser (KAM) Theory).  Fractals and Multifractals Fractals generalize Euclidean geometrical objects to non-integer dimensions and allow us, for the first time, to delve into the study of complex systems, disorder, and chaos (e.g. see articles on  Fractals Meet Chaos,  Dynamics on Fractals).  Non-linear Ordinary Differential Equations are n-dimensional ODEs of the form dy/dx D F(x; y) that are not linear. Poincaré developed both quantitative and qualitative methods and his approach has shaped the analysis of non-linear

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Preface









ODEs in the period that followed becoming part of the topic of dynamical systems (see article on  Lyapunov– Schmidt Method for Dynamical Systems). Non-Linear Partial Differential Equations include the Euler and Navier–Stokes equations in fluid dynamics and the Boltzmann equation in gas dynamics. Other fundamental models include reaction-diffusion, porous media, nonlinear Schrödinger, Klein–Gordon, Burger and conservation laws, nonlinear wave Korteweg–de Vries (e.g. see articles on  Hamilton–Jacobi Equations and Weak KAM Theory, and  Scaling Limits of Large Systems of Non-linear Partial Differential Equations). Perturbation Theory involves nonlinear mathematical problems which are not exactly solvable–either due to an inherent impossibility or due to insufficient mathematics. Perturbation theory is often the only way to approach realistic nonlinear systems (e.g. see  Hamiltonian Perturbation Theory (and Transition to Chaos),  Kolmogorov– Arnold–Moser (KAM) Theory, and  Non-linear Dynamics, Symmetry and Perturbation Theory in). Solitons are spatially localized waves in a medium that can interact strongly with other solitons but will afterwards regain original form (e.g. see articles on  Partial Differential Equations that Lead to Solitons,  Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, and  Solitons and Compactons). Systems and Control Theory has a simple and elegant answer in the case of linear systems. However, while much also is known for nonlinear systems, many challenges remain in both the finite dimensional setting (see  Finite Dimensional Controllability) and in the setting of distributed systems modeled by partial differential equations (see  Control of Non-linear Partial Differential Equations).

The complete listing of articles and section editors is presented on pages VII to IX. The articles are written for an audience of advanced university undergraduate and graduate students, professors, and professionals in a wide range of fields who must manage complexity on scales ranging from the atomic and molecular to the societal and global. Each article was selected and peer reviewed by one of our 8 Section Editors with advice and consultation provided by our Board Members: Pierre-Louis Lions, Benoit Mandelbrot, Stephen Wolfram, Jerrod Marsden and Joseph Kung; and the Editor-in-Chief. This level of coordination assures that the reader can have a level of confidence in the relevance and accuracy of the information far exceeding that generally found on the World Wide Web. Accessibility is also a priority and for this reason each article includes a glossary of important terms and a concise definition of the subject. Robert A. Meyers Editor-in-Chief Larkspur, California July 2011

Sections

EiC Selections, Section Editor: Robert A. Meyers Catastrophe Theory Infinite Dimensional Controllability Philosophy of Science, Mathematical Models in Ergodic Theory, Section Editor: Bryna Kra Chaos and Ergodic Theory Entropy in Ergodic Theory Ergodic Theorems Ergodic Theory on Homogeneous Spaces and Metric Number Theory Ergodic Theory, Introduction to Ergodic Theory: Basic Examples and Constructions Ergodic Theory: Fractal Geometry Ergodic Theory: Interactions with Combinatorics and Number Theory Ergodic Theory: Non-singular Transformations Ergodic Theory: Recurrence Ergodic Theory: Rigidity Ergodicity and Mixing Properties Isomorphism Theory in Ergodic Theory Joinings in Ergodic Theory Measure Preserving Systems Pressure and Equilibrium States in Ergodic Theory Smooth Ergodic Theory Spectral Theory of Dynamical Systems Symbolic Dynamics Topological Dynamics Fractals and Multifractals, Section Editor: Daniel ben-Avraham and Shlomo Havlin Anomalous Diffusion on Fractal Networks Dynamics on Fractals Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks Fractal and Multifractal Time Series Fractal and Transfractal Scale-Free Networks Fractal Geometry, A Brief Introduction to Fractal Growth Processes Fractal Structures in Condensed Matter Physics Fractals and Economics Fractals and Multifractals, Introduction to Fractals and Percolation

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Sections

Fractals and Wavelets: What can we Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences? Fractals in Biology Fractals in Geology and Geophysics Fractals in the Quantum Theory of Spacetime Fractals Meet Chaos Phase Transitions on Fractals and Networks Reaction Kinetics in Fractals Non-Linear Ordinary Differential Equations and Dynamical Systems, Section Editor: Ferdinand Verhulst Center Manifolds Dynamics of Parametric Excitation Existence and Uniqueness of Solutions of Initial Value Problems Hyperbolic Dynamical Systems Lyapunov–Schmidt Method for Dynamical Systems Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to Numerical Bifurcation Analysis Periodic Orbits of Hamiltonian Systems Periodic Solutions of Non-autonomous Ordinary Differential Equations Relaxation Oscillations Stability Theory of Ordinary Differential Equations Non-Linear Partial Differential Equations, Section Editor: Italo Capuzzo Dolcetta Biological Fluid Dynamics, Non-linear Partial Differential Equations Control of Nonlinear Partial Differential Equations Dispersion Phenomena in Partial Differential Equations Hamilton-Jacobi Equations and Weak KAM Theory Hyperbolic Conservation Laws Navier-Stokes Equations: A Mathematical Analysis Non-linear Partial Differential Equations, Introduction to Non-linear Partial Differential Equations, Viscosity Solution Method in Non-linear Stochastic Partial Differential Equations Scaling Limits of Large Systems of Nonlinear Partial Differential Equations Vehicular Traffic: A Review of Continuum Mathematical Models Perturbation Theory, Section Editor: Giuseppe Gaeta Diagrammatic Methods in Classical Perturbation Theory Hamiltonian Perturbation Theory (and Transition to Chaos) Kolmogorov-Arnold-Moser (KAM) Theory n-Body Problem and Choreographies Nekhoroshev Theory Non-linear Dynamics, Symmetry and Perturbation Theory in Normal Forms in Perturbation Theory Perturbation Analysis of Parametric Resonance Perturbation of Equilibria in the Mathematical Theory of Evolution Perturbation of Systems with Nilpotent Real Part Perturbation Theory Perturbation Theory and Molecular Dynamics Perturbation Theory for Non-smooth Systems Perturbation Theory for PDEs

Sections

Perturbation Theory in Celestial Mechanics Perturbation Theory in Quantum Mechanics Perturbation Theory, Introduction to Perturbation Theory, Semiclassical Perturbative Expansions, Convergence of Quantum Bifurcations Solitons, Section Editor: Mohamed A. Helal Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory Inverse Scattering Transform and the Theory of Solitons Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the Korteweg–de Vries Equation (KdV) History, Exact N-Soliton Solutions and Further Properties Non-linear Internal Waves Partial Differential Equations that Lead to Solitons Shallow Water Waves and Solitary Waves Soliton Perturbation Solitons and Compactons Solitons Interactions Solitons, Introduction to Solitons, Tsunamis and Oceanographical Applications of Solitons: Historical and Physical Introduction Water Waves and the Korteweg–de Vries Equation Systems and Control Theory, Section Editor: Matthias Kawski Chronological Calculus in Systems and Control Theory Discrete Control Systems Finite Dimensional Controllability Hybrid Control Systems Learning, System Identification, and Complexity Maximum Principle in Optimal Control Mechanical Systems: Symmetries and Reduction Nonsmooth Analysis in Systems and Control Theory Observability (Deterministic Systems) and Realization Theory Robotic Networks, Distributed Algorithms for Stability and Feedback Stabilization Stochastic Noises, Observation, Identification and Realization with System Regulation and Design, Geometric and Algebraic Methods in Systems and Control, Introduction to

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About the Editor-in-Chief

Robert A. Meyers President: RAMTECH Limited Manager, Chemical Process Technology, TRW Inc. Post-doctoral Fellow: California Institute of Technology Ph. D. Chemistry, University of California at Los Angeles B. A., Chemistry, California State University, San Diego

Biography Dr. Meyers has worked with more than 25 Nobel laureates during his career. Research Dr. Meyers was Manager of Chemical Technology at TRW (now Northrop Grumman) in Redondo Beach, CA and is now President of RAMTECH Limited. He is co-inventor of the Gravimelt process for desulfurization and demineralization of coal for air pollution and water pollution control. Dr. Meyers is the inventor of and was project manager for the DOE-sponsored Magnetohydrodynamics Seed Regeneration Project which has resulted in the construction and successful operation of a pilot plant for production of potassium formate, a chemical utilized for plasma electricity generation and air pollution control. Dr. Meyers managed the pilot-scale DoE project for determining the hydrodynamics of synthetic fuels. He is a co-inventor of several thermo-oxidative stable polymers which have achieved commercial success as the GE PEI, Upjohn Polyimides and Rhone-Polenc bismaleimide resins. He has also managed projects for photochemistry, chemical lasers, flue gas scrubbing, oil shale analysis and refining, petroleum analysis and refining, global change measurement from space satellites, analysis and mitigation (carbon dioxide and ozone), hydrometallurgical refining, soil and hazardous waste remediation, novel polymers synthesis, modeling of the economics of space transportation systems, space rigidizable structures and chemiluminescence-based devices. He is a senior member of the American Institute of Chemical Engineers, member of the American Physical Society, member of the American Chemical Society and serves on the UCLA Chemistry Department Advisory Board. He was a member of the joint USA-Russia working group on air pollution control and the EPA-sponsored Waste Reduction Institute for Scientists and Engineers.

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About the Editor-in-Chief

Dr. Meyers has more than 20 patents and 50 technical papers. He has published in primary literature journals including Science and the Journal of the American Chemical Society, and is listed in Who’s Who in America and Who’s Who in the World. Dr. Meyers’ scientific achievements have been reviewed in feature articles in the popular press in publications such as The New York Times Science Supplement and The Wall Street Journal as well as more specialized publications such as Chemical Engineering and Coal Age. A public service film was produced by the Environmental Protection Agency of Dr. Meyers’ chemical desulfurization invention for air pollution control. Scientific Books Dr. Meyers is the author or Editor-in-Chief of 12 technical books one of which won the Association of American Publishers Award as the best book in technology and engineering. Encyclopedias Dr. Meyers conceived and has served as Editor-in-Chief of the Academic Press (now Elsevier) Encyclopedia of Physical Science and Technology. This is an 18-volume publication of 780 twenty-page articles written to an audience of university students and practicing professionals. This encyclopedia, first published in 1987, was very successful, and because of this, was revised and reissued in 1992 as a second edition. The Third Edition was published in 2001 and is now online. Dr. Meyers has completed two editions of the Encyclopedia of Molecular Cell Biology and Molecular Medicine for Wiley VCH publishers (1995 and 2004). These cover molecular and cellular level genetics, biochemistry, pharmacology, diseases and structure determination as well as cell biology. His eight-volume Encyclopedia of Environmental Analysis and Remediation was published in 1998 by John Wiley & Sons and his 15-volume Encyclopedia of Analytical Chemistry was published in 2000, also by John Wiley & Sons, all of which are available on-line.

Editorial Board Members

PIERRE-LOUIS LIONS 1994 Fields Medal Current interests include: nonlinear partial differential equations and applications

STEPHEN W OLFRAM Founder and CEO, Wolfram Research Creator, Mathematica® Author, A New Kind of Science

PROFESSOR BENOIT B. MANDELBROT Sterling Professor Emeritus of Mathematical Sciences at Yale University 1993 Wolf Prize for Physics and the 2003 Japan Prize for Science and Technology Current interests include: seeking a measure of order in physical, mathematical or social phenomena that are characterized by abundant data but wild variability.

JERROLD E. MARSDEN Professor of Control & Dynamical Systems California Institute of Technology

JOSEPH P. S. KUNG Professor Department of Mathematics University of North Texas

Section Editors

Ergodic Theory

Fractals and Multifractals

BRYNA KRA Professor Department of Mathematics Northwestern University

SHLOMO HAVLIN Professor Department of Physics Bar Ilan University and

DANIEL BEN-AVRAHAM Professor Department of Physics Clarkson University

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Section Editors

Non-linear Ordinary Differential Equations and Dynamical Systems

FERDINAND VERHULST Professor Mathematisch Institut University of Utrecht

Solitons

MOHAMED A. HELAL Professor Department of Mathematics Faculty of Science University of Cairo

Non-linear Partial Differential Equations

Systems and Control Theory

ITALO CAPUZZO DOLCETTA Professor Dipartimento di Matematica “Guido Castelnuovo” Università Roma La Sapienza

MATTHIAS KAWSKI Professor, Department of Mathematics and Statistics Arizona State University

Perturbation Theory

GIUSEPPE GAETA Professor in Mathematical Physics Dipartimento di Matematica Università di Milano, Italy

Table of Contents

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory Abdul-Majid Wazwaz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Anomalous Diffusion on Fractal Networks Igor M. Sokolov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Biological Fluid Dynamics, Non-linear Partial Differential Equations Antonio DeSimone, François Alouges, Aline Lefebvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Catastrophe Theory Werner Sanns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Center Manifolds George Osipenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Chaos and Ergodic Theory Jérôme Buzzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Chronological Calculus in Systems and Control Theory Matthias Kawski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Control of Non-linear Partial Differential Equations Fatiha Alabau-Boussouira, Piermarco Cannarsa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

Diagrammatic Methods in Classical Perturbation Theory Guido Gentile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

Discrete Control Systems Taeyoung Lee, Melvin Leok, Harris McClamroch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

Dispersion Phenomena in Partial Differential Equations Piero D’Ancona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

Dynamics on Fractals Raymond L. Orbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

Dynamics of Parametric Excitation Alan Champneys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

Entropy in Ergodic Theory Jonathan L. F. King . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

Ergodicity and Mixing Properties Anthony Quas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225

Ergodic Theorems Andrés del Junco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

Ergodic Theory: Basic Examples and Constructions Matthew Nicol, Karl Petersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

264

Ergodic Theory: Fractal Geometry Jörg Schmeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

288

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Table of Contents

Ergodic Theory on Homogeneous Spaces and Metric Number Theory Dmitry Kleinbock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302

Ergodic Theory: Interactions with Combinatorics and Number Theory Tom Ward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313

Ergodic Theory, Introduction to Bryna Kra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

Ergodic Theory: Non-singular Transformations Alexandre I. Danilenko, Cesar E. Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329

Ergodic Theory: Recurrence Nikos Frantzikinakis, Randall McCutcheon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

Ergodic Theory: Rigidity Viorel Ni¸tic˘a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369

Existence and Uniqueness of Solutions of Initial Value Problems Gianne Derks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383

Finite Dimensional Controllability Lionel Rosier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395

Fractal Geometry, A Brief Introduction to Armin Bunde, Shlomo Havlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409

Fractal Growth Processes Leonard M. Sander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks Sidney Redner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

446

Fractal and Multifractal Time Series Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

Fractals in Biology Sergey V. Buldyrev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

488

Fractals and Economics Misako Takayasu, Hideki Takayasu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

512

Fractals in Geology and Geophysics Donald L. Turcotte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

532

Fractals Meet Chaos Tony Crilly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537

Fractals and Multifractals, Introduction to Daniel ben-Avraham, Shlomo Havlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

557

Fractals and Percolation Yakov M. Strelniker, Shlomo Havlin, Armin Bunde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

559

Fractals in the Quantum Theory of Spacetime Laurent Nottale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

571

Fractal Structures in Condensed Matter Physics Tsuneyoshi Nakayama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

591

Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences? Alain Arneodo, Benjamin Audit, Edward-Benedict Brodie of Brodie, Samuel Nicolay, Marie Touchon, Yves d’Aubenton-Carafa, Maxime Huvet, Claude Thermes . . . . . . . . . . . . . . . . . . . . . . . . . . .

606

Fractal and Transfractal Scale-Free Networks Hernán D. Rozenfeld, Lazaros K. Gallos, Chaoming Song, Hernán A. Makse . . . . . . . . . . . . . . . . .

637

Table of Contents

Hamiltonian Perturbation Theory (and Transition to Chaos) Henk W. Broer, Heinz Hanßmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

657

Hamilton–Jacobi Equations and Weak KAM Theory Antonio Siconolfi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

683

Hybrid Control Systems Andrew R. Teel, Ricardo G. Sanfelice, Rafal Goebel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

704

Hyperbolic Conservation Laws Alberto Bressan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

729

Hyperbolic Dynamical Systems Vitor Araújo, Marcelo Viana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

740

Infinite Dimensional Controllability Olivier Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

755

Inverse Scattering Transform and the Theory of Solitons Tuncay Aktosun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

771

Isomorphism Theory in Ergodic Theory Christopher Hoffman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

783

Joinings in Ergodic Theory Thierry de la Rue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

796

Kolmogorov–Arnold–Moser (KAM) Theory Luigi Chierchia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

810

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the Yu-Jie Ren . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

837

Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the Yi Zang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

884

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the Do˘gan Kaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

890

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the Mustafa Inc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

908

Learning, System Identification, and Complexity M. Vidyasagar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

924

Lyapunov–Schmidt Method for Dynamical Systems André Vanderbauwhede . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

937

Maximum Principle in Optimal Control Velimir Jurdjevic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

953

Measure Preserving Systems Karl Petersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

964

Mechanical Systems: Symmetries and Reduction Jerrold E. Marsden, Tudor S. Ratiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

981

Navier–Stokes Equations: A Mathematical Analysis Giovanni P. Galdi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1009

n-Body Problem and Choreographies Susanna Terracini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1043

Nekhoroshev Theory Laurent Niederman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1070

Non-linear Dynamics, Symmetry and Perturbation Theory in Giuseppe Gaeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1082

XIX

XX

Table of Contents

Non-linear Internal Waves Moustafa S. Abou-Dina, Mohamed A. Helal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1102

Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to Ferdinand Verhulst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1111

Non-linear Partial Differential Equations, Introduction to Italo Capuzzo Dolcetta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1113

Non-linear Partial Differential Equations, Viscosity Solution Method in Shigeaki Koike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1115

Non-linear Stochastic Partial Differential Equations Giuseppe Da Prato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1126

Nonsmooth Analysis in Systems and Control Theory Francis Clarke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1137

Normal Forms in Perturbation Theory Henk W. Broer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1152

Numerical Bifurcation Analysis Hil Meijer, Fabio Dercole, Bart Oldeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1172

Observability (Deterministic Systems) and Realization Theory Jean-Paul André Gauthier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1195

Partial Differential Equations that Lead to Solitons Do˘gan Kaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1205

Periodic Orbits of Hamiltonian Systems Luca Sbano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1212

Periodic Solutions of Non-autonomous Ordinary Differential Equations Jean Mawhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1236

Perturbation Analysis of Parametric Resonance Ferdinand Verhulst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1251

Perturbation of Equilibria in the Mathematical Theory of Evolution Angel Sánchez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1265

Perturbation of Systems with Nilpotent Real Part Todor Gramchev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1276

Perturbation Theory Giovanni Gallavotti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1290

Perturbation Theory in Celestial Mechanics Alessandra Celletti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1301

Perturbation Theory, Introduction to Giuseppe Gaeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1314

Perturbation Theory and Molecular Dynamics Gianluca Panati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1317

Perturbation Theory for Non-smooth Systems Marco Antônio Teixeira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1325

Perturbation Theory for PDEs Dario Bambusi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1337

Perturbation Theory in Quantum Mechanics Luigi E. Picasso, Luciano Bracci, Emilio d’Emilio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1351

Perturbation Theory, Semiclassical Andrea Sacchetti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1376

Table of Contents

Perturbative Expansions, Convergence of Sebastian Walcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1389

Phase Transitions on Fractals and Networks Dietrich Stauffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1400

Philosophy of Science, Mathematical Models in Zoltan Domotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1407

Pressure and Equilibrium States in Ergodic Theory Jean-René Chazottes, Gerhard Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1422

Quantum Bifurcations Boris Zhilinskií . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1438

Reaction Kinetics in Fractals Ezequiel V. Albano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1457

Relaxation Oscillations Johan Grasman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1475

Robotic Networks, Distributed Algorithms for Francesco Bullo, Jorge Cortés, Sonia Martínez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1489

Scaling Limits of Large Systems of Non-linear Partial Differential Equations D. Benedetto, M. Pulvirenti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1505

Shallow Water Waves and Solitary Waves Willy Hereman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1520

Smooth Ergodic Theory Amie Wilkinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1533

Soliton Perturbation Ji-Huan He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1548

Solitons and Compactons Ji-Huan He, Shun-dong Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1553

Solitons: Historical and Physical Introduction François Marin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1561

Solitons Interactions Tarmo Soomere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1576

Solitons, Introduction to Mohamed A. Helal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1601

Solitons, Tsunamis and Oceanographical Applications of M. Lakshmanan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1603

Spectral Theory of Dynamical Systems Mariusz Lema´nczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1618

Stability and Feedback Stabilization Eduardo D. Sontag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1639

Stability Theory of Ordinary Differential Equations Carmen Chicone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1653

Stochastic Noises, Observation, Identification and Realization with Giorgio Picci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1672

Symbolic Dynamics Brian Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1689

System Regulation and Design, Geometric and Algebraic Methods in Alberto Isidori, Lorenzo Marconi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1711

XXI

XXII

Table of Contents

Systems and Control, Introduction to Matthias Kawski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1724

Topological Dynamics Ethan Akin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1726

Vehicular Traffic: A Review of Continuum Mathematical Models Benedetto Piccoli, Andrea Tosin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1748

Water Waves and the Korteweg–de Vries Equation Lokenath Debnath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1771

List of Glossary Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1811

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1819

Contributors

ABOU-DINA , MOUSTAFA S. Cairo University Giza Egypt

AUDIT, BENJAMIN ENS-Lyon CNRS Lyon Cedex France

AKIN, ETHAN The City College New York City USA

BAMBUSI , DARIO Università degli Studi di Milano Milano Italia

AKTOSUN, TUNCAY University of Texas at Arlington Arlington USA

BEN -AVRAHAM, D ANIEL Clarkson University Potsdam USA

ALABAU-BOUSSOUIRA , FATIHA Université de Metz Metz France

BENEDETTO, D. Dipartimento di Matematica, Università di Roma ‘La Sapienza’ Roma Italy

ALBANO, EZEQUIEL V. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA) CCT La Plata La Plata Argentina ALOUGES, FRANÇOIS Université Paris-Sud Orsay cedex France ARAÚJO, VITOR CMUP Porto Portugal IM-UFRJ Rio de Janeiro Brazil ARNEODO, ALAIN ENS-Lyon CNRS Lyon Cedex France

BRACCI , LUCIANO Università di Pisa Pisa Italy Sezione di Pisa Pisa Italy BRESSAN, ALBERTO Penn State University University Park USA BRODIE OF BRODIE, EDWARD-BENEDICT ENS-Lyon CNRS Lyon Cedex France BROER, HENK W. University of Groningen Groningen The Netherlands

XXIV

Contributors

BULDYREV, SERGEY V. Yeshiva University New York USA

CORTÉS, JORGE University of California San Diego USA

BULLO, FRANCESCO University of California Santa Barbara USA

CRILLY, TONY Middlesex University London UK

BUNDE, ARMIN Justus-Liebig-Universität Giessen Germany BUZZI , JÉRÔME C.N.R.S. and Université Paris-Sud Orsay France CANNARSA , PIERMARCO Università di Roma “Tor Vergata” Rome Italy CELLETTI , ALESSANDRA Università di Roma Tor Vergata Roma Italy CHAMPNEYS, ALAN University of Bristol Bristol United Kingdom CHAZOTTES, JEAN-RENÉ CNRS/École Polytechnique Palaiseau France CHICONE, CARMEN University of Missouri-Columbia Columbia USA

D’ANCONA , PIERO Unversità di Roma “La Sapienza” Roma Italy DANILENKO, ALEXANDRE I. Ukrainian National Academy of Sciences Kharkov Ukraine DA PRATO, GIUSEPPE Scuola Normale Superiore Pisa Italy D’AUBENTON -CARAFA , Y VES CNRS Gif-sur-Yvette France

DEBNATH, LOKENATH University of Texas – Pan American Edinburg USA DE LA R UE , T HIERRY CNRS – Université de Rouen Saint Étienne du Rouvray France

CHIERCHIA , LUIGI Università “Roma Tre” Roma Italy

D’E MILIO, E MILIO Università di Pisa Pisa Italy Sezione di Pisa Pisa Italy

CLARKE, FRANCIS Institut universitaire de France et Université de Lyon Lyon France

DERCOLE, FABIO Politecnico di Milano Milano Italy

Contributors

DERKS, GIANNE University of Surrey Guildford UK

GLASS, OLIVIER Université Pierre et Marie Curie Paris France

DESIMONE, ANTONIO SISSA-International School for Advanced Studies Trieste Italy

GOEBEL, RAFAL Loyola University Chicago USA

DOLCETTA , ITALO CAPUZZO Sapienza Universita’ di Roma Rome Italy

GRAMCHEV, TODOR Università di Cagliari Cagliari Italy

DOMOTOR, Z OLTAN University of Pennsylvania Philadelphia USA

GRASMAN, JOHAN Wageningen University and Research Centre Wageningen The Netherlands

FRANTZIKINAKIS, N IKOS University of Memphis Memphis USA

HANSSMANN, HEINZ Universiteit Utrecht Utrecht The Netherlands

GAETA , GIUSEPPE Università di Milano Milan Italy

HAVLIN, SHLOMO Bar–Ilan University Ramat–Gan Israel

GALDI , GIOVANNI P. University of Pittsburgh Pittsburgh USA

HE, JI -HUAN Donghua University Shanghai China

GALLAVOTTI , GIOVANNI Università di Roma I “La Sapienza” Roma Italy

HELAL, MOHAMED A. Cairo University Giza Egypt

GALLOS, LAZAROS K. City College of New York New York USA

HEREMAN, W ILLY Colorado School of Mines Golden USA

GAUTHIER, JEAN-PAUL ANDRÉ University of Toulon Toulon France

HOFFMAN, CHRISTOPHER University of Washington Seattle USA

GENTILE, GUIDO Università di Roma Tre Roma Italy

HUVET, MAXIME CNRS Gif-sur-Yvette France

XXV

XXVI

Contributors

INC, MUSTAFA Fırat University Elazı˘g Turkey

KRA , BRYNA Northwestern University Evanston USA

ISIDORI , ALBERTO University of Rome La Sapienza Italy

LAKSHMANAN, M. Bharathidasan University Tiruchirapalli India

JUNCO, ANDRÉS DEL University of Toronto Toronto Canada

LEE, TAEYOUNG University of Michigan Ann Arbor USA

JURDJEVIC, VELIMIR University of Toronto Toronto Canada

LEFEBVRE, ALINE Université Paris-Sud Orsay cedex France

KANTELHARDT, JAN W. Martin-Luther-University Halle-Wittenberg Halle Germany

´ , MARIUSZ LEMA NCZYK Nicolaus Copernicus University Toru´n Poland

KAWSKI , MATTHIAS Arizona State University Tempe USA

LEOK, MELVIN Purdue University West Lafayette USA

˘ KAYA , DO GAN Firat University Elazig Turkey

MAKSE, HERNÁN A. City College of New York New York USA

KELLER, GERHARD Universität Erlangen-Nürnberg Erlangen Germany

MARCONI , LORENZO University of Bologna Bologna Italy

KING, JONATHAN L. F. University of Florida Gainesville USA

MARCUS, BRIAN University of British Columbia Vancouver Canada

KLEINBOCK, DMITRY Brandeis University Waltham USA

MARIN, FRANÇOIS Laboratoire Ondes et Milieux Complexes, Fre CNRS 3102 Le Havre Cedex France

KOIKE, SHIGEAKI Saitama University Saitama Japan

MARSDEN, JERROLD E. California Institute of Technology Pasadena USA

Contributors

MARTÍNEZ, SONIA University of California San Diego USA

N OTTALE, LAURENT Paris Observatory and Paris Diderot University Paris France

MAWHIN, JEAN Université Catholique de Louvain Maryland USA

OLDEMAN, BART Concordia University Montreal Canada

MCCLAMROCH, HARRIS University of Michigan Ann Arbor USA

ORBACH, RAYMOND L. University of California Riverside USA

MCCUTCHEON, RANDALL University of Memphis Memphis USA MEIJER, HIL University of Twente Enschede The Netherlands N AKAYAMA , TSUNEYOSHI Toyota Physical and Chemical Research Institute Nagakute Japan N ICOLAY, SAMUEL Université de Liège Liège Belgium N ICOL, MATTHEW University of Houston Houston USA N IEDERMAN, LAURENT Université Paris Paris France IMCCE Paris France ˘ , VIOREL N I TIC ¸ A West Chester University West Chester USA Institute of Mathematics Bucharest Romania

OSIPENKO, GEORGE State Polytechnic University St. Petersburg Russia PANATI , GIANLUCA Università di Roma “La Sapienza” Roma Italy PETERSEN, KARL University of North Carolina Chapel Hill USA PICASSO, LUIGI E. Università di Pisa Pisa Italy Sezione di Pisa Pisa Italy PICCI , GIORGIO University of Padua Padua Italy PICCOLI , BENEDETTO Consiglio Nazionale delle Ricerche Rome Italy PULVIRENTI , M. Dipartimento di Matematica, Università di Roma ‘La Sapienza’ Roma Italy

XXVII

XXVIII

Contributors

QUAS, ANTHONY University of Victoria Victoria Canada

SANFELICE, RICARDO G. University of Arizona Tucson USA

RATIU, TUDOR S. École Polytechnique Fédérale de Lausanne Lausanne Switzerland

SANNS, W ERNER University of Applied Sciences Darmstadt Germany

REDNER, SIDNEY Boston University Boston USA

SBANO, LUCA University of Warwick Warwick UK

REN, YU-JIE Shantou University Shantou People’s Republic of China Dalian Polytechnic University Dalian People’s Republic of China Beijing Institute of Applied Physics and Computational Mathematics Beijing People’s Republic of China ROSIER, LIONEL Institut Elie Cartan Vandoeuvre-lès-Nancy France ROZENFELD, HERNÁN D. City College of New York New York USA SACCHETTI , ANDREA Universitá di Modena e Reggio Emilia Modena Italy SÁNCHEZ, ANGEL Universidad Carlos III de Madrid Madrid Spain Universidad de Zaragoza Zaragoza Spain SANDER, LEONARD M. The University of Michigan Ann Arbor USA

SCHMELING, JÖRG Lund University Lund Sweden SICONOLFI , ANTONIO “La Sapienza” Università di Roma Roma Italy SILVA , CESAR E. Williams College Williamstown USA SOKOLOV, IGOR M. Humboldt-Universität zu Berlin Berlin Germany SONG, CHAOMING City College of New York New York USA SONTAG, EDUARDO D. Rutgers University New Brunswick USA SOOMERE, TARMO Tallinn University of Technology Tallinn Estonia STAUFFER, DIETRICH Cologne University Köln Germany

Contributors

STRELNIKER, YAKOV M. Bar–Ilan University Ramat–Gan Israel

VANDERBAUWHEDE, ANDRÉ Ghent University Gent Belgium

TAKAYASU, HIDEKI Sony Computer Science Laboratories Inc Tokyo Japan

VERHULST, FERDINAND University of Utrecht Utrecht The Netherlands

TAKAYASU, MISAKO Tokyo Institute of Technology Tokyo Japan

VIANA , MARCELO IMPA Rio de Janeiro Brazil

TEEL, ANDREW R. University of California Santa Barbara USA TEIXEIRA , MARCO ANTÔNIO Universidade Estadual de Campinas Campinas Brazil TERRACINI , SUSANNA Università di Milano Bicocca Milano Italia THERMES, CLAUDE CNRS Gif-sur-Yvette France TOSIN, ANDREA Consiglio Nazionale delle Ricerche Rome Italy TOUCHON, MARIE CNRS Paris France Université Pierre et Marie Curie Paris France TURCOTTE, DONALD L. University of California Davis USA

VIDYASAGAR, M. Software Units Layout Hyderabad India W ALCHER, SEBASTIAN RWTH Aachen Aachen Germany W ARD, TOM University of East Anglia Norwich UK W AZWAZ, ABDUL-MAJID Saint Xavier University Chicago USA W ILKINSON, AMIE Northwestern University Evanston USA Z ANG, YI Zhejiang Normal University Jinhua China Z HILINSKIÍ , BORIS Université du Littoral Dunkerque France Z HU, SHUN-DONG Zhejiang Lishui University Lishui China

XXIX

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory ABDUL-MAJID W AZWAZ Department of Mathematics, Saint Xavier University, Chicago, USA Article Outline Glossary Definition of the Subject Introduction Adomian Decomposition Method and Adomian Polynomials Modified Decomposition Method and Noise Terms Phenomenon Solitons, Peakons, Kinks, and Compactons Solitons of the KdV Equation Kinks of the Burgers Equation Peakons of the Camassa–Holm Equation Compactons of the K(n,n) Equation Future Directions Bibliography Glossary Solitons Solitons appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The solitons are localized highly stable waves that retains its identity: shape and speed, upon interaction, and resemble particle like behavior. In the case of a collision, solitons undergo a phase shift. Types of solitons Solitary waves, which are localized traveling waves, are asymptotically zero at large distances and appear in many structures such as solitons, kinks, peakons, cuspons, and compactons, among others. Solitons appear as a bell-shaped sech profile. Kink waves rise or descend from one asymptotic state to another. Peakons are peaked solitary-wave solutions. Cuspons exhibit cusps at their crests. In the peakon structure, the traveling-wave solutions are smooth except for a peak at a corner of its crest. Peakons are the points at which spatial derivative changes sign so that peakons have a finite jump in the first derivative of the solution u(x; t). Unlike peakons, where the derivatives at the peak differ only by a sign, the derivatives at the jump of a cuspon diverge. Compactons are solitons with compact spatial support such that each compacton is a soliton confined to a finite core or a soliton without exponential tails. Com-

pactons are generated due to the delicate interaction between the effect of the genuine nonlinear convection and the genuinely nonlinear dispersion. Adomian method The Adomian decomposition method approaches linear and nonlinear, and homogeneous and inhomogeneous differential and integral equations in a unified way. The method provides the solution in a rapidly convergent series with terms that are elegantly determined in a recursive manner. The method can be used to obtain closed-form solutions, if such solutions exist. A truncated number of the obtained series can be used for numerical purposes. The method was modified to accelerate the computational process. The noise terms phenomenon, which may appear for inhomogeneous cases, can give the exact solution in two iterations only. Definition of the Subject Nonlinear phenomena play a significant role in many branches of applied sciences such as applied mathematics, physics, biology, chemistry, astronomy, plasma, and fluid dynamics. Nonlinear dispersive equations that govern these phenomena have the genuine soliton property. Solitons are pulses that propagate without any change of its identity, i. e., shape and speed, during their travel through a nonlinear dispersive medium [1,5,34]. Solitons resemble properties of a particle, hence the suffix on is used [19,20]. Solitons exist in many scientific branches, such as optical fiber photonics, fiber lasers, plasmas, molecular systems, laser pulses propagating in solids, liquid crystals, nonlinear optics, cosmology, and condensed-matter physics. Based on its importance in many fields, a huge amount of research work has been conducted during the last four decades to make more progress in understanding the soliton phenomenon. A variety of very powerful algorithms has been used to achieve this goal. The Adomian decomposition method introduced in [2,6,21,22,23, 24,25,26], which will be used in this work, is one of the reliable methods that has been used recently. Introduction The aim of this work is to apply the Adomian decomposition method to derive specific types of soliton solutions. Solitons were discovered experimentally by John Scott Russell in 1844. Korteweg and de Vries in 1895 investigated analytically the soliton concept [10], where they derived the pioneer equation of solitons, well known as the KdV equation, that models the height of the surface of shallow water in the presence of solitary waves. Moreover,

1

2

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Zabusky and Kruskal [35] investigated this phenomenon analytically in 1965. Since 1965, a huge number of research works have been conducted on nonlinear dispersive and dissipative equations. The aim of these works has been to study thoroughly the characteristics of soliton solutions and to study various types of solitons that appear as a result of these equations. Several reliable methods were used in the literature to handle nonlinear dispersive equations. Hirota’s bilinear method [8,9] has been used for single- and multiple-soliton solutions. The inverse scattering method [1] has been widely used. For single-soliton solutions, several methods, such as the tanh method [13,14,15], the pseudospectral method, and the truncated Painlevé expansion, are used. However, in this work, the decomposition method, introduced by Adomian in 1984, will be applied to derive the desired types of soliton solutions. The method approaches all problems in a unified way and in a straightforward manner. The method computes the solution in a fast convergent series with components that are elegantly determined. Unlike other methods, the initial or boundary conditions are necessary for the use of the Adomian method. Adomian Decomposition Method and Adomian Polynomials The Adomian decomposition method, developed by George Adomian in 1984, has been receiving much attention in applied mathematics in general and in the area of initial value and boundary value problems in particular. Moreover, it is also used in the area of series solutions for numerical purposes. The method is powerful and effective and can be used in linear or nonlinear, ordinary or partial differential equations, and integral equations. The decomposition method demonstrates fast convergence of the solution and provides numerical approximation with a high level of accuracy. The method handles applied problems directly and in a straightforward manner without using linearization, perturbation, or any other restrictive assumption that might change the physical behavior of the physical model under investigation. The method is effectively addressed and thoroughly used by many researchers in the literature [1,21,22,23,24, 25,26]. It is important to indicate that well-known methods, such as Backlund transformation, inverse scattering method, Hirota’s bilinear formalism, the tanh method, and many others, can handle problems without using initial or boundary value conditions. For the Adomian decomposition method, the initial or boundary conditions are necessary to conduct the determination of the components recursively. However, some of the standard methods require

huge calculations, whereas the Adomian method minimizes the volume of computational work. The Adomian decomposition method consists in decomposing the unknown function u(x; t) of any equation into a sum of an infinite number of components given by the decomposition series u(x; t) D

1 X

u n (x; t) ;

(1)

nD0

where the components u n (x; t); n  0 are to be determined in a recursive manner. The decomposition method concerns itself with the determination of the components u0 (x; t); u1 (x; t); u2 (x; t); : : : individually. The determination of these components can be obtained through a recursive relation that usually involves evaluation of simple integrals. We now give a clear overview of Adomian decomposition method. Consider the linear differential equation written in an operator form by Lu C Ru D f ;

(2)

where L is the lower-order derivative, which is assumed to be invertible, R is a linear differential operator of order greater than L, and f is a source term. We next apply the inverse operator L1 to both sides of Eq. (2) and use the given initial or boundary condition to get u(x; t) D g  L1 (Ru) ;

(3)

where the function g represents the terms arising from integrating the source term f and from using the given conditions; all are assumed to be prescribed. Substituting the infinite series of components u(x; t) D

1 X

u n (x; t)

(4)

nD0

into both sides of (3) yields 1 X

un D g  L

1

nD0

R

X 1

! un

:

(5)

nD0

The decomposition method suggests that the zeroth component u0 is usually defined by all terms not included under the inverse operator L1 , which arise from the initial data and from integrating the inhomogeneous term. This in turn gives the formal recursive relation u0 (x; t) D g ; u kC1 (x; t) D L1 (R(u k )) ;

k  0;

(6)

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Adomian assumed that the nonlinear term F(u) can be expressed by an infinite series of the so-called Adomian polynomials An given in the form

or, equivalently, u0 (x; t) D g ; u1 (x; t) D L1 (R(u0 )) ; u2 (x; t) D L1 (R(u1 )) ; u3 (x; t) D L

1

(7)

(R(u2 )) ;

:: :: The differential equation under consideration is now reduced to integrals that can be easily evaluated. Having determined these components u0 (x; t); u1 (x; t); u2 (x; t); : : : , we then substitute the obtained components into (4) to obtain the solution in a series form. The determined series may converge very rapidly to a closed-form solution, if an exact solution exists. For concrete problems, where a closed-form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. Few terms of the truncated series give an approximation with a high degree of accuracy. The convergence concept of the decomposition series is investigated thoroughly in the literature. Several significant studies were conducted to compare the performance of the Adomian method with other methods, such as Picard’s method, Taylor series method, finite differences method, perturbation techniques, and others. The conclusions emphasized the fact that the Adomian method has many advantages and requires less computational work compared to existing techniques. The Adomian method and many others applied this method to many deterministic and stochastic problems. However, the Adomian method, like some other methods, suffers if the zeroth component u0 (x; t) D 0 and makes the integrand of the right side in (7) u1 (x; t) D 0. If the right side integrand in (7) does not vanish, if it is of a form such as eu 0 or ln(˛ C u0 ); ˛ > 0, then the method works effectively. As stated before, the Adomian method decomposes the unknown function u(x; t) into an infinite number of components. However, for nonlinear functions of u(x; t) such as u2 ; u3 ; ln(1 C u); cos u; eu ; uu x , etc., a special representation for nonlinear terms was developed by Adomian and others. Adomian introduced a formal algorithm to establish the proper representation for all forms of nonlinear functions of u(x; t). This representation of nonlinear terms is necessary to apply the Adomian method properly. Several alternative algorithms have been introduced in the literature by researchers to calculate Adomian polynomials. However, the Adomian algorithm remains the most commonly applied because it is simple and practical; therefore, it will be used in this work.

F(u) D

1 X

A n (u0 ; u1 ; u2 ; : : : ; u n ) ;

(8)

nD0

where An can be evaluated for all forms of nonlinearity. Adomian polynomials An for the nonlinear term F(u) can be evaluated using the following expression: " !# n X 1 dn i An D  ui ; n D 0; 1; 2;    : (9) F n! dn iD0

0

The general formula (9) can be simplified as follows. Assuming that the nonlinear function is F(u), using (9), Adomian polynomials are given by A0 D F(u0 ) ; A1 D u1 F 0 (u0 ) ; A2 D u2 F 0 (u0 ) C

1 2 00 u F (u0 ) ; 2! 1

1 A3 D u3 F 0 (u0 ) C u1 u2 F 00 (u0 ) C u13 F 000 (u0 ) ; 3!   1 2 u2 C u1 u3 F 00 (u0 ) A4 D u4 F 0 (u0 ) C (10) 2! 1 1 C u12 u2 F 000 (u0 ) C u14 F (iv) (u0 ) ; 2! 4! A5 D u5 F 0 (u0 ) C (u2 u3 C u1 u4 )F 00 (u0 )   1 1 C u1 u22 C u12 u3 F 000 (u0 ) 2! 2! 1 1 3 C u1 u2 F (iv) (u0 ) C u15 F (v) (u0 ) : 3! 5! Other polynomials can be generated in a similar manner. It is clear that A0 depends only on u0 , A1 depends only on u0 and u1 , A2 depends only on u0 ; u1 , and u2 , and so on. For F(u) D u2 , we find A0 D F(u0 ) D u02 ; A1 D u1 F 0 (u0 ) D 2u0 u1 ; 1 A2 D u2 F 0 (u0 ) C u12 F 00 (u0 ) D 2u0 u2 C u12 ; 2! 1 0 A3 D u3 F (u0 ) C u1 u2 F 00 (u0 ) C u13 F 000 (u0 ) 3! D 2u0 u3 C 2u1 u2 :

(11)

Modified Decomposition Method and Noise Terms Phenomenon A reliable modification of the Adomian decomposition method [22] was developed by Wazwaz in 1999. The modification will further accelerate the convergence of the se-

3

4

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

ries solution. As presented earlier, the standard decomposition method admits the use of the recursive relation u0 D g ; u kC1 D  L1 (Ru k ) ;

k 0:

(12)

The modified decomposition method introduces a slight variation to the recursive relation (12) that will lead to the determination of the components of u in a faster and easier way. For specific cases, the function g can be set as the sum of two partial functions, g 1 and g 2 . In other words, we can set g D g1 C g2 :

(13)

This assumption gives a slight qualitative change in the formation of the recursive relation (12). To reduce the size of calculations, we identify the zeroth component u0 by one part of g, that is, g 1 or g 2 . The other part of g can be added to the component u1 among other terms. In other words, the modified recursive relation can be defined as u0 D g 1 ; u1 D g2  L1 (Ru0 ) ; u kC1 D  L

1

(Ru k ) ;

(14) k 1:

The change occurred in the formation of the first two components u0 and u1 only [22,24,25]. Although this variation in the formation of u0 and u1 is slight, it plays a major role in accelerating the convergence of the solution and in minimizing the size of calculations. It is interesting to point out that by selecting the parts g 1 and g 2 properly, the exact solution u(x; t) may be obtained by using very few iterations, and sometimes by evaluating only two components. Moreover, if g consists of one term only, the standard decomposition method should be employed in this case. Another useful feature of the Adomian method is the noise terms phenomenon. The noise terms may appear for inhomogeneous problems only. This phenomenon was addressed by Adomian in 1994. In 1997, Wazwaz investigated the necessary conditions for the appearance of noise terms in the decomposition series. Noise terms are defined as identical terms with opposite signs [21] that arise in the components u0 and u1 particularly, and in other components as well. By canceling the noise terms between u0 and u1 , even though u1 contains other terms, the remaining noncanceled terms of u0 may give the exact solution of the equation. Therefore, it is necessary to verify that the noncanceled terms of u0 satisfy the equation. The noise terms, if they exist in the components u0 and u1 , will provide the solution in a closed form with only two successive iterations.

It was formally proved by Wazwaz in 1997 that a necessary condition for the appearance of the noise terms is required. The conclusion made is that the zeroth component u0 must contain the exact solution u, among other terms [21]. Moreover, it was shown that the nonhomogeneity condition does not always guarantee the appearance of the noise terms. Solitons, Peakons, Kinks, and Compactons There are many types of solitary waves. Solitons, which are localized traveling waves, are asymptotically zero at large distances [1,5,7,21,22,23,24,25,26,27,28,29,30,31,32, 33]. Solitons appear as a bell-shaped sech profile. Soliton solution u() results as a balance between nonlinearity and dispersion, where u(); u0 (); u00 ();    ! 0 as  ! ˙1, where  D x  ct, and c is the speed of the wave prorogation. The soliton solution either decays exponentially as in the KdV equation, or it converges to a constant at infinity such as the kinks of the sine-Gordon equation. This means that the soliton solutions appear as sech˛ or arctan(e˛(xc t) ). Moreover, one soliton interacts with other solitons preserving its permanent form. Another type of solitary wave is the kink wave, which rises or descends from one asymptotic state to another [18]. The Burgers equation and the sine-Gordon equation are examples of nonlinear wave equations that exhibit kink solutions. It is to be noted that the kink u() converges to ˙˛, where ˛ is a constant. However, u0 (); u00 ();    ! 0 as  ! ˙1. Peakons are peaked solitary-wave solutions and are another type of solitary-wave solution. In this structure, the traveling wave solutions are smooth except for a peak at a corner of its crest. Peakons are the points at which a spatial derivative changes signs so that peakons have a finite jump in the first derivative of the solution u(x; t) [3,4,11,12,32]. This means that the first derivative of u(x; t) has identical values with opposite signs around the peak. A significant type of soliton is the compacton, which is a soliton with compact spatial support such that each compacton is a soliton confined to a finite core or a soliton without exponential tails [16,17]. Compactons were formally derived by Rosenau and Hyman in 1993 where a special kind of the KdV equation was used to derive this new discovery. Unlike a soliton that narrows as the amplitude increases, a compacton’s width is independent of the amplitude [16,17,27,28,29,30]. Classical solitons are analytic solutions, whereas compactons are nonanalytic solutions [16,17]. As will be shown by a graph below, compactons are solitons that are free of exponential wings or

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

tails. Compactons arise as a result of the delicate interaction between genuine nonlinear terms and the genuinely nonlinear dispersion, as is the case with the K(n,n) equation.

Applying L1 t to both sides of (16) and using the initial condition we find 2c 2 e cx u(x; t) D C L1 (20) t (u x x x  6uu x ) : (1 C e cx )2

Solitons of the KdV Equation

Notice that the right-hand side contains a linear term u x x x and a nonlinear term uu x . Accordingly, Adomian polynomials for this nonlinear term are given by

In 1895, Korteweg, together with his Ph.D student de Vriesf, derived analytically a nonlinear partial differential equation, well known now by the KdV equation given in its simplest form by u t C 6uu x C u x x x

2c 2 e cx D 0; u(x; 0) D : (1 C e cx )2

(15)

@ ; @t

D u0 x u3 C u1 x u2 C u2 x u1 C u3 x u0 : Recall that the Adomian decomposition method suggests that the linear function u may be represented by a decomposition series uD

1 X

un ;

(22)

nD0

whereas nonlinear term F(u) can be expressed by an infinite series of the so-called Adomian polynomials An given in the form 1 X A n (u0 ; u1 ; u2 ; : : : ; u n ) : (23) F(u) D nD0

Using the decomposition identification for the linear and nonlinear terms in (20) yields ! 1 1 X X 2c 2 ecx 1 u n (x; t) D  Lt u n (x; t) (1 C ecx )2 nD0 nD0 xxx ! 1 X  6L1 A n : (24) t nD0

(16)

(17)

The components u n ; n  0 can be elegantly calculated by

L1 t

assuming that the integral operator exists and may be regarded as a onefold definite integral defined by Z t (:)dt : (18) L1 t (:) D 0

This means that L1 t L t u(x; t) D u(x; t)  u(x; 0) :

A3 D 12 L x (2u0 u3 C 2u1 u2 )

The Adomian method allows for the use of the recursive relation 2c 2 ecx ; u0 (x; t) D (1 C ecx )2 (25)   1 (A ) ; k  0 : u kC1 (x; t) D  L1 u  6L k k t t xxx

where the differential operator Lt is Lt D

A1 D 12 L x (2u0 u1 ) D u0x u1 C u0 u1x ; A2 D 12 L x (2u0 u2 C u12 ) D u0x u2 C u1x u1 C u2x u0 ; (21)

The KdV equation is the simplest equation embodying both nonlinearity and dispersion [5,34]. This equation has served as a model for the development of solitary-wave theory. The KdV equation is a completely integrable biHamiltonian equation. The KdV equation is used to model the height of the surface of shallow water in the presence of solitary waves. The nonlinearity represented by uu x tends to localize the wave, while the linear dispersion represented by u x x x spreads it out. The balance between the weak nonlinearity and the linear dispersion gives solitons that consist of single humped waves. The equilibrium between the nonlinearity uu x and the dispersion u x x x of the KdV equation is stable. In 1965, Zabusky and Kruskal [35] investigated numerically the nonlinear interaction of a large solitary wave overtaking a smaller one and discovered that solitary waves underwent nonlinear interactions following the KdV equation. Further, the waves emerged from this interaction retaining their original shape and amplitude, and therefore conserved energy and mass. The only effect of the interaction was a phase shift. To apply the Adomian decomposition method, we first write the KdV Eq. (15) in an operator form: L t u D u x x x  6uu x ;

A0 D F(u0 ) D u0 u0x ;

(19)

2c 2 ecx ; (1 C ecx )2   u1 (x; t) D  L1 u0x x x  6L1 t t (A0 )

u0 (x; t) D

2c 5 e cx (ecx  1) t; (1 C ecx )3   u2 (x; t) D  L1 u1x x x  6L1 t t (A1 ) D

D

c 8 ecx (e2cx  4e cx C 1) 2 t ; (1 C e cx )4

5

6

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

  u2x x x  6L1 u3 (x; t) D  L1 t t (A2 ) c 11 ecx (e3cx  11e2cx C 11ecx  1) 3 t ; 3(1 C e cx )5   u3x x x  6L1 u4 (x; t) D  L1 t t (A3 ) D

D

c 14 ecx (e4cx 26e3cx C66e2cx 26ecx C1) 4 t ; 12(1 C e cx )6 (26)

and so on. The series solution is thus given by 2c 2 ecx u(x; t) D (1 C ecx )2  c 3 (e cx  1) c 6 (e2cx  4e cx C 1) 2  1C t t C (1 C ecx ) 2(1 C e cx )2  c 9 (e3cx  11e2cx C 11ecx  1) 3 C t C    ; 6(1 C e cx )3 (27) and in a closed form by u(x; t) D 

2c 2 ec(xc 1C

2 t)

2 ec(xc 2 t)

:

(28)

The last equation emphasizes the fact that the dispersion relation is ! D c 3 . Moreover, the exact solution (28) can be rewritten as hc  i c2 x  c2 t : u(x; t) D (29) sech2 2 2 This in turn gives the bell-shaped soliton solution of the KdV equation. A typical graph of a bell-shaped soliton is given in Fig. 1. The graph shows that solitons are characterized by exponential wings or tails. The graph also confirms that solitons become asymptotically zero at large distances.

Burgers (1895–1981) introduced one of the fundamental model equations in fluid mechanics [18] that demonstrates coupling between nonlinear advection uu x and linear diffusion u x x . The Burgers equation appears in gas dynamics and traffic flow. Burgers introduced this equation to capture some of the features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and diffusion. The standard form of the Burgers equation is given by

t > 0;

the equation is called an inviscid Burgers equation. The inviscid Burgers equation will not be examined in this work. It is the goal of this work to apply the Adomian decomposition method to the Burgers equation; therefore we write (30) in an operator form L t u D u x x  uu x ;

 ¤0;

(30)

where u(x; t) is the velocity and  is a constant that defines the kinematic viscosity. If the viscosity  D 0, then

(31)

where the differential operator Lt is Lt D

Kinks of the Burgers Equation

u t C uu x D u x x ; 2c ; u(x; 0) D cx (1 C e  )

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 1 Figure 1 shows the soliton graph u(x; t) D sech2 (x  ct); c D 1; 5  x; t  5

@ ; @t

(32)

and as a result Z t (:) D (:)dt : L1 t

(33)

0

This means that L1 t L t u(x; t) D u(x; t)  u(x; 0) :

(34)

Applying L1 t to both sides of (31) and using the initial condition we find u(x; t) D

2c cx

(1 C e  )

C L1 t (u x x  uu x ) :

(35)

Notice that the right-hand side contains a linear term and a nonlinear term uu x . The Adomian polynomials for this nonlinear term uu x are the same as in the KdV equation.

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Using the decomposition identification for the linear and nonlinear terms in (35) yields 1 X

2c

u n (x; t) D

cx

(1 C e  )

nD0

C

L1 t



nD0 1 X

 L1 t

!

1 X

u n (x; t) xx

! :

An

(36)

nD0

The Adomian method allows for the use of the recursive relation 2c

u0 (x; t) D

; cx (1 C e n )   u k x x x  L1 u kC1 (x; t) D L1 t t (A k ) ;

(37) k 0:

The components u n ; n  0 can be elegantly calculated by 2c

; cx (1 C e  )   u0x x x  L1 u1 (x; t) D L1 t t (A0 ) u0 (x; t) D

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 2 Figure 2 shows the kink graph u(x; t) D tanh(x  ct); c D 1; 10  x; t  10

cx

2c 3 e 

D

t; cx (1 C e  )2   u2 (x; t) D L1 u1x x x  L1 t t (A1 ) cx cx c 5 e  (e 

 1)

t2 ; e )3   u2x x x  L1 u3 (x; t) D L1 t t (A2 ) D

cx

D

c 7 e  (e

2cx 

3 3 (1 C e )4   u3x x x  L1 u4 (x; t) D L1 t t (A3 ) D

c 9 e (e

3cx 

 11e

12 4 (1

2cx 

Ce

t3 ;

Peakons of the Camassa–Holm Equation

C 11e cx 

cx 

 1)

)5

Camassa and Holm [4] derived in 1993 a completely integrable wave equation

t4 ;

u t C 2ku x  u x x t C 3uu x D 2u x u x x C uu x x x

and so on. The series solution is thus given by cx

u(x; t) D

2ce 

cx

(1 C e  ) cx

c2

cx

 e  C

cx

(1 C e  ) C

c 6 (e

2cx 

tC

c 4 (e   1) cx

2 2 (1 C e  )2

t2

cx

 4e  C 1) cx

6 3 (1 C e  )3

t3 C   

! ;

(39)

and in a closed form by 2c

u(x; t) D 

c

1 C e  (xc t)

;

(41)

Figure 2 shows a kink graph. The graph shows that the kink converges to ˙1 as  ! ˙1.

cx

 4e  C 1) cx 

cx 

(38)

cx 

 2 (1 C

or equivalently i  h c (x  ct) : u(x; t) D c 1  tanh 2

 ¤0;

(40)

(42)

by retaining two terms that are usually neglected in the small-amplitude shallow-water limit [3]. The constant k is related to the critical shallow-water wave speed. This equation models the unidirectional propagation of water waves in shallow water. The CH Eq. (42) differs from the well-known regularized long wave (RLW) equation only through the two terms on the right-hand side of (42). Moreover, this equation has an integrable bi-Hamiltonian structure and arises in the context of differential geometry, where it can be seen as a reexpression for geodesic flow on an infinite-dimensional Lie group. The CH equation admits a second-order isospectral problem and allows for peaked solitary-wave solutions, called peakons.

7

8

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

In this work, we will apply the Adomian method to the CH equation u t u x x t C3uu x D 2u x u x x Cuu x x x ;

u(x; 0) D cejxj ; (43)

The Adomian scheme allows the use of the recursive relation u0 (x; t) D cex ; u kC1 (x; t) D L1 t (u k (x; t)) x x t C L1 t (3 A k C 2B k C C k ) ;

or equivalently

k 0: (51)

u t D u x x t 3uu x C2u x u x x Cuu x x x ;

u(x; 0) D cejxj ; (44)

where we set k D 0. Proceeding as before, the CH Eq. (44) in an operator form is as follows: L t u D u x x t 3uu x C2u x u x x Cuu x x x ;

u(x; 0) D cejxj ; (45)

where the differential operator Lt is as defined above. It is important to point out that the CH equation includes three nonlinear terms; therefore, we derive the following three sets of Adomian polynomials for the terms uu x ; u x u x x , and uu x x x : A 0 D u0 u0 x ; A 1 D u0 x u1 C u0 u1 x ;

(46)

A 2 D u0 x u2 C u1 x u1 C u2 x u0 ; B 0 D u0 x u0 x x ; B 1 D u0 x x u1 x C u0 x u1 x x ;

(47)

B 2 D u0 x x u2 x C u1 x u1 x x C u0 x u2 x x ;

The components u n ; n  0 can be elegantly computed by u0 (x; t) D cex ; 1 u1 (x; t) D L1 t (u0 (x; t)) x x t C L t (3 A0 C 2B0 C C0 )

D c 2 ex t ; 1 u2 (x; t) D L1 t (u1 (x; t)) x x t C L t (3 A1 C 2B1 C C1 ) 1 D c 3 ex t 2 ; 2! 1 u3 (x; t) D L1 t (u2 (x; t)) x x t C L t (3 A2 C 2B2 C C2 ) 1 D c 4 ex t 3 ; 3! 1 u4 (x; t) D L1 t (u3 (x; t)) x x t C L t (3 A3 C 2B3 C C3 ) 1 (52) D c 4 ex t 4 ; 4!

and so on. The series solution for x > 0 is given by u(x; t) D cex   1 1 1  1 C ct C (ct)2 C (ct)3 C (ct)4 C    ; 2! 3! 4! (53) and in a closed form by

and

u(x; t) D ce(xc t) :

C0 D u0 u0 x x x ; (48)

C1 D u0 x x x u1 C u0 u1 x x x ; C2 D u0 x x x u2 C u1 u1 x x x C u0 u2 x x x :

Applying the inverse operator L1 t to both sides of (45) and using the initial condition we find u(x; t) D cejxj CL1 t (u x x t  3uu x C 2u x u x x C uu x x x ) : (49) Case 1. For x > 0, the initial condition will be u(x; 0) D ex . Using the decomposition series (1) and Adomian polynomials in (49) yields ! 1 1 X X x 1 u n (x; t) D ce C L t u n (x; t) nD0

C

nD0

L1 t

3

1 X nD0

An C 2

1 X nD0

Bn C

xxt 1 X nD0

!

Cn

:

(50)

(54)

Case 2. For x < 0, the initial condition will be u(x; 0) D e x . Proceeding as before, we obtain u0 (x; t) D ce x ; 1 u1 (x; t) D L1 t (u0 (x; t)) x x t C L t (3 A0 C 2B0 C C0 )

D  c 2 ex t ; 1 u2 (x; t) D L1 t (u1 (x; t)) x x t C L t (3 A1 C 2B1 C C1 ) 1 D c 3 ex t 2 ; 2! 1 u3 (x; t) D L1 t (u2 (x; t)) x x t C L t (3 A2 C 2B2 C C2 ) 1 D  c4 ex t3 ; 3! 1 u4 (x; t) D L1 t (u3 (x; t)) x x t C L t (3 A3 C 2B3 C C3 ) 1 D c 4 ex t 4 ; 4! (55)

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

KdV equation, named K(m,n), of the form u t C (u m )x C (u n )x x x D 0 ;

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 3 Figure 3 shows the peakon graph u(x; t) D ej(xct)j ; c D 1; 2  x; t  2

and so on. The series solution for x < 0 is given by u(x; t) D ce x   1 1 1 2 3 4  1  ct C (ct)  (ct) C (ct) C    ; 2! 3! 4! (56) and in a closed form by u(x; t) D ce(xc t) :

(57)

Combining the results for both cases gives the peakon solution u(x; t) D cejxc tj :

(58)

Figure 3 shows a peakon graph. The graph shows that a peakon with a peak at a corner is generated with equal first derivatives at both sides of the peak, but with opposite signs. Compactons of the K(n,n) Equation The K(n,n) equation [16,17] was introduced by Rosenau and Hyman in 1993. This equation was investigated experimentally and analytically. The K(m,n) equation is a genuinely nonlinear dispersive equation, a special type of the

n>1:

(59)

Compactons, which are solitons with compact support or strict localization of solitary waves, have been investigated thoroughly in the literature. The delicate interaction between the effect of the genuine nonlinear convection (u n )x and the genuinely nonlinear dispersion of (u n )x x x generates solitary waves with exact compact support that are called compactons. It was also discovered that solitary waves may compactify under the influence of nonlinear dispersion, which is capable of causing deep qualitative changes in the nature of genuinely nonlinear phenomena [16,17]. Unlike solitons that narrow as the amplitude increases, a compacton’s width is independent of the amplitude. Compactons such as drops do not possess infinite wings; hence they interact among themselves only across short distances. Compactons are nonanalytic solutions, whereas classical solitons are analytic solutions. The points of nonanalyticity at the edge of a compacton correspond to points of genuine nonlinearity for the differential equation and introduce singularities in the associated dynamical system for the traveling waves [16,17,27,28,29,30]. Compactons were proved to collide elastically and vanish outside a finite core region. This discovery was studied thoroughly by many researchers who were involved with identical nonlinear dispersive equations. It is to be noted that solutions were obtained only for cases where m D n. However, for m ¤ n, solutions have not yet been determined. Without loss of generality, we will examine the special K(3,3) initial value problem p   1 6c u t C (u3 )x C (u3 )x x x D 0 ; u(x; 0) D cos x : 2 3 (60) We first write the K(3,3) Eq. (60) in an operator form: p   1 6c 3 3 cos x ; L t u D (u )x  (u )x x x ; u(x; 0) D 2 3 (61) where the differential operator Lt is as defined above. The differential operator Lt is defined by Lt D

@ : @t

(62)

Applying L1 t to both sides of (61) and using the initial condition we find p    3  6c 1 (u )x C (u3 )x x x : (63) u(x; t) D cos x  L1 t 2 3

9

10

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Notice that the right-hand side contains two nonlinear terms (u3 )x and (u3 )x x x . Accordingly, the Adomian polynomials for these terms are given by A0 D (u03 )x ; A1 D (3u02 u1 )x ;

(64)

A2 D (3u02 u2 C 3u0 u12 )x ; A3 D (3u02 u3 C 6u0 u1 u2 C u13 )x ;

B0 D (u03 )x x x ; B1 D (3u02 u1 )x x x ;

(65)

B2 D (3u02 u2 C 3u0 u12 )x x x ; B3 D (3u02 u3 C 6u0 u1 u2 C u13 )x x x ;

(69)

u(x; t) p          1 1 1 1 6c D cos x cos ct Csin x sin ct ; 2 3 3 3 3 (70) or equivalently

u(x; t) D

respectively. Proceeding as before, Eq. (63) becomes

nD0

u(x; t) ! p       1 6c 1 ct 4 1 ct 2 C C cos x 1 D 2 3 2! 3 4! 3 ! p       1 6c ct 1 ct 3 1 ct 5 C C C   : sin x  2 3 3 3! 3 5! 5 This is equivalent to

and

1 X

and so on. The series solution is thus given by

8n p  o < 6c cos 1 (x  ct) ; 2 3

j x  ct j

:0

otherwise :

3 2

;

(71)

! p   1 X 1 6c 1 u n (x; t) D An C Bn : cos x L t 2 3 nD0 (66)

This in turn gives the compacton solution of the K(3,3) equation. Figure 4 shows a compacton confined to a finite core without exponential wings. The graph shows a compacton: a soliton free of exponential wings.

This gives the recursive relation p

  1 6c cos x ; 2 3  1  L t (A k C B k ) ; u kC1 (x; t) D  L1 t u0 (x; t) D

(67) k 0:

The components u n ; n  0 can be recursively determined as p   1 6c cos x ; u0 (x; t) D 2 3 p   1 c 6c u1 (x; t) D sin x t; 6 3 p   1 c 2 6c u2 (x; t) D  cos x t2 ; 36 3 (68) p   1 c 3 6c 3 u3 (x; t) D  sin x t ; 324 3 p   1 c 4 6c u4 (x; t) D cos x t4 ; 3888 3 p   1 c 5 6c u5 (x; t) D sin x t5 ; 58320 3

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 4 1

Figure 4 shows the compacton graph u(x; t) D cos 2 (x  ct); c D 1; 0  x; t  1

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

It is interesting to point out that the generalized solution of the K(n,n) equation, n > 1, is given by the compactons u(x; t) 8n o 1 < 2cn cos2  n1 (x  ct) n1 ; (nC1) 2n D : 0

j x  ct j 

 2

;

otherwise ; (72)

and u3 (x; t) 8n o 1 < 2cn sin2  n1 (x  ct) n1 ; (nC1) 2n D : 0

j x  ct j

 

;

otherwise : (73)

Future Directions The most significant advantage of the Adomian method is that it attacks any problem without any need for a transformation formula or any restrictive assumption that may change the physical behavior of the solution. For singular problems, it was possible to overcome the singularity phenomenon and to attain practically a series solution in a standard way. Moreover, for problems on unbounded domains, combining the obtained series solution by the Adomian method with Padé approximants provides a promising tool to handle boundary value problems. The Padé approximants, which often show superior performance over series approximations, provide a promising tool for use in applied fields. As stated above, unlike other methods such as Hirota and the inverse scattering method, where solutions can be obtained without using prescribed conditions, the Adomian method requires such conditions. Moreover, these conditions must be of a form that does not provide zero for the integrand of the first component u1 (x; t). Such a case should be addressed to enhance the performance of the method. Another aspect that should be addressed is Adomian polynomials. The existing techniques require tedious work to evaluate it. Most importantly, the Adomian method guarantees only one solution for nonlinear problems. This is an issue that should be investigated to improve the performance of the Adomian method compared to other methods. On the other hand, it is important to further examine the N-soliton solutions by simplified forms, such as the form given by Hereman and Nuseir in [7]. The bilinear form of Hirota is not that easy to use and is not always attainable for nonlinear models.

Bibliography Primary Literature 1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge 2. Adomian G (1984) A new approach to nonlinear partial differential equations. J Math Anal Appl 102:420–434 3. Boyd PJ (1997) Peakons and cashoidal waves: travelling wave solutions of the Camassa–Holm equation. Appl Math Comput 81(2–3):173–187 4. Camassa R, Holm D (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71(11):1661–1664 5. Drazin PG, Johnson RS (1996) Solitons: an introduction. Cambridge University Press, Cambridge 6. Helal MA, Mehanna MS (2006) A comparison between two different methods for solving KdV-Burgers equation. Chaos Solitons Fractals 28:320–326 7. Hereman W, Nuseir A (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math Comp Simul 43:13–27 8. Hirota R (1971) Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192–1194 9. Hirota R (1972) Exact solutions of the modified Korteweg– de Vries equation for multiple collisions of solitons. J Phys Soc Jpn 33(5):1456–1458 10. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos Mag 5th Ser 36:422–443 11. Lenells J (2005) Travelling wave solutions of the Camassa– Holm equation. J Differ Equ 217:393–430 12. Liu Z, Wang R, Jing Z (2004) Peaked wave solutions of Camassa–Holm equation. Chaos Solitons Fractals 19:77–92 13. Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654 14. Malfliet W, Hereman W (1996) The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys Scr 54:563–568 15. Malfliet W, Hereman W (1996) The tanh method: II. Perturbation technique for conservative systems. Phys Scr 54:569–575 16. Rosenau P (1998) On a class of nonlinear dispersive-dissipative interactions. Phys D 230(5/6):535–546 17. Rosenau P, Hyman J (1993) Compactons: solitons with finite wavelengths. Phys Rev Lett 70(5):564–567 18. Veksler A, Zarmi Y (2005) Wave interactions and the analysis of the perturbed Burgers equation. Phys D 211:57–73 19. Wadati M (1972) The exact solution of the modified Korteweg– de Vries equation. J Phys Soc Jpn 32:1681–1687 20. Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5/6):841–847 21. Wazwaz AM (1997) Necessary conditions for the appearance of noise terms in decomposition solution series. Appl Math Comput 81:265–274 22. Wazwaz AM (1998) A reliable modification of Adomian’s decomposition method. Appl Math Comput 92:1–7 23. Wazwaz AM (1999) A comparison between the Adomian decomposition method and the Taylor series method in the series solutions. Appl Math Comput 102:77–86

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24. Wazwaz AM (2000) A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl Math Comput 111(1):33–51 25. Wazwaz AM (2001) Exact specific solutions with solitary patterns for the nonlinear dispersive K(m,n) equations. Chaos, Solitons and Fractals 13(1):161–170 26. Wazwaz AM (2002) General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n,n) equations in higher dimensional spaces. Appl Math Comput 133(2/3):229–244 27. Wazwaz AM (2003) An analytic study of compactons structures in a class of nonlinear dispersive equations. Math Comput Simul 63(1):35–44 28. Wazwaz AM (2003) Compactons in a class of nonlinear dispersive equations. Math Comput Model l37(3/4):333–341 29. Wazwaz AM (2004) The tanh method for travelling wave solutions of nonlinear equations. Appl Math Comput 154(3):713– 723 30. Wazwaz AM (2005) Compact and noncompact structures for variants of the KdV equation. Int J Appl Math 18(2):213– 221 31. Wazwaz AM (2006) New kinds of solitons and periodic solutions to the generalized KdV equation. Numer Methods Partial Differ Equ 23(2):247–255 32. Wazwaz AM (2006) Peakons, kinks, compactons and solitary patterns solutions for a family of Camassa–Holm equations by using new hyperbolic schemes. Appl Math Comput 182(1):412–424 33. Wazwaz AM (2007) The extended tanh method for new soli-

tons solutions for many forms of the fifth-order KdV equations. Appl Math Comput 184(2):1002–1014 34. Whitham GB (1999) Linear and nonlinear waves. Wiley, New York 35. Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243

Books and Reviews Adomian G (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston Burgers JM (1974) The nonlinear diffusion equation. Reidel, Dordrecht Conte R, Magri F, Musette M, Satsuma J, Winternitz P (2003) Lecture Notes in Physics: Direct and Inverse methods in Nonlinear Evolution Equations. Springer, Berlin Hirota R (2004) The Direct Method in Soliton Theory. Cambridge University Press, Cambridge Johnson RS (1997) A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge Kosmann-Schwarzbach Y, Grammaticos B, Tamizhmani KM (2004) Lecture Notes in Physics: Integrability of Nonlinear Systems. Springer, Berlin Wazwaz AM (1997) A First Course in Integral Equations. World Scientific, Singapore Wazwaz AM (2002) Partial Differential Equations: Methods and Applications. Balkema, The Netherlands

Anomalous Diffusion on Fractal Networks

Anomalous Diffusion on Fractal Networks IGOR M. SOKOLOV Institute of Physics, Humboldt-Universität zu Berlin, Berlin, Germany Article Outline Glossary Definition of the Subject Introduction Random Walks and Normal Diffusion Anomalous Diffusion Anomalous Diffusion on Fractal Structures Percolation Clusters Scaling of PDF and Diffusion Equations on Fractal Lattices Further Directions Bibliography Glossary Anomalous diffusion An essentially diffusive process in which the mean squared displacement grows, however, not as hR2 i / t like in normal diffusion, but as hR2 i / t ˛ with ˛ ¤ 0, either asymptotically faster than in normal diffusion (˛ > 1, superdiffusion) or asymptotically slower (˛ < 1, subdiffusion). Comb model A planar network consisting of a backbone (spine) and teeth (dangling ends). A popular simple model showing anomalous diffusion. Fractional diffusion equation A diffusion equation for a non-Markovian diffusion process, typically with a memory kernel corresponding to a fractional derivative. A mathematical instrument which adequately describes many processes of anomalous diffusion, for example the continuous-time random walks (CTRW). Walk dimension The fractal dimension of the trajectory of a random walker on a network. The walk dimension is defined through the mean time T or the mean number of steps n which a walker needs to leave for the first time the ball of radius R around the starting point of the walk: T / R d w . Spectral dimension A property of a fractal structure which substitutes the Euclidean dimension in the expression for the probability to be at the origin at time t, P(0; t) / t d s /2 . Defines the behavior of the network Laplace operator. Alexander–Orbach conjecture A conjecture that the spectral dimension of the incipient infinite cluster in percolation is equal to 4/3 independently on the Eu-

clidean dimension. The invariance of spectral dimension is only approximate and holds within 2% accuracy even in d D 2. The value of 3/4 is achieved in d > 6 and on trees. Compact visitation A property of a random walk to visit practically all sites within the domain of the size of the mean squared displacement. The visitation is compact if the walk dimension dw exceeds the fractal dimension of the substrate df . Definition of the Subject Many situations in physics, chemistry, biology or in computer science can be described within models related to random walks, in which a “particle” (walker) jumps from one node of a network to another following the network’s links. In many cases the network is embedded into Euclidean space which allows for discussing the displacement of the walker from its initial site. The displacement’s behavior is important e. g. for description of charge transport in disordered semiconductors, or for understanding the contaminants’ behavior in underground water. In other cases the discussion can be reduced to the properties which can be defined for a whatever network, like return probabilities to the starting site or the number of different nodes visited, the ones of importance for chemical kinetics or for search processes on the corresponding networks. In the present contribution we only discuss networks embedded in Euclidean space. One of the basic properties of normal diffusion is the fact that the mean squared displacement of a walker grows proportionally to time, R2 / t, and asymptotic deviations from this law are termed anomalous diffusion. Microscopic models leading to normal diffusion rely on the spatial homogeneity of the network, either exact or statistical. If the system is disordered and its local properties vary from one its part to another, the overall behavior at very large scales, larger than a whatever inhomogeneity, can still be diffusive, a property called homogenization. The ordered systems or systems with absent longrange order but highly homogeneous on larger scales do not exhaust the whole variety of relevant physical systems. In many cases the system can not be considered as homogeneous on a whatever scale of interest. Moreover, in some of these cases the system shows some extent of scale invariance (dilatation symmetry) typical for fractals. Examples are polymer molecules and networks, networks of pores in some porous media, networks of cracks in rocks, and so on. Physically all these systems correspond to a case of very strong disorder, do not homogenize even at largest relevant scales, and show fractal properties. The diffusion in such systems is anomalous, exhibiting large deviations

13

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Anomalous Diffusion on Fractal Networks

from the linear dependence of the mean squared displacement on time. Many other properties, like return probabilities, mean number of different visited nodes etc. also behave in a way different from the one they behave in normal diffusion. Understanding these properties is of primary importance for theoretical description of transport in such systems, chemical reactions in them, and in devising efficient search algorithms or in understanding ones used in natural search, as taking place in gene expression or incorporated into the animals’ foraging strategies. Introduction The theory of diffusion processes was initiated by A. Fick who was interested in the nutrients’ transport in a living body and put down the phenomenological diffusion equation (1855). The microscopic, probabilistic discussion of diffusion started with the works of L. Bachelier on the theory of financial speculation (1900), of A. Einstein on the motion of colloidal particles in a quiescent fluid (the Brownian motion) (1905), and with a short letter of K. Pearson to the readers of Nature in the same year motivated by his work on animal motion. These works put forward the description of diffusion processes within the framework of random walk models, the ones describing the motion as a sequence of independent bounded steps in space requiring some bounded time to be completed. These models lead to larger scales and longer times to a typical diffusion behavior as described by Fick’s laws. The most prominent property of this so-called normal diffusion is the fact that the mean squared displacement of the diffusing particle from its initial position at times much longer than the typical time of one step grows as hx 2 (t)i / t. Another prominent property of normal diffusion is the fact that the distribution of the particle’s displacements tends to a Gaussian (of width hx 2 (t)i1/2 ), as a consequence of the independence of different steps. The situations described by normal diffusion are abundant, but looking closer into many other processes (which still can be described within a random walk picture, however, with only more or less independent steps, or with steps showing a broad distribution of their lengths or times) revealed that these processes, still resembling or related to normal diffusion, show a vastly different behavior of the mean squared displacement, e. g. hx 2 (t)i / t ˛ with ˛ ¤ 1. In this case one speaks about anomalous diffusion [11]. The cases with ˛ < 1 correspond to subdiffusion, the ones with ˛ > 1 correspond to superdiffusion. The subdiffusion is typical for the motion of charge carriers in disordered semiconductors, for contaminant transport by underground water or for proteins’ motion in cell

membranes. The models of subdiffusion are often connected either with the continuous-time random walks or with diffusion in fractal networks (the topic of the present article) or with combinations thereof. The superdiffusive behavior, associated with divergent mean square displacement per complete step, is encountered e. g. in transport in some Hamiltonian models and in maps, in animal motion, in transport of particles by flows or in diffusion in porous media in Knudsen regime. The diffusion on fractal networks (typically subdiffusion) was a topic of extensive investigation in 1980s–1990s, so that the most results discussed here can be considered as well-established. Therefore most of the references in this contribution are given to the review articles, and only few of them to original works (mostly to the pioneering publications or to newer works which didn’t find their place in reviews). Random Walks and Normal Diffusion Let us first turn to normal diffusion and concentrate on the mean squared displacement of the random walker. Our model will correspond to a random walk on a regular lattice or in free space, where the steps of the random walker will be considered independent, identically distributed random variables (step lengths s i ) following a distribution with a given symmetric probability density function (PDF) p(s), so that hsi D 0. Let a be the root mean square displacement in one step, hs2i i D a2 . The squared displacement of the walker after n steps is thus * hr2n i

D

n X iD1

!2 + si

D

n n n X X X hs2i i C 2 hs i s j i : iD1

iD1 jDiC1

The first sum is simply na2 , and the second one vanishes if different˝ steps are uncorrelated and have zero mean. ˛ Therefore r2n D a 2 n: The mean squared displacement in a walk is proportional to the number of steps. Provided the mean time  necessary to complete a step exists, this expression can be translated into the temporal dependence of the displacement hr2 (t)i D (a 2 /)t, since the mean number of steps done during the time t is n D t/. This is the time-dependence of the mean squared displacement of a random walker characteristic for normal diffusion. The prefactor a 2 / in the hr2 (t)i / t dependence is connected with the usual diffusion coefficient K, defined through hr2 (t)i D 2dKt, where d is the dimension of the Euclidean space. Here we note that in the theory of random walks one often distinguishes between the situations termed as genuine “walks” (or velocity models) where the particle moves, say, with a constant velocity and

Anomalous Diffusion on Fractal Networks

changes its direction at the end of the step, and “random flight” models, when steps are considered to be instantaneous (“jumps”) and the time cost of a step is attributed to the waiting time at a site before the jump is made. The situations considered below correspond to the flight picture. The are three important postulates guaranteeing that random walks correspond to a normal diffusion process:  The existence of the mean squared displacement per step, hs2 i < 1  The existence of the mean time per step (interpreted as a mean waiting time on a site)  < 1 and  The uncorrelated nature of the walk hs i s j i D hs2 iı i j . These are exactly the three postulates made by Einstein in his work on Brownian motion. The last postulate can be weakened: The persistent (i. e. correlated) random walks P still lead to normal diffusion provided 1 jDi hs i s j i < 1. In the case when the steps are independent, the central limit theorem immediately states that the limiting distribution of the displacements will be Gaussian, of the form   1 dr2 P(r; t) D exp  : 4Kt (4 Kt)d/2 This distribution scales as a whole: rescaling the distance by a factor of , x ! x and of the time by the factor of 2 , t ! 2 t, does not change the form of this distribution. TheRmoments of the displacement scale according to hr k i D P(r; t)r k d d r k D const(k)t k/2 . Another important property of the the PDF is the fact that the return probability, i. e. the probability to be at time t at the origin of the motion, P(0; t), scales as P(0; t) / t d/2 ;

(1)

i. e. shows the behavior depending on the substrate’s dimension. The PDF P(r; t) is the solution of the Fick’s diffusion equation @ P(r; t) D kP(r; t) ; @t

(2)

where  is a Laplace operator. Equation 2 is a parabolic partial differential equation, so that its form is invariant under the scale transformation t ! 2 t, r ! r discussed above. This invariance of the equation has very strong implications. If follows, for example, that one can invert the scaling relation for the mean squared displacement R2 / t and interpret this inverse one T / r2

as e. g. a scaling rule governing the dependence of a characteristic passage time from some initial site 0 to the sites at a distance r from this one. Such scaling holds e. g. for the mean time spent by a walk in a prescribed region of space (i. e. for the mean first passage time to its boundary). For an isotropic system Eq. (2) can be rewritten as an equation in the radial coordinate only:

 @ 1 @ d1 @ P(r; t) D K d1 r P(r; t) : (3) @t @r @r r Looking at the trajectory of a random walk at scales much larger than the step’s length one infers that it is self-similar in statistical sense, i. e. that its whatever portion (of the size considerably larger then the step’s length) looks like the whole trajectory, i. e. it is a statistical, random fractal. The fractal (mass) dimension of the trajectory of a random walk can be easily found. Let us interthe number of steps as the “mass” M of the object and ˝pret ˛1/2 r2n as its “size” L. The mass dimension D is then defined as M / L D , it corresponds to the scaling of the mass of a solid D-dimensional object with its typical size. The scaling of the mean squared displacement of a simple random walk with its number of steps suggests that the corresponding dimension D D dw for a walk is dw D 2. The large-scale properties of the random walks in Euclidean space are universal and do not depend on exact form of the PDF of waiting times and of jumps’ lengths (provided the first possesses at least one and the second one at least two moments). This fact can be used for an interpretation of several known properties of simple random walks. Thus, random walks on a one-dimensional lattice (d D 1) are recurrent, i. e. each site earlier visited by the walk is revisited repeatedly afterwards, while for d  3 they are nonrecurrent. This property can be rationalized through arguing that an intrinsically two-dimensional object cannot be “squeezed” into a one-dimensional space without self-intersections, and that it cannot fill fully a space of three or more dimensions. The same considerations can be used to rationalize the behavior of the mean number of different sites visited by a walker during n steps hS n i. hS n i typically goes as a power law in n. Thus, on a one-dimensional lattice hS n i / n1/2 , in lattices in three and more dimensions hS n i / n, and in a marginal two-dimensional situation hS n i / n/ log n. To understand the behavior in d D 1 and in d D 3, it is enough to note that in a recurrent one-dimensional walk all sites within the span of the walk are visited by a walker at least once, so that the mean number of different sites visited by a one-dimensional walk (d D 1) grows as a span of the walk, which itself is proportional to a mean squared displacement, hS n i / L D hr2n i1/2 , i. e.

15

16

Anomalous Diffusion on Fractal Networks

hS n i / n1/2 D n1/d w . In three dimensions (d D 3), on the contrary, the mean number of different visited sites grows as the number of steps, hS n i / n, since different sites are on the average visited only finite number of times. The property of random walks on one-dimensional and twodimensional lattices to visit practically all the sites within the domain of the size of their mean square displacement is called the compact visitation property. The importance of the number of different sites visited (and its moments like hS n i or hS(t)i) gets clear when one considers chemical reactions or search problems on the corresponding networks. The situation when each site of a network can with the probability p be a trap for a walker, so that the walker is removed whenever it visits such a marked site (reaction scheme ACB ! B, where the symbol A denotes the walker and symbol B the “immortal” and immobile trap) corresponds to a trapping problem of reaction kinetics. The opposite situation, corresponding to the evaluation of the survival probability of an immobile A-particle at one of the network’s nodes in presence of mobile B-walkers (the same A C B ! B reaction scheme, however now with immobile A and mobile B) corresponds to the target problem, or scavenging. In the first case the survival probability ˚(N) for the A-particle after N steps goes for p small as ˚(t) / hexp(pS N )i, while for the second case it behaves as ˚(t) / exp(phS N i). The simple A C B ! B reaction problems can be also interpreted as the ones of the distribution of the times necessary to find one of multiple hidden “traps” by one searching agent, or as the one of finding one special marked site by a swarm of independent searching agents. Anomalous Diffusion The Einstein’s postulates leading to normal diffusion are to no extent the laws of Nature, and do not have to hold for a whatever random walk process. Before discussing general properties of such anomalous diffusion, we start from a few simple examples [10].

consider the random walk to evolve in discrete time; the time t and the number of steps n are simply proportional to each other. Diffusion on a Comb Structure The simplest example of how the subdiffusive motion can be created in a more or less complex (however yet non-fractal) geometrical structure is delivered by the diffusion on a comb. We consider the diffusion of a walker along the backbone of a comb with infinite “teeth” (dangling ends), see Fig. 1. The walker’s displacement in x-direction is only possible when it is on the backbone; entering the side branch switches off the diffusion along the backbone. Therefore the motion on the backbone of the comb is interrupted by waiting times. The distribution of these waiting times is given by the power law (t) / t 3/2 corresponding to the first return probability of a one-dimensional random walk in a side branch to its origin. This situation – simple random walk with broad distribution of waiting times on sites – corresponds to a continuous-time random walk model. In this model, for the case of the power-law waiting time distributions of the form (t) / t 1˛ with ˛ < 1, the mean number of steps hn(t)i (in our case: the mean number of steps over the backbone) as a function of time goes as hn(t)i / t ˛ . Since the mean squared displacement along the backbone is proportional to this number of steps, it is ˝ ˛ also governed by x 2 (t) / t ˛ , i. e. in our ˝ case ˛ we have to do with strongly subdiffusive behavior x 2 (t) / t 1/2 . The PDF of the particle’s displacements in CTRW with the waiting time distribution (t) is governed by a nonMarkovian generalization of the diffusion equation Z 1 @ @ M(t  t 0 )P(x; t 0 ) ; P(x; t) D a 2  @t @t 0 with the memory kernel M(t) which has a physical meaning of the time-dependent density of steps and is given by ˜ the inverse Laplace transform of M(u) D ˜ (u)/[1  ˜ (u)] (with ˜ (u) being the Laplace-transform of (t)). For the

Simple Geometric Models Leading to Subdiffusion Let us first discuss two simple models of geometrical structures showing anomalous diffusion. The discussion of these models allows one to gain some intuition about the emergence and properties of anomalous diffusion in complex geometries, as opposed to Euclidean lattices, and discuss mathematical notions appearing in description of such systems. In all cases the structure on which the walk takes place will be considered as a lattice with the unit spacing between the sites, and the walk corresponds to jumps from a site to one of its neighbors. Moreover, we

Anomalous Diffusion on Fractal Networks, Figure 1 A comb structure: Trapping in the dangling ends leads to the anomalous diffusion along the comb’s backbone. This kind of anomalous diffusion is modeled by CTRW with power-law waiting time distribution and described by a fractional diffusion equation

Anomalous Diffusion on Fractal Networks

case of a power-law waiting-time probability density of the type (t) / t 1˛ the structure of the right-hand side of the equation corresponds to the fractional Riemann– Liouvolle derivative 0 Dˇt , an integro-differential operator defined as Z 1 d t 0 f (t 0 ) ˇ dt ; (4) 0 D t f (t) D  (1  ˇ) dt 0 (t  t 0 )ˇ (here only for 0 < ˇ < 1), so that in the case of the diffusion on a backbone of the comb we have @ P(x; t) ; P(x; t) D K0 D1˛ t @t

(5)

(here with ˛ D 1/2). An exhaustive discussion of such equations is provided in [13,14]. Similar fractional diffusion equation was used in [12] for the description of the diffusion in a percolation system (as measured by NMR). However one has to be cautious when using CTRW and corresponding fractional equations for the description of diffusion on fractal networks, since they may capture some properties of such diffusion and fully disregard other ones. This question will be discussed to some extent in Sect. “Anomalous Diffusion on Fractal Structures”. Equation (5) can be rewritten in a different but equivalent form  ˛ 0 Dt

@ P(x; t) D KP(x; t) ; @t

with the operator 0 D˛t , a Caputo derivative, conjugated to the Riemann–Liouville one, Eq. (4), and different from Eq. (4) by interchanged order of differentiation and integration (temporal derivative inside the integral). The derivation of both forms of the fractional equations and their interpretation is discussed in [21]. For the sake of transparency, the corresponding integrodifferential operators 0 Dˇt and 0 Dˇt are simply denoted as a fractional partial derivative @ˇ /@t ˇ and interpreted as a Riemann– Liouville or as a Caputo derivative depending whether they stand on the left or on the right-hand side of a generalized diffusion equation. Random Walk on a Random Walk and on a Self-Avoiding Walk Let us now turn to another source of anomalous diffusion, namely the nondecaying correlations between the subsequent steps, introduced by the structure of the possible ways on a substrate lattice. The case where the anomalies of diffusion are due to the tortuosity of the path between the two sites of a fractal “network” is illustrated by the examples of random walks on polymer chains, being rather simple, topologically one-dimensional objects. Considering lattice models

for the chains we encounter two typical situations, the simpler one, corresponding to the random walk (RW) chains and a more complex one, corresponding to self-avoiding ones. A conformation of a RW-chain of N monomers corresponds to a random walk on a lattice, self-intersections are allowed. In reality, RW chains are a reasonable model for polymer chains in melts or in -solutions, where the repulsion between the monomers of the same chain is compensated by the repulsion from the monomers of other chains or from the solvent molecules, so that when not real intersections, then at least very close contacts between the monomers are possible. The other case corresponds to chains in good solvents, where the effects of steric repulsion dominate. The end-to-end distance in a random walk chain grows as R / l 1/2 (with l being its contour length). In a self-avoiding-walk (SAW) chain R / l  , with  being a so-called Flory exponent (e. g.   3/5 in 3d). The corresponding chains can be considered as topologically onedimensional fractal objects with fractal dimensions df D 2 and df D 1/  5/3 respectively. We now concentrate on an walker (excitation, enzyme, transcription factor) performing its diffusive motion on a chain, serving as a “rail track” for diffusion. This chain itself is embedded into the Euclidean space. The contour length of the chain corresponds to the chemical coordinate along the path, in each step of the walk at a chain the chemical coordinate changes by ˙1. Let us first consider the situation when a particle can only travel along this chemical path, i. e. cannot change its track at a point of a self-intersection of the walk or at a close contact between the two different parts of the chain. This walk starts with its first step at r D 0. Let K be the diffusion coefficient of the walker along the chain. The typical displacement of a walker along the chain after time t will then be l / t 1/2 D t 1/d w . For a RW-chain this displacement along the chain is translated into the displacement in Euclidean space according to R D hR2 i1/2 / l 1/2 , so that the typical displacement in Euclidean space after time t goes as R / t 1/4 , a strongly subdiffusive behavior known as one of the regimes predicted by a repetition theory of polymer dynamics. The dimension of the corresponding random walk is, accordingly, dw D 4. The same discussion for the SAW chains leads to dw D 1/2. After time t the displacement in the chemical space is given by a Gaussian distribution   l2 1 exp  : pc (l; t) D p 4Kt 4 Kt The change in the contour length l can be interpreted as the number of steps along the random-walk “rail”, so that the Euclidean displacement for the given l is distributed

17

18

Anomalous Diffusion on Fractal Networks

according to  pE (r; l) D



  d dr2 exp  : 2 a2 jlj 2a2 jlj

(6)

We can, thus, obtain the PDF of the Euclidean displacement: Z 1 P(r; l) D pE (r; l)pc (l; t)dl : (7) 0

The same discussion can be pursued for a self-avoiding walk for which

 r   1  r 1/(1)  pE (r; l) '  exp const  ; (8) l l with being the exponent describing the dependence of the number of realizations of the corresponding chains on their lengths. Equation (6) is a particular case of Eq. (8) with D 1 and  D 1/2. Evaluating the integral using the Laplace method, we get " (11/d w )1 #  r P(r; t) ' exp const 1/d t w up to the preexponential. We note that in this case the reason for the anomalous diffusion is the tortuosity of the chemical path. The topologically one-dimensional structure of the lattice allowed us for discussing the problem in considerable detail. The situation changes when we introduce the possibility for a walker to jump not only to the nearest-neighbor in chemical space by also to the sites which are far away along the chemical sequence but close to each other in Euclidean space (say one lattice site of the underlying lattice far). In this case the structure of the chain in the Euclidean space leads to the possibility to jump very long in chemical sequence of the chain, with the jump length distribution going as p(l) / l 3/2 for an RW and p(l) / l 2:2 for a SAW in d D 3 (the probability of a close return of a SAW to its original path). The corresponding distributions lack the second moment (and even the first moment for RW chains), and therefore one might assume that also the diffusion in a chemical space of a chain will be anomalous. It indeed shows considerable peculiarities. If the chain’s structure is frozen (i. e. the conformational dynamics of a chain is slow compared to the diffusion on the chain), the situation in both cases corresponds to “paradoxical diffusion” [22]: Although the PDF of displacements in the chemical space lacks the second (and higher) moments, the width of the PDF (described e. g. as its interquartile distance) grows, according to W / t 1/2 . In the Euclidean

space we have here to do with the diffusion on a static fractal structure, i. e. with a network of fractal dimension df coinciding with the fractal dimension of the chain. Allowing the chain conformation changing in time (i. e. considering the opposite limiting case when the conformation changes are fast enough compared to the diffusion along the chain) leads to another, superdiffusive, behavior, namely to Lévy flights along the chemical sequence. The difference between the static, fixed networks and the transient ones has to be borne in mind when interpreting physical results. Anomalous Diffusion on Fractal Structures Simple Fractal Lattices: the Walk’s Dimension As already mentioned, the diffusion on fractal structures is often anomalous, hx 2 (t)i / t ˛ with ˛ ¤ 1. Parallel to the situation with random walks in Euclidean space, the trajectory of a walk is typically self similar (on the scales exceeding the typical step length), so that the exponent ˛ can be connected with the fractal dimension of the walk’s trajectory dw . Assuming the mean time cost per step to be finite, i. e. t / n, one readily infers that ˛ D 2/dw . The fractal properties of the trajectory can be rather easily checked, and the value of dw can be rather easily obtained for simple regular fractals, the prominent example being a Sierpinski gasket, discussed below. Analytically, the value of dw is often obtained via scaling of the first passage time to a boundary of the region (taken to be a sphere of radius r), i. e. using not the relation R / t 1/d w but the inverse one, T / r d w . The fact that both values of dw coincide witnesses strongly in favor of the self-similar nature of diffusion on fractal structures. Diffusion on a Sierpinski Gasket Serpinski gasket (essentially a Sierpinski lattice), one of the simplest regular fractal network is a structure obtained by iterating a generator shown in Fig. 2. Its simple iterative structure and the ease of its numerical implementation made it a popular toy model exemplifying the properties of fractal networks. The properties of diffusion on this network were investigated in detail, both numerically and analytically. Since the structure does not possess dangling ends, the cause of the diffusion anomalies is connected with tortuosity of the typical paths available for diffusion, however, differently from the previous examples, these paths are numerous and non-equivalent. The numerical method allowing for obtaining exact results on the properties of diffusion is based on exact enumeration techniques, see e. g. [10]. The idea here is to calculate the displacement probability distribution based on the exact number of ways Wi;n the particle, starting at

Anomalous Diffusion on Fractal Networks

Anomalous Diffusion on Fractal Networks, Figure 2 A Sierpinski gasket: a its generator b a structure after 4th iteration. The lattice is rescaled by the factor of 4 to fit into the picture. c A constriction obtained after triangle-star transformation used in the calculation of the walk’s dimension, see Subsect. “The Spectral Dimension”

a given site 0 can arrive to a given site i at step n. For a given network the number Wi;n is simply a sum of the numbers of ways Wj;n1 leading from site 0 to the nearest neighbors j of the site i, Wj;n1 . Starting from W0;0 D 1 and Wi;0 D 0 for i ¤ 0 one simply updates the array P of Wi;n according to the rule Wi;n D jDnn(i) Wj;n1 , where nn(i) denotes the nearest neighbors of the node i in the network. The probability Pi;n is proportional to the number of these equally probably ways, and is obtained from Wi;n by normalization. This is essentially a very old method used to solve numerically a diffusion equation in the times preceding the computer era. For regular lattices this gives us the exact values of the probabilities. For statistical fractals (like percolation clusters) an additional averaging over the realizations of the structure is necessary, so that other methods of calculation of probabilities and exponents might be superior. The value of dw can then be obtained by plotting ln n vs. lnhr2n i1/2 : the corresponding points fall on a straight line with slope dw . The dimension of the walks on a Sierpinski gasket obtained by direct enumeration coincides with its theoretical value dw D ln 5/ ln 2 D 2:322 : : : , see Subsect. “The Spectral Dimension”. Loopless Fractals (Trees) Another situation, the one similar the the case of a comb, is encountered in loopless fractals (fractal trees), as exemplified by a Vicsek’s construction shown in Fig. 3, or by diffusion-limited aggregates (DLA). For the structure depicted in Fig. 3, the walk’s dimension is dw D log 15/ log 3 D 2:465 : : : [3]. Parallel to the case of the comb, the main mechanism leading to subdiffusion is trapping in dangling ends, which, different from the case of the comb, now themselves have a fractal structure. Parallel to the case of the comb, the time of travel

Anomalous Diffusion on Fractal Networks, Figure 3 A Vicsek fractal: the generator (left) and its 4th iteration. This is a loopless structure; the diffusion anomalies here are mainly caused by trapping in dangling ends

from one site of the structure to another one is strongly affected by trapping. However, trapping inside the dangling end does not lead to a halt of the particle, but only to confining its motion to some scale, so that the overall equations governing the PDF might be different in the form from the fractional diffusion ones. Diffusion and Conductivity The random walks of particles on a network can be described by a master equation (which is perfectly valid for the exponential waiting time distribution on a site and asymptotically valid in case of all other waiting time distributions with finite mean waiting time ): Let p(t) be the vector with elements p i (t) being the probabilities to find a particle at node i at time t. The master equation d p D Wp dt

(9)

gives then the temporal changes in this probability. A similar equation with the temporal derivative changed to d/dn

19

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Anomalous Diffusion on Fractal Networks

describes the n-dependence for the probabilities p i;n in a random walk as a function of the number of steps, provided n is large enough to be considered as a continuous variable. The matrix W describes the transition probabilities between the nodes of the lattice or network. The nondiagonal elements of the corresponding matrix are wij , the transition probabilities from site i to site j per unit time or in one step. The diagonal elements are the sums of all non-diagonal elements in the corresponding column P taken with the opposite sign: w i i D  j w ji , which represents the probability conservation law. The situation of unbiased random walks corresponds to a symmetric matrix W: w i j D w ji . Considering homogeneous networks and putting all nonzero wij to unity, one sees that the difference operator represented by each line of the matrix is a symmetric difference approximation to a Laplacian. The diffusion problem is intimately connected with the problem of conductivity of a network, as described by Kirchhoff’s laws. Indeed, let us consider a stationary situation in which the particles enter the network at some particular site A at constant rate, say, one per unit time, and leave it at some site B (or a given set set of sites B). Let us assume that after some time a stationary distribution of the particles over the lattice establishes, and the particles’ concentration on the sites will be described by the vector proportional to the vector of stationary probabilities satisfying Wp D 0 :

(10)

Calculating the probabilities p corresponds then formally to calculating the voltages on the nodes of the resistor network of the same geometry under given overall current using the Kirchhoff’s laws. The conductivities of resistors connecting nodes i and j have to be taken proportional to the corresponding transition probabilities wij . The condition given by Eq. (10) corresponds then to the second Kirchhoff’s law representing the particle conservation (the fact that the sum of all currents to/from the node i is zero), and the fact that the corresponding probability current is proportional to the probability difference is replaced by the Ohm’s law. The first Kirchhoff’s law follows from the uniqueness of the solution. Therefore calculation of the dimension of a walk can be done by discussing the scaling of conductivity with the size of the fractal object. This typically follows a power law. As an example let us consider a Sierpinski lattice and calculate the resistance between the terminals A and B (depicted by thick wires outside of the triangle in Fig. 2c) of a fractal of the next generation, assuming that the conductivity between the corresponding nodes of the lattice

of the previous generation is R D 1. Using the trianglestar transformation known in the theory of electric circuits, i. e. passing to the structure shown in Fig. 2c by thick lines inside the triangle, with the conductivity of each bond r D 1/2 (corresponding to the same value of the conductivity between the terminals), we get the resistance of a renormalized structure R0 D 5/3. Thus, the dependence of R on the spacial scale L of the object is R / L with  D log(5/3)/ log 2. The scaling of the conductivity G is correspondingly G / L . Using the flow-over-population approach, known in calculation of the mean first passage time in a system, we get I / N/hti, where I is the probability current through the system (the number of particles entering A per unit time), N is the overall stationary number of particles within the system and hti is the mean time a particle spends inside the system, i. e. the mean first passage time from A to B. The mean number of particles inside the system is proportional to a typical concentration (say to the probability pA to find a particle at site A for the given current I) and to the number of sites. The first one, for a given current, scales as the system’s resistivity, pA / R / L , and the second one, clearly, as Ld f where L is the system’s size. On the other hand, the mean first passage time scales according to hti / Ld w (this time corresponds to the typical number of steps during which the walk transverses an Euclidean distance L), so that dw D df C  :

(11)

Considering the d-dimensional generalizations of Sierpinski gaskets and using analogous considerations we get  D log[(d C 3)/(d C 1)]/ log 2. Combining this with the fractal dimension df D log(d C 1)/ log 2 of a gasket gives us for the dimension of the walks dw D log(d C 3)/ log 2. The relation between the scaling exponent of the conductivity and the dimension of a random walk on a fractal system can be used in the opposite way, since dw can easily be obtained numerically for a whatever structure. On the other hand, the solutions of the Kirchhoff’s equations on complex structures (i. e. the solution of a large system of algebraic equations), which is typically achieved using relaxation algorithms, is numerically much more involved. We note that the expression for  can be rewritten as  D df (2/ds  1), where ds is the spectral dimension of the network, which will be introduced in Subsect. “The Spectral Dimension”. The relation between the walk dimension and this new quality therefore reads dw D

2df : ds

(12)

Anomalous Diffusion on Fractal Networks

A different discussion of the same problem based on the Einstein’s relation between diffusion coefficient and conductivity, and on crossover arguments can be found in [10,20]. The Spectral Dimension In mathematical literature spectral dimension is defined through the return probability of the random walk, i. e. the probability to be at the origin after n steps or at time t. We consider a node on a network and take it to be an origin of a simple random walk. We moreover consider the probability P(0; t) to be at this node at time t (a return probability). The spectral dimension defines then the asymptotic behavior of this probability for n large: P(0; t) / t d s /2 ;

for t ! 1 :

(13)

The spectral dimension ds is therefore exactly a quantity substituting the Euclidean dimension d is Eq. (1). In the statistical case one should consider an average of p(t) over the origin of the random walk and over the ensemble of the corresponding graphs. For fixed graphs the spectral dimension and the fractal (Hausdorff) dimension are related by 2df  ds  df ; 1 C df

(14)

provided both exist [5]. The same relation is also shown true for some random geometries, see e. g. [6,7] and references therein. The bounds are optimal, i. e. both equalities can be realized in some structures. The connection between the walk dimension dw and the spectral dimension ds of a network can be easily rationalized by the following consideration. Let the random walk have a dimension dw , so that after time t the position of the walker can be considered as more or less homogeneously spread within the spatial region of the linear size R / n1/d w or R / t 1/d w , with the overall number of nodes N / R d f / t d f /d w . The probability to be at one particular node, namely at 0, goes then as 1/N, i. e. as t d f /d w . Comparing this with the definition of the spectral dimension, Eq. (13) we get exactly Eq. (12). The lower bound in the inequality Eq. (14) follows from Eq. (13) and from the the observation that the value of  in Eq. (11) is always smaller or equal to one (the value of  D 1 would correspond to a one-dimensional wire without shunts). The upper bound can be obtained using Eq. (12) and noting that the walks’ dimension never gets less than 2, its value for the regular network: the random walk on a whatever structure is more compact than one on a line.

We note that the relation Eq. (12) relies on the assumption that the spread of the walker within the r-domain is homogeneous, and needs reinterpretation for strongly anisotropic structures like combs, where the spread in teeth and along the spine are very different (see e. g. [2]). For the planar comb with infinite teeth df D 2, and ds as calculated through the return time is ds D 3/2, while the walk’s dimension (mostly determined by the motion in teeth) is dw D 2, like for random walks in Euclidean space. The inequality Eq. (14) is, on the contrary, universal. Let us discuss another meaning of the spectral dimension, the one due to which it got its name. This one has to do with the description of random walks within the master equation scheme. From the spectral (Laplace) representation of the solution of the master equation, Eq. (9), we can easily find the probability P(0; t) that the walker starting at site 0 at t D 0 is found at the same site at time t. This one reads 1 X a i exp( i t) ; P(0; t) D iD1

where i is the ith eigenvalue of the matrix W and ai is the amplitude of its ith eigenvector at site 0. Considering the lattice as infinite, we can pass from discrete eigenvalue decomposition to a continuum Z 1 N ( )a( ) exp( t)d : P(0; t) D 0

For long times, the behavior of P(0; t) is dominated by the behavior of N ( ) for small values of . Here N ( ) is the density of states of a system described by the matrix W. The exact forms of such densities are well known for many Euclidean lattices, since the problem is equivalent to the calculating of spectrum in tight-binding approximation used in the solid state physics. For all Euclidean lattices N ( ) / d/21 for ! 0, which gives us the forms of famous van Hove singularities of the spectrum. Assuming a( ) to be nonsingular at ! 0, we get P(0; t) / t d/2 . For a fractal structure, the value of d changed for the one of the spectral dimension ds . This corresponds to the density of states N ( ) / d s /21 which describes the properties of spectrum of a fractal analog of a Laplace operator. The dimension ds is also often called fracton dimension of the structure, since the corresponding eigenvectors of the matrix (corresponding to eigenstates in tight-binding model) are called fractons, see e. g. [16]. In the examples of random walks on quenched polymer chains the dimension dw was twice the fractal dimension of the underlying structures, so that the spectral dimension of the corresponding structures (without intersections) was exactly 1 just like their topological dimension.

21

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Anomalous Diffusion on Fractal Networks

The spectral dimension of the network governs also the behavior of the mean number of different sites visited by the random walk. A random walk of n steps having a property of compact visitation typically visits all sites within the radius of the order of its typical displacement R n / n1/d w . The number of these sites is S n / R dnf where df is the fractal dimension of the network, so that S n / nd s /2 (provided ds  2, i. e. provided the random walk is recurrent and shows compact visitation). The spectral dimension of a structure plays important role in many other applications [16]. This and many other properties of complex networks related to the spectrum of W can often be easier obtained by considering random walk on the lattice than by obtaining the spectrum through direct diagonalization. Percolation Clusters Percolation clusters close to criticality are one of the most important examples of fractal networks. The properties of these clusters and the corresponding fractal and spectral dimensions are intimately connected to the critical indices in percolation theory. Thus, simple crossover arguments show that the fractal dimension of an incipient infinite percolation cluster in d  6 dimensions is df D d  ˇ/, where ˇ is the critical exponent of the density of the infinite cluster, P1 / (p  pc )ˇ and  is the critical exponent of the correlation length,  / jp  pc j , and stagnates at df D 4 for d > 6, see e. g. [23]. On the other hand, the critical exponent t, describing the behavior of the resistance of a percolation system close to percolation threshold, / (p pc )t , is connected with the exponent  via the following crossover argument. The resistance of the system is the one of the infinite cluster. For p > pc this cluster can be considered as fractal at scales L <  and as homogeneous at larger scales. Our value for the fractal dimension of the cluster follows exactly from this argument by matching the density of the cluster P(L) / L(d f d) at L D  / (p pc ) to its density P1 / (ppc )ˇ at larger scales. The infinite cluster for p > pc is then a dense regular assembly of subunits of size  which on their turn are fractal. If the typical resistivity of each fractal subunit is R then the resistivity of the overall assembly goes as R / R (L/)2d . This is a well-known relation showing that the resistivity of a wire grows proportional to its length, the resistivities of similar flat figures are the same etc. This relation holds in the homogeneous regime. On the other hand, in the fractal regime R L / L , so that overall R /  Cd2 / (p  pc )(Cd2) giving the value of the critical exponent t D ( C d  2). The

value of  can in its turn be expressed through the values of spectral and fractal dimensions of a percolation cluster. The spectral dimension of the percolation cluster is very close to 4/3 in any dimension larger then one. This surprising finding lead to the conjecture by Alexander and Orbach that this value of ds D 4/3 might be exact [1] (it is exact for percolation systems in d  6 and for trees). Much effort was put into proving or disproving the conjecture, see the discussion in [10]. The latest, very accurate simulations of two-dimensional percolation by Grassberger show that the conjecture does not hold in d D 2, and the prediction ds D 4/3 is off by around 2% [9]. In any case it can be considered as a very useful mnemonic rule. Anomalous diffusion on percolation clusters corresponds theoretically to the most involved case, since it combines all mechanisms generating diffusion anomalies. The infinite cluster of the percolation system consists of a backbone, its main current-carrying structure, and the dangling ends (smaller clusters on all scales attached to the backbone through only one point). The anomalous diffusion on the infinite cluster is thus partly caused by trapping in this dangling ends. If one considers a shortest (chemical) way between the two nodes on the cluster, this way is tortuous and has fractal dimension larger than one. The Role of Finite Clusters The exponent t of the conductivity of a percolation system is essentially the characteristic of an infinite cluster only. Depending on the physical problem at hand, one can consider the situations when only the infinite cluster plays the role (like in the experiments of [12], where only the infinite cluster is filled by fluid pressed through the boundary of the system), and situations when the excitations can be found in infinite as well as in the finite clusters, as it is the case for optical excitation in mixed molecular crystals, a situation discussed in [24]. Let us concentrate on the case p D pc . The structure of large but finite clusters at lower scales is indistinguishable from those of the infinite cluster. Thus, the mean squared displacement of a walker on a cluster of size L (one with N / Ld f sites) grows as R2 (t) / t 2/d w until is stagnates at the value of R of the order of the cluster’s size, R2 / N 2/d f . At a given time t we can subdivide all clusters into two classes: those whose size is small compared to t 1/d w (i. e. the ones with N < t d s /2 ), on which the mean square displacement stagnates, and those of the larger size, on which it still grows: ( 2

R (t) /

t 2/d w ;

t 1/d w < N 1/ f f

N 2/d f

otherwise:

;

Anomalous Diffusion on Fractal Networks

The characteristic size Ncross (t) corresponds to the crossover between these two regimes for a given time t. The probability that a particle starts at a cluster of size N is proportional to the number of its sites and goes as P(N) / N p(N) / N 1 with  D (2d ˇ)/(d ˇ), see e. g. [23]. Here p(N) is the probability to find a cluster of N sites among all clusters. The overall mean squared displacement is then given by averaging over the corresponding cluster sizes: hr2 (t)i /

N cross X(t)

N 2/d f N 1 C

ND1

1 X

t 2/d w N 1 :

N cross (t)

Introducing the expression for Ncross (t), we get hr2 (t)i / t 2/d w C(2)d f /d w D t (1/d w )(2d f ˇ /) : A similar result holds for the number of distinct sites visited, where one has to perform the analogous averaging with ( t d s /2 ; t 1/d w < N 1/ ff S(t) / N; otherwise giving S(t) / t (d s /2)(1ˇ/d f ) . This gives us the effective spectral dimension of the system d˜s D (ds /2)(1  ˇ/df ). Scaling of PDF and Diffusion Equations on Fractal Lattices If the probability density of the displacement of a particle on a fractal scales, the overall scaling form of this displacement can be obtained rather easily: Indeed, the typical displacement R during the time t goes as r / t 1/d w , so that the corresponding PDF has to scale as P(r; t) D

r d f d f t d s /2



r t 1/d w

 :

(15)

The prefactor takes into account the normalization of the probability density on a fractal structure (denominator) and also the scaling of the density of the fractal structure in the Euclidean space (enumerator). The simple scaling assumes that all the moments ofR the distance scale P(r; t)r k d d r k D in the same way, so that hr k i D k/d w . The overall scaling behavior of the PDF was const(k)t confirmed numerically for many fractal lattices like Sierpinski gaskets and percolation clusters, and is corroborated in many other cases by the coincidence of the values of dw obtained via calculation of the mean squared displacement and of the mean first passage time (or resistivity of the network). The existence of the scaling form

leads to important consequences e. g. for quantification of anomalous diffusion in biological tissues by characterizing the diffusion-time dependence of the magnetic resonance signal [17], which, for example, allows for differentiation between the healthy and the tumor tissues. However, even in the cases when the scaling form Eq. (15) is believed to hold, the exact form of the scaling function f (x) is hard to get; the question is not yet resolved even for simple regular fractals. For applications it is often interesting to have a kind of a phenomenological equation roughly describing the behavior of the PDF. Such an approach will be analogous to putting down Richardson’s equation for the turbulent diffusion. Essentially, the problem here is to find a correct way of averaging or coarse-graining the microscopic master equation, Eq. (9), to get its valid continuous limit on larger scales. The regular procedure to do this is unknown; therefore, several phenomenological approaches to the problem were formulated. We are looking for an equation for the PDF of the walker’s displacement from its initial position and assume the system to be isotropic on the average. We look for @ an analog of the classical Fick’s equation, @t P(r; t) D rK(r)rP(r; t) which in spherical coordinates and for K D K(r) takes the form of Eq. (3):

 @ 1 @ d1 @ P(r; t) D d1 r K(r) P(r; t) : @t @r @r r On fractal lattices one changes from the Euclidean dimension of space to the fractal dimension of the lattice df , and takes into account that the effective diffusion coefficient K(r) decays with distance as K(r) ' Kr with D dw 2 to capture the slowing down of anomalous diffusion on a fractal compared to the Euclidean lattice situation. The corresponding equation put forward by O’Shaughnessy and Procaccia [18] reads:

 1 @ d f d w C1 @ @ P(r; t) D K d 1 r P(r; t) : (16) @t @r r f @r This equation was widely used in description of anomalous diffusion on fractal lattices and percolation clusters, but can be considered only as a rough phenomenological approximation. Its solution ! r d f d rdw P(r; t) D A d /2 exp B ; t t s (with B D 1/Kdw2 and A being the corresponding normalization constant) corresponds exactly to the type of scaling given by Eq. (15), with the scaling function f (x) D

23

24

Anomalous Diffusion on Fractal Networks

exp(x d w ). This behavior of the scaling function disagrees e. g. with the results for random walks on polymer chains, for which case we had f (x) D exp(x  ) with  D (1  1/dw )1 for large enough x. In literature, several other proposals (based on plausible assumptions but to no extent following uniquely from the models considered) were made, taking into account possible non-Markovian nature of the motion. These are the fractional equations of the type i @1/d w 1 @ h (d s 1)/2 P(r; t) D K P(r; t) ; (17) r 1 @t 1/d w r(d s 1)/2 @r resembling “half” of the diffusion equation and containing a fractional derivative [8], as well as the “full” fractional diffusion equation

 1 @ d s 1 @ @2/d w P(r; t) D K r P(r; t) ; 2 d 1 @r @t 2/d w r s @r

(18)

as proposed in [15]. All these equation are invariant under the scale transformation t ! t ;

r ! 1/d w r ;

and lead to PDFs showing correct overall scaling properties, see [19]. None of them reproduces correctly the PDF of displacements on a fractal (i. e. the scaling function f (x)) in the whole range of distances. Ref. [19] comparing the corresponding solutions with the results of simulation of anomalous diffusion an a Sierpinski gasket shows that the O’Shaughnessy–Procaccia equation, Eq. (16), performs the best for the central part of the distribution (small displacements), where Eq. (18) overestimates the PDF and Eq. (17) shows an unphysical divergence. On the other hand, the results of Eqs. (17) and (18) reproduce equally well the PDF’s tail for large displacements, while Eq. (16) leads to a PDF decaying considerably faster than the numerical one. This fact witnesses favor of strong non-Markovian effects in the fractal diffusion, however, the physical nature of this non-Markovian behavior observed here in a fractal network without dangling ends, is not as clear as it is in a comb model and its more complex analogs. Therefore, the question of the correct equation describing the diffusion in fractal system (or different correct equations for the corresponding different classes of fractal systems) is still open. We note also that in some cases a simple fractional diffusion equation (with the corresponding power of the derivative leading to the correct scaling exponent dw ) gives a reasonable approximation to experimental data, as the ones of the model experiment of [12].

Further Directions The physical understanding of anomalous diffusion due to random walks on fractal substrates may be considered as rather deep and full although it does not always lead to simple pictures. For example, although the spectral dimension for a whatever graph can be rather easily calculated, it is not quite clear what properties of the graph are responsible for its particular value. Moreover, there are large differences with respect to the degree of rigor with which different statements are proved. Mathematicians have recognized the problem, and the field of diffusion on fractal networks has become a fruitful field of research in probability theory. One of the examples of recent mathematical development is the deep understanding of spectral properties of fractal lattices and a proof of inequalities like Eq. (14). A question which is still fully open is the one of the detailed description of the diffusion within a kind of (generalized) diffusion equation. It seems clear that there is more than one type of such equations depending on the concrete fractal network described. However even the classification of possible types of such equations is still missing. All our discussion (except for the discussion of the role of finite clusters in percolation) was pertinent to infinite structures. The recent work [4] has showed that finite fractal networks are interesting on their own and opens a new possible direction of investigations. Bibliography Primary Literature 1. Alexander S, Orbach RD (1982) Density of states on fractals– fractons. J Phys Lett 43:L625–L631 2. Bertacci D (2006) Asymptotic behavior of the simple random walk on the 2-dimensional comb. Electron J Probab 45:1184– 1203 3. Christou A, Stinchcombe RB (1986) Anomalous diffusion on regular and random models for diffusion-limited aggregation. J Phys A Math Gen 19:2625–2636 4. Condamin S, Bénichou O, Tejedor V, Voituriez R, Klafter J (2007) First-passage times in complex scale-invariant media. Nature 450:77–80 5. Coulhon T (2000) Random Walks and Geometry on Infinite Graphs. In: Ambrosio L, Cassano FS (eds) Lecture Notes on Analysis on Metric Spaces. Trento, CIMR, (1999) Scuola Normale Superiore di Pisa 6. Durhuus B, Jonsson T, Wheather JF (2006) Random walks on combs. J Phys A Math Gen 39:1009–1037 7. Durhuus B, Jonsson T, Wheather JF (2007) The spectral dimension of generic trees. J Stat Phys 128:1237–1260 8. Giona M, Roman HE (1992) Fractional diffusion equation on fractals – one-dimensional case and asymptotic-behavior. J Phys A Math Gen 25:2093–2105; Roman HE, Giona M, Fractional diffusion equation on fractals – 3-dimensional case and scattering function, ibid., 2107–2117

Anomalous Diffusion on Fractal Networks

9. Grassberger P (1999) Conductivity exponent and backbone dimension in 2-d percolation. Physica A 262:251–263 10. Havlin S, Ben-Avraham D (2002) Diffusion in disordered media. Adv Phys 51:187–292 11. Klafter J, Sokolov IM (2005) Anomalous diffusion spreads its wings. Phys World 18:29–32 12. Klemm A, Metzler R, Kimmich R (2002) Diffusion on randomsite percolation clusters: Theory and NMR microscopy experiments with model objects. Phys Rev E 65:021112 13. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77 14. Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A Math Gen 37:R161–R208 15. Metzler R, Glöckle WG, Nonnenmacher TF (1994) Fractional model equation for anomalous diffusion. Physica A 211:13–24 16. Nakayama T, Yakubo K, Orbach RL (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 17. Özarslan E, Basser PJ, Shepherd TM, Thelwall PE, Vemuri BC, Blackband SJ (2006) Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal. J Magn Res 183:315–323 18. O’Shaughnessy B, Procaccia I (1985) Analytical solutions for diffusion on fractal objects. Phys Rev Lett 54:455–458 19. Schulzky C, Essex C, Davidson M, Franz A, Hoffmann KH (2000) The similarity group and anomalous diffusion equations. J Phys A Math Gen 33:5501–5511 20. Sokolov IM (1986) Dimensions and other geometrical critical exponents in the percolation theory. Usp Fizicheskikh Nauk 150:221–255 (1986) translated in: Sov. Phys. Usp. 29:924 21. Sokolov IM, Klafter J (2005) From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion. Chaos 15:026103 22. Sokolov IM, Mai J, Blumen A (1997) Paradoxical diffusion in chemical space for nearest-neighbor walks over polymer chains. Phys Rev Lett 79:857–860

23. Stauffer D (1979) Scaling theory of percolation clusters. Phys Rep 54:1–74 24. Webman I (1984) Diffusion and trapping of excitations on fractals. Phys Rev Lett 52:220–223

Additional Reading The present article gave a brief overview of what is known about the diffusion on fractal networks, however this overview is far from covering all the facets of the problem. Thus, we only discussed unbiased diffusion (the effects of bias may be drastic due to e. g. stronger trapping in the dangling ends), and considered only the situations in which the waiting time at all nodes was the same (we did not discuss e. g. the continuous-time random walks on fractal networks), as well as left out of attention many particular systems and applications. Several review articles can be recommended as a further reading, some of them already mentioned in the text. One of the best-known sources is [10] being a reprint of the text from the “sturm und drang” period of investigation of fractal geometries. A lot of useful information on random walks models in general an on walks on fractals is contained in the review by Haus and Kehr from approximately the same time. General discussion of the anomalous diffusion is contained in the work by Bouchaud and Georges. The classical review of the percolation theory is given in the book of Stauffer and Aharony. Some additional information on anomalous diffusion in percolation systems can be found in the review by Isichenko. A classical source on random walks in disordered systems is the book by Hughes. Haus JW, Kehr K (1987) Diffusion in regular and disordered lattices. Phys Rep 150:263–406 Bouchaud JP, Georges A (1990) Anomalous diffusion in disordered media – statistical mechanisms, models and physical applications. Phys Rep 195:127–293 Stauffer D, Aharony A (2003) Introduction to Percolation Theory. Taylor & Fransis, London Isichenko MB (1992) Percolation, statistical topography, and transport in random-media. Rev Mod Phys 64:961–1043 Hughes BD (1995) Random Walks and random Environments. Oxford University Press, New York

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

Biological Fluid Dynamics, Non-linear Partial Differential Equations ANTONIO DESIMONE1 , FRANÇOIS ALOUGES2 , ALINE LEFEBVRE2 1 SISSA-International School for Advanced Studies, Trieste, Italy 2 Laboratoire de Mathématiques, Université Paris-Sud, Orsay cedex, France Article Outline Glossary Definition of the Subject Introduction The Mathematics of Swimming The Scallop Theorem Proved Optimal Swimming The Three-Sphere Swimmer Future Directions Bibliography Glossary Swimming The ability to advance in a fluid in the absence of external propulsive forces by performing cyclic shape changes. Navier–Stokes equations A system of partial differential equations describing the motion of a simple viscous incompressible fluid (a Newtonian fluid)   @v C (v  r)v D r p C v

@t div v D 0 where v and p are the velocity and the pressure in the fluid, is the fluid density, and  its viscosity. For simplicity external forces, such as gravity, have been dropped from the right hand side of the first equation, which expresses the balance between forces and rate of change of linear momentum. The second equation constrains the flow to be volume preserving, in view of incompressibility. Reynolds number A dimensionless number arising naturally when writing Navier–Stokes equations in nondimensional form. This is done by rescaling position and velocity with x  D x/L and v  D v/V , where L and V are characteristic length scale and velocity associated with the flow. Reynolds number (Re) is defined by Re D

VL V L D  

where  D / is the kinematic viscosity of the fluid, and it quantifies the relative importance of inertial versus viscous effects in the flow. Steady Stokes equations A system of partial differential equations arising as a formal limit of Navier–Stokes equations when Re ! 0 and the rate of change of the data driving the flow (in the case of interest here, the velocity of the points on the outer surface of a swimmer) is slow v C r p D 0 div v D 0 : Flows governed by Stokes equations are also called creeping flows. Microscopic swimmers Swimmers of size L D 1 μm moving in water ( 1 mm2 /s at room temperature) at one body length per second give rise to Re 106 . By contrast, a 1 m swimmer moving in water at V D 1 m/s gives rise to a Re of the order 106 . Biological swimmers Bacteria or unicellular organisms are microscopic swimmers; hence their swimming strategies cannot rely on inertia. The devices used for swimming include rotating helical flagella, flexible tails traversed by flexural waves, and flexible cilia covering the outer surface of large cells, executing oar-like rowing motion, and beating in coordination. Self propulsion is achieved by cyclic shape changes described by time periodic functions (swimming strokes). A notable exception is given by the rotating flagella of bacteria, which rely on a submicron-size rotary motor capable of turning the axis of an helix without alternating between clockwise and anticlockwise directions. Swimming microrobots Prototypes of artificial microswimmers have already been realized, and it is hoped that they can evolve into working tools in biomedicine. They should consist of minimally invasive, small-scale self-propelled devices engineered for drug delivery, diagnostic, or therapeutic purposes. Definition of the Subject Swimming, i. e., being able to advance in a fluid in the absence of external propulsive forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers (Re). This is the regime of interest for micro-organisms and micro-robots or nano-robots, where hydrodynamics is governed by Stokes equations. Thus, besides the rich mathematics it generates, low Re propulsion is of great interest in biology (How do microorganism swim? Are their strokes optimal and, if so, in which sense? Have

Biological Fluid Dynamics, Non-linear Partial Differential Equations

these optimal swimming strategies been selected by evolutionary pressure?) and biomedicine (can small-scale selfpropelled devices be engineered for drug delivery, diagnostic, or therapeutic purposes?). For a microscopic swimmer, moving and changing shape at realistically low speeds, the effects of inertia are negligible. This is true for both the inertia of the fluid and the inertia of the swimmer. As pointed out by Taylor [10], this implies that the swimming strategies employed by bacteria and unicellular organism must be radically different from those adopted by macroscopic swimmers such as fish or humans. As a consequence, the design of artificial microswimmers can draw little inspiration from intuition based on our own daily experience. Taylor’s observation has deep implications. Based on a profound understanding of low Re hydrodynamics, and on a plausibility argument on which actuation mechanisms are physically realizable at small length scales, Berg postulated the existence of a sub-micron scale rotary motor propelling bacteria [5]. This was later confirmed by experiment.

cal simplification is obtained by looking at axisymmetric swimmers which, when advancing, will do so by moving along the axis of symmetry. Two such examples are the three-sphere-swimmer in [7], and the push-me–pull-you in [3]. In fact, in the axisymmetric case, a simple and complete mathematical picture of low Re swimming is now available, see [1,2]. The Mathematics of Swimming This article focuses, for simplicity, on swimmers having an axisymmetric shape ˝ and swimming along the axis of symmetry, with unit vector E{ . The configuration, or state s of the system is described by N C 1 scalar parameters: s D fx (1) ; : : : ; x (NC1) g. Alternatively, s can be specified by a position c (the coordinate of the center of mass along the symmetry axis) and by N shape parameters  D f (1) ; : : : ;  (N) g. Since this change of coordinates is invertible, the generalized velocities u(i) : D x˙ (i) can be represented as linear functions of the time derivatives of position and shape: (u(1) ; : : : ; u(NC1) ) t D A( (1) ; : : : ;  (N) )(˙(1) ; : : : ; ˙(N) ; c˙) t

Introduction In his seminal paper Life at low Reynolds numbers [8], Purcell uses a very effective example to illustrate the subtleties involved in microswimming, as compared to the swimming strategies observable in our mundane experience. He argues that at low Re, any organism trying to swim adopting the reciprocal stroke of a scallop, which moves by opening and closing its valves, is condemned to the frustrating experience of not having advanced at all at the end of one cycle. This observation, which became known as the scallop theorem, started a stream of research aiming at finding the simplest mechanism by which cyclic shape changes may lead to effective self propulsion at small length scales. Purcell’s proposal was made of a chain of three rigid links moving in a plane; two adjacent links swivel around joints and are free to change the angle between them. Thus, shape is described by two scalar parameters (the angles between adjacent links), and one can show that, by changing them independently, it is possible to swim. It turns out that the mechanics of swimming of Purcell’s three-link creature are quite subtle, and a detailed understanding has started to emerge only recently [4,9]. In particular, the direction of the average motion of the center of mass depends on the geometry of both the swimmer and of the stroke, and it is hard to predict by simple inspection of the shape of the swimmer and of the sequence of movements composing the swimming stroke. A radi-

(1) where the entries of the N C 1  N C 1 matrix A are independent of c by translational invariance. Swimming describes the ability to change position in the absence of external propulsive forces by executing a cyclic shape change. Since inertia is being neglected, the total drag force exerted by the fluid on the swimmer must also vanish. Thus, since all the components of the total force in directions perpendicular to E{ vanish by symmetry, self-propulsion is expressed by Z 0D  n  E{ (2) @˝

where  is the stress in the fluid surrounding ˝, and n is the outward  unit normal to @˝. The stress  D  rv C (rv) t  pId is obtained by solving Stokes equation outside ˝ with prescribed boundary data v D v¯ on @˝. In turn, v¯ is the velocity of the points on the boundary @˝ of the swimmer, which moves according to (1). By linearity of Stokes equations, (2) can be written as 0D

NC1 X iD1 t

' (i) ( (1) ; : : : ;  (N) )u(i)

D A ˚  (˙(1) ; : : : ; ˙(N) ; c˙) t

(3)

where ˚ D (' (1) ; : : : ; ' (N) ) t , and we have used (1). Notice that the coefficients ' (i) relating drag force to velocities are

27

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

Biological Fluid Dynamics, Non-linear Partial Differential Equations, Figure 1 A mirror-symmetric scallop or an axisymmetric octopus

independent of c because of translational invariance. The coefficient of c˙ in (3) represents the drag force corresponding to a rigid translation along the symmetry axis at unit speed, and it never vanishes. Thus (3) can be solved for c˙, and we obtain c˙ D

N X

Vi ( (1) ; : : : ;  (N) )˙(i) D V ()  ˙ :

(4)

iD1

Equation (4) links positional changes to shape changes through shape-dependent coefficients. These coefficients encode all hydrodynamic interactions between ˝ and the surrounding fluid due to shape changes with rates ˙(1) ; : : : ; ˙(N) . A stroke is a closed path in the space S of admissible shapes given by [0; T] 3 t 7! ( (1) ; : : :  (N1) ). Swimming requires that Z

N TX

0 ¤ c D 0

Vi ˙(i) dt

(5)

iD1

i. e., that the differential form

PN

iD1 Vi

d (i) is not exact.

The Scallop Theorem Proved Consider a swimmer whose motion is described by a parametrized curve in two dimensions (N D 1), so that (4) becomes ˙ ; c˙(t) D V((t))(t)

t 2R;

(6)

and assume that V 2 L1 (S) is an integrable function in the space of admissible shapes and  2 W 1;1 (R; S) is a Lipschitz-continuous and T-periodic function for some T > 0, with values in S. Figure 1 is a sketch representing concrete examples compatible with these hypotheses. The axisymmetric case consists of a three-dimensional cone with axis along E{ and opening angle  2 [0; 2] (an axisymmetric octopus). A non-axisymmetric example is also allowed in

this discussion, consisting of two rigid parts (valves), always maintaining mirror symmetry with respect to a plane (containing E{ and perpendicular to it) while swiveling around a joint contained in the symmetry plane and perpendicular to E{ (a mirror-symmetric scallop), and swimming parallel to E{ . Among the systems that are not compatible with the assumptions above are those containing helical elements with axis of rotation E{ , and capable of rotating around E{ always in the same direction (call the rotation angle). Indeed, a monotone function t 7! (t) is not periodic. The celebrated “scallop theorem” [8] states that, for a system like the one depicted in Fig. 1, the net displacement of the center of mass at the end of a periodic stroke will always vanish. This is due to the linearity of Stokes equation (which leads to symmetry under time reversals), and to the low dimensionality of the system (a one-dimensional periodic stroke is necessarily reciprocal). Thus, whatever forward motion is achieved by the scallop by closing its valves, it will be exactly compensated by a backward motion upon reopening them. Since the low Re world is unaware of inertia, it will not help to close the valves quickly and reopen them slowly. A precise statement and a rigorous short proof of the scallop theorem are given below. Theorem 1 Consider a swimmer whose motion is described by ˙ ; c˙(t) D V((t))(t)

t 2 R;

(7)

with V 2 L1 (S). Then for every T-periodic stroke  2 W 1;1 (R; S), one has Z

T

c˙(t)dt D 0 :

c D

(8)

0

Proof Define the primitive of V by Z s  (s) D V( ) d 0

(9)

Biological Fluid Dynamics, Non-linear Partial Differential Equations

so that  0 () D V(). Then, using (7), Z

T

c D

Substituting (13) in (11), the expended power becomes a quadratic form in ˙

˙ V ((t))(t)dt

Z

0

Z

T

@˝

d  ((t))dt dt

D 0

Z

by the T-periodicity of t 7! (t).

G i j () D

Optimal Swimming A classical notion of swimming efficiency is due to Lighthill [6]. It is defined as the inverse of the ratio between the average power expended by the swimmer during a stroke starting and ending at the shape 0 D (0(1) ; : : : ; 0(N) ) and the power that an external force would spend to translate the system rigidly at the same average speed c¯ D c/T : Eff

D

1 T

RTR

R1R n  v n  v D 0 @˝ 6L c¯2 6L(c)2

0

@˝

(10)

@˝

At a point p 2 @˝, the velocity v(p) accompanying a change of state of the swimmer can be written as a linear combination of the u(i) v(p) D

NC1 X

V i (p; )u(i)

(12)

W i (p; )˙(i) :

(13)

iD1

D

N X iD1

Indeed, the functions V i are independent of c by translational invariance, and (4) has been used to get (13) from the line above.

@˝

DN(W i (p; ))  W j (p; )dp :

(15)

Strokes of maximal efficiency may be defined as those producing a given displacement c of the center of mass with minimal expended power. Thus, from (10), maximal efficiency is obtained by minimizing Z

Z

1Z 0

@˝

1

n  v D

˙ ) ˙ (G();

(16)

0

subject to the constraint Z

where  is the viscosity of the fluid, L D L(0 ) is the effective radius of the swimmer, and time has been rescaled to a unit interval to obtain the second identity. The expression in the denominator in (10) comes from a generalized version of Stokes formula giving the drag on a sphere of radius L moving at velocity c¯ as 6L c¯. Let DN: H 1/2 (@˝) ! H 1/2 (@˝) be the Dirichlet to Neumann map of the outer Stokes problem, i. e., the map such that  n D DNv, where  is the stress in the fluid, evaluated on @˝, arising in response to the prescribed velocity v on @˝, and obtained by solving the Stokes problem outside ˝. The expended power in (10) can be written as Z Z n  v D DN(v)  v : (11) @˝

(14)

where the symmetric and positive definite matrix G() is given by

D  ((T))   ((0)) D 0

1

˙ ) ˙  n  v D (G();

c D

1

V()  ˙

(17)

0

among all closed curves  : [0; 1] ! S in the set S of admissible shapes such that (0) D (1) D 0 . The Euler–Lagrange equations for this optimization problem are 0

@G ˙ ˙ ;  @ (1) :: :

1

B C C   d ˙ 1B B C  (G )C B C C r V  rt V ˙ D 0 dt 2 B C @ @G A ˙ ˙ ; (N) @ (18) where r V is the matrix (r V ) i j D @Vi /@ j , rt V is its transpose, and  is the Lagrange multiplier associated with the constraint (17). Given an initial shape  0 and an initial position c0 , the solutions of (18) are in fact sub-Riemannian geodesics joining the states parametrized by (0 ; c0 ) and (0 ; c0 Cc) in the space of admissible states X , see [1]. It is well known, and easy to prove using (18), that along such geodesics ˙ is constant. This has interesting consequences, (G( ) ˙ ; ) because swimming strokes are often divided into a power phase, where jG( )j is large, and a recovery phase, where jG( )j is smaller. Thus, along optimal strokes, the recovery phase is executed quickly while the power phase is executed slowly.

29

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

Biological Fluid Dynamics, Non-linear Partial Differential Equations, Figure 2 Swimmer’s geometry and notation

The Three-Sphere Swimmer For the three-sphere-swimmer of Najafi and Golestanian [7], see Fig. 2, ˝ is the union of three rigid disjoint balls B(i) of radius a, shape is described by the distances x and y, the space of admissible shapes is S D (2a; C1)2 , and the kinematic relation (1) takes the form 1 u(1) D c˙  (2x˙ C y˙) 3 1 u(2) D c˙ C (x˙  y˙) 3 1 (3) u D c˙ C (2 y˙ C x˙ ) : 3

(19)

Consider, for definiteness, a system with a D 0:05 mm, swimming in water. Calling f (i) the total propulsive force on ball B(i) , we find that the following relation among forces and ball velocities holds 0 (1) 1 0 (1) 1 f u @ f (2) A D R(x; y) @ u(2) A (20) f (3) u(3) where the symmetric and positive definite matrix R is known as the resistance matrix. From this last equation, using also (19), the condition for self-propulsion f (1) C f (2) C f (3) D 0 is equivalent to c˙ D Vx (x; y)x˙ C Vy (x; y) y˙ ;

(21)

where

Biological Fluid Dynamics, Non-linear Partial Differential Equations, Table 1 Energy consumption (10–12 J) for the three strokes of Fig. 3 inducing the same displacement c D 0:01 mm in T D 1 s Optimal stroke Small square stroke Large square stroke 0.229 0.278 0.914

which is guaranteed, in particular, if curl V is bounded away from zero. Strokes of maximal efficiency for a given initial shape (x0 ; y0 ) and given displacement c are obtained by solving Eq. (18). For N D 2, this becomes   1 (@x G ˙ ; ˙ ) d C curlV( ) ˙ ? D 0 (25)  (G ˙ ) C ˙ dt 2 (@ y G ˙ ; ) where @x G and @ y G stand for the x and y derivatives of the 2  2 matrix G(x; y). It is important to observe that, for the three-sphere swimmer, all hydrodynamic interactions are encoded in the shape dependent functions V(x; y) and G(x; y). These can be found by solving a two-parameter family of outer Stokes problems, where the parameters are the distances x and y between the three spheres. In [1], this has been done numerically via the finite element method: a representative example of an optimal stroke, compared to two more naive proposals, is shown in Fig. 3. Future Directions

Re c  (e c  e y ) Vx (x; y) D Re c  (e x  e y ) Re c  (e c  e x ) Vy (x; y) D  : Re c  (e x  e y )

(22) (23)

Moreover, e x D (1; 1; 0) t , e y D (0; 1; 1) t , e c D (1/3; 1/3; 1/3) t . Given a stroke D @! in the space of admissible shapes, condition (5) for swimming reads Z T   Vx x˙ C Vy y˙ dt 0 ¤ c D Z0 curlV(x; y)dxdy (24) D !

The techniques discussed in this article provide a head start for the mathematical modeling of microscopic swimmers, and for the quantitative optimization of their strokes. A complete theory for axisymmetric swimmers is already available, see [2], and further generalizations to arbitrary shapes are relatively straightforward. The combination of numerical simulations with the use of tools from sub-Riemannian geometry proposed here may prove extremely valuable for both the question of adjusting the stroke to global optimality criteria, and of optimizing the stroke of complex swimmers. Useful inspiration can come from the sizable literature on the related field dealing with control of swimmers in a perfect fluid.

Biological Fluid Dynamics, Non-linear Partial Differential Equations

3. Avron JE, Kenneth O, Oakmin DH (2005) Pushmepullyou: an efficient micro-swimmer. New J Phys 7:234-1–8 4. Becker LE, Koehler SA, Stone HA (2003) On self-propulsion of micro-machines at low Reynolds numbers: Purcell’s three-link swimmer. J Fluid Mechanics 490:15–35 5. Berg HC, Anderson R (1973) Bacteria swim by rotating their flagellar filaments. Nature 245:380–382 6. Lighthill MJ (1952) On the Squirming Motion of Nearly Spherical Deformable Bodies through Liquids at Very Small Reynolds Numbers. Comm Pure Appl Math 5:109–118 7. Najafi A, Golestanian R (2004) Simple swimmer at low Reynolds numbers: Three linked spheres. Phys Rev E 69:062901-1–4 8. Purcell EM (1977) Life at low Reynolds numbers. Am J Phys 45:3–11 9. Tan D, Hosoi AE (2007) Optimal stroke patterns for Purcell’s three-link swimmer. Phys Rev Lett 98:068105-1–4 10. Taylor GI (1951) Analysis of the swimming of microscopic organisms. Proc Roy Soc Lond A 209:447–461 Biological Fluid Dynamics, Non-linear Partial Differential Equations, Figure 3 Optimal stroke and square strokes which induce the same displacement c D 0:01 mm in T D 1 s, and equally spaced level curves of curl V. The small circle locates the initial shape 0 D (0:3 mm; 0:3 mm)

Bibliography Primary Literature 1. Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number swimmers: an example. J Nonlinear Sci 18:277–302 2. Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number axisymmetric swimmers. Preprint SISSA 61/2008/M

Books and Reviews Agrachev A, Sachkov Y (2004) Control Theory from the Geometric Viewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87, Control Theory and Optimization. Springer, Berlin Childress S (1981) Mechanics of swimming and flying. Cambridge University Press, Cambridge Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Nijhoff, The Hague Kanso E, Marsden JE, Rowley CW, Melli-Huber JB (2005) Locomotion of Articulated Bodies in a Perfect Fluid. J Nonlinear Sci 15: 255–289 Koiller J, Ehlers K, Montgomery R (1996) Problems and Progress in Microswimming. J Nonlinear Sci 6:507–541 Montgomery R (2002) A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys and Monographs, vol 91. American Mathematical Society, Providence

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Catastrophe Theory

Catastrophe Theory W ERNER SANNS University of Applied Sciences, Darmstadt, Germany Article Outline Glossary Definition of the Subject Introduction Example 1: The Eccentric Cylinder on the Inclined Plane Example 2: The Formation of Traffic Jam Unfoldings The Seven Elementary Catastrophes The Geometry of the Fold and the Cusp Further Applications Future Directions Bibliography Glossary Singularity Let f : Rn ! Rm be a differentiable map defined in some open neighborhood of the point p 2 Rn and J p f its Jacobian matrix at p, consisting of the partial derivatives of all components of f with respect to all variables. f is called singular in p if rank J p f < min fn; mg. If rank J p f D min fm; ng, then f is called regular in p. For m D 1 (we call the map f : Rn ! R a differentiable function) the definition implies: A differentiable function f : Rn ! R is singular in p 2 Rn , if grad f (p) D 0. The point p, where the function is singular, is called a singularity of the function. Often the name “singularity” is used for the function itself if it is singular at a point p. A point where a function is singular is also called critical point. A critical point p of a function f is called a degenerated critical point if the Hessian (a quadratic matrix containing the second order partial derivatives) is singular at p; that means its determinant is zero at p. Diffeomorphism A diffeomorphism is a bijective differentiable map between open sets of Rn whose inverse is differentiable, too. Map germ Two continuous maps f : U ! R k and g : V ! R k , defined on neighborhoods U and V of p 2 Rn , are called equivalent as germs at p if there exists a neighborhood W U \ V on which both coincide. Maps or functions, respectively, that are equivalent as germs can be considered to be equal regarding local features. The equivalence classes of this equivalence relation are called germs. The set of all germs of

differentiable maps Rn ! R k at a point p is named " p (n; k). If p is the origin of Rn , one simply writes "(n; k) instead of "0 (n; k). Further, if k D 1, we write "(n) instead of "(n; 1) and speak of function germs (also simplified as “germs”) at the origin of Rn . "(n; k) is a vector space and "(n) is an algebra, that is, a vector space with a structure of a ring. The ring "(n) contains a unique maximal ideal (n) D f f 2 "(n)j f (0) D 0g. The ideal (n) is generated by the germs of the coordinate functions x1 ; : : : ; x n . We use the form (n) D hx1 ; : : : ; x n i"(n) to emphasize, that this ideal is generated over the ring "(n). So a function germ in (n) is of P the form niD1 a i (x)  x i , with certain function germs a i (x) 2 "(n). r-Equivalence Two function germs f ; g 2 "(n) are called r-equivalent if there exists a germ of a local diffeomorphism h : Rn ! Rn at the origin, such that g D f ı h. Unfolding An unfolding of a differentiable function germ f 2 (n) is a germ F 2 (nCr) with FjRn D f (here j means the restriction. The number r is called an unfolding dimension of f . An unfolding F of a germ f is called universal if every other unfolding of f can be received by suitable coordinate transformations, “morphisms of unfoldings” from F, and the number of unfolding parameters of F is minimal (see “codimension”). Unfolding morphism Suppose f 2 "(n) and F : Rn  R k ! R and G : Rn  Rr ! R be unfoldings of f . A right-morphism from F to G, also called unfolding morphism, is a pair (˚; ˛), with ˚ 2 "(n C k; n C r) and ˛ 2 (k), such that: 1. ˚jRn D id(Rn ), that is ˚(x; 0) D (x; 0), 2. If ˚ D (; ), with  2 "(n C k; n), k; r), then 2 "(k; r),

2 "(n C

3. For all (x; u) 2 Rn  R k we get F(x; u) D G(˚(x; u)) C ˛(u). Catastrophe A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. Codimension The codimension of a singularity f is given by codim( f ) D dimR (n)/h@x f i (quotient space). Here h@x f i is the Jacobian ideal generated by the partial derivatives of f and (n) D f f 2 "(n)j f (0) D 0g. The codimension of a singularity gives of the minimal number of unfolding parameters needed for the universal unfolding of the singularity. Potential function Let f : Rn ! Rn be a differentiable map (that is, a differentiable vector field). If there exists a function ' : Rn ! R with the property that

Catastrophe Theory

grad ' D f , then f is called a gradient vector field and ' is called a potential function of f . Definition of the Subject Catastrophe theory is concerned with the mathematical modeling of sudden changes – so called “catastrophes” – in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral. You can approach catastrophe theory from the point of view of differentiable maps or from the point of view of dynamical systems, that is, differential equations. We use the first case, where the theory is developed in the mathematical language of maps and functions (maps with range R). We are interested in those points of the domain of differentiable functions where their gradient vanishes. Such points are called the “singularities” of the differentiable functions. Assume that a system’s behavior can be described by a potential function (see Sect. “Glossary”). Then the singularities of this function characterize the equilibrium points of the system under consideration. Catastrophe theory tries to describe the behavior of systems by local properties of corresponding potentials. We are interested in local phenomena and want to find out the qualitative behavior of the system independent of its size. An important step is the classification of catastrophe potentials that occur in different situations. Can we find any common properties and unifying categories for these catastrophe potentials? It seems that it might be impossible to establish any reasonable criteria out of the many different natural processes and their possible catastrophes. One of the merits of catastrophe theory is the mathematical classification of simple catastrophes where the model does not depend on too many parameters. Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions. Many mathematical questions arise in catastrophe theory, but there are also other kinds of interesting problems, for example, historical themes, didactical questions and even social or political ones. How did catastrophe theory come up? How can we teach catastrophe theory to the stu-

dents at our universities? How can we make it understandable to non-mathematicians? What can people learn from this kind of mathematics? What are its consequences or insights for our lives? Let us first have a short look at mathematical history. When students begin to study a new mathematical field, it is always helpful to learn about its origin in order to get a good historical background and to get an overview of the dependencies of inner-mathematical themes. Catastrophe theory can be thought of as a link between classical analysis, dynamical systems, differential topology (including singularity theory), modern bifurcation theory and the theory of complex systems. It was founded by the French mathematician René Thom (1923–2002) in the sixties of the last century. The name ‘catastrophe theory’ is used for a combination of singularity theory and its applications. In the year 1972 Thom’s famous book “Stabilité structurelle et morphogénèse” appeared. Thom’s predecessors include Marston Morse, who developed his theory of the singularities of differentiable functions (Morse theory) in the thirties, Hassler Whitney, who extended this theory in the fifties to singularities of differentiable maps of the plane into the plane, and John Mather (1960), who introduced algebra, especially the theory of ideals of differentiable functions, into singularity theory. Successors to Thom include Christopher Zeeman (applications of catastrophe theory to physical systems), Martin Golubitsky (bifurcation theory), John Guckenheimer (caustics and bifurcation theory), David Schaeffer (shock waves) and Gordon Wassermann (stability of unfoldings). From the point of view of dynamical systems, the forerunners of Thom are Jules Henry Poincaré (1854–1912) who looked at stability problems of dynamical systems, especially the problems of celestial mechanics, Andrei Nikolaevic Kolmogorov, Vladimir I. Arnold and Jürgen Moser (KAM), who influenced catastrophe theory with their works on stability problems (KAM-theorem). From the didactical point of view, there are two main positions for courses in catastrophe theory at university level: Trying to teach the theory as a perfect axiomatic system consisting of exact definitions, theorems and proofs or trying to teach mathematics as it can be developed from historical or from natural problems (see [9]). In my opinion the latter approach has a more lasting effect, so there is a need to think about simple examples that lead to the fundamental ideas of catastrophe theory. These examples may serve to develop the theory in a way that starts with intuitive ideas and goes forward with increasing mathematical precision. When students are becoming acquainted with catastrophe theory, a useful learning tool is the insight that con-

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tinuous changes in influencing system parameters can lead to catastrophes in the system behavior. This phenomenon occurs in many systems. Think of the climate on planet earth, think of conflicts between people or states. Thus, they learn that catastrophe theory is not only a job for mathematical specialists, but also a matter of importance for leading politicians and persons responsible for guiding the economy. Introduction Since catastrophe theory is concerned with differentiable functions and their singularities, it is a good idea to start with the following simple experiment: Take a piece of wire, about half a meter in length, and form it as a ‘curve’ as shown in Fig. 1. Take a ring or screw-nut and thread the wire through it so that the object can move along the wire easily. The wire represents the model of the graph of a potential function, which determines the reactions of a system by its gradient. This means the ring’s horizontal position represents the actual value of a system parameter x that characterizes the state of the system, for example, x might be its position, rotation angle, temperature, and so on. This parameter x is also called the state parameter. The form of the function graph determines the behavior of the system, which tries to find a state corresponding to a minimum of the function (a zero of the gradient of f ) or to stay in

such a ‘stable’ position. (Move the ring to any position by hand and then let it move freely.) Maxima of the potential function are equilibrium points too, but they are not stable. A small disturbance makes the system leave such an equilibrium. But if the ring is in a local minimum, you may disturb its position by slightly removing it from that position. If this disturbance is not too big, the system will return to its original position at the minimum of the function when allowed to move freely. The minimum is a stable point of the function, in the sense that small disturbances of the system are corrected by the system’s behavior, which tries to return to the stable point. Observe that you may disturb the system by changing its position by hand (the position of the ring) or by changing the form of the potential function (the form of the wire). If the form of the potential function is changed, this corresponds to the change of one or more parameters in the defining equation for the potential. These parameters are called external parameters. They are usually influenced by the experimenter. If the form of the potential (wire) is changed appropriately (for example, if you pull upward at one side of the wire), the ring can leave the first disappearing minimum and fall into the second stable position. This is the way catastrophes happen: changes of parameters influence the potentials and thus may cause the system to suddenly leave a stable position. Some typical questions of catastrophe theory are:  How can one find a potential function that describes the system under consideration and its catastrophic jumps?  What does this potential look like locally? How can we describe the changes in the potential’s parameters?  Can we classify the potentials into simple categories?  What insights can such a classification give us? Now we want to consider two simple examples of systems where such sudden changes become observable. They are easy to describe and lead to the simplest forms of catastrophes which Thom listed in the late nineteen-sixties. The first is a simple physical model of an eccentric cylinder on an inclined plane. The second example is a model for the formation of a traffic jam. Example 1: The Eccentric Cylinder on the Inclined Plane

Catastrophe Theory, Figure 1 Ring on the wire

We perform some experiments with a cylinder or a wide wheel on an inclined plane whose center of gravity is eccentric with respect to its axes (Fig. 2). You can build such a cylinder from wood or metal by taking two disks of about 10 cm in radius. Connect them

Catastrophe Theory

Catastrophe Theory, Figure 2 The eccentric cylinder on the inclined plane. The inclination of the plane can be adjusted by hand. The position of the disc center and the center of gravity are marked by colored buttons

Catastrophe Theory, Figure 3 Two equilibrium positions (S1 and S2 ) are possible with this configuration. If the point A lies “too close” to M, there is no equilibrium position possible at all when releasing the wheel, since the center of gravity can never lie vertically over P

with some sticks of about 5 cm in length. The eccentric center of gravity is generated by fixing a heavy piece of metal between the two discs at some distance from the axes of rotation. Draw some lines on the disks, where you can read or estimate the angle of rotation. (Hint: It could be interesting for students to experimentally determine the center of gravity of the construction.) If mass and radius are fixed, there are two parameters which determine the system: The position of the cylinder can be specified by the angle x against a zero level (that is, against the horizontal plane; Fig. 4). This angle is the “inner” variable of the system, which means that it is the “answer” of the cylinder to the experimenter’s changes of

the inclination u of the plane, which is the “outer” parameter. In this example you should determine experimentally the equilibrium positions of the cylinder, that is, the angle of rotation where the cylinder stays at rest on the inclined plane (after releasing it cautiously and after letting the system level off a short time). We are interested in the stable positions, that is, the positions where the cylinder returns after slight disturbances (slightly nudging it or slightly changing the inclination of the plane) without rolling down the whole plane. The search for the equilibrium position x varies with the inclination angle u. You can make a table and a graphic showing the dependence of x and u.

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Catastrophe Theory, Figure 4 The movement of S is a composite of the movement of the cylinder represented by the movement of point M and the rotation movement with angle x

The next step is to find an analytical approach. Refer to Fig. 3: Point P is the supporting point of the cylinder on the plane. Point A marks the starting position of the center of gravity. When the wheel moves and the distance between M and A is big enough, this center can come into two equilibrium positions S1 and S2 , which lie vertically over P. Now, what does the potential function look like? The variables are: R: radius of the discs, u: angle of the inclined plane against the horizontal plane measured in radians, x: angle of rotation of the cylinder measured against the horizontal plane in radians, r: distance between the center of gravity S of the cylinder from the axes marked by point M, x0 (resp. u0 ): angles where the cylinder suddenly starts to roll down the plane. Refer to Fig. 4. If the wheel is rolled upward by hand from the lower position on the right to the upper position, the center of gravity moves upward with the whole wheel, or as the midpoint M or the point P. But point S also moves downward by rotation of the wheel with an amount of r  sin(x). The height gained is h D R  x  sin(u)  r  sin(x) :

(1)

From physics one knows the formula for the potential energy Epot of a point mass. It is the product m  g  h, where m D mass, g D gravitational constant, h D height, where the height is to be expressed by the parameters x and u and the given data of the cylinder.

To a fixed angle u1 of the plane, by formula (1), there corresponds the following potential function: f (x) D Epot D m  g  h D m  g (R x sin(u1 )r sin(x)) : If the inclination of the plane is variable, the angle u of inclination serves as an outer parameter and we obtain a whole family of potential functions as a function F in two variables x and u: F(x; u) D Fu (x) D m  g  (R  x  sin(u)  r  sin(x)) : Thus, the behavior of the eccentric cylinder is described by a family of potential functions. The extrema of the single potentials characterize the equilibria of the system. The saddle point characterizes the place where the cylinder suddenly begins to roll down due to the slightest disturbance, that is, where the catastrophe begins (see Fig. 5). Example. The data are m D 1 kg, g D 10 m/sec2 , R D 0:1 m, r D 0:025 m. For constant angle of inclination u, each of the functions f (x) D F(x; u D const) is a section of the function F. To determine the minima of each of the section graphs, we calculate the zeros of the partial derivative of F with respect to the inner variable x. In our particular example above, the result is @x F D 0:25 cos(x) C sin(u). The solution for @x F D 0 is u D arcsin(0:25 cos(x)) (see Fig. 6). Calculating the point in x–u-space at which both partial derivatives @x F and @x x F vanish (saddle point of the section) and using only positive values of angle u gives the point (x0 ; u0 ) D (0; 0:25268). In general: x0 D 0 and u0 D arcsin(r/R). In our example this means the catastrophe begins at an inclination angle of u0 D 0:25268 (about

Catastrophe Theory

Catastrophe Theory, Figure 6 Curve in x–u-space, for which F(x; u) has local extrema

Catastrophe Theory, Figure 5 Graph of F(x; u) for a concrete cylinder

14.5 degrees). At that angle, sections of F with u > u0 do not possess any minimum where the system could stably rest. Making u bigger than u0 means that the cylinder must leave its stable position and suddenly roll down the plane. The point u0 in the u-parameter space is called catastrophe point. What is the shape of the section of F at u0 ? Since in our example f (x) D F(x; u0 ) D 0:25x  0:25 sin(x) ; the graph appears as shown in Fig. 7. The point x D 0 is a degenerated critical point of f (x), since f 0 (x) D f 00 (x) D 0. If we expand f into its Taylor polynomial t(x) around that point we get t(x) D 0:0416667x 3  0:00208333x 5 C 0:0000496032x 7 6:88933  107 x 9 C O[x]11 :

Catastrophe Theory, Figure 7 Graph of the section F(x; u0 )

Catastrophe theory tries to simplify the Taylor polynomials of the potentials locally about their degenerated critical points. This simplification is done by coordinate transformations, which do not change the structure of the singularities. These transformations are called diffeomorphisms (see Sect. “Glossary”). A map h : U ! V, with U and V open in Rn , is a local diffeomorphism if the Jacobian determinant det(J p h) ¤ 0 at all points p in U. Two functions f and g, both defined in some neighborhood of the origin of Rn , are called r-equivalent if there exists a local diffeomorphism h : U ! V at the origin, such that g D f ı h.

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Catastrophe Theory, Figure 8 Characteristic surface S, initial density, fold curve and cusp curve

The Taylor polynomial t(x) in our example starts with a term of 3rd order in x. Thus, there exists a diffeomorphism (a non-singular coordinate transformation), such that in the new coordinates t is of the form x3 . To see this, we write t(x) D x 3  a(x), with a(0) ¤ 0. We define h(x) D x a(x)1/3 . Note that the map h is a local diffeomorphism near the origin since h0 (0) D a(0)1/3 ¤ 0. (Hint: Modern computer algebra systems, such as Mathematica, are able to expand expressions like a(x)1/3 and calculate the derivative of h easily for the concrete example.) Thus for the eccentric cylinder the intersection of the graph of the potential F(x; u) with the plane u D u0 is the graph of a function, whose Taylor series at its critical point x D 0 is t D x 3 ı h, in other words: t is r-equivalent to x3 . Before we continue with this example and its relation to catastrophe theory, let us introduce another example which leads to a potential of the form x4 . Then we will have two examples which lead to the two simplest catastrophes in René Thom’s famous list of the seven elementary catastrophes. Example 2: The Formation of Traffic Jam In the first example we constructed a family F of potential functions directly from the geometric properties of the model of the eccentric cylinder. We then found its critical points by calculating the derivatives of F. In this second example the method is slightly different. We start with a partial differential equation which occurs in traffic flow modeling. The solution surface of the initial value problem

will be regarded as the surface of zeros of the derivative of a potential family. From this, we then calculate the family of potential functions. Finally, we will see that the Taylor series of a special member of this family, the one which belongs to its degenerated critical point, is equivalent to x4 . Note: When modeling a traffic jam we use the names of variables that are common in literature about this subject. Later we will rename the variables into the ones commonly used in catastrophe theory. Let u(x; t) be the (continuous) density of the traffic at a street point x at time t and let a(u) be the derivative of the traffic flow f (u). The mathematical modeling of simple traffic problems leads to the well known traffic flow equation: @u(x; t) @u(x; t) C a(u) D0: @t @x We must solve the Cauchy problem for this quasilinear partial differential equation of first order with given initial density u0 D u(x; 0) by the method of characteristics. The solution is constant along the (base) characteristics, which are straight lines in xt-plane with slope a(u0 (y)) against the t-axes, if y is a point on the x-axes, where the characteristic starts at t D 0. For simplicity we write “a” instead of a(u0 (y)). Thus the solution of the initial value problem is u(x; t) D u0 (x  a  t) and the characteristic surface S D f(x; t; u)ju  u0 (x  a  t) D 0g is a surface in xtuspace. Under certain circumstances S may not lie uniquely above the xt-plane but may be folded as is shown in Fig. 8.

Catastrophe Theory

There is another curve to be seen on the surface: The border curve of the fold. Its projection onto the xt-plane is a curve with a cusp, specifying the position and the beginning time of the traffic jam. The equations used in the example, in addition to the traffic flow equation, can be deduced in traffic jam modeling from a simple parabolic model for a flow-density relation (see [5]). The constants in the following equations result from considerations of maximal possible density and an assumed maximal allowed velocity on the street. In our example a model of a road of 2 km in length and maximal traffic velocity of 100 km/h is chosen. Maximal traffic density is umax D 0:2. The equations are f (u) D 27:78u  138:89u2 and u(x; 0) D u0 (x) D 0:1 C 0:1  Exp(((x  2000)/700)2 ). The latter was constructed to simulate a slight increase of the initial density u0 along the street at time t D 0. The graph of u0 can be seen as the front border of the surface. To determine the cusp curve shown in Fig. 8, consider the following parameterization ˚ of the surface S and the projection map  : S ! R2 : ˚ : R  R0 ! S R

3

(x; t) 7! (x C a(u0 (x))  t; t; u0 (x)) : The Jacobian matrix of  ı ˚ is 0 1 @( ı ˚ )1 @( ı ˚)1 B C @x @t J( ı ˚)(x; t) D @ A @( ı ˚ )2 @( ı ˚)2 @x @t   0 1 C a(u0 (x))  t a(u0 (x)) : D 0 1 It is singular (that is, its determinate vanishes) if 1 C (a(u0 (x)))0  t D 0. From this we find the curve c : R ! R2   1 c(x) D x;  0 a (u0 (x))  u00 (x) which is the pre-image of the cusp curve and the cusp curve in the Fig. 8 is the image of c under  ı ˚. Whether the surface folds or not depends on both the initial density u0 and the properties of the flow function f (u). If such folding happens, it would mean that the density would have three values at a given point in xt-space, which is a physical impossibility. Thus, in this case, no continuous solution of the flow equation can exist in the whole plane. “Within” the cusp curve, which is the boundary for the region where the folding happens, there exists a curve with its origin in the cusp point. Along this curve the surface must be “cut” so that a discontinuous solution occurs. The curve within the cusp region is called

Catastrophe Theory, Figure 9 The solution surface is “cut” along the shock curve, running within the cusp region in the xt-plane

a shock curve and it is characterized by physical conditions (jump condition, entropy condition). The density makes a jump along the shock. The cusp, together with the form of the folding of the surface, may give an association to catastrophe theory. One has to find a family of potential functions for this model, that is, functions F(x;t) (u), whose negative gradient, that is, its derivative by u (the inner parameter), describes the characteristic surface S by its zeroes. The family of potentials is given by   xa(u)t Z Z u0 (x)dx F(x; t; u) D t  u  a(u)  a(u)du  0

(see [6]). Since gradFx;t (u) D 

@F D 0 , u  u0 (x  a(u)  t) D 0 @u

the connection of F and S is evident. One can show that a member f of the family F, the function f (u) D F(; ; u), is locally in a traffic jam formation point (; ) of the form f (u) D u4 (after Taylor expansion and after suitable coordinate transformations, similar to the first example). The complicated function terms of f and F thus can be replaced in qualitative investigations by simple polynomials. From the theorems of catastrophe theory, more properties of the models under investigation can be discovered. Now we want to use the customary notation of catastrophe theory. We shall rename the variables: u is the inner variables, which in catastrophe theory usually is x, and x, t (the outer parameters) get the names u and v respectively.

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Thus F D F(x; u; v) is a potential family with one inner variable (the state variable x) and two outer variables (the control variables u and v). Unfoldings In the first examples above we found a family F(x; u) of potential functions for the eccentric cylinder. For u fixed we found single members of that family. The member which belonged to the degenerated critical point of F, that is, to the point (x; u) where @x F D @x x F D 0, turned out to be equivalent to x3 . In the second example the corresponding member is equivalent to x4 (after renaming the variables). These two singularities are members of potential families which also can be transformed into simple forms. In order to learn how such families, in general, arise from a single function germ, let us look at the singularity f (x) D x 3 . If we add to it a linear “disturbance” ux, where the factor (parameter) u may assume different values, we qualitatively have one of the function graphs in Fig. 10. The solid curve in Fig. 10 represents the graph of f (x) D x 3 . The dashed curve is the graph of g(x) D x 3  u  x; u > 0. The dotted line is h(x) D x 3 C u  x; u > 0. Please note: While for positive parameter values, the disturbed function has no singularity, in the case of negative u-values a relative maximum and a relative minimum exist. The origin is a singular point only for u D 0, that is, for the undisturbed function f . We can think of f (x) D x 3 as a member of a function family, which contains disturbances with linear terms. The function f (x) D x 3 changes the type of singularity with the addition of an appropriate small disturbing function, as we have seen. Therefore f is called “structurally unstable”. But we have also learned that f (x) D x 3 can be seen as a member of a whole family of functions F(x; u) D x 3 C u  x, because F(x; 0) D f (x). This fam-

Catastrophe Theory, Figure 10 Perturbations of x 3

ily is stable in the sense that all kinds of singularities of the perturbed function are included by its members. This family is only one of the many possibilities to “unfold” the function germ f (x) D x 3 by adding disturbing terms. For example F(x; u; v) D x 3 C u  x 2 C v  x would be another such unfolding (see Sect. “Glossary”) of f . But this unfolding has more parameters than the minimum needed. The example of a traffic jam leads to an unfolding of f (x) D x 4 which is F(x; u; v) D x 4 C u  x 2 C v  x. How can we find unfoldings, which contain all possible disturbances (all types of singularities), that is, which are stable and moreover have a minimum number of parameters? Examine the singularity x4 (Fig. 12). It has a degenerated minimum at the origin. We disturb this function by a neighboring polynomial of higher degree, for example, by u  x 5 , where the parameter u with small values (here and in what follows “small” means small in amount), ensures that the perturbed function is “near” enough to x4 .

Catastrophe Theory, Figure 11 The function f(x) D x 3 embedded in a function family

Catastrophe Theory

without a term of degree n  1. Indeed, the expression F(x; u; v) D x 4 C u  x 2 C v  x is, as we shall see later, the “universal” unfolding of x4 . Here an unfolding F of a germ f is called universal, if every other unfolding of f can be received by suitable coordinate transformations “morphism of unfoldings” from F, and the number of unfolding parameters of F is minimal. The minimal number of unfolding parameters needed for the universal unfolding F of the singularity f is called the codimension of f . It can be computed as follows: Catastrophe Theory, Figure 12 The function f(x) D x4 C u  x5, u D 0 (solid), u D 0:2 (dotted) and u D 0:3 (dashed)

For f (x) D x 4 C u  x 5 , we find that the equation D 0 has the solutions x1 D x2 D x3 D 0, x4 D 4/(5u). To the threefold zero of the derivative, that is, the threefold degenerated minimum, another extremum is added which is arbitrarily far away from the origin for values of u that are small enough in amount due to the term 4/(5u). This is so, because the term 4/(5u) increases as u decreases. (This is shown in Fig. 12 with u D 0, u D 0:2 and u D 0:3). The type of the singularity at the origin thus is not influenced by neighboring polynomials of the form u  x5. If we perturb the polynomial x4 by a neighboring polynomial of lower degree, for example, u  x 3 , then for small amounts of u a new minimum is generated arbitrarily close to the existing singularity at the origin. If disturbed by a linear term, the singularity at the origin can even be eliminated. The only real solution of h0 (x) D 0, where h(x) D x 4 C ux, is x D (u/4)(1/3) . Note: Only for u D 0, that is, for a vanishing disturbance, is h singular at the origin. For each u ¤ 0 the linear disturbance ensures that no singularity at the origin occurs. The function f (x) D x 4 is structurally unstable. To find a stable unfolding for the singularity f (x) D x 4 , we must add terms of lower order. But we need not take into account all terms x 4 C u  x 3 C v  x 2 C w  x C k, since the absolute term plays no role in the computation of singular points, and each polynomial of degree 4 can be written by a suitable change of coordinates without a cubic term. Similarly, a polynomial of third degree after a coordinate transformation can always be written without a quadratic term. The “Tschirnhaus-transformation”: x 7! x  a1 /n, if a1 is the coefficient of x n1 in a polynomial p(x) D x n C a1 x n1 C : : : C a n1 x C a n of degree n, leads to a polynomial q(x) D x n C b1 x n2 C : : : C b n2 x C b n1 f 0 (x)

codim( f ) D

dimR (n) : h@x f i

Here h@x f i is the Jacobian ideal generated by the partial derivatives of f and (n) D f f 2 "(n)j f (0) D 0g. "(n) is the vector space of function germs at the origin of Rn . The quotient (n)/h@x f i is the factor space. What is the idea behind that factor space? Remember, that the mathematical description of a plane (a two dimensional object) in space (3 dimensional) needs only one equation, while the description of a line (1 dimensional) in space needs two equations. The number of equations that are needed for the description of these geometrical objects is determined by the difference between the dimensions of the surrounding space and the object in it. This number is called the codimension of the object. Thus the codimension of the plane in space is 1 and the codimension of the line in space is 2. From linear algebra it is known that the codimension of a subspace W of a finite dimensional vector space V can be computed as codim W D dim V  dim W. It gives the number of linear independent vectors that are needed to complement a basis of W to get a basis of the whole space V. Another well known possibility for the definition is codimW D dimV/W. This works even for infinite dimensional vector spaces and agrees in finite dimensional cases with the previous definition. In our example V D (n) and W should be the set O of all functions germs, which are r-equivalent to f (O is the orbit of f under the action of the group of local diffeomorphisms preserving the origin of Rn ). But this is an infinite dimensional manifold, not a vector space. Instead, we can use its tangent space T f O in the point f , since spaces tangent to manifolds are vector spaces with the same dimension as the manifold itself and this tangent space is contained in the space tangent to (n) in an obvious way. It turns out, that the tangent space T f O is h@x f i. The space tangent to (n) agrees with (n), since it is a vector space. The details for the computations together with some examples can be found in the excellent article by M. Golubitsky (1978).

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Catastrophe Theory

The Seven Elementary Catastrophes Elementary catastrophe theory works with families (unfoldings) of potential functions F : Rn  R k ! R (with k  4) (x; u) 7! F(x; u) : is called the state space, R k is called the parameter space or control space. Accordingly x D (x1 ; : : : ; x n ) is called a state variable or endogenous variable and u D (u1 ; : : : ; u k ) is called a control variable (exogenous variable, parameter). We are interested in the singularities of F with respect to x and the dependence of F with respect to the control variables u. We regard F as the unfolding of a function germ f and want to classify F. In analogy to the r-equivalence of function germs, we call two unfoldings F and G of the same function germ r-equivalent as unfoldings if there exists an unfoldingmorphism (see Sect. “Glossary”) from F to G. An unfolding F of f is called versal (or stable), if each other unfolding of f is right-equivalent as unfolding to F, that is, if there exists a right-morphism between the unfoldings. A versal unfolding with minimal unfolding dimensions is called universal unfolding of f . Thus we have the following theorem (all proofs of the theorems cited in this article can be found in [1,2] or [12].) Rn

Theorem 1 (Theorem on the existence of a universal unfolding) A singularity f has a universal unfolding iff codim( f ) D k < 1. Examples (see also the following theorem with its examples): If f (x) D x 3

it follows

F(x; u; v) D x 3 C ux 2 C vx 3

F(x; u) D x C ux 3

F(x; u) D x C ux

is versal;

is universal; 2

is not versal :

The following theorem states how one can find a universal unfolding of a function germ: Theorem 2 (Theorem on the normal form of universal unfoldings) Let f 2 (n) be a singularity with codim( f ) D k < 1. Let u D (u1 ; : : : ; u k ) be the parameters of the unfolding. Let b i (x), i D 1; : : : ; k, be the elements of (n), whose cosets modulo h@x f i generate the vector space (n)/h@x f i. Then F(x; u) D f (x) C

k X

u i b i (x)

iD1

is a universal unfolding of f .

Examples: 1. Consider f (x) D x 4 . Here n D 1 and (n) D (1) D hxi. The derivative of f is 4x 3 , so h@x f i D hx 3 i and we get hxi/hx 3 i D hx; x 2 i. Thus F(x; u1 ; u2 ) D x 4 C u1 x 2 C u2 x is the universal unfolding of f . 2. Consider f (x; y) D x 3 C y 3 . Here n D 2 and (n) D (2) D hx; yi. The partial derivatives of f with respect to x and y are 3x 2 and 3y 2 , thus h@x f i D hx 2 ; y 2 i and we get hx; yi/hx 2 ; y 2 i D hx; x y; yi. Therefore F(x; y; u1 ; u2 ; u3 ) D x 3 C y 3 C u1  x  y C u2  x C u3  y is the universal unfolding of f . Hint: There exists a useful method for the calculation of the quotient space which you will find in the literature as “Siersma’s trick” (see for example [7]). We now present René Thom’s famous list of the seven elementary catastrophes. Classification Theorem (Thom’ s List) Up to addition of a non degenerated quadratic form in other variables and up to multiplication by ˙1 a singularity f of codimension k (1  k  4) is right-equivalent to one of the following seven: f x3 x4 x5 x3 C y3

codim f 1 2 3 3

x3  xy2 3 x6 4 x2 y C y4 4

universal unfolding x3 C ux x4 C ux2 C vx x5 C ux3 C vx2 C wx x3 C y3 C uxy C vx C wy

name fold cusp swallowtail hyperbolic umbilic x3  xy2 C u(x2 C y2 ) elliptic C vx C wy umbilic x6 C ux4 C vx3 C wx2 C tx butterfly x2 y C y4 C ux2 C vy2 parabolic C wx C ty umbilic

In a few words: The seven elementary catastrophes are the seven universal unfoldings of singular function germs of codimension k (1  k  4). The singularities itself are called organization centers of the catastrophes. We will take a closer look at the geometry of the simplest of the seven elementary catastrophes: the fold and the cusp. The other examples are discussed in [9]. The Geometry of the Fold and the Cusp The fold catastrophe is the first in Thom’s list of the seven elementary catastrophes. It is the universal unfolding of the singularity f (x) D x 3 which is described by the equation F(x; u) D x 3 C u  x. We are interested in the set of singular points of F relative to x, that is, for those points where the first partial derivative with respect to x (the sys-

Catastrophe Theory

tem variable) vanishes. This gives the parabola u D 3x 2 . Inserting this into F gives a space curve. Figure 13 shows the graph of F(x; u). The curve on the surface joins the extrema of F and the projection curve represents their x; u values. There is only one degenerated critical point of F (at the vertex of the Parabola), that is, a critical point where the second derivative by x also vanishes. The two branches of the parabola in the xu-plane give the positions of the maxima and the minima of F, respectively. At the points belonging to a minimum, the system is stable, while in a maximum it is unstable. Projecting x–u space, and with it the parabola, onto parameter space (u-axis), one gets a straight line shown within the interior of the parabola, which represents negative parameter values of u. There, the system has a stable minimum (and an unstable maximum, not observable in nature). For parameter values u > 0 the system has no stability points at all. Interpreting the parameter u as time, and “walking along” the u-axis, the point u D 0 is the beginning or the end of a stable system behavior. Here the system shows catastrophic behavior. According to the interpretation of the external parameter as time or space, the morphology of the fold is a “beginning,” an “end” or a “border”, where something new occurs. What can we say about the fold catastrophe and the modeling of our first example, the eccentric cylinder? We have found that the potential related to the catastrophe point of the eccentric cylinder is equivalent to x3 , and F(x; u) D x 3 C ux is the universal unfolding. This is independent of the size of our ‘machine’. The behavior of the machine should be qualitatively the same as that of other machines which are described by the fold catastrophe as a mechanism. We can expect that the begin (or end) of a catastrophe depends on the value of a single outer parameter. Stable states are possible (for u < 0 in the standard fold. In the untransformed potential function of our model of the eccentric cylinder, this value is u D 0:25268) corresponding to the local minima of F(:; u). This means that it is possible for the cylinder to stay at rest on the inclined plane. There are unstable equilibria (local maxima of F(:; u) and the saddle point, too). The cylinder can be turned such that the center of gravity lies in the upper position over the supporting point (see Fig. 3). But a tiny disturbance will make the system leave this equilibrium. Let us now turn to the cusp catastrophe. The cusp catastrophe is the universal unfolding of the function germ f (x) D x 4 . Its equation is F(x; u; v) D x 4 Cu  x 2 Cv  x. In our second example (traffic jam), the function f is equivalent to the organization center of the cusp catastrophe, the second in René Thom’s famous list of the seven elementary catastrophes. So the potential family is equivalent to

Catastrophe Theory, Figure 13 Graph of F(x; u) D x 3 C u  x

the universal unfolding F(x; u; v) D x 4 Cux 2 Cvx of the function x4 . We cannot draw such a function in three variables in three-space, but we can try to draw sections giving u or v certain constant values. We want to look at its catastrophe surface S, that is, the set of those points in (x; u; v)space, where the partial derivative of F with respect to the system variable x is zero. It is a surface in three-space called catastrophe manifold, stability surface or equilibrium surface, since the surface S describes the equilibrium points of the system. If S is not folded over a point in u; v-space there is exactly one minimum of the potential F. If it is folded in three sheets, there are two minima (corresponding to the upper and lower sheet of the fold) and a maximum (corresponding to the middle sheet). Stable states of the system belong to the stable minima, that is, to the upper and lower sheets of the folded surface. Other points in three-space that do not lie on the surface S correspond to the states of the system that are not equilibrium. The system does not rest there. Catastrophes do occur when the

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Catastrophe Theory

Catastrophe Theory, Figure 14 The catastrophe manifold of the cusp catastrophe

system jumps suddenly from one stable state to another stable state, that is, from one sheet to the other. There are two principal possibilities, called “conventions”, where this jump can happen. The “perfect delay convention” says that jumps happen at the border of the folded surface. The “Maxwell convention” says that jumps can happen along a curve in uv-space (a “shock-curve”) which lays inside the cusp area. This curve consists of those points (u; v) in parameter space, where F(:; u; v) has two critical points with the same critical value. In our traffic jam model a shock curve is a curve in xt-space that we can interpret dynamically as the movement of the jam formation front along the street (see Fig. 9). The same figure shows what we call the morphology of the cusp: according to the convention, the catastrophe manifold with its fold or its cut can be interpreted as a fault or slip (as in geology) or a separation (if one of the parameter represents time). In our traffic jam model, there is a “line” (actually a region) of separation (shock wave) between the regions of high and low traffic density. Besides the catastrophe manifold (catastrophe surface) there are two other essential terms: catastrophe set and bifurcation set (see Fig. 15). The catastrophe set can be viewed as the curve which goes along the border of the fold on the catastrophe manifold. It is the set of degenerated critical points and is described mathematically as the set f(x; u; v)j@x F D @2x F D 0g. Its projection into the uv-parameter space is the cusp curve, which is called the bifurcation set of the cusp catastrophe.

Catastrophe Theory, Figure 15 The catastrophe manifold (CM), catastrophe set (CS) and bifurcation set (BS) of the cusp catastrophe

The cusp catastrophe has some special properties which we discuss now with the aid of some graphics. If the system under investigation shows one or more of these properties, the experimenter should try to find a possible cusp potential that accompanies the process. The first property is called divergence. This property can be shown by the two curves (black and white) on the catastrophe manifold which describes the system’s equilibrium points. Both curves initially start at nearby positions on the surface, that is, they start at nearby stable system states. But the development of the system can proceed quite differently. One of the curves runs on the upper sheet of the surface while the other curve runs on the lower sheet. The system’s stability positions are different and thus its behavior is different. The next property is called hysteresis. If the system development is running along the path shown in Fig. 17 from P1 to P2 or vice versa, the jumps in the system behavior, that is, the sudden changes of internal system variables, occur at different parameter constellations, depending on the direction the path takes, from the upper to the lower sheet or vice versa. Jumps upward or downward happen

Catastrophe Theory

Catastrophe Theory, Figure 16 The divergence property of the cusp catastrophe

Catastrophe Theory, Figure 18 The bifurcation property of the cusp catastrophe

Further Applications

Catastrophe Theory, Figure 17 The hysteresis property of the cusp catastrophe

at the border of the fold along the dotted lines. The name of this property of the cusp comes from the characteristic hysteresis curve in physics, occurring for example at the investigation of the magnetic properties of iron. Figure 18 shows the bifurcation property of the cusp catastrophe: the number of equilibrium states of the system splits from one to three along the path beginning at the starting point (SP) and running from positive to negative u values as shown. If the upper and the lower sheet of the fold correspond to the local minima of F and the middle sheet corresponds to the local maxima, then along the path shown, one stable state splits into two stable states and one unstable state.

Many applications of catastrophe theory are attributable to Christopher Zeeman. For example, his catastrophe machine, the “Zeeman wheel,” is often found in literature. This simple model consists of a wheel mounted flat against a board and able to turn freely. Two elastics are attached at one point (Fig. 19, point B) close to the periphery of the wheel. One elastic is fixed with its second end on the board (point A). The other elastic can be moved with its free end in the plane (point C). Moving the free end smoothly, the wheel changes its angle of rotation smoothly almost everywhere. But at certain positions of C, which can be marked with a pencil, the wheel suddenly changes this angle dramatically. Joining the marked points where these jumps occur, you will find a cusp curve in the plane. In order to describe this behavior, the two coordinates of the position of point C serve as control parameters in the catastrophe model. The potential function for this model results from Hook’s law and simple geometric considerations. Expanding the potential near its degenerated critical point into its Taylor series, and transforming the series with diffeomorphisms similar to the examples we have already calculated, this model leads to the cusp catastrophe. Details can be found in the book of Poston and Stewart [7] or Saunders (1986). Another example of an application of catastrophe theory is similar to the traffic jam model we used here. Catastrophe theory can describe the formation of shockwaves in hydrodynamics. Again the cusp catastrophe describes

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Catastrophe Theory

Catastrophe Theory, Figure 19 The Zeeman wheel

this phenomenon. The mathematical frame is given in the works of Lax [6], Golubitsky and Schaeffer[3] and Guckenheimer [4]. In geometrical optics light rays are investigated which are reflected on smooth surfaces. Caustics are the envelopes of the light rays. They are the bright patterns of intensive light, which can be seen, for example, on a cup of coffee when bright sunlight is reflected on the border of the cup. Catastrophe theory can be applied to light caustics, because the light rays obey a variational principle. According to Fermat’s principle, light rays travel along geodesics and the role of the potential functions in catastrophe theory is played by geodesic distance functions. Caustics are their bifurcation sets. The calculations are given for example in the work of Sinha [11]. A purely speculative example, given by Zeeman, is an aggression model for dogs. Suppose that fear and rage of a dog can be measured in some way from aspects of its body language (its attitude, ears, tail, mouth, etc.). Fear and rage are two conflicting parameters, so that a catastrophe in the dog’s behavior can happen. Look at Fig. 20. The vertical axis shows the dog’s aggression potential, the other axes show its fear and rage. Out of the many different ways the dog can behave, we choose one example. Suppose we start at point A on the catastrophe manifold (behavioral surface). Maybe the dog is dozing in the sun without any thought of aggression. The dog’s aggression behavior is neutral (point A). Another dog is coming closer and our dog, now awakened, is becoming more and more angry by this breach of his territory. Moving along the path on the behavior surface, the dog is on a trajectory to attack his opponent. But it suddenly notices that the enemy is very big (point B). So its fear is growing, its

Catastrophe Theory, Figure 20 A dog’s aggression behavior depending on its rage and fear

rage diminishes. At point C the catastrophe happens: Attack suddenly changes to flight (point D). This flight continues until the dog notices that the enemy does not follow (point E). Fear and rage both get smaller until the dog becomes calm again. The example can be modified a little bit to apply to conflicts between two states. The conflicting factors can be the costs and the expected gain of wars, and so on. Among the variety of possible examples, those worthy of mention include investigations of the stability of ships, the gravitational collapse of stars, the buckling beam (Euler beam), the breaking of ocean waves, and the development of cells in biology. The latter two problems are examples for the occurrence of higher catastrophes, for example

Catastrophe Theory

the hyperbolic umbilic (see Thom’s List of the seven elementary catastrophes). Future Directions When catastrophe theory came up in the nineteen-sixties, much enthusiasm spread among mathematicians and other scientists about the new tool that was expected to explain many of the catastrophes in natural systems. It seemed to be the key for the explanation of discontinuous phenomena in all sciences. Since many applications were purely qualitative and speculative, a decade later this enthusiasm ebbed away and some scientists took this theory for dead. But it is far from that. Many publications in our day show that it is still alive. The theory has gradually become a ‘number producing’ theory, so that it is no longer perceived as purely qualitative. It seems that it will be applied the sciences increasingly, producing numerical results. Thom’s original ideas concerning mathematical biology seem to have given the basis for the trends in modern biology. This is probably the most promising field of application for catastrophe theory. New attempts are also made, for example, in the realm of decision theory or in statistics, where catastrophe surfaces may help to clarify statistical data. From the point of view of teaching and learning mathematics, it seems that catastrophe theory is becoming and increasingly popular part of analysis courses at our universities. Thus, students of mathematics meet the basic ideas of catastrophe theory within their first two years of undergraduate studies. Perhaps in the near future its basic principles will be taught not only to the mathematical specialists but to students of other natural sciences, as well. Bibliography Primary Literature 1. Bröcker T (1975) Differentiable germs and catastrophes. London Mathem. Soc. Lecture Notes Series. Cambridge Univ Press, Cambridge 2. Golubitsky M, Guillemin V (1973) Stable mappings and their singularities. Springer, New York, GTM 14 3. Golubitsky M, Schaeffer DG (1975) Stability of shock waves for a single conservation law. Adv Math 16(1):65–71 4. Guckenheimer J (1975) Solving a single conservation law. Lecture Notes, vol 468. Springer, New York, pp 108–134 5. Haberman R (1977) Mathematical models – mechanical vibrations, population dynamics, and traffic flow. Prentice Hall, New Jersey 6. Lax P (1972) The formation and decay of shockwaves. Am Math Monthly 79:227–241 7. Poston T, Stewart J (1978) Catastrophe theory and its applications. Pitman, London

8. Sanns W (2000) Catastrophe theory with mathematica – a geometric approach. DAV, Germany 9. Sanns W (2005) Genetisches Lehren von Mathematik an Fachhochschulen am Beispiel von Lehrveranstaltungen zur Katastrophentheorie. DAV, Germany 10. Saunders PT (1980) An introduction to catastrophe theory. Cambridge University Press, Cambridge. Also available in German: Katastrophentheorie. Vieweg (1986) 11. Sinha DK (1981) Catastrophe theory and applications. Wiley, New York 12. Wassermann G (1974) Stability of unfoldings. Lect Notes Math, vol 393. Springer, New York

Books and Reviews Arnold VI (1984) Catastrophe theory. Springer, New York Arnold VI, Afrajmovich VS, Il’yashenko YS, Shil’nikov LP (1999) Bifurcation theory and catastrophe theory. Springer Bruce JW, Gibblin PJ (1992) Curves and singularities. Cambridge Univ Press, Cambridge Castrigiano D, Hayes SA (1993) Catastrophe theory. Addison Wesley, Reading Chillingworth DRJ (1976) Differential topology with a view to applications. Pitman, London Demazure M (2000) Bifurcations and catastrophes. Springer, Berlin Fischer EO (1985) Katastrophentheorie und ihre Anwendung in der Wirtschaftswissenschaft. Jahrb f Nationalök u Stat 200(1):3–26 Förster W (1974) Katastrophentheorie. Acta Phys Austr 39(3):201– 211 Gilmore R (1981) Catastrophe theory for scientists and engineers. Wiley, New York Golubistky M (1978) An introduction to catastrophe theory. SIAM 20(2):352–387 Golubitsky M, Schaeffer DG (1985) Singularities and groups in bifurcation theory. Springer, New York Guckenheimer J (1973) Catastrophes and partial differential equations. Ann Inst Fourier Grenoble 23:31–59 Gundermann E (1985) Untersuchungen über die Anwendbarkeit der elementaren Katastrophentheorie auf das Phänomen “Waldsterben”. Forstarchiv 56:211–215 Jänich K (1974) Caustics and catastrophes. Math Ann 209:161–180 Lu JC (1976) Singularity theory with an introduction to catastrophe theory. Springer, Berlin Majthay A (1985) Foundations of catastrophe theory. Pitman, Boston Mather JN (1969) Stability of C1 -mappings. I: Annals Math 87:89– 104; II: Annals Math 89(2):254–291 Poston T, Stewart J (1976) Taylor expansions and catastrophes. Pitman, London Stewart I (1977) Catastrophe theory. Math Chronicle 5:140–165 Thom R (1975) Structural stability and morphogenesis. Benjamin Inc., Reading Thompson JMT (1982) Instabilities and catastrophes in science and engineering. Wiley, Chichester Triebel H (1989) Analysis und mathematische Physik. Birkhäuser, Basel Ursprung HW (1982) Die elementare Katastrophentheorie: Eine Darstellung aus der Sicht der Ökonomie. Lecture Notes in Economics and Mathematical Systems, vol 195. Springer, Berlin Woodcock A, Davis M (1978) Catastrophe theory. Dutton, New York Zeeman C (1976) Catastrophe theory. Sci Am 234(4):65–83

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Center Manifolds

Center Manifolds GEORGE OSIPENKO State Polytechnic University, St. Petersburg, Russia Article Outline Glossary Definition of the Subject Introduction Center Manifold in Ordinary Differential Equations Center Manifold in Discrete Dynamical Systems Normally Hyperbolic Invariant Manifolds Applications Center Manifold in Infinite-Dimensional Space Future Directions Bibliography Glossary Bifurcation Bifurcation is a qualitative change of the phase portrait. The term “bifurcation” was introduced by H. Poincaré. Continuous and discrete dynamical systems A dynamical system is a mapping X(t; x), t 2 R or t 2 Z, x 2 E which satisfies the group property X(t C s; x) D X(t; X(s; x)). The dynamical system is continuous or discrete when t takes real or integer values, respectively. The continuous system is generated by an autonomous system of ordinary differential equations x˙ 

dx D F(x) dt

(1)

as the solution X(t, x) with the initial condition X(0; x) D x. The discrete system generated by a system of difference equations x mC1 D G(x m )

(2)

as X(n; x) D G n (x). The phase space E is the Euclidean or Banach. Critical part of the spectrum Critical part of the spectrum for a differential equation x˙ D Ax is  c = {eigenvalues of A with zero real part}. Critical part of the spectrum for a diffeomorphism x ! Ax is  c = {eigenvalues of A with modulus equal to 1}. Eigenvalue and spectrum If for a matrix (linear mapping) A the equality Av D v, v ¤ 0 holds then v and  are called eigenvector and eigenvalue of A. The set of the eigenvalues is the spectrum of A. If there exists k such that (A  I) k v D 0, v is said to be generating vector.

Equivalence of dynamical systems Two dynamical systems f and g are topologically equivalent if there is a continuous one-to-one correspondence (homeomorphism) that maps trajectories (orbits) of f on trajectories of g: It should be emphasized that the homeomorphism need not be differentiable. Invariant manifold In applications an invariant manifold arises as a surface such that the trajectories starting on the surface remain on it under the system evolution. Local properties If F(p) D 0, the point p is an equilibrium of (1). If G(q) D q; q is a fixed point of (2). We study dynamics near an equilibrium or a fixed point of the system. Thus we consider the system in a neighborhood of the origin which is supposed to be an equilibrium or a fixed point. In this connection we use terminology local invariant manifold or local topological equivalence. Reduction principle In accordance with this principle a locally invariant (center) manifold corresponds to the critical part of the spectrum of the linearized system. The behavior of orbits on a center manifold determines the dynamics of the system in a neighborhood of its equilibrium or fixed point. The term “reduction principle” was introduced by V. Pliss [59]. Definition of the Subject Let M be a subset of the Euclidean space Rn . Definition 1 A set M is said to be smooth manifold of dimension d  n if for each point p 2 M there exists a neighborhood U M and a smooth mapping g : U ! U0 R d such that there is the inverse mapping g –1 and its differential Dg –1 (matrix of partial derivatives) is an injection i. e. its maximal rang is d. For sphere in R3 which is a two-dimensional smooth manifold the introduction of the described neighborhoods is demonstrated in [77]. A manifold M is Ck -smooth, k  1 if the mappings g and g –1 have k continuous derivatives. Moreover, chosing the mappings g and g –1 as C 1 smooth or analytical, we obtain the manifold with the same smoothness. Consider a continuous dynamical system x˙ D F(x) ; x 2 R n ; Rn

Rn

is a Ck -smooth vector field,

(3)

! k  1; i. e. where F : the mapping F has k continuous derivatives. Let us denote the solution of (3) passing through the point p 2 R n at t D 0 by X(t, p). Suppose that the solution is determined for all t 2 R. By the fundamental theorems of the differential equations theory, the hypotheses imposed on F guarantee the existence, uniqueness, and smoothness of X(t, p)

Center Manifolds

for all p 2 R n . In this case the mapping X t (p) D X(t; p) under fixed t is a diffeomorphism of Rn . Thus, F generates the smooth flow X t : R n ! R n , X t (p) D X(t; p). The differential DX t (0) is a fundamental matrix of the linearized system v˙ D DF(0)v. A trajectory (an orbit) of the system (3) through x0 is the set T(x0 ) D fx D X(t; x0 ); t 2 Rg. Definition 2 A manifold M is said to be invariant under (3) if for any p 2 M the trajectory through p lies in M. A manifold M is said to be locally invariant under (3) if for every p 2 M there exists, depending on p, an interval T1 < 0 < T2 , such that X t (p) 2 M as t 2 (T1 ; T2 ). It means that the invariant manifold is formed by trajectories of the system and the locally invariant manifold consists of arcs of trajectories. An equilibrium is 0-dimensional invariant manifold and a periodic orbit is 1-dimensional one. The concept of invariant manifold is a useful tool for simplification of dynamical systems. Definition 3 A manifold M is said to be invariant in T a neighborhood U if for any p 2 M U the moving point X t (p) remains on M as long as X t (p) 2 U. In this case the manifold M is locally invariant. Near an equilibrium point O the system (3) can be rewritten in the form x˙ D Ax C f (x) ;

(4)

It is evident that the dynamics of (4) and (6) is the same on U D fjxj < /2g. If the constructed system (6) has T an invariant manifold M then M U is locally invariant manifold for the initial system (4) or, more precisely, the manifold M is invariant in U. It should be noted that for the case of infinite-dimensional phase space the described construction is more delicate (see details below). Let us consider a Ck -smooth mapping (k  1) G : R n ! R n which has the inverse Ck -smooth mapping G–1 . The mapping G generates the discrete dynamical system of the form x mC1 D G(x m ) ;

m D : : : ; 2; 1; 0; 1; 2; : : : : (7)

Each continuous system X(t, p) gives rise to the discrete system G(x) D X(1; x) which is the shift operator on unit time. Such a system preserves the orientation of the phase space,where as an arbitrary discrete system may change it. Hence, the space of discrete systems is more rich than the space of continuous ones. Moreover, usually investigation of a discrete system is, as a rule, simpler than study of a differential equations one. In many instances the investigation of a differential equation may be reduced to study of a discrete system by Poincaré (first return) mapping. An orbit of (7) is the set T D fx D x m ; m 2 Zg; where xm satisfies (7). Near a fixed point O the mapping x ! G(x) can be rewritten in the form x ! Ax C f (x) ;

where O D fx D 0g, A D DF(O), f is second-order at the origin, i. e. f (0) D 0 and D f (0) D 0. Let us show that the system (4) near O can be considered as a perturbation of the linearized system x˙ D Ax :

(5)

For this we construct a Ck -smooth mapping g which coincides with f in a sufficiently small neighborhood of the origin and is C1 -close to zero. To construct the mapping g one uses the C 1 -smooth cut-off function ˛ : RC ! RC 8 r < 1/2 ; ˆ <1; ˛(r) D > 0; 1/2  r  1 ; ˆ : 0; r >1: Then set g(x) D ˛(jxj/ ) f (x) where " is a parameter. If jxj > , g(x) D 0 and if jxj < /2, g(x) D f (x) and g is C1 -close to zero. Consider the system x˙ D Ax C g(x) :

(6)

(8)

where O D fx D 0g, A D DG(O), f is second-order at the origin. It is shown above that near O (8) may be considered as a perturbation of the linearized system x ! Ax :

(9)

Motivational Example Consider the discrete dynamical system     x X(x; y) D x C x y ! : (10) y Y(x; y) D 12 y C 12 x 2 C 2x 2 y C y 3 The origin (0; 0) is fixed point. Our task is to examine the stability of the fixed point. The linearized system at 0 is defined by the matrix   1 0 0 1/2 which has two eigenvalues 1 and 1/2. Hence, the first approximation system contracts along y-axis, whereas its action along x-axis is neutral. So the stability of the fixed

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Center Manifolds, Figure 1 Dynamics on the invariant curve y D x 2

point depends on nonlinear terms. Let us show that the curve y D x 2 is invariant for the mapping (10). It is enough to check that if a point (x, y) is on the curve then the image (X, Y) is on the curve, i. e. Y D X 2 as y D x 2 . We have Yj yDx 2 D 12 y C 12 x 2 C 2x 2 y C y 3 j yDx 2 D x 2 C 2x 4 C x 6 ; X 2 j yDx 2 D (x C x y)2 j yDx 2

(11)

D x 2 C 2x 4 C x 6 : Thus, the curve y D x 2 is an invariant one-dimensional manifold which is a center manifold W c for the system (10), see Fig. 1. The fixed point O is on W c . The restriction of the system on the manifold is x ! x C x yj yDx 2 D x(1 C x 2 ) :

(12)

It follows that the fixed point 0 is unstable and x m ! 1 as m ! 1, see Fig. 1. Hence, the origin is unstable fixed point for the discrete system (10). It turns out the system (10) near O is topologically equivalent to the system x ! x(1 C x 2 ) ; y ! 12 y

(13)

which is simpler than (10). The center manifold y D x 2 is tangent to the x-axis at the origin and W c near 0 can be considered as a perturbed manifold of fy D 0g. Introduction In his famous dissertation “General Problem on Stability of Motion” [41] published at 1892 in Khar’kov (Ukraine) A.M. Lyapunov proved that the equilibrium O of the system (4) is stable if all eigenvalues of matrix A have negative real parts and O is unstable if there exists an eigenvalue with positive real part. He studied the case when

some eigenvalues of A have negative real part and the rest of them have zero real one. Lyapunov proved that if the matrix A has a pair of pure imaginary eigenvalues and the other eigenvalues have negative real parts then there exists a two-dimensional invariant surface M through O, and the equilibrium O is stable if O is stable for the system restricted on M. Speaking in modern terms, Lyapunov proved the existence of “center manifold” and formulated “reduction principle” under the described conditions. Moreover, he found the stability condition by using power series expansions, which is an extension of his first method to evaluate stability of systems whose eigenvalues have zero real part. Now we use the same method to check stability [10,25,31,36,43,76]. The general reduction principle in stability theory was established by V. Pliss [59] in 1964, the term “center manifold” was introduced by A. Kelley at 1967 in the paper [38] where the existence of a family of invariant manifolds through equilibrium was proved in general case. As we see the center manifold can be considered as a perturbation of the center subspace of linearized system. H. Poincaré [61] was probably the first who perceived the importance of the perturbation problem and began to study conditions ensuring the preservation of the equilibriums and periodic orbits under a perturbation of differential equations. Hadamard [26] and Perron [57] proved the existence of stable and unstable invariant manifolds for a hyperbolic equilibrium point and, in fact, showed their preservation. The conditions necessary and sufficient for the preservation of locally invariant manifolds passing through an equilibrium have been obtained in [51]. Many results on the center manifold follow from theory of normal hyperbolicity [77] which studies dynamics of a system near a compact invariant manifold. They will be considered below. The books by Carr [10], Guckenheimer and Holmes [25], Marsden and McCracken [43], Iooss [36], Hassard, Kazarinoff and Wan [31], and Wiggins [76] are popular sources of the information about center manifolds. Center Manifold in Ordinary Differential Equations Consider a linear system of differential equations v˙ D Lv ;

(14)

where v 2 R n and L is a matrix. Divide the eigenvalues of the matrix L into three parts: stable  s = {eigenvalues with negative real part}, unstable  u = {eigenvalues with positive real part} and central (neutral, critical)  c = {eigenvalues with zero real part}. If the matrix does not have the eigenvalues with zero real part, the system is called

Center Manifolds

hyperbolic. The matrix L has three eigenspaces: the stable subspace Es , the unstable subspace Eu , and the central subspace Ec correspond to these parts of spectrum. The subspaces Es , Eu , and Ec are invariant under the system (14), that is, the solution starting on the subspace stays on it [33]. The solutions on Es have exponential tending to 0, the solutions on Eu have exponential growth. The subspaces E s ; E c ; E u meet in pairs only at the origin and E s C E c C E u D R n , i. e. we have the invariant decomposition R n D E s ˚ E c ˚ E u . There exists a linear change of coordinates (see [33]) transforming (14) to the form x˙ D Ax ; y˙ D By ;

(15)

Center Manifolds, Figure 2 The classical five invariant manifolds

z˙ D Cz ; where the matrix A has eigenvalues with zero real part, the matrix B has eigenvalues with negative real part, and the matrix C has eigenvalues with positive real part. So, (14) decomposes into three independent systems of differential equations. It is known [1,23,24] that the system (15) is topologically equivalent to the system x˙ D Ax ; y˙ D y ;

(16)

z˙ D z : Thus, the dynamics of the system (14) is determined by the system x˙ D Ax which is the restriction of (14) on the center subspace Ec . Our goal is to justify a similar “reduction principle” for nonlinear systems of differential equations. Summarizing the results of V. Pliss [59,60], A. Kelley [38], N. Fenichel [19], and M. Hirsch, C. Pugh, M. Shub [34] we obtain the following theorem. Theorem 1 (Existence Theorem) Consider a Ck -smooth (1  k < 1) system of differential equations v˙ D F(v), F(O) D 0 and L D DF(O). Let E s ; E u and E c be the stable, unstable, and central eigenspaces of L. Then near the equilibrium O there exist the following five Ck -smooth invariant manifolds (see Fig. 2): The center manifold W c , tangent to E c at O; The stable manifold W s , tangent to E s at O; The unstable manifold W u , tangent to E u at O; The center-stable manifold W cs , tangent to E c C E s at O;  The center-unstable manifold W cu , tangent to E c C E u at O;

   

The stable and unstable manifolds are unique, but center, center-stable, and center-unstable manifolds are not necessarily unique.

Solutions on W s have exponential tending to O as t ! 1. Solutions on W u have exponential tending to O as t ! 1.

Remark 1 The expression “near the equilibrium O” means that there exists a neighborhood U(O), depending on F and k, where the statements of the Existence Theorem hold. Remark 2 The linearized system v˙ D DF(O)v has the invariant subspace E s C E u whereas the complete system v˙ D F(v) may not have any smooth invariant manifolds with the tangent space E s C E u at O, see [5,30]. Representation of the Center Manifold At first we suppose that our system does not have the unstable eigenspace and is transformed by the linear change of coordinates mentioned above to the form x˙ D Ax C f (x; y) ; y˙ D By C g(x; y) ;

(17)

where E c D f(x; 0)g, E s D f(0; y)g. Since the center manifold W c is tangent to E c D f(x; 0)g at the origin O D (0; 0), it can be represented near O in the form W c D f(x; y) j jxj < "; y D h(x)g : In other words the center manifold is represented as the graph of a smooth mapping h : V Rc ! Rs , where c and s are dimensions of the center and stable subspaces, see Fig. 3. Since W c goes through the origin and is tangent to Es , we have the equalities h(0) D 0 ;

Dh(0) D 0 :

(18)

51

52

Center Manifolds

Center Manifolds, Figure 3 Representation of a center manifold

The invariance of W c means that if an initial point (x, y) is in W c , i. e. y D h(x) then the solution (X(t; x; y); Y(t; x; y)) of (17) is in W c , i. e. Y(t; x; y)) D h(X(t; x; y)). Thus, we get the invariance condition Y(t; x; h(x))) D h(X(t; x; h(x))) :

(19)

Differentiating (19) with respect to t and putting t D 0 we get Bh(x) C g(x; h(x)) D Dh(x)(Ax C f (x; h(x))) : (20) Thus, the mapping h have to be a solution of the partial differential equation (20) and satisfy (18). Suppose that the system has the form x˙ D Ax C f (x; y; z) ; y˙ D By C g 1 (x; y; z) ;

(21)

z˙ D Cz C g2 (x; y; z) ; where A has the eigenvalues with zero real part, B has the eigenvalues with negative real part, C has the eigenvalues with positive real part. Analogously to the previous case we can show that the center manifold is represented in the form W c D f(x; y; z) j jxj < ; y D h1 (x); z D h2 (x)g

Center Manifolds, Figure 4 Nonuniqueness of the center manifold

Uniqueness and Nonuniqueness of the Center Manifold The center manifold is nonunique in general. Let us consider the illustrating example from [38]. The system has the form x˙ D x 2 ;

(24)

y˙ D y ;

where (x; y) 2 R2 : It is obviously that the origin (0; 0) is the equilibrium with single stable manifold W s D f(0; y)g: There is a center manifold of the form W c D f(x; 0)g. Moreover, there are the center manifolds that can be obtained when solving the equation dy y D 2 : dx x

(25)

The solution of (24) has the form   1 y(x) D a exp x for x ¤ 0 and any constant a 2 R. It follows that

where the mappings h1 and h2 satisfy the equations W c (a) D f(x; y)jy D a exp

Bh1 (x) C g1 (x; h1 (x); h2 (x)) D Dh1 (x)(Ax C f (x; h1 (x); h2 (x))) Ch2 (x) C g2 (x; h1 (x); h2 (x))

(22)

D Dh2 (x)(Ax C f (x; h1 (x); h2 (x))) and the conditions h1 (0) D 0 ;

Dh1 (0) D 0

h2 (0) D 0 ;

Dh2 (0) D 0 :

(26)

(23)

  1 x

as x < 0 and y D 0 as x  0g is center manifold for each a, being C 1 -smooth and pasting together the curve fy D a exp(1/x)g as x < 0 and the straight line fy D 0g as x  0:, see Fig. 4. However, the center manifolds possess the following weak uniqueness property. Let U be a neighborhood of the W c : It turns out that the maximal invariant set I in U

Center Manifolds

that the center manifold has to be given by the series y P n D 1 nD2 (n  1)!x which diverges when x ¤ 0. If the system is C 1 -smooth then the Existence Theorem guaranties the existence of a Ck -smooth center manifold for any k < 1. However, the center manifold may not be C 1 -smooth. Van Strien S.J. [71] showed that the system x˙ D x 2 C 2 ; y˙ D y  (x 2  2 ) ;

(28)

˙ D0: 

Center Manifolds, Figure 5 Weak uniqueness of the center manifold

must be in W c , that follows from the Reduction Theorem, see below. Consequently, all center manifolds contact on I. In this case the center manifold is called locally maximal. Thus, the center manifolds may differ on the trajectories leaving the neighborhood as t ! 1 or 1. For example, suppose that a center manifold is twodimensional and a limit cycle is generated from the equilibrium through the Hopf bifurcation [31,35]. In this case the invariant set I is a disk bounded by the limit cycle and all center manifolds will contain this disk, see Fig. 5. Smoothness M. Hirsch, C. Pugh, M. Shub [34] proved that the center manifold is Ck -smooth if the system is Ck -smooth and k < 1. Moreover if the kth derivative of the vector field is ˛Hölder or Lipschitz mapping, the center manifold is the same. However, if the system is C 1 or analytic then the center manifold is not necessary the same. First consider the analytic case. It is clear that if the analytic center manifold exists then it is uniquely determined by applying the Taylor power series expansion. Consider the illustrating example [31] x˙ D x 2 ; y˙ D y C x 2

(27)

which does not have an analytic center manifold. In fact, applying the Taylor power series expansions we obtain

does not have a C 1 -smooth center manifold. In fact if the system is C 1 -smooth then for each k there is a neighborhood U k (O) of a Ck -smooth central manifold W c : There are the systems for which the sequence fU k g can shrink to O as k ! 1 [25,71]. The results of the papers [19,34] show that the smoothness of the invariant manifold depends on the relation between Lyapunov exponents on the center manifold and on the normal subspace (stable plus unstable subspaces). This relation  is included in the concept of the normal hyperbolicity. At an equilibrium  D min(Ren /Rec ), where Ren /Rec > 0; c is the eigenvalue on the center subspace and n is the eigenvalue on the normal subspace. The condition  > 1 is necessary for a persistence of smooth invariant manifold. One can guarantee degree of smoothness k of the manifold provided k <  . Moreover, there exists examples showing that the condition k <  is essential. It means that the system has to contract (expand) along E s (E u )k-times stronger than along the center manifold. If a center manifold to the equilibrium O exists then Re c D 0 and  D 1 at O, but near O may be other equilibrium (or other orbits) on the center manifold where  < 1, and as a consequence, the center manifold may not be C 1 -smooth. Let us consider the illustrating example [25] x˙ D x  x 3 ; y˙ D y C x 4 ;

(29)

˙ D0:  The point O D (0; 0; 0) is equilibrium, the system linearized at O has the form x˙ D 0 ; y˙ D y ;

(30)

˙ D0:  The system (30) has the following invariant subspace: E s D f0; 0; 0g, E c D fx; 0; g, E u D f0; y; 0g. The -axis

53

54

Center Manifolds

consists of the equilibriums (0; 0; 0 ) for the system (29). The system linearized at (0; 0 ; 0) is the following x˙ D 0 x ; y˙ D y ;

(31)

˙ D0:  The eigenvalues on the center subspace are 0 and 0 , the eigenvalue on the normal (unstable) subspace is 1. Therefore the degree of smoothness is bounded by  D 1/0 : Detailed information and examples are given in [10,25,31,36,43,76].

tigation of the dynamics near a nonhyperbolic fixed point to the study of the system on the center manifold. Construction of the Center Manifold The center manifold may be calculated by using simple iterations [10]. However, this method does not have wide application and we consider a method based on the Taylor power series expansion. First it was proposed by A. Lyapunov in [41]. Suppose that the unstable subspace is 0, that is the last equation of the system (32) is absent. Such systems have a lot of applications in stability theory. So we consider a system

Reduction Principle

x˙ D Ax C f (x; y) ;

As we saw above, a smooth system of differential equations near an equilibrium point O can be written in the form

y˙ D By C g(x; y) ;

The center manifold is the graph of the mapping y D h(x) which satisfies the equation

x˙ D Ax C f (x; y; z) ; y˙ D By C g(x; y; z) ;

(32) Dh(x)(Ax C f (x; h(x)))  Bh(x)  g(x; h(x)) D 0 (36)

z˙ D Cz C q(x; y; z) ; where O D (0; 0; 0), A has eigenvalues with zero real part, B has eigenvalues with negative real part and C has eigenvalues with positive real part; f (0; 0; 0) D g(0; 0; 0) D q(0; 0; 0) D 0 and D f (0; 0; 0) D Dg(0; 0; 0) D Dq(0; 0; 0) D 0. In this case the invariant subspaces at O are E c D f(x; 0; 0)g, E s D f(0; y; 0)g, and E u D f(0; 0; z)g. The center manifold has the form ˚ W c D (x; y; z) j x 2 V Rc ;

y D h1 (x); z D h2 (x) ;

(33)

where c is the dimension of the center subspace Ec , the mappings h1 (x) and h2 (x) are Ck -smooth, k  1, h1 (0) D h2 (0) D 0 and Dh1 (0) D Dh2 (0) D 0. The last equalities mean that the manifold W c goes through O and is tangent to Ec at O. Summarizing the results of V. Pliss [59,60], A. Shoshitaishvili [68,69], A. Reinfelds [63], K. Palmer [55], Pugh C.C., Shub M. [62], Carr J. [10], Grobman D.M. [24], and F. Hartman [29] we obtain the following theorem. Theorem 2 (Reduction Theorem) The system of differential Equations (32) near the origin is topologically equivalent to the system x˙ D Ax C f (x; h1 (x); h2 (x)) ; y˙ D y ;

(35)

(34)

and the conditions h(0) D 0 and Dh(0) D 0: Let us try to solve the equation by applying the Taylor power series expansion. Denote the left part of (36) by N(h(x)). J. Carr [10] and D. Henry [32] proved the following theorem. Theorem 3 Let  : V(0) Rc ! Rs be a smooth mapping, (0) D 0 and D(0) D 0 such that N((x)) D o(jxjm ) for some m > 0 as jxj ! 0 then h(x)  (x) D o(jxj m ) as jxj ! 0 where r(x) D o(jxj m ) means that r(x)/jxj m ! 0 as jxj ! 0. Thus, if we solve the Eq. (36) with a desired accuracy we construct h with the same accuracy. Theorem 3 substantiates the application of the described method. Let us consider a simple example from [76] x˙ D x 2 y  x 5 ;

(x; y) 2 R2 , the equilibrium is at the origin (0; 0). The eigenvalues of the linearized system are 0 and 1. According to the Existence Theorem the center manifold is locally represented in the form ˚ W c D (x; y) j y D h(x); jxj < ı;

h(0) D 0; Dh(0) D 0) ;

z˙ D z : The first equation is the restriction of the system on the center manifold. The theorem allows to reduce the inves-

(37)

y˙ D y C x 2 ;

where ı is sufficiently small. It follows that h has the form h D ax 2 C bx 3 C    :

(38)

Center Manifolds

The equation for the center manifold is given by Dh(x)(Ax C f (x; h(x)))  Bh(x)  g(x; h(x)) D 0 ; (39)

their Taylor expansions coincide up to all existing orders. For example, the system (24) has the center manifolds of the form ˚   W c (a) D (x; y)jy D a exp x1

where A D 0, B D 1, f (x; y) D x 2 y  x 5 , g(x; y) D x 2 . Substituting (38) in (39) we obtain the equality (2ax C 3bx 2 C    )(ax 4 C bx 5  x 5 C    ) C ax 2 C bx 3  x 2 C    D 0 :

: :

a  1 D 0; ) a D 1 b D 0; :: ::

for any a. However, each manifold has null-expansion at 0.

y˙ D y ;

(42)

(43)

(44)

Hence the equilibrium is unstable. It should be noted that the calculation of bx3 is unnecessary, since substituting h(x) D x 2 C bx 3 C    in the first equation of the system (43), we also obtain (44). Thus, it is enough to compute the first term ax2 of the mapping h. This example brings up the following question. The system (37) is topologically equivalent to the system x˙ D x 4 C    ; y˙ D y ;

(46)

where v 2 R n , L is a matrix and G is second-order at the origin. Without loss of generality we consider the system (46) as a perturbation of the linear system

where h(x) D x 2 C 0x 3 C    : Substituting h we obtain the equation x˙ D x 4 C : : : :

v ! Lv C G(v) ;

(41)

In this connection we have to decide how many terms (powers of x) have be computed? The solution depends on our goal. For example, suppose that we study the Lyapunov stability of the equilibrium (0; 0) for (37). According to the Reduction Theorem the system (37) is topologically equivalent to the system x˙ D x 2 h(x)  x 5 ;

Center Manifold in Discrete Dynamical Systems Consider a dynamical system generated by the mapping

Therefore we have h(x) D x 2 C 0x 3 C    :



(40)

Equating coefficients on each power of x to zero we obtain x2 x3 :: :

as x < 0 and y D 0 as x  0

(45)

which has many center manifolds. From this it follows that the initial system has a lot of center manifolds. Which of center manifolds is actually being found when approximating the center manifold via power series expansion? It turns out [10,70,74] that any two center manifolds differ by transcendentally small terms, i. e. the terms of

x ! Lx :

(47)

Divide the eigenvalues of L into three parts: stable s D feigenvalues with modulus less than 1g, unstable u D feigenvalues with modulus greater than 1g, critical c D feigenvalues with modulus equal to 1g. The matrix L has three eigenspaces: the stable subspace Es , the unstable subspace Eu , and the central subspace Ec , which correspond to the mentioned spectrum parts respectively, being E s ˚ E u ˚ E c D R n . The next theorem follows from the theorem on perturbation of -hyperbolic endomorphism on a Banach space [34]. Theorem 4 (Existence Theorem) Consider a Ck -smooth (1  k < 1) discrete system v ! Lv C G(v) ; G(0) D 0 and G is C1 -close to zero. Let E s ; E u and Ec be the stable, unstable, and central eigenspaces of the linear mapping L. Then near the fixed point O there exist the following Ck -smooth invariant manifolds:     

The center manifold W c tangent to E c at O, The stable manifold W s tangent to E s at O, The unstable manifold W u tangent to E u at O, The center-stable manifold W cs tangent to E c C E s at O, The center-unstable manifold W cu tangent to E c C E u at O.

The stable and unstable manifolds are unique, but center, center-stable, and center-unstable manifolds may not be unique.

55

56

Center Manifolds

The orbits on W s have exponential tending to O as m ! 1. The orbits on W u have exponential tending to O as m ! 1. There exists a linear change of coordinates transforming (46) to the form 1 1 0 x Ax C f (x; y; z) @ y A ! @ By C g1 (x; y; z) A ; Cz C g2 (x; y; z) z 0

(48)

where A has eigenvalues with modulus equal 1, B has eigenvalues with modulus less than 1, C has eigenvalues with modulus greater than 1. The system (48) at the origin has the following invariant eigenspaces: E c D f(x; 0; 0)g, E s D f(0; y; 0)g, and E u D f(0; 0; z)g. Since the center manifold W c is tangent to Ec at the origin O D (0; 0), it can be represented near O in the form W c D f(x; y; z) j jxj < ; y D h1 (x); z D h2 (x)g ; where h1 and h2 are second-order at the origin, i. e. h1;2 (0) D 0; Dh1;2 (0) D 0: The invariance of W c means that if a point (x; y; z) 2 W c ; i. e. y D h1 (x); z D h2 (x), then (Ax C f (x; y; z); By C g1 (x; y; z); Cz C g2 (x; y; z)) 2 W c ; i. e. By C g1 (x; y; z) D h1 (Ax C f (x; y; z)); Cz C g2 (x; y; z) D h2 (Ax C f (x; y; z)). Thus, we get the invariance property Bh1 (x) C g1 (x; h1 (x); h2 (x)) D h1 (Ax C f (x; h1 (x); h2 (x)) ; Ch2 (x) C g2 (x; h1 (x); h2 (x))

(49)

D h2 (Ax C f (x; h1 (x); h2 (x)) : Results on smoothness and uniqueness of center manifold for discrete systems are the same as for continuous systems. Theorem 5 (Reduction Theorem [55,62,63,64]) The discrete system (48) near the origin is topologically equivalent to the system x ! Ax C f (x; h1 (x); h2 (x)) ; y ! By ;

(50)

z ! Cz : The first mapping is the restriction of the system on the center manifold. The theorem reduces the investigation of dynamics near a nonhyperbolic fixed point to the study of the system on the center manifold.

Normally Hyperbolic Invariant Manifolds As it is indicated above, a center manifold can be considered as a perturbed invariant manifold for the center subspace of the linearized system. The normal hyperbolicity conception arises as a natural condition for the persistence of invariant manifold under a system perturbation. Informally speaking, an f -invariant manifold M, where f -diffeomorphism, is normally hyperbolic if the action of Df along the normal space of M is hyperbolic and dominates over its action along the tangent one. (the rigorous definition is below). Many of the results concerning center manifold follows from the theory of normal hyperbolic invariant manifolds. In particular, the results related to center eigenvalues of Df with jj ¤ 1 may be obtained using this theory. The problem of existence and preservation for invariant manifolds has a long history. Initial results on invariant (integral) manifolds of differential equations were obtained by Hadamard [26] and Perron [57], Bogoliubov, Mitropolskii [8] and Lykova [48], Pliss [60], Hale [28], Diliberto [16] and other authors (for references see [39,77]). In the late 1960s and throughout 1970s the theory of perturbations of invariant manifolds began to assume a very general and well-developed form. Sacker [66] and Neimark [50] proved (independently and by different methods) that a normally hyperbolic compact invariant manifold is preserved under C1 perturbations. Final results on preservation of invariant manifolds were obtained by Fenichel [19,20] and Hirsch, Pugh, Shub [34]. The linearization theorem for a normally hyperbolic manifold was proved in [62]. Let f : R n ! R n be a Cr -diffeomorphism, 1  r < 1, and a compact manifold M is f -invariant. Definition 4 An invariant manifold M is called r-normally hyperbolic if there exists a continuous invariant decomposition TR n j M D TM ˚ E s ˚ E u

(51)

tangent bundle TRn

into a direct sum of subbundles of the TM; E s ; E u and constants a;  > 0 such that for 0  k  r and all p 2 M, jD s f n (p)j j(D0 f n (p))1 j k  a exp(n) u n

0 n

1 k

jD f (p)j j(D f (p)) j  a exp(n)

for n  0 ; for n  0 : (52)

Here D0 f ; Ds f and Du f are restrictions of Df on TM; E s and Eu , respectively. The invariance of the bundle E  means that for every p 2 M0 , the image of the fiber E  (p) at p under Df (p)

Center Manifolds

is the fiber E  (q) at q D f (p): The bundles Es and Eu are called stable and unstable, respectively. In other words, M is normally hyperbolic if the differential Df contracts (expands) along the normal (to M) direction and this contraction (expansion) is stronger in r times than a conceivable contraction (expansion) along M: Summarizing the results of [19,20,34,50,62,66] we obtain the following theorem. Theorem 6 Let a Cr diffeomorphism f be r-normally hyperbolic on a compact invariant manifold M with the decomposition TR n j M D TM ˚ E s ˚ E u . Then  there exist invariant manifolds W s and W u near M, which are tangent at M to TM ˚ E s and TM ˚ E u ;  the manifolds M, W s and W u are Cr -smooth;  if g is another Cr -diffeomorphism Cr -close to f ; then there exists the unique manifold M g which is invariant and r-normally hyperbolic for g;

The reduced system is of the form x˙ D ax 3 C : : : : Thus, if a < 0, the origin is stable. The book [25] contains other examples of investigation of stability. Bifurcations Consider a parametrized system of differential equations x˙ D A()x C f (x; ) ;

(55)

where  2 R k , f is C1 -small when  is small. To study the bifurcation problem near  D 0 it is useful to deal with the extended system of the form x˙ D A()x C f (x; ) ; ˙ D0: 

(56)

Similarly result takes place for flows. Mañé [44] showed that a locally unique, preserved, compact invariant manifold is necessarily normally hyperbolic. Local uniqueness means that near M, an invariant set I of the perturbation g is in M g . Conditions for the preservation of locally nonunique compact invariant manifolds and linearization were found by G. Osipenko [52,53].

Suppose that (55) has a n-dimensional center manifold for  D 0. Then the extended system (56) has a n C k-dimensional center manifold. The reduction principle guarantees that all bifurcations lie on the center manifold. Moreover, the center manifold has the form y D h(x; 0 ) for every ˙ D 0 and the map0 which is a solution of the equation  ping h may be presented as a power series in . Further interesting information regarding to recent advancements in the approximation and computation of center manifolds is in [37]. The book [31] deals with Hopf bifurcation and contains a lot of examples and applications. Invariant manifolds and foliations for nonautonomous systems are considered in [3,13]. Partial linearization for noninvertible mappings is studied in [4].

Applications

Center Manifold in Infinite-Dimensional Space

Stability

Center manifold theory is a standard tool to study the dynamics of discrete and continuous dynamical systems in infinite dimensional phase space. We start with the discrete dynamical system generated by a mapping of a Banach space. The existence of the center manifold of the mappings in Banach space was proved by Hirsch, Pugh and Shub [34]. Let T : E ! E be a linear endomorphism of a Banach space E.

 near M f is topologically equivalent to D f j E s˚E u , which in local coordinates has the form (x; y; z) ! ( f (x); D f (x)y; D f (x)z) ; x2M;

y 2 E s (x) ;

z 2 E u (x) :

(53)

As it was mentioned above, it was Lyapunov who pioneered in applying the concept of center manifold to establish the stability of equilibrium. V. Pliss [59] proved the general reduction principle in stability theory. According to this principle the equilibrium is stable if and only if it is stable on the center manifold. We have considered the motivating example where the center manifold concept was used. The system x˙ D x y ; y˙ D y C ax 2 ;

(54)

has the center manifold y D h(x): Applying Theorem 3, we obtain h(x) D ax 2 C    :

Definition 5 A linear endomorphism T is said to be -hyperbolic, > 0, if no eigenvalue of T has modulus , i. e., its spectrum Spect(T) has no points on the circle of radius

in the complex plane C. A linear 1-hyperbolic isomorphism T is called hyperbolic. For a -hyperbolic linear endomorphism there exists a T-invariant splitting of E D E1 ˚ E2 such that the spectrum of the restriction T1 D Tj E 1 lies outside of the disk of

57

58

Center Manifolds

the radius , where as the spectrum of T2 D Tj E 2 lies inside it. So T 1 is automorphism and T11 exists. It is known (for details see [34]) that one can define norms in E1 and E2 such that associated norms of T11 and T 2 may be estimated in the following manner jT11 j <

1

;

jT2 j < :

(57)

Conversely, if T admits an invariant splitting T D T1 ˚ T2 with jT11 j  a; jT2 j  b; ab < 1 then Spect(T1 ) lies outside the disk f 2 C : jj < 1/ag, and Spect(T2 ) lies in the disk f 2 C : jj  bg: Thus, T is -hyperbolic with b < < 1/a: Theorem 7 Let T be a -hyperbolic linear endomorphism,  1. Assume that f : E ! E is C r ; r  1; f D T C g; f (0) D 0 and there is ı > 0 such that if g is a Lipschitz mapping and Lip(g) D Lip( f  T)  ı ;

(58)

then there exists a manifold Wf ; which is a graph of a C1 map ' : E1 ! E2 , i. e. Wf D fx C y j y D '(x); x 2 E1 ; y 2 E2 g ; with the following properties:  W f is f -invariant;  if kT11 k j kT2 k < 1; j D 1; : : : ; r then W f is Cr and depends continuously on f in the Cr topology;  WT D E1 ;  for x 2 Wf j f 1 (x)j  (a C 2") jxj ;

(59)

where a < 1/ , and " is small provided ı is small;  if D f (0) D T then W f is tangent to E1 at 0. Suppose that the spectrum of T is contained in A1 where A1 D fz 2 C; jzj  1g ;

S

A2

A2 D fz 2 C; jzj  a < 1g :

Corollary 1 If f : E ! E is C r ; 1  r < 1; f (0) D 0 and Lip( f  T)  is small, then there exists a centerunstable manifold W cu D Wf which is the graph of a Cr function E1 ! E2 : The center-unstable manifold is attractor, i. e. for any x 2 E the distance between f n (x) and W cu tends to zero as n ! 1. The manifold Wf D W c is center manifold when A1 D fz 2 C; jzj D 1g. Apply this theorem to a mapping of the general form u ! Au C f (u) ;

(60)

where E is a Banach space, u 2 E, A: E ! E is a linear operator, f is second-order at the origin. The question arises of whether there exists a smooth mapping g such that g coincides with f in a sufficiently small neighborhood of the origin and g is C1 -close to zero? The answer is positive if E admits C 1 -norm, i. e. the mapping u ! kuk is C 1 -smooth for u ¤ 0. It is performed for Hilbert space. The desired mapping g may be constructed with the use of a cut-off function. To apply Theorem 7 to (60) we have to suppose Hypothesis on C 1 -norm: the Banach space E admits C 1 -norm. Thus, Theorem 7 together with Hypothesis on C 1 norm guarantee an existence of center manifold for the system (60) in a Banach space. A. Reinfelds [63,64] prove a reduction theorem for homeomorphism in a metric space from which an analog of Reduction Theorem 5 for Banach space follows. Flows in Banach Space Fundamentals of center manifold theory for flows and differential equations in Banach spaces can be found in [10,21,43,73]. In Euclidean space a flow is generated by a solution of a differential equation and the theory of ordinary differential equations guarantees the clear connection between flows and differential equations. In infinite dimensional (Banach) space this connection is more delicate and at first, we consider infinite dimensional flows and then (partial) differential equations. Definition 6 A flow (semiflow) on a domain D is a continuous family of mappings F t : D ! D where t 2 R (t  0) such that F0 D I and F tCs D F t Fs for t; s 2 R (t; s  0): Theorem 8 (Center Manifold for Infinite Dimensional Flows [43]) Let the Hypothesis on C 1 -norm be fulfilled and F t : E ! E be a semiflow defined near the origin for 0  t  , Ft (0) D 0. Suppose that the mapping F t (x) is C0 in t and C kC1 in x and the spectrum of the linear semigroup DF t (0) is of the form e t(1 [2 ) ; where e t(1 ) is on the unit circle (i. e. 1 is on the imaginary axis) and e t(2 ) is inside the unit circle with a positive distance from one as t > 0 (i. e. 2 is in the left half-plane). Let E D E1 ˚ E2 be the DF t (0)-invariant decomposition corresponding to the decomposition of the spectrum e t(1 [2 ) and dimE1 D d < 1. Then there exists a neighborhood U of the origin and Ck -submanifold W c U of the dimension d such that  W c is invariant for F t in U, 0 2 W c and W c is tangent to E1 at 0;  W c is attractor in U, i. e. if F tn (x) 2 U; n D 0; 1; 2; : : : then F tn (x) ! W c as n ! 1.

Center Manifolds

In infinite dimensional dynamics it is also popular notion of “inertial manifold” which is an invariant finite dimensional attracting manifold corresponding to W c of the Theorem 8. Partial Differential Equations At the present the center manifold method takes root in theory of partial differential equations. Applying center manifold to partial differential equations leads to a number of new problems. Consider an evolution equation u˙ D Au C f (u) ;

(61)

where u 2 E, E is a Banach space, A is a linear operator, f is second-order at the origin. Usually A is a differential operator defined on a domain D E; D ¤ E. To apply Theorem 8 we have to construct a semiflow F t defined near the origin of E. As a rule, an infinite dimensional phase space E is a functional space where the topology may be chosen by the investigator. A proper choice of functional space and its topology may essentially facilitate the study. As an example we consider a nonlinear heat equation u0t D u C f (u) ;

(62)

where u is defined on a bounded domain ˝ R n with Dirichlet condition on the boundary uj@˝ D 0, u D (@2 /@x12 C    C @2 /@x n2 )u is the Laplace operator defined on the set C02 D fu 2 C 2 (˝); u D 0 on @˝g : According to [49] we consider a HilbertR space E D L2 (˝) with the inner product < u; v >D ˝ uvdx. The operator  may be extended to a self-conjugate operator A: D A ! L2 (˝) with the domain DA that is the closure of C02 in L2 (˝). Consider the linear heat equation u0t D u and its extension u0t D Au;

u 2 DA :

(63)

What do we mean by a flow generated by (63)? Definition 7 A linear operator A is the infinitisemal generator of a continuous semigroup U(t); t  0 if: U(t) is a linear mapping for each t; U(t) is a semigroup and kU(t)u  uk ! 0 as t ! C0; U(t)u  u t!C0 t

Au D lim

(64)

whenever the limit exists. For Laplace operator we have < u; u > 0 for u 2 C02 ; hence the operator A is dissipative. It guarantees [6] the existence of the semigroup U(t) with the infinitesimal gen-

erator A. The mapping U(t) is the semigroup DF t (0) in Theorem 8. The next problem is the relation between the linearized system u0 D Au and full nonlinear Equation (61). To consider the mapping f in (62) as C1 -small perturbation, the Hypothesis on C 1 -norm has to be fulfilled. To prove the existence of the solution of the nonlinear equation with u(0) D u0 one uses the Duhamel formula Zt u(t) D U t u0 C

U ts f (u(s))ds 0

and the iterative Picard method. By this way we construct the semiflow required to apply Theorem 8. The relation between the spectrum of a semigroup U(t) and the spectrum of its infinitisemal generator gives rise to an essential problem. All known existence theorems are formulated in terms of the semigroup as in Theorem 8. The existence of the spectrum decompositions of a semigroup U(t) and the corresponding estimations on the appropriate projections are supposed. This framework is inconvenient for many applications, especially in partial differential equations. In finite dimensions, a linear system x˙ D Ax has a solution of the form U(t)x D e At x and  is an eigenvalue of A if and only if et is an eigenvalue of U(t). We have the spectral equality Spect(U(t)) D eSpect(A)t : In infinite dimensions, relating the spectrum of the infinitesimal generator A to that of the semigroup U(t) is a spectral mapping problem which is often nontrivial. The spectral inclusion Spect(U(t)) eSpect(A)t always holds and the inverse inclusion is a problem that is solved by the spectral mapping theorems [12,18,22,65]. Often an application of center manifold theory needs to prove an appropriate version of the reduction theorem [7,11,12,14,15,18]. Pages 1–5 of the book [40] give an extensive list of the applications of center manifold theory to infinite dimensional problems. Here we consider a few of typical applications. Reaction-diffusion equations are typical models in chemical reaction (Belousov– Zhabotinsky reaction), biological systems, population dynamics and nuclear reaction physics. They have the form u0t D (K() C D)u C f (u; ) ;

(65)

where K is a matrix depending on a parameter, D is a symmetric, positive semi-definite, often diagonal matrix,  is the Laplace operator,  is a control parameter. The pa-

59

60

Center Manifolds

pers [2,27,31,67,78] applied center manifold theory to the study of the reaction-diffusion equation. Invariant manifolds for nonlinear Schrodinger equations are studied in [22,40]. In the elliptic quasilinear equations the spectrum of the linear operator is unbounded in unstable direction and in this case the solution does not generate even a semiflow. Nevertheless, center manifold technique is successfully applied in this case as well [45]. Henry [32] proved the persistence of normally hyperbolic closed linear subspace for semilinear parabolic equations. It follows the existence of the center manifold which is a perturbed manifold for the center subspace. I. Chueshov [14,15] considered a reduction principle for coupled nonlinear parabolic-hyperbolic partial differential equations that has applications in thermoelastisity. Mielke [47] has developed the center manifold theory for elliptic partial differential equations and has applied it to problems of elasticity and hydrodynamics. The results for non-autonomous system in Banach space can be found in [46]. The reduction principle for stochastic partial differential equations is considered in [9,17,75]. Future Directions Now we should realize that principal problems in the center manifolds theory of finite dimensional dynamics have been solved and we may expect new results in applications. However the analogous theory for infinite dimensional dynamics is far from its completion. Here we have few principal problems such as the reduction principle (an analogue of Theorem 2 or 5) and the construction of the center manifold. The application of the center manifold theory to partial differential equations is one of promising and developing direction. In particular, it is very important to investigate the behavior of center manifolds when Galerkin method is applied due to a transition from finite dimension to infinite one. As it was mentioned above each application of center manifold methods to nonlinear infinite dimensional dynamics is nontrivial and gives rise to new directions of researches. Bibliography Primary Literature 1. Arnold VI (1973) Ordinary Differential Equations. MIT, Cambridge 2. Auchmuty J, Nicolis G (1976) Bifurcation analysis of reactiondiffusion equation (III). Chemical Oscilations. Bull Math Biol 38:325–350 3. Aulbach B (1982) A reduction principle for nonautonomous differential equations. Arch Math 39:217–232 4. Aulbach B, Garay B (1994) Partial linearization for noninvertible mappings. J Appl Math Phys (ZAMP) 45:505–542

5. Aulbach B, Colonius F (eds) (1996) Six Lectures on Dynamical Systems. World Scientific, New York 6. Balakrishnan AV (1976) Applied Functional Analysis. Springer, New York, Heidelberg 7. Bates P, Jones C (1989) Invariant manifolds for semilinear partial differential equations. In: Dynamics Reported 2. Wiley, Chichester, pp 1–38 8. Bogoliubov NN, Mitropolsky YUA (1963) The method of integral manifolds in non-linear mechanics. In: Contributions Differential Equations 2. Wiley, New York, pp 123–196 9. Caraballo T, Chueshov I, Landa J (2005) Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations. Nonlinearity 18:747–767 10. Carr J (1981) Applications of Center Manifold Theory. In: Applied Mathematical Sciences, vol 35. Springer, New York 11. Carr J, Muncaster RG (1983) Applications of center manifold theory to amplitude expansions. J Diff Equ 59:260–288 12. Chicone C, Latushkin Yu (1999) Evolution Semigroups in Dynamical Systems and Differential Equations. Math Surv Monogr 70. Amer Math Soc, Providene 13. Chow SN, Lu K (1995) Invariant manifolds and foliations for quasiperiodic systems. J Diff Equ 117:1–27 14. Chueshov I (2007) Invariant manifolds and nonlinear masterslave synchronization in coupled systems. In: Applicable Analysis, vol 86, 3rd edn. Taylor and Francis, London, pp 269–286 15. Chueshov I (2004) A reduction principle for coupled nonlinesr parabolic-hyperbolic PDE. J Evol Equ 4:591–612 16. Diliberto S (1960) Perturbation theorem for periodic surfaces I, II. Rend Cir Mat Palermo 9:256–299; 10:111–161 17. Du A, Duan J (2006) Invariant manifold reduction for stochastic dynamical systems. http://arXiv:math.DS/0607366 18. Engel K, Nagel R (2000) One-parameter Semigroups for Linear Evolution Equations. Springer, New York 19. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Ind Univ Math 21:193–226 20. Fenichel N (1974) Asymptotic stability with rate conditions. Ind Univ Math 23:1109–1137 21. Gallay T (1993) A center-stable manifold theorem for differential equations in Banach space. Commun Math Phys 152:249–268 22. Gesztesy F, Jones C, Latushkin YU, Stanislavova M (2000) A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations. Ind Univ Math 49(1):221–243 23. Grobman D (1959) Homeomorphism of system of differential equations. Dokl Akad Nauk SSSR 128:880 (in Russian) 24. Grobman D (1962) The topological classification of the vicinity of a singular point in n-dimensional space. Math USSR Sbornik 56:77–94; in Russian 25. Guckenheimer J, Holmes P (1993) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors Fields. Springer, New York 26. Hadamard J (1901) Sur l’etaration et les solution asymptotiques des equations differentielles. Bull Soc Math France 29:224–228 27. Haken H (2004) Synergetics: Introduction and Advanced topics. Springer, Berlin 28. Hale J (1961) Integral manifolds of perturbed differential systems. Ann Math 73(2):496–531 29. Hartman P (1960) On local homeomorphisms of Euclidean spaces. Bol Soc Mat Mex 5:220

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30. Hartman P (1964) Ordinary Differential Equations. Wiley, New York 31. Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge 32. Henry D (1981) Geometric theory of semilinear parabolic equations. Lect Notes Math 840:348 33. Hirsch M, Smale S (1974) Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, Orlando 34. Hirsch M, Pugh C, Shub M (1977) Invariant manifolds. Lect Notes Math 583:149 35. Hopf E (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber Verh Sachs Akad Wiss Leipzig Math-Nat 94:3–22 36. Iooss G (1979) Bifurcation of Maps and Application. N-Holl Math Stud 36:105 37. Jolly MS, Rosa R (2005) Computation of non-smooth local centre manifolds. IMA J Numer Anal 25(4):698–725 38. Kelley A (1967) The stable, center stable, center, center unstable and unstable manifolds. J Diff Equ 3:546–570 39. Kirchgraber U, Palmer KJ (1990) Geometry in the Neighborhood of Invariant Manifolds of the Maps and Flows and Linearization. In: Pitman Research Notes in Math, vol 233. Wiley, New York 40. Li C, Wiggins S (1997) Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrodinger Equations. Springer, New York 41. Lyapunov AM (1892) Problémé Générale de la Stabilité du Mouvement, original was published in Russia 1892, transtalted by Princeton Univ. Press, Princeton, 1947 42. Ma T, Wang S (2005) Dynamic bifurcation of nonlinear evolution equations and applications. Chin Ann Math 26(2): 185–206 43. Marsden J, McCracken M (1976) Hopf bifurcation and Its Applications. Appl Math Sci 19:410 44. Mañé R (1978) Persistent manifolds are normally hyperbolic. Trans Amer Math Soc 246:261–284 45. Mielke A (1988) Reduction of quasilinear elliptic equations in cylindrical domains with application. Math Meth App Sci 10:51–66 46. Mielke A (1991) Locally invariant manifolds for quasilinear parabolic equations. Rocky Mt Math 21:707–714 47. Mielke A (1996) Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability. In: Pitman Research Notes in Mathematics Series, vol 352. Longman, Harlow, pp 277 48. Mitropolskii YU, Lykova O (1973) Integral Manifolds in Nonlinear Mechanics. Nauka, Moscow 49. Mizohata S (1973) The Theory of Partial Differential Equations. Cambridge University Press, Cambridge 50. Neimark Y (1967) Integral manifolds of differential equations. Izv Vuzov, Radiophys 10:321–334 (in Russian) 51. Osipenko G, Ershov E (1993) The necessary conditions of the preservation of an invariant manifold of an autonomous system near an equilibrium point. J Appl Math Phys (ZAMP) 44:451–468 52. Osipenko G (1996) Indestructibility of invariant non-unique manifolds. Discret Contin Dyn Syst 2(2):203–219 53. Osipenko G (1997) Linearization near a locally non-unique invariant manifold. Discret Contin Dyn Syst 3(2):189–205

54. Palis J, Takens F (1977) Topological equivalence of normally hyperbolic dynamical systems. Topology 16(4):336–346 55. Palmer K (1975) Linearization near an integral manifold. Math Anal Appl 51:243–255 56. Palmer K (1987) On the stability of center manifold. J Appl Math Phys (ZAMP) 38:273–278 57. Perron O (1928) Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystem. Math Z 29:129–160 58. Pillet CA, Wayne CE (1997) Invariant manifolds for a class of dispersive. Hamiltonian, partial differential equations. J Diff Equ 141:310–326 59. Pliss VA (1964) The reduction principle in the theory of stability of motion. Izv Acad Nauk SSSR Ser Mat 28:1297–1324; translated (1964) In: Soviet Math 5:247–250 60. Pliss VA (1966) On the theory of invariant surfaces. In: Differential Equations, vol 2. Nauka, Moscow pp 1139–1150 61. Poincaré H (1885) Sur les courbes definies par une equation differentielle. J Math Pure Appl 4(1):167–244 62. Pugh C, Shub M (1970) Linearization of normally hyperbolic diffeomorphisms and flows. Invent Math 10:187–198 63. Reinfelds A (1974) A reduction theorem. J Diff Equ 10:645–649 64. Reinfelds A (1994) The reduction principle for discrete dynamical and semidynamical systems in metric spaces. J Appl Math Phys (ZAMP) 45:933–955 65. Renardy M (1994) On the linear stability of hyperbolic PDEs and viscoelastic flows. J Appl Math Phys (ZAMP) 45:854–865 66. Sacker RJ (1967) Hale J, LaSalle J (eds) A perturbation theorem for invariant Riemannian manifolds. Proc Symp Diff Equ Dyn Syst Univ Puerto Rico. Academic Press, New York, pp 43–54 67. Sandstede B, Scheel A, Wulff C (1999) Bifurcations and dynamics of spiral waves. J Nonlinear Sci 9(4):439–478 68. Shoshitaishvili AN (1972) Bifurcations of topological type at singular points of parameterized vector fields. Func Anal Appl 6:169–170 69. Shoshitaishvili AN (1975) Bifurcations of topological type of a vector field near a singular point. Trudy Petrovsky seminar, vol 1. Moscow University Press, Moscow, pp 279–309 70. Sijbrand J (1985) Properties of center manifolds. Trans Amer Math Soc 289:431–469 71. Van Strien SJ (1979) Center manifolds are not C 1 . Math Z 166:143–145 72. Vanderbauwhede A (1989) Center Manifolds, Normal Forms and Elementary Bifurcations. In: Dynamics Reported, vol 2. Springer, Berlin, pp 89–169 73. Vanderbauwhede A, Iooss G (1992) Center manifold theory in infinite dimensions. In: Dynamics Reported, vol 1. Springer, Berlin, pp 125–163 74. Wan YH (1977) On the uniqueness of invariant manifolds. J Diff Equ 24:268–273 75. Wang W, Duan J (2006) Invariant manifold reduction and bifurcation for stochastic partial differential equations. http://arXiv: math.DS/0607050 76. Wiggins S (1992) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York 77. Wiggins S (1994) Normally Hyperbolic Invariant Manifolds of Dynamical Systems. Springer, New York 78. Wulff C (2000) Translation from relative equilibria to relative periodic orbits. Doc Mat 5:227–274

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Books and Reviews Bates PW, Lu K, Zeng C (1998) Existence and persistence of invariant manifolds for semiflows in Banach spaces. Mem Amer Math Soc 135:129 Bates PW, Lu K, Zeng C (1999) Persistence of overflowing manifolds for semiflow. Comm Pure Appl Math 52(8):983–1046 Bates PW, Lu K, Zeng C (2000) Invariant filiations near normally hyperbolic invariant manifolds for semiflows. Trans Amer Math Soc 352:4641–4676 Babin AV, Vishik MI (1989) Attractors for Evolution Equations. Nauka. Moscow; English translation (1992). Elsevier Science, Amsterdam Bylov VF, Vinograd RE, Grobman DM, Nemyskiy VV (1966) The Theory of Lyapunov Exponents. Nauka, Moscow (in Russian) Chen X-Y, Hale J, Tan B (1997) Invariant foliations for C 1 semigroups in Banach spaces. J Diff Equ 139:283–318 Chepyzhov VV, Goritsky AYU, Vishik MI (2005) Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation. Russ J Math Phys 12(1): 17–39 Chicone C, Latushkin YU (1997) Center manifolds for infinite dimensional nonautonomous differential equations. J Diff Equ 141:356–399 Chow SN, Lu K (1988) Invariant manifolds for flows in Banach spaces. J Diff Equ 74:285–317 Chow SN, Lin XB, Lu K (1991) Smooth invariant foliations in infinite dimensional spaces. J Diff Equ 94:266–291 Chueshov I (1993) Global attractors for non-linear problems of mathematical physics. Uspekhi Mat Nauk 48(3):135–162; English translation in: Russ Math Surv 48:3 Chueshov I (1999) Introduction to the Theory of Infinite-Dimensional Dissipative Systems. Acta, Kharkov (in Russian); English translation (2002) http://www.emis.de/monographs/ Chueshov/. Acta, Kharkov Constantin P, Foias C, Nicolaenko B, Temam R (1989) Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Appl Math Sci, vol 70. Springer, New York Gonçalves JB (1993) Invariant manifolds of a differentiable vector field. Port Math 50(4):497–505 Goritskii AYU, Chepyzhov VV (2005) Dichotomy property of solu-

tions of quasilinear equations in problems on inertial manifolds. SB Math 196(4):485–511 Hassard B, Wan Y (1978) Bifurcation formulae derived from center manifold theory. J Math Anal Appl 63:297–312 Hsia C, Ma T, Wang S (2006) Attractor bifurcation of three-dimensional double-diffusive convection. http://arXiv:nlin.PS/ 0611024 Knobloch HW (1990) Construction of center manifolds. J Appl Math Phys (ZAMP) 70(7):215–233 Latushkin Y, Li Y, Stanislavova M (2004) The spectrum of a linearized 2D Euler operator. Stud Appl Math 112:259 Leen TK (1993) A coordinate independent center manifold reduction. Phys Lett A 174:89–93 Li Y (2005) Invariant manifolds and their zero-viscosity limits for Navier–Stokes equations. http://arXiv:math.AP/0505390 Osipenko G (1989) Examples of perturbations of invariant manifolds. Diff Equ 25:675–681 Osipenko G (1985, 1987, 1988) Perturbation of invariant manifolds I, II, III, IV. Diff Equ 21:406–412, 21:908–914, 23:556–561, 24:647–652 Podvigina OM (2006) The center manifold theorem for center eigenvalues with non-zero real parts. http://arXiv:physics/ 0601074 Sacker RJ, Sell GR (1974, 1976, 1978) Existence of dichotomies and invariant splitting for linear differential systems. J Diff Equ 15:429-458, 22:478–522, 27:106–137 Scarpellini B (1991) Center manifolds of infinite dimensional. Main results and applications. J Appl Math Phys (ZAMP) 43: 1–32 Sell GR (1983) Vector fields on the vicinity of a compact invariant manifold. Lect Notes Math 1017:568–574 Swanson R (1983) The spectral characterization of normal hyperbolicity. Proc Am Math Soc 89(3):503–508 Temam R (1988) Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin Zhenquan Li, Roberts AJ (2000) A flexible error estimate for the application of center manifold theory. http://arXiv.org/abs/math. DS/0002138 Zhu H, Campbell SA, Wolkowicz (2002) Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J Appl Math 63:636–682

Chaos and Ergodic Theory

Chaos and Ergodic Theory JÉRÔME BUZZI C.N.R.S. and Université Paris-Sud, Orsay, France

Article Outline Glossary Definition of the Subject Introduction Picking an Invariant Probability Measure Tractable Chaotic Dynamics Statistical Properties Orbit Complexity Stability Untreated Topics Future Directions Acknowledgments Bibliography

Glossary For simplicity, definitions are given for a continuous or smooth self-map or diffeomorphism T of a compact manifold M. Entropy, measure-theoretic (or: metric entropy) For an ergodic invariant probability measure , it is the smallest exponential growth rates of the number of orbit segments of given length, with respect to that length, after restriction to a set of positive measure. We denote it by h(T; ). See  Entropy in Ergodic Theory and Subsect. “Local Complexity” below. Entropy, topological It is the exponential growth rates of the number of orbit segments of given length, with respect to that length. We denote it by htop ( f ). See  Entropy in Ergodic Theory and Subsect. “Local Complexity” below. Ergodicity A measure is ergodic with respect to a map T if given any measurable subset S which is invariant, i. e., such that T 1 S D S, either S or its complement has zero measure. Hyperbolicity A measure is hyperbolic in the sense of Pesin if at almost every point no Lyapunov exponent is zero. See  Smooth Ergodic Theory. Kolmogorov typicality A property is typical in the sense of Kolmogorov for a topological space F of parametrized families f D ( f t ) t2U , U being an open subset of Rd for some d  1, if it holds for f t for Lebesgue almost every t and topologically generic f 2 F .

Lyapunov exponents The Lyapunov exponents ( Smooth Ergodic Theory) are the limits, when they exist, lim n!1 n1 log k(T n )0 (x):vk where x 2 M and v is a non zero tangent vector to M at x. The Lyapunov exponents of an ergodic measure is the set of Lyapunov exponents obtained at almost every point with respect to that measure for all non-zero tangent vectors. Markov shift (topological, countable state) It is the set of all infinite or bi-infinite paths on some countable directed graph endowed with the left-shift, which just translates these sequences. Maximum entropy measure It is a measure  which maximizes the measured entropy and, by the variational principle, realized the topological entropy. Physical measure It is a measure  whose basin, P fx 2 M : 8 : M ! R continuous limn!1 n1 n1 kD0 R ( f k x) D  dg has nonzero volume. Prevalence A property is prevalent in some complete metric, separable vector space X if it holds outside of a set N such that, for some Borel probability measure  on X: (A C v) D 0 for all v 2 X. See [76,141,239]. Sensitivity on initial conditions T has sensitivity to initial conditions on X 0 X if there exists a constant

> 0 such that for every x 2 X 0 , there exists y 2 X, arbitrarily close to x, and n  0 such that d(T n y; T n x) > . Sinai–Ruelle–Bowen measures It is an invariant probability measure which is absolutely continuous along the unstable foliation (defined using the unstable manifolds of almost every x 2 M, which are the sets W u (x), of points y such that limn!1 n1 log d(T n y; T n x) < 0). Statistical stability T is statistically stable if the physical measures of nearby deterministic systems are arbitrarily close to the convex hull of the physical measures of T. Stochastic stability T is stochastically stable if the invariant measures of the Markov chains obtained from T by adding a suitable, smooth noise with size ! 0 are arbitrarily close to the convex hull of the physical measures of T. Structural stability T is structurally stable if any S close enough to T is topologically the same as T: there exists a homeomorphism h : M ! M such that hı T D S ı h (orbits are sent to orbits). Subshift of finite type It is a closed subset ˙F of ˙ D AZ or ˙ D AN where A is a finite set satisfying: ˙F D fx 2 ˙ : 8k < ` : x k x kC1 : : : x` … Fg for some finite set F.

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Topological genericity Let X be a Baire space, e. g., a complete metric space. A property is (topologically) generic in a space X (or holds for the (topologically) generic element of X) if it holds on a nonmeager set (or set of second Baire category), i. e., on a dense Gı subset.

T : X ! X on a compact metric space whose distance is denoted by d: T has sensitivity to initial conditions on X 0 X if there exists a constant > 0 such that for every x 2 X 0 , there exists y 2 X, arbitrarily close to x with a finite separating time: 9n  0 such that d(T n y; T n x) > :

Definition of the Subject Chaotic dynamical systems are those which present unpredictable and/or complex behaviors. The existence and importance of such systems has been known at least since Hadamard [126] and Poincaré [208], however it became well-known only in the sixties. We refer to [36,128,226,236] and [80,107,120,125,192,213] for the relevance of such dynamics in other fields, mathematical or not (see also  Ergodic Theory: Interactions with Combinatorics and Number Theory,  Ergodic Theory: Fractal Geometry). The numerical simulations of chaotic dynamics can be difficult to interpret and to plan, even misleading, and the tools and ideas of mathematical dynamical system theory are indispensable. Arguably the most powerful set of such tools is ergodic theory which provides a statistical description of the dynamics by attaching relevant probability measures. In opposition to single orbits, the statistical properties of chaotic systems often have good stability properties. In many cases, this allows an understanding of the complexity of the dynamical system and even precise and quantitative statistical predictions of its behavior. In fact, chaotic behavior of single orbits often yields global stability properties. Introduction The word chaos, from the ancient Greek ˛o, “shapeless void” [131] and “raw confused mass” [199], has been used [˛o also inspired Van Helmont to create the word “gas” in the seventeenth century and this other thread leads to the molecular chaos of Boltzmann in the nineteenth century and therefore to ergodic theory itself .] since a celebrated paper of Li and Yorke [169] to describe evolutions which however deterministic and defined by rather simple rules, exhibit unpredictable or complex behavior. Attempts at Definition We note that, like many ideas [237], this is not captured by a single mathematical definition, despite several attempts (see, e. g., [39,112,158,225] for some discussions as well as the monographs on chaotic dynamics [15,16,58,60,78,104, 121,127,215,250,261]). Let us give some of the most wellknown definitions, which have been given mostly from the topological point of view, i. e., in the setting of a self-map

In other words, any uncertainty on the exact value of the initial condition x makes T n (x) completely unknown for n large enough. If X is a manifold, then sensitivity to initial conditions in the sense of Guckenheimer [120] means that the previous phenomenon occurs for a set X 0 with nonzero volume. T is chaotic in the sense of Devaney [94] if it admits a dense orbit and if the periodic points are dense in X. It implies sensitivity to initial conditions on X. T is chaotic in the sense of Li and Yorke [169] if there exists an uncountable subset X 0 X of points, such that, for all x ¤ y 2 X 0 , lim inf d(T n x; T n y) D 0 and n!1

lim sup d(T n x; T n y) > 0 : n!1

T has generic chaos in the sense of Lasota [206] if the set f(x; y) 2 X  X : lim inf d(T n x; T n y) D 0 n!1

< lim sup d(T n x; T n y)g n!1

is topologically generic (see glossary.) in X  X. Topological chaos is also sometimes characterized by nonzero topological entropy ( Entropy in Ergodic Theory): there exist exponentially many orbit segments of a given length. This implies chaos in the sense of Li and Yorke by [39]. As we shall see ergodic theory describes a number of chaotic properties, many of them implying some or all of the above topological ones. The main such property for a smooth dynamical system, say a C 1C˛ -diffeomorphism of a compact manifold, is the existence of an invariant probability measure which is: 1. Ergodic (cannot be split) and aperiodic (not carried by a periodic orbit); 2. Hyperbolic (nearby orbits converge or diverge at a definite exponential rate); 3. Sinai–Ruelle–Bowen (as smooth as it is possible). (For precise definitions we refer to  Smooth Ergodic Theory or to the discussions below.) In particular such

Chaos and Ergodic Theory

a situation implies nonzero entropy and sensitivity to initial condition of a set of nonzero Lebesgue measure (i. e., positive volume). Before starting our survey in earnest, we shall describe an elementary and classical example, the full tent map, on which the basic phenomena can be analyzed in a very elementary way. Then, in Sect. “Picking an Invariant Probability Measure”, we shall give some motivations for introducing probability theory in the description of chaotic but deterministic systems, in particular the unpredictability of their individual orbits. We define two of the most relevant classes of invariant measures: the physical measures and those maximizing entropy. It is unknown in which generality these measures exist and can be analyzed but we describe in Sect. “Tractable Chaotic Dynamics” the major classes of dynamics for which this has been done. In Sect. “Statistical Properties” we describe some of the finer statistical properties that have been obtained for such good chaotic systems: sums of observables along orbits are statistically undistinguishable from sums of independent and identically distributed random variables. Sect. “Orbit Complexity” is devoted to the other side of chaos: the complexity of these dynamics and how, again, this complexity can be analyzed, and sometimes classified, using ergodic theory. Sect “Stability” describes perhaps the most striking aspect of chaotic dynamics: the unstability of individual orbit is linked to various forms of stability of the global dynamics. Finally we conclude by mentioning some of the most important topics that we could not address and we list some possible future directions. Caveat. The subject-matter of this article is somewhat fuzzy and we have taken advantage of this to steer our path towards some of our favorite theorems and to avoid the parts we know less (some of which are listed below). We make no pretense at exhaustivity neither in the topics nor in the selected results and we hope that our colleagues will excuse our shortcomings. Remark 1 In this article we only consider compact, smooth and finite-dimensional dynamical systems in discrete time, i. e., defined by self-maps. In particular, we have omitted the natural and important variants applying to flows, e. g., evolutions defined by ordinary differential equations but we refer to the textbooks (see, e. g.,[15,128,148]) for these. Elementary Chaos: A Simple Example We start with a toy model: the full tent map T of Fig. 1. Observe that for any point x 2 [0; 1], T n (x) D f((k; n)x C k)2n : k D 0; 1; : : : ; 2n 1g, where (k; n) D ˙1.

Chaos and Ergodic Theory, Figure 1 The graph of the full tent map f(x) D 1  j1  2xj over [0; 1]

S Hence n0 T n (x) is dense in [0; 1]. It easily follows that T exhibits sensitive dependence to initial conditions. Even worse in this example, the qualitative asymptotic behavior can be completely changed by this arbitrarily small perturbation: x may have a dense orbit whereas y is eventually mapped to a fixed point! This is Devaney chaos [94]. This kind of unstability was first discovered by J. Hadamard [126] in his study of the geodesic flow (i. e., the frictionless movement of a point mass constrained to remain on a surface). At that time, such an unpredictability was considered a purely mathematical pathology, necessarily devoid of any physical meaning [Duhem qualified Hadamard’s result as “an example of a mathematical deduction which can never be used by physics” (see pp. 206–211 in [103])!]. Returning to out tent map, we can be more quantitative. At any point x 2 [0; 1] whose orbit never visits 1/2, the Lyapunov exponent limn!1 n1 log j(T n )0 (x)j is log 2. (See the glossary.) Such a positive Lyapunov exponent corresponds to infinitesimally close orbits getting separated exponentially fast. This can be observed in Fig. 2. Note how this exponential speed creates a rather sharp transition. It follows in particular that experimental or numerical errors can grow very quickly to size 1, [For simple precision arithmetic the uncertainty is 1016 which grows to size 1 in 38 iterations of T.] i. e., the approximate orbit may contain after a while no information about the true orbit. This casts a doubt on the reliability of simulations. Indeed, a simulation of T on most computers will suggest that all orbits quickly converge to 0, which is completely false [Such a collapse to 0 does really occurs but only for a countable

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Chaos and Ergodic Theory, Figure 2 jT n (x)  T n (y)j for T(x) D 1  j1  2xj, jx  yj D 1012 and 0  n  100. The vertical red line is at n D 28 and shows when jT n x  T n yj  0:5  104 for the first time

subset of initial conditions in [0; 1] whereas the points with dense orbit make a subset of [0; 1] with full Lebesgue measure (see below). This artefact comes from the way numbers are represented – and approximated – on the computer: multiplication by even integers tends to “simplify” binary representations. Thus the computations involved in drawing Fig. 2 cannot be performed too naively.]. Though somewhat atypical in its dramatic character, this failure illustrates the unpredictability and unstability of individual orbits in chaotic systems. Does this mean that all quantitative predictions about orbits of T are to be forfeited? Not at all, if we are ready to change our point of view and look beyond a single orbit. This can be seen easily in this case. Let us start with such a global analysis from the topological point of view. Associate to x 2 [0; 1], a sequence i(x) D i D i0 i1 i2 : : : of 0s and 1s according to i k D 0 if T k x  1/2, i k D 1 otherwise. One can check that [Up to a countable set of exceptions.] fi(x) : x 2 [0; 1]g is the set ˙2 :D f0; 1gN of all infinite sequences of 0s and 1s and that at most one x 2 [0; 1] can realize a given sequence as i(x). Notice how the transformation f becomes trivial in this representation:

This can be a very powerful tool. Observe for instance how here it makes obvious that we have complete combinatorial freedom over the orbits of T: one can easily build orbits with various asymptotic behaviors: if a sequence of ˙ 2 contains all the finite sequences of 0s and 1s, then the corresponding point has a dense orbit; if the sequence is periodic, then the corresponding point is itself periodic, to give two examples of the richness of the dynamics. More quantitatively, the number of distinct subsequences of length n appearing in sequences i(x), x 2 [0; 1], is 2n . It follows that the topological entropy ( Entropy in Ergodic Theory) of T is htop (T) D log 2. [For the coincidence of the entropy and the Lyapunov exponent see below.] The positivity of the topological entropy can be considered as the signature of the complexity of the dynamics and considered as the definition, or at least the stamp, of a topologically chaotic dynamics. Let us move on to a probabilistic point of view. Pick x 2 [0; 1] randomly according to, say, the uniform law in [0; 1]. It is then routine to check that i(x) follows the (1/2; 1/2)-Bernoulli law: the probability that, for any given k, i k (x) D 0 is 1/2 and the ik s are independent. Thus i(x), seen as a sequence of random 0 and 1 when x is subject to the uniform law on [0; 1], is statistically undistinguishable from coin tossing! This important remark leads to quantitative predictions. For instance, the strong law of large numbers implies that, for Lebesgue-almost every x 2 [0; 1] (i. e., for all x 2 [0; 1] except in a set of Lebesgue measure zero), the fraction of the time spent in any dyadic interval I D [k  2N ; `  2N ] [0; 1], k; `; N 2 N, by the orbit of x, lim

n!1

1 #f0  k < n : T k x 2 Ig n

(1)

exists and is equal to the length, 2N , of that interval. [Eq. (1) in fact holds for any interval I. This implies that the orbit of almost every x 2 [0; 1] visits all subintervals of [0; 1], i. e., the orbit is dense: in complete contradiction with the above mentioned numerical simulation!] More generally, we shall see that, if  : [0; 1] ! R is any continuous function, then, for Lebesgue almost every-x, Z n1 1X k (T x) exists and is equal to (x) dx : lim n!1 n kD0

i( f (x)) D i1 i2 i3 : : :

if i(x) D i0 i1 i2 i3 : : :

Thus f is represented by the simple and universal “leftshift” on sequences, which is denoted by . This representation of a rather general dynamical system by the leftshift on a space of sequences is called symbolic dynamics ( Symbolic Dynamics), [171].

(2) Using strong mixing properties ( Ergodicity and Mixing Properties) of the Lebesgue measure under T, one can prove further properties, e. g., sensitivity on initial conditions in the sense of Guckenheimer [The Lebesgue measure is weak-mixing: Lebesgue-almost all couples of points (x; y) 2 [0; 1]2 get separated. Note that it is not true

Chaos and Ergodic Theory

of every couple off the diagonal: Counter-examples can be found among couples (x; y) with T n x D T n y arbitrarily close to 1.] and study the fluctuations of the averages 1 Pn1 kD0 (Tx) by the way of limit theorems. n The above analysis relied on the very special structure of T but, as we shall explain, the ergodic theory of differentiable dynamical systems shows that all of the above (and much more) holds in some form for rather general classes of chaotic systems. The different chaotic properties are independent in general (e. g., one may have topological chaos whereas the asymptotic behavior of almost all orbits is periodic) and the proofs can become much more difficult. We shall nonetheless be rewarded for our efforts by the discovery of unexpected links between chaos and stability, complexity and simplicity as we shall see. Picking an Invariant Probability Measure One could think that dynamical systems, such as those defined by self-maps of manifolds, being completely deterministic, have nothing to do with probability theory. There are in fact several motivations for introducing various invariant probability measures. Statistical Descriptions An abstract goal might be to enrich the structure: a smooth self-map is a particular case of a Borel self-map, hence one can canonically attach to this map its set of all invariant Borel probability measures [From now on all measures will be Borel probability measures except if it is explicitly stated otherwise.], or just the set of ergodic [A measure is ergodic if all measurable invariant subsets have measure 0 or 1. Note that arbitrary invariant measures are averages of ergodic ones, so many questions about invariant measures can be reduced to ergodic ones.] ones. By the Krylov–Bogoliubov theorem (see, e. g., [148]), this set is non-empty for any continuous self-map of a compact space. By the following fundamental theorem  Ergodic Theorems, each such measure is the statistical description of some orbit: Birkhoff Pointwise Ergodic Theorem Let (X; F ; ) be a space with a -field and a probability measure. Let f : X ! X be a measure-preserving map, i. e., f 1 (F )

F and  ı f 1 D . Assume ergodicity of  (See the glossary.) and (absolute) integrability of  : X ! R with respect to . Then for -almost every x 2 X, n1 1X lim ( f k x) exists and is n!1 n kD0

Z  d :

This theorem can be interpreted as saying that “time averages” coincide almost surely with “ensemble averages” (or “phase space average”), i. e., that Boltzmann’s Ergodic Hypothesis of statistical mechanics [110] holds for dynamical systems that cannot be split in a measurable and non trivial way. [This indecomposability is however often difficult to establish. For instance, for the hard ball model of a gas it is known only under some generic assumption (see [238] and the references therein).] We refer to [161] for background. Remark 2 One should observe that the existence of the above limit is not at all obvious. In fact it often fails from other points of view. One can show that for the full tent map T(x) D 1  j1  2xj analyzed above and many functions , the set of points for which it fails is large both from the topological point of view (it contains a dense Gı set) and from the dimension point of view (it has Hausdorff dimension 1 [28]). This is an important point: the introduction of invariant measures allows one to avoid some of the wilder pathologies. To illustrate this let us consider the full tent map T(x) D 1  j1  2xj again and the two ergodic invariant measures: ı 0 (the Dirac measure concentrated at the fixed point 0) and the Lebesgue measure dx. In the first case, we obtain a complex proof of the obvious fact that the time average at x D 0 (some set of full measure!) and the ensemble average with respect to ı 0 are both equal to f (0). In the second case, we obtain a very general proof of the above Eq. (2). Another type of example is provided by the contracting map S : [0; 1] ! [0; 1], S(x) D x/2. S has a unique invariant probability measure, ı 0 . For Birkhoff theorem the situation is the same as that of T and ı 0 : it asserts only that the orbit of 0 is described by ı 0 . One can understand Birkhoff theorem as a (first and rather weak) stability result: the time averages are independent of the initial condition, almost surely with respect to . Physical Measures In the above silly example S, much more is true than the conclusion of Birkhoff Theorem: all points of [0; 1] are described by ı 0 . This leads to the definition of the basin of a probability measure  for a self-map f of a space M: ( B() :D

x 2 M : 8 :

) Z n1 1X k ( f x) D  d : M ! R continuous lim n!1 n kD0

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If M is a manifold, then there is a notion of volume and one can make the following definition. A physical measure is a probability measure whose basin has nonzero volume in M. Say that a dynamical system f : M ! M on a manifold has a finite statistical description if there exists finitely many invariant probability measures 1 ; : : : ; n the union of whose basins is the whole of M, up to a set of zero Lebesgue measure. Physical measures are among the main subject of interest as they are expected to be exactly those that are “experimentally visible”. Indeed, if x0 2 B() and 0 > 0 is small enough, then, by Lebesgue density theorem, a point x picked according to, say, the uniform law in the ball B(x0 ; 0 ) of center x0 and radius 0 , will be in B() with probability almost 1 and therefore its ergodic averages will be described by . Hence “experiments” can be expected to follow the physical measures and this is what is numerically observed in most of the situations (see however the caveat in the discussion of the full tent map). The existence of a finite statistical description (or even of a physical measure) is, as we shall see, not automatic nor routine to prove. Attracting periodic points as in the above silly example provide a first type of physical measures. Birkhoff ergodic theorem asserts that absolutely continuous ergodic invariant measures, usually obtained from some expansion property, give another class of physical measures. These contracting and expanding types can be combined in the class of Sinai–Ruelle–Bowen measures [166] which are the invariant measures absolutely continuous “along expanding directions” (see for the precise but technical definition  Smooth Ergodic Theory). Any ergodic Sinai–Ruelle–Bowen measure which is ergodic and without zero Lyapunov exponent [That is, the set of points x 2 M such that lim n!1 n1 log k( f n )0 (x):vk D 0 for some v 2 Tx M has zero measure.] is a physical measure. Conversely, “most” physical measures [For counter-examples see [136].] are of this type [243,247]. Measures of Maximum Entropy For all parameters t 2 [3:96; 4], the quadratic maps Q t (x) D tx(1  x), Q t : [0; 1] ! [0; 1], have nonzero topological entropy [91] and exponentially many periodic points [134]: lim

n!1

#fx 2 [0; 1] : Q tn (x) D xg e n htop (Q t )

D1:

On the other hand, by a deep theorem [117,178] there is an open and dense subset of t 2 [0; 4], such that Qt has a unique physical measure concentrated on a periodic or-

bit! Thus the physical measures can completely miss the topological complexity (and in particular the distribution of the periodic points). Hence one must look at other measures to get a statistical description of the complexity of such Qt . Such a description is often provided by measures of maximum entropy  M whose measured entropy [The usual phrases are “measure-theoretic entropy”, “metric entropy”.] ( Entropy in Ergodic Theory) satisfies: h( f ;  M ) D sup h( f ; ) D1 htop ( f ) : 2M( f )

M( f ) is the set of all invariant measures. [One can restrict this to the ergodic invariant measures without changing the value of the supremum ( Entropy in Ergodic Theory).] Equality 1 above is the variational principle: it holds for all continuous self-maps of compact metric spaces. One can say that the ergodic complexity (the complexity of f as seen by its invariant measures) captures the full topological complexity (defined by counting all orbits). Remark 3 The variational principle implies the existence of “complicated invariant measures” as soon as the topological entropy is nonzero (see [47] for a setting in which this is of interest). Maximum entropy measures do not always exist. However, if f is C 1 smooth, then maximum entropy measures exist by a theorem of Newhouse [195] and they indeed describe the topological complexity in the following sense. Consider the probability measures: n; :D

n1 1X n

X

ıf kx

kD0 x2E(n; )

where E(n; ) is an arbitrary ( ; n)-separated subset [See  Entropy in Ergodic Theory: 8x; y 2 E(n; ) x ¤ y H) 90  k < n d(T k x; T k y)  .] of M with maximum cardinality. Then accumulation points for the weak star topology on the space of probability measures on M of n; when n ! 1 and then ! 0 are maximum entropy measures [185]. Let us quote two important additional properties, discovered by Margulis [179], that often hold for the maximum entropy measures:  The equidistribution of periodic points with respect to some maximum entropy measure  M : X 1 ıx :  M D lim n!1 #fx 2 X : x D f n xg n xD f x

 The holonomy invariance which can be loosely interpreted by saying that the past and the future are independent conditionally on the present.

Chaos and Ergodic Theory

Other Points of View Many other invariant measures are of interest in various contexts and we have made no attempt at completeness: for instance, invariant measures maximizing dimension [111,204], or pressure in the sense of the thermodynamical formalism [148,222], or some energy [9,81,146], or quasi-physical measures describing the dynamics around saddle-type invariant sets [104] or in systems with holes [75]. Tractable Chaotic Dynamics The Palis Conjecture There is, at this point, no general theory allowing the analysis of all dynamical systems or even of most of them despite many recent and exciting developments in the theory of generic C1 -diffeomorphisms [51,84]. In particular, the question of the generality in which physical measures exist remains open. One would like generic systems to have a finite statistical description (see Subsect. “Physical Measures”). This fails in some examples but these look exceptional and the following question is asked by Palis [200]: Is it true that any dynamical system defined by a Cr diffeomorphism on a compact manifold can be transformed by an arbitrarily small Cr -perturbation to another dynamical system having a finite statistical description? This is completely open though widely believed [Observe, however, that such a statement is false for conservative diffeomorphisms with high order smoothness as KAM theory implies stable existence of invariant tori foliating a subset of positive volume.]. Note that such a good description is not possible for all systems (see, e. g., [136,194]). Note that one would really like to ask about unperturbed “typical” [The choice of the notion of typicality is a delicate issue. The Newhouse phenomenon shows that among C2 -diffeomorphisms of multidimensional compact manifolds, one cannot use topological genericity and get a positive answer. Popular notions are prevalence and Kolmogorov genericity – see the glossary.] dynamical systems in a suitable sense, but of course this is even harder. One is therefore led to make simplifying assumptions: typically of small dimension, uniform expansion/ contraction or geometry. Uniformly Expanding/Hyperbolic Systems The most easily analyzed systems are those with uniform expansion and/or contraction, namely the uniformly

expanding maps and uniformly hyperbolic diffeomorphisms, see  Smooth Ergodic Theory. [We require uniform hyperbolicity on the so-called chain recurrent set. This is equivalent to the usual Axiom-A and no-cycle condition.] An important class of example is obtained as follows. Consider A: Rd ! Rd , a linear map preserving Zd (i. e., A is a matrix with integer coefficients in the canonical basis) so ¯ T d ! T d on the torus. If there is that it defines a map A: a constant  > 1 such that for all v 2 Rd , kA:vk  kvk then A¯ is a uniformly expanding map. If A has determinant ˙1 and no eigenvalue on the unit circle, then A¯ is a uniformly hyperbolic diffeomorphism ( Smooth Ergodic Theory) (see also [60,148,215,233]). Moreover all C1 -perturbations of the previous examples are again uniformly expanding or uniformly hyperbolic. [One can define uniform hyperbolicity for flows and an important class of examples is provided by the geodesic flow on compact manifolds with negative sectional curvature [148].] These uniform systems are sometimes called “strongly chaotic”. Remark 4 Mañé Stability Theorem (see below) shows that uniform hyperbolicity is a very natural notion. One can also understand on a more technical level uniform hyperbolicity as what is needed to apply an implicit function theorem in some functional space (see, e. g., [233]). The existence of a finite statistical description for such systems has been proved since the 1970s by Bowen, Ruelle and Sinai [54,218,235] (the expanding case is much simpler [162]). Theorem 1 Let f : M ! M be a C 1C˛ map of a compact manifold. Assume f to be (i) a uniformly expanding map on M or (ii) a uniformly hyperbolic diffeomorphism.  f admits a finite statistical description by ergodic and hyperbolic Sinai–Ruelle–Bowen measures (absolutely continuous in case (i)).  f has finitely many ergodic maximum entropy measures, each of which makes f isomorphic to a finite state Markov chain. The periodic points are uniformly distributed according to some canonical average of these ergodic maximum entropy measures.  f is topologically conjugate [Up to some negligible subset.] to a subshift of finite type (See the glossary.) The construction of absolutely continuous invariant measures for a uniformly expanding map f can be done in a rather direct way by considering the pushed forP k ward measures n1 n1 kD0 f Leb and taking weak star limits while preventing the appearance of singularities, by, e. g., bounding some Hölder norm of the density using expansion and distortion of f .

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The classical approach to the uniformly hyperbolic dynamics [52,222,233] is through symbolic dynamics and coding. Under the above hypothesis one can build a finite partition of M which is tailored to the dynamics (a Markov partition) so that the corresponding symbolic dynamics has a very simple structure: it is a full shift f1; : : : ; dgZ , like in the example of the full tent map, or a subshifts of finite type. The above problems can then be solved using the thermodynamical formalism inspired from the statistical mechanics of one-dimensional ferromagnets [217]: ergodic properties are obtained through the spectral properties of a suitable transfer operator acting on some space of regular functions, e. g., the Hölder-continuous functions defined over the symbolic dynamics with respect to the P n distance d(x; y) :D n2Z 2 1 x n ¤y n where 1s¤t is 1 if s ¤ t, 0 otherwise. A recent development [24,43,116] has been to find suitable Banach spaces to apply the transfer operator technique directly in the smooth setting, which not only avoids the complication of coding (or rather replace them with functional analytic preliminaries) but allows the use of the smoothness beyond Hölder-continuity which is important for finer ergodic properties. Uniform expansion or hyperbolicity can easily be obstructed in a given system: a “bad” point (a critical point or a periodic point with multiplier with an eigenvalue on the unit circle) is enough. This leads to the study of other systems and has motivated many works devoted to relaxing the uniform hyperbolicity assumptions [51]. Pesin Theory The most general such approach is Pesin theory. Let f be a C 1C˛ -diffeomorphism [It is an important open problem to determine to which extent Pesin theory could be generalized to the C1 setting.] f with an ergodic invariant measure . By Oseledets Theorem  Smooth Ergodic Theory, for almost every x with respect to any invariant measure, the behavior of the differential Tx f n for n large is described by the Lyapunov exponents 1 ; : : : ; d , at x. Pesin is able to build charts around almost every orbit in which this asymptotic linear behavior describes that of f at the first iteration. That is, there are diffeomorphisms ˚x : U x M ! Vx Rd with a “reasonable dependence on x” such that the differential of ˚ f n x ı f n ı ˚x1 at any point where it is defined is close to a diagonal matrix with entries (e(1 ˙ )n ; e(2 ˙ )n ; : : : ; e(d ˙ )n ). In this full generality, one already obtains significant results:  The entropy is bounded by the expansion: h( f ; )  Pd C iD1  i () [219]

 At almost every point x, there are strong stable resp. unstable manifolds W ss (x), resp. W uu (x), coinciding with the sets of points y such that d(T n x; T n y) ! 0 exponentially fast when n ! 1, resp. n ! 1. The corresponding holonomies are absolutely continuous (see, e. g., [59]) like in the uniform case. This allows Ledrappier’s definition of Sinai–Ruelle–Bowen measures [166] in that setting.  Equality in the above formula holds if and only if  is a Sinai–Ruelle–Bowen measure [167]. More generP ally the entropy can be computed as diD1 i ()C i () where the i are some fractal dimensions related to the exponents. Under the only assumption of hyperbolicity (i. e., no zero Lyapunov exponent almost everywhere), one gets further properties:  Existence of an hyperbolic measure which is not periodic forces htop ( f ) > 0 [However h( f ; ) can be zero.] by [147].   is exact dimensional [29,256]: the limit limr!0 log (B(x; r))/ log r exists -almost everywhere and is equal to the Hausdorff dimension of  (the infimum of the Hausdorff dimension of the sets with full -measure). This is deduced from a more technical “asymptotic product structure” property of any such measure. For hyperbolic Sinai–Ruelle–Bowen measures , one can then prove, e. g.,:  local ergodicity [202]:  has at most countably many ergodic components and -almost every point has a neighborhood whose Lebesgue-almost every point are contained in the basin of an ergodic component of .  Bernoulli [198]: Each ergodic component of  is conjugate in a measure-preserving way, up to a period, to a Bernoulli shift, that is, a full shift f1; : : : ; NgZ equipped with a product measure. This in particular implies mixing and sensitivity on initial conditions for a set of positive Lebesgue measure. However, establishing even such a weak form of hyperbolicity is rather difficult. The fragility of this condition can be illustrated by the result [44,45] that the topologically generic area-preserving surface C1 -diffeomorphism is either uniformly hyperbolic or has Lebesgue almost everywhere vanishing Lyapunov exponents, hence is never non-uniformly hyperbolic! (but this is believed to be very specific to the very weak C1 topology). Moreover, such weak hyperbolicity is not enough, with the current techniques, to build Sinai–Ruelle–Bowen measures or ana-

Chaos and Ergodic Theory

lyze maximum entropy measures only assuming non-zero Lyapunov exponents. Let us though quote two conjectures. The first one is from [251] [We slightly strengthened Viana’s statement for expository reasons.]: Conjecture 1 Let f be a C 1C -diffeomorphism of a compact manifold. If Lebesgue-almost every point x has welldefined Lyapunov exponents in every direction and none of these exponents is zero, then there exists an absolutely continuous invariant -finite positive measure. The analogue of this conjecture has been proved for C3 interval maps with unique critical point and negative Schwarzian derivative by Keller [150], but only partial results are available for diffeomorphisms [168]. We turn to measures of maximum entropy. As we said, C 1 smoothness is enough to ensure their existence but this is through a functional-analytic argument (allowed by Yomdin theory [254]) which says nothing about their structure. Indeed, the following problem is open: Conjecture 2 Let f be a C 1C -diffeomorphism of a compact surface. If the topological entropy of f is nonzero then f has at most countably many ergodic invariant measures maximizing entropy. The analogue of this conjecture has been proved for C 1C interval maps [64,66,70]. In the above setting a classical result of Katok shows the existence of uniformly hyperbolic compact invariant subsets with topological entropy arbitrarily close to that of f implying the existence of many periodic points: lim sup n!1

1 log #fx 2 M : f n (x) D xg  htop ( f ) : n

The previous conjecture would follow from the following one: Conjecture 3 Let f be a C 1C -diffeomorphism of a compact manifold. There exists an invariant subset X M, carrying all ergodic measures with maximum entropy, such that the restriction f jX is conjugate to a countable state topological Markov shift (See the glossary.). Systems with Discontinuities We now consider stronger assumptions to be able to build the relevant measures. The simplest step beyond uniformity is to allow discontinuities, considering piecewise expanding maps. The discontinuities break the rigidity of the uniformly expanding situation. For instance, their symbolic dynamics are usually no longer subshifts of finite type though they still retain some “simplicity” in good cases (see [68]).

To understand the problem in constructing the absolutely continuous invariant measures, it is instructive to consider the pushed forwards of a smooth measure. Expansion tends to keep the measure smooth whereas discontinuities may pile it up, creating non-absolute continuity in the limit. One thus has to check that expansion wins. In dimension 1, a simple fact resolves the argument: under a high enough iterate, one can make the expansion arbitrarily large everywhere, whereas a small interval can be chopped into at most two pieces. Lasota and Yorke [165] found a suitable framework. They considered C2 interval maps with j f 0 (x)j  const > 1 except at finitely many points. They used the Ruelle transfer operator directly on the interval. Namely they studied X (y) (L)(x) D jT 0 (y)j 1 y2T

x

acting on functions  : [0; 1] ! R with bounded variation and obtained the invariant density as the eigenfunction associated to the eigenvalue 1. One can then prove a Lasota– Yorke inequality (which might more accurately be called Doeblin–Fortet since it was introduced in the theory of Markov chains much earlier): kLkBV  ˛kkBV C ˇkk1

(3)

where k  kBV , k  k1 are a strong and a weak norm, respectively and ˛ < 1 and ˇ < 1. One can then apply general theorems [143] or [196] (see [21] for a detailed presentation of this approach and its variants). Here ˛ can essentially be taken as 2 (reflecting the locally simple discontinuities) divided by the minimum expansion: so ˛ < 1, perhaps after replacing T with an iterate. In particular, the existence of a finite statistical description then follows (see [61] for various generalizations and strengthenings of this result on the interval). The situation in higher dimension is more complex for the reason explained above. One can obtain inequalities such as (3) on suitable if less simple functional spaces (see, e. g., [231]) but proving ˛ < 1 is another matter: discontinuities can get arbitrarily complex under iteration. [67,241] show that indeed, in dimension 2 and higher, piecewise uniform expansion (with a finite number of pieces) is not enough to ensure a finite statistical description if the pieces of the map have only finite smoothness. In dimension 2, resp. 3 or more, piecewise real-analytic, resp. piecewise affine, is enough to exclude such examples [65,240], resp. [242]. [82] has shown that, for any r > 1, an open and dense subset of piecewise Cr and expanding maps have a finite statistical description.

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Piecewise hyperbolic diffeomorphisms are more difficult to analyze though several results (conditioned on technical assumptions that can be checked in many cases) are available [22,74,230,257]. Interval Maps with Critical Points A more natural but also more difficult situation is a map for which the uniformity of the expansion fails because of the existence of critical points. [Note that, by a theorem of Mañé a circle map without critical points or indifferent periodic point is either conjugate to a rotation or uniformly expanding [181].] A class which has been completely analyzed at the level of the above conjecture is that of real-analytic families of maps of the interval f t : [0; 1] ! [0; 1], t 2 I, with a unique critical point, the main example being the quadratic family Q t (x) D tx(1  x) for 0  t  4. It is not very difficult to find quadratic maps with the following two types of behavior: (stable) the orbit of Lebesgue-almost every x 2 [0; 1] tends to an attracting periodic orbit; (chaotic) there is an absolutely continuous invariant probability measure  whose basin contains Lebesguealmost every x 2 [0; 1]. To realize the first it is enough to arrange the critical point to be periodic. One can easily prove that this stable behavior occurs on an open set of parameters – thus it is stable with respect to the parameter or the dynamical system. p The second occurs for Q4 with  D dx/  x(1  x). It is much more difficult to show that this chaotic behavior occurs for a set of parameters of positive Lebesgue measure. This is a theorem of Jakobson [145] for the quadratic family (see for a recent variant [265]). Let us sketch two main ingredients of the various proofs of this theorem. The first is inducing: around Lebesgue-almost every point x 2 [0; 1] one tries to find a time (x) and an interval J(x) such that f (x) : J(x) ! f (x) (J(x)) is a map with good expansion and distortion properties. This powerful idea appears in many disguises in the non-uniform hyperbolic theory (see for instance [133,262]). The second ingredient is parameter exclusion: one removes the parameters at which a good inducing scheme cannot be built. More precisely one proceeds inductively, performing simultaneously the inducing and the exclusion, the good properties of the early stage of the inducing allowing one to control the measure of the parameters that need to be excluded to continue [30,145]. Indeed, the expansion established at a given stage allows to transfer estimates in the dynamical space to the parameter space. Using methods from complex analysis and renormalization theory one can go much further and prove the fol-

lowing difficult theorems (actually the product of the work of many people, including Avila, Graczyk, Kozlovski, Lyubich, de Melo, Moreira, Shen, van Strien, Swiatek), which in particular solves Palis conjecture in this setting: Theorem 2 [117,160,178] Stable maps (that is, such that Lebesgue almost every orbit converges to one of finitely many periodic orbits) form an open and dense set among Cr interval maps, for any r  2. [In fact this is even true for polynomials.]. The picture has been completed in the unimodal case (that is, with a unique critical point): Theorem 3 [19,20,117,159,178] Let f t : [0; 1] ! [0; 1], t 2 [t0 ; t1 ], be a real-analytic family of unimodal maps. Assume that it is not degenerate [ f t0 and f t1 are not conjugate]. Then:  The set of t such that f t is chaotic in the above sense has positive Lebesgue measure;  The set of t such that f t is stable is open and dense;  The remaining set of parameters has zero Lebesgue measure. [This set of “strange parameters” of zero Lebesgue measure has however positive Hausdorff dimension according to work of Avila and Moreira. In particular each of the following situations is realized on a set of parameters t of positive Hausdorff dimension: non-existence of the Birkhoff limit at Lebesgue-almost every point, the physical measure is ı p for p a repelling fixed point, the physical measure is non-ergodic.] We note that the theory underlying the above theorem yields much more results, including a very paradoxical rigidity of typical analytic families as above. See [19]. Non-uniform Expansion/Contraction Beyond the dimension 1, only partial results are available. The most general of those assume uniform contraction or expansion along some direction, restricting the non-uniform behavior to an invariant sub-bundle often one-dimensional, or “one-dimensional-like”. A first, simpler situation is when there is a dominated decomposition with a uniformly expanding term: there is a continuous and invariant splitting of the tangent bundle, T M D E uu ˚ E cs , over some  an attracting set: for all unit vectors v u 2 E uu , v c 2 E cs , k( f n )0 (x):v u k  Cn n 0

and

k( f ) (x):v k  C k( f n )0 (x):v u k : c

n

Standard techniques (pushing the Riemannian volume of a piece of unstable leaf and taking limits) allow the construction of Gibbs u-states as introduced by [205].

Chaos and Ergodic Theory

Theorem 4 (Alves–Bonatti–Viana [8]) [A slightly different result is obtained in [49].] Let f : M ! M be a C2 diffeomorphism with an invariant compact subset . Assume that there is a dominated splitting T M D E u ˚ E cs such that, for some c > 0, n1

lim sup n!1

Y 1 k f 0 ( f x )jE cs k  c < 0 log n kD0

on a subset of  of positive Lebesgue measure. Then this subset is contained, up to a set of zero Lebesgue measure, in the union of the basins of finitely many ergodic and hyperbolic Sinai – Ruelle – Bowen measures. The non-invertible, purely expansive version of the above theorem can be applied in particular to the following maps of the cylinder (d  16, a is properly chosen close to 2 and ˛ is small): f ( ; x) D (dx mod 2; a  x 2 C sin( )) which are natural examples of maps with multidimensional expansion and critical lines considered by Viana [249]. A series of works have shown that the above maps fit in the above non-uniformly expanding setting with a proper control of the critical set and hence can be thoroughly analyzed through variants of the above theorem [4,249] and the references in [5]. For a; b properly chosen close to 2 and small , the following maps should be even more natural examples: f (x; y) D (a  x 2 C y; b  y 2 C x) :

(4)

However the inexistence of a dominated splitting has prevented the analysis of their physical measures. See [66,70] for their maximum entropy measures. Cowieson and Young have used completely different techniques (thermodynamical formalism, Ledrappier– Young formula and Yomdin theory on entropy and smoothness) to prove the following result (see [13] for related work): Theorem 5 (Cowieson–Young [83]) Let f : M ! M be a C 1 diffeomorphism of a compact manifold. Assume that f admits an attractor  M on which the tangent bundle has an invariant continuous decomposition T M D E C ˚ E  such that all vectors of E C n f0g, resp. E  n f0g, have positive, resp. negative, Lyapunov exponents. Then any zero-noise limit measure  of f is a Sinai–Ruelle– Bowen measure and therefore, if it is ergodic and hyperbolic, a physical measure. One can hope, that typically, the latter ergodicity and hyperbolicity assumptions are satisfied (see, e. g., [27]).

By pushing classical techniques and introducing new ideas for generic maps with one expanding and one weakly contracting direction, Tsujii has been able to prove the following generic result (which can be viewed as a 2-dimensional extension of some of the one-dimensional results above: one adds a uniformly expanding direction): Theorem 6 (Tsujii [243]) Let M be a compact surface. Consider the space of C20 self-maps f : M ! M which admits directions that are uniformly expanded [More precisely, there exists a continuous, forward invariant cone field which is uniformly expanded under the differential of f .] Then existence of a finite statistical description is both topologically generic and prevalent in this space of maps. Hénon-Like Maps and Rank One Attractors In 1976, Hénon [130] observed that the diffeomorphism of the plane H a;b (x; y) D (1  ax 2 C y; bx) seemed to present a “strange attractor” for a D 1:4 and b D 0:3, that is, the points of a numerically simulated orbit seemed to draw a set looking locally like the product of a segment with a Cantor set. This attractor seems to be supported by the unstable manifold W u (P) of the hyperbolic fixed point P with positive absciss. On the other hand, the Plykin classification [207] excluded the existence of a uniformly hyperbolic attractor for a dissipative surface diffeomorphism. For almost twenty years the question of the existence of such an attractor (by opposition to an attracting periodic orbit with a very long period) remained open. Indeed, one knew since Newhouse that, for many such maps, there exist infinitely many such periodic orbits which are very difficult to distinguish numerically. But in 1991 Benedicks and Carleson succeeded in proposing an argument refining (with considerable difficulties) their earlier proof of Jakobson one-dimensional theorem and established the first part of the following theorem: Theorem 7 (Benedicks–Carleson [31]) For any > 0, for jbj small enough, there is a set A with Leb(A) > 0 satisfying: for all a 2 A, there exists z 2 W u (P) such that  The orbit of z is dense in W u (P);  lim infn!1 n1 log k( f n )0 (z)k > 0. Further properties were then established, especially by Benedicks, Viana, Wang, Young [32,34,35,264]. Let us quote the following theorem of Wang and Young which includes the previous results:

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Theorem 8 [264] Let Tab : S 1  [1; 1] ! S 1  [1; 1] be such that  Ta0 (S 1  [1; 1]) S 1  f0g;  For b > 0, Tab is a diffeomorphism on its image with c 1 b  j det Tab (x; y)j  c  b for some c > 1 and all (x; y) 2 S 1  [1; 1] and all (a; b). Let f a : S 1 ! S 1 be the restriction of Ta0 . Assume that f D f0 satisfies:  Non-degenerate critical points: f 0 (c) D 0 ) f 00(c) ¤ 0;  Negative Schwarzian derivative: for all x 2 S 1 non-critical, f 000 (x)/ f 0 (x)  3/2( f 00 (x)/ f 0 (x))2 < 0;  No indifferent or attracting periodic point, i. e., x such that f n (x) D x and j( f )0 (x)j  1;  Misiurewicz condition: d( f n c; d) > c > 0 for all n  1 and all critical points c; d. Assume the following transversality condition on f at a D 0: for every critical point c, ( d/da)( f a (c a )  p a ) ¤ 0 if ca is the critical point of f a near c and pa is the point having the same itinerary under f a as f (c) under c. Assume the following non-degeneracy of T : f00 (c) D 0 H) @T00 (c; 0)/@y ¤ 0.  Tab restricted to a neighborhood of S 1  f0g has a finite statistical description by a number of hyperbolic Sinai– Ruelle–Bowen measures bounded by the number of critical points of f ;  There is exponential decay of correlations and a Central Limit Theorem (see below) – except, in an obvious way, if there is a periodic interval with period > 1;  There is a natural coding of the orbits that remains for ever close to S 1 f0g by a closed invariant subset of a full shift. Very importantly, the above dynamical situation has been shown to occur near typical homoclinic tangencies: [190] proved that there is an open and dense subset of the set of all C3 families of diffeomorphisms unfolding a first homoclinic tangency such that the above holds. However [201] shows that the set of parameters with a Henon-like attractor has zero Lebesgue density at the bifurcation itself, at least under an assumption on the so-called stable and unstable Hausdorff dimensions. [95] establishes positive density for another type of bifurcation. Furthermore [191] has related the Hausdorff dimensions to the abundance of uniformly hyperbolic dynamics near the tangency. [248] is able to treat situations with more than one contracting direction. More recently [266] has proposed

a rather general framework, with easily checkable assumptions in order to establish the existence of such dynamics in various applications. See also [122,252] for applications. Statistical Properties The ergodic theorem asserts that time averages of integrable functions converge to phase space averages for any ergodic system. The speed of convergence is quite arbitrary in that generality [161] (only upcrossing inequalities seem to be available [38,132]), however many results are available under very natural hypothesis as we are going to explain in this section. The underlying idea is that for sufficiently chaotic dynamics T and reasonably smooth observables , the time averages A n (x) :D

n1 1X  ı T k (x) n kD0

should behave as averages of independent and identically distributed random variables and therefore satisfy the classical limit theorems of probability theory. The dynamical systems which are amenable to current technology are in a large part [But other approaches are possible. Let us quote the work [98] on partially hyperbolic systems, for instance.] those that can be reduced to the following type: Definition 1 Let T : X ! X be a nonsingular map on a probability, metric space (X; B; ; d) with bounded diameter, preserving the probability measure . This map is said to be Gibbs–Markov if there exists a countable (measurable) partition ˛ of X such that: 1. For all a 2 ˛, T is injective on a and T(a) is a union of elements of ˛. 2. There exists  > 1 such that, for all a 2 ˛, for all points x; y 2 a, d(Tx; Ty)  d(x; y). 3. Let Jac be the inverse of the Jacobian of T. There exists C > 0 such that, for all a 2 ˛, for all points x; y 2 a, j1  Jac(x)/Jac(y)j  Cd(Tx; Ty). 4. The map T has the “big image property”: inf a2˛ (Ta) > 0. Some piecewise expanding and C2 maps are obviously Gibbs–Markov but the real point is that many dynamics can be reduced to that class by the use of inducing and tower constructions as in [262], in particular. This includes possibly piecewise uniformly hyperbolic diffeomorphisms, Collet–Eckmann maps of the interval [21] (typical chaotic maps in the quadratic family), billiards with convex scatterers [262], the stadium billiard [71], Hénon-like mappings [266].

Chaos and Ergodic Theory

We note that in many cases one is led to first analyze mixing properties through decay of correlations, i. e., to prove inequalities of the type [21]: ˇZ ˇ ˇ  ˇ

ı T n d 

X

Z

ˇ ˇ dˇˇ  kkk kw a n

Z  d X

X

(5) where (a n )n1 is some sequence converging to zero, e. g., a n D en , 1/n˛ , . . . and k  k, k  kw a strong and a weak norm (e. g., the variation norm and the L1 norm). These rates of decay are often linked with return times statistics [263]. Rather general schemes have been developed to deduce various limit theorems such as those presented below from sufficiently quick decay of correlations (see notably [175] based on a dynamical variant of [113]). Probabilistic Limit Theorems The foremost limit property is the following: Definition 2 A class C of functions  : X ! R is said to satisfy the Central Limit Theorem if the following holds: for all  2 C , there is a number  D () > 0 such that: R

 lim 

n!1

x 2 X:



A n (x)   d t  n1/2 Z t dx 2 2 D ex /2 p 2 1

except for the degenerate case when (x) D (x) C const.

(6)

(Tx) 

The Central Limit Theorem can be seen in many cases as essentially a by-product of fast decay of correlations [175], P i. e., if n0 a n < 1 in the notations of Eq. (5). It has been established for Hölder-continuous observables for many systems together with their natural invariant measures including: uniformly hyperbolic attractors, piecewise expanding maps of the interval [174], Collet–Eckmann unimodal maps on the interval [152,260], piecewise hyperbolic maps [74], billiards with convex scatterers [238], Hénon-like maps [35]. Remark 5 The classical Central Limit Theorem holds for square-integrable random variables [193]. For maps exhibiting intermittency (e.g, interval maps like f (x) D x C x 1C˛ mod 1 with an indifferent fixed point at 0) the invariant density has singularities and the integrability condition is non longer automatic for smooth functions. One can then observe convergence to stable laws, instead of the normal law [114].

A natural question is the speed of the convergence in (6). The Berry–Esseen inequality: R ˇ  ˇ  Z t ˇ ˇ A n (x)   d x 2 /2 2 dx ˇ ˇ  t  e p ˇ 1/2 n 2 ˇ 1 C  ı/2 n for some ı > 0. It holds with ı D 1 in the classical, probabilistic setting. The Local Limit Theorem looks at a finer scale, asserting that for any finite interval [a; b], any t 2 R, n p p lim n x 2 X : nA n (x) 2 [a; b] C nt n!1

Z Cn

2 2 o et /2 :  d D jb  aj p 2

Both the Berry–Esseen inequality and the local limit theorem have been shown to hold for non-uniformly expanding maps [115] (also [62,216]). Almost Sure Results It is very natural to try and describe the statistical properties of the averages A n (x) for almost every x, instead of the weaker above statements in probability over x. An important such property is the almost sure invariance principle. It asks for the discrete random walk defined by the increments of  ı T n (x) to converge, after a suitable renormalization, to a Brownian motion. This has been proved for systems with various degrees of hyperbolicity [92,98,135,183]. Another one is the almost sure Central Limit Theorem. In the independent case (e. g., if X1 ; X2 ; : : : are independent and identically distributed random variables in L2 with zero average and unit variance), the almost sure Central Limit Theorem states that, almost surely: n 1 X1 ıPk1 p log n k jD0 X j / k kD1

converges in law to the normal distribution. This implies, that, almost surely, for any t 2 R: n 1 X1 n ı Pk1 p o n!1 log n k jD0 X j / kt kD1 Z t dx 2 2 D ex /2 p 2 1

lim

compare to (6).

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A general approach is developed in [73], covering Gibbs–Markov maps and those that can be reduced to it. They show in particular that the dynamical properties needed for the classical Central Limit Theorem in fact suffice to prove the above almost invariance principle and even the almost sure version of the Central Limit Theorem (using general probabilistic results, see [37,255]). Other Statistical Properties Essentially all the statistical properties of sums of independent identically distributed random variables can be established for tractable systems. Thus one can also prove large deviations [156,172,259], iterated law of the logarithm, etc. We note that the monograph [78] contains a nice introduction to the current work in this area. Orbit Complexity The orbit complexity of a dynamical system f : M ! M is measured by its topological and measured entropies. We refer to  Entropy in Ergodic Theory for detailed definitions. The Variational Principle Bowen–Dinaburg and Katok formulae can be interpreted as meaning that the topological entropy counts the number of arbitrary orbits whereas the measured entropy counts the number of orbits relevant for the given measure. In most situations, and in particular for continuous self-map of compact metric spaces, the following variational principle holds: htop ( f ) D sup h( f ; ) 2M( f )

where M( f ) is the set of all invariant probability measures. This is all the more striking in light of the fact that for many systems, the set of points which are typical from the point of view of ergodic theory [for instance, those x P k such that limn!1 n1 n1 kD0 (T x) exists for all continuous functions .] is topologically negligible [A subset of a countable union of closed sets with empty interior, that is, meager or first Baire category.]. Strict Inequality For a fixed invariant measure, one can only assert that h( f ; )  htop ( f ). One should be aware that this inequality may be strict even for a measure with full support. For instance, it is not difficult to check that the full tent map with htop ( f ) D log 2, admits ergodic invariant measures with full support and zero entropy.

There are also examples of dynamical systems preserving Lebesgue measure which have simultaneously positive topological entropy and zero entropy with respect to Lebesgue measure. That this occurs for C1 surface diffeomorphisms preserving area is a simple consequence of a theorem of Bochi [44] according to which, generically in the C1 topology, such a diffeomorphism is either uniformly hyperbolic or with Lyapunov exponents Lebesguealmost everywhere zero. [Indeed, it is easy to build such a diffeomorphism having both a uniformly hyperbolic compact invariant subset which will have robustly positive topological entropy and a non-degenerate elliptic fixed point which will prevent uniform hyperbolicity and therefore force all Lyapunov exponents to be zero. But Ruelle–Margulis inequality then implies that the entropy with respect to Lebesgue measure is zero.] Smooth examples also exist [46]. Remark 6 Algorithmic complexity [170] suggests another way to look at orbit complexity. One obtains in fact in this way another formula for the entropy. However this point of view becomes interesting in some settings, like extended systems defined by partial differential equations in unbounded space. Recently, [47] has used this approach to build interesting invariant measures. Orbit Complexity on the Set of Measures We have considered the entropies of each invariant measure separately, sharing only the common roof of topological entropy. One may ask how these different complexity sit together. A first answer is given by the following theorem in the symbolic and continuous setting. Let . Theorem 9 (Downarowicz–Serafin [102]) Let K be a Choquet simplex and H : K ! R be a convex function. Say that H is realized by a self-map f : X ! X and its set M( f ) of f -invariant probability measures equipped with the weak star topology if the following holds. There exists an affine homeomorphism  : M( f ) ! K such that, if h : M( f ) ! [0; 1] is the entropy function, H D h ı  . Then  H is realized by some continuous self-map of a compact space if and only if it is an increasing limit of upper semicontinuous and affine functions.  H is realized by some subshift on a finite alphabet, i. e., by the left shift on a closed invariant subset ˙ of f1; 2; : : : ; NgZ for some N < 1, if and only if it is upper semi-continuous Thus, in both the symbolic and continuous settings it is possible to have a unique invariant measure with any prescribed entropy. This stands in contrast to surface C 1C -

Chaos and Ergodic Theory

diffeomorphisms for which the set of the entropies of ergodic invariant measures is always the interval [0; htop ( f )] as a consequence of [147]. Local Complexity Recall that the topological entropy can be computed as: htop ( f ) D lim !0 htop ( f ; ) where: htop ( f ; ) :D lim

n!1

hloc ( f ) D lim sup h( f ; )  h( f ; ; ı) ı!0

1 log s(ı; n; X) n

x ¤ y H) 90  k < n d( f k x; f k y)  (see Bowen’s formula of the topological entropy  Entropy in Ergodic Theory). Likewise, the measure-theoretic entropy h(T; ) of an ergodic invariant probability measure  is lim !0 h( f ; ; ) where: 1 h(T; ) :D lim log r(ı; n; ) n!1 n where r(ı; n; ) is the minimum cardinality of C X such that ˚  x 2 X : 9y 2 C such that  80  k < n d( f k x; f k y) < > 1/2 : One can ask at which scales does entropy arise for a given dynamical system?, i. e., how the above quantities h(T; ), h(T; ; ) converge when ! 0. An answer is provided by the local entropy. [This quantity was introduced by Misiurewicz [186] under the name conditional topological entropy and is called tail entropy by Downarowicz [100].] For a continuous map f of a compact metric space X, it is defined as: hloc ( f ) :D lim hloc ( f ; ) with !0

hloc ( f ; ) :D sup hloc ( f ; ; x) and hloc ( f ; ; x) :D lim lim sup ı!0 n!1

 ˚ 1 log s ı; n; y 2 X : n

8k  0 d( f k y; f k x) <



D sup lim sup h( f ; )  h( f ; )

where s(ı; n; E) is the maximum cardinality of a subset S of E such that:

x2X

h( f ; ) D limı!0 h( f ; ; ı). An exercise in topology shows that the local entropy therefore also bounds the defect in upper semicontinuity of  7! h( f ; ). In fact, by a result of Downarowicz [100] (extended by David Burguet to the non-invertible case), there is a local variational principle: !



Clearly from the above formulas: htop ( f )  htop ( f ; ı) C hloc ( f ; ı) and h( f ; )  h( f ; ; ı) C hloc ( f ; ı) : Thus the local entropy bounds the defect in uniformity with respect to the measure of the pointwise limit



!

for any continuous self-map f of a compact metric space. The local entropy is easily bounded for smooth maps using Yomdin’s theory: Theorem 10 ([64]) For any Cr map f of a compact manifold, hloc ( f )  dr log supx k f 0 (x)k. In particular, hloc ( f ) D 0 if r D 1. Thus C 1 smoothness implies the existence of a maximum entropy measure (this was proved first by Newhouse) and the existence of symbolic extension: a subshift over a finite alphabet  : ˙ ! ˙ and a continuous and onto map  : ˙ ! M such that  ı  D f ı  . More precisely, Theorem 11 (Boyle, Fiebig, Fiebig [56]) Given a homeomorphism f of a compact metric space X, there exists a principal symbolic extension  : ˙ ! ˙ , i. e., a symbolic extension such that, for every  -invariant probability measure , h(; ) D h( f ;  ı  1 ), if and only if hloc ( f ) D 0. We refer to [55,101] for further results, including a realization theorem showing that the continuity properties of the measured entropy are responsible for the properties of symbolic extensions and also results in finite smoothness. Global Simplicity One can marvel at the power of mathematical analysis to analyze such complex evolutions. Of course another way to look at this is to remark that this analysis is possible once this evolution has been fitted in a simple setting: one had to move focus away from an individual, unpredictable orbit, of, say, the full tent map to the set of all the orbits of that map, which is essentially the set of all infinite sequences over two symbols: a very simple set indeed corresponding to full combinatorial freedom [[253] describes a weakening of this which holds for all positive entropy symbolic dynamics.]. The complete description of a given typical orbit requires an infinite amount of information, whereas the set of all orbits has a finite and very tractable definition. The complexity of the individual orbits is seen now as coming from purely random choices inside a simple structure.

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The classical systems, namely uniformly expanding maps or hyperbolic diffeomorphisms of compact spaces, have a simple symbolic dynamics. It is not necessarily a full shift like for the tent map, but it is a subshift of finite type, i. e., a subshift obtained from a full shift by forbidding finitely many finite subwords. What happens outside of the uniform setting? A fundamental example is provided by piecewise monotone maps, i. e., interval maps with finitely many critical points or discontinuities. The partition cut by these points defines a symbolic dynamics. This subshift is usually not of finite type. Indeed, the topological entropy taking arbitrary finite nonnegative values [For instance, the topological entropy of the ˇ-transformation, x 7! ˇx mod 1, is log ˇ for ˇ  1.], a representation that respects it has to use an uncountable class of models. In particular models defined by finite data, like the subshifts of finite type, cannot be generally adequate. However there are tractable “almost finite representations” in the following senses: Most symbolic dynamics ˙ (T) of piecewise monotone maps T can be defined by finitely many infinite sequences, the kneading invariants of Milnor and Thurston:  0C ; 1 ; 1C ; : : : ; dC1 2 f0; : : : ; dgN if d is the number of critical/discontinuity points. [The kneading invariants are the suitably defined (left and right) itineraries of the critical/discontinuity points and endpoints.] Namely, ˚ ˙ (T) D ˛ 2 f0; : : : ; dgN : 8n  0 ˛Cn  n ˛ ˛n C1



where is a total order on f0; : : : ; dgN making the coding x 7! ˛ non-decreasing. Observe how the kneading invariants determine ˙ (T) in an effective way: knowing their first n symbols is enough to know the sequences of length n which begin sequences of ˙ (T). We refer to [91] for the wealth of information that can be extracted from these kneading invariants following Milnor and Thurston [184]. This form of global simplicity can be extended to other classes of non-uniformly expanding maps, including those like Eq. (4) using the notions of subshifts and puzzles of quasi-finite type [68,69]. This leads to the notion and analysis of entropy-expanding maps, a new open class of nonuniformly expanding maps admitting critical hypersurfaces, defined purely in terms of entropies including the otherwise untractable examples of Eq. (4). A generalization of the representation of uniform systems by subshifts of finite type is provided by strongly positive recurrent countable state Markov shifts, a subclass of Markov shifts (see glossary.) which shares many properties with the subshifts of finite type [57,123,124,224,229].

These “simple” systems admit a classification result which in particular identifies their measures with entropy close to the maximum [57]. Such a classification generalizes [164]. The “ideology” here is that complexity of individual orbits in a simple setting must come from randomness, but purely random systems are classified by their entropy according to Ornstein [197]. Stability By definition, chaotic dynamical systems have orbits which are unstable and numerically unpredictable. It is all the more surprising that, once one accepts to consider their dynamics globally, they exhibit very good stability properties. Structural Stability A simple form of stability is structural stability: a system f : M ! M is structurally Cr -stable if any system g sufficiently Cr -close to f is topologically the same as f , formally: g is topologically conjugate, i. e., there is some homeomorphism [If h were C1 , the conjugacy would imply, among other things, that for every p-periodic point: det(( f p )0 (x)) D det((g p )0 (h(p))), a much too strong requirement.] h : M ! M mapping the orbits of f to those of g, i. e., g ı h D h ı f . Andronov and Pontryaguin argued in the 1930’s that only such structurally stable systems are physically relevant. Their idea was that the model of a physical system is always known only to some degree of approximation, hence mathematical model whose structure depends on arbitrarily small changes should be irrelevant. A first question is: What are these structurally stable systems? The answer is quite striking: Theorem 12 (Mañé [182]) Let f : M ! M be a C1 diffeomorphism of a compact manifold. f is structurally stable among C1 -diffeomorphisms of M if and only if f is uniformly hyperbolic on its chain recurrent set. [A point x is chain recurrent if, for all > 0, there exists a finite sequence x0 ; x1 ; : : : ; x n such that x0 D x n D x and d( f (x k ); x kC1 ) < . The chain recurrent set is the set of all chain recurrent points.] A basic idea in the proof of the theorem is that failure of uniform hyperbolicity gives the opportunity to make an arbitrarily small perturbation contradicting the structural stability. In higher smoothness the required perturbation lemmas (e. g., the closing lemma [14,148,180]) are not available. We note that uniform hyperbolicity without invertibility does not imply C1 -stability [210].

Chaos and Ergodic Theory

A second question is: are these stable systems dense? (So that one could offer structurally stable models for all physical situations). A deep discovery around 1970 is that this is not the case: Theorem 13 (Abraham–Smale, Simon [3,234]) For any r  1 and any compact manifold M of dimension  3, the set of uniformly hyperbolic diffeomorphisms is not dense in the space of Cr diffeomorphisms of M. [They use the phenomenon called “heterodimensional homoclinic intersections”.] Theorem 14 (Newhouse [194]) For any r  2, for any compact manifold M of dimension  2, the set of uniformly hyperbolic diffeomorphisms is not dense in the space of Cr diffeomorphisms of M. More precisely, there exists a nonempty open subset in this space which contains a dense Gı subset of diffeomorphisms with infinitely many periodic sinks. [So these diffeomorphisms have no finite statistical description.] Observe that it is possible that uniform hyperbolicity could be dense among surface C1 -diffeomorphisms (this is the case for C1 circle maps by a theorem of Jakobson [145]). In light of Mañé C1 -stability theorem this implies that structurally stable systems are not dense, thus one can robustly see behaviors that are topologically modified by arbitrarily small perturbations (at least in the C1 -topology)! So one needs to look beyond these and face that topological properties of relevant dynamical system are not determined from “finite data”. It is natural to ask whether the dynamics is almost determined by “sufficient data”. Continuity Properties of the Topological Dynamics Structural stability asks the topological dynamics to remain unchanged by a small perturbation. It is probably at least as interesting to ask it to change continuously. This raises the delicate question of which topology should be put on the rather wild set of topological conjugacy classes. It is perhaps more natural to associate to the system a topological invariant taking value in a more manageable set and ask whether the resulting map is continuous. A first possibility is Zeeman’s Tolerance Stability Conjecture. He associated to each diffeomorphism the set of all the closures of all of its orbits and he asked whether the resulting map is continuous on a dense Gı subset of the class of Cr -diffeomorphisms for any r  0. This conjecture remains open, we refer to [85] for a discussion and related progress. A simpler possibility is to consider our favorite topological invariant, the topological entropy, and thus ask whether the dynamical complexity as measured by the

entropy is a stable phenomenon. f 7! htop ( f ) is lower semicontinuous for f among C0 maps of the interval [On the set of interval maps with a bounded number of critical points, the entropy is continuous [188]. Also t 7! htop (Q t ) is non-decreasing by complex arguments [91], though it is a non-smooth function.] [187] and for f among C 1C diffeomorphisms of a compact surface [147]. [It is an important open question whether this actually holds for C1 diffeomorphisms. It fails for homeomorphisms [214].] In both cases, one shows the existence of structurally stable invariant uniformly expanding or hyperbolic subsets with topological entropy close to that of the whole dynamics. On the other hand f 7! htop ( f ) is upper semi-continuous for C 1 maps [195,254]. Statistical Stability Statistical stability is the property that deterministic perturbations of the dynamical system cause only small changes in the physical measures, usually with respect to the weak star topology on the space of measures. When the physical measure  g is uniquely defined for all systems g near f , statistic stability is the continuity of the map g 7!  g thus defined. Statistical stability is known in the uniform setting and also in the piecewise uniform case, provided the expansion is strong enough [25,42,151] (otherwise there are counterexamples, even in the one-dimensional case). It also holds in some non-uniform settings without critical behavior, in particular for maps with dominated splitting satisfying robustly a separation condition between the positive and negative Lyapunov exponents [247] (see also [7]). However statistical unstability occurs in dynamics with critical behaviors [18]: Theorem 15 Consider the quadratic family Q t (x) D tx(1  x), t 2 [0; 4]. Let I [0; 4] be the full measure subset of [0; 4] of good parameters t such that in particular Qt admits a physical measure (necessarily unique). For t 2 I, let  t be this physical measure. Lebesgue almost every t 2 I such that  t is not carried by a periodic orbit [At such parameters, statistical stability is easily proved.], is a discontinuity point of the map t 7!  t [M([0; 1]) is equipped with the vague topology.]. However, for Lebesgue almost every t, there is a subset I t I for which t is a Lebesgue density point [t is a Lebesgue density point if for all r > 0, the Lebesgue measure m(I t \ [t  r; t C r]) > 0 and lim !0 m(I t \ [t  ; t C ])/2 D 1.] and such that  : I t ! M([0; 1]) is continuous at t.

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Stochastic Stability A physically motivated and a technically easier approach is to study stability of the physical measure under stochastic perturbations. For simplicity let us consider a diffeomorphism of a compact subset of Rd allowing for a direct definition of additive noise. Let (x)dx be an absolutely continuous probability law with compact support. [Sometimes additional properties are required of the density, e. g., (x) D (x)1B where 1B is the characteristic function of the unit ball and C 1  (x)  C for some C < 1.] For > 0, consider the Markov chain f with state space M and transition probabilities:   Z y  f (x) dy P (x; A) D : d A The evolution of measures is given by: ( f )(A) D R P (x; A) d. Under rather weak irreducibility assump M tions on f , f has a unique invariant measure  (contrarily to f ) and  is absolutely continuous. When f has a unique physical measure , it is said to be stochastically stable if lim !0  D  in the appropriate topology (the weak star topology unless otherwise specified). It turns out that stochastic stability is a rather common property of Sinai–Ruelle–Bowen measures. It holds not only for uniformly expanding maps or hyperbolic diffeomorphisms [153,258], but also for most interval maps [25], for partially hyperbolic systems of the type E u ˚ E cs or Hénon-like diffeomorphisms [6,13,33,40]. We refer to the monographs [21,41] for more background and results. Untreated Topics For reasons of space and time, many important topics have been left out of this article. Let us list some of them. Other phenomena related to chaotic dynamics have been studied: entrance times [77,79], spectral properties and dynamical zeta functions [21], escape rates [104], dimension [204], differentiability of physical measures with respect to parameters [99,223], entropies and volume or homological growth rates [118,139,254]. As far as the structure of the setting is concerned, one can go beyond maps or diffeomorphisms or flows and study: more general group actions [128]; holomorphic and meromorphic structures [72] and the references therein; symplectic or volume-preserving [90,137] and in particular the Pugh–Shub program around stable ergodicity of partially hyperbolic systems [211]; random iterations [17, 154,155]. A number of important problems have motivated the study of special forms of chaotic dynamics: equidistribution in number theory [105,109,138] and geometry [236];

quantum chaos [10,125]; chaotic control [227]; analysis of algorithms [245]. We have also omitted the important problem of applying the above results. Perhaps because of the lack of a general theory, this can often be a challenge (see for instance [244] for the already complex problem of verifying uniform hyperbolicity for a singular flow). Liverani has shown how theoretical results can lead to precise and efficient estimates for the toy model of piecewise expanding interval maps [176]. Ergodic theory implies that, in some settings at least, adding noise may make some estimates more precise (see [157]). We refer to [106] and the reference therein. Future Directions We conclude this article by a (very partial) selection of open problems. General Theory In dimension 1, we have seen that the analogue of the Palis conjecture (see above) is established (Theorem 2). However the description of the typical dynamics in Kolmogorov sense is only known in the unimodal, non-degenerate case by Theorem 3. Indeed, results like [63] suggest that the multimodal picture could be more complex. In higher dimensions, our understanding is much more limited. As far as a general theory is concerned, a deep problem is the paucity of results on the generic dynamics in Cr smoothness with r > 1. The remarkable current progress in generic dynamics (culminating in the proof of the weak Palis conjecture, see [84], the references therein and [51] for background) seems restricted to the C1 topology because of the lack of fundamental tools (e. g., closing lemmas) in higher smoothness. But Pesin theory requires higher smoothness at least technically. This is not only a hard technical issue but generic properties of physical measures, when they have been analyzed are often completely different between the C1 case and higher smoothness [45]. Physical Measures In higher dimensions, Benedicks and Carleson analysis of the Hénon map has given rise to a rather general theory of Hénon-like maps and more generally of the dynamical phenomena associated to homoclinic tangencies. However, the proofs are extremely technical. Could they be simplified? Current attempts like [266] center on the introduction of a simpler notion of critical points, possibly a non-inductive one [212].

Chaos and Ergodic Theory

Can this Hénon theory be extended to the weakly dissipative situation? to the conservative situation (for which the standard map is a well-known example defying analysis)? In the strongly dissipative setting, what are the typical phenomena on the complement of the Benedicks– Carleson set of parameters? From a global perspective one of the main questions is the following: Can infinitely many sinks coexist for a large set of parameters in a typical family or is Newhouse phenomenon atypical in the Kolmogorov or prevalent sense? This seems rather unlikely (see however [11]). Away from such “critical dynamics”, there are many results about systems with dominated splitting satisfying additional conditions. Can these conditions be weakened so they would be satisfied by typical systems satisfying some natural conditions (like robust transitivity)? For instance: Could one analyze the physical measures of volumehyperbolic systems? A more specific question is whether Tsujii’s striking analysis of surface maps with one uniformly expanding direction can be extended to higher dimensions? can one weaken the uniformity of the expansion? The same questions for the corresponding invertible situation is considered in [83]. Maximum Entropy Measures and Topological Complexity As we explained, C 1 smoothness, by a Newhouse theorem, ensures the existence of maximum entropy measures, making the situation a little simpler than with respect to physical measures. This existence results allow in particular an easy formulation of the problem of the typicality of hyperbolicity: Are maximum entropy ergodic measures of systems with positive entropy hyperbolic for most systems? A more difficult problem is that of the finite multiplicity of the maximum entropy measures. For instance: Do typical systems possess finitely many maximum entropy ergodic measures? More specifically, can one prove intrinsic ergodicity (ie, uniqueness of the measure of maximum entropy) for an

isolated homoclinic class of some diffeomorphisms (perhaps C1 -generic)? Can a generic C1 -diffeomorphism carry an infinite number of homoclinic classes, each with topological entropy bounded away from zero? A perhaps more tractable question, given the recent progress in this area: Is a C1 -generic partially hyperbolic diffeomorphisms, perhaps with central dimension 1 or 2, intrinsically ergodic? We have seen how uniform systems have simple symbolic dynamics, i. e., subshifts of finite type, and how interval maps and more generally entropy-expanding maps keep some of this simplicity, defining subshifts or puzzles of quasi-finite type [68,69]. [264] have defined symbolic dynamics for topological Hénon-like map which seems close to that of that of a one-dimensional system. Can one describe Wang and Young symbolic dynamics of Hénon-like attractors and fit it in a class in which uniqueness of the maximum entropy measure could be proved? More generally, can one define nice combinatorial descriptions, for surface diffeomorphisms? Can one formulate variants of the entropy-expansion condition [For instance building on our “entropy-hyperbolicity”.] of [66,70], that would be satisfied by a large subset of the diffeomorphisms? Another possible approach is illustrated by the pruning front conjecture of [87] (see also [88,144]). It is an attempt to build a combinatorial description by trying to generalize the way that, for interval maps, kneading invariants determine the symbolic dynamics by considering the bifurcations from a trivial dynamics to an arbitrary one. We hope that our reader has shared in our fascination with this subject, the many surprising and even paradoxical discoveries that have been made and the exciting current progress, despite the very real difficulties both in the analysis of such non-uniform systems as the Henon map and in the attemps to establish a general (and practical) ergodic theory of chaotic dynamical systems. Acknowledgments I am grateful for the advice and/or comments of the following colleagues: P. Collet, J.-R. Chazottes, and especially S. Ruette. I am also indebted to the anonymous referee. Bibliography Primary Literature 1. Aaronson J, Denker M (2001) Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch Dyn 1:193–237

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Chronological Calculus in Systems and Control Theory MATTHIAS KAWSKI Department of Mathematics, Arizona State University, Tempe, USA Article Outline Glossary Definition of the Subject Introduction, History, and Background Fundamental Notions of the Chronological Calculus Systems That Are Affine in the Control Future Directions Acknowledgments Bibliography Glossary Controllability A control system is controllable if for every pair of points (states) p and q there exists an admissible control such that the corresponding solution curve that starts at p ends at q. Local controllability about a point means that all states in some open neighborhood can be reached. Pontryagin Maximum Principle of optimal control Optimality of a control-trajectory pair geometrically is a property dual to local controllability in the sense that an optimal trajectory (endpoint) lies on the boundary of the reachable sets (after possibly augmenting the state of the system by the running cost). The maximum principle is a necessary condition for optimality. Geometrically it is based on analyzing the effect of families of control variations on the endpoint map. The chronological calculus much facilitates this analysis. E (M) D C 1 (M) The algebra of smooth functions on a finite dimensional manifold M, endowed with the topology of uniform convergence of derivatives of all orders on compact sets.  1 (M) The space of smooth vector fields on the manifold M. Chronological calculus An approach to systems theory based on a functional analytic operator calculus, that replaces nonlinear objects such as smooth manifolds by infinite dimensional linear ones, by commutative algebras of smooth functions. Chronological algebra A linear space with a bilinear product F that satisfies the identity a ? (b ? c)  b ? (a ? c) D (a ? b) ? c  (b ? a) ? c. This structure arises

Rt naturally via the product ( f ? g) t D 0 [ f s ; g s0 ]ds of time-varying vector fields f and g in the chronological calculus. Here [ ; ] denotes the Lie bracket. Zinbiel algebra A linear space with a bilinear product that satisfies the identity a (b c) D (a b) c C (b a) c. This structure arises naturally in the special case ofR affine control systems for the product t (U V)(t) D 0 U(s)V 0 (s)ds of absolutely continuous scalar valued functions U and V. The name Zinbiel is Leibniz read backwards, reflecting the duality with Leibniz algebras, a form of noncommutative Lie algebras. There has been some confusion in the literature with Zinbiel algebras incorrectly been called chronological algebras. IIF (U Z ) For a suitable space U of time-varying scalars, e. g. the space of locally absolutely continuous real-valued functions defined on a fixed time interval, and an indexing set Z, IIF (U Z ) denotes the space of iterated integral functionals from the space of Z-tuples with values in U to the space U. Definition of the Subject The chronological calculus is a functional analytic operator calculus tool for nonlinear systems theory. The central idea is to replace nonlinear objects by linear ones, in particular, smooth manifolds by commutative algebras of smooth functions. Aside from its elegance, its main virtue is to provide tools for problems that otherwise would effectively be untractable, and to provide new avenues to investigate the underlying geometry. Originally conceived to investigate problems in optimization and control, specifically for extending Pontryagin’s Maximum Principle, the chronological calculus continues to establish itself as the preferred language of geometric control theory, and it is spawning new sets of problems, including its own set of algebraic structures that are now studied in their own right. Introduction, History, and Background This section starts with a brief historical survey of some landmarks that locate the chronological calculus at the interface of systems and control theory with functional analysis. It is understood that such brief survey cannot possibly do justice to the many contributors. Selected references given are meant to merely serve as starting points for the interested reader. Many problems in modern systems and control theory are inherently nonlinear and e. g., due to conserved quantities or symmetries, naturally live on manifolds rather than on Euclidean spaces. A simple example is the problem of stabilizing the attitude of a satellite via feedback con-

Chronological Calculus in Systems and Control Theory

trols. In this case the natural state space is the tangent bundle T SO(3) of a rotation group. The controlled dynamics are described by generally nonautonomous nonlinear differential equations. A key characteristic of their flows is their general lack of commutativity. Solutions of nonlinear differential equations generally do not admit closed form expressions in terms of the traditional sets of elementary functions and symbols. The chronological calculus circumvents such difficulties by reformulating systems and control problems in a different setting which is infinite dimensional, but linear. Building on well established tools and theories from functional analysis, it develops a new formalism and a precise language designed to facilitate studies of nonlinear systems and control theory. The basic plan of replacing nonlinear objects by linear ones, in particular smooth manifolds by commutative algebras of smooth functions, has a long history. While its roots go back even further, arguably this approach gained much of its momentum with the path-breaking innovations by John von Neumann’s work on the “Mathematical Foundations of Quantum Mechanics” [104] and Marshall Stone’s seminal work on “Linear Transformations in Hilbert Space” [89], quickly followed by Israel Gelfand’s dissertation on “Abstract Functions and Linear Operators” [34] (published in 1938). The fundamental concept of maximal ideal justifies this approach of identifying manifolds with commutative normed rings (algebras), and vice versa. Gelfand’s work is described as uniting previously uncoordinated facts and revealing the close connections between classical analysis and abstract functional analysis [102]. In the seventy years since, the study of Banach algebras, C -algebras and their brethren has continued to develop into a flourishing research area. In the different arena of systems and control theory, major innovations at formalizing the subject were made in the 1950s. The Pontryagin Maximum Principle [8] of optimal control theory went far beyond the classical calculus of variations. At roughly the same time Kalman and his peers introduced the Kalman filter [51] for extracting signals from noisy observations, and pioneered statespace approaches in linear systems theory, developing the fundamental concepts of controllability and observability. Linear systems theory has bloomed and grown into a vast array of subdisciplines with ubiquitous applications. Via well-established transform techniques, linear systems lend themselves to be studied in the frequency domain. In such settings, systems are represented by linear operators on spaces of functions of a complex variable. Starting in the 1970s new efforts concentrated on rigorously extending linear systems and control theory to nonlinear settings. Two complementary major threads emerged

that rely on differential geometric methods, and on operators represented by formal power series, respectively. The first is exemplified in the pioneering work of Brockett [10,11], Haynes and Hermes [43], Hermann [44], Hermann and Krener [45], Jurdjevic and Sussmann [50], Lobry [65], and many others, which focuses on state-space representations of nonlinear systems. These are defined by collections of vector fields on manifolds, and are analyzed using, in particular, Lie algebraic techniques. On the other side, the input-output approach is extended to nonlinear settings primarily through a formal power series approach as initiated by Fliess [31]. The interplay between these approaches has been the subject of many successful studies, in which a prominent role is played by Volterra series and the problem of realizing such input-output descriptions as a state-space system, see e. g. Brockett [11], Crouch [22], Gray and Wang [37], Jakubczyk [49], Krener and Lesiak [61], and Sontag and Wang [87]. In the late 1970s Agrachëv and Gamkrelidze introduced into nonlinear control theory the aforementioned abstract functional analytic approach, that is rooted in the work of Gelfand. Following traditions from the physics community, they adopted the name chronological calculus. Again, this abstract approach may be seen as much unifying what formerly were disparate and isolated pieces of knowledge and tools in nonlinear systems theory. While originally conceived as a tool for extending Pontryagin’s Maximum Principle in optimal control theory [1], the chronological calculus continues to yield a stream of new results in optimal control and geometry, see e. g. Serres [85], Sigalotti [86], Zelenko [105] for very recent results utilizing the chronological calculus for studying the geometry. The chronological calculus has led to a very different way of thinking about control systems, epitomized in e. g. the monograph on geometric control theory [5] based on the chronological calculus, or in such forceful advocacies for this approach for e. g. nonholonomic path finding by Sussmann [95]. Closely related are studies of nonlinear controllability, e. g. Agrachëv and Gamkrelidze [3,4], Tretyak [96,97,98], and Vakhrameev [99], including applications to controllability of the Navier–Stokes equation by Agrachëv and Sarychev [6]. The chronological calculus also lends itself most naturally to obtaining new results in averaging theory as in Sarychev [83], while Cortes and Martinez [20] used it in motion control of mechanical systems with symmetry and Bullo [13,68] for vibrational control of mechanical systems. Noteworthy applications include locomotion of robots by Burdick and Vela [100], and even robotic fish [71]. Instrumental is its interplay with series expansion as in Bullo [12,21] that uti-

89

90

Chronological Calculus in Systems and Control Theory

lize affine connections of mechanical systems. There are further applications to stability and stabilization Caiado and Sarychev [15,82], while Monaco et.al. [70] extended this approach to discrete-time dynamics and Komleva and Plotnikov used it in differential game theory [60]. Complementing such applications are new subjects of study such as the abstract algebraic structures that underlie the chronological calculus. The chronological algebra itself has been the subject of study as early as Agrachëv and Gamkrelidze [2]. The closely related structure of Zinbiel structures has recently found more attention, see work by Dzhumadil’daev [25,26], Kawski and Sussmann [57] and Kawski [54,55]. Zinbiel algebras arise in the special case when the dynamics (“time-varying vector fields”) splits into a sum of products of time-varying coefficients and autonomous vector fields. There is an unfortunate confusion of terms in the literature as originally Zinbiel algebras had also been called chronological algebras. Recent usage disentangles these closely related, but distinct, structures and reflects the primacy of the latter term coined by Loday [66,67] who studies Leibniz algebras (which appear in cyclic homology). Zinbiel is simply Leibniz spelled backwards, a choice which reflect that Leibniz and Zinbiel are dual operands in the sense of Koszul duality as investigated by Ginzburg and Kapranov [36]. Fundamental Notions of the Chronological Calculus From a Manifold to a Commutative Algebra This and the next sections very closely follow the introductory exposition of Chapter 2 of Agrachëv and Sachkov [5], which is also recommended for the full technical and analytical details regarding the topology and convergence. The objective is to develop the basic tools and formalism that facilitate the analysis of generally nonlinear systems that are defined on smooth manifolds. Rather than primarily considering points on a smooth manifold M, the key idea is to instead focus on the commutative algebra of E (M) D C 1 (M; R) of real-valued smooth functions on M. Note that E (M) not only has the structure of a vector space over the field R, but it also inherits the structure of a commutative ring under pointwise addition and multiplication from the codomain R. Every point p 2 M gives rise to a functional pˆ : E (M) 7! R defined by pˆ(') D '(p). This functional is linear and multiplicative, and is a homomorphism of the algebras E (M) and R: For every p 2 M, for '; 2 E (M) and t 2 R the following hold pˆ(' C and

) D pˆ' C pˆ'; pˆ('  pˆ (t') D t  ( pˆ ') :

) D ( pˆ ')  ( pˆ ) ;

Much of the power of this approach derives from the fact that this correspondence is invertible: For a nontrivial multiplicative linear functional : E (M) 7! R consider its kernel ker D f' 2 E (M) : ' D 0g. A critical observation is that this is a maximal ideal, and that it must be of the form f' 2 E (M) : '(p) D 0g for some, uniquely defined, p 2 M. For the details of a proof see appendix A.1. of [5]. Proposition 1 For every nontrivial multiplicative linear functional : E (M) 7! R there exists p 2 M such that D pˆ. Note on the side, that there may be maximal ideals in the space of all multiplicative linear functionals on E (M) that do not correspond to any point on M – e. g. the ideal of all linear functionals that vanish on every function with compact support. But this does not contradict the stated proposition. Not only can one recover the manifold M as a set from the commutative ring E (M), but using the weak topology on the space of linear functionals on E (M) one also recovers the topology on M p n ! p if and only if 8 f 2 E (M); pˆn f ! pˆ f : (1) The smooth structure on M is recovered from E (M) in a trivial way: A function g on the space of multiplicative linear functionals pˆ : p 2 M is smooth if and only if there exists f 2 E (M) such that for every pˆ; g( pˆ ) D pˆ f . In modern differential geometry it is routine to identify tangent vectors to a smooth manifold with either equivalence classes of smooth curves, or with first order partial differential operators. In this context, tangent vectors at a point q 2 M are derivations of E (M), that is, linear functionals fˆq on E (M) that satisfy the Leibniz rule. ˆ this means for every '; 2 E (M), Using q, ˆ Xˆ q ) : Xˆ q (' ) D ( Xˆ q ')(qˆ ) C (q')(

(2)

Smooth vector fields on M correspond to linear functionals fˆ : E (M) 7! E (M) that satisfy for all '; 2 E (M) fˆ(' ) D ( fˆ') 

C '  ( fˆ ) :

(3)

Again, the correspondence between tangent vectors and vector fields and the linear functionals as above is invertible. Write  1 (M) for the space of smooth vector fields on M. Finally, there is a one-to-one correspondence between smooth diffeomorphisms ˚ : M 7! M and automorphisms of E (M). The map ˚ˆ : E (M) 7! E (M) defined for ˆ D '(˚(p)) clearly has p 2 M and ' 2 E (M) by ˚(')(p)

Chronological Calculus in Systems and Control Theory

the desired properties. For the reverse direction, suppose  : E (M) 7! E (M) is an automorphism. Then for every p 2 M the map pˆ ı  : E (M) 7! R is a nontrivial linear multiplicative functional, and hence equals qˆ for some q 2 M. It is easy to see that the map ˚ : M 7! M is indeed a diffeomorphism, and  D ˚ˆ . In the sequel we shall omit the hats, and simply write, say, p for the linear functional pˆ. The role of each object is usually clear from the order. For example, for a point p, a smooth function ', smooth vector  fields f ; g; h, with flow e th and its tangent map e th  , what in traditional   format might be expressed as ( e th  g)'(e f (p)) is simply written as pe f ge th ', not requiring any parentheses. Frechet Space and Convergence The usual topology on the space E (M) is the one of uniform convergence of all derivatives on compact sets, i. e., a sequence of functions f' k g1 kD1  E (M) converges to  2 E (M) if for every finite sequence f1 ; f2 ; : : : ; f s of smooth vector fields on M and every compact set K  M the sequence f f s : : : f2 f1 ' k g1 kD1 converges uniformly on K to f s : : : f2 f1 '. This topology is also obtained by a countable family of semi-norms k  ks;K defined by k'ks;K D supf jp f s : : : f2 f1 'j : p 2 K; f i 2  1 (M); s 2 ZC g

(4)

where K ranges over a countable collection of compact subsets whose union is all of M. In an analogous way define semi-norms of smooth vector fields f 2  1 (M) k f ks;K D supf k f 'ks;K : k f 'ksC1;K D 1 g :

(5)

Finally, for every smooth diffeomorphism ˚ of M, s 2 ZC and K  M compact there exist Cs;K;˚ 2 R such that for all ' 2 E (M) k˚ 'ks;K  Cs;K;˚ k'ks;'(K) :

(6)

Regularity properties of one-parameter families of vector fields and diffeomorphisms (“time-varying vector fields and diffeomorphisms”) are understood in the weak sense. In particular, for a family of smooth vector fields f t 2  1 (M), its integral and derivative (if they exist) are defined as the operators that satisfy for every ' 2 E (M) 

The Chronological Exponential This section continues to closely follow [5] which contains full details and complete proofs. On a manifold M consider generally time-varying differential equations of the d form dt q t D f t (q t ). To assure existence and uniqueness of solutions to initial value problems, make the typical regularity assumptions, namely that in every coordinate chart U  M ; x : U 7! Rn the vector field (x f t ) is (i) measurable and locally bounded with respect to t for every fixed x and (ii) smooth with locally bounded partial derivatives with respect to x for every fixed t. For the purposes of this article and for clarity of exposition, also assume that vector fields are complete. This means that solutions to initial value problems are defined for all times t 2 R. This is guaranteed if, for example, all vector fields considered vanish identically outside a common compact subset of M. As in the previous sections, for each fixed t interpret q t : E (M) 7! R as a linear functional on E (M), and note that this family satisfies the time-varying, but linear differential equation, suggestively written as q˙ t D q t ı f t :

d dt

   Z t Z ! ! exp f d D exp 0

0

d dt

 Z  exp

a

 f t ' dt :



t

f d

0

(9)

 Z  D f t ı exp



t 0

f d

: (10)

t

q() ı f d

Z t (7)

ı ft :

f d

Formally, one obtains a series expansion for the chronological exponentials by rewriting the differential equation as an integral equation and solving it by iteration

0





t

Analogously the left chronological exponential satisfies

q t D q0 C b

(8)

on the space of linear functionals on E (M). It may be shown that under the above assumptions it has a unique solution, called the right chronological exponential of the vector field f t as the corresponding flow. Formally, it satisfies for almost all t 2 R

Z

 d d   f t ' and ft ' D dt dt ! Z b Z f t dt ' D a

The convergence of series expansions of vector fields and of diffeomorphisms encountered in the sequel are to be interpreted in the analogous weak sense.

D q0 C

Z

0





q0 C 0

(11)

q( ) ı f d

ı f d

(12)

91

92

Chronological Calculus in Systems and Control Theory

to eventually, formally, obtain the expansion !

Z

t

f d Id C

exp 0

1 Z tZ X kD1 0

tk

Z 

0

Variation of Parameters and the Chronological Logarithm

t2

0

 fk ı    ı f2 ı f1 d k : : : d2 d1

(13)

and analogously for the left chronological exponential 

Z

t

f d Id C

exp 0

1 Z tZ X kD1 0

tk

Z 

0

t2

0

 f1 ı f2 ı    ı fk d k : : : d2 d1 :

(14)

While this series never converges, not even in a weak sense, it nonetheless has an interpretation as an asymptotic expansion. In particular, for any fixed function  2 E (M) and any semi-norm as in the previous section, on any compact set one obtains an error estimate for the remainder after truncating the series at any order N which establishes that the truncation error is of order O(t N ) as t ! 0. When one considers the restrictions to any f t -invariant normed linear subspace L  E (M), i. e. for all t, f t (L)  L, on which f t is bounded, then the asymptotic series converges to the chronological exponential and it satisfies for every '2L    Z Rt   ! t  exp 0 k f  k d k' k : f d '    e 

(15)

0

In the case of analytic vector fields f and analytic functions ' one obtains convergence of the series for sufficiently small t. While in general there is no reason why the vector fields f t should commute at different times, the chronological exponentials nonetheless still share some of the usual properties of exponential and flows, for example the composition of chronological exponentials satisfies for all t i 2 R  Z ! exp

t2 t1

  Z ! f d ı exp



t3

f d

t2

 Z ! D exp



t3 t1

f d

:

Moreover, the left and right chronological exponentials are inverses of each other in the sense that for all t0 ; t1 2 R 1

t1 t0

f d

!

Z

t0



Z

t1 t1

D exp D exp t0

f d (17) ( f ) d :

q˙ D f t (q) C g t (q) :

(18)

The objective is to obtain a formula for the flow of the field ( f t C g t ) as a perturbation of the flow of f t . Writing !

Z

t

˚ t D exp 0

!

f d and  t D exp

Z

t 0

( f Cg ) d; (19)

this means one is looking for a family of operators  t such that for all t t D t ı ˚t :

(20)

Differentiating and using the invertibility of the flows one obtains a differential equation in the standard form ˙ (t) D  t ı ˚ t ı g t ı ˚ t1 D  t ı (Ad˚ t ) g t   Z t ! D  t ı exp ad f d g t ; (21) 0

which has the unique solution !

 t D exp

Z t Z ! exp 0



 0

ad f d

g d ;

(22)

and consequently one has the variations formula !

Z

t

exp 0

(16)

 Z ! exp

In control one rarely considers only a single vector field, and, instead, for example, is interested in the interaction of a perturbation or control vector field with a reference or drift vector field. In particular, in the nonlinear setting, this is where the chronological calculus very much facilitates the analysis. For simplicity consider a differential equation defined by two, generally time varying, vector fields (with the usual regularity assumptions)

 Z t Z ! ! ( f C g ) d D exp exp 0

 Z  !  g d ı exp



 0

ad f d

t 0

f d

 :

(23)

This formula is of fundamental importance and used ubiquitously for analyzing controllability and optimality: One typically considers f t as defining a reference dynamical system and then considers the effect of adding a control term g t . A special application is in the theory of averaging where g t is considered a perturbation of the field f t , which in the most simple form may be assumed to be time-periodic. The monograph [81] by Sanders and Verhulst is the

Chronological Calculus in Systems and Control Theory

classical reference for the theory and applications of averaging using traditional language. The chronological calculus, in particular the above variations formula, have been used successfully by Sarychev [83] to investigate in particular higher order averaging, and by Bullo [13] for averaging in the context of mechanical systems and its interplay with mechanical connections. Instrumental to the work [83] is the notion of the chronological logarithm, which is motivated by rewriting the defining Eqs. (9) and (10) [2,80] as  1   Z t Z d ! t ! f d ı exp f d f t D exp dt 0 0   1  Z t Z t d   D exp f d ı exp f d : (24) dt 0 0 While in general the existence of the logarithm is a delicate question [62,83], under suitable hypotheses it is instrumental for the derivation of the nonlinear Floquet theorem in [83]. Theorem 1 (Sarychev [83]) Suppose f t D f tC1 is a period-one smooth vector field generating the flow ˚ t and  D log ˚1 , then there exists a period-one family of diffeomorphisms  t D  tC1 such that for all t ˚ t D e t ı  t . The logarithm of the chronological exponential of a timevarying vector field " f t may be expanded as a formal chronological series Z 1  1 X ! " f d D " j j : (25) log exp 0

jD1

The first three terms of which are given explicitly [83] as  Z 1 Z Z  1 1 1 2 f d ;  D f d ; f d;  D 2 0 0 0 1 and 3 D  [1 ; 2 ] 2  Z  Z 1 1 2 ad f d f d : (26) C 3 0 0 Aside from such applications as averaging, the main use of the chronological calculus has been for studying the dual problems of optimality and controllability, in particular controllability of families of diffeomorphisms [4]. In particular, this approach circumvents the limitations encountered by more traditional approaches that use parametrized families of control variations that are based on a finite number of switching times. However, since [52] it is known that a general theory must allow also, e. g., for increasing numbers of switchings and other more general families of control variations. The chronological calculus is instrumental to developing such general theory [4].

Chronological Algebra When using the chronological calculus for studying controllability, especially when differentiating with respect to a parameter, and also in the aforementioned averaging theory, the following chronological product appears almost everywhere: For two time-varying vector fields f t and g t that are absolutely continuous with respect to t, define their chronological product as Z t [ f s ; g s0 ]ds (27) ( f ? g) t D 0

where [; ] denotes the Lie bracket. It is easily verified that this product is generally neither associative nor satisfies the Jacobi identity, but instead it satisfies the chronological identity x ? (y ? z)  y ? (x ? z) D (x ? y) ? z  (y ? x) ? z : (28) One may define an abstract chronological algebra (over a field k) as a linear space that is equipped with a bilinear product that satisfies the chronological identity (28). More compactly, this property may be written in terms of the left translation  that associates with any element x 2 A of a not-necessarily associative algebra A the map x : y 7! x y. Using this , the chronological identity (28) is simply [x;y] D [x ;  y ] ;

(29)

i. e., the requirement that the map x 7! x is a homomorphism of the algebra A with the commutator product (x; y) 7! x y  yx into the Lie algebra of linear maps from A into A under its commutator product. Basic general properties of chronological algebras have been studied in [2], including a discussion of free chronological algebras (over a fixed generating set S). Of particular interest are bases of such free chronological algebras as they allow one to minimize the number of terms in series expansions by avoiding redundant terms. Using the graded structure, it is shown in [2] that an ordered basis of a free chronological algebra over a set S may be obtained recursively from products of the form b 1 b 2    b k s where s 2 S is a generator and b1 4 b2 4    4 b k are previously constructed basis elements. According to [2], the first elements of such a basis over the single-element generating set S D fsg may be chosen as (using juxtaposition for multiplication) b1 D s ; b 2 D b 1 s D s 2 ; b3 D b 2 s D s 2 s ; b4 D b 1 b 1 s D ss 2 ; b5 D b 3 s D (s 2 s)s ; b6 D b 4 s D (ss 2 )s ; b7 D b 1 b 2 s D s(s 2 s) ; b8 D b 2 b 1 s D s(ss 2 ) ; : : :

(30)

93

94

Chronological Calculus in Systems and Control Theory

Using only this algebraic and combinatorial structure, one may define exponentials (and logarithms) and analyze the group of formal flows [2]. Of particular interest systems and control is an explicit formula that allows one to express products (compositions) of two or more exponentials (flows) as a single exponential. This is tantamount to a formula for the logarithm of a product of exponentials of two noncommuting indeterminates X and Y. The classical version is known as the Campbell–Baker–Hausdorff formula [16], the first few terms being 1

1

1

e X  eY D e XCYC 2 [X;Y]C 12 [XY;[X;Y]] 24 [X;[Y;[X;Y]]]::: : (31) A well-known short-coming of the classical formula is that the higher-order iterated Lie brackets appearing in this formula are always linearly dependent and hence the coefficients are never well-defined. Hence one naturally looks for more compact, minimal formulas which avoid such redundancies. The chronological algebra provides one such alternative and has been developed and utilized in [2]. The interested reader is referred to the original reference for technical details. Systems That Are Affine in the Control Separating Time-Varying Control and Geometry The chronological calculus was originally designed for work with nonstationary (time-varying) families of vector fields, and it arguably reaps the biggest benefits in that setting where there are few other powerful tools available. Nonetheless, the chronological calculus also much streamlines the analysis of autonomous vector fields. The best studied case is that of affine control systems in which the time-varying control can be separated from the geometry determined by autonomous vector fields. In classical notation such control systems are written in the form x˙ D u1 (t) f1 (x) C    C u m (t) f m (x) :

(32)

where f i are vector fields on a manifold M and u is a measurable function of time taking values typically in a compact subset U  Rm . This description allows one to also accommodate physical systems that have an uncontrolled drift term by simply fixing u1  1. For the sake of clarity, and for best illustrating the kind of results available, assume that the fields f i are smooth and complete. Later we specialize further to real analytic vector fields. This set-up does not necessarily require an interpretation of the ui as controls: they may equally well be disturbances, or it may simply be a dynamical system which splits in this convenient way.

As a starting point consider families of piecewise constant control functions u : [0; T] 7! U  Rm . On each interval [t j ; t jC1 ] on which the control is constant u[t j ;t jC1] D u( j) , the right hand side of (32) is a fixed vector ( j)

( j)

field g j D u1 f1 C : : : u m f m . The endpoint of the solution curve starting from x(0) D p is computed as a directed product of ordinary exponentials (flows) of vector fields pe(t1 t0 )g 1 ı e(t2 t1 )g 2 ı    e(Tt s1 )g s :

(33)

But even in this case of autonomous vector fields the chronological calculus brings substantial simplifications. Basically this means that rather than considering the expression (33) as a point on M, consider this product as a functional on the space E (M) of smooth functions on M. To reiterate the benefit of this approach [5,57] consider the simple example of a tangent vector to a smooth curve

: ("; ") 7! M which one might want to define as 1

˙ (0) D lim ( (t)  (0)) : t!0 t

(34)

Due to the lack of a linear structure on a general manifold, with a classical interpretation this does not make any sense at all. Nonetheless, when interpreting (t) ; (0) ; 0 (0), and 1t ( (t) (0)) as linear functionals on the space E (M) of smooth functions on M this is perfectly meaningful. The meaning of the limit can be rigorously justified as indicated in the preceding sections, compare also [1]. A typical example to illustrate how this formalism almost trivializes important calculations involving Lie brackets is the following, compare [57]. Suppose f and g are smooth vector fields on the manifold M, p 2 M, and t 2 R is sufficiently small in magnitude. Using that d tf tf dt pe D pe f and so on, one immediately calculates d  t f t g t f t g  pe e e e dt D pe t f f e t g et f et g C pe t f e t g get f et g  pe t f e t g et f f et g  pe t f e t g et f et g g :

(35)

In particular, at t D 0 this expression simplifies to p f C pg  p f  pg D 0. Analogously the second derivative is calculated as d2  t f t g t f t g  pe e e e dt 2 D pe t f f 2 e t g et f et g C pe t f f  e t g get f et g  pe t f f et g et f f et g  pe t f f e t g et f  et g g C pe t f f e t g get f et g C pe t f e t g g 2 et f et g  pe t f e t g get f f et g  pe t f e t g get f et g g  pe t f f

Chronological Calculus in Systems and Control Theory

 e t g et f f et g  pe t f e t g get f f et g C pe t f e t g et f 2 t g

f e

t f t g t f

C pe e e

t f tg

 pe e ge

t f t g

e

t g

fe

t g t f t g

tf

g  pe f e e

t f t g t f

g C pe e e

t g

fe

g C pe

t g t f t g 2

e e

e

e

g

g

tf

(36)

which at t D 0 evaluates to p f 2 C p f g  p f 2  p f g C p f g C pg 2  pg f  pg 2  p f 2  pg f C p f 2 C p f g  p f g  pg 2 C p f g C pg 2

where the sum ranges over all multi-indices I D (i1 ; : : : ; i s ); s  0 with i j 2 f1; 2 : : : mg. This series gives rise to an asymptotic series for solution curves of the system (32). For example, in the analytic setting, following [91], one has: Theorem 2 Suppose f i are analytic vector fields on Rn  : Rn 7! R is analytic and U Rm is compact. Then for every compact set K  Rn , there exists T > 0 such that the series

D 2p f g  2pg f D 2p[ f ; g] : While this simple calculation may not make sense on a manifold with a classical interpretation in terms of points and vectors on a manifold, it does so in the context of the chronological calculus, and establishes the familiar formula for the Lie bracket as the limit of a composition of flows pe t f e t g et f et g D pCt 2 p[ f ; g]CO(t 2 ) as t ! 0: (37) Similar examples may be found in [5] (e. g. p. 36), as well as analogous simplifications for the variations formula for autonomous vector fields ([5] p. 43). Asymptotic Expansions for Affine Systems Piecewise constant controls are a starting point. But a general theory requires being able to take appropriate limits and demands completeness. Typically, one considers measurable controls taking values in a compact subspace U  Rm , and uses L1 ([0; t]; U) as the space of admissible controls. Correspondingly the finite composition of exponentials as in (33) is replaced by continuous formulae. These shall still separate the effects of the time-varying controls from the underlying geometry determined by the autonomous vector fields. This objective is achieved by the Chen–Fliess series which may be interpreted in a number of different ways. Its origins go back to the 1950s when K. T. Chen [18], studying geometric invariants of curves in Rn , associated to each smooth curve a formal noncommutative power series. In the early 1970s, Fliess [30,31] recognized the utility of this series for studying control systems. Using a set X 1 ; X2 ; : : : ; X m of indeterminates, the Chen– Fliess series of a measurable control u 2 L1 ([0; t]; U) as above is the formal power series SCF (T; u) D

XZ 0

I

Z  0

Z u i s (ts )

T

Z t3    u i 2 (t2 )

t s1 0

0

t2

u i 1 (t1 ) dt1 : : : dts  X i 1 : : : X i s

(38)

SCF; f (T; u; p)(') D Z  0

Z u i s (ts )

XZ

T

0

I

0

Z t3    u i 2 (t2 )

t s1

0

t2

u i 1 (t1 ) dt1 : : : dts  p f i 1 : : : f i s '

(39)

converges uniformly to the solution of (32) for initial conditions x(0) D p 2 K and u : [0; T] 7! U measurable. Here the series SCF; f (T; u; p) is, for every fixed triple ( f ; u; p), interpreted as an element of the space of linear functionals on E (M). But one may abstract further. In particular, for each fixed multi-index I as above, the iterated integral coefficient is itself a functional that takes as input a control function u 2 Um and maps it to the corresponding iterated integral. R t It is convenient to work with the primitives U j : t 7! 0 u j (s) ds of the control functions uj rather than the controls themselves. More specifically, if e. g. U D AC([0; T]; [1; 1]) then for every multi-index I D (i1 ; : : : ; i s ) 2 f1; : : : ; mgs as above one obtains the iterated integral functional I : Um 7! U defined by Z

T

I : U 7! 0

U i0s (ts )

Z

t s1

Z

 0

0

Z

t3

U i02 (t2 )

t2

 0

U i01 (t1 ) dt1 dt2 : : : dts :

(40)

Denoting by IIF (U Z ) the linear space of iterated integral functionals spanned by the functionals of the above form, the map  maps multi-indices to IIF (U Z ). The algebraic and combinatorial nature of these spaces and this map will be explored in the next section. On the other side, there is the map F that maps a multi-index (i1 ; i2 ; : : : ; i s ) to the partial differential operator f i 1 : : : f i s : E (M) 7! E (M), considered as a linear transformation of E (M). In the analytic case one may go further, since as a basic principle, the relations between the iterated Lie brackets of the vector fields f j completely determine the geometry and dynamical behavior [90,91]. Instead of considering actions on E (M), it suffices to consider the action of these partial differential operators, and

95

96

Chronological Calculus in Systems and Control Theory

Zinbiel Algebra and Combinatorics

w a D wa and w (za) D (w z C z w)a (41) and extending bilinearly to AC (Z)  AC (Z). One easily verifies that this product satisfies the Zinbiel identity for all r; s; t 2 AC (Z) r (s t) D (r s) t C (s r) t :

(42)

The symmetrization of this product is the better known associative shuffle product w z D w z C z w which may be defined on all of A(Z)  A(Z). Algebraically, the shuffle product is characterized as the transpose of the coproduct  : A(Z) 7! A(Z) ˝ A(Z) which is the concatenation product homomorphism that on letters a 2 Z is defined as  : a 7! a ˝ 1 C 1 ˝ a :

(43)

This relation between this coproduct and the shuffle is that for all u; v; w 2 A(Z) 9

h(w); u ˝ vi D hw; u

vi :

(44)

After this brief side-trip into algebraic and combinatorial objects we return to the control systems. The shuffle product has been utilized for a long time in this and related contexts. But the Zinbiel product structure was only recently recognized as encoding important information. Its most immediate role is seen in the fact that the aforementioned map  from multi-indices to iterated integral functionals, now easily extends to a map (using the same name)  : A(Z) 7! IIF (U Z ) is a Zinbiel algebra homomorphism when the latter is equipped with a pointwise product induced by the product on U D AC([0; T]; [1; 1]) defined by Z t U(s)V 0 (s)ds : (45) (U V)(t) D 9

Just as the chronological calculus of time-varying vector fields is intimately connected to chronological algebras, the calculus of affine systems gives rise to its own algebraic and combinatorial structure. This, and the subsequent section, give a brief look into this algebra structure and demonstrate how, together with the chronological calculus, it leads to effective solution formulas for important control problems and beyond. It is traditional and convenient to slightly change the notation and nomenclature. For further details see [78] or [57]. Rather than using small integers 1; 2; : : : ; m to index the components of the control and the vector fields in (32), use an arbitrary index set Z whose elements will be called letters and denoted usually by a; b; c; : : : For the purposes of this note the alphabet Z will be assumed to be finite, but much of the theory can be developed for infinite alphabets as well. Multi-indices, i. e. finite sequences with values in Z are called words, and they have a natural associative product structure denoted by juxtaposition. For example, if w D a1 a2 : : : ar and z D b1 b2 : : : bs then wz D a1 a2 : : : ar b1 b2 : : : bs . The empty word is denoted e n or 1. The set of all words of length n is Zn , Z  D [1 nD0 Z C 1 n and Z D [nD1 Z are the sets of all words and all nonempty words. Endow the associative algebra A(Z), generated by the alphabet Z with coefficients in the field k D R, with an inner product h; i so that Z  is an orthonormal ˆ basis. Write A(Z) for the completion of A(Z) with respect to the uniform structure in which a fundamental system of basic neighborhoods of a polynomial p 2 A(Z) is the family of subsets fq 2 A(Z) : 8w 2 W; hw; p  qi D 0g indexed by all finite subsets W Z  .

The smallest linear subspace of the associative algebra A(Z) that contains the set Z and is closed under the commutator product (w; z) 7! [w; z] D wzzw is isomorphic to the free Lie algebra L(Z) over the set Z. On the other side, one may equip AC (Z) with a Zinbiel algebra structure by defining the product : AC (Z)  AC (Z 7! AC (Z) for a 2 Z and w; z 2 Z C by

9

of the dynamical system (32) on a set of polynomial functions. In a certain technical sense, an associative algebra (of polynomial functions) is dual to the Lie algebra (generated by the analytic vector fields f j ). However, much further simplifications and deeper insights are gained by appropriately accounting for the intrinsic noncommutative structure of the flows of the system. Using a different product structure, the next section will lift the system (32) to a universal system that has no nontrivial relations between the iterated Lie brackets of the formal vector fields and which is modeled on an algebra of noncommutative polynomials (and noncommutative power series) rather than the space E (M) of all smooth functions on M. An important feature of this approach is that it does not rely on polynomial functions with respect to an a-priori chosen set of local coordinates, but rather uses polynomials which map to intrinsically geometric objects.

0

It is straightforward to verify that this product indeed equips U with a Zinbiel structure, and hence also IIF (U Z ). On the side, note that the map  maps the shuffle product of words (noncommuting polynomials) to the associative pointwise product of scalar functions. The connection of the Zinbiel product with differential equations and the Chen–Fliess series is immediate. First lift

Chronological Calculus in Systems and Control Theory

the system (32) from the manifold M to the algebra A(Z), roughly corresponding to the linear space of polynomial functions in an algebra of iterated integral functionals. Formally consider curves S : [0; T] 7! A(Z) that satisfy S˙ D S

X

au a (t) :

(46)

a2Z

the Zinbiel product (45), and writing U a (t) D RUsing t u () d for the primitives of the controls, the inte0 a grated form of the universal control system (46) Z S(t) D 1 C

t

S()F 0 () d with F D

0

X

U a ; (47)

a2Z

is most compactly written as S D1CS F :

(48)

Iteration yields the explicit series expansion S D 1 C (1 C S F) F D 1 C F C ((1 C S F) F) F D 1 C F C (F F) C (((1 C S F) F) F) F D 1 C F C (F F) C ((F F) F) C ((((1 C S F) F) F) F) F :: : D 1 C F C (F F) C ((F F) F) C (((F F) F) F) : : : Using intuitive notation for Zinbiel powers this solution formula in the form of an infinite series is compactly written as SD

1 X

necessary conditions and sufficient conditions for local controllability and optimality, compare e. g. [53,88,94], this series expansion has substantial shortcomings. For example, it is clear from Ree’s theorem [77] that the series is an exponential Lie series. But this is not at all obvious from the series as presented (38). Most detrimental for practical purposes is that finite truncations of the series never correspond to any approximating systems. Much more convenient, in particular for path planning algorithms [47,48,63,64,75,95], but also in applications for numerical integration [17,46] are expansions as directed infinite products of exponentials or as an exponential of a Lie series. In terms of the map F that substitutes for each letter a 2 Z the corresponding vector field f a , one finds that the Chen–Fliess series (39) is simply the image of a natural object under the map  ˝ F . Indeed, under the usual identification of the space Hom(V ; W) of linear maps between vector spaces V and W with the product V  ˝ W, and noting that Z is an orthonormal basis for A(Z), the identity map from A(Z) to A(Z) is identified with the series X ˆ w ˝ w 2 A(Z) ˝ A(Z) : (50) IdA(Z)  w2Z 

Thus, any rewriting of the combinatorial object on the right hand side will, via the map  ˝ F , give an alternative presentation of the Chen–Fliess series. In particular, from elementary consideration, using also the Hopf-algebraic structures of A(Z), it is a-priori clear that there exist expansions in the forms ! X X w ˝ w D exp  h ˝ [h] w2Z 

h2H

D F

n

2

D 1C F C F C F

3

4

CF CF

5

6

C F C  

nD0

(49) The reader is encouraged to expand all of the terms and match each step to the usual computations involved in obtaining Volterra series expansions. Indeed this formula (49) is nothing more than the Chen–Fliess series (38) in very compact notation. For complete technical details and further discussion we refer the interested reader to the original articles [56,57] and the many references therein. Application: Product Expansions and Normal Forms While the Chen–Fliess series as above has been extremely useful to obtain precise estimates that led to most known

Y

exp ( h ˝ [h])

(51)

h2H

where H indexes an ordered basis f[h] : H 2 H g of the free Lie algebra L(Z) and for each h 2 H ;  h and  h are polynomials in A(Z) that are mapped by  to corresponding iterated integral functionals. The usefulness of such expression depends on the availability of simple formulas for H ,  h and  h . Bases for free Lie algebras are well-known since Hall [42], and have been unified by Viennot [101]. Specifically, a Hall set over a set Z is any strictly ordered subset ˜  T (Z) from the set M(Z) labeled binary trees (with H leaves labeled by Z) that satisfies ˜ (i) Z  H ˜ iff t 0 2 H ˜ ; t0 < a (ii) Suppose a 2 Z. Then (t; a) 2 H 0 and a < (t ; a).

97

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Chronological Calculus in Systems and Control Theory

˜ . Then (t 0 ; (t 000; t 0000 )) 2 H ˜ (iii) Suppose u; v; w; (u; v) 2 H iff t 000  t 0  (t 000 ; t 0000 ) and t 0 < (t 0 ; (t 000 ; t 0000 )). There are natural mappings : T (Z) 7! L(Z)  A(Z) and # : T (Z) 7! A(Z) (the foliage map) that map labeled binary trees to Lie polynomials and to words, and which are defined for a 2 Z, (t 0 ; t 00 ) 2 T (Z) by (a) D #(a) D a and recursively ((t 0 ; t 00 )) D [ (t 0 ); (t 00 )] and #((t 0 ; t 00 )) D #(t 0 )#(t 00 ). The image of a Hall set under the map is a basis for the free Lie algebra L(Z). Moreover, the restriction of the map # to a Hall-set is one-to-one which leads to a fundamental unique factorization property of words into products of Hall-words, and a unique way of recovering Hall trees from the Hall words that are their images under # [101]. This latter property is very convenient as it allows one to use Hall words rather than Hall trees for indexing various objects. The construction of Hall sets, as well as the proof that they give rise to bases of free Lie algebras, rely in an essential way on the process of Lazard elimination [9] which is intimately related to the solution of differential Eqs. (32) or (46) by a successive variation of parameters [93]. As a result, using Hall sets it is possible to obtain an extremely simple and elegant recursive formula for the coefficients  h in (51), whereas formulas for the  h are less immediate, although they may be computed by straightforward formulas in terms of natural maps of the Hopf algebra structure on A(Z) [33]. Up to a normalization factor  in terms of multi-factorials, the formula for the  h is extremely simple in terms of the Zinbiel product on A(Z). For letters a 2 Z and Hall words h; k; hk 2 H one has a D a

and

 hk D  hk   h  k :

(52)

Using completely different notation, this formula appeared originally in the work of Schützenberger [84], and was later rediscovered in various settings by Grayson and Grossman [38], Melancon and Reutenauer [69] and Sussmann [93]. In the context of control, it appeared first in [93]. To illustrate the simplicity of the recursive formula, which is based on the factorization of Hall words, consider the following normal form for a free nilpotent system (of rank r D 5) using a typical Hall set over the alphabet Z D f0; 1g. This is the closest nonlinear analogue to the Kalman controller normal form of a linear system, which is determined by Kronecker indices that classify the lengths of chains of integrators. For practical computations nilpotent systems are commonly used as approximations of general nonlinear systems – and every nilpotent system (that is, a system of form (32) whose vector fields f a generate a nilpotent Lie algebra) is the image of a free

nilpotent system as below under some projection map. For convenience, the example again uses the notation xh instead of  h . x˙0 D u0 x˙1 D u1 x˙01 D x0  x˙1 D x0 u1 x˙001 D x0  x˙01 D x02 u1 using # 1 (001) D (0(01)) x˙101 D x1  x˙01 D x1 x0 u1 using # 1 (101) D (1(01)) x˙0001 D x0  x˙001 D x03 u1 using # 1 (0001) D (0(0(01))) x˙1001 D x1  x˙001 D x1 x02 u1 using # 1 (1001) D (1(0(01))) x˙1101 D x1  x˙101 D x12 x0 u1 using # 1 (1101) D (1(1(01))) x˙00001 D x0  x˙001 D x04 u1 using # 1 (00001) D (0(0(0(01)))) x˙10001 D x1  x˙0001 D x1 x03 u1 using # 1 (10001) D (1(0(0(01)))) x˙11001 D x1  x˙1001 D x12 x02 u1 using # 1 (11001) D (1(1(0(01)))) x˙01001 D x01  x˙001 D x01 x03 u1 using # 1 (01001) D ((01)(0(01))) x˙01101 D x01  x˙101 D x01 x12 x0 u1 using # 1 (01101) D ((01)(1(01))) : This example may be interpreted as a system of form (32) with the xh being a set of coordinate functions on the manifold M, or as the truncation of the lifted system (46) with the xh being a basis for a finite dimensional subspace of the linear space E (M). For more details and the technical background see [56,57]. Future Directions The chronological calculus is still a young methodology. Its intrinsically infinite dimensional character demands a certain sophistication from its users, and thus it may take some time until it reaches its full potential. From a different point of view this also means that there are many opportunities to explore its advantages in ever new areas of applications. A relatively straightforward approach simply checks classical topics of systems and control for whether

Chronological Calculus in Systems and Control Theory

they are amenable to this approach, and whether this will be superior to classical techniques. The example of [70] which connects the chronological calculus with discrete time systems is a model example for such efforts. There are many further opportunities, some more speculative than others. Control The chronological calculus appears ideally suited for the investigation of fully nonlinear systems – but in this area there is still much less known than in the case of nonlinear control systems that are affine in the control, or the case of linear systems. Specific conditions for the existence of solutions, controllability, stabilizability, and normal forms are just a few subjects that, while studies have been initiated [1,2,4], deserve further attention. Computation In recent years much effort has been devoted to understanding the foundations of numerical integration algorithms in nonlinear settings, and to eventually design more efficient algorithms and eventually prove their superiority. The underlying mathematical structures, such as the compositions of noncommuting flows are very similar to those studies in control, as observed in e. g. [23], one of the earlier publications in this area. Some more recent work in this direction is [17,46,72,73,74]. This is also very closely related to the studies of the underlying combinatorial and algebraic structures – arguably [27,28,29,76] are some of the ones most closely related to the subject of this article. There appears to be much potential for a two-way exchange of ideas and results. Geometry Arguably one of the main thrusts will continue to be the use of the chronological calculus for studying the differential geometric underpinnings of systems and control theory. But it is conceivable that such studies may eventually abstract further – just like after the 1930s – to studies of generalizations of algebras that stem from systems on classical manifolds, much in the spirit of modern noncommutative geometry with [19] being the best-known proponent. Algebra and combinatorics In general very little is known about possible ideal and subalgebra structures of chronological and Zinbiel algebras. This work was initiated in [2], but relatively little progress has been made since. (Non)existence of finite dimensional (nilpotent) Zinbiel algebras has been established, but only over a complex field [25,26]. Bases of free chronological algebras are discussed in [2], but many other aspects of this algebraic structure remain unexplored. This also intersects with the aforementioned efforts in foundations of computation – on one side there are

Hopf-algebraic approaches to free Lie algebras [78] and nonlinear control [33,40,41], while the most recent breakthroughs involve dendriform algebras and related subjects [27,28,29]. This is an open field for uncovering the connections between combinatorial and algebraic objects on one side and geometric and systems objects on the other side. The chronological calculus may well serve as a vehicle to elucidate the correspondences. Acknowledgments This work was partially supported by the National Science Foundation through the grant DMS 05-09030 Bibliography 1. Agrachëv A, Gamkrelidze R (1978) Exponential representation of flows and chronological calculus. Math Sbornik USSR (Russian) 107(N4):487–532. Math USSR Sbornik (English translation) 35:727–786 2. Agrachëv A, Gamkrelidze R (1979) Chronological algebras and nonstationary vector fields. J Soviet Math 17:1650–1675 3. Agrachëv A, Gamkrelidze R, Sarychev A (1989) Local invariants of smooth control systems. Acta Appl Math 14:191–237 4. Agrachëv A, Sachkov YU (1993) Local controllability and semigroups of diffeomorphisms. Acta Appl Math 32:1–57 5. Agrachëv A, Sachkov YU (2004) Control Theory from a Geometric Viewpoint. Springer, Berlin 6. Agrachëv A, Sarychev A (2005) Navier Stokes equations: controllability by means of low modes forcing. J Math Fluid Mech 7:108–152 7. Agrachëv A, Vakhrameev S (1983) Chronological series and the Cauchy–Kowalevski theorem. J Math Sci 21:231–250 8. Boltyanski V, Gamkrelidze R, Pontryagin L (1956) On the theory of optimal processes (in Russian). Doklady Akad Nauk SSSR, vol.10, pp 7–10 9. Bourbaki N (1989) Lie Groups and Lie algebras. Springer, Berlin 10. Brockett R (1971) Differential geometric methods in system theory. In: Proc. 11th IEEE Conf. Dec. Cntrl., Berlin, pp 176–180 11. Brockett R (1976) Volterra series and geometric control theory. Autom 12:167–176 12. Bullo F (2001) Series expansions for the evolution of mechanical control systems. SIAM J Control Optim 40:166–190 13. Bullo F (2002) Averaging and vibrational control of mechanical systems. SIAM J Control Optim 41:542–562 14. Bullo F, Lewis A (2005) Geometric control of mechanical systems: Modeling, analysis, and design for simple mechanical control systems. Texts Appl Math 49 IEEE 15. Caiado MI, Sarychev AV () On stability and stabilization of elastic systems by time-variant feedback. ArXiv:math.AP/0507123 16. Campbell J (1897) Proc London Math Soc 28:381–390 17. Casas F, Iserles A (2006) Explicit Magnus expansions for nonlinear equations. J Phys A: Math General 39:5445–5461 18. Chen KT (1957) Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Ann Math 65:163– 178

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control problems with general boundary conditions. J Dyn Control Syst 11:91–123 Sontag E, Wang Y (1992) Generating series and nonlinear systems: analytic aspects, local realizability and i/o representations. Forum Math 4:299–322 Stefani G (1985) Polynomial approximations to control systems and local controllability. In: Proc. 25th IEEE Conf. Dec. Cntrl., pp 33–38, New York Stone M (1932) Linear Transformations in Hilbert Space. Amer Math Soc New York Sussmann H (1974) An extension of a theorem of Nagano on transitive Lie algebras. Proc Amer Math Soc 45:349–356 Sussmann H (1983) Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J Cntrl Opt 21:686–713 Sussmann H (1983) Lie brackets, real analyticity, and geometric control. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control. pp 1–116, Birkhauser Sussmann H (1986) A product expansion of the Chen series. In: Byrnes C, Lindquist A (eds) Theory and Applications of Nonlinear Control Systems. Elsevier, North-Holland, pp 323– 335 Sussmann H (1987) A general theorem on local controllability. SIAM J Control Opt 25:158–194 Sussmann H (1992) New differential geometric methods in nonholonomic path finding. In: Isidori A, Tarn T (eds) Progr Systems Control Theory 12. Birkhäuser, Boston, pp 365– 384 Tretyak A (1997) Sufficient conditions for local controllability and high-order necessary conditions for optimality. A differential-geometric approach. J Math Sci 85:1899–2001 Tretyak A (1998) Chronological calculus, high-order necessary conditions for optimality, and perturbation methods. J Dyn Control Syst 4:77–126 Tretyak A (1998) Higher-order local approximations of smooth control systems and pointwise higher-order optimality conditions. J Math Sci 90:2150–2191 Vakhrameev A (1997) A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems. Dynamic systems 4. J Math Sci 85:2002–2016 Vela P, Burdick J (2003) Control of biomimetic locomotion via averaging theory. In: Proc. IEEE Conf. Robotics and Automation. pp 1482–1489, IEEE Publications, New York Viennot G (1978) Algèbres de Lie Libres et Monoïdes Libres. Lecture Notes in Mathematics, vol 692. Springer, Berlin Visik M, Kolmogorov A, Fomin S, Shilov G (1964) Israil Moiseevich Gelfand, On his fiftieth birthday. Russ Math Surv 19:163– 180 Volterra V (1887) Sopra le funzioni che dipendono de altre funzioni. In: Rend. R Academia dei Lincei. pp 97–105, 141– 146, 153–158 von Neumann J (1932) Mathematische Grundlagen der Quantenmechanik. Grundlehren Math. Wissenschaften 38. Springer, Berlin Zelenko I (2006) On variational approach to differential invariants of rank two distributions. Diff Geom Appl 24:235–259

101

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Control of Non-linear Partial Differential Equations FATIHA ALABAU-BOUSSOUIRA 1 , PIERMARCO CANNARSA 2 1 L.M.A.M., Université de Metz, Metz, France 2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy Article Outline Glossary Definition of the Subject Introduction Controllability Stabilization Optimal Control Future Directions Bibliography Glossary R denotes the real line, Rn the n-dimensional Euclidean space, x  y stands for the Euclidean scalar product of x; y 2 Rn , and jxj for the norm of x. State variables quantities describing the state of a system; in this note they will be denoted by u; in the present setting, u will be either a function defined on a subset of R  Rn , or a function of time taking its values in an Hilbert space H. Space domain the subset of Rn on which state variables are defined. Partial differential equation a differential equation containing the unknown function as well as its partial derivatives. State equation a differential equation describing the evolution of the system of interest. Control function an external action on the state equation aimed at achieving a specific purpose; in this note, control functions they will be denoted by f ; f will be used to denote either a function defined on a subset of R  Rn , or a function of time taking its values in an Hilbert space F. If the state equation is a partial differential equation of evolution, then a control function can be: 1. distributed if it acts on the whole space domain; 2. locally distributed if it acts on a subset of the space domain; 3. boundary if it acts on the boundary of the space domain; 4. optimal if it minimizes (together with the corresponding trajectory) a given cost;

5. feedback if it depends, in turn, on the state of the system. Trajectory the solution of the state equation uf that corresponds to a given control function f . Distributed parameter system a system modeled by an evolution equation on an infinite dimensional space, such as a partial differential equation or a partial integro-differential equation, or a delay equation; unlike systems described by finitely many state variables, such as the ones modeled by ordinary differential equations, the information concerning these systems is “distributed” among infinitely many parameters. 1A denotes the characteristic function of a set A Rn , that is, ( 1 x2A 1A (x) D 0 x 2 Rn n A @ t ; @x i denote partial derivatives with respect to t and xi , respectively. L2 (˝) denotes the Lebesgue space of all real-valued square integrable functions, where functions that differ on sets of zero Lebesgue measure are identified. H01 (˝) denotes the Sobolev space of all real-valued functions which are square integrable together with their first order partial derivatives in the sense of distributions in ˝, and vanish on the boundary of ˝; similarly H 2 (˝) denotes the space of all functions which are square integrable together with their second order partial derivatives. H 1 (˝) denotes the dual of H01 (˝). H n1 denotes the (n  1)-dimensional Hausdorff measure. H denotes a normed spaces over R with norm k  k, as well as an Hilbert space with the scalar product h; i and norm k  k. L2 (0; T; H) is the space of all square integrable functions f : [0; T] ! H; C([0; T]; H) (continuous functions) and H 1 (0; T; H) (Sobolev functions) are similarly defined . Given Hilbert spaces F and H, L(F; H) denotes the (Banach) space of all bounded linear operators  : F ! H with norm kk D supkxkD1 kxk (when F D H, we use the abbreviated notation L(H));  : H ! F denotes the adjoint of  given by h u; i D hu; i for all u 2 H,  2 F. Definition of the Subject Control theory (abbreviated, CT) is concerned with several ways of influencing the evolution of a given system by an external action. As such, it originated in the nine-

Control of Non-linear Partial Differential Equations

teenth century, when people started to use mathematics to analyze the perfomance of mechanical systems, even though its roots can be traced back to the calculus of variation, a discipline that is certainly much older. Since the second half of the twentieth century its study was pursued intensively to address problems in aerospace engineering, and then economics and life sciences. At the beginning, CT was applied to systems modeled by ordinary differential equations (abbreviated, ODE). It was a couple of decades after the birth of CT—in the late sixties, early seventies—that the first attempts to control models described by a partial differential equation (abbreviated, PDE) were made. The need for such a passage was unquestionable: too many interesting applications, from diffusion phenomena to elasticity models, from fluid dynamics to traffic flows on networks and systems biology, can be modeled by a PDE. Because of its peculiar nature, control of PDE’s is a rather deep and technical subject: it requires a good knowledge of PDE theory, a field of enormous interest in its own right, as well as familiarity with the basic aspects of CT for ODE’s. On the other hand, the effort put into this research direction has been really intensive. Mathematicians and engineers have worked together in the construction of this theory: the results—from the stabilization of flexible structures to the control of turbulent flows—have been absolutely spectacular. Among those who developed this subject are A. V. Balakrishnan, H. Fattorini, J. L. Lions, and D. L. Russell, but many more have given fundamental contributions. Introduction The basic examples of controlled partial differential equations are essentially two: the heat equation and the and the wave equation. In a bounded open domain ˝ Rn with sufficiently smooth boundary  the heat equation @ t u D u C f

: in Q T D (0; T)  ˝

(1)

describes the evolution in time of the temperature u(t; x) at any point x of the body ˝. The term u D @2x 1 u C    C @2x n u, called the Laplacian of u, accounts for heat diffusion in ˝, whereas the additive term f represents a heat source. In order to solve the above equation uniquely one needs to add further data, such as the initial distribution u0 and the temperature of the boundary surface  of ˝. The fact that, for any given data u0 2 L2 (˝) and f 2 L2 (Q T ) Eq. (1) admits a unique weak solution uf satisfying the boundary condition : u D 0 on ˙T D (0; T)  

(2)

and the initial condition u(0; x) D u0 (x) 8x D (x1 ; : : : ; x n ) 2 ˝

(3)

is well-known. So is the maximal regularity result ensuring that     u f 2 H 1 0; T; L2 (˝) \ C [0; T]; H01 (˝)   \ L2 0; T; H 2 (˝)

(4)

whenever u0 2 H01 (˝). If problem (1)–(3) possesses a unique solution which depends continuously on data, then we say that the problem is well-posed. Similarly, the wave equation @2t u D u C f

in

QT

(5)

describes the vibration of an elastic membrane (when n D 2) subject to a force f . Here, u(t; x) denotes the displacement of the membrane at time t in x. The initial condition now concerns both initial displacement and velocity: ( u(0; x) D u0 (x) 8x 2 ˝ (6) @ t u(0; x) D u1 (x) : It is useful to treat the above problems as a first order evolution equation in a Hilbert space H u0 (t) D Au(t) C B f (t)

t 2 (0; T) ;

(7)

where f (t) takes its valued in another Hilbert space F, and B 2 L(F; H). In this abstract set-up, the fact that (7) is related to a PDE translates into that the closed linear operator A is not defined on the whole space but only on a (dense) subspace D(A) H, called the domain of A; such a property is often referred to as the unboundedness of A. For instance, in the case of the heat equation (1), H D L2 (˝) D F, and A is defined as ( D(A) D H 2 (˝) \ H01 (˝) (8) Au D u ; 8u 2 D(A) ; whereas B D I. As for the wave equation, since it is a second order differential equation with respect to t, the Hilbert space H should be given by the product H01 (˝)  L2 (˝). Then, problem (5) is turned into the first order equation U 0 (t) D AU(t) C B f (t)

t 2 (0; T) ;

where  UD

u v



 ;

BD

0 I

 ;

F D L2 (˝) :

103

104

Control of Non-linear Partial Differential Equations

Accordingly, A : D(A) H ! H is given by 8   D(A) D H 2 (˝) \ H01 (˝)  H01 (˝) ˆ ˆ < ! ! 0 I v ˆ ˆ UD 8U 2 D(A) ; :AU D A 0 Au where A is taken as in (8). Another advantage of the abstract formulation (7) is the possibility of considering locally distributed or boundary source terms. For instance, one can reduce to the same set-up the equation @ t u D u C 1! f

in Q T ;

(9)

where 1! denotes the characteristic function of an open set ! ˝, or the nonhomogeneus boundary condition of Dirichlet type uD f

on ˙T ;

(10)

or Neumann type @u D f @

on ˙T ;

(11)

where  is the outward unit normal to  . For Eq. (9), B reduces to multiplication by 1! —a bounded operator on L2 (˝); conditions (10) and (11) can also be associated to suitable linear operators B—which, in this case, turn out to be unbounded. Similar considerations can be adapted to the wave equation (5) and to more general problems. Having an efficient way to represent a source term is essential in control theory, where such a term is regarded as an external action, the control function, exercised on the state variable u for a purpose, of which there are two main kinds:  positional: u(t) is to approach a given target in X, or attain it exactly at a given time t > 0;  optimal: the pair (u; f ) is to minimize a given functional. The first criterion leads to approximate or exact controllability problems in time t, as well as to stabilization problems as t ! 1. Here, the main tools will be provided by certain estimates for partial differential operators that allow to study the states that can be attained by the solution of a given controlled equation. These issues will be addressed in Sects. “Controllability” and “Stabilization” for linear evolution equations. Applications to the heat and wave equations will be discussed in the same sections. On the other hand, optimal control problems require analyzing the typical issues of optimizations: existence results, necessary conditions for optimality, sufficient conditions, robustness. Here, the typical problem that has been

successfully studied is the Linear Quadratic Regulator that will be discussed in Sect. “Linear Quadratic Optimal Control”. Control problems for nonlinear partial differential equations are extremely interesting but harder to deal with, so the literature is less rich in results and techniques. Nevertheless, among the problems that received great attention are those of fluid dynamics, specifically the Euler equations @ t u C (u  r)u C r p D 0 and the Navier–Stokes equations @ t u  u C (u  r)u C r p D 0 subject to a boundary control and to the incompressibility condition div u D 0. Controllability We now proceed to introduce the main notions of controllability for the evolution equation (7). Later on in this section we will give interpretations for the heat and wave equations. In a given Hilbert space H, with scalar product h; i and norm k  k, let A: D(A) H ! H be the infinitesimal generator of a strongly continuous semigroup etA , t  0, of bounded linear operators on X. Intu: itively, this amounts to saying that u(t) D e tA u0 is the unique solution of the Cauchy problem ( u0 (t) D Au(t) t  0 u(0) D u0 ; in the classical sense for u0 2 D(A), and in a suitable generalized sense for all u0 2 H. Necessary and sufficient conditions in order for an unbounded operator A to be the infinitesimal generator of a strongly continuous semigroup are given by the celebrated Hille–Yosida Theorem, see, e. g. [99] and [55]. Abstract Evolution Equations Let F be another Hilbert space (with scalar product and norm denoted by the same symbols as for H), the so-called control space, and let B : F ! H be a linear operator, that we will assume to be bounded for the time being. Then, given T > 0 and u0 2 H, for all f 2 L2 (0; T; F) the Cauchy problem ( u0 (t) D Au(t) C B f (t) t  0 (12) u(0) D u0

Control of Non-linear Partial Differential Equations

has a unique mild solution u f 2 C([0; T]; H) given by Z t e(ts)A B f (s) 8t  0 (13) u f (t) D e tA u0 C 0

Note 1 Boundary control problems can be reduced to the same abstract form as above. In this case, however, B in (12) turns out to be an unbounded operator related to suitable fractional powers of A, see, e. g., [22]. For any t  0 let us denote by  t : L2 (0; t; F) ! H the bounded linear operator

Such a characterization is the object of the following theorem. Notice that the above identity and (14) yield 

T u(s) D B e(Ts)A u Theorem 1 System (7) is:

 exactly controllable in time T if and only if there is a constant C > 0 such that Z T   tA 2 B e u dt  Ckuk2 8u 2 H ; (15) 0

Z

t

t f D

e(ts)A B f (s) ds

8 f 2 L2 (0; t; F) :

(14)

0

The attainable (or reachable) set from u0 at time t, A(u0 ; t) is the set of all points in H of the form u f (t) for some control function f , that is : A(u0 ; t) D e tA u0 C  t L2 (0; t; F) :

 null controllable in time T if and only if there is a constant C > 0 such that Z T     tA 2 B e u dt  C eTA u2 8u 2 H ; (16) 0

 approximately controllable in time T if and only if, for every u 2 H, 

We introduce below the main notions of controllability for (7). Let T > 0. Definition 1 System (7) is said to be:  exactly controllable in time T if A(u0 ; T) D H for all u0 2 H, that is, if for all u0 ; u1 2 H there is a control function f 2 L2 (0; T; F) such that u f (T) D u1 ;  null controllable in time T if 0 2 A(u0 ; T) for all u0 2 H, that is, if for all u0 2 H there is a control function f 2 L2 (0; T; F) such that u f (T) D 0;  approximately controllable in time T if A(u0 ; T) is dense in H for all u0 2 H, that is, if for all u0 ; u1 2 H and for any " > 0 there is a control function f 2 L2 (0; T; F) such that ku f (T)  u1 k < ". Clearly, if a system is exactly controllable in time T, then it is also null and approximately controllable in time T. Although these last two notions of controllability are strictly weaker than strong controllability, for specific problems—like when A generates a strongly continuous group—some of them may coincide. Since controllability properties concern, ultimately, the range of the linear operator T defined in (14), it is not surprising that they can be characterized in terms of the adjoint operator T : H ! L2 (0; T; F), which is defined by Z 0

T

˝

8s 2 [0; T] :

T u(s); f (s)ids D hu0 ; T f i 8u 2 H ; 8 f 2 L2 (0; T; F) :

B e tA u D 0

t 2 [0; T] a.e. H) u D 0 :

(17)

To benefit the reader who is more familiar with optimization theory than abstract functional analysis, let us explain, by a variational argument, why estimate (16) implies null controllability. Consider, for every " > 0, the penalized problem ˚ min J" ( f ) : f 2 L2 (0; T; H) ; where 1 J" ( f ) D 2

Z

T

k f (t)k2 dt C

0

1 ku f (T)k2 2" 8 f 2 L2 (0; T; H) :

Since J" is strictly convex, it admits a unique minimum point f" . Set u" D u f" . Recalling (13) we have, By Fermat’s rule, 0 D J"0 ( f" )g D

Z

T 0

h f" (t); g(t)i dt

1 C hu" (T); T gi "

8g 2 L2 (0; T; H) :

(18)

Therefore, passing to the adjoint of T , Z

T 0

D

f" (t) C

E  1  T u" (T) (t); g(t) dt D 0 " 8g 2 L2 (0; T; H) ;

105

106

Control of Non-linear Partial Differential Equations

Taking

whence, owing to (14), f" (t) D 

 1   u" (T) (t) D B v" (t) " T 8t 2 [0; T] ;

H D L2 (˝) D F ; (19)

 : where v" (t) D "1 e(Tt)A u" (T) is the solution of the dual problem ( t 2 [0; T] v 0 C A v D 0

v(T) D 1" u" (T) : Z

T

0

1 k f" (t)k2 dt C ku" (T)k2  Cku0 k2 " 8" > 0

0

hence Z 0

T

  2 B v"  dt D hu0 ; v" (0)i :

Now, apply estimate (16) with u D  v" (T  t) D e tA u ""(T) to obtain Z 0

T

(22)

on ˙T x2˝

8i D 1; : : : ; n :

(23)

Notice that the above property already suffices to explain why the heat equation cannot be exactly controllable: it is impossible to attain a state u1 2 L2 (˝) which is not compatible with (23). On the other hand, null controllability holds true in any positive time.

(21)

“

2

Z

j f j dxdt  C T u " (T) "

in Q T

Theorem 2 Let T > 0 and let ! be an open subset of ˝ such that ! ˝. Then the heat equation (9) with homogeneous Dirichlet boundary conditions is null controllable in time T, i. e., for every initial condition u0 2 L2 (˝) there is a control function f 2 L2 (Q T ) such that the solution uf of (22) satisfies u f (T; )  0. Moreover,



 2 d hu" ; v" i C B v"  dt D 0 ; dt

1 ku" (T)k2 C "

8 ˆ <@ t u D u C 1! f uD0 ˆ : u(0; x) D u0 (x)

@x i u 2 L2 (Q T )

So, T

and A as in (8), one obtains that, for any u0 2 L2 (˝) and f 2 L2 (Q T ), the initial-boundary value problem

(20)

for some positive constant C. Indeed, observe that, in view of (19), (˝ ˛ u"0  Au" C BB v" ; v" D 0 ; u" (0) D u0 ˝ 0 ˛ v" C A v" ; u" D 0 ; v" (T) D 1" u" (T) : Z

8 f 2 L2 (˝)

has a unique mild solution u f 2 C([0; T]; L2 (˝)). Moreover, multiplying both sides of equation (9) by u and integrating by parts, it is easy to see that

It turns out that 1 2

B f D 1! f

QT

and note that

   B v" (t)2 dt  C kv" (0)k2

for some positive constant C. Hence, (20) follows from (21) and (19). Finally, from (20) one deduces the existence of a weakly convergent subsequence f" j in L2 (0; T; F). Then, called f 0 the weak limit of f" j , u" j (t) ! u f0 (t) for all t 2 [0; T]. So, owing to (20), u f0 (T) D 0. Heat Equation It is not hard to see that the heat equation (9) with Dirichlet boundary conditions (2) fails to be exactly controllable. On the other hand, one can show that it is null controllable in any time T > 0, hence approximately controllable. Let ! be an open subset of ˝ such that ! ˝.

˝

ju0 j2 dx

for some positive constant CT . The above property is a consequence of the abstract result in Theorem 1 and of concrete estimates for solutions of parabolic equations. Indeed, in order to apply Theorem 1 one has to translate (16) into an estimate for the heat operator. Now, observing that both A and B are self-adjoint, one promptly realizes that (16) reduces to Z

T 0

Z !

jv(t; x)j2 dxdt  C

Z ˝

jv(T; x)j2 dx

for every solution v of the problem ( @ t v D v in Q T vD0

on ˙T :

(24)

(25)

Estimate (24) is called an observability inequality for the heat operator for obvious reasons: problem (25) is not

Control of Non-linear Partial Differential Equations

well-posed since the initial condition is missing. Nevertheless, if, “observing” a solution v of such a problem on the “small” cylinder (0; T)  !, you find that it vanishes, then you can conclude that v(T; )  0 in the whole domain ˝. Thus, v(0; )  0 by backward uniqueness. In conclusion, as elegant as the abstract approach to null controllability may be, one is confronted by the difficult task of proving observability estimates. In fact, for the heat operator there are several ways to prove inequality (24). One of the most powerful, basically due to Fursikov and Imanuvilov [65], relies on global Carleman estimates. Compared to other methods that can be used to derive observability, such a technique has the advantage of applying to second order parabolic operators with variable coefficients, as well as to more general operators. Global Carleman estimates are a priori estimates in weighted norms for solutions of the problem ( @ t v D v C f in Q T (26) vD0 on ˙T : regardless of initial conditions. The weight function is usually of the form   2rkk : 1;˝  er(x) (t; x) 2 Q T ; (27) r (t; x) D (t) e where r is a positive constant,  is a given function in C 2 (˝) such that r(x) ¤ 0 8x 2 ˝ ;

(28)

and : (t) D

1 t(T  t)

0< t< T:

Note that > 0; r

> 0;

(t) ! 1 r (t; x)

are positive constants r; s0 and C such that, for any s > s0 , “ s3 3 (t)jv(t; x)j2 e2s r dxdt QT

“

2 2s

j f (t; x)j e

C QT

Z 



Z r

T

dxdt C Cs

ˇ ˇ2 ˇ @v ˇ @ ˇ ˇ (x) ˇ (t; x)ˇ e2s ˇ @ ˇ @

(t)dt 0

r

dH n1 (x)

(29)

It is worth underlying that, thanks to the singular behavior of near 0 and T, the above result is independent of the initial value of v. Therefore, it can be applied, indifferently, to any solution of (26) as well as to any solution of the backward problem ( @ t v C v D f in Q T vD0

on ˙T :

Moreover, inequality (29) can be completed adding first and second order terms to its left-hand side, each with its own adapted power of s and . Instead of trying to sketch the proof of Theorem 3, which would go beyond the scopes of this note, it is interesting to explain how it can be used to recover the observability inequality (24), which is what is needed to show that the heat equation is null controllable. The reasoning—not completely straightforward—is based on the following topological lemma, proved in [65]. Lemma 1 Let ˝ Rn be a bounded domain with boundary  of class Ck , for some k  2, and let ! ˝ be an open set such that ! ˝. Then there is function  2 C k (˝) such that ( (x) < 0 8x 2  (i) (x) D 0 and @ @ (30) (ii) fx 2 ˝jr(x) D 0g ! :

t ! 0;T

!1

t # 0;t " T :

Using the above notations, a typical global Carleman estimate for the heat operator is the following result obtained in [65]. Let us denote by (x) the outword unit normal to  at a point x 2  , and by @ (x) D r(x)  (x) @ the normal derivative of  at x. Theorem 3 Let ˝ be a bounded domain of Rn with boundary of class C2 , let f 2 L2 (Q T ), and let  be a function satisfying (28). Let v be a solution of (26). Then there

Now, given a solution v of (25) and an open set ! such that ! ˝, let ! 0

! 00

! be subdomains with smooth boundary. Then the above lemma ensures the existence of a function  such that fx 2 ˝jr(x) D 0g ! 0 : : “Localizing” problem (25) onto ˝ 0 D ˝ n ! 0 by a cutoff function  2 C 1 (Rn ) such that 0    1;

1

on Rn n ! 00 ;

0

that is, taking w D v, gives ( : @ t w D w C h in Q T0 D (0; T)  ˝ 0 w(t; ) D 0

on @˝ 0 D @˝ [ @! 0 ;

on ! 0 ;

(31)

107

108

Control of Non-linear Partial Differential Equations

with h :D v C 2r  ru. Since r ¤ 0 on ˝ 0 , Theorem 3 can be applied to w on Q T0 to obtain s3

“

3 jwj2 e2s

Q 0T

C Z

2 2s

Q 0T

jhj e

Z

T

C Cs

dt 0

Z



Z

T

C Cs

dt

r

dxdt

r

dxdt

ˇ ˇ2 ˇ @w ˇ 2s r ˇ ˇ e d H n1 ˇ @ ˇ ˇ ˇ @ ˇˇ @w ˇˇ2 2s r e d H n1 @ ˇ @ ˇ “ C jhj2 e2s r dxdt

for s sufficiently large. On the other hand, for any 0 < T0 < T1 < T, “

3 jwj2 e2s

Q 0T

 s3

Z

Z

T1

dt

˝n!

T0

r

dxdt

3 jwj2 e2s r dxdt Z

Z

T1



dt

˝n!

T0

jvj2 dx

Therefore, recalling the definition of h, Z

T1

dt Z

˝n!

Z

T

C

dt Z

0

dt 0

! 00 n! 0

Z

T

C

jvj2 dx  C

!

“

jhj2 e2s

Q 0T

Z

r

dxdt

dt

Z dt

0

! 00 n! 0

Z

jrvj2 e2s

2 2s

jrvj e

! 00 n! 0

r

r

Z

Z

T

dt 0

to conclude that Z 2T/3 Z Z dt jvj2 dx  C ˝n!

dt

!

jvj2 dx

0

!

dt

Z

Z

2T/3

dt T/3

Z

˝ T

v 2 (t; x) dx Z dt

0

!

v 2 (t; x) dx ;

Wave Equation Compared to the heat equation, the wave equation (5) exhibits a quite different behavior from the point of view of exact controllability. Indeed, on the one hand, there is no obstruction to exact controllability since no regularizing effect is connected with wave propagation. On the other hand, due to the finite speed of propagation, exact controllability cannot be expected to hold true in arbitrary time, as null controllability does for the heat equation. In fact, a typical result that holds true for the wave equation is the following, where a boundary control of Dirichlet type acts on a part 1  , while homogeneous boundary conditions are imposed on 0 D  n 1 : 8 2 in Q T ˆ <@ t u D u u D f 11 on ˙T ˆ : u(0; x) D u0 (x) ; @ t u(0; x) D u1 (x) x 2 ˝

jvj2 e2s

r

dx ;

u0 2L2 (˝) ;

(32)

u1 2 H 1 (˝)

f 2L2 (0; T; L2 ( )) u 2C([0; T]; L2 (˝)) \ C 1 ([0; T]; H 1 (˝)) : Theorem 4 Let ˝ be a bounded domain of Rn with boundary of class C2 and suppose that, for some point x0 2 Rn , ( (x  x0 )  (x) > 0 8x 2 1 (x  x0 )  (x)  0 8x 2 0 : Let

Z

T

3 v (T; x) dx  T ˝ 2

which is exactly (24).

dx :

dx

C

T/3

Z

T

Observe that problem (32) is well-posed taking

0

Now, fix T0 D T/3 ; T1 D 2T/3 and use Caccioppoli’s inequality (a well-known estimate for solution of elliptic and parabolic PDE’s) T

Z 0

3  (1 C C) T

T



Z

Z

 2 2 2 jr j v C jrj2jrvj2 e2s r dx

jvj2 dx C C

˝

jvj2 dx  (1 C C)

for some constant C. Then, the R dissipativity of the heat operator (that is, the fact that ˝ jv(t; x)j2 dx is decreasing with respect to t) implies that

Q 0T

T0

Z

2T/3 T/3

@ @

@! 0

0

Z

Z

dt “

s3

or

!

jvj2 dx

R D sup jx  x0 j : x2˝

Control of Non-linear Partial Differential Equations

If T > 2R, then, for all (u0 ; u1 ); (v0 ; v1 ) 2 L2 (˝)  H 1 (˝) there is a control function f 2 L2 (0; T; L2 ( )) such that the solution uf of (32) satisfies u f (T; x) D v0 (x) ;

@ t u f (T; x) D v1 (x) :

As we saw for abstract evolution equations, the above exact controllability property is proved to be equivalent to an observability estimate for the dual homogeneous problem using, for instance, the Hilbert Uniqueness Method (HUM) by J.-L. Lions [86]. Bibliographical Comments The literature on controllability of parabolic equations and related topics is so huge, that no attempt to provide a comprehensive account of it would fit within the scopes of this note. So, the following comments have to be taken as a first hint for the interested reader to pursue further bibliographical research. The theory of exact controllability for parabolic equations was initiated by the seminal paper [58] by Fattorini and Russell. Since then, it has experienced an enormous development. Similarly, the multiplier method to obtain observability inequalities for the wave equation was developed in [17,73,74,77,86]. Some fundamental early contributions were surveyed by Russell [108]. The next essential progress was made in the work by Lebeau and Robbiano [83] and then by Fursikov and Imanuvilov in a series of papers. In [65] one can find an introduction to global Carleman estimates, as well as applications to the controllability of several ODE’s. In particular, the presentation of this paper as for observability inequalities and Carleman estimates for the heat operator is inspired by the last monograph. General perspectives for the understanding of global Carleman estimates and their applications to unique continuation and control problems for PDE’s can be found in the works by Tataru [113,114,115,116]. Usually, the above approach requires coefficients to be sufficiently smooth. Recently, however, interesting adaptations of Carleman estimates to parabolic operators with discontinuous coefficients have been obtained in [21,82]. More recently, interest has focussed on control problems for nonlinear parabolic equations. Different approaches to controllability problems have been proposed in [57] and [44]. Then, null and approximate controllability results have been improved by Fernandez–Cara and Zuazua [61,62]. Techniques to produce insensitizing controls have been developed in [117]. These techniques have been successfully applied to the study of Navier–Stokes equations by several authors, see e. g. [63].

Fortunately, several excellent monographs are now available to help introduce the reader to this subject. For instance, the monograph by Zabczyk [121] could serve as a clean introduction to control and stabilization for finiteand infinite-dimensional systems. Moreover, [22,50,51], as well as [80,81] develop all the basic concepts of control and system theory for distributed parameter systems with special emphasis on abstract formulation. Specific references for the controllability of the wave equation by HUM can be found in [86] and [74]. More recent results related to series expansion and Ingham type methods can be found in [75]. For the control of Navier–Stokes equations the reader is referred to [64], as well as to the book by Coron [43], which contains an extremely rich collection of classical results and modern developments.

Stabilization Stabilization of flexible structures such as beams, plates, up to antennas of satellites, or of fluids as, for instance, in aeronautics, is an important part of CT. In this approach, one wants either to derive feedback laws that will allow the system to autoregulate once they are implemented, or study the asymptotic behavior of the stabilized system i. e. determine whether convergence toward equilibrium states as times goes to infinity holds, determine its speed of convergence if necessary or study how many feedback controls are required in case of coupled systems. Different mathematical tools have been introduced to handle such questions in the context of ODE’s and then of PDE’s. Stabilization of ODE’s goes back to the work of Lyapunov and Lasalle. The important property is that trajectories decay along Lyapunov functions. If trajectories are relatively compact in appropriate spaces and the system is autonomous, then one can prove that trajectories converge to equilibria asymptotically. However, the construction of Lyapunov functions is not easy, in general. This section will be concerned with some aspects of the stabilization of second order hyperbolic equations, our model problem being represented by the wave equation with distributed damping 8 ˆ <@ t t u  u C a(x)u t D 0 in ˝  R ; uD0 on ˙ D (0; 1)   (33) ˆ : 0 1 on ˝ ; (u; @ t u)(0) D (u ; u ) in a bounded domain ˝ Rn with a smooth boundary  . For n D 2, u(t; x) represents the displacement of point x of the membrane at time t. Therefore, equation (33) describes an elastic system. The energy of such

109

110

Control of Non-linear Partial Differential Equations

a system is given by E(t) D

1 2

Z ˝

Geometrical Aspects

 ju t (t; x)j2 C jru(t; x)j2 dx :

When a  0, the feedback term a(x)u t models friction: it produces a loss of energy through a dissipation phenomenon. More precisely, multiplying the equation in (33) by ut and integrating by parts on ˝, it follows that E 0 (t) D 

Z ˝

a(x)ju t j2 dx  0 ;

8t  0 :

(34)

On the other hand, if a  0, then the system is conservative, i. e., E(t) D E(0) for all t  0. Another well-investigated stabilization problem for the wave equation is when the feedback is localized on a part  0 of the boundary  , that is, 8 ˆ @ t t u  u D 0 ˆ ˆ ˆ < @u C u D 0

in ˝  R

on ˙0 D (0; 1)  0 t @ ˆ u D 0 on ˙1 D (0; 1)  ( n 0 ) ˆ ˆ ˆ :(u; @ u)(0) D (u0 ; u1 ) t

(35) In this case, the dissipation relation (34) takes the form 0

E (t) D 

Z 0

ju t j2 dH n1  0 ;

8t  0 :

In many a situation—such as to improve the quality of an acoustic hall—one seeks to reduce vibrations to a minimum: this is why stabilization is an important issue in CT. We note that the above system has a unique stationary solution—or, equilibrium—given by u  0. Stabilization theory studies all questions related to the convergence of solutions to such an equilibrium: existence of the limit, rate of convergence, different effects of nonlinearities in both displacement and velocity, effects of geometry, coupled systems, damping effects due to memory in viscoelastic materials, and so on. System (33) is said to be:  strongly stable if E(t) ! 0 as t ! 1;  (uniformly) exponentially stable if E(t)  Ce˛t E(0) for all t  0 and some constants ˛ > 0 and C  0, independent of u0 ; u1 . This note will focus on some of the above issues, such as geometrical aspects, nonlinear damping, indirect damping for coupled systems and memory damping.

A well-known property of the wave equation is the socalled finite speed of propagation, which means that, if the initial conditions u0 ; u1 have compact support, then the support of u(t; ) evolves in time at a finite speed. This explains why, for the wave equation, the geometry of ˝ plays an essential role in all the issues related to control and stabilization. The size and localization of the region in which the feedback is active is of great importance. In this paper such a region, denoted by !, is taken as a subset of ˝ of positive Lebesgue measure. More precisely, a is assumed to be continuous on ˝ and such that a0

on

˝

and

a  a0

on

!;

(36)

for some constant a0 > 0. In this case, the feedback is said to be distributed. Moreover, it is said to be globally distributed if ! D ˝ and locally distributed if ˝ n ! has positive Lebesgue measure. Two main methods have been used or developed to study stabilization, namely the multiplier method and microlocal analysis. The one that gives the sharpest results is based on microlocal analysis. It goes back to the work of Bardos, Lebeau and Rauch [17], giving geodesics sufficient conditions on the region of active control for exact controllability to hold. These conditions say that each ray of geometric optics should meet the control region. Burq and Gérard [25] showed that these results hold under weaker regularity assumptions on the domain and coefficients of the operators (see also [26,27]). These geodesics conditions are not explicit, in general, but they allow to get decay estimates of the energy under very general hypotheses. The multiplier method is an explicit method, based on energy estimates, to derive decay rates (as well as observability and exact controllability results). For boundary control and stabilization problems it was developed in the works of several authors, such as Ho [38,73], J.-L. Lions [86], Lasiecka–Triggiani, Komornik–Zuazua [76], and many others. Zuazua [123] gave an explicit geometric condition on ! for a semilinear wave equation subject to a locally distributed damping. Such a condition was then relaxed K. Liu [87] (see also [93]) who introduced the so-called piecewise multiplier method. Lasiecka and Triggiani [80,81] introduced a sharp trace regularity method which allows to estimate boundary terms in energy estimates. There also exist intermediate results between the geodesics conditions of Bardos–Lebeau–Rauch and the multiplier method, obtained by Miller [95] using differentiable escape functions.

Control of Non-linear Partial Differential Equations

Zuazua’s multiplier geometric condition can be described as follows. If a subset O of ˝ is given, one can define an "-neighborhood of O in ˝ as the subset of points of ˝ which are at distance at most " of O. Zuazua proved that if the set ! is such that there exists a point x0 2 Rn —an observation point—for which ! contains an "neighborhood of  (x 0 ) D fx 2 @˝ ; (x  x 0 )  (x)  0g, then the energy decays exponentially. In this note, we refer to this condition as (MGC). If a vanishes for instance in a neighborhood of the two poles of a ball ˝ in Rn , one cannot find an observation point x0 such that (MGC) holds. K. Liu [87] (see also [93]) introduced a piecewise multiplier method which allows to choose several observation points, and therefore to handle the above case. Introduce disjoint lipschitzian domains ˝ j of ˝, j D 1; : : : ; J, and observation points x j 2 R N , j D 1; : : : ; J and define

j (x j ) D fx 2 @˝ j ; (x  x j )   j (x)  0g Here  j stands for the unit outward normal vector to the boundary of ˝ j . Then the piecewise multiplier geometrical condition for ! is:    (PWMGC) !  N" [ JjD1 j (x j ) [ ˝n [ JjD1 ˝ j It will be denoted by (PWMGC) condition in the sequel. Assume now that a vanishes in a neighborhood of the two poles of a ball in Rn . Then, one can choose two subsets ˝ 1 and ˝ 2 containing, respectively, the two regions where a vanishes and apply the piecewise multiplier method with J D 2 and with the appropriate choices of two observation points and ". The multiplier method consists of integrating by parts expressions of the form Z TZ  t

˝

@2t u

otherwise. One can then prove that the energy satisfies an estimate of the form Z

T

E(s) ds t

Z

Z

T

 cE(t) C

2

˝

t

 u C a(x)u t Mu dx dt D 0 80  t  T ;

where u stands for a (strong) solution of (33), with an appropriate choice of Mu. Multipliers have generally the form Mu D (m(x)  ru C c u) (x) ; where m depends on the observation points and is a cut-off function. Other multipliers of the form Mu D 1 (ˇu), where ˇ is a cut-off function and 1 is the inverse of the Laplacian operator with homogeneous Dirichlet boundary conditions, have also used. The geometric conditions (MGC) or (PWMGC) serve to bound above by zero terms which cannot be controlled

!

ju t j

2

ds

8t  0 :

(37)

Once this estimate is proved, one can use the dissipation relation to prove that the energy satisfies integral inequalities of Gronwall type. This is the subject of the next section. Decay Rates, Integral Inequalities and Lyapunov Techniques The Linear Feedback Case Using the dissipation relation (34), one has Z TZ ˝

t

aju t j2 dx ds 

Z

T

E 0 (s) ds  E(t)

t

8 0  t  T: On the other hand, thanks to assumption (36) on a Z TZ !

t

u2t dx ds 

1 a0

Z TZ ˝

t



aju t j2 dx ds 1 E(t) a0

80  t  T :

By the above inequalities and (37), E satisfies Z

T

E(s) ds  cE(t) ;



a(x)ju t j C

!

Z

80  t  T :

(38)

t

Since E is a nonincreasing function and thanks to this integral inequality, Haraux [71] (see also Komornik [74]) proved that E decays exponentially at infinity, that is  E(t)  E(0) exp 1  t/c) ;

8t  c :

This proof is as follows. Define Z

1

(t) D exp(t/c)

E(s) ds

8t  0 :

t

Thanks to (38)  is nonincreasing on [0; 1), so that Z

1

(t)  (0) D

E(s) ds : 0

(39)

111

112

Control of Non-linear Partial Differential Equations

Using once again (38) with t D 0 in this last inequality and the definition of , one has Z 1 E(s) ds  cE(0) exp(t/c) 8t  0 :

one can prove that the energy E of solutions satisfies the following inequality for all 0  t  T Z

t

E (pC1)/2 (s) dt

t

Since E is a nonnegative and nonincreasing function Z t Z 1 cE(t)  E(s) ds  E(s) ds tc

T

 cE (pC1)/2 (t) C c

Z

T t

tc

 cE(0) exp((t  c)/c) ; so that (39) is proved. An alternative method is to introduce a modified (or perturbed) energy E" which is equivalent to the natural one for small values of the parameter " as in Komornik and Zuazua [76]. Then one shows that this modified energy satisfies a differential Gronwall inequality so that it decays exponentially at infinity. The exponential decay of the natural energy follows then at once. In this case, the modified energy is indeed a Lyapunov function for the PDE. The natural energy cannot be in general such a Lyapunov function due to the finite speed of propagation (consider initial data which have compact support compactly embedded in ˝n!). There are also very interesting approaches using the frequency domain approach, or spectral analysis such as developed by K. Liu [87] Z. Liu and S. Zheng [88]. In the sequel, we concentrate on the integral inequality method. This method has been generalized in several directions and we present in this note some results concerning extensions to  nonlinear feedback  indirect or single feedback for coupled system  memory type feedbacks Generalizations to Nonlinear Feedbacks Assume now that the feedback term a(x)u t in (33) is replaced by a nonlinear feedback a(x) (u t ) where is a smooth, increasing function satisfying v (v)  0 for v 2 R, linear at 1 and with polynomial growth close to zero, that is: (v) D jvj p for jvj  1 where p 2 (1; 1). Assume moreover that ! satisfies Zuazua’s multiplier geometric condition (MGC) or Liu’s piecewise multiplier method (PWMGC). Then using multipliers of the space and time variables defined as E(s)(p1)/2 Mu(x) where Mu(x) are multipliers of the form described in section 5.1 and integrating by parts expressions of the form Z T E(s)(p1)/2 t Z   2  @ t u  u C a(x) (u t ) Mu(x) dx ds D 0 ; ˝

E (p1)/2 (s) Z  Z 2 2

(u t ) C ju t j :  ˝

!

One can remark than an additional multiplicative weight in time depending on the energy has to be taken. This weight is E (p1)/2 . Then as in the linear case, but in a more involved way, thanks to the dissipation relation E 0 (t) D 

Z ˝

a(x)u t (u t ) ;

(40)

one can prove that E satisfies the following nonlinear integral inequality Z

T

E (pC1)/2 (s) ds  cE(t) ;

80  t  T :

t

Thanks to the fact that E is nonincreasing, a wellknown result by Komornik [74] shows that E is polynomially decaying, as t 2/(p1) at infinity. The above type results have been obtained by many authors under weaker form (see also [40,41,71,98,122]). Extensions to nonlinear feedbacks without growth conditions close to zero have been studied by Lasiecka and Tataru [78], Martinez [93], W. Liu and Zuazua [89], Eller Lagnese and Nicaise [56] and Alabau–Boussouira [5]. We present the results obtained in this last reference since they provide optimal decay rates. The method is as follows. Define respectively the linear and nonlinear kinetic energies (R

R!

˝

ju t j2 dx ; ja(x) (u t )j2 dx ;

and use a weight function in time f (E(s)) which is to be determined later on in an optimal way. Integrating by parts expressions of the form Z

Z

T

f (E(s)) t

 ˝

 @2t uuCa(x) (u t ) Mu(x) dx ds D 0 ;

one can prove that the energy E of solutions satisfies the

Control of Non-linear Partial Differential Equations

following inequality for all 0  t  T Z

T t

Z T E(s) f (E(s)) ds  c f (E(t)) C c f (E(s)) t Z  Z 2 2  ja(x) (u t )j C ju t j : ˝

!

types of the feedback, one can assume convexity of H only in a neighborhood of 0. One can prove from (41) that there exists an (explicit) T0 > 0 such that for all initial data, E satisfies the following nonlinear integral inequality (41)

The difficulty is to determine the optimal weight under general growth conditions on the feedback close to 0, in particular for cases for which the feedback decays to 0 faster than polynomials. Assume now that the feedback satisfies g(jvj)  j (v)j  Cg 1 (jvj) ;

8jvj  1 ;

(42)

Z

E(s) f (E(s)) ds  T0 E(t)

then, define a function F as follows: 8  ˆ < Hˆ (y) if y 2 (0; C1) ; F(y) D y ˆ :0 if y D 0 ; ˆ that is where Hˆ  stands for the convex conjugate of H, ˆ ˆ  (y) D sup fx y  H(x)g : H

Z ˝t

where ˇ is of the form max(1 ; 2 E(0)), 1 and 2 being explicit positive constants. One can prove that the above formulas make sense, and in particular that F is invertible and smooth. More precisely, F is twice continuously differentiable strictly increasing, one-to-one function from [0; C1) onto [0; r02 ). Note that since the feedback is supposed to be linear at infinity, if one wants to obtain results for general growth

ja(x) (u t )j2 dx  1 (t)Hˆ 1   Z 1  a(x)u t (u t ) dx

1 (t) ˝

In a similar way, one proves that Z

ju t j2 dx  2 (t)Hˆ 1



1

2 (t)



Z ˝

a(x)u t (u t ) dx

where ˝ t and ! t are time-dependent sets of respective Lebesgue measures 1 (t) and 2 (t) on which the velocity u t (t; x) is sufficiently small. Using the above two estimates, together with the linear growth of at infinity, one proves Z

Z

T

f (E(s)) t

˝

Z

T

 t

ja(x) (u t )j2 C

Z !

ju t j2



  Z 1 f (E(s))Hˆ 1 a(x)u t (u t ) dx c ˝

Using then Young’s inequality, together with the dissipation relation (40) in the above inequality, one obtains Z

Z

T

f (E(s)) t

x2R

Then the optimal weight function f is determined in the following way   f (s) D F 1 (s/2ˇ) s 2 0; 2ˇr02 ;

(43)

This inequality is proved thanks to convexity arguˆ one can ments as follows. Thanks to the convexity of H, use Jensen’s inequality and (42), so that

!t

Moreover, is assumed to have a linear growth at infinity. We define the optimal weight function f as follows. We first extend H to a function Hˆ define on all R (  H(x) if x 2 0; r02 ; ˆ H(x) D C1 otherwise ;

80  t  T :

t

where g is continuously differentiable on R strictly increasing with g(0) D 0 and 8 2 ˆ < g 2 C ([0; r0 ]) ; r0 sufficiently small ;  p p H(:) D :g( :) is strictly convex on 0; r02 ; ˆ : g is odd :

T

˝ T

Z

ja(x) (u t )j2 C

Hˆ

 C1

 ?

Z !

ju t j2



 f (E(s) ds C C2 E(t) ;

(44)

t

where C i > 0 i D 1; 2 is a constant independent of the initial data. Using the dissipation relation (40) in the above inequality, this gives for all 0  t  T Combining this last inequality with (41) gives Z

Z

T

T

E(s) f (E(s)) ds  ˇ t

  ˆ ? f (E(s) ds C C2 E(t) (H)

t

where ˇ is chosen of the form max(1 ; 2 E(0)), 1 and 2 being explicit positive constants to guarantee that the argument E of f stays in the domain of definition of f .

113

114

Control of Non-linear Partial Differential Equations

Thus (43) is proved, thanks to the fact that the weight function has been chosen so that ˆ ? ( f (E(s)) D ˇH

1 E(s) f (E(s)) 2

80  s:

Therefore E satisfies a nonlinear integral inequality with a weight function f (E) which is defined in a semi-explicit way in general cases of feedback growth. The last step is to prove that a nonincreasing and nonnegative absolutely continuous function E satisfying a nonlinear integral inequality of the form (43) is decaying at infinity, and to establish at which rate this holds. For this, one proceeds as in [5]. Let  > 0 and T0 > 0 be fixed given real numbers and F be a strictly increasing function from [0; C1) on [0; ), with F(0) D 0 and lim y!C1 F(y) D . For any r 2 (0; ), we define a function K r from (0; r] on [0; C1) by Z r dy ; (45) K r () D 1  yF (y) and a function r which is a strictly increasing function from [ F 11 (r) ; C1) onto [ F 11 (r) ; C1) by r (z)

1 D z C K r (F( ))  z; z

8z 

1 ; F 1 (r)

(46)

One can prove that if E is a nonincreasing, absolutely continuous function from [0; C1) on [0; C1), satisfying 0 < E(0) <  and the inequality Z T E(s)F 1 (E(s)) ds  T0 E(S) ; 8 0  t  T ; (47) t

then E satisfies the following estimate: ! T0 1 ; 8t  1 ; E(t)  F 1 ( t ) F (r) r T

(48)

0

where r is any real such that Z C1 1 E()F 1 (E()) d  r   : T0 0 Thus, one can apply the above result to E with  D r02 and show that lim t ! C1E(t) D 0, the decay rate being given by estimate (48). If g is polynomial close to zero, one gets back that the 2 energy E(t) decays as t p1 at infinity. If g(v) behaves as exp(1/jvj) close to zero, then E(t) decays as 1/(ln(t))2 at infinity. The usefulness of convexity arguments has been first pointed out by Lasiecka and Tataru [78] using Jensen’s

inequality and then in different ways by Martinez [93] (the weight function does not depend on the energy) and W. Liu and Zuazua [89] and Eller Lagnese and Nicaise [56]. Optimal decay rates have been obtained by Alabau–Boussouira [5,6] using a weight function determined through the theory of convex conjugate functions and Young’s (named also as Fenchel–Moreau’s) inequality. This argument was also used by W. Liu and Zuazua [89] in a slightly different way and combined to a Lyapunov technique. Optimality of estimates in [5] is proved in one-dimensional situation and for boundary dampings applying optimality results of Vancostenoble [119] (see also Martinez and Vancostenoble [118]). Indirect Damping for Coupled Systems Many complex phenomena are modeled through coupled systems. In stabilizing (or controlling) energies of the vector state, one has very often access only to some components of this vector either due to physical constraints or to cost considerations. In this case, the situation is to stabilize a full system of coupled equation through a reduced number of feedbacks. This is called indirect damping. This notion has been introduced by Russell [109] in 1993. As an example, we consider the following system: (

@2t u  u C @ t u C ˛v D 0 @2t v  v C ˛u D 0 in ˝  R ;

uD0Dv

on @˝  R :

(49)

Here, the first equation is damped through a linear distributed feedback, while no feedback is applied to the second equation. The question is to determine if this coupled system inherits any kind of stability for nonzero values of the coupling parameter ˛ from the stabilization of the first equation only. In the finite dimensional case, stabilization (or control) of coupled ODE’s can be analyzed thanks to a powerful rank type condition named Kalman’s condition. The situation is much more involved in the case of coupled PDE’s. One can show first show that the above system fails to be exponentially stable (see also [66] for related results). More generally, one can study the stability of the system ( u00 C A1 u C Bu0 C ˛v D 0 (50) v 00 C A2 v C ˛u D 0 in a separable Hilbert space H with norm jj, where A1 ; A2 and B are self-adjoint positive linear operators in H. Moreover, B is assumed to be a bounded operator. So, our analysis applies to systems with internal damping supported

Control of Non-linear Partial Differential Equations

in the whole domain ˝ such as (49); the reader is referred to [1,2] for related results concerning boundary stabilization problems (see also Beyrath [23,24] for localized indirect dampings). In light of the above observations, system (50) fails to be exponentially stable, at least when H is infinite dimensional and A1 has a compact resolvent as in (49). Indeed it is shown in Alabau, Cannarsa and Komornik [8] that the total energy of sufficiently smooth solutions of (50) decays polynomially at infinity whenever j˛j is small enough but nonzero. From this result we can also deduce that any solution of (50) is strongly stable regardless of its smoothness: this fact follows by a standard density argument since the semigroup associated with (50) is a contraction semigroup. A brief description of the key ideas of the approach developed in [2,8] is as follows. Essentially, one uses a finite iteration scheme and suitable multipliers to obtain an estimate of the form Z

j

T

E(u(t); v(t))dt  c 0

X kD0

(51)

where j is a positive integer and E denotes the total energy of the system  1  1/2 2 jA1 uj C ju0 j2 2  1  1/2 2 jA2 vj C jv 0 j2 C ˛hu; vi : C 2 Once (51) is proved, an abstract lemma due to Alabau [1,2] shows that E(u(t); v(t)) decays polynomially at 1. This abstract lemma can be stated as follows. Let A be the infinitesimal generator of a continuous semi-group exp(tA) on an Hilbert space H , and D(A) its domain. For U 0 in H we set in all the sequel U(t) D exp(tA)U 0 and assume that there exists a functional E defined on C([0; C1); H ) such that for every U 0 in H , E(exp(:A)) is a non-increasing, locally absolutely continuous function from [0; C1) on [0; C1). Assume moreover that there exist an integer k 2 N ? and nonnegative constants cp for p D 0; : : : k such that E(u; v) D

Z

T

E(U(t))dt  S

k X

c p E(U (p) (S))

pD0

80  S  T ; 8U 0 2 D(Ak ) :

T

E(U()) S

(52) U0

in Then the following inequalities hold for every D(Akn ) and all 0  S  T where n is any positive integer:

kn X   (  S)n1 E U (p) (S) ; (53) d  c (n  1)! pD0

and

E(U(t))  c

kn X   E U (p) (0) t n pD0

8t > 0 ;

  8U 0 2 D Akn ;

where c is a constant which depends on n. First (53) is proved by induction on n. For n D 1, it reduces to the hypothesis (52). Assume now that (53) holds for n and let U 0 be given in D(Ak(nC1) ). Then we have Z TZ

T

E(U()) S

c

t kn Z X pD0 S

E(u(k) (0); v (k) (0)) 8T  0;

Z

T

(  t)n1 ddt (n  1)!

  E U (p) (t) dt   8 0  S  T ; 8 U 0 2 D Akn :

Since U 0 is in D(Ak(nC1) ) we deduce that U (p) (0) D A p U 0 is in D(Ak ) for p 2 f0; : : : kng. Hence we can apply the assumption (52) to the initial data U (p) (0). This together with Fubini’s Theorem applied on the left hand side of the above inequality give (53) for n C 1. Using the property that E(U(t)) is non increasing in (53) we easily obtain the last desired inequality. Applications to wave-wave, wave-Petrowsky equations and various concrete examples hold. The above results have been studied later on by Batkai, Engel, Prüss and Schnaubelt [18] using very interesting resolvent and spectral criteria for polynomial stability of abstract semigroups. The above abstract lemma in [2] has also been generalized using interpolation theory. One should note that this integral inequality involving higher order energies of solutions is not of differential nature, unlike Haraux’s and Komornik’s integral inequalities. Another approach based on decoupling techniques and for slightly different abstract systems have been introduced by Ammar Khodja Bader and Ben Abdallah [12]. Spectral conditions have also been studied by Z. Liu [88] and later on by Z. Liu and Rao [90], Loreti and Rao [92] for peculiar abstract systems and in general for coupled equations only of the same nature (wave-wave for instance), so that a dispersion relation for the eigenvalues of the coupled system can be derived. Also these last results are given for internal stabilization only. Because of the above limitations, Z. Liu–Rao and Loreti–Rao’s results

115

116

Control of Non-linear Partial Differential Equations

are less powerful in generality than the ones given by Alabau, Cannarsa and Komornik [8] and Alabau [2]. Moreover results through energy type estimates and integral inequalities can be generalized to include nonlinear indirect dampings as shown in [7]. On the other side spectral methods are very useful to obtain optimal decay rates provided that one can determine at which speed the eigenvalues approach the imaginary axis for high frequencies.

in a Hilbert space X, where A: D(A) X ! X is an accretive self-adjoint linear operator with dense domain, and rF denotes the gradient of a Gâteaux differentiable functional F : D(A1/2 ) ! R. In particular, equation (54) fits into this framework as well as several other classical equations of mathematical physics such as the linear elasticity system. We consider the following assumptions.

Memory Dampings

Assumptions (H1)

We consider the following model problem

1. A is a self-adjoint linear operator on X with dense domain D(A), satisfying

8 Rt ˆ ˆ ˆu t t (t; x)  u(t; x) C 0 ˇ(t  s)u(s; x) ds D ˆ < ju(t; x)j u(t; x) ˆ u(t; )j@˝ D 0 ˆ ˆ ˆ :(u(0; ); u (0; )) D (u ; u ) t 0 1 (54) 2 where 0 <  N2 holds. The second member is a source term. The damping

Z

hAx; xi  Mkxk2

8x 2 D(A)

(56)

for some M > 0. 2. ˇ : [0; 1) ! [0; 1) is a locally absolutely continuous function such that Z

1

ˇ(t)dt < 1 ˇ(0) > 0

ˇ 0 (t)  0

0

for a.e. t  0 :

t

ˇ(t  s)u(s; x) ds

3. F : D(A1/2 ) ! R is a functional such that 1. F is Gâteaux differentiable at any point x 2 D(A1/2 ); 2. for any x 2 D(A1/2 ) there exists a constant c(x) > 0 such that

0

is of memory type. The energy is defined by Eu (t) D

1 ku t (t)k2L 2 (˝) dx 2   Z t 1 ˇ(s) ds kru(t)k2L 2 (˝) 1 C 2 0 1  C2  ku(t)kL  C2 (˝)

C2 Z 1 t C ˇ(t  s)kru(t)  ru(s)k2L 2 (˝) ds 2 0

The damping term produces dissipation of the energy, that is (for strong solutions) 1 Eu0 (t) D  ˇ(t)kru(t)k2 2 Z 1 t 0 C ˇ (t)kru(s)  ru(t)k2 ds  0 2 0 One can consider more general abstract equations of the form u00 (t) C Au(t) 

Z

t

jDF(x)(y)j  c(x)kyk;

where DF(x) denotes the Gâteaux derivative of F in x; consequently, DF(x) can be extended to the whole space X (and we will denote by rF(x) the unique vector representing DF(x) in the Riesz isomorphism, that is, hrF(x); yi D DF(x)(y), for any y 2 X); 3. for any R > 0 there exists a constant C R > 0 such that krF(x)  rF(y)k  C R kA1/2 x  A1/2 yk for all x; y 2 D(A1/2 ) satisfying kA1/2 xk ; kA1/2 yk  R. Assumptions (H2) 1. There exist p 2 (2; 1] and k > 0 such that ˇ 0 (t)  kˇ

ˇ(t  s)Au(s) ds D rF(u(t))

for any y 2 D(A1/2 );

1C 1p

(t) for a.e. t  0

0

t 2 (0; 1)

(55)

(here we have set

1 p

D 0 for p D 1).

Control of Non-linear Partial Differential Equations

2. F(0) D 0, rF(0) D 0, and there is a strictly increasing continuous function : [0; 1) ! [0; 1) such that (0) D 0 and jhrF(x); xij 

1/2

(kA

1/2

xk)kA

2

xk

T

m p

Z

t

Eu (t) S

8x 2 D(A ) :

ˇ(t  s)kA1/2 u(s)  A1/2 u(t)k2 dsdt

0 p pCm

 CEu

Z

T

(S)

1C mp Eu (t)'m (t)dt

!

m pCm

(58)

S

for some constant C > 0. Suppose that, for some m  1, the function ' m defined in (57) is bounded. Then, for any S0 > 0 there is a positive constant C such that Z

1

1C mp

Eu

 m m  p p (t)dt  C Eu (0) C k'm k1 Eu (S)

S

8 S  S0 :

(59)

One uses this last result first with m D 2 noticing that ' 2 is bounded and  D E 2/p . This gives a first energy decay rate as (t C1)p/2 . This estimate shows that ' 1 is bounded. Then one applies once again the last result with m D 1 and  D E 1/p . One deduces then that E decays as (t C 1)p which is the optimal decay rate expected.

0

where the multipliers Mu are of the form (s)(c1 (ˇ u)(s) C c2 (s)u) with  which is a differentiable, nonincreasing and nonnegative function, and c1 being a suitable constant, whereas c2 may be chosen dependent on ˇ. Integrating by parts the resulting relations and performing some involved estimates, one can prove that for all t0 > 0 and all T  t  t0 Z T Z T (s)E(s) ds  C(0)E(t) C (s) t t Z s  2  ˇ(s  ) A1/2 u(s)  A1/2 u() d ds ; 0

If p D 1, that is if the kernel ˇ decays exponentially, one can easily bound the last term of the above estimate by cE(t) thanks to the dissipation relation. If p 2 (2; 1), one has to proceed differently since the term Z T Z s  2 (s) ˇ(s  )  A1/2 u(s)  A1/2 u() d ds t

Z

1/2

Under these assumptions, global existence for sufficiently small (resp. all) initial data in the energy space can be proved for nonvanishing (resp. vanishing) source terms. It turns out that the above energy methods based on multiplier techniques combined with linear and nonlinear integral inequalities can be extended to handle memory dampings and applied to various concrete examples such as wave, linear elastodynamic and Petrowsky equations for instance. This allows to show in [10] that exponential as well as polynomial decay of the energy holds if the kernel decays respectively exponentially or polynomially at infinity. The method is as follows. One evaluates expressions of the form Z T Z t hu00 (s) C Au(s)  ˇ Au(s)  rF(u(s); Mui ds t

Then, we have for any T  S  0

0

cannot be directly estimated thanks to the dissipation relation. To bound this last term, one can generalize an argument of Cavalcanti and Oquendo [37] as follows. Define, for any m  1, Z t 1 'm (t) :D ˇ 1 m (t  s)kA1/2 u(s)  A1/2 u(t)k2 ds ; 0

t  0:

(57)

Bibliographical Comments For an introduction to the multiplier method, we refer the interested reader to the books of J.-L. Lions [86], Komornik [74] and the references therein. The celebrated result of Bardos Lebeau and Rauch is presented in [86]. A general abstract presentation of control problems for hyperbolic and parabolic equations can be found in the book of Lasiecka and Triggiani [80,81]. Results on spectral methods and the frequency domain approach can be found in the book of Z. Liu [88]. There also exists an interesting approach developed for bounded feedback operators by Haraux and extended to the case of unbounded feedbacks by Ammari and Tucsnak [11]. In this approach, the polynomial (or exponential) stability of the damped system is proved thanks to the corresponding observability for the undamped (conservative) system. Such observability results for weakly coupled undamped systems have been obtained for instance in [3]. Many other very interesting issues have been studied in connection to semilinear wave equations, see [34,123] and the references therein, and damped wave equations with nonlinear source terms [39]. Well-posedness and asymptotic properties for PDE’s with memory terms have first been studied by Dafermos [53,54] for convolution kernels with past history (convolution up to t D 1), by Prüss [103] and Prüss and Propst [102] in which the efficiency of different models of dampings are compared to experiments (see also

117

118

Control of Non-linear Partial Differential Equations

Londen Petzeltova and Prüss [91]). Decay estimates for the energy of solutions using multiplier methods combined with Lyapunov type estimates for an equivalent energy are proved in Munoz Rivera [97], Munoz Rivera and Salvatierra [96], Cavalcanti and Oquendo [37] and Giorgi Naso and Pata [67] and many other papers. Optimal Control As for positional control, also for optimal control problems it is convenient to adopt the abstract formulation introduced in Sect. “Abstract Evolution Equations”. Let the state space be represented by the Hilbert space H, and the state equation be given in the form (12), that is (

u0 (t) D Au(t) C B f (t)

t 2 [0; T]

u(0) D u0 :

(60)

Recall that A is the infinitesimal of a strongly continuous semigroup, etA , in H, B is a (bounded) linear operator from F (the control space) to H, and uf stands for the unique (mild) solution of (60) for a given control function f 2 L2 (0; T; H). A typical optimal control problem of interest for PDE’s is the Bolza problem which consists in 8 the cost functional ˆ < minimizing   : RT J( f ) D 0 L(t; u f (t); f (t))dt C ` u f (T) ˆ : over all controls f 2 L2 (0; T; F) :

(61)

Here, T is a positive number, called the horizon, whereas L and ` are given functions, called the running cost and final cost, respectively. Such functions are usually assumed to be bounded below. A control function f 2 L2 (0; T; F) at which the above minimum is attained is called an optimal control for problem (61) and the corresponding solution u f of (60) is said to be an optimal trajectory. Alltogether, fu f ; f g is called an optimal (trajectory/control) pair. For problem (61) the following issues will be addressed in the sections below:  the existence of controls minimizing functional J;  necessary conditions that a candidate solution must satisfy;  sufficient conditions for optimality provided by the dynamic programming method. Other problems of particular interest to CT for PDE’s are problems with an infinite horizon (T D 1), problems with a free horizon T and a final target, and problems with constraints on both control variables and state variables.

Moreover, the study of nonlinear variants of (60), including semilinear problems of the form ( u0 (t) D Au(t) C h(t; u(t); f (t)) t 2 [0; T] (62) u(0) D u0 ; is strongly motivated by applications. The discussion of all these variants, however, will not be here pursued in detail. Traditionally, in optimal control theory, state variables are denoted by the letters x; y; : : : , whereas u; v; : : : are reserved for control variables. For notational consistency, in this section u() will still denote the state of a given system and f () a control function, while  will stand for a fixed element of control space F. Existence of Optimal Controls From the study of finite dimensional optimization it is a familiar fact that the two essential ingredients to guarantee the existence of minima are compactness and lower semicontinuity. Therefore, it is clear that, in order to obtain a solution of the optimal control problem (60)–(61), one has to make assumptions that allow to recover such properties. The typical hypotheses that are made for this purpose are the following:  coercivity: there exist constants c0 > 0 and c1 2 R such that `()  c1

and L(t; u; )  c0 kk2 C c1 8(t; u; ) 2 [0; T]  H  F

(63)

 convexity: for every (t; u) 2 [0; T]  H  7! L(t; u; )

is convex on

F:

(64)

Under the above hypotheses, assuming lower semicontinuity of ` and of the map L(t; ; ), it is not hard to show that problem (60)–(61) has at least one solution. Indeed, assumption (63) allows to show that any minimizing sequence of controls f f k g is bounded in L2 (0; T; H). So, it admits a subsequence, still denoted by f f k g which converges weakly in L2 (0; T; H) to some function f . Then, by linearity, u f k (t) converges to u f (t) for every t 2 [0; T]. So, using assumption (64), it follows that f is a solution of (60)–(61). The problem becomes more delicate when the Tonelli type coercivity condition (63) is relaxed, or the state equation is nonlinear as in (62). Indeed, the convergence of u f k (t) is no longer ensured, in general. So, in order to recover compactness, one has to make further assumptions, such as the compactness of etA , or structural properties of L

Control of Non-linear Partial Differential Equations

and h. For further reading, one may consult the monographs [22,85], and [79], for problems where the running and final costs are given by quadratic forms (the so-called Linear Quadratic problem), or [84] and [59] for more general optimal control problems. Necessary Conditions Once the existence of a solution to problem (60)–(61) has been established, the next important step is to provide conditions to detect a candidate solution, possibly showing that it is, in fact, optimal. By and large the optimality conditions of most common use are the ones known as Pontryagin’s Maximum Principle named after the Russian mathematician L.S. Pontryagin who greatly contributed to the development of control theory, see [100,101]. So, suppose fu ; f g, where u D u f is a candidate optimal pair and consider the so-called adjoint system 8 0  ˆ <p (t) D A p(t) C @u L(t; u (t); f (t)) D 0 ˆ :

t 2 [0; T] a.e.

p(T) D @`(u (T)) ;

where @u L(t; u; ) and @`(u) denote the Fréchet gradients of the maps L(t; ; ) and ` at u, respectively. Observe that the above is a backward linear Cauchy problem with terminal condition, which can obviously be reduced to a forward one by the change of variable t ! T  t. So, it admits a unique mild solution, labeled p , which is called the adjoint state associated with fu ; f g. Pontryagin’s Maximum Principle states that, if fu ; f g is optimal, then hp (t); B f (t)i C L(t; u (t); f (t)) D  min hp (t); Bi C L(t; u (t); ) 2F

t 2 [0; T] a.e.

(65)

The name Maximum Principle rather than Minimum Principle, as it would be more appropriate, is due to the fact that, traditionally, attention was focussed on the maximization—instead of minimization—of the functional in (61). Even today, in most models from economics, one is interested in maximizing payoffs, such as revenues, utility, capital and so on. In that case, (65) would still be true, with a “max” instead of a “min”. At first glance, it might be hard to understand the revelance of (65) to problem (61). To explain this, introduce the function, called the Hamiltonian,  H (t; u; p) D min hp; Bi C L(t; u; ) 2F

(t; u; p) 2 [0; T]  H  H :

(66)

Then, Fermat’s rule yields B p C @ L(t; u; ) D 0 at every  2 F at which the minimum in (66) is attained. Therefore, from (65) it follows that B p (t)C@ L(t; u (t); f (t)) D 0

t 2 [0; T] a.e. (67)

which provides a much-easier-to-use optimality condition. There is a vast literature on necessary conditions for optimality for distributed parameter systems. The set-up that was considered above can be generalized in several ways: one can consider nonlinear state equations as in (62), nonsmooth running and finals costs, constraints on both state and control, problems with infinite horizon or exit times. Further reading and useful references on most of these extensions can be found in the aforementioned monographs [22,79,84,85], and in [59] which is mainly concerned with time optimal control problems. Dynamic Programming Though useful as it may be, Pontryagin’s Maximum Principle remains a necessary condition. So, without further information, it does not suffice to prove the optimality of a give trajectory/control pair. Moreover, even when the map  7! @ L(t; u; ) turns out to be invertible, the best result identity (67) can provide, is a representation of f (t) in terms of u (t) and p (t): not enough to determine f (t), in general. This is why other methods to construct optimal controls have been proposed over the years. One of the most interesting ones is the so-called dynamic programming method (abbreviated, DP), initiated by the work of R. Bellman [20]. Such a method will be briefly described below in the set-up of distributed parameter systems. Fix T > 0, s such that 0  s  T, and consider the optimal control problem to minimize Z J s;v ( f ) D

T s

    s;v L t; u s;v (t); f (t) dt C ` u (T) f f (68)

over all control functions f 2 L2 (s; T; F), where u s;v f (t) is the solution of the controlled system ( u0 (t) D Au(t) C B f (t) t 2 [s; T] (69) u(s) D v : The value function U associated to (68)-(69) is the realvalued function defined by U(s; v) D

inf

f 2L 2 (s;T;F)

J s;v ( f ) 8(s; v) 2 [0; T]  H : (70)

119

120

Control of Non-linear Partial Differential Equations

A fundamental step of DP is the following result, known as Bellman’s optimality principle. Theorem 5 For any (s; v) 2 [0; T]  H and any f 2 L2 (s; T; F) Z

r

U(s; v)  s



   s;v L t; u s;v (t); f (t) dt C U r; u (r) f f 8r 2 [s; T] :

Moreover, f  () is optimal if and only if

Now, suppose there is a control function f 2 L2 (s; T; F) such that, for all t 2 [s; T], h@v W(t; u (t)); B f (t)i C L(t; u (t); f (t)) D H (t; u (t); @v W(t; u (t))) ;

(73)

where u () D u s;v f  (). Then, from (71) and (73) it follows that d W(t; u (t)) D L(t; u(t); f (t)) ; dt whence

Z

r

U(s; v) D s



   s;v L t; u s;v f (t); f (t) dt C U r; u f (r) 8r 2 [s; T] :

The connection between DP and optimal control is based on the properties of the value function. Indeed, applying Bellman’s optimality principle, one can show that, if U is Fréchet differentiable, then 8 ˆ <@s U(s; v) C hAv; @v U(s; v)i C H (s; v; @v U(s; v)) D 0 (s; v) 2 (0; T)  D(A) ˆ : U(T; v) D `(v) v 2 H where H is the Hamiltonian defined in (66). The above equation is the celebrated Hamilton–Jacobi equation of DP. To illustrate its connections with the original optimal control problem, a useful formal argument—that can, however, be made rigorous—is the following. Consider a sufficiently smooth solution W of the above problem and let (s; v) 2 (0; T)  D(A). Then, for any trajectory/control pair fu; f g, d W(t; u(t)) D @s W(t; u(t)) C h@v W(t; u(t)); Au(t) dt C B f (t)i D h@v W(t; u(t)); B f (t)i  H (t; u(t); @v W(t; u(t)))  L(t; u(t); f (t))

(71)

W(s; v) D J s;v ( f )  U(s; v) : From the above inequality and (72) it follows that W(s; v) D U(s; v) for all (s; v) 2 (0; T)  D(A), hence for all (s; v) 2 (0; T)  H since D(A) is dense in H. So, f is an optimal control. Note 2 The above considerations lead to the following procedure to obtain optimal an optimal trajectory:  find a smooth solution of the Hamilton–Jacobi equation;  for every (t; v) 2 (0; T)  D(A) provide a feedback f (t; v) such that h@v W(t; v); B f (t; v)i C L(t; v; f (t; v)) D H (t; v; @v W(t; v))  solve the so-called closed loop equation ( u0 (t) D Au(t) C B f (t; u(t)) t 2 [s; T] u(s) D v Notice that not only is trajectory u optimal, but the corresponding control f is given in feedback form as well. Linear Quadratic Optimal Control One of the most successful applications of DP is the socalled Linear Quadratic optimal control problem. Consider problem (68)–(69) with costs L and ` given by L(t; u; ) D hM(t)u; ui C hN(t); i

by the definition of H . Therefore, integrating from s to T, Z

T

`(u(T))  W(s; v)  

and

L(t; u(t); f (t))dt; s

`(u) D hDu; ui

whence J s;v ( f )  W(s; v). Thus, taking the infimum over all f 2 L2 (s; T; F), W(s; v)  U(s; v)

8(s; v) 2 (0; T)  D(A) :

8(T; u; ) 2 [0; T]  H  F

(72)

8u 2 H ;

where  M : [0; T] ! L(H) is continuous, M(t) is symmetric and hM(t)u; ui  0 for every (t; u) 2 [0; T]  H;

Control of Non-linear Partial Differential Equations

 N : [0; T] ! F is continuous, N(t) is symmetric and hN(t); i  c0 jj2 for every (t; ) 2 [0; T]  F and some constant c0 > 0;  D 2 L(H) is symmetric and hDu; ui  0 for every u 2 H. Then, assumptions (63) and (64) are satisfied. So, a solution to (68)–(69) does exist. Moreover, it is unique because of the strict convexity of functional J s;v . In order to apply DP, one computes the Hamiltonian h i H (t; u; p) D min hp; Bi C hM(t)u; ui C hN(t); i

So, solving the closed loop equation ( u0 (t) D [A  BN 1 (t)B P(t)]u(t)

t 2 (s; T)

u(s) D v one obtains the unique optimal trajectory of problem (68)–(69). In sum, by DP one reduces the original Linear Quadratic optimal control problem to the problem of finding the solution of the Riccati equation, which is easier to solve than the Hamilton–Jacobi equation.

2F

1 D hM(t)u; ui  hBN 1 (t)B p; pi ; 4

Bibliographical Comments

where the above minimum is attained at 1  (t; p) D  N 1 (t)B p : 2

(74)

Therefore, the Hamilton–Jacobi equation associated to the problem is 8 ˆ ˆ ˆ@s W(s; v) C hAv; @v W(s; v)i C hM(s)v; vi ˆ <  1 hBN 1 (s)B @ W(s; v); @ W(s; v)i D 0 v v 4 ˆ 8(s; v) 2 (0; T)  D(A) ˆ ˆ ˆ :w(T; v) D hDv; vi 8v 2 H It is quite natural to search a solution of the above problem in the form W(s; v) D hP(s)v; vi

8(s; v) 2 [0; T]  H ;

with P : [0; T] ! L(H) continuous, symmetric and such that hP(t)u; ui  0. Substituting into the Hamilton–Jacobi equation yields 8 0  ˆ ˆhP (s)v; vi C h[A P(s) C P(s)A]v; vi C hM(s)v; vi ˆ ˆ < hBN 1 (s)B P(s)v; P(s)vi D 0 ˆ 8(s; v) 2 (0; T)  D(A) ˆ ˆ ˆ :hP(T)v; vi D hDv; vi 8v 2 H Therefore, P must be a solution of the so-called Riccati equation 8 0  ˆ

P(s)BN 1 (s)B P(s) D 0

8s 2 (0; T)

P(T) D D

Once a solution P() of the Riccati equation is known, the procedure described in Note 2 can be applied. Indeed, recalling (74) and the fact that @v W(t; v) D 2P(t)v, one concludes that f (t; v) D N 1 (t)B P(t)v is a feedback law.

Different variants of the Riccati equation have been successfully studied by several authors in connection with different state equations and cost functionals, including boundary control problems and problems for other functional equations, see [22,79] and the references therein. Sometimes, the solution of the Riccati equation related to a linearized model provides feedback stabilization for nonlinear problems as in [104]. Unfortunately, the DP method is hard to implement for general optimal control problems, because of several obstructions: nonsmoothness of solutions to Hamilton– Jacobi equations, selection problems that introduce discontinuities, unboundedness of the coefficients, numerical complexity. Besides the Linear Quadratic case, the so-called Linear Convex case is the other example that can be studied by DP under fairly general conditions, see [14]. For nonlinear optimal control problems some of the above difficulties have been overcome extending the notion of viscosity solutions to infinite dimensional spaces, see [45,46,47,48,49], see also [28,29,30,31,32,33] and [112]. Nevertheless, finding additional ideas to make a generalized use of DP for distributed parameter systems possible, remains a challenging problem for the next generations. Future Directions In addition to all considerations spread all over this article on promising developments of recent—as well as established—research lines, a few additional topics deserve to be mentioned. The one subject that has received the highest attention, recently, is that of numerical approximation of control problems, from the point of view of both controllability and optimal control. Here the problem is that, due to high frequency spurious numerical solutions, stable algorithms for solving initial-boundary value problems do not necessarily yield convergent algorithms for computing

121

122

Control of Non-linear Partial Differential Equations

controls. This difficulty is closely related to the existence of concentrated numerical solutions that escape the observation mechanisms. Nevertheless, some interesting results have been obtained so far, see, e. g., [124,125]. Several interesting results for nonlinear control problems have been obtained by the return method, developed initially by Coron [42] for a stabilization problem. This and other techniques have then been applied to fluid models ([68,69]), the Korteweg–de Vries equation ([105,106,107]), and Schrödinger type equations ([19]), see also [43] and the references therein. It seems likely that these ideas, possibly combined with other techniques like Carleman estimates as in [70], will lead to new exiting results in the years to come. A final comment on null controllability for degenerate parabolic equations is in order. Indeed, many problems that are relevant for applications are described by parabolic equation equations in divergence form @ t u D r (A(x)ru)Cb(x)ruCc(t; x)uC f

in

QT ;

or in the general form @ t u D Tr [A(x)r 2 u]Cb(x)ruCc(t; x)uC f

4.

5.

6.

7.

8. 9.

10.

11.

12.

in

QT ; 13.

where A(x) is a symmetric matrix, positive definite in ˝ but possibly singular on  . For instance, degenerate parabolic equations arise in fluid dynamics as suitable transformations of the Prandtl equations, see, e. g., [94]. They can also be obtained as Kolmogorov equations of diffusions processes on domains that are invariant for stochastic flows, see, e. g., [52]. The latter interpretation explains why they have been applied to biological problems, such as gene frequency models for population genetics (see, e. g., the Wright–Fischer model studied in [111]). So far, null controllability properties of degenerate parabolic equations have been fully understood only in dimension one: for some kind of degeneracy, null controllability holds true (see [36] and [9]), but, in general, one can only expect regional null controllability (see [35]). Since very little is known on null controllability for degenerate parabolic equations in higher space dimensions, it is conceivable that such a topic will provide interesting problems for future developments. Bibliography 1. Alabau F (1999) Stabilisation frontière indirecte de systèmes faiblement couplés. C R Acad Sci Paris Sér I 328:1015–1020 2. Alabau F (2002) Indirect boundary stabilization of weakly coupled systems. SIAM J Control Optim 41(2):511–541 3. Alabau-Boussouira F (2003) A two-level energy method for indirect boundary observability and controllability of weakly

14. 15.

16.

17.

18.

19. 20. 21.

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Diagrammatic Methods in Classical Perturbation Theory GUIDO GENTILE Dipartimento di Matematica, Università di Roma Tre, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Examples Trees and Graphical Representation Small Divisors Multiscale Analysis Resummation Generalizations Conclusions and Future Directions Bibliography Glossary Dynamical system Let W  R N be an open set and f : W  R ! R N be a smooth function. The ordinary differential equation x˙ D f (x; t) on W defines a continuous dynamical system. A discrete dynamical system on W is defined by a map x ! x0 D F(x), with F depending smoothly on x. Hamiltonian system Let A  Rd be an open set and H : A  Rd  R ! R be a smooth function (A  Rd is called the phase space). Consider the system of ordinary differential equations q˙k D @H (q; p; t)/@p k , p˙ k D @H (q; p; t)/@q k , for k D 1; : : : ; d. The equations are called Hamilton equations, and H is called a Hamiltonian function. A dynamical system described by Hamilton equations is called a Hamiltonian system. Integrable system A Hamiltonian system is called integrable if there exists a system of coordinates (˛; A) 2 T d  Rd , called angle-action variables, such that in these coordinates the motion is (˛; A) ! (˛ C !(A)t; A), for some smooth function !(A). Hence in these coordinates the Hamiltonian function H depends only on the action variables, H D H0 (A). Invariant torus Given a continuous dynamical system we say that the motion occurs on an invariant d-torus if it takes place on a d-dimensional manifold and its position on the manifold is identified through a coordinate in T d . In an integrable Hamiltonian system all phase space is filled by invariant tori. In a quasi-integrable

system the KAM theorem states that most of the invariant tori persist under perturbation, in the sense that the relative Lebesgue measure of the fraction of phase space filled by invariant tori tends to 1 as the perturbation tends to disappear. The persisting invariant tori are slight deformations of the unperturbed invariant tori. Quasi-integrable system A quasi-integrable system is a Hamiltonian system described by a Hamiltonian function of the form H D H0 (A) C " f (˛; A), with (˛; A) angle-action variables, " a small real parameter and f periodic in its arguments ˛. Quasi-periodic motion Consider the motion ˛ ! ˛ C ! t on T 2 , with ! D (!1 ; !2 ). If !1 /!2 is rational, the motion is periodic, that is there exists T > 0 such that !1 T D !2 T D 0 mod 2. If !1 /!2 is irrational, the motion never returns to its initial value. On the other hand it densely fills T 2 , in the sense that it comes arbitrarily close to any point of T 2 . We say in that case that the motion is quasi-periodic. The definition extends to T d , d > 2: a linear motion ˛ ! ˛ C ! t on T d is quasi-periodic if the components of ! are rationally independent, that is if ! D !1 1 C  C!d d D 0 for  2 Zd if and only if  D 0 (a  b is the standard scalar product between the two vectors a, b). More generally we say that a motion on a manifold is quasi-periodic if, in suitable coordinates, it can be described as a linear quasi-periodic motion. The vector ! is usually called the frequency or rotation vector. Renormalization group By renormalization group one denotes the set of techniques and concepts used to study problems where there are some scale invariance properties. The basic mechanism consists in considering equations depending on some parameters and defining some transformations on the equations, including a suitable rescaling, such that after the transformation the equations can be expressed, up to irrelevant corrections, in the same form as before but with new values for the parameters. Torus The 1-torus T is defined as T D R/2Z, that is the set of real numbers defined modulo 2 (this means that x is identified with y if x  y is a multiple of 2). So it is the natural domain of an angle. One defines the d-torus T d as a product of d 1-tori, that is T d D T      T . For instance one can imagine T 2 as a square with the opposite sides glued together. Tree A graph is a collection of points, called nodes, and of lines which connect the nodes. A walk on the graph is a sequence of lines such that any two successive lines in the sequence share a node; a walk is nontrivial if it

Diagrammatic Methods in Classical Perturbation Theory

contains at least one line. A tree is a planar graph with no closed loops, that is, such that there is no nontrivial walk connecting any node to itself. An oriented tree is a tree with a special node such that all lines of the tree are oriented toward that node. If we add a further oriented line connecting the special node to another point, called the root, we obtain a rooted tree (see Fig. 1 in Sect. “Trees and Graphical Representation”). Definition of the Subject Recursive equations naturally arise whenever a dynamical system is considered in the regime of perturbation theory; for an introductory article on perturbation theory see  Perturbation Theory. A classical example is provided by Celestial Mechanics, where perturbation series, known as Lindstedt series, are widely used; see Gallavotti [21] and  Perturbation Theory in Celestial Mechanics. A typical problem in Celestial Mechanics is to study formal solutions of given ordinary differential equations in the form of expansions in a suitable small parameter, the perturbation parameter. In the case of quasi-periodic solutions, the study of the series, in particular of its convergence, is made difficult by the presence of the small divisors – which will be defined later on. Under some non-resonance condition on the frequency vector, one can show that the series are well-defined to any order. The first proof of such a property was given by Poincaré [53], even if the convergence of the series remained an open problem up to the advent of KAM theory – an account can be found in Gallavotti [17] and in Arnold et al. [2]; see also  Kolmogorov–Arnold–Moser (KAM) Theory. KAM is an acronym standing for Kolmogorov [47], Arnold [1] and Moser [51], who proved in the middle of last century the persistence of most of invariant tori for quasi-integrable systems. Kolmogorov and Arnold proofs apply to analytic Hamiltonian systems, while Moser’s approach deals also with the differentiable case; the smoothness condition on the Hamiltonian function was thereafter improved by Pöschel [54]. In the analytic case, the persisting tori turn out to be analytic in the perturbation parameter, as explicitly showed by Moser [52]. In particular, this means that the perturbation series not only are well-defined, but also converge. However, a systematic analysis with diagrammatic techniques started only recently after the pioneering, fundamental works by Eliasson [16] and Gallavotti [18], and were subsequently extended to many other problems with small divisors, including dynamical systems with infinitely many degrees of freedom, such as nonlinear partial differential equations, and non-Hamiltonian systems.

Some of these extensions will be discussed in Sect. “Generalizations”. From a technical point of view, the diagrammatic techniques used in classical perturbation theory are strongly reminiscent of the Feynman diagrams used in quantum field theory: this was first pointed out by Gallavotti [18]. Also the multiscale analysis used to control the small divisors is typical of renormalization group techniques, which have been successfully used in problems of quantum field theory, statistical mechanics and classical mechanics; see Gallavotti [20] and Gentile & Mastropietro [38] for some reviews. Note that there exist other renormalization group approaches to the study of dynamical systems, and of KAMlike problems in particular, different from that outlined in this article. By confining ourselves to the framework of problems of KAM-type, we can mention the paper by Bricmont et al. [11], which also stressed the similarity of the technique with quantum field theory, and the so called dynamical renormalization group method – see MacKay [50] – which recently produced rigorous proofs of persistence of quasi-periodic solutions; see for instance Koch [46] and Khanin et al. [45]. Introduction Consider the ordinary differential equation on Rd Du D G(u) C "F(u) ;

(1)

where D is a pseudo-differential operator and G, F are real analytic functions. Assume that (1) admits a solution u(0) (t) for " D 0, that is Du(0) D G(u(0) ). The problem we are interested in is to investigate whether there exists a solution of (1) which reduces to u(0) as " ! 0. For simplicity assume G D 0 in the following. The first attempt one can try is to look for solutions in the form of power series in ", u(t) D

1 X

" k u(k) (t) ;

(2)

kD0

which, inserted into (1), when equating the left and right hand sides order by order, gives the list of recursive equations Du(0) D 0, Du(1) D F(u(0) ), Du(2) D @u F(u(0) )u(1) , and so on. In general to order k  1 one has Du(k) D

k1 X 1 s @u F(u(0) ) s! sD0

X

u(k 1 ) : : : u(k s ) ;

(3)

k 1 CCk s Dk1 k i 1

where @su F, the sth derivative of F, is a tensor with s C 1 indices (s must be contracted with the vectors

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u(k 1 ) ; : : : ; u(k s ) ), and the term with s D 0 in the sum has to be interpreted as F(u(0) ) and appears only for k D 1. For instance for F(u) D u3 the first orders give Du(1) D u(0)3 ; Du(2) D 3u(0)2 u(1) ; Du(3) D 3u(0)2 u(2) C 3u(0) u(1)2 ;

(4)

Du(4) D 3u(0)2 u(3) C 6u(0) u(1) u(2) C u(1)3 ; as is easy to check. If the operator D can be inverted then the recursions (3) provide an algorithm to compute the functions u(k) (t). In that case we say that (2) defines a formal power series: by this we mean that the functions u(k) (t) are well-defined for all k  0. Of course, even if this can be obtained, there is still the issue of the convergence of the series that must be dealt with. Examples In this section we consider a few paradigmatic examples of dynamical systems which can be described by equations of the form (1). A Class of Quasi-integrable Hamiltonian Systems Consider the Hamiltonian system described by the Hamiltonian function H (˛; A) D 12 A2 C " f (˛) ;

Td

(5)

Rd

where (˛; A) 2  are angle-action variables, with T D R/2Z, f is a real analytic function, 2-periodic in each of its arguments, and A2 D A  A, if (here and henceforth)  denotes the standard scalar product in Rd , that is a  b D a1 b1 C    C ad bd . Assume also for simplicity that f is a trigonometric polynomial of degree N. The corresponding Hamilton equations are (we shorten @x D @/@x )

 (k) ˛¨ (k) D  " f (˛) :D 

k1 X 1 sC1 @˛ f (! t) s! sD0

X

˛ (k 1 ) : : : ˛ (k s ) : (7)

k 1 CCk s Dk1 k i 1

We look for a quasi-periodic solution of (6), that is a solution of the form ˛(t) D ! t C h(! t), with h(! t) D O("). We call h the conjugation function, as it “conjugates” (that is, maps) the perturbed solution ˛(t) to the unperturbed solution ! t. In terms of the function h (6) becomes h¨ D "@˛ f (! t C h) ;

(8)

where @˛ denotes derivative with respect to the argument. Then (8) can be more conveniently written in Fourier space, where the operator D acts as a multiplication operator. If we write h(! t) D

X

e i!t h ;

2Z d

h D

1 X

" k h(k) ;

which can be written as an equation involving only the angle variables: (6)

which is of the form (1) with u D ˛, G D 0, F D @˛ f , and D D d2 /dt 2 . For " D 0, ˛ (0) (t) D ˛0 C! t is a solution of (6) for any choice of ˛0 2 T d and ! 2 Rd . Take for simplicity ˛0 D

(9)

kD1

and insert (9) into (8) we obtain  (k) (!  )2 h(k) D "@˛ f (˛)  :D

k1 X

X

X

sD0 k 1 CCk s Dk1 0 C1 CC s D  i 2Z n k i 1

 (i0 )sC1 f0 h(k11 ) : : : h(ks s ) :

˛˙ D @A H (˛; A) D A ; A˙ D @˛ H (˛; A) D "@˛ f (˛) ;

˛¨ D "@˛ f (˛) ;

0: we shall see that this choice makes sense. We say that for " D 0 the Hamiltonian function (5) describes a system of d rotators. We call ! the frequency vector, and we say that ! is irrational if its components are rationally independent, that is if !   D 0 for  2 Zd if and only if  D 0. For irrational ! the solution ˛ (0) (t) describes a quasi-periodic motion with frequency vector !, and it densely fills T d . Then (3) becomes

1 s! (10)

These equations are well-defined to all orders provided ["@˛ f (˛)](k)  D 0 for all  such that !   D 0. If ! is an irrational vector we need ["@˛ f (˛)](k) 0 D 0 for the equations to be well-defined. In that case the coefficients h0(k) are left undetermined, and we can fix them arbitrarily to vanish (which is a convenient choice). We shall see that under some condition on ! a quasiperiodic solution ˛(t) exists, and densely fills a d-dimensional manifold. The analysis carried out above for ˛0 D 0 can be repeated unchanged for all values of ˛0 2 T d : ˛0

Diagrammatic Methods in Classical Perturbation Theory

represents the initial phase of the solution, and by varying ˛0 we cover all the manifold. Such a manifold can be parametrized in terms of ˛0 , so it represents an invariant torus for the perturbed system. A Simplified Model with No Small Divisors Consider the same equation as (8) with D D d2 /dt 2 replaced by 1, that is h D "@˛ f (! t C h) :

(11)

Of course in this case we no longer have a differential equation; still, we can look again for quasi-periodic solutions h(! t) D O(") with frequency vector !. In such a case in Fourier space we have  (k) h(k) D "@˛ f (˛)  :D

k1 X

X

X

sD0 k 1 CCk s Dk1 0 C1 CC s D  i 2Z n k i 1

1 (i0 )sC1 s!

 f0 h(k11 ) : : : h(ks s ) :

(12)

For instance if d D 1 and f (˛) D cos ˛ the equation, which is known as the Kepler equation, can be explicitly solved by the Lagrange inversion theorem [55], and gives

h(k)

D

8 ˆ < ˆ :

jj  k;

i(1) kC(k)/2 ; 2 ((k  )/2)!((k C )/2)!

 C k even ;

0;

otherwise :

k

(13) We shall show in Sect. “Small Divisors” that a different derivation can be provided by using the forthcoming diagrammatic techniques. The Standard Map Consider the finite difference equation D˛ D " sin ˛ ;

(14)

on T , where now D is defined by D˛( ) :D 2˛( )  ˛( By writing ˛ D Dh D " sin(

C !)  ˛(

k1

h(k) D

X 1 2 4 sin (!/2) sD0

X

X

k 1 CCk s Dk1 0 C1 CC s D  i 2Z k i 1

 (i0 )sC1 f0 h(k11 ) : : : h(ks s ) ;

1 s!

(18)

where 0 D ˙1 and f˙1 D 1/2. Note that (17) is a discrete dynamical system. However, when passing to Fourier space, (18) acquires the same form as for the continuous dynamical systems previously considered, simply with a different kernel for D. In particular if we replace D with 1 we recover the Kepler equation. The number ! is called the rotation number. We say that ! is irrational if the vector (2; !) is irrational according to the previous definition. Trees and Graphical Representation Take ! to be irrational. We study the recursive equations (  (k) h(k) D g(!  ) "@˛ f (˛)  ;  ¤ 0 ; (19) (k)   D0; "@˛ f (˛) 0 D 0 ; where the form of g depends on the particular model we are investigating. Hence one has either g(!  ) D (!  )2 or g(!  ) D 1 or g(!  ) D (2 sin(!/2))2 according to models described in Sect. “Examples”. For  ¤ 0 we have equations which express the co0 efficients h(k) ,  2 Zd , in terms of the coefficients h(k ) ,  2 Zd , with k 0 < k, provided the equations for  D 0 are satisfied for all k  1. Recursive equations, such as (19), naturally lead to a graphical representation in terms of trees. Trees

 !) :

(15)

C h( ), (14) becomes C h) ;

In other words, by writing x D C h( ) and y D ! C h( ) h( !), with ( ; !) solving (17) for " D 0, that is ( 0 ; ! 0 ) D ( C !; !), we obtain a closed-form equation for h, which is exactly (16). In Fourier space the operator D acts as D : e i ! 4 sin2 (!/2)e i , so that, by expanding h according to (9), we can write (16) as

(16)

which is the functional equation that must be solved by the conjugation function of the standard map ( x 0 D x C y C " sin x ; (17) y 0 D y C " sin x :

A connected graph G is a collection of points (nodes) and lines connecting all of them. Denote with N(G ) and L(G ) the set of nodes and the set of lines, respectively. A path between two nodes is the minimal subset of L(G ) connecting the two nodes. A graph is planar if it can be drawn in a plane without graph lines crossing. A tree is a planar graph G containing no closed loops. Consider a tree G with a single special node v0 : this introduces a natural partial ordering on the set of lines and nodes, and one can imagine that each line carries an arrow

129

130

Diagrammatic Methods in Classical Perturbation Theory

pointing toward the node v0 . We add an extra oriented line `0 exiting the special node v0 ; the added line will be called the root line and the point it enters (which is not a node) will be called the root of the tree. In this way we obtain a tree defined by N( ) D N(G ) and L( ) D L(G ) [ `0 . A labeled tree is a rooted tree together with a label function defined on the sets L( ) and N( ). We call equivalent two rooted trees which can be transformed into each other by continuously deforming the lines in the plane in such a way that the lines do not cross each other. We can extend the notion of equivalence also to labeled trees, by considering equivalent two labeled trees if they can be transformed into each other in such a way that the labels also match. In the following we shall deal mostly with nonequivalent labeled trees: for simplicity, where no confusion can arise, we call them just trees. Given two nodes v; w 2 N( ), we say that w  v if v is on the path connecting w to the root line. We can identify a line ` through the node v it exits by writing ` D `v . We call internal nodes the nodes such that there is at least one line entering them, and end-points the nodes which have no entering line. We denote with L( ), V ( ) and E( ) the set of lines, internal nodes and end-points, respectively. Of course N( ) D V( ) [ E( ). The number of unlabeled trees with k nodes (and hence with k lines) is bounded by 22k , which is a bound on the number of random walks with 2k steps [38]. For each node v denote by S(v) the set of the lines entering v and set sv D jS(v)j. Hence sv D 0 if v is an endnode, and sv  1 if v is an internal node. One has X X sv D sv D k  1 ; (20) v2N( )

v2V ( )

this can be easily checked by induction on the order of the tree. An example of unlabeled tree is represented in Fig. 1. For further details on graphs and trees we refer to the literature; cf. for instance Harary [43]. Labels and Diagrammatic Rules We associate with each node v 2 N( ) a mode label v 2 Zd , and with each line ` 2 L( ) a momentum label ` 2 Zd , with the constraint X X w D v C ` ; (21) `v D w2N( ) wv

`2S(v)

which represents a conservation rule for each node. Call T k; the set of all trees with k nodes and momentum  associated with the root line. We call k and  the order and the momentum of , respectively.

Diagrammatic Methods in Classical Perturbation Theory, Figure 1 An unlabeled tree with 17 nodes

Diagrammatic Methods in Classical Perturbation Theory, Figure 2 Graph element

Diagrammatic Methods in Classical Perturbation Theory, Figure 3 Graphical representation of the recursive equations

We want to show that trees naturally arise when studying (19). Let h(k) be represented with the graph element in Fig. 2 as a line with label  exiting from a ball with label (k). Then we can represent (19) graphically as depicted in Fig. 3. Simply represent each factor h(ki i ) on the right hand side as a graph element according to Fig. 2. The lines of all such graph elements enter the same node v0 . This is a graphical expedient to recall the conservation rule: the momentum  of the root line is the sum of the mode label 0 of the node v0 plus the sum of the momenta of the lines entering v0 . The first few orders k  4 are as depicted in Fig. 4. For each node the conservation rule (21) holds: for instance for k D 2 one has  D 1 C 2 , for k D 3 one has  D 1 C `1 and `1 D 2 C 3 in the first tree and  D 1 C 2 C 3 in

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 4 Trees of lower orders

the second tree, and so on. Moreover one has to sum over all possible choices of the labels v , v 2 N( ), which sum up to . Given any tree 2 T k; we associate with each node v 2 N( ) a node factor F v and with each line ` 2 L( ) a propagator g` , by setting Fv :D

1 (iv )s v C1 fv ; sv !

g` :D g(!  ` ) ;

and define the value of the tree as  Y   Y Fv g` : Val( ) :D v2N( )

(22)

(23)

`2L( )

The propagators g` are scalars, whereas each F v is a tensor with sv C 1 indices, which can be associated with the sv C 1 lines entering or exiting v. In (23) the indices of the tensors F v must be contracted: this means that if a node v is connected to a node v0 by a line ` then the indices of F v and Fv 0 associated with ` are equal to each other, and eventually one has to sum over all the indices except that associated with the root line. For instance the value of the tree in Fig. 4 contributing to h(2) is given by Val( ) D (i1 )2 f1 (i2 ) f2 g(!  )g(!  2 ) ;

with 1 C 2 D , while the value of the last tree in Fig. 4 contributing to h(4) is given by Val( ) D

(i1 )4 f1 (i2 ) f2 (i3 ) f3 (i4 ) f4 3!  g(!  )g(!  2 )g(!  3 )g(!  4 ) ;

with 1 C 2 C 3 C 4 D . It is straightforward to prove that one can write X h(k) D Val( ) ;  ¤0; k  1:

(24)

2T k;

This follows from the fact that the recursive equations (19) can be graphically represented through Fig. 3: one iterates the graphical representation of Fig. 3 until only graph elements of order k D 1 appear, and if is of order 1 (cf. Fig. 4) then Val( ) D (i) f g(!  ). Each line ` 2 L( ) can be seen as the root line of the tree consisting of all nodes and lines preceding `. The choice h0(k) D 0 for all k  1 implies that no line can have zero momentum: in other words we have ` ¤ 0 for all ` 2 L( ). Therefore in order to prove that (9) with h(k) given by (24) solves formally, that is order by order, the Eqs. (19),

131

132

Diagrammatic Methods in Classical Perturbation Theory

we have only to check that ["@˛ f (! t C h(! t))](k) 0 D 0 for all k  1. If we define g` D 1 for ` D 0, then also the second relation in (19) can be graphically represented as in Fig. 3 by setting  D 0 and requiring h0(k) D 0, which yields that the sum of the values of all trees on the right hand side must vanish. Note that this is not an equation to solve, but just an identity that has to be checked to hold at all orders. For instance for k D 2 (the case k D 1 is trivial) the identity ["@˛ f (! t C h(! t))](2) 0 D 0 reads (cf. the second line in Fig. 4) X (i1 )2 f1 (i2 ) f2 g(!  2 ) D 0 ; 1 C2 D0

which is found to be satisfied because the propagators are even in their arguments. Such a cancellation can be graphically interpreted as follows. Consider the tree with mode labels 1 and 2 , with 1 C 2 D 0: its value is (i1 )2 f1 (i2 ) f2 g(!  2 ). One can detach the root line from the node with mode label 1 and attach it to the node with mode label 2 , and reverse the arrow of the other line so that it points toward the new root line. In this way we obtain a new tree (cf. Fig. 5): the value of the new tree is (i1 ) f1 (i2 )2 f2 g(!  1 ), where g(!  1 ) D g(!  2 ) D g(!  2 ), so that the values of the two trees contain a common factor (i1 ) f1 (i2 ) f2 g(!  2 ) times an extra factor which is (i1 ) for the first tree and (i2 ) for the second tree. Hence the sum of the two values gives zero. The cancellation mechanism described above can be generalized to all orders. Given a tree one considers all trees which can be obtained by detaching the root line and attaching to the other nodes of the tree, and by reversing the arrows of the lines (when needed) to make them point toward the root line. Then one sums together the values of all the trees so obtained: such values contain a common factor times a factor iv , if v is the node which the root line exits (the only nontrivial part of the proof is to check that the combinatorial factors match each other: we refer to Gentile & Mastropietro [37] for details). Hence the sum gives zero, as the sum of all the mode labels vanishes. For instance for k D 3 the cancellation operates by considering the three trees in Fig. 5: such trees can be considered to be obtained from each other by shifting the root line and consistently reversing the arrows of the lines.

In such a case the combinatorial factors of the node factors are different, because in the second tree the node factor associated with the node with mode label 2 contains a factor 1/2: on the other hand if 1 ¤ 3 there are two nonequivalent trees with that shape (with the labels 1 and 3 exchanged between themselves), whereas if 1 D 3 there is only one such tree, but then the first and third trees are equivalent, so that only one of them must be counted. Thus, by using that 1 C 2 C 3 D 0 – which implies g(!  (2 C 3 )) D g(!  1 ) and g(!  (1 C 2 )) D g(!  3 ) – in all cases we find that the sum of the values of the trees gives a common factor (i1 ) f1 (i2 )2 f2 (i3 ) f3 g(!  3 )g(!  1 ) times a factor 1 or 1/2 times i(1 C2 C3 ), and hence vanishes: once more the property that g is even is crucial. Small Divisors We want to study the convergence properties of the series h(! t) D

X

e i!t h ;

h D

2Z d

" k h(k) ;

(25)

kD1

which has been shown to be well–defined as a formal power series for the models considered in Sect. “Examples”. Recall that the number of unlabeled trees of order k is bounded by 22k . To sum over the labels we can confine ourselves to the mode labels, as the momenta are uniquely determined by the mode labels. If f is a trigonometric polynomial of degree N, that is f D 0 for all  such that jj :D j1 j C    C jd j > N, we have that h(k) D 0 for all jj > kN (which can be easily proved by induction), and moreover we can bound the sum over the mode labels of any tree of order k by (2N C 1)d k . Finally we can bound Y Y jv js vC1  N s vC1  N 2k ; (26) v2N( )

v2N( )

because of (20). For the model (11), where g` D 1 in (22), we can bound ˇ ˇ ˇ X ˇˇ ˇ ˇ (k) ˇ ˇh(k) ˇ  22k (2N C 1)d k N 2k ˚ k ; ˇh ˇ  2Z d

(27) ˚ D max j f j ; jjN

which shows that the series (25) converges for " small enough, more precisely for j"j < "0 , with "0 :D C0 (4N 2 ˚(2N C 1)d )1 ;

Diagrammatic Methods in Classical Perturbation Theory, Figure 5 Trees to be considered together to prove that ["@f(˛)]20 D 0

1 X

(28)

where C0 D 1. Hence the function h(! t) in that case is analytic in ". For d D 1 and f (˛) D cos ˛, we can eas-

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 6 Trees to be considered together to prove that ["@f(˛)]30 D 0

ily provide an exact expression for the coefficients h(k) : all the computational difficulties reduce to a combinatorial check, which can be found in Gentile & van Erp [42], and the formula (13) is recovered. However for the models where g` ¤ 1, the situation is much more involved: the propagators can be arbitrarily close to zero for  large enough. This is the so-called small divisor problem. The series (25) is formally well–defined, assuming only an irrationality condition on !. But to prove the convergence of the series, we need a stronger condition. For instance one can require the standard Diophantine condition j!  j >

jj

8 ¤ 0 ;

(29)

for suitable positive constants and . For fixed  > d 1, the sets of vectors which satisfy (29) for some constant

> 0 has full Lebesgue measure in Rd [17]. We can also impose a weaker condition, known as the Bryuno condition, which can be expressed by requiring B(!) :D

1 X 1 1 log <1: k min0<jj2k j!  j 2

(30)

kD0

We call Diophantine vectors and Bryuno vectors the vectors satisfying (29) and (30), respectively. Note that any Diophantine vector satisfies (30). If we assume only analyticity on f – that is, we remove the assumption that f be a trigonometric polynomial, – then we need a Diophantine condition, such as (29) or (30), also to show that the series (25) are well-defined as formal power series: the condition is needed, as one can easily check, in order to sum over the mode labels. In the case of the standard map j!  j must be replaced with min p2Z j!  pj, and the function B(!) can be expressed in terms of the best approximants of the number ! [15]. For the models with small divisors considered in Sect. “Examples” convergence of the series (25) can be proved for ! satisfying (29) or (30). But this requires more work, and, in particular, a detailed discussion of the product of propagators in (24). In the following we shall consider the case of Diophantine vectors, by referring to the

bibliography for the Bryuno vectors (cf. Sect. “Generalizations”).

Multiscale Analysis Consider explicitly the case g(!  ) D (!  )2 in (19) and ! satisfying (29). We can introduce a label characterizing the size of the propagator: we say that  2 Zd is on scale (

n 1;

if 2n  j!  j < 2(n1) ;

n D0;

if  j!  j ;

(31)

where is the constant appearing in (29), and we say that a line ` has a scale label n` D n if ` is on scale n. If Nn ( ) denotes the number of lines ` 2 L( ) with scale n` D n, then we can bound in (23) 1 ˇ Y ˇ Y ˇ ˇ g` ˇ  2k 22nNn ( ) ; ˇ `2L( )

(32)

nD0

so that the problem is reduced to bounding N n ( ). The product of propagators gives problems when the small divisors “accumulate”. To make more precise the idea of accumulation we introduce the notion of cluster. Once all lines of a tree have been given their scale labels, for any n  0 we can identify the maximal connected sets of lines with scale not larger than n. If at least one among such lines has scale equal to n we say that the set is a cluster on scale n. For instance consider the tree in Fig. 1, and assign the mode labels to the nodes: this uniquely fixes the momenta, and hence the scale labels, of the lines. Suppose that we have found the scales as in Fig. 7a. Then a cluster decomposition as in Fig. 7b follows. Given a cluster T call L(T) the set of lines of contained in T, and denote by N(T) the set of nodes connected by such lines. We define k T D jN(T)j the order of the cluster T. Any cluster has either one or no exiting line, and can have an arbitrary number of entering lines. We call selfenergy clusters the clusters which have one exiting line and only one entering line and are such that both lines have the same momentum. This means that if T is a self-energy

133

134

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 7 Example of clusters with their scales

cluster and `1 and `2 are the lines entering and exiting T, respectively, then `1 D `2 , so that X

v D 0 :

(33)

v2N(T)

By definition of scale, both lines `1 and `2 have the same scale, say n, and that, by definition of cluster, one has n` < n for all ` 2 L(T). We define the value of the self-energy cluster T whose entering line has momentum  as the matrix VT (!  ) :D

 Y v2N(T)

Fv

 Y

 g` ;

(34)

`2L(T)

where all the indices of the node factors must be contracted except those associated with the line `1 entering T and with the line `2 exiting T. We can extend the notion of self-energy clusters also to a single node, by saying that v is a self-energy cluster if sv D 1 and the line entering v has the same momentum as the exiting line. In that case (34) has to be interpreted as VT (!  ) D Fv : in particular it is independent of !  . The simplest self-energy cluster one can think of consists of only one node v, but then (33) implies v D 0,

Diagrammatic Methods in Classical Perturbation Theory, Figure 8 Self-energy clusters of order k D 2

so that Fv D 0, see (22), hence the corresponding value is zero. Thus the simplest non-trivial self-energy clusters contain at least two nodes, and are represented by the clusters T 1 , T 2 and T 3 of Fig. 8. In all cases one has 1 C2 D 0. By using the definition (34) one has VT1 (x) D (i1 )2 f1 (i2 )2 f2 g(!  2 C x) and VT2 (x) D VT3 (x) D (i1 )3 f1 (i2 ) f2 g(!  2 )/2, with x D !  . Hence for x D 0 the sum of the three values gives zero. It is not difficult to see that also the first derivatives of the values of all self-energy clusters of order 2 sum up to zero, that is the sum of the values of all possible self-energy clusters of order k D 2 gives zero up to order x2 . (The only caveat is that, if we want to derive VT (x), the sharp multiscale decomposition in (31) can be a little annoying, hence it can be more convenient to replace it with a smooth decomposition through C 1 compact support functions; we refer to the literature for details). This property generalizes to all orders k, and the underlying cancellation mechanism is essentially the same that ensures the validity of the second relation in (19). The reason why it is important to introduce the selfenergy clusters is that if we could neglect them then the product of small divisors would be controlled. Indeed, let us denote by Rn ( ) the number of lines on scale n which do exit a self-energy cluster, and set Nn ( ) D N n ( ) 

Diagrammatic Methods in Classical Perturbation Theory

Rn ( ). Then an important result, known as the Siegel– Bryuno lemma, is that Nn ( )  c2n/ k ;

(35)

for some constant c, where k is the order of and  is the Diophantine exponent in (29). The bound (35) follows from the fact that if Nn ( ) ¤ 0 then Nn ( )  E(n; k) :D 2N k2(n2)/  1, which can be proved by induction on k as follows. Given a tree let `0 be its root line, let `1 ; : : : ; `m , m  0, be the lines on scales  n which are the closest to `0 , and let 1 ; : : : ; m the trees with root lines `1 ; : : : ; `m , respectively (cf. Fig. 9 – note that by construction all lines ` in the subgraph T have scales n` < n, so that if n`0  n then T is necessarily a cluster). If either `0 is not on scale n or it is on scale n but exits a self-energy cluster then Nn ( ) D Nn ( 1 ) C    C Nn ( m ) and the bound Nn ( )  E(n; k) follows by the inductive hypothesis. If `0 does not exit a self-energy cluster and n`0 D n then Nn ( ) D 1 C Nn ( 1 ) C    C Nn ( m ), and the lines `1 ; : : : ; `m enter a cluster T with k T D k  (k1 C    C k m ), where k1 ; : : : ; k m are the orders of 1 ; : : : ; m , respectively. If m  2 the bound Nn ( )  E(n; k) follows once more by the inductive hypothesis. If m D 0 then Nn ( ) D 1; on the other hand for `0 to be on scale n`0 D n one must have j!  `0 j < 2nC1 (see (31)), which, by the Diophantine condition (29), implies N k  j`0 j > 2(n1)/ , hence E(n; k) > 1. If m D 1 call 1 and 2 the momenta of the lines `0 and `1 , respectively. By construction T cannot be a self-energy cluster, hence 1 ¤ 2 , so that, by the Diophantine condition (29), 2nC2  j!  1 j C j!  2 j  j!  (1  2 )j

; > j1  2 j because n`0 D n and n`1  n. Thus, one has X N kT  jv j  j1  2 j > 2(n2)/ ;

hence T must contain “many nodes”. In particular, one finds also in this case Nn ( ) D 1 C Nn ( 1 )  1 C E(n; k1 )  1 C E(n; k)  E(n; k T )  E(n; k), where we have used that E(n; k T )  1 by (37). The argument above shows that small divisors can accumulate only by allowing self-energy clusters. That accumulation really occurs is shown by the example in Fig. 10, where a tree of order k containing a chain of p self-energy clusters is depicted. Assume for simplicity that k/3 is an integer: then if p D k/3 the subtree 1 with root line ` is of order k/3. If the line ` entering the rightmost self-energy cluster T p has momentum , also the lines exiting the p self-energy clusters have the same momentum . Suppose that jj  N k/3 and j!  j  /jj (this is certainly possible for some ). Then the value of the tree grows like a1k (k!) a 2 , for some constants a1 and a2 : a bound of this kind prevents the convergence of the perturbation series (25). If no self-energy clusters could occur (so that Rn ( ) D 0) the Siegel–Bryuno lemma would allow us to bound in (32) 1 Y

22nNn ( ) D

nD0

1 Y



22nNn ( )

nD0 1   X  exp C1 k n2n/  C2k ;

(38)

nD0

for suitable constants C1 and C2 . In that case convergence of the series for j"j < "0 would follow, with "0 defined as in (26) with C0 D 2 /C2 . However, there are selfenergy clusters and they produce factorials, as the example in Fig. 10 shows, so that we have to deal with them.

(36) Resummation (37)

v2N(T)

Let us come back to the Eq. (10). If we expand g(!  )["@˛ f (˛)](k)  in trees according to the diagrammatic rules described in Sect. “Trees and Graphical Representation”, we can distinguish between contributions in which the root line exits a self-energy cluster T, that we can write as X g(!  ) VT (!  )h(kk T ) ; (39) T : k T
Diagrammatic Methods in Classical Perturbation Theory, Figure 9 Construction for the proof of the Siegel–Bryuno lemma

and all the other contributions, that we denote by ["@˛ f (˛)](k)  . We can shift the contributions (39) to the left hand side of (10) and divide by g(!  ), so to obtain X VT (!  )h(kk T ) D(!  )h(k)  T : k T
D ["@˛ f (! t C h)](k)  :

(40)

135

136

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 10 Example of accumulation of small divisors because of the self-energy clusters

where D(!  ) D 1/g(!  ) D (!  )2 . By summing over k and setting M(!  ; ") D

1 X kD1

"k

X

VT (!  ) ;

(41)

T : k T Dk

 1 g` :D g [n] (!; ") D D(!  )  M[n] (!  ; ") ; (44)

then (40) gives D(!  )h D ["@˛ f (! t C h)] ;

with

D(!  ) :D D(!  )  M(!  ; ") :

(42)

The motivation for proceeding in this way is that, at the price of changing the operator D into D, hence of changing the propagators, lines exiting self-energy clusters no longer appear. Therefore, in the tree expansion of the right hand side of the equation, we have eliminated the selfenergy clusters, that is the source of the problem of accumulation of small divisors. Unfortunately the procedure described above has a problem: M(!  ; ") itself is a sum of self-energy clusters, which can still contain some other self-energy clusters on lower scales. So finding a good bound for M(!  ; ") could have the same problems as for the values of the trees. To deal with such a difficulty we modify the prescription by proceeding recursively, in the following sense. Let us start from the momenta  which are on scale n D 0. Since there are no self-energy clusters with exiting line on scale n D 0, for such  one has M(!  ; ") D 0. Next, we consider the momenta  which are on scale n D 1, and we can write (42), where now all self-energy clusters T whose values contribute to M(!  ; ") cannot contain any self-energy clusters, because the lines ` 2 T are on scale n` D 0. Then, we consider the momenta  which are on scale n D 2: again all the self-energy clusters contributing to M(!  ; ") do not contain any self-energy clusters, because the lines on scale n D 0; 1 cannot exit self-energy clusters by the construction of the previous step, and so on. The conclusion is that we have obtained a different expansion for h(! t), that we call a resummed series, h(! t) D

X 2Z d

e i!t h ;

where the self-energy clusters do not appear any more in the tree expansion and the propagators must be defined recursively: the propagator g` of a line ` on scale n` D n and momentum ` D  is the matrix

h D

1 X kD1

" k h[k] (") ; (43)

X

M[n] (!  ; ") :D

" k T VT (!  ) ;

(45)

T : n T
where the value VT (!  ) is written in accord with (34), with all the lines `0 2 L(T) on scales n`0 < n and the corresponding propagators g`0 expressed in terms of matrices M[n `0 ] (! `0 ; ") as in (44). By construction, the new propagators depend on ", so that the coefficients h[k] (") depend explicitly on ": hence (43) is not a power series expansion. The coefficients h[k] (") still admit a tree expansion h[k] (") D

X

Val( );

R 2T k;

Val( ) :D

 Y v2N( )

Fv

 ¤ 0; k  1 ;

 Y

 (46) g [n ` ] (!  ` ; ") ;

`2L( )

R which replaces (24). In particular T k; is defined as the set of renormalized trees of order k and momentum , where “renormalized” means that the trees do not contain any self-energy clusters. Now the propagators are matrices: the contractions of the indices in (46) yields that if ` connects v to v0 the indices of F v and Fv 0 associated with ` must be equal to the column and row indices of g` , respectively. R Since for any tree 2 T k; one has Nn ( ) D  Nn ( ), we can bound the product of propagators according to (36) provided the propagators on scale n can still be bounded proportionally to 22n . This is certainly not obvious, because of the extra term M[n] (!  ; ") appearing in (44). It is a remarkable cancellation that M[n] (x; ") vanishes up to second order, that is M[n] (x; ") D O(x 2 ), so that, by

Diagrammatic Methods in Classical Perturbation Theory

taking into account also that VT (x) ¤ 0 requires k T  2, we can write, for some constant C, [n] M[n] (x; ") D "2 x 2 M (x; ") ;

   [  M n](x; ")  C ; (47)

where k  k denotes – say – the uniform norm. The cancellation leading to (47) can be proved as discussed in Sect. “Multiscale Analysis” – where it has been explicitly proved for k D 2 in the absence of resummation. Now the propagators are matrices, but one can prove (by induction on n) that M[n] (x; ") D (M[n] (x; "))T D (M[n] (x; ")) , with T and † denoting transposition and adjointness, respectively, and this is enough to see that the same cancellation mechanism applies [23]. Thus (47) implies that  1     [n]   g (x; ") D  x 2  "2 x 2 M[n] (x; ")   2 ; (48)  x2  and the same argument as used in Sect. “Small Divisors” implies that the series in (43) for h converges for j"j < "0 , where "0 is defined as in (26) with C0 D 2 2 /C2 . Moreover (47) also implies that h(! t) is analytic in " (notwithstanding that the expansion is not a power expansion), so that we can say a posteriori that the original power series (25) also converges. Generalizations The case where the function G(u) in (1) does not vanish is notationally more involved, and it is explicitly discussed in Gentile [28]. Extensions of the results described above to Hamiltonian functions more general than (5) require only some notational complications, and can be found in Gentile & Mastropietro [37] for anisochronous systems and in Bartuccelli & Gentile [3] for isochronous systems. The case (5) with f a real analytic function was first discussed by using trees in Chierchia & Falcolini [12]. Lower-Dimensional Tori A tree formalism for hyperbolic lower-dimensional tori was introduced and studied in Gallavotti [19], Gentile [26] and Gallavotti, Gentile & Mastropietro [25], for systems consisting of a set of rotators interacting with a pendulum. In that case also the stable and unstable manifolds (whiskers) of the tori were studied with diagrammatic techniques. For Hamiltonian functions of the form (5) solutions of the form (43) describe quasi-periodic motions with

frequency vector ! on a d-dimensional manifold. If we fix t, say t D 0, and keep ˛0 as a parameter, we obtain a parametrization of the manifold in terms of ˛0 2 T d , hence the manifold is an invariant torus. We say that the torus is a maximal torus. We can study the problem of persistence of lowerdimensional tori by considering unperturbed solutions ˛(t) D ˛0 C ! t, where the components of ! are rationally dependent. For instance we can imagine that there exist s linearly independent vectors b 1 ; : : : ;b s 2 Zd such that !  b  k D 0 for k D 1; : : : ; s. In that case, we say that the unperturbed torus is a resonant torus of order s. We can imagine performing a linear change of variables which transforms the frequency vector into a new vector, that we ¯ 0), where !¯ 2 Rr still denote with !, such that ! D (!; s and 0 2 R , with r C s D d, and !¯ is irrational. This nat¯ ˇ), with ˛¯ 2 T r and urally suggests that we write ˛ D (˛; s ˇ 2 T . More generally, here and henceforth in this subsection for any vector v 2 Rd we denote by v¯ the vector in Rr whose components are the first r components of v. For instance for the initial phase ˛0 we write ˛0 D (˛¯ 0 ; ˇ0 ). In general the resonant torus is destroyed by the perturbation, and only some lower-dimensional tori persist under perturbation. We assume on !¯ one of the Diophantine conditions of Sect. “Small Divisors” in Rr , for instance j!¯  ¯ j > /j¯j for all ¯ 2 Zr , ¯ ¤ 0. To prove the existence of a maximal torus for the system with Hamiltonian function (5) we needed no condition on the perturbation f . On the contrary to prove existence of lower-dimensional tori, we need some nondegeneracy condition: by defining Z f0 (ˇ) D

Tr

d˛¯ ¯ ˇ) ; f (˛; (2)r

(49)

if @ˇ f0 (ˇ ) D 0 for some ˇ then we assume that the matrix @2ˇ f0 (ˇ ) is positive definite (more generally one could assume it to be non-singular, that is det @2ˇ f0 (ˇ ) ¤ 0). The formal analysis can be carried out as in Sect. “Trees and Graphical Representation”, with the only difference being that now the compatibility conditions ["@˛ f (˛)](k)  D 0 have to be imposed for all  such that ¯ D 0, because !   D !¯  ¯ . It turns out to be an identity only for the first r components (this is trivial for k D 1, whereas it requires some work for k > 1). For the last s components, for k D 1 it reads @ˇ f0 (ˇ0 ) D 0, hence it fixes ˇ0 to be a stationary point for f0 (ˇ), while for higher values of k it fixes the corrections of higher order of these values (to do this we need the non-degeneracy condition). Thus, we are free to choose only ˛¯ 0 as a free parameter, since the last s components of ˛0 have to be fixed.

137

138

Diagrammatic Methods in Classical Perturbation Theory

Clusters and self-energy clusters are defined as in Sect. “Multiscale Analysis”. Note that only the first r components ¯ of the momenta  intervene in the definition of the scales – again because !   D !¯  ¯ . In particular, in the definition of self-energy clusters, in (33) we must replace v with ¯ v . Thus, already to first order the value of a self-energy cluster can be non-zero: for k T D 1, that is for T containing only a node v with mode label (¯v ; ˜v ) D (0; ˜ v ), the matrix VT (x; ") is of the form  0 VT (x) D 0 (b˜ v ) i; j D e



0 ; b˜ v

i ˜ v ˇ0

(50)

(i˜v;i )(i˜v; j ) f(0;˜v ) ;

with i; j D r C 1; : : : ; d. If we sum over ˜ v 2 Zs and multiply times ", we obtain 

 0 0 ; 0 "B (51) X ˜ D e i ˇ0 (i˜ i )(i˜ j ) f(0;˜) D @ˇ i @ˇ j f0 (ˇ0 ) :

M0 :D B i; j

˜ 2Z s

The s  s block B is non-zero: in fact, the non-degeneracy condition yields that it is invertible. To higher orders one finds that the matrix M[n] (x; "), with x D !   D !¯  ¯ , is a self-adjoint matrix and M[n] (x; ") D (M[n] (x; "))T , as in the case of maximal tori. Moreover the corresponding eigenvalues [n] i (x; ") 2 x 2 ) for i D 1; : : : ; r and [n] (x; (x; ") D O(" satisfy [n] i i ") D O(") for i D r C 1; : : : ; d; this property is not trivial because of the off-diagonal blocks (which in general do not vanish at orders k  2), and to prove it one has to use the self-adjointness of the matrix M[n] (x; "). More 2 precisely one has [n] i (x; ") D "a i C O(" ) for i > r, where a rC1 ; : : : ; ad are the s eigenvalues of the matrix B in (51). From this point on the discussion proceeds in a very different way according to the sign of " (recall that we are assuming that a i > 0 for all i > r). 2 For " < 0 one has [n] i (x; ") D a i " C O(" ) < 0 for i > r, so that we can bound the last s eigenvalues of x 2  M[n] (x; ") with x2 , and the first r with x 2 /2 by the same argument as in Sect. “Resummation”. Hence we obtain easily the convergence of the series (43); of course, analyticity at the origin is prevented because of the condition " < 0. We say in that case that the lower-dimensional tori are hyperbolic. We refer to Gallavotti & Gentile [22] and Gallavotti et al. [23] for details. The case of elliptic lower-dimensional tori – that is " > 0 when a i > 0 for all i > r – is more difficult. Essentially the idea is as follows (we only sketch the strat-

egy: the details can be found in Gentile & Gallavotti [36]). One has to define the scales recursively, by using a variant, first introduced in Gentile [27], of the resummation technique described in Sect. “Resummation”. We say that  is on scale 0 if j!¯  ¯ j  and on scale [ 1] otherwise: for  on scale 0 we write (42) with M D M0 , as given in (51). This defines the propagators of the lines ` on scale n D 0 as 1  g` D g [0] (!  ` ) D (!¯  ¯ ` )2  M0 :

(52)

Denote by  i the eigenvalues of M 0 : given  on scale [ 1] wepsay that  is on scale 1 if 21  min iD1;:::;d pj(!¯  ¯ )2   i j, and on scale [ 2] if min iD1;:::;d j(!¯  ¯ )2   i j < 21 . For  on scale 1 we write (42) with M replaced by M[0] (!  ¯ ; "), which is given by M 0 plus the sum of the values of all self-energy clusters T on scale n T D 0. Then the propagators of the lines ` on scale n` D 1 is defined as  1 g` D g [1] (! ` ) D (!¯  ¯ ` )2  M[0] (!¯  ¯ ` ; ") : (53) [n] Call [n] i (x; ") the eigenvalues of M (x; "): given  on scale [ 2]qwe say that  is on scale 2 if 22 

¯  ¯ ; ")j, and on scale [ 3] j(!¯  ¯ )2  [0] i (! q ¯  ¯ ; ")j < 22 . For  if min iD1;:::;d j(!¯  ¯ )2  [0] i (! on scale 2 we write (42) with M replaced by M[1] (!¯  ¯ ; "), which is given by M[0] (!¯  ¯ ; ") plus the sum of the values of all self-energy clusters T on scale n T D 1. Thus, the propagators of the lines ` on scale n` D 2 will be defined as min iD1;:::;d

 1 ; (54) g` D g [2] (! ` ) D (!¯  ¯ ` )2  M[1] (!¯  ¯ ` ; ") and so on. The propagators are self-adjoint matrices, hence their norms can be bounded through the corresponding eigenvalues. In order to proceed as in Sects. “Multiscale Analysis” and “Resummation” we need some Diophantine conditions on these eigenvalues. We can assume for some  0 >  ˇ ˇ q ˇ ˇ ˇj!¯  ¯ j  j[n] (!¯  ¯ ; ")jˇ >

i ˇ ˇ j¯j0

8¯ ¤ 0 ; (55)

for all i D 1; : : : ; d and n  0. These are known as the first Melnikov conditions. Unfortunately, things do not proceed so plainly. In order to prove a bound like (35), possibly with a different  0 replacing , we need to compare the propagators of the

Diagrammatic Methods in Classical Perturbation Theory

lines entering and exiting clusters T which are not selfenergy clusters. This requires replacing (36) with 2nC2  ˇ ˇ q q ˇ ˇ ˇ!¯  (¯1 ¯2 ) ˙ j[n] (!¯  ¯ 1 ; ")j ˙ j[n] (!¯  ¯ 2 ; ")jˇ i j ˇ ˇ

; > j¯1  ¯ 2 j0 (56) for all i; j D 1; : : : ; d and choices of the signs ˙, and hence introduces further Diophantine conditions, known as the second Melnikov conditions. The conditions in (56) turn out to be too many, because for all n  0 and all ¯ 2 Zr such that ¯ D ¯ 1  ¯ 2 there are infinitely many conditions to be considered, one per pair (¯1 ; ¯2 ). However we can impose both the condi¯  ¯ ; "), tions (55) and (56) not for the eigenvalues [n] i (! ¯ (") independent of  and then but for some quantities [n] i use the smoothness of the eigenvalues in x to control ¯  ¯ ; ") in terms of (!¯  ¯ )2  [n] (!¯  ¯ )2  [n] i (! i ("). Even¯ we have to tually, beside the Diophantine condition on !, impose the Melnikov conditions ˇ ˇ q ˇ ˇ ˇj!¯  ¯ j  j[n] (")jˇ > 0 ; i ˇ ˇ j¯j ˇ ˇ (57) q q ˇ ˇ ˇj!¯  ¯ j ˙ j[n] (")j ˙ j[n] (")jˇ > 0 ; i j ˇ ˇ j¯j for all ¯ ¤ 0 and all n  0. Each condition in (57) leads us to eliminate a small interval of values of ". For the values of " which are left define h(! t) according to (43) and (46), with the new definition of the propagators. If " is small enough, say j"j < "0 , then the series (43) converges. Denote by E [0; "0 ] the set of values of " for which the conditions (57) are satisfied. One can prove that E is a Cantor set, that is a perfect, nowhere dense set. Moreover E has large relative Lebesgue measure, in the sense that lim

"!0

meas(E \ [0; "]) D1; "

(58)

provided  0 in (57) is large enough with respect to . The property (58) yields that, notwithstanding that we are eliminating infinitely many intervals, the measure of the union of all these intervals is small. If a i < 0 for all i > r we reason in the same way, simply exchanging the role of positive and negative ". On the contrary if a i D 0 for some i > r, the problem becomes much more difficult. For instance if s D 1 and arC1 D 0, then in general perturbation theory in " is not possible, not even at a formal level. However, under some conditions,

one can still construct fractional series in ", and prove that the series can be resummed [24]. Other Ordinary Differential Equations The formalism described above extends to other models, such as skew-product systems [29] and systems with strong damping in the presence of a quasi-periodic forcing term [32]. As an example of skew-product system  one can con sider the linear differential equation x˙ D A C " f (! t) x on SL(2; R), where  2 R, " is a small real parameter, ! 2 Rn is an irrational vector, and A; f 2 sl(2; R), with A is a constant matrix and f an analytic function periodic in its arguments. Trees for skew-products were considered by Iserles and Nørsett [44], but they used expansions in time, hence not suited for the study of global properties, such as quasi-periodicity. Quasi-periodically forced one-dimensional systems with strong damping are described by the ordinary differential equations x¨ C x˙ C g(x) D f (! t), where x 2 R, " D 1/ is a small real parameter, ! 2 Rn is irrational, and f ,g are analytic functions (g is the “force”), with f periodic in its arguments. We refer to the bibliography for details and results on the existence of quasi-periodic solutions. Bryuno Vectors The diagrammatic methods can be used to prove the any unperturbed maximal torus with frequency vector which is a Bryuno vector persists under perturbation for " small enough [31]. One could speculate whether the Bryuno condition (30) is optimal. In general the problem is open. However, in the case of the standard map – see Sect. “The Standard Map”, – one can prove [6,15] that, by considering the radius of convergence "0 of the perturbation series as a function of !, say "0 D 0 (!), then there exists a universal constant C such that jlog 0 (!) C 2B(!)j  C :

(59)

In particular this yields that the invariant curve with rotation number ! persists under perturbation if and only if ! is a Bryuno number. The proof of (59) requires the study of a more refined cancellation than that discussed in Sect. “Resummation”. We refer to Berretti & Gentile [6] and Gentile [28] for details. Extensions to Bryuno vectors for lower-dimensional tori can also be found in Gentile [31]. For the models considered in Sect. “Other Ordinary Differential Equations” we refer to Gentile [29] and Gentile et al. [33].

139

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Diagrammatic Methods in Classical Perturbation Theory

Partial Differential Equations Existence of quasi-periodic solutions in systems described by one-dimensional nonlinear partial differential equations (finite-dimensional tori in infinite-dimensional systems) was first studied by Kuksin [48], Craig and Wayne [14] and Bourgain [9]. In these systems, even the case of periodic solutions yields small divisors, and hence requires a multiscale analysis. The study of persistence of periodic solutions for nonlinear Schrödinger equations and nonlinear wave equations, with the techniques discussed here, can be found in Gentile & Mastropietro [39], Gentile et al. [40], and Gentile & Procesi [41]. The models are still described by (1), with G(u) D 0, but now D is given by D D @2t   C  in the case of the wave equation and by D D i@ t   C  in the case of the Schrödinger equation, where  is the Laplacian and  2 R. In dimension 1, one has  D @2x . If we look for periodic solutions with frequency ! it can be convenient to pass to Fourier space, where the operator D acts as   D : e i!ntCi mx ! ! 2 n2 C m2 C  e i!ntCi mx ; (60) for the wave equation; a similar expression holds for the Schrödinger equation. Therefore the kernel of D can be arbitrarily close to zero for n and m large enough. Then one can consider, say, (1) for x 2 [0; ] and F(u) D u3 , with Dirichlet boundary conditions u(0) D u() D 0, and study the existence of periodic solutions with frequency ! close to some of the unperturbed frequencies. We refer to the cited bibliography for results and proofs.

Conclusions and Future Directions The diagrammatic techniques described above have been applied also in cases where no small divisors appear; cf. Berretti & Gentile [4] and Gentile et al. [34]. Of course, such problems are much easier from a technical point of view, and can be considered as propaedeutic examples to become familiar with the tree formalism. Also the study of lower-dimensional tori becomes easy for r D 1 (periodic ¯ j  j!j ¯ for all ¯ ¤ 0, so solutions): in that case one has j!¯ ¯ 2k , that the product of the propagators is bounded by j!j and one can proceed as in Sect. “Small Divisors” to obtain analyticity of the solutions. In the case of hyperbolic lower-dimensional tori, if ! is a two-dimensional Diophantine vector of constant type (that is, with  D 1) the conjugation function h can be proved to be Borel summable [13]. Analogous considerations hold for the one-dimensional systems in the presence

of friction and of a quasiperiodic forcing term described in Sect. “Other Ordinary Differential Equations”; in that case one has Borel summability also for one-dimensional !, that is for periodic forcing [33]. It would be interesting to investigate whether Borel summability could be obtained for higher values of . Recently existence of finite-dimensional tori in the nonlinear Schrödinger equation in higher dimensions was proved by Bourgain [10]. It would be interesting to investigate how far the diagrammatic techniques extend to deal with such higher dimensional generalizations. The main problem is that (the analogues of) the second Melnikov conditions in (57) cannot be imposed. In certain cases the tree formalism was extended to non-analytic systems, such as some quasi-integrable systems of the form (5) with f in a class of Cp functions for some finite p [7,8]. However, up to exceptional cases, the method described here seems to be intrinsically suited in cases in which the vector fields are analytic. The reason is that in order to exploit the expansion (3), we need that F be infinitely many times differentiable and we need a bound on the derivatives. It is a remarkable property that the perturbation series can be given a meaning also in cases where the solutions are not analytic in ". An advantage of the diagrammatic method is that it allows rather detailed information about the solutions, hence it could be more convenient than other techniques to study problems where the underlying structure is not known or too poor to exploit general abstract arguments. Another advantage is the following. If one is interested not only in proving the existence of the solutions, but also in explicitly constructing them with any prefixed precision, this requires performing analytical or numerical computations with arbitrarily high accuracy. Then high perturbation orders have to be reached, and the easiest and most direct way to proceed is just through perturbation theory: so the approach illustrated here allows a unified treatment for both theoretical investigations and computational ones. The resummation technique described in Sect. “Resummation” can also be used for computational purposes. With respect to the naive power series expansion it can reduce the computation time required to approximate the solution within a prefixed precision. It can also provide accurate information on the analyticity properties of the solution. For instance, for the Kepler equation, Levi-Civita at the beginning of the last century described a resummation rule (see [49]), which gives immediately the radius of convergence of the perturbation series. Of course, in the case of small divisor problems, everything becomes much more complicated.

Diagrammatic Methods in Classical Perturbation Theory

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21. Gallavotti G (2006) Classical mechanics. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 1. Elsevier, Oxford 22. Gallavotti G, Gentile G (2002) Hyperbolic low-dimensional invariant tori and summations of divergent series. Comm Math Phys 227(3):421–460 23. Gallavotti G, Bonetto F, Gentile G (2004) Aspects of ergodic, qualitative and statistical theory of motion. Texts and Monographs in Physics. Springer, Berlin 24. Gallavotti G, Gentile G, Giuliani A (2006) Fractional Lindstedt series. J Math Phys 47(1):1–33 25. Gallavotti G, Gentile G, Mastropietro V (1999) A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems. J Math Phys 40(12):6430–6472 26. Gentile G (1995) Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dyn Stab Syst 10(3):269–308 27. Gentile G (2003) Quasi-periodic solutions for two-level systems. Comm Math Phys 242(1-2):221–250 28. Gentile G (2006) Brjuno numbers and dynamical systems. Frontiers in number theory, physics, and geometry. Springer, Berlin 29. Gentile G (2006) Diagrammatic techniques in perturbation theory. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 2. Elsevier, Oxford 30. Gentile G (2006) Resummation of perturbation series and reducibility for Bryuno skew-product flows. J Stat Phys 125(2): 321–361 31. Gentile G (2007) Degenerate lower-dimensional tori under the Bryuno condition. Ergod Theory Dyn Syst 27(2):427–457 32. Gentile G, Bartuccelli MV, Deane JHB (2005) Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms. J Math Phys 46(6):1–21 33. Gentile G, Bartuccelli MV, Deane JHB (2006) Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J Math Phys 47(7):1–10 34. Gentile G, Bartuccelli MV, Deane JHB (2007) Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies. Rev Math Phys 19(3):307–348 35. Gentile G, Cortez DA, Barata JCA (2005) Stability for quasiperiodically perturbed Hill’s equations. Comm Math Phys 260(2):403–443 36. Gentile G, Gallavotti G (2005) Degenerate elliptic tori. Comm Math Phys 257(2):319–362 37. Gentile G, Mastropietro V (1996) Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev Math Phys 8(3):393–444 38. Gentile G, Mastropietro V (2001) Renormalization group for one-dimensional fermions. A review on mathematical results. Renormalization group theory in the new millennium III. Phys Rep 352(4–6):273–437 39. Gentile G, Mastropietro V (2004) Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method. J Math Pures Appl 83(8):1019–1065 40. Gentile G, Mastropietro V, Procesi M (2005) Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Comm Math Phys 256(2):437– 490 41. Gentile G, Procesi M (2006) Conservation of resonant peri-

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Books and Reviews Berretti A, Gentile G (2001) Renormalization group and field theoretic techniques for the analysis of the Lindstedt series. Regul Chaotic Dyn 6(4):389–420 Gentile G (1999) Diagrammatic techniques in perturbations theory, and applications. Symmetry and perturbation theory. World Science, River Edge

Discrete Control Systems

Discrete Control Systems TAEYOUNG LEE1 , MELVIN LEOK2 , HARRIS MCCLAMROCH1 1 Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA 2 Department of Mathematics, Purdue University, West Lafayette, USA Article Outline Glossary Definition of the Subject Introduction Discrete Lagrangian and Hamiltonian Mechanics Optimal Control of Discrete Lagrangian and Hamiltonian Systems Controlled Lagrangian Method for Discrete Lagrangian Systems Future Directions Acknowledgments Bibliography Glossary Discrete variational mechanics A formulation of mechanics in discrete-time that is based on a discrete analogue of Hamilton’s principle, which states that the system takes a trajectory for which the action integral is stationary. Geometric integrator A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such as symplecticity, momentum preservation, and the structure of the configuration space. Lie group A differentiable manifold with a group structure where the composition is differentiable. The corresponding Lie algebra is the tangent space to the Lie group based at the identity element. Symplectic A map is said to be symplectic if given any initial volume in phase space, the sum of the signed projected volumes onto each position-momentum subspace is invariant under the map. One consequence of symplecticity is that the map is volume-preserving as well. Definition of the Subject Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader

field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control systems aims to address this deficiency by restricting the approximation to the choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure. Introduction The dynamics of Lagrangian and Hamiltonian systems have unique geometric properties; the Hamiltonian flow is symplectic, the total energy is conserved in the absence of non-conservative forces, and the momentum maps associated with symmetries of the system are preserved. Many interesting dynamics evolve on a non-Euclidean space. For example, the configuration space of a spherical pendulum is the two-sphere, and the configuration space of rigid body attitude dynamics has a Lie group structure, namely the special orthogonal group. These geometric features determine the qualitative behavior of the system, and serve as a basis for theoretical study. Geometric numerical integrators are numerical integration algorithms that preserve structures of the continuous dynamics such as invariants, symplecticity, and the configuration manifold (see [14]). The exact geometric properties of the discrete flow not only generate improved qualitative behavior, but also provide accurate and efficient numerical techniques. In this article, we view a geometric integrator as an intrinsically discrete dynamical system, instead of concentrating on the numerical approximation of a continuous trajectory.

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Numerical integration methods that preserve the simplecticity of a Hamiltonian system have been studied (see [28,36]). Coefficients of a Runge–Kutta method are carefully chosen to satisfy a simplecticity criterion and order conditions to obtain a symplectic Runge–Kutta method. However, it can be difficult to construct such integrators, and it is not guaranteed that other invariants of the system, such as a momentum map, are preserved. Alternatively, variational integrators are constructed by discretizing Hamilton’s principle, rather than discretizing the continuous Euler–Lagrange equation (see [31,34]). The resulting integrators have the desirable property that they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long times (see [2]). Lie group methods are numerical integrators that preserve the Lie group structure of the configuration space (see [18]). Recently, these two approaches have been unified to obtain Lie group variational integrators that preserve the geometric properties of the dynamics as well as the Lie group structure of the configuration space without the use of local charts, reprojection, or constraints (see [26,29,32]). Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. Discrete optimal control problems involve finding a control input for a discrete dynamic system such that an optimality objective is achieved with prescribed constraints. Optimality conditions are derived from the discrete equations of motion, described by a set of discrete equations. This approach is in contrast to traditional techniques where a discretization appears at the last stage to solve the optimality condition numerically. Discrete mechanics and optimal control approaches determine optimal control inputs and trajectories more accurately with less computational load (see [19]). Combined with an indirect optimization technique, they are substantially more efficient (see [17,23,25]). The geometric approach to mechanics can provide a theoretical basis for innovative control methodologies in geometric control theory. For example, these techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion. While the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. By applying geometric control algorithms to discrete mechanics that preserve geometric properties, we

obtain an exact numerical implementation of the geometric control theory. In particular, the method of controlled Lagrangian systems is based on the idea of adopting a feedback control to realize a modification of either the potential energy or the kinetic energy, referred to as potential shaping or kinetic shaping, respectively. These ideas are applied to construct a real-time digital feedback controller that stabilizes the inverted equilibrium of the cart-pendulum (see [10,11]). In this article, we will survey discrete Lagrangian and Hamiltonian mechanics, and their applications to optimal control and feedback control theory. Discrete Lagrangian and Hamiltonian Mechanics Mechanics studies the dynamics of physical bodies acting under forces and potential fields. In Lagrangian mechanics, the trajectory of the object is derived by finding the path that extremizes the integral of a Lagrangian over time, called the action integral. In many classical problems, the Lagrangian is chosen as the difference between kinetic energy and potential energy. The Legendre transformation provides an alternative description of mechanical systems, referred to as Hamiltonian mechanics. Discrete Lagrangian and Hamiltonian mechanics has been developed by reformulating the theorems and the procedures of Lagrangian and Hamiltonian mechanics in a discrete time setting (see, for example, [31]). Therefore, discrete mechanics has a parallel structure with the mechanics described in continuous time, as summarized in Fig. 1 for Lagrangian mechanics. In this section, we describe discrete Lagrangian mechanics in more detail, and we derive discrete Euler–Lagrange equations for several mechanical systems. Consider a mechanical system on a configuration space Q, which is the space of possible positions. The Lagrangian depends on the position and velocity, which are elements of the tangent bundle to Q, denoted by TQ. Let L : TQ ! R be the Lagrangian of the system. The discrete Lagrangian, Ld : Q  Q ! R is an approximation to the exact discrete Lagrangian, Z h (q ; q ) D L(q01 (t); q˙01 (t)) dt ; (1) Lexact 0 1 d 0

where q01 (0) D q0 , q01 (h) D q1 ; and q01 (t) satisfies the Euler–Lagrange equation in the time interval (0; h). A discrete action sum Gd : Q NC1 ! R, analogous to the action integral, is given by Gd (q0 ; q1 ; : : : ; q N ) D

N1 X kD0

Ld (q k ; q kC1 ) :

(2)

Discrete Control Systems

Discrete Control Systems, Figure 1 Procedures to derive continuous and discrete equations of motion

The discrete Hamilton’s principle states that ıGd D 0 for any ıq k , which yields the discrete Euler–Lagrange (DEL) equation, D2 Ld (q k1 ; q k ) C D1 Ld (q k ; q kC1 ) D 0 :

(3)

This yields a discrete Lagrangian flow map (q k1 ; q k ) 7! (q k ; q kC1 ). The discrete Legendre transformation, which from a pair of positions (q0 ; q1 ) gives a position-momentum pair (q0 ; p0 ) D (q0 ; D1 Ld (q0 ; q1 )) provides a discrete Hamiltonian flow map in terms of momenta. The discrete equations of motion, referred to as variational integrators, inherit the geometric properties of the continuous system. Many interesting Lagrangian and Hamiltonian systems, such as rigid bodies evolve on a Lie group. Lie group variational integrators preserve the nonlinear structure of the Lie group configurations as well as geometric properties of the continuous dynamics (see [29,32]). The basic idea for all Lie group methods is to express the update map for the group elements in terms of the group operation, g1 D g0 f0 ;

(4)

where g0 ; g1 2 G are configuration variables in a Lie group G, and f0 2 G is the discrete update represented

by a right group operation on g 0 . Since the group element is updated by a group operation, the group structure is preserved automatically without need of parametrizations, constraints, or re-projection. In the Lie group variational integrator, the expression for the flow map is obtained from the discrete variational principle on a Lie group, the same procedure presented in Fig. 1. But, the infinitesimal variation of a Lie group element must be carefully expressed to respect the structure of the Lie group. For example, it can be expressed in terms of the exponential map as ˇ d ˇˇ ıg D g exp  D g ; d ˇ D0 for a Lie algebra element  2 g. This approach has been applied to the rotation group SO(3) and to the special Euclidean group SE(3) for dynamics of rigid bodies (see [24,26,27]). Generalizations to arbitrary Lie groups gives the generalized discrete Euler–Poincaré (DEP) equation,   Te L f0  D2 Ld (g0 ; f0 )  Adf0  Te L f1  D2 Ld (g1 ; f1 ) (5) CTe L g 1  D1 Ld (g1 ; f1 ) D 0 ; for a discrete Lagrangian on a Lie group, Ld : G G ! R. Here L f : G ! G denotes the left translation map given by L f g D f g for f ; g 2 G, Tg L f : Tg G ! T f g G is the

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tangential map for the left translation, and Ad g : g ! g is the adjoint map. A dual map is denoted by a superscript  (see [30] for detailed definitions). We illustrate the properties of discrete mechanics using several mechanical systems, namely a mass-spring system, a planar pendulum, a spherical pendulum, and a rigid body. Example 1 (Mass-spring System) Consider a mass-spring system, defined by a rigid body that moves along a straight frictionless slot, and is attached to a linear spring. Continuous equation of motion: The configuration space is Q D R, and the Lagrangian L : R  R ! R is given by ˙ D L(q; q)

1 2 1 2 m q˙  q ; 2 2

(6)

where q 2 R is the displacement of the body measured from the point where the spring exerts no force. The mass of the body and the spring constant are denoted by m;  2 R, respectively. The Euler–Lagrange equation yields the continuous equation of motion. m q¨ C q D 0 :

(7)

Discrete equation of motion: Let h > 0 be a discrete time step, and a subscript k denotes the kth discrete variable at t D kh. The discrete Lagrangian Ld : RR ! R is an approximation of the integral of the continuous Lagrangian (6) along the solution of (7) over a time step. Here, we choose the following discrete Lagrangian. 

 q k C q kC1 q kC1  q k Ld (q k ; q kC1 ) D hL ; 2 h 1 h D m(q kC1  q k )2  (q k C q kC1 )2 : 2h 8 (8) Direct application of the discrete Euler–Lagrange equation to this discrete Lagrangian yields the discrete equations of motion. We develop the discrete equation of motion using the discrete Hamilton’s principle to illustrate the principles more explicitly. Let Gd : R NC1 ! R be the discrete P N1 action sum defined as Gd D kD0 Ld (q k ; q kC1 ), which approximates the action integral. The infinitesimal variation of the action sum can be written as ıGd D

 m h (q kC1  q k )  (q k C q kC1 ) h 4 kD0

 m h C ıq k  (q kC1  q k )  (q k C q kC1 ) : h 4 N1 X

ıq kC1

Since ıq0 D ıq N D 0, the summation index can be rewritten as ıGd D

N1 X kD1

n m ıq k  (q kC1  2q k C q k1 ) h 

o h (q kC1 C 2q k C q k1 ) : 4

From discrete Hamilton’s principle, ıGd D 0 for any ıq k . Thus, the discrete equation of motion is given by m (q kC1  2q k C q k1 ) h h (q kC1 C 2q k C q k1 ) D 0 : C 4

(9)

For a given (q k1 ; q k ), we solve the above equation to obtain q kC1 . This yields a discrete flow map (q k1 ; q k ) 7! (q k ; q kC1 ), and this process is repeated. The discrete Legendre transformation provides the discrete equation of motion in terms of the velocity as     h2  h2  ˙ q kC1 D h q k C 1  qk ; 1C 4m 4m h h q˙ kC1 D q˙ k  qk  q kC1 : 2m 2m

(10) (11)

For a given (q k ; q˙ k ), we compute q kC1 and q˙ kC1 by (10) and (11), respectively. This yields a discrete flow map (q k ; q˙k ) 7! (q kC1 ; q˙kC1 ). It can be shown that this variational integrator has second-order accuracy, which follows from the fact that the discrete action sum is a second-order approximation of the action integral. Numerical example: We compare computational properties of the discrete equations of motion given by (10) and (11) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg,  D 1 kg/s2 so that the natural p frequency is 1 rad/s. The initial conditions are q0 D 2 m, q˙0 D 0, and the total energy is E D 1 Nm. The simulation time is 200 sec, and the stepsize h D 0:035 of the discrete equations of motion is chosen such that the CPU times are the same for both methods. Figure 2a shows the computed total energy. The variational integrator preserves the total energy well. The mean variation is 2:73271013 Nm. But, there is a notable dissipation of the computed total energy for the Runge–Kutta method Example 2 (Planar Pendulum) A planar pendulum is a mass particle connected to a frictionless, one degree-offreedom pivot by a rigid massless link under a uniform gravitational potential. The configuration space is the one-

Discrete Control Systems

Discrete Control Systems, Figure 2 Computed total energy (RK45: blue, dotted, VI: red, solid)

sphere S1 D fq 2 R2 j kqk D 1g. While it is common to parametrize the one-sphere by an angle, we develop parameter-free equations of motion in the special orthogonal group SO(2), which is a group of 22 orthogonal matrices with determinant 1, i. e. SO(2) D fR 2 R22 j RT R D I22 ; det[R] D 1g. SO(2) is diffeomorphic to the one-sphere. It is also possible to develop global equations of motion on the one-sphere directly, as shown in the next example, but here we focus on the special orthogonal group in order to illustrate the key steps to develop a Lie group variational integrator. We first exploit the basic structures of the Lie group SO(2). Define a hat map ˆ, which maps a scalar ˝ to a 22 skew-symmetric matrix ˝ˆ as ˝ˆ D



0 ˝

 ˝ : 0

The set of 22 skew-symmetric matrices forms the Lie algebra so(2). Using the hat map, we identify so(2) with R. An inner product on so(2) can be induced from the inner product on R as h˝ˆ 1 ; ˝ˆ 2 i D 1/2tr[˝ˆ 1T ˝ˆ 2 ] D ˝1 ˝2 for any ˝1 ; ˝2 2 R. The matrix exponential is a local diffeomorphism from so(2) to SO(2) given by

cos ˝ exp ˝ˆ D sin ˝

 sin ˝ cos ˝



ˆ D 1 ml 2 ˝ 2 C mg l e T Re2 L(R; ˝) 2 2 (13) 1 2 ˆ ˆ D ml h˝; ˝i C mg l e2T Re2 ; 2 where the constant g 2 R is the gravitational acceleration. The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The second expression is used to define a discrete Lagrangian later. We choose the bases of the inertial frame and the body-fixed frame such that the unit vector along the gravity direction in the inertial frame, and the unit vector along the pendulum axis in the body-fixed frame are represented by the same vector e2 D [0; 1] 2 R2 . Thus, for example, the hanging attitude is represented by R D I22 . Here, the rotation matrix R 2 SO(2) represents the linear transformation from a representation of a vector in the body-fixed frame to the inertial frame. Since the special orthogonal group is not a linear vector space, the expression for the variation should be carefully chosen. The infinitesimal variation of a rotation matrix R 2 SO(2) can be written in terms of its Lie algebra element as ˇ ˇ ˇ d ˇˇ R exp ˆ D Rˆ exp ˆ ˇˇ D Rˆ ; (14) ıR D d ˇ D0

:

The kinematics equation for R 2 SO(2) can be written in terms of a Lie algebra element as R˙ D R˝ˆ :

as

(12)

Continuous equations of motion: The Lagrangian for a planar pendulum L : SO(2)so(2) ! R can be written

D0

where  2 R so that ˆ 2 so(2). The infinitesimal variation of the angular velocity is induced from this expression and (12) as ı ˝ˆ D ıRT R˙ C RT ı R˙ ˆ˙ ˆ T R˙ C RT (R˙ ˆ C R) D (R)

(15)

D ˆ ˝ˆ C ˝ˆ ˆ C ˆ˙ D ˆ˙ ; where we used the equality of mixed partials to compute ı R˙ as d/dt(ıR).

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RT ˆ Define the action integral to be G D 0 L(R; ˝)dt. The infinitesimal variation of the action integral is obtained by using (14) and (15). Hamilton’s principle yields the following continuous equations of motion. g ˝˙ C e2T Re1 D 0 ; (16) l R˙ D R˝ˆ :

(17)

where we use the fact that F ˆ F T D ˆ for any F 2 SO(2) and ˆ 2 so(2). Define an action sum Gd : SO(2) NC1 ! R as Gd D P N1 kD0 Ld (R k ; F k ). Using (21) and (22), the variation of the action sum is written as ıGd D

N1 X kD1

If we parametrize the rotation matrix as

 cos  sin RD ; sin cos

  1 T ml 2 F k1  F k1 2h 

these equations are equivalent to g (18) ¨ C sin D 0 : l Discrete equations of motion: We develop a Lie group variational integrator on SO(2). Similar to (4), define F k 2 SO(2) such that R kC1 D R k F k :

(19)

Thus, F k D RTk R kC1 represents the relative update between two integration steps. If we find F k 2 SO(2), the orthogonal structure is preserved through (19) since multiplication of orthogonal matrices is also orthogonal. This is a key idea of Lie group variational integrators. Define the discrete Lagrangian Ld : SO(2)SO(2) ! R to be 1 ml 2 hF k  I22 ; F k  I22 i Ld (R k ; F k ) D 2h h h C mg l e2T R k e2 C mg l e2T R kC1 e2 2 2 (20) 1 h 2 D ml tr[I22  F k ] C mg l e2T R k e2 2h 2 h C mg l e2T R kC1 e2 ; 2 which is obtained by an approximation h˝ˆ k ' RT (R kC1

2

  1  F k  F kT  hmg l e2T R k e1 ; ˆ k : 2h

From the discrete Hamilton’s principle, ıGd D 0 for any ˆ k . Thus, we obtain the Lie group variational integrator on SO(2) as 

3

   2h2 g T T  F k  F kT  F kC1  F kC1 e R kC1 e1 D 0; (23) l 2 R kC1 D R k F k :

(24)

For a given (R k ; F k ) and R kC1 D R k F k , (23) is solved to find F kC1 . This yields a discrete map (R k ; F k ) 7! (R kC1 ; F kC1 ). If we parametrize the rotation matrices R and F with and  and if we assume that   1, these equations are equivalent to 1 hg ( kC1  2 k C k ) C sin k D 0 : h l The discrete version of the Legendre transformation provides the discrete Hamiltonian map as follows. F k  F kT D 2h˝ˆ 

2

h2 g T e R k e1 ; l 2

R kC1 D R k F k ; ˝ kC1 D ˝ k 

(25) (26)

hg T hg T e2 R k e1  e R kC1 e1 : 2l 2l 2

(27)

k

 R k ) D F k  I22 , applied to the continuous Lagrangian given by (13). As for the continuous time case, expressions for the infinitesimal variations should be carefully chosen. The infinitesimal variation of a rotation matrix is the same as (14), namely ıR k D R k ˆ k ;

(21)

for  k 2 R, and the constrained variation of F k is obtained from (19) as ıF k D ıRTk R kC1 C RTk ıR kC1 D ˆ k F k C F k ˆ kC1 D F k (ˆ  kC1  F kT ˆ k F k ) (22)  kC1  ˆ k ) ; D F k (ˆ

For a given (R k ; ˝ k ), we solve (25) to obtain F k . Using this, (R kC1 ; ˝ kC1 ) is obtained from (26) and (27). This yields a discrete map (R k ; ˝ k ) 7! (R kC1 ; ˝ kC1 ). Numerical example: We compare the computational properties of the discrete equations of motion given by (25)–(27) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg, l D 9:81 m. The initial conditions are 0 D /2 rad, ˝ D 0, and the total energy is E D 0 Nm. The simulation time is 1000 sec, and the step-size h D 0:03 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 2b shows the computed total energy for both methods. The variational integrator preserves the total energy well. There is no drift in the computed total energy, and the

Discrete Control Systems

mean variation is 1:0835102 Nm. But, there is a notable dissipation of the computed total energy for the Runge– Kutta method. Note that the computed total energy would further decrease as the simulation time increases.

Since q˙ D !  q for some angular velocity ! 2 R3 satisfying !  q D 0, this can also be equivalently written as

Example 3 (Spherical Pendulum) A spherical pendulum is a mass particle connected to a frictionless, two degree-offreedom pivot by a rigid massless link. The mass particle acts under a uniform gravitational potential. The configuration space is the two-sphere S2 D fq 2 R3 j kqk D 1g. It is common to parametrize the two-sphere by two angles, but this description of the spherical pendulum has a singularity. Any trajectory near the singularity encounters numerical ill-conditioning. Furthermore, this leads to complicated expressions involving trigonometric functions. Here we develop equations of motion for a spherical pendulum using the global structure of the two-sphere without parametrization. In the previous example, we develop equations of motion for a planar pendulum using the fact that the one-sphere S1 is diffeomorphic to the special orthogonal group SO(2). But, the two-sphere is not diffeomorphic to a Lie group. Instead, there exists a natural Lie group action on the two-sphere. That is the 3-dimensional special orthogonal group SO(3), a group of 33 orthogonal matrices with determinant 1, i. e. SO(3) D fR 2 R33 j RT R D I33 ; det[R] D 1g. The special orthogonal group SO(3) acts on the two-sphere in a transitive way; for any q1 ; q2 2 S2 , there exists a R 2 SO(3) such that q2 D Rq1 . Thus, the discrete update for the two-sphere can be expressed in terms of a rotation matrix as (19). This is a key idea to develop a discrete equations of motion for a spherical pendulum. Continuous equations of motion: Let q 2 S2 be a unit vector from the pivot point to the point mass. The Lagrangian for a spherical pendulum can be written as

(32)

1 ˙ D ml 2 q˙  q˙ C mg l e3  q ; L(q; q) 2

(28)

where the gravity direction is assumed to be e3 D [0; 0; 1] 2 R3 . The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The infinitesimal variation of the unit vector q can be written in terms of the vector cross product as ıq D   q ;

(29)

where  2 R3 is constrained to be orthogonal to the unit vector, i. e.  q D 0. Using this expression for the infinitesimal variation, Hamilton’s principle yields the following continuous equations of motion. g ˙ C (q  (q  e3 )) D 0 : q¨ C (q˙  q)q l

(30)

g q  e3 D 0 ; l q˙ D !  q :

!˙ 

(31)

These are global equations of motion for a spherical pendulum; these are much simpler than the equations expressed in term of angles, and they have no singularity. Discrete equations of motion: We develop a variational integrator for the spherical pendulum defined on S2 . Since the special orthogonal group acts on the two-sphere transitively, we can define the discrete update map for the unit vector as q kC1 D F k q k

(33)

for F k 2 SO(3). The rotation matrix F k is not uniquely defined by this condition, since exp(qˆ k ), which corresponds to a rotation about the qk direction fixes the vector qk . Consequently, if F k satisfies (33), then F k exp(qˆ k ) does as well. We avoid this ambiguity by requiring that F k does not rotate about qk , which can be achieved by letting F k D exp( fˆk ), where f k  q k D 0. Define a discrete Lagrangian Ld : S2  S2 ! R to be Ld (q k ; q kC1 ) D

1 ml 2 (q kC1  q k )  (q kC1  q k ) 2h h h C mg l e3  q k C mg l e3  q kC1 : 2 2

The variation of qk is the same as (29), namely ıq k D  k  q k

(34)

for  k 2 R3 with a constraint  k  q k D 0. Using this discrete Lagrangian and the expression for the variation, discrete Hamilton’s principle yields the following discrete equations of motion for a spherical pendulum.   h2 g q k  e3  q k q kC1 D h! k C 2l 2 !1/2    h2 g  C 1   h! k C qk ; q k  e3   2l hg hg ! kC1 D ! k C q k  e3 C q kC1  e3 : 2l 2l

(35) (36)

Since an explicit solution for F k 2 SO(3) can be obtained in this case, the rotation matrix F k does not appear in the equations of motion. This variational integrator on S2 exactly preserves the unit length of qk , the constraint

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Discrete Control Systems, Figure 3 Numerical simulation of a spherical pendulum (RK45: blue, dotted, VI: red, solid)

q k  ! k D 0, and the third component of the angular velocity ! k  e3 which is conserved since gravity exerts no moment along the gravity direction e3 . Numerical example: We compare the computational properties of the discrete equations of motion given by (35) and (36) with a 4(5)th order variable step size Runge–Kutta method for (31) and (32). We choose m D 1p kg, l D 9:81 m. The p initial conditions are q0 D [ 3/2; 0; 1/2], !0 D 0:1[ 3; 0; 3] rad/sec, and the total energy is E D 0:44 Nm. The simulation time is 200 sec, and the step-size h D 0:05 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 3 shows the computed total energy and the unit length errors. The variational integrator preserves the total energy and the structure of the two-sphere well. The mean total energy deviation is 1:5460104 Nm, and the mean unit length error is 3:2476  1015 . But, there is a notable dissipation of the computed total energy for the Runge–Kutta method. The Runge–Kutta method also fails to preserve the structure of the two-sphere. The mean unit length error is 1:0164102 . Example 4 (Rigid Body in a Potential Field) Consider a rigid body under a potential field that is dependent on the position and the attitude of the body. The configuration space is the special Euclidean group, which is a semidirect product of the special orthogonal group and Eus R3 . clidean space, i. e. SE(3) D SO(3)  Continuous equations of motion: The equations of motion for a rigid body can be developed either from Hamilton’s principle (see [26]) in a similar way as Example 2, or directly from the generalized discrete Euler–Poincaré equation given at (5). Here, we summarize the results. Let m 2 R and J 2 R33 be the mass and the moment of inertia matrix of a rigid body. For (R; x) 2 SE(3), the linear

transformation from the body-fixed frame to the inertial frame is denoted by the rotation matrix R 2 SO(3), and the position of the mass center in the inertial frame is denoted by a vector x 2 R3 . The vectors ˝; v 2 R3 are the angular velocity in the body-fixed frame, and the translational velocity in the inertial frame, respectively. Suppose that the rigid body acts under a configuration-dependent potential U : SE(3) ! R. The continuous equations of motion for the rigid body can be written as R˙ D R˝ˆ ;

(37)

x˙ D v ;

(38)

J ˝˙ C ˝  J˝ D M ;

(39)

@U ; @x

(40)

m˙v D 

where the hat map ˆ is an isomorphism from R3 to 3  3 skew-symmetric matrices so(3), defined such that xˆ y D x  y for any x; y 2 R3 . The moment due to the potential M 2 R3 is obtained by the following relationship: T ˆ D @U R  RT @U : M @R @R

(41)

The matrix @U/@R 2 R33 is defined such that [@U/@R] i j D@U/@[R] i j for i; j 2 f1; 2; 3g, where the i, jth element of a matrix is denoted by [] i j . Discrete equations of motion: The corresponding discrete equations of motion are given by

b

h J ˝k C

h2 ˆ M k D F k Jd  Jd F kT ; 2

R kC1 D R k F k ;

(42) (43)

Discrete Control Systems

Discrete Control Systems, Figure 4 Numerical simulation of a dumbbell rigid body (LGVI: red, solid, RK45 with rotation matrices: blue, dash-dotted, RK45 with quaternions: black, dotted)

h2 @U k ; 2m @x k h h D F kT J˝ k C F kT M k C M kC1 ; 2 2 h Uk h @U kC1 D mv k   ; 2 @x k 2 @x kC1

x kC1 D x k C hv k 

(44)

J˝ kC1

(45)

mv kC1

(46)

where Jd 2 R33 is a non-standard moment of inertia matrix defined as Jd D1/2tr[J]I33  J. For a given (R k ; x k ; ˝ k ; v k ), we solve the implicit Eq. (42) to find F k 2 SO(3). Then, the configuration at the next step R kC1 ;x kC1 is obtained by (43) and (44), and the moment and force M kC1;(@U kC1 )/(@x kC1 ) can be computed. Velocities ˝ kC1 ;v kC1 are obtained from (45) and (46). This defines a discrete flow map, (R k ; x k ; ˝ k ; v k ) 7! (R kC1 ; x kC1 ; ˝ kC1 ; v kC1 ), and this process can be repeated. This Lie group variational integrator on SE(3) can be generalized to multiple rigid bodies acting under their mutual gravitational potential (see [26]). Numerical example: We compare the computational properties of the discrete equations of motion given by (42)–(46) with a 4(5)th order variable step size Runge– Kutta method for (37)–(40). In addition, we compute the attitude dynamics using quaternions on the unit threesphere S3 . The attitude kinematics Eq. (37) is rewritten in terms of quaternions, and the corresponding equations are integrated by the same Runge–Kutta method. We choose a dumbbell spacecraft, that is two spheres connected by a rigid massless rod, acting under a central gravitational potential. The resulting system is a restricted full two body problem. The dumbbell spacecraft model has an analytic expression for the gravitational potential, resulting in a nontrivial coupling between the attitude dynamics and the orbital dynamics. As shown in Fig. 4a, the initial conditions are chosen such that the resulting motion is a near-circular orbit

combined with a rotational motion. Figure 4b and c show the computed total energy and the orthogonality errors of the rotation matrix. The Lie group variational integrator preserves the total energy and the Lie group structure of SO(3). The mean total energy deviation is 2:5983104 , and the mean orthogonality error is 1:8553  1013 . But, there is a notable dissipation of the computed total energy and the orthogonality error for the Runge–Kutta method. The mean orthogonality errors for the Runge– Kutta method are 0.0287 and 0.0753, respectively, using kinematics equation with rotation matrices, and using the kinematics equation with quaternions. Thus, the attitude of the rigid body is not accurately computed for Runge– Kutta methods. It is interesting to see that the Runge– Kutta method with quaternions, which is generally assumed to have better computational properties than the kinematics equation with rotation matrices, has larger total energy error and orthogonality error. Since the unit length of the quaternion vector is not preserved in the numerical computations, orthogonality errors arise when converted to a rotation matrix. This suggests that it is critical to preserve the structure of SO(3) in order to study the global characteristics of the rigid body dynamics. The importance of simultaneously preserving the symplectic structure and the Lie group structure of the configuration space in rigid body dynamics can be observed numerically. Lie group variational integrators, which preserve both of these properties, are compared to methods that only preserve one, or neither, of these properties (see [27]). It is shown that the Lie group variational integrator exhibits greater numerical accuracy and efficiency. Due to these computational advantages, the Lie group variational integrator has been used to study the dynamics of the binary near-Earth asteroid 66391 (1999 KW 4 ) in joint work between the University of Michigan and the Jet Propulsion Laboratory, NASA (see [37]).

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Optimal Control of Discrete Lagrangian and Hamiltonian Systems Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. The equations of motion for a system are constrained by Lagrange multipliers, and necessary conditions for optimality is obtained by the calculus of variations. The solution for the corresponding two point boundary value problem provides the optimal control input. Alternatively, a sub-optimal control law is obtained by approximating the control input history with finite data points. Discrete optimal control problems involve finding a control input for a given system described by discrete Lagrangian and Hamiltonian mechanics. The control inputs are parametrized by their values at each discrete time step, and the discrete equations of motion are derived from the discrete Lagrange–d’Alembert principle [21], ı

N1 X kD0

Ld (q k ; q kC1 ) D

N1 X

[Fd (q k ; q kC1 )  ıq k

kD0

C FdC (q k ; q kC1 )  ıq kC1 ] ; which modifies the discrete Hamilton’s principle by taking into account the virtual work of the external forces. Discrete optimal control is in contrast to traditional techniques such as collocation, wherein the continuous equations of motion are imposed as constraints at a set of collocation points, since this approach induces constraints on the configuration at each discrete timestep. Any optimal control algorithm can be applied to the discrete Lagrangian or Hamiltonian system. For an indirect method, our approach to a discrete optimal control problem can be considered as a multiple stage variational problem. The discrete equations of motion are derived by the discrete variational principle. The corresponding variational integrators are imposed as dynamic constraints to be satisfied by using Lagrange multipliers, and necessary conditions for optimality, expressed as discrete equations on multipliers, are obtained from a variational principle. For a direct method, control inputs can be optimized by using parameter optimization tools such as a sequential quadratic programming. The discrete optimal control can be characterized by discretizing the optimal control problem from the problem formulation stage. This method has substantial computational advantages when used to find an optimal control law. As discussed in the previous section, the discrete dynamics are

more faithful to the continuous equations of motion, and consequently more accurate solutions to the optimal control problem are obtained. The external control inputs break the Lagrangian and Hamiltonian system structure. For example, the total energy is not conserved for a controlled mechanical system. But, the computational superiority of the discrete mechanics still holds for controlled systems. It has been shown that the discrete dynamics is more reliable even for controlled system as it computes the energy dissipation rate of controlled systems more accurately (see [31]). For example, this feature is extremely important in computing accurate optimal trajectories for long term spacecraft attitude maneuvers using low energy control inputs. The discrete dynamics does not only provide an accurate optimal control input, but also enables us to find it efficiently. For the indirect optimal control approach, optimal solutions are usually sensitive to a small variation of multipliers. This causes difficulties, such as numerical ill-conditioning, when solving the necessary conditions for optimality expressed as a two point boundary value problem. Sensitivity derivatives along the discrete necessary conditions do not have numerical dissipation introduced by conventional numerical integration schemes. Thus, they are numerically more robust, and the necessary conditions can be solved computationally efficiently. For the direct optimal control approach, optimal control inputs can be obtained by using a larger discrete step size, which requires less computational load. We illustrate the basic properties of the discrete optimal control using optimal control problems for the spherical pendulum and the rigid body model presented in the previous section. Example 5 (Optimal Control of a Spherical Pendulum) We study an optimal control problem for the spherical pendulum described in Example 3. We assume that an external control moment u 2 R3 acts on the pendulum. Control inputs are parametrized by their values at each time step, and the discrete equations of motion are modified to include the effects of the external control inputs by using the discrete Lagrange–d’Alembert principle. Since the component of the control moment that is parallel to the direction along the pendulum has no effect, we parametrize the control input as u k D q k w k for w k 2 R3 . The objective is to transfer the pendulum from a given initial configuration (q0 ; !0 ) to a desired configuration (qd ; ! d ) during a fixed maneuver time N, while minimizing the square of the weighted l2 norm of the control moments. N N X X h T h min J D uk uk D (q k  w k )T (q k  w k ) : wk 2 2 kD0

kD0

Discrete Control Systems

Discrete Control Systems, Figure 5 Optimal control of a spherical pendulum

We solve this optimal control problem by using a direct numerical optimization method. The terminal boundary condition is imposed as an equality constraint, and the 3(N C 1) control input parameters fw k g N kD0 are numerically optimized using sequential quadratic programming. This method is referred to as a DMOC (Discrete Mechanics and Optimal Control) approach (see [19]). Figure 5 shows a optimal solution transferring the spherical pendulum from a hanging configuration given by (q0 ; !0 ) D (e3 ; 031 ) to an inverted configuration (qd ; ! d ) D (e3 ; 031 ) during 1 second. The time step size is h D 0:033. Experiment have shown that the DMOC approach can compute optimal solutions using larger step sizes than typical Runge–Kutta methods, and consequently, it requires less computational load. In this case, using a second-order accurate Runge–Kutta method, the same optimization code fails while giving error messages of inaccurate and singular gradient computations. It is presumed that the unit length errors of the Runge– Kutta method, shown in Example 3, cause numerical instabilities for the finite-difference gradient computations required for the sequential quadratic programming algorithm. Example 6 (Optimal Control of a Rigid Body in a Potential Field) We study an optimal control problem of a rigid body using a dumbbell spacecraft model described in Example 4 (see [25] for detail). We assume that external control forces u f 2 R3 , and control moment u m 2 R3 act on the dumbbell spacecraft. Control inputs are parametrized by their values at each time step, and the Lie group variational integrators are modified to include the effects of the external control inputs by using discrete Lagrange– d’Alembert principle. The objective is to transfer the dumbbell from a given initial configuration (R0 ; x0 ; ˝0 ; v0 ) to a desired configuration (R d ; x d ; ˝ d ; v d ) during a fixed maneuver

time N, while minimizing the square of the l2 norm of the control inputs. N1 X

min J D

u kC1

kD0

T h  f T h f Wm u m u kC1 Wf u kC1C u m kC1 kC1 ; 2 2

where Wf ; Wm 2 R33 are symmetric positive definite matrices. Here we use a modified version of the discrete equations of motion with first order accuracy, as it yields a compact form for the necessary conditions. Necessary conditions for optimality: We solve this optimal control problem by using an indirect optimization method, where necessary conditions for optimality are derived using variational arguments, and a solution of the corresponding two-point boundary value problem provides the optimal control. This approach is common in the optimal control literature; here the optimal control problem is discretized at the problem formulation level using the Lie group variational integrator presented in Sect. “Discrete Lagrangian and Hamiltonian Mechanics”. Ja D

N1 X kD0

h  f T f f h  m T m m u W u kC1 C W u kC1 u 2 kC1 2 kC1

C 1;T k fx kC1 C x k C hv k g

 @U kC1 f 2;T mv kC1 C mv k  h C hu kC1 C k @x kC1  _ logm(F k  RTk R kC1 ) C 3;T k  ˚  C 4;T J˝ kC1 C F kT J˝ k C h M kC1 C u m kC1 ; k where 1k ; 2k ; 3k ; 4k 2 R3 are Lagrange multipliers. The matrix logarithm is denoted by logm : SO(3) ! so(3) and the vee map _ : so(3) ! R3 is the inverse of the hat map introduced in Example 4. The logarithm form of (43) is used, and the constraint (42) is considered implicitly using constrained variations. Using similar expressions for the

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Discrete Control Systems, Figure 6 Optimal control of a rigid body

variation of the rotation matrix and the angular velocity given in (14) and (15), the infinitesimal variation can be written as N1 n o X f ;T f hıu k W f u k C 2k1 ıJa D kD1

˚ 4 C hıu m;T Wm u m k C  k1 k ˚ C zTk  k1 C ATk  k ; where  k D [1k ; 2k ; 3k ; 4k ] 2 R12 , and z k 2 R12 represents the infinitesimal variation of (R k ; x k ; ˝ k ; v k ), given by z k D [logm(RTk ıR k )_ ; ıx k ; ı˝ k ; ıv k ]. The matrix A k 2 R1212 is defined in terms of (R k ; x k ; ˝ k ; v k ). Thus, necessary conditions for optimality are given by f

u kC1 D Wf1 2k ;

(47)

1 4 um kC1 D Wm  k ;

(48)

 k D ATkC1  kC1

(49)

together with the discrete equations of motion and the boundary conditions. Computational approach: Necessary conditions for optimality are expressed in terms of a two point boundary problem. The problem is to find the optimal discrete flow, multipliers, and control inputs to satisfy the equations of motion, optimality conditions, multiplier equations, and boundary conditions simultaneously. We use a neighboring extremal method (see [12]). A nominal solution satisfying all of the necessary conditions except the boundary conditions is chosen. The unspecified initial multiplier is updated by successive linearization so as to satisfy the specified terminal boundary conditions in the limit. This is also referred to as the shooting method. The main advantage of the neighboring extremal method is that the number of iteration variables is small.

The difficulty is that the extremal solutions are sensitive to small changes in the unspecified initial multiplier values. The nonlinearities also make it hard to construct an accurate estimate of sensitivity, thereby resulting in numerical ill-conditioning. Therefore, it is important to compute the sensitivities accurately to apply the neighboring extremal method. Here the optimality conditions (47) and (48) are substituted into the equations of motion and the multiplier equations, which are linearized to obtain sensitivity derivatives of an optimal solution with respect to boundary conditions. Using this sensitivity, an initial guess of the unspecified initial conditions is iterated to satisfy the specified terminal conditions in the limit. Any type of Newton iteration can be applied. We use a line search with backtracking algorithm, referred to as Newton–Armijo iteration (see [22]). Figure 6 shows optimized maneuvers, where a dumbbell spacecraft on a reference circular orbit is transferred to another circular orbit with a different orbital radius and inclination angle. Figure 6a shows the violation of the terminal boundary condition according to the number of iterations on a logarithmic scale. Red circles denote outer iterations in the Newton–Armijo iteration to compute the sensitivity derivatives. The error in satisfaction of the terminal boundary condition converges quickly to machine precision after the solution is close to the local minimum at around the 20th iteration. These convergence results are consistent with the quadratic convergence rates expected of Newton methods with accurately computed gradients. The neighboring extremal method, also referred to as the shooting method, is numerically efficient in the sense that the number of optimization parameters is minimized. But, in general, this approach may be prone to numerical ill-conditioning (see [3]). A small change in the initial multiplier can cause highly nonlinear behavior of the terminal attitude and angular momentum. It is difficult

Discrete Control Systems

to compute the gradient for Newton iterations accurately, and the numerical error may not converge. However, the numerical examples presented in this article show excellent numerical convergence properties. The dynamics of a rigid body arises from Hamiltonian mechanics, which have neutral stability, and its adjoint system is also neutrally stable. The proposed Lie group variational integrator and the discrete multiplier equations, obtained from variations expressed in the Lie algebra, preserve the neutral stability property numerically. Therefore the sensitivity derivatives are computed accurately. Controlled Lagrangian Method for Discrete Lagrangian Systems The method of controlled Lagrangians is a procedure for constructing feedback controllers for the stabilization of relative equilibria. It relies on choosing a parametrized family of controlled Lagrangians whose corresponding Euler–Lagrange flows are equivalent to the closed loop behavior of a Lagrangian system with external control forces. The condition that these two systems are equivalent results in matching conditions. Since the controlled system can now be viewed as a Lagrangian system with a modified Lagrangian, the global stability of the controlled system can be determined directly using Lyapunov stability analysis. This approach originated in Bloch et al. [5] and was then developed in Auckly et al. [1]; Bloch et al. [6,7,8,9]; Hamberg [15,16]. A similar approach for Hamiltonian controlled systems was introduced and further studied in the work of Blankenstein, Ortega, van der Schaft, Maschke, Spong, and their collaborators (see, for example, [33,35] and related references). The two methods were shown to be equivalent in Chang et al. [13] and a nonholonomic version was developed in Zenkov et al. [40,41], and Bloch [4]. Since the method of controlled Lagrangians relies on relating the closed-loop dynamics of a controlled system with the Euler–Lagrange flow associated with a modified Lagrangian, it is natural to discretize this approach through the use of variational integrators. In Bloch et al. [10,11], a discrete theory of controlled Lagrangians was developed for variational integrators, and applied to the feedback stabilization of the unstable inverted equilibrium of the pendulum on a cart. The pendulum on a cart is an example of an underactuated control problem, which has two degrees of freedom, given by the pendulum angle and the cart position. Only the cart position has control forces acting on it, and the stabilization of the pendulum has to be achieved indirectly through the coupling between the pendulum and the cart. The controlled Lagrangian is obtained by modifying the

kinetic energy term, a process referred to as kinetic shaping. Similarly, it is possible to modify the potential energy term using potential shaping. Since the pendulum on a cart model involves both a planar pendulum, and a cart that translates in one-dimension, the configuration space is a cylinder, S1 R. Continuous kinetic shaping: The Lagrangian has the form kinetic minus potential energy ˙ D L(q; q)

1  ˙2 ˛ C 2ˇ( ) ˙ s˙ C s˙2  V(q); 2

(50)

and the corresponding controlled Euler–Lagrange dynamics is d @L @L D0;  dt @ ˙ @ d @L Du; dt @˙s

(51) (52)

where u is the control input. Since the potential energy is translation invariant, i. e., V(q) D V( ), the relative equilibria D e , s˙ D const are unstable and given by non-degenerate critical points of V( ). To stabilize the relative equilibria D e , s˙ D const with respect to , kinetic shaping is used. The controlled Lagrangian in this case is defined by 1 ˙ D L( ; ˙ ; s˙ C ( ) ˙ ) C  (( ) ˙ )2 ; (53) L; (q; q) 2 where ( ) D ˇ( ). This velocity shift corresponds to a new choice of the horizontal space (see [8] for details). The dynamics is just the Euler–Lagrange dynamics for controlled Lagrangian (53), d @L; @L;  D0; dt @ ˙ @ d @L; D0: dt @˙s

(54) (55)

The Lagrangian (53) satisfies the simplified matching conditions of Bloch et al. [9] when the kinetic energy metric coefficient in (50) is constant.  Setting u D d ( ) ˙ /dt defines the control input, makes Eqs. (52) and (55) identical, and results in controlled momentum conservation by dynamics (51) and (52). Setting  D 1/  makes Eqs. (51) and (54) identical when restricted to a level set of the controlled momentum. Discrete kinetic shaping: Here, we adopt the following notation: q kC1/2 D

q k C q kC1 ; 2 q k D q kC1  q k ;

q k D ( k ; s k ) :

155

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Discrete Control Systems

Then, a second-order accurate discrete Lagrangian is given by, Ld (q k ; q kC1 ) D hL(q kC1/2 ; q k /h) : The discrete dynamics is governed by the equations @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D0; @ k @ k @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D u k ; @s k @s k

(56) (57)

where u k is the control input. Similarly, the discrete controlled Lagrangian is, ; (q kC1/2 ; q k /h) ; L; d (q k ; q kC1 ) D hL

with discrete dynamics given by, @L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0; @ k @ k

@L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0: @s k @s k

Numerical example: Simulating the behavior of the discrete controlled Lagrangian system involves viewing Eqs. (58)–(59) as an implict update map (q k2 ; q k1 ) 7! (q k1 ; q k ). This presupposes that the initial conditions are given in the form (q0 ; q1 ); however it is generally preferable to specify the initial conditions as (q0 ; q˙0 ). This is achieved by solving the boundary condition,

(58) (59)

@L (q0 ; q˙0 ) C D1 Ld (q0 ; q1 ) C Fd (q0 ; q1 ) D 0 ; @q˙ for q1 . Once the initial conditions are expressed in the form (q0 ; q1 ), the discrete evolution can be obtained using the implicit update map. We first consider the case of kinetic shaping on a level surface, when  is twice the critical value, and without dissipation. Here, h D 0:05 sec, m D 0:14 kg, M D 0:44 kg, and l D 0:215 m. As shown in Fig. 7, the dynamics is stabilized, but since there is no dissipation, the oscillations are sustained. The s dynamics exhibits both a drift and oscillations, as potential shaping is necessary to stabilize the translational dynamics.

Equation (59) is equivalent to the discrete controlled momentum conservation: Future Directions pk D  ; where @L; d (q k ; q kC1 ) @s k (1 C )ˇ( kC1/2 ) k C s k D : h

pk D 

Setting uk D 

 k ( kC1/2 )   k1 ( k1/2 ) h

makes Eqs. (57) and (59) identical and allows one to represent the discrete momentum equation (57) as the discrete momentum conservation law p k D p. The condition that (56)–(57) are equivalent to (58)– (59) yield the discrete matching conditions. The dynamics determined by Eqs. (56)–(57) restricted to the momentum level p k D p is equivalent to the dynamics of Eqs. (58)– (59) restricted to the momentum level p k D  if and only if the matching conditions  D hold.

1 ;



D

p ; 1 C 

Discrete Receding Horizon Optimal Control: The existing work on discrete optimal control has been primarily focused on constructing the optimal trajectory in an open loop sense. In practice, model uncertainty and actuation errors necessitate the use of feedback control, and it would be interesting to extend the existing work on optimal control of discrete systems to the feedback setting by adopting a receding horizon approach. Discrete State Estimation: In feedback control, one typically assumes complete knowledge regarding the state of the system, an assumption that is often unrealistic in practice. The general problem of state estimation in the context of discrete mechanics would rely on good numerical methods for quantifying the propagation of uncertainty by solving the Liouville equation, which describes the evolution of a phase space distribution function advected by a prescribed vector field. In the setting of Hamiltonian systems, the solution of the Liouville equation can be solved by the method of characteristics (Scheeres et al. [38]). This implies that a collocational approach (Xiu [39]) combined with Lie group variational integrators, and interpolation based on noncommutative harmonic analysis on Lie groups could yield an efficient means of propagating uncertainty, and serve as the basis of a discrete state estimation algorithm.

Discrete Control Systems

Discrete Control Systems, Figure 7 Discrete controlled dynamics with kinetic shaping and without dissipation. The discrete controlled system stabilizes the  motion about the equilibrium, but the s dynamics is not stabilized; since there is no dissipation, the oscillations are sustained

Forced Symplectic–Energy-Momentum Variational Integrators: One of the motivations for studying the control of Lagrangian systems using the method of controlled Lagrangians is that the method provides a natural candidate Lyapunov function to study the global stability properties of the controlled system. In the discrete theory, this approach is complicated by the fact that the energy of a discrete Lagrangian system is not exactly conserved, but rather oscillates in a bounded fashion. This can be addressed by considering the symplectic-energy-momentum [20] analogue to the discrete Lagrange-d’Alembert principle, ı

N1 X kD0

Ld (q k ; q kC1 ; h k ) D

N1 X

[Fd (q k ; q kC1 ; h k )  ıq k

kD0

C FdC (q k ; q kC1 ; h k )  ıq kC1 ] ; where the timestep hk is allowed to vary, and is chosen to satisfy the variational principle. The variations in hk yield an Euler–Lagrange equation that reduces to the conservation of discrete energy in the absence of external forces. By developing a theory of controlled Lagrangians around a geometric integrator based on the symplecticenergy-momentum version of the Lagrange–d’Alembert principle, one would potentially be able to use Lyapunov techniques to study the global stability of the resulting numerical control algorithms.

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cations to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications. Toronto, Canada, 28–31 August 2005, pp 962–967 Lee T, Leok M, McClamroch NH (2006) Optimal control of a rigid body using geometrically exact computations on SE(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 2710–2715 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem. Comput Method Appl Mech Eng 196:2907–2924 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem in orbital mechanics. Celest Mech Dyn Astron 98(2):121–144 Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Leok M (2004) Foundations of Computational Geometric Mechanics. Ph D thesis, California Instittute of Technology Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer, New York Marsden JE, West M (2001) Discrete mechanics and variational integrators. In: Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Marsden JE, Pekarsky S, Shkoller S (1999) Discrete Euler–Poincarée and Lie–Poisson equations. Nonlinearity 12(6):1647–1662 Maschke B, Ortega R, van der Schaft A (2001) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Autom Contr 45:1498–1502 Moser J, Veselov AP (1991) Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun Math Phys 139:217–243 Ortega R, Spong MW, Gómez-Estern F, Blankenstein G (2002) Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans Autom Contr 47:1218–1233 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286 Scheeres DJ, Fahnestock EG, Ostro SJ, Margot JL, Benner LAM, Broschart SB, Bellerose J, Giorgini JD, Nolan MC, Magri C, Pravec P, Scheirich P, Rose R, Jurgens RF, De Jong EM, Suzuki S (2006) Dynamical configuration of binary near-Earth asteroid (66391) 1999 KW4. Science 314:1280–1283 Scheeres DJ, Hsiao F-Y, Park RS, Villac BF, Maruskin JM (2006) Fundamental limits on spacecraft orbit uncertainty and distribution propagation. J Astronaut Sci 54:505–523 Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Comm Comput Phys 2:293–309 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2000) Matching and stabilization of low-dimensional nonholonomic systems. In: Proceedings of the IEEE Conference on Decision and Control. Sydney Convention and Exhibition Centre, Sydney, NSW Australia; 12–15 December 2000, pp 1289–1295 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2002) Flat nonholonomic matching. In: Proceedings of the American Control Conference. Hilton Anchorage, Anchorage, AK, 8–10 May 2002, pp 2812–2817

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Books and Reviews Bloch AM (2003) Nonholonomic Mechanics and Control. Interdisciplinary Appl Math, vol 24. Springer Bullo F, Lewis AD (2005) Geometric control of mechanical systems. Texts in Applied Mathematics, vol 49. Springer, New York Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365

Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer Marsden JE, West M (2001) Discrete mechanics and variational integrators. In Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286

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Dispersion Phenomena in Partial Differential Equations PIERO D’ANCONA Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Roma, Italy

  kukH s D (1  )s/2 uL q : Recall that this definition does not reduce to the preceding one when q D 1. We shall also use the homogeneous space H˙ qs , with norm   kukH˙ s D ()s/2 u L q :

Article Outline Glossary Definition of the Subject Introduction The Mechanism of Dispersion Strichartz Estimates The Nonlinear Wave Equation The Nonlinear Schrödinger Equation Future Directions Bibliography Glossary Notations Partial derivatives are written as ut or @ t u, @˛ D @˛x 11  @˛x nn , the Fourier transform of a function is defined as Z F f () D b f () D ei x f (x)dx and we frequently use the mute constant notation A . B to mean A  CB for some constant C (but only when the precise dependence of C from the other quantities involved is clear from the context). Evolution equations Partial differential equations describing physical systems which evolve in time. Thus, the variable representing time is distinguished from the others and is usually denoted by t. Cauchy problem A system of evolution equations, combined with a set of initial conditions at an initial time t D t0 . The problem is well posed if a solution exists, is unique and depends continuously on the data in suitable norms adapted to the problem. Blow up In general, the solutions to a nonlinear evolution equation are not defined for all times but they break down after some time has elapsed; usually the L1 norm of the solution or some of its derivatives becomes unbounded. This phenomenon is called blow up of the solution. Sobolev spaces we shall use two instances of Sobolev space: the space W k;1 with norm kukW k;1 D

and the space H qs with norm

X j˛jk

k@˛ ukL 1

Dispersive estimate a pointwise decay estimate of the form ju(t; x)j  Ct ˛ , usually for the solution of a partial differential equation. Definition of the Subject In a very broad sense, dispersion can be defined as the spreading of a fixed amount of matter, or energy, over a volume which increases with time. This intuitive picture suggests immediately the most prominent feature of dispersive phenomena: as matter spreads, its size, defined in a suitable sense, decreases at a certain rate. This effect should be contrasted with dissipation, which might be defined as an actual loss of energy, transferred to an external system (heat dissipation being the typical example). This rough idea has been made very precise during the last 30 years for most evolution equations of mathematical physics. For the classical, linear, constant coefficient equations like the wave, Schrödinger, Klein–Gordon and Dirac equations, the decay of solutions can be measured in the Lp norms, and sharp estimates are available. In addition, detailed information on the profile of the solutions can be obtained, producing an accurate description of the evolution. For nonlinear equations, the theory now is able to explain and quantify several complex phenomena such as the splitting of solutions into the sum of a train of solitons, plus a radiation part which disperses and decays as time increases. Already in the 1960s it was realized that precise information on the decay of free waves could be a powerful tool to investigate the effects of linear perturbations (describing interactions with electromagnetic fields) and even nonlinear perturbations of the equations. One of the first important applications was the discovery that a nonlinear PDE may admit global small solutions, provided the rate of decay is sufficiently strong compared with the concentration effect produced by the nonlinear terms. This situation is very different from the ODE setting, where dispersion is absent. Today, dispersive and Strichartz estimates represent the backbone of the theory of nonlinear evolution equations. Their applications include local and global existence

Dispersion Phenomena in Partial Differential Equations

results for nonlinear equations, existence of low regularity solutions, scattering, qualitative study of evolution equations on manifolds, and many others. Introduction To a large extent, mathematical physics is the study of qualitative and quantitative properties of the solutions to systems of differential equations. Phenomena which evolve in time are usually described by nonlinear evolution equations, and their natural setting is the Cauchy problem: the state of the system is assigned at an initial time, and the following evolution is, or should be, completely determined by the equations. The most basic question concerns the local existence and uniqueness of the solution; we should require that, on one hand, the set of initial conditions is not too restrictive, so that a solution exists, and on the other hand it is not too lax, so that a unique solution is specified. For physically meaningful equations it is natural to require in addition that small errors in the initial state propagate slowly in time, so that two solutions corresponding to slightly different initial data remain close, at least for some time (continuous dependence on the data). This set of requirements is called local well posedness of the system. Global well posedness implies the possibility of extending these properties to arbitrary large times. However, local well posedness is just the first step in understanding a nonlinear equation; basically, it amounts to a check that the equation and its Cauchy problem are meaningful from the mathematical and physical point of view. But the truly interesting questions concern global existence, asymptotic behavior and more generally the structure and shape of solutions. Indeed, if an equation describes a microscopic system, all quantities measured in an experiment correspond to asymptotic features of the solution, and local properties are basically useless in this context. The classical tool to prove local well posedness is the energy method; it can be applied to many equations and leads to very efficient proofs. But the energy method is intrinsically unable to give answers concerning the global existence and behavior of solutions. Let us illustrate this point more precisely. Most nonlinear evolution equations are of the form Lu D N(u) with a linear part L (e. g., the d’Alembert or Schrödinger operator) and a nonlinear term N(u). The two terms are in competition: the linear flow is well behaved, usually with

several conserved quantities bounded by norms of the initial data, while the nonlinear term tends to concentrate the peaks of u and make them higher, increasing the singularity of the solution. At the same time, a small function u is made even smaller by a power-like nonlinearity N(u). The energy method tries to use only the conserved quantities, which remain constant during the evolution of the linear flow. Essentially, this means to regard the equation as an ODE of the form y 0 (t) D N(y(t)). This approach is very useful if the full (nonlinear) equation has a positive conserved quantity. But in general the best one can prove using the energy method is local well posedness, while a blow up in finite time can not be excluded, even if the initial data are assumed to be small, as the trivial example y 0 D y 2 clearly shows. To improve on this situation, the strength of the linear flow must be used to a full extent. This is where dispersive phenomena enter the picture. A finer study of the operator L shows that there are quantities, different from the standard L2 type norms used in the energy estimates, which actually decay during the evolution. Hence the term L has an additional advantage in the competition with N which can lead to global existence of small solutions. See Sect. “The Nonlinear Wave Equation” for more details. This circle of ideas was initiated at the beginning of the 1970s and rapidly developed during the 1980s in several papers, (see [26,27,31,32,33,34,38,40,41,42,50] to mention just a few of the fundamental contributions on the subject). Global existence of small solutions was proved for many nonlinear evolution equations, including the nonlinear wave, Klein–Gordon and Schrödinger equations. The proof of the stability of vacuum for the Einstein equations [15,35] can be regarded as an extreme development of this direction of research. Another early indication of the importance of Lp estimates came from the separate direction of harmonic analysis. In 1970, at a time when general agreement in the field was that the L2 framework was the only correct one for the study of the wave equation, R. Strichartz ([44,45], see also [10,11,46]) proved the estimate u t t  u D F ;

u(0; x) D u t (0; x) D 0

H)

kukL p (RnC1 ) . kFkL p0 (RnC1 )

for p D 2(n C 1)/(n  1). This was a seminal result, both for the emphasis on the Lp approach, and for the idea of regarding all variables including time on the same level, an idea which would play a central role in the later developments. At the same time, it hinted at a deep connection

161

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between dispersive properties and the techniques of harmonic analysis. The new ideas were rapidly incorporated in the mainstream theory of partial differential equations and became an important tool ([22,23,52]); the investigation of Strichartz estimates is still in progress. The freedom to work in an Lp setting was revealed as useful in a wide variety of contexts, including linear equations with variable coefficients, scattering, existence of low regularity solutions, and several others. We might say that this point of view created a new important class of dispersive equations, complementing and overlapping with the standard classification into elliptic, hyperbolic, parabolic equations. Actually this class is very wide and contains most evolution equations of mathematical physics: the wave, Schrödinger and Klein–Gordon equations, Dirac and Maxwell systems, their nonlinear counterparts including wave maps, Yang–Mills and Einstein equations, Korteweg de Vries, Benjamin–Ono, and many others. In addition, many basic physical problems are described by systems obtained by coupling these equations; among the most important ones we mention the Dirac–Klein–Gordon, Maxwell–Klein–Gordon, Maxwell– Dirac, Maxwell–Schrödinger and Zakharov systems. Finally, a recent line of research pursues the extension of the dispersive techniques to equations with variable coefficients and equations on manifolds. The main goal of this article is to give a quick overview of the basic dispersive techniques, and to show some important examples of their applications. It should be kept in mind however that the field is evolving at an impressive speed, mainly due to the contribution of ideas from Fourier and harmonic analysis, and it would be difficult to give a self-contained account of the recent developments, for which we point at the relevant bibliography. In Sect. “The Mechanism of Dispersion” we analyze the basic mechanism of dispersion, with special attention given to two basic examples (Schrödinger and wave equation). Section “Strichartz Estimates” is a coincise exposition of Strichartz estimates, while Sects. “The Nonlinear Wave Equation” and “The Nonlinear Schrödinger Equation” illustrate some typical applications of dispersive techniques to problems of global existence for nonlinear equations. In the last Sect. “Future Directions” we review some of the more promising lines of research in this field.

The same mechanism lies behind dispersive phenomena, and indeed two dual explanations are possible: a) in physical space, dispersion is characterized by a finite speed of propagation of the signals; data concentrated on some region tend to spread over a larger volume following the evolution flow. Correspondingly, the L1 and other norms of the solution tend to decrease; b) in Fourier space, dispersion is characterized by oscillations in the Fourier variable , which increase for large jj and large t. This produces cancelation and decay in suitable norms. We shall explore these dual points of view in two fundamental cases: the Schrödinger and the wave equation. These are the most important and hence most studied dispersive equations, and a fairly complete theory is available for both. The Schrödinger Equation in Physical Space The solution u(t; x) of the Schrödinger equation iu t C u D 0 ;

u(0; x) D f (x)

admits several representations which are not completely equivalent. From spectral theory we have an abstract representation as a group of L2 isometries u(t; x) D e i t f ;

(1)

then we have an explicit representation using the fundamental solution 2 1 i jxj 4t f e (4 it)n/2 Z jx yj2 1 i 4t e f (y)dy ; D (4 it)n/2 Rn

u(t; x) D

finally, we can represent the solution via Fourier transform as Z 2 n f ()d : (3) e i(tjj Cx)b u(t; x) D (2) Notice for instance that the spectral and Fourier representations are well defined for f 2 L2 , while it is not immediately apparent how to give a meaning to (2) if f 62 L1 . From (1) we deduce immediately the dispersive estimate for the Schrödinger equation, which has the form 1

The Mechanism of Dispersion

je i t f j  (4)n/2

The Fourier transform puts in duality oscillations and translations according to the elementary rule

Since the L2 norm of u is constant in time

F ( f (x C h)) D ei xh F f :

(2)

t n/2

ke i t f kL 2 D k f kL 2 ;

k f kL 1 :

(4)

(5)

Dispersion Phenomena in Partial Differential Equations

this might suggest that the constant L2 mass spreads on a region of volume t n i. e. of diameter t. Apparently, this is in contrast with the notion that the Schrödinger equation has an infinite speed of propagation. Indeed, this is a correct but incomplete statement. To explain this point with an explicit example, consider the evolution of a wave packet 1

from zero, and this already suggests a decay of the order t n/2 . This argument can be made precise and gives an alternative proof of (4). We also notice that the Fourier transform in both space and time of u(t; x) D e i t f is a temperate distribution in S0 (RnC1 ) which can be written, apart from a constant, ˜ ) D ı(  jj2) u(;

2

x 0 ;0 D e 2 jxx 0 j e i(xx 0 )0 : A wave packet is a mathematical representation of a particle localized near position x0 , with frequency localized near  0 ; notice indeed that its Fourier transform is 1

2

b  x 0 ;0 D ce 2 j0 j e ix 0 for some constant c. Exact localization both in space and frequency is of course impossible by Heisenberg’s principle; however the approximate localization of the wave packet is a very good substitute, since the gaussian tails decay to 0 extremely fast. The evolution of a wave packet is easy to compute explicitly:   1 (x  x0  2t0 )2 i i t  n2 e x 0 ;0 D (1C2it) exp  e 2 (1 C 4t 2 )

and is supported on the paraboloid  D jj2. This representation of the solution gives additional insight to its behavior in certain frequency regimes, and has proved to be of fundamental importance when studying the existence of low regularity solutions to the nonlinear Schrödinger equation [6,9]. The Wave Equation in Physical Space For the wave equation, the situation is reversed: the most efficient proof of the dispersive estimate is obtained via the Fourier representation of the solution. This is mainly due to the complexity of the fundamental solution; we recall the relevant formulas briefly. The solution to the homogeneous wave equation

where the real valued function  is given by  D

t(x  x0 )2 C (x  x0  t0 )  0 : (1 C 4t 2 )

We notice that the particle “moves” with velocity 20 and the height of the gaussian spreads precisely at a rate t n/2 . The mass is essentially concentrated in a ball of radius t, as suspected. This simple remark is at the heart of one of the most powerful techniques to have appeared in harmonic analysis in recent years, the wave packet decomposition, which has led to breakthroughs in several classical problems (see e. g. [7,8,48,51]).

w t t  w D 0 ;

2

(t; x; ) D tjj C x  

H)

r  D 2t C x

so that for each (t; x) we have a unique stationary point  D x/2t; the n curvatures at that point are different

w t (0; x) D g(x)

can be written as u(t; x) D cos(tjDj) f C

sin(tjDj) g jDj

where we used the symbolic notations cos(tjDj) f D F 1 cos(tjj)F f ; sin(tjDj) sin(tjj) Fg g D F 1 jDj jj (recall that F denotes the Fourier transform in space variables). Notice that

The Schrödinger Equation in Fourier Space In view of the ease of proof of (4), it may seem unnecessary to examine the solution in Fourier space, however a quick look at this other representation is instructive and prepares the discussion for the wave equation. If we examine (3), we see that the integral is a typical oscillatory integral which can be estimated using the stationary phase method. The phase here is

w(0; x) D f (x) ;

cos(tjDj) f D

@ sin(tjDj) f; @t jDj

hence the properties of the first operator can be deduced from the second. Moreover, by Duhamel’s formula, we can express the solution to the nonhomogeneous wave equation w t t  w D F(t; x) ;

w(0; x) D 0 ;

w t (0; x) D 0

as Z

t

u(t; x) D 0

sin((t  s)jDj) F(u(s; ))ds : jDj

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Dispersion Phenomena in Partial Differential Equations

of the solution remains constant, in view of the energy estimate

In other words, a study of the operator S(t)g D

sin(tjDj) g jDj

ku t (t; )k2L 2 Ckru(t; )k2L 2 D ku t (0; )k2L 2 Ckru(0; )k2L 2 ;

is sufficient to deduce the behavior of the complete solution to the nonhomogeneous wave equation with general data. The operator S(t) can be expressed explicitly using the fundamental solution of the wave equation. We skip the one-dimensional case since then from the expression of the general solution u(t; x) D (t C x) C

(t  x)

it is immediately apparent that no dispersion can occur. In odd space dimension n  3 we have !   n3 Z 2 1 sin(tjDj) 1 g(x  y)d y ; g D cn @t jDj t t jyjDt

on the other hand, the velocity of propagation for the wave equation is exactly 1, so that the constant L2 energy is spread on a volume which increases at a rate t n ; as a consequence, we would expect the same rate of decay t n/2 as for the Schrödinger equation. However, a more detailed study of the shape of the solution reveals that it tends to concentrate in a shell of constant thickness around the light cone. For instance, in odd dimension larger than 3 initial data with support in a ball B(0; R) produce a solution which at time t is supported in the shell B(0; t C R) n B(0; t  R). Also in even dimension the solution tends to concentrate along the light cone jtj D jxj, with a faster decreasing tail far from the cone. Thus, the energy spreads over a volume of size t n1 , in accordance with the rate of decay in the dispersive estimate.

and in even space dimension n  2 sin(tjDj) g D cn jDj



1 @t t

 n2 2

1 t

Z jyj
g(x  y) p dy t 2  jyj2

!

for some constant cn . These formulas can be expanded and the resulting integrals estimated in order to compute the explicit decay rate of the solution; this approach was followed in [50] (see also [43] for a more concise proof based on the fundamental solution). We shall prove the dispersive estimate using Fourier methods in the next section, but here it is instructive to consider the particularly simple case n D 3, when the above representation reduces to Kirchhoff’s formula Z sin(tjDj) c g(x  y)d y : gD jDj t jyjDt Using the divergence theorem it is easy to deduce that ˇ ˇ ˇ sin(tjDj) ˇ 1 ˇ ˇ . kr f kL 1 : g ˇ jDj ˇ t Recall that the rate of decay for the Schrödinger equation in dimension 3 is t 3/2 ; for the wave equation the rate of decay is slower, in addition the estimate shows a loss of one derivative with respect to the initial data. Actually this estimate is optimal, as it is evident from the explicit representation of the solution. In dimension n the correct n1 rate of decay is t  2 and the loss of derivative is of order (n  1)/2, as we shall see in the next section. This result is in apparent contrast with our physical space picture. Indeed the L2 norm of the first derivatives

The Wave Equation in Fourier Space Since 2i

e i tjDj ei tjDj sin(tjDj) gD g g; jDj jDj jDj

2 cos(tjDj) f D e i tjDj f C ei tjDj f ; we can unify the study of the various components of the solution by considering the operator Z f d : e i tjDj f D F 1 e i tjj F f D (2)n e i(xCtjj)b The phase of the last integral is (t; x; ) D tjj C x  

H)

r  D t

 Cx jj

and we notice immediately that the structure of stationary points is more complex than for Schrödinger: if jtj D jxj, i. e. if the point (t; x) is on the light cone, the phase has a half line of critical points jtj D jxj

H)

 x D jj t

which of course are degenerate since only n  1 curvatures are different from zero; on the other hand, when (t; x) is not on the light cone, the phase is nondegenerate at all points. In the nondegenerate case jtj ¤ jxj, we notice that r  D t

 Cx jj

H)

jr j  jjtj  jxjj

Dispersion Phenomena in Partial Differential Equations

hence by direct integration by parts we obtain ˇ ˇZ ˇ ˇ ˇ  C( f )jjtj  jxjj(n1) : ˇ e i(xCtjj)b f d ˇ ˇ

(6)

Notice that the singularity jj of the phase makes it impossible to perform the integration by parts more than n  1 times, but this is largely sufficient for our purposes here. In the degenerate case jtj D jxj, we can eliminate the degeneracy by passing to polar coordinates: Z 1 Z i(xCtjj)b f d D e i t n1 e 0 Z f ( !)d!d :  e i x !b j!jD1

Now the phase in the inner integral has exactly two isolated critical points for each value of x; by a standard application of the stationary phase method we can obtain an estimate of the form ˇZ ˇ ˇ ˇ ˇ e i(xCtjj)b ˇ  C( f )jxj n1 2 : f d (7) ˇ ˇ The power (n  1)/2 is directly related to the dimension n  1 of the unit sphere. From the above computations it is possible to obtain a very detailed description of the shape of the solution (see e. g. [20]). However, here we are only interested in the rate of decay of the L1 norm of the solution. Thus, using estimate (6) when jxj < jtj/2 and (7) when jxj > jtj/2 we obtain je i tjDj f j  C( f )jtj

n1 2

:

It is also possible to make precise the form of the constant C( f ) appearing in the estimate. This can be done in several ways; one of the most efficient exploits the scaling properties of the wave equation (see [10,23]). In order to state the optimal estimate we briefly recall the definition of some function spaces. The basic ingredient is the homogeneous Paley–Littlewood partition of unity which has the form X  j () D 1 ;  j () D 0 (2 j ) ; j2Z

sup  j D f : 2 j1  jj  2 jC1 g for a fixed 0 2 C01 (Rn ) (the following definitions are independent of the precise choice of 0 ). To each symbol  j () we associate by the standard calculus the operator  j (D) D F 1  j ()F . Now the homogeneous Besov norm s can be defined as follows: B˙ 1;1 X s D 2 js k j (D) f kL 1 (8) k f kB˙ 1;1 j2Z

and the corresponding Besov space can be obtained by completing C01 (see [4] for a thorough description of these definitions). Then the optimal dispersive estimate for the wave equation is je i tjDj f j  Ck f k

n1

B˙ 1;12

jtj

n1 2

:

(9)

We might add that in odd space dimension a slightly better estimate can be proved involving only the standard ˙ n1 2 instead of the Besov norm. Sobolev norm W The singularity in t at the right hand side of (9) can be annoying in applications, thus it would be natural to try to replace jtj by (1 C jtj) by some different proof. However, this is clearly impossible since both sides have the same scaling with respect to the transformation u(t; x) ! u(t; x) which takes a solution of the homogeneous wave equation into another solution. Moreover, if such an estimate were true, taking the limit as t ! 0 we would obtain a false Sobolev embedding. Thus, the only way to get a nonsingular estimate is to replace the Besov norm of the data by some stronger norm. Indeed, for jtj < 1 we can use the L2 conservation which implies, for any > 0, je i tjDj f j  ke i tjDj f kH n/2C D k f kH n/2C : Combining the two estimates, we arrive at   n1 je i tjDj f j . k f k n1 C k f k (1Cjtj) 2 : (10) n/2C H 2 B˙ 1;1

Other Dispersive Equations Dispersive estimates can be proved for many other equations; see for instance [3] for general results concerning the equations of the form iu t D P(D)u ;

P polynomial

which correspond to integrals of the form Z e i(xx iCP())b f ()d : Here we would like to add some more details on two equations of particular importance for physics. The Klein– Gordon equation u t t  u C u D 0 ;

u(0; x) D f (x) ;

u t (0; x) D g(x)

shows the same decay rate as the Schrödinger equation, although for most other properties it is very close to the wave

165

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Dispersion Phenomena in Partial Differential Equations

equation. This phenomenon can be easily explained. In physical space, we notice that the solution has finite speed of propagation less or equal to 1, so that for compactly supported initial data the solution spreads uniformly on a volume of order tn , and there is no concentration along the light cone as for the wave equation. On the other hand, the Fourier representation of the solution is sin(thDi) u(t; x) D cos(thDi) f C g hDi

cos(thDi) f D F 1 cos(tjhij)F f ;

with hi D (1 C i. e. hDi D (1  Thus, we see that, similarly to the wave equation, the study of the solutions can be reduced to the basic operator Z f ()d : e i thDi f D F 1 e i thi F f D (2)n e i(thiCx)b It is easy to check that the phase has an isolated stationary point if and only if jtj < jxj, and no stationary point if jtj  jj:  Cx; hi

in accord with the claimed dispersion rate of order t n/2 . However, the proof is not completely elementary (for a very detailed study, see Section 7.2 of [25]). The optimal estimate is now je

i thDi

f j  Ck f k

n

2 B 1;1

jtj

 n2

sup  j D f : 2

and the 4  4 Dirac matrices can be written     I 0 k 0 ˛k D ; ˇD 2 ; k 0 0 I2 in terms of the Pauli matrices    1 0 0 I2 D ; 1 D 0 1 1    0 i 1 2 D ; 3 D i 0 0

j  0;

 jj  2 jC1 g ;

sup 1 D fjxj  1g for some 1 ; 0 2 C01 (Rn ). A second important example is the Dirac system. From a mathematical point of view (and setting the values of the

1 0

k D 1; 2; 3

 ;

 0 : 1

Then the solution u(t; x) D e i t D f of the massless Dirac system with initial value u(0; x) D f (x) satisfies the dispersive estimate ju(t; x)j 

C k f kB˙ 2 1;1 t

while in the massive case we have ju(t; x)j 

j2N

j1

3 1X ˛k @k i

(11)

(see the Appendix to [18]). Here we used the nonhomoges which is defined exactly as the honeous Besov norm B1;1 mogeneous one (8) by using the inhomogeneous version of the Palye–Littlewood partition of unity: X 1 () C  j () D 1 ;  j () D 0 (2 j ) for

iu t C Du C ˇu D 0

kD1

)1/2 .

r  D t

in the massless case, and

DD

sin(thDi) sin(tjhij) Fg g D F 1 hDi jhij

(t; x; ) D thi C x   ;

iu t C Du D 0

in the massive case. Here u : R t  R3x ! C 4 , the operator D is defined as

where we used the symbolic notations

jj2)1/2

speed of light and Planck’s constant equal to 1 for simplicity of notation) this is a 4  4 constant coefficient system of the form

C 3

t2

kfk

5

2 B 1;1

(see Sect. 7 of [16]). Strichartz Estimates Since the Schrödinger flow e i t f is an L2 isometry, it may be tempting to regard it as some sort of “rotation” of the Hilbert space L2 . This picture is far from true: there exist small subspaces of L2 which contain the flow for almost all times. This surprising phenomenon, which holds in greater generality for any unbounded selfadjoint operator on an abstract Hilbert space, is known as Kato smoothing and was discovered by T. Kato in 1966 [29], see also [39]. This fact is just a corollary of a quantitative estimate, the Kato smoothing estimate, which in the case of the Laplace

Dispersion Phenomena in Partial Differential Equations

operator takes the following form: there exist closed unbounded operators A on L2 such that kAe

i t

f kL 2 L 2 . k f kL 2 :

As a consequence, the flow belongs for a. e. t to the domain of the operator A, which can be a very small subspace of L2 . Two simple examples of smoothing estimates are the following: V(x) 2 L2 \ L n \ L nC H)

kV(x)e i t f kL 2 L 2 . k f kL 2

(12)

and kjxj1 e i t f kL 2 L 2 . k f kL 2 :

(13)

On the other hand, Strichartz proved in 1977 [46] that pD

2(n C 2) n H)

2n

L 2 L n2

ke

i t

f kL p (RnC1 ) . k f kL 2

(14)

. k f kL 2

(15)

while (13) would follow from the slightly stronger ke i t f k

2n

L 2 L n2 ;2

The homogeneous Strichartz estimates have the general form ke i t f kL p L q . k f kL 2 for suitable values of the couple (p; q). Notice that we can restrict the L p L q norm at the right hand side to an arbitrary time interval I (i. e. to L p (I; L q (Rn ))). Notice also that by the invariance of the flow under the scaling u(t:x) 7! u(2 t; x) the possible couples (p; q) are bound to satisfy n n 2 C D : p q 2

. k f kL 2

(16)

where L p;2 denotes a Lorentz space. Both (15) and (16) are true, although their proof is far from elementary. Such estimates are now generally called Strichartz estimates. They can be deduced from the dispersive estimates, hence they are in a sense weaker and contain less information. However, for this same reason they can be obtained for a large number of equations, even with variable coefficients, and their flexibility and wide range of applications makes them extremely useful. The full set of Strichartz estimates, with the exception of the endpoint (see below) was proved in [22], where it was also applied to the problem of global well posedness for the nonlinear Schrödinger equation. The corresponding estimates for the wave equation were obtained in [23], however they turned out to be less useful because of the loss of derivatives. A fundamental progress was made in [30], where not only the most difficult endpoint case was solved, but it also outlined a general procedure to deduce the Strichartz estimates from the dispersive estimates, with applications to any evolution equation. In the following we shall focus on the two main examples, the Schrödinger and the wave equation, and we refer to the original papers for more details and the proofs.

(17)

The range of possible values of q is expressed by the inequalities 2q

thus showing that the smoothing effect could be measured also in the Lp norms. If we try to reformulate (12) in terms of an Lp norm, we see that it would follow from ke i t f k

The Schrödinger Equation

2n n2

2  q  1 if

if

n  2;

n D1;

q ¤1;

(18) (19)

the index p varies accordingly between 1 and 2 (between 4 and 2 in one space dimension). Such couples (p; q) are called (Schrödinger) admissible. It is easy to visualize the admissible couples by plotting the usual diagram in the (1/p; 1/q) space (see Fig 1–3). Note that the choice (p; q) D (1; 2), the point E in the diagrams, corresponds to the conservation of the L2 norm, while at the other extreme we have for n  2 point P corresponding to the couple   2n (p; q) D 2; : (20) n2 This is usually called the endpoint and is excluded in dimension 2, allowed in dimension  3. The original Strichartz estimate corresponds to the case p D q and is denoted by S. After the proof in [22] of the general Strichartz estimates, it was not clear if the endpoint case P could be reached or not; the final answer was given in [30] where the endpoint estimates were proved to be true for all dimension larger than three. For the two-dimensional case and additional estimates, see [47]. Notice that all admissible couples can be obtained by interpolation between the endpoint and the L2 conservation, thus in a sense the endpoint contains the maximum of information concerning the Lp smoothing properties of the flow. There exist an inhomogeneous form of the estimates, which is actually equivalent to the homogeneous one, and can be stated as follows:  Z t   i(ts)   e F(s)ds . kFkL p˜ 0 L q˜0 (21)   0

L p Lq

167

168

Dispersion Phenomena in Partial Differential Equations

Dispersion Phenomena in Partial Differential Equations, Figure 3 Schrödinger admissible couples, n  3 Dispersion Phenomena in Partial Differential Equations, Figure 1 Schrödinger admissible couples, n D 1

neous global estimates are optimal since other combination of indices are excluded by scale invariance; however, the local Strichartz estimates can be slightly improved allowing for indices outside the admissible region (see [21]). We also mention that a much larger set of estimates can be deduced from the above ones by combining them with Sobolev embeddings; in the above diagrams, all couples in the region bounded by the line of admissibility, the axes and the line 1/p D 2 can be reached, while the external region is excluded by suitable counterexamples (see [30]). Strichartz Estimates for the Wave Equation

Dispersion Phenomena in Partial Differential Equations, Figure 2 Schrödinger admissible couples, n D 2

Most of the preceding section has a precise analogue for the wave equation. Since the decay rate for the dispersive estimates is t (n1)/2 for the wave equation instead of t n/2 as for the Schrödinger equation, this causes a shift n ! n 1 in the numerology of indices. A more serious difference is the loss of derivatives: this was already apparent in the original estimate by Strichartz [44], which can be written ke i tjDj f k

˜ which can be for all admissible couples (p; q) and ( p˜ ; q), chosen independently. Here p0 denotes the conjugate exponent to p. These estimates continue to be true in their local version, with the norms at both sides restricted to a fixed time interval I R. Notice that the inhomoge-

L

2(nC1) n1

. k f kH˙ 1/2 ;

n  2:

The general homogeneous Strichartz estimates for the wave equation take the form ke i tjDj f kL p L q . k f k

1 nC1 n1

˙p H

;

Dispersion Phenomena in Partial Differential Equations

now the (wave) admissible couples of indices must satisfy the scaling condition 2 n1 n1 C D p q 2

(22)

and the constraints 2  p  1, 2q

2(n  1) n3

2q1

if

n  3;

if

q¤1

(23)

n D2;

(24)

The Nonlinear Wave Equation In this section we shall illustrate, using the semilinear wave equation as a basic example, what kind of improvements can be obtained over the classical energy method via the additional information given by dispersive techniques. From the linear theory it is well known that the correct setting of the Cauchy problem for the wave equation u t t  u D F(t; x) ;

requires two initial data at t D 0: u(0; x) D u0 (x) ;

while p varies accordingly between 2 and 1 (between 4 and 1 in dimension 2). Recall that in dimension 1 the solutions of the wave equation do not decay since they can be decomposed into traveling waves. We omit the graphic representation of these sets of conditions since it is completely analogous to the Schrödinger case (apart from the shift n ! n  1). The inhomogeneous estimates may be written  Z t   i(ts)   e F(s)ds   0

nC1  1p n1

˙q LpH

. kFk

nC1  1 n1

0 ˙ p˜ L p˜ H q˜0

(25)

u(t; x) : R  Rn ! C

u t (0; x) D u1 (x) :

For this equation the energy method is particularly simple. If we multiply the equation by u t we can recast it in the form   @ t ju t j2 C jruj2 D 2
. ku1 kL 2 C kru0 kL 2 C kFkL 1 L 2 T

in terms of the homogeneous Sobolev norms kgkH˙ sq D kjDjs gkL q ;

jDjs D (1  )s/2 :

Other Equations The general procedure of Keel and Tao can be applied to a large variety of equations. For the convenience of the reader we list here the precise form of the homogeneous Strichartz estimates in some important cases. For the Klein–Gordon equation we have ke i thDi f kL p L q . k f k

1 nC2 n

where we are using the shorthand notation p

L T L q D L p (0; T; L q (Rn )) : Estimate (26) is the basic energy estimate, which is enough to prove the existence of a global solution u 2 C(R; H 1 (Rn )) \ C 1 (R; L2 (Rn )) of the linear wave equation for data in (u0 ; u1 ) 2 H 1  L2 and a forcing term in F 2 L1T L2 for all T. Uniqueness and continuous dependence on the data are simple consequences of the energy estimate. Moreover, if we differentiate the equation k times, the same computations lead to the higher order energy estimates

˙p H

with (p; q) Schrödinger admissible (recall that the rate of dispersion is t n/2 for Klein–Gordon). For the massless Dirac system we have ke i t D f kL p L q . k f k

(26)

2

˙p H

;

n D3;

T

5

H 3p

;

nD3

with (p; q) Schrödinger admissible in dimension 3.

T

. ku1 kH k Cku0 kH kC1 CkFkL 1 H k ; T

k D 0; 1; 2; : : : (27)

Consider now the nonlinear wave equation u t t  u D F(u)

with (p; q) wave admissible in dimension 3, while for the massive Dirac system the estimate takes the form ke i t(DCˇ ) f kL p L q . k f k

ku t kL 1 H k C kukL 1 H kC1

(28)

for some function F(s) sufficiently smooth. The classical approach to solve it is to look for a fixed point of the mapping u D ˚ (v), where u(t; x) solves the linearized equation u t t  u D F(v) :

(29)

169

170

Dispersion Phenomena in Partial Differential Equations

If we apply the energy estimate to the difference of two solutions u1 D ˚(v1 ) and u2 D ˚(v2 ) with the same initial data, we obtain k@ t (u1  u2 )kL 1 H k C ku1  u2 kL 1 H kC1 T

T

. kF(v1 )  F(v2 )kL 1 H k : T

Using a standard Moser-type estimate (see e. g. [25,49]) k f (v1 )  f (v2 )kH k . C(kv1 kL 1 C kv2 kL 1 )kv1  v2 kH k we arrive at the inequality k@ t (u1  u2 )kL 1 H k C ku1  u2 kL 1 H kC1 T

T

. C(kv1 kL 1 1 C kv2 k L 1 L 1 )  kv1  v2 k L 1 H k : T L T T

k In particular, writing X T D L1 T H , and noticing that

kGkL 1 H k  kGk X T  T ; T

Actually, the two difficulties are related to each other. Indeed, from the above argument one can check that the lifespan of the local solution is a function of the H k norm of the initial data. Most equations with a physical origin possess conserved quantities, which are essentially H k norms of the solution with k low, e. g. k D 0 or 1. Assume now that we can prove the local existence for data say in H 1 , on some interval [0; T0 ] with T 0 depending on the H 1 norm of the data. Since this norm is conserved, we can take T 0 as a new initial time and construct a solution on [T0 ; 2T0 ], and so on. This is the essence of the continuation method. As a consequence, the local existence of low regularity solution with the addition of suitable conservation laws can lead to global existence. We shall now apply the dispersive information to improve on the preceding result and obtain the global existence of small solution, under suitable assumptions on the nonlinear term. Consider a free wave w(t; x), solution of w t t  w D 0 ;

w(0; x) D f (x) ;

w t (0; x) D g(x) :

we obtain Recalling the dispersive estimate (10), here the following simplified version

ku1  u2 k X T . C(kv1 kL 1 1 C kv2 k L 1 L 1 )  kv1  v2 k X T  T : T L T

jw(t; x)j  C(1 C t)

n1 2

L1

norm of vj , which can be We still have to handle the controlled by the H k norm provided k > n/2. In conclusion k>

n 2

H)

ku1  u2 k X T

. C(kv1 k X T C kv2 k X T )  kv1  v2 k X T  T : At this point the philosophy of the method should be clear: if we agree to differentiate n/2 times the equation, we can prove a nonlinear estimate, with a coefficient which can be made small for T small. Thus, for T  1 the mapping v 7! u is a contraction, and we obtain a local (in time) solution of the nonlinear equation. The energy method is quite elementary, yet it has the remarkable property that it can be adapted to a huge number of equations: it can be applied to equations with variable coefficients, and even to fully nonlinear equations. An outstanding example of its flexibility is the proof of the local solvability of Einstein’s equations ([14]). The main drawbacks of the energy method are: 1) A high regularity of the data and of the solution is required; 2) The nonlinear nature of the estimate leads to a local existence result, leaving completely open the question of global existence.

k f kW N;1 ;

N>n

will be sufficient. This suggests modification of the energy norm by adding a term that incorporates the decay information. To this end, we introduce the time-dependent quantity M(t) D sup

n

0t

kD

hni 2

ku(; )kH kC1 C (1 C )

n1 2

o ku(; )kL 1 ;

:

Consider now the nonlinear Eq. (28) with initial data u0 ; u1 . We already know that a local solution exists, provided the initial data are smooth enough; we shall now try to extend this solution to a global one. Using Duhamel’s principle we rewrite the equation as Z u(t; x) D w(t; x) C 0

t

sin((t  s)jDj) F(u(s; ))ds jDj

where we are using the symbolic notations of Sect. “Other Dispersive Equations”. F being the space Fourier transform. Note that the above dispersive estimate can be applied in particular to the last term; we obtain ˇ ˇ ˇ sin((t  s)jDj) ˇ ˇ ˇ  C(1 C t  s) n1 2 kFk N;1 : F(s) W ˇ ˇ jDj

Dispersion Phenomena in Partial Differential Equations

By this inequality and the energy estimate already proved, in a few steps we arrive at the estimate n1 2

M(T) . C0 C kFkL 1 H k C (1 C T) T Z T n1 (1 C T  s) 2 kF(u(s))kW N;1 ds 0

where C0 D ku0 kW N;1 C ku1 kW N;1 and the mute constant is independent of T. By suitable nonlinear estimates of Moser type, we can bound the derivatives of the nonlinear term using only the L1 and the H k norm of u, i. e., the quantity M(t) itself. If we assume that F(u) juj for small u, we obtain

Consider the Cauchy problem iu t  u D juj ;

u(0; x) D f (x)

(31)

for some  1. We want to study the local and global solvability in L2 of this Cauchy problem. Recall that this is a difficult question, still not completely solved; however, at least in the subcritical range, a combination of dispersive techniques and the contraction mapping principle gives the desired result easily. Instead of Eq. (31), we are actually going to solve the corresponding integral equation Z t e i(ts) ju(s)j ds : u(t; x) D e i t f C i 0

n1

M(T) . C0 C M(T) (1 C T) 2 Z T n1 n1 (1 C T  s) 2 (1 C s) 2 ( 2) ds : 0

Our standard approach is to look for a fixed point of the mapping u D ˚(v) defined by Z t ˚(v) D e i t f C i e i(ts) jv(s)j ds ; 0

Now we have Z

t

(1 C t  )

n1 2

 (1 C )

n1 2 ( 2)

d

0

 C(1 C t)

n1 2

k˚(v)kL p L q . k f kL 2 C kj˚(v)j kL p˜ 0 L q˜0

provided

> 2C

it is easy to check, using Strichartz estimates, that the mapping ˚ is well defined, provided the function v is in the suitable L p L q . Indeed we can write, for all admissible cou˜ ples (p; q) and ( p˜; q)

2 n1

so that, for such values of , we obtain M(T) . C0 CM(T) : For small initial data, which means small values of the constant C0 , (30) implies that M(T) is uniformly bounded as long as the smooth solution exists. By a simple continuation argument, this implies that the solution is in fact global, provided the initial data are small enough.

and by Hölder’s inequality this implies  0 0 L  p˜ L  q˜

k˚(v)kL p L q  C1 k f kL 2 C C1 kvk

:

(32)

Thus, we see that the mapping ˚ operates on the space L p L q , provided we can satisfy the following set of conditions: (30) p˜0 D p ; q˜0 D q; 2 n n n n 2˜ C D ; C D ; ˜ p q˜ p q 2 2

 2n p; p˜ 2 [2; 1] ; q˜ 2 2; : n2 Notice that from the first two identities we have

The Nonlinear Schrödinger Equation Strichartz estimates for the Schrödinger equation are expressed entirely in terms of Lp norms, without loss of derivatives, in contrast with the case of the wave equations. For this reason they represent a perfect tool to handle semilinear perturbations. Using Strichartz estimates, several important results can be proved with minimal effort. As an example, we chose to illustrate in this Section the critical and subcritical well posedness in L2 in the case of power nonlinearities (see [13] for a systematic treatment).

n 2

n

2 C D˛ Cn p˜ q˜ p q and by the admissibility conditions this implies n n

D2Cn ; 2 2 which forces the value of to be

D1C

4 : n

This value of is called L2 critical.

(33)

171

172

Dispersion Phenomena in Partial Differential Equations

Let us assume for the moment that is exactly the critical exponent (33). Then it is easy to check that there exist ˜ which satisfy all the above conchoices of (p; q) and ( p˜ ; q) straints. To prove that ˚ is a contraction on the space

p˜0 < p ;

we take any v1 ; v2 2 X and compute   k˚(v1 )  ˚(v2 )k X . jv1 j  jv2 j  L p˜ 0 L q˜0   . jv1  v2 j(jv1 j 1 C jv2 j 1 )

0

L p˜ L q˜

 1

k˚ (v1 )  ˚(v2 )k X  C2 kv1  v2 k X kjv1 j C jv2 jk X

0

;

(34)

for some constant C2 . Now assume that

 C2 T  kv1  v2 k X T kjv1 j C jv2 jkX1 : T

provided "; ı are so small that

k f kL 2 <

On the other hand, by (34) we get 1 kv1 v2 k X 2

provided " is so small that 1 : 2

In conclusion, if the initial data are small enough, the mapping ˚ is a contraction on a ball B(0; ") X and the unique fixed point is a global small solution of the critical L2 equation 4

iu t  u D juj1C n : Notice that in estimate (32) the first couple can also be taken equal to (1; 2), thus the solution we have constructed belongs to L1 L2 . A limit argument (dominated convergence) in the identity t

(35)

(36)

Now let M be so large and T so small that

" C1 " < : 2 

Z

and proceeding as above and using Hölder inequality in time, we obtain the inequality. Indeed, choosing the indices as above and applying Hölder’s inequality in time we have

k˚ (v1 )  ˚(v2 )k X T

k˚(v)k X  C1 (ı C " ) < "

u D e i t f C i

q˜0 D q

for some strictly positive  > 0. Analogously, estimate (34) will be replaced by

using (32) we can write

C2 (2") 1 <

p˜0 < p ;

k˚(v)k X T  C1 k f kL 2 C C1 T  kvk X T

k f kL 2 < ı ;

k˚(v1 )˚(v2 )k X  C2 kv1 v2 k X (2") 1 <

q˜0 < q :

˜ so that It is easy to check that we can shift the point ( p˜ ; q)

the same computation as above gives

" C1 ı < ; 2

p

X T D L T L q D L p (0; T; L q (Rn )) : If we follow the steps of the preceding proof and choose the same indices, we have now

X D L p Lq

kv i k X < " ;

We briefly sketch a proof of the local existence in the subcritical case 1 < < 1 C n4 . Consider the local space

e i(ts) ju(s)j ds ;

0

proves that we have actually u 2 C(R; L2 (Rn )).

M ; 2C1

C1 T  <

M 2

we obtain from (35) that ˚ takes the ball B(0; M) X T into itself; if in addition we have also, by possibly taking T smaller, C2 T  (2M) 1 <

1 2

we see by (36) that ˚ is a contraction, and in conclusion we obtain the existence of a unique local solution to the subcritical equation. By the same argument as above this solution is in C([0; T]; L2 (Rn )). We notice that in the above proof the quantity M is only determined by the L2 norm of f , and hence also the lifespan of the local solution is a function of k f kL 2 . If for some reason we know a priori that the L2 norm of the solution is conserved by the evolution, this remark implies immediately that the solution is in fact global, by an elementary continuation argument. This is the case for instance of the gauge invariant equation iu t  u D ˙juj 1 u for which any solution continuous with values in L2 satisfies ku(t)kL 2 D ku(0)kL 2 . The above computations apply without modification and we obtain the global well posedness for all subcritical exponents .

Dispersion Phenomena in Partial Differential Equations

Future Directions The dispersive properties of constant coefficient equations are now fairly well understood. In view of the wide range of applications, it would be very useful to extend the techniques to more general classes of equations, in particular for equations with variable coefficients and on manifolds. These problems have been the subject of intense interest in recent years and are actively pursued. Strichartz estimates have been extended to very general situations, including the fully variable coefficients case, under suitable smoothness, decay and spectral assumptions on the coefficients (see e. g. [36,37] and, for singular coefficients, [18]). The results concerning dispersive estimates are much less complete. They have been proved for perturbations of order zero iu t  u C V(x)u D 0 ;

u t t  u C V (x)u D 0

with a potential V(x) sufficiently smooth and decaying at infinity (see e. g. [2,19,28,53,54,55]) while for perturbation of order one iu t  u C a(x)  ru x j C V(x)u D 0 ; u t t  u C a(x)  ru x j C V(x)u D 0 very few results are available, (see [16] for the threedimensional wave and Dirac equations, and [1,17] for the one-dimensional case where quite general results are available). A closely connected problem is the study of the dispersive properties of equations on manifolds. This part of the theory is advancing rapidly and will probably be a very interesting direction of research in the next years (see e. g. [12] and related papers for the case of compact manifolds, and [5,24] for noncompact manifolds). Bibliography 1. Artbazar G, Yajima K (2000) The Lp -continuity of wave operators for one dimensional Schrödinger operators. J Math Sci Univ Tokyo 7(2):221–240 2. Beals M (1994) Optimal L1 decay for solutions to the wave equation with a potential. Comm Partial Diff Eq 19(7/8):1319– 1369 3. Ben-Artzi M, Trèves F (1994) Uniform estimates for a class of evolution equations. J Funct Anal 120(2):264–299 4. Bergh J, Löfström J (1976) Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften, No 223. Springer, Berlin 5. Bouclet J-M, Tzvetkov N (2006) On global strichartz estimates for non trapping metrics. 6. Bourgain J (1993) Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom Funct Anal 3(2):107–156

7. Bourgain J (1995) Some new estimates on oscillatory integrals. In: (1991) Essays on Fourier analysis in honor of Elias M. Stein. Princeton Math Ser vol. 42. Princeton Univ Press, Princeton, pp 83–112 8. Bourgain J (1995) Estimates for cone multipliers. In: Geometric aspects of functional analysis. Oper Theory Adv Appl, vol 77. Birkhäuser, Basel, pp 41–60 9. Bourgain J (1998) Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int Math Res Not 5(5):253–283 10. Brenner P (1975) On Lp  Lp0 estimates for the wave-equation. Math Z 145(3):251–254 11. Brenner P (1977) Lp  Lp0 -estimates for Fourier integral operators related to hyperbolic equations. Math Z 152(3):273–286 12. Burq N, Gérard P, Tzvetkov N (2003) The Cauchy problem for the nonlinear Schrödinger equation on a compact manifold. J Nonlinear Math Phys 10(1):12–27 13. Cazenave T (2003) Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol 10. New York University Courant Institute of Mathematical Sciences, New York 14. Choquet-Bruhat Y (1950) Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique. CR Acad Sci Paris 230:618–620 15. Christodoulou D, Klainerman S (1989) The nonlinear stability of the Minkowski metric in general relativity. In: Bordeaux (1988) Nonlinear hyperbolic problems. Lecture Notes in Math, vol 1402. Springer, Berlin, pp 128–145 16. D’Ancona P, Fanelli L (2006) Decay estimates for the wave and dirac equations with a magnetic potential. Comm Pure Appl Anal 29:309–323 17. D’Ancona P, Fanelli L (2006) Lp – boundedness of the wave operator for the one dimensional Schrödinger operator. Comm Math Phys 268:415–438 18. D’Ancona P, Fanelli L (2008) Strichartz and smoothing estimates for dispersive equations with magnetic potentials. Comm Partial Diff Eq 33(6):1082–1112 19. D’Ancona P, Pierfelice V (2005) On the wave equation with a large rough potential. J Funct Anal 227(1):30–77 20. D’Ancona P, Georgiev V, Kubo H (2001) Weighted decay estimates for the wave equation. J Diff Eq 177(1):146–208 21. Foschi D (2005) Inhomogeneous Strichartz estimates. J Hyperbolic Diff Eq 2(1):1–24 22. Ginibre J, Velo G (1985) The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann Inst Poincaré H Anal Non Linéaire 2(4):309–327 23. Ginibre J, Velo G (1995) Generalized Strichartz inequalities for the wave equation. J Funct Anal 133(1):50–68 24. Hassell A, Tao T, Wunsch J (2006) Sharp Strichartz estimates on nontrapping asymptotically conic manifolds. Am J Math 128(4):963–1024 25. Hörmander L (1997) Lectures on nonlinear hyperbolic differential equations. Mathématiques & Applications (Mathematics & Applications), vol 26. Springer, Berlin 26. John F (1979) Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr Math 28(1–3):235– 268 27. John F, Klainerman S (1984) Almost global existence to nonlinear wave equations in three space dimensions. Comm Pure Appl Math 37(4):443–455 28. Journé J-L, Soffer A, Sogge CD (1991) Decay estimates for Schrödinger operators. Comm Pure Appl Math 44(5):573–604

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43. Shatah J, Struwe M (1998) Geometric wave equations. In: Courant Lecture Notes in Mathematics, vol 2. New York University Courant Institute of Mathematical Sciences, New York 44. Strichartz RS (1970) Convolutions with kernels having singularities on a sphere. Trans Am Math Soc 148:461–471 45. Strichartz RS (1970) A priori estimates for the wave equation and some applications. J Funct Anal 5:218–235 46. Strichartz RS (1977) Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. J Duke Math 44(3):705–714 47. Tao T (2000) Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm Part Diff Eq 25(7–8):1471–1485 48. Tao T (2003) Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J Diff Eq 189(2):366–382 49. Taylor ME (1991) Pseudodifferential operators and nonlinear PDE. Progr Math 100. Birkhäuser, Boston 50. von Wahl W (1970) Über die klassische Lösbarkeit des CauchyProblems für nichtlineare Wellengleichungen bei kleinen Anfangswerten und das asymptotische Verhalten der Lösungen. Math Z 114:281–299 (in German) 51. Wolff T (2001) A sharp bilinear cone restriction estimate. Ann Math (2) 153(3):661–698 52. Yajima K (1987) Existence of solutions for Schrödinger evolution equations. Comm Math Phys 110(3):415–426 53. Yajima K (1995) The W k;p -continuity of wave operators for Schrödinger operators. J Math Soc Japan 47(3):551–581 54. Yajima K (1995) The W k;p -continuity of wave operators for schrödinger operators. III. even-dimensional cases m  4. J Math Sci Univ Tokyo 2(2):311–346 55. Yajima K (1999) Lp -boundedness of wave operators for two-dimensional schrödinger operators. Comm Math Phys 208(1):125–152

Dynamics on Fractals

˜

Dynamics on Fractals RAYMOND L. ORBACH Department of Physics, University of California, Riverside, USA Article Outline Glossary Definition of the Subject Introduction Fractal and Spectral Dimensions Nature of Dynamics on Fractals – Localization Mapping of Physical Systems onto Fractal Structures Relaxation Dynamics on Fractal Structures Transport on Fractal Structures Future Directions Bibliography Glossary Fractal Fractal structures are of two types: deterministic fractals and random fractals. The former involves repeated applications of replacing a given structural element by the structure itself. The process proceeds indefinitely, leading to dilation symmetry: if we magnify part of the structure, the enlarged portion looks the same as the original. Examples are the Mandelbrot– Given fractal and the Sierpinski gasket. A random fractal obeys the same properties (e. g. dilation symmetry), but only in terms of an ensemble average. The stereotypical example is the percolating network. Fractals can be constructed in any dimension, d, with for example, d D 6 being the mean-field dimension for percolating networks. Fractal dimension The fractal dimension represents the “mass” dependence upon length scale (measuring length). It is symbolized by Df , with the number of sites on a fractal as a function of the measurement length L being proportional to L D f , in analogy to a homogeneous structure embedded in a dimension d having mass proportional to the volume spanned by a length L proportional to Ld . Spectral (or fracton) dimension The spectral (or fracton) dimension refers to the dynamical properties of fractal networks. It is symbolized by d˜s and can be most easily thought of in terms of the density of states of a dynamical fractal structure (e. g. vibrations of a fractal network). Thus, if the excitation spectrum is measured as a function of energy !, the density of states for excitations of a fractal network would be propor-

tional to ! (d s 1) , in analogy to a homogeneous structure embedded in a dimension d having a density of states proportional to ! (d1) . Localization exponent Excitations on a fractal network are in general strongly localized in the sense that wave functions fall off more rapidly than a simple exponential,  (r) exp[f[r/(!)]d  g] where (!) is an energy dependent localization length, and the exponent d is in general greater than unity. Definition of the Subject The dynamical properties of fractal networks are very different from homogeneous structures, dependent upon strongly localized excitations. The thermal properties of fractals depend upon a “spectral dimension” d˜s less than the “Euclidean” or embedding dimension d. Fractal randomness introduces statistical properties into relaxation dynamics. Transport takes place via hopping processes, reminiscent of Mott’s variable range-rate hopping transport for electronic impurity states in semi-conductors. Fractal dynamics serve as a guide for the behavior of random structures where the short length scale excitations are localized. Introduction The study of the dynamics on and of fractal networks [29, 30,31] is not an arcane investigation, with little application to real physical systems. Rather, there are many examples of real materials that exhibit fractal dynamics. Examples are the vibrational properties of silica aerogels [21,45], and magnetic excitations in diluted antiferromagnets [18,19,43]. Beyond these explicit physical realizations, one can learn from the very nature of the excitations on fractal structures how localized excitations in homogeneous materials behave. That is, the dynamics of fractal networks serve as a calculable model for transport of localized excitations in random structures that are certainly not mass fractals. Examples are thermal transport in glasses above the so-called “plateau” temperature [2,20,26], and the lifetime of high-energy lattice vibrations in a-Si [32,33,36,37]. As we shall show in subsequent sections, the former is an example of vibrational hopping-type transport (similar in nature to Mott’s variable-range hopping [22,23] for localized electronic states), while the latter exhibits counter-intuitive energy dependences for vibrational lifetimes. The following Sect. “Fractal and Spectral Dimensions” describes in brief terms the concept of the fractal (or mass) dimension, Df [7,8,9,10,15,40,41,46], and the spectral (or fracton) dimension d˜s [5,14,17,34,35]. The latter is re-

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lated to the anomalous diffusion characteristics of fractals (the so-called “ant in a labyrinth” introduced by de Gennes [8]), and is at the heart of the thermal properties of fractal structures. A conjecture introduced by Alexander and Orbach [5] suggested that the mean-field value (exact at d D 6) for d˜s for percolating networks, d˜s D 4/3, might be universal, independent of dimension. It is now generally regarded that this conjecture is only approximate, though a very good approximation for percolating networks in 2 D d < 6. Were it to be exact, it would mean that the dynamical properties of percolating networks could be expressed in terms of their geometrical properties. Section “Nature of Dynamics on Fractals–Localization” introduces the nature of the wave function for an excitation embedded in a fractal structure. In general, random self-similar structures generate “super-localized” wave functions, falling off faster than exponential. We will characterize the envelope of the wave function by the functional form [1]  (r) exp[f[r/(!)]d  g] ; where (!) is an energy-dependent localization length, and the exponent d is in general greater than unity. For excitations on fractal structures, the localization length (!) can be calculated analytically, allowing for explicit computation of physical phenomena. This allows analytic expressions for scattering lifetimes and excitation transport. Mapping of the diffusion problem onto the secular equation for scalar lattice vibrations allows the extraction of the density of vibrational states [5]. It is given ˜ by the simple relationship D(!) ! (d s 1) displaying the utility of the spectral dimension d˜s . A similar though slightly more complicated mapping can be applied to diluted Heisenberg antiferromagnets. The silica aerogels [13,16,42,44,45], and Mnx Zn1x F2 [43] and RbMnx Mg1x F3 [18,19] are realizations of such systems, respectively, and have been extensively investigated experimentally. The random nature of fractal structures introduces statistical complexity into relaxation processes. The direct (single vibration) [38] and Raman (scattering of two vibrations) [24,39] spin-lattice relaxation processes are discussed for random fractal networks in Sect. “Relaxation Dynamics on Fractal Structures”. The theoretical results are complex, related to the “Devil’s staircase” for Raman relaxation processes. Experiments probing these rather bizarre behaviors would be welcome. Given the “super-localization” of excitations on fractal networks [1,42], transport properties need to be ex-

pressed according to the same concepts as introduced by Mott [22,23] for the insulating phase of doped semiconductors—the so-called “variable range-rate hopping”. This is developed in Sect. “Transport on Fractal Structures”, and applied to the thermal transport of fractal structures [2,20,26]. The “lessons learned” from these applications suggest an interpretation of the thermal transport of glassy materials above the so-called “plateau” temperature range. This instance of taking the insights from fractal structures, and applying it to materials that are certainly not fractal yet possess localized vibrational states, is an example of the utility of the analysis of fractal dynamics. Finally, some suggestions, hinted at above, for future experimental investigations are briefly discussed in Sect. “Future Directions”. Fractal and Spectral Dimensions Fractals can be defined by the nature of the symmetry they exhibit: self-similar geometry or equivalently dilation symmetry [7,8,9,10,15,40,41,46]. This symmetry is most assuredly not translational, leading to a very different set of behaviors for physical realizations that shall be explored in subsequent Sections. Stated most simply, self-similar structures “look the same” on any length scale. Conversely, one cannot extract a length scale from observation of a self-similar structure. There are clearly limits to self-similarity for physical realizations. For example, at the atomic level, dilation symmetry must come to an end. And for finite structures, there will be a largest length scale. There are other length scales that bracket the self-similar regime. Thus, for percolating networks, an example of a random fractal [38,39], the percolation correlation length, (p), where p is the concentration of occupied sites or bonds, defines an upper limit to the regime of fractal geometry. In general, for percolating networks, the range of length scales ` over which the structure is fractal is sandwiched between the atomic length a, and the percolation correlation length (p) : a < ` < (p). Fractal structures can be deterministic or random. The former is a result of applying a structural rule indefinitely, replacing an element of a design by the design itself. Examples are the Mandelbrot–Given fractal and the Sierpinski gasket. The latter arises from a set of probabilistic rules. A simple example is the percolating network where a site or bond (e. g. in d D 2, a square grid) is occupied randomly with probability p. The resulting clusters contain a random number of sites or bonds, where the distribution function for the number of finite clusters of size s is denoted by ns (p). There exists a critical value p D pc where a connected cluster extends over all space (the “in-

Dynamics on Fractals

finite” cluster). For p < pc only finite clusters exist, with the largest cluster spanned by the length scale (p). (p) diverges at p D pc as jp  pc j . For p > pc finite clusters and the infinite cluster coexist, the number of finite clusters vanishing at p approaches unity. The probability that a site belongs to the infinite cluster varies with occupation probability p as (p  pc )ˇ . The characteristic length (p) for p > pc is a measure of the largest finite cluster size. Hence, for length scales ` > (p) the systems looks homogeneous. Thus, the fractal regime is sandwiched between the lattice constant a and the percolation correlation length (p). The remarkable utility of the percolating network becomes clear from these properties: there is a crossover length scale between fractal and homogeneous structural behavior that can be chosen at will by choosing a value for the site or bond occupation probability p > pc . Excitations of structures that map onto percolating networks can change their nature from fractal to continuous as a function of their length scale `, the transition occurring when `c (p). This property makes the percolating network the “fruit fly” of random structures. The flexibility of adjusting the crossover length `c enables a fit between those properties calculated for a percolating network with the properties of those physical systems that map onto percolating networks. The fractal dimension, Df , is defined from the dependence of the “mass” on length scale. For fractals, the number of points or bonds on the infinite cluster within a length scale ` (hence, the “mass”), for `  (p), depends upon ` as M(`) ` D f . From above it is straight forward to show Df D d  (ˇ/). For a percolation network [29] in d D 2; Df D 91/48. For d D 3; Df D 2:48 ˙ 0:09. In mean field, d D 6; Df D 4. The spectral dimension, d˜s , follows from the analysis of diffusion on a fractal network, as first postulated by de Gennes [14]: “An ant parachutes down onto an occupied site of the infinite cluster of a percolating network. At every time unit, the ant makes one attempt to jump to one of its adjacent sites. If that site is occupied, it moves there. If it is empty, the ant stays at its original site. What is the ensemble-averaged square distance that the ant travels in time t?” Gefen et al. [17] found that hr2 (t)i t 2C where depends upon the scaling exponent  (conductivity exponent), the probability that a site belongs to the infinite cluster scaling exponent ˇ, and the scaling exponent  for (p) : D (  ˇ)/. Alexander and Orbach [5] defined the spectral (or fracton) dimension as d˜s D 2Df /(2 C ). The importance of d˜s can be seen from the calculation of the probability of finding a diffusing particle at the starting point at time t, P0 (t), which for compact diffusion is the inverse of the number of visited sites in

˜

time t, V(t) t (d s /2) . For scalar elasticity, the vibrational density of states ˜

D(!) D (2!/)Im P˜0 (! 2 C i0C ) ! (d s 1) ; where P˜0 (!) is the Laplace transform of P0 (t). Noting that for homogeneous systems, D(!) ! (d1) , the name spectral dimension for d˜s becomes evident. Alexander and Orbach [5] named the vibrational excitations of fractal structures, fractons, in analogy with the vibrational excitations of homogeneous structures, phonons. Hence, they termed d˜s the fracton dimension. Alexander and Orbach noted that for percolation structures in the mean-field limit, d D 6, d˜s D 4/3 precisely. At that time (1982), values for d˜s for d < 6 appeared close to that value, and they conjectured that d˜s D 4/3 for 2  d  6. We now know that this is only approximate (but remarkably close [24]): d˜s D 1:325 ˙ 0:002 in d D 2, and 1:317 ˙ 0:003 in d D 3. Were the conjecture exact, a numerical relationship between structure and transport would be exact [ D (3Df /2)  2], dictated solely by fractal geometry. For percolating networks, the structure appears fractal for length scales `  (p) and homogeneous for `  (p). The vibrational excitations in the fractal regime are termed fractons, and in the homogeneous regime phonons. We define the crossover frequency !c as that for which the excitation length scale equals (p). Then !c (p  ˜ pc )( D f /d s ) . Continuity leads to a phonon velocity v(p) (p  pc ) [(Df /d˜s )  1]. This then leads to the dispersion relations: [`  (p); !  !c ; phonon regime]! v(p)k ; and ˜

[`  (p); !  !c ; fracton regime]! k (D f /d s ) : In the latter case, as we shall show in the next Section, vibrational excitations in the fractal regime are localized, so that k should not be thought of as a wave vector but rather the inverse of the localization length (!). Nature of Dynamics on Fractals – Localization The conductance of a d dimensional percolating network of size L; G(L), can be shown to be proportional to Lˇ where the exponent ˇ D (Df /d˜s )(d˜s  2). In the Anderson sense [6], localization occurs when ˇ  0 (marginal at ˇ D 0). The Alexander–Orbach conjecture [5], d˜s  4/3, leads to ˇ well less than zero for all embedding dimensions d. One can think of this form of localization as geometrical, as opposed to the scattering localization that traditionally is associated with Anderson localization. The lo-

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calization is strong, the wave function falling off faster than simple exponential. The wave function has the form [1]: h (r)i exp[ f[r/(!)]d  g]. The exponent d is a geometrical exponent describing the fact that an exponential decay along the fractal will be distorted when viewed in real space. In general, 1  d  dmin , where dmin is defined by ` R d min , with ` being the shortest path along the network between two points separated by a Pythagorean distance R. Bunde and Roman [11] have found that for percolating networks d D 1. The decay length (!) is the localization length ˜ from Sect. “Introduction”, (!) ! (d s /D f ) . For a random system such as a percolating network, a given realization of  (r) is badly behaved, depending upon the particular choice of origin taken for the position coordinate r. The ensemble average takes the average over all realizations of  (r), and is denoted by h (r)i. This has its dangers, as the calculation of physical properties should be performed using a specific realization for  (r), and then ensemble averaged. Calculation of physical properties utilizing an ensemble averaged wave function h (r)i will not yield the same result, and could be very misleading. Specific realizations for  (r) for two-dimensional bond percolation vibrational networks have been calculated by Nakayama and Yakubo [25,28,47]. The fracton “core” (or largest amplitude) possesses very clear boundaries for the edges of the excitation, with an almost steplike character and a long tail. The tail extends over large distances and the amplitude oscillates in sign. This is required for orthogonality to the uniform translational mode with ! D 0. Clearly these modes do not look like simple exponentials, hence the warning about the use of h (r)i as compared to  (r) for the calculation of matrix elements. Mapping of Physical Systems onto Fractal Structures The foregoing description of dynamics of fractal networks would be interesting in and of itself, but its applicability to physical systems makes it of practical importance. Further, the nature of localization, and computation of “hopping-type” transport on fractal structures, has application to materials that are not fractal. Thus, it is worthwhile to consider what systems are fractal, the nature of dynamics on such systems, and the lessons we can learn from them for structures that possess similar characteristics but are not fractal. Specifically, Mott’s [22,23] variable range-rate hopping for localized electronic impurity states has direct application to transport of vibrational energy via fracton hopping on fractal networks [2], and we shall argue to

thermal transport in glassy materials above the “plateau” temperature [20,26]. The fundamental equations for fractal dynamics spring from the equation for diffusion of a random walker P on a network: dPi /dt D j¤I w i j (P j  Pi ) where Pi is the probability that the ith site is occupied, wij is the probability per unit time for hopping from site i to site j, and w i j D w ji . Introducing new quantities Wi j D w i j for P i ¤ j, and Wi i D  j¤I w i j , one can rewrite the difP fusion equation as dPi /dt D j Wi j P j with the defined P relation j Wi j D 0. For scalar elasticity, the equation of motion for atomic P vibrations is d2 u i /dt 2 D j K i j u j , with ui the atomic displacement at position i, and K i j (i ¤ j) is given by K i j D k i j /m i with kij the force constant connecting two sites i and j, and mi the mass of the ith site. The diagonal element W ii is obtained from the balancing of forces at P the ith site: j K i j D 0. The only difference between these two equations is the order of the time derivatives, leading to the equivalence of the eigenvalue problems. We have used this equivalence to extract the density of vibrational states for scalar elasticity, and the definition of the spectral (or fracton) dimension in Sect. “Introduction”. The most direct physical example of a vibrating fractal network is the silica aerogel [13,16,42,44]. Different preparation conditions can change the density of these materials to very small values, indicative of the open porous structure of these materials. Two length scales are present: the fundamental building block of the structure (the silica), and the correlation length of the gel. In between, the clusters possess a fractal structure, while at length scales beyond the correlation length the gel behaves as a homogeneous porous glass. This crossover from fractal to homogeneous is the physical example of our previous discussion for percolating networks when the length scale of excitations passes through the percolation correlation length (p). The thermal, transport, and scattering properties of aerogels have been well studied, and are the classic example of fractal behavior. There are other examples of structures that map onto the eigenvalue spectrum of the diffusion equation. Ferromagnetic and antiferromagnetic materials can be diluted by non-magnetic impurities. This leads to random site dilution, mapping directly onto the site-diluted percolating network for magnetic interactions. This can be seen from the Heisenberg Hamiltonian for spin systems: P H D 1/2 i; j J i j S i S j , where S i is the vector spin at the ith site, and J ij the isotropic exchange interaction between sites i and j. The equation of motion for the spin operx x ator S C i D S i C iS i becomes for diluted ferromagnets

Dynamics on Fractals

P z C z C i„@S C j¤i J i j (S i S j  S j S i ). For low levels of i /@t D spin wave excitations, S zi  S zj  S, so that one obP tains the linear equation of motion i„@S C j¤I J i j i /@t D C C (S j  S i ). This is the same form as that for diffusion of a random walker on a network, so that, in analogy with the scalar vibrational network, the eigenvalue spectrum is the same. This mapping allows the spin wave spectrum to exhibit magnons and fracton waves obeying the same properties that phonons and fractons exhibit for scalar vibrations. For antiferromagnets, the change in sign of S zi for the two sublattices complicates the linearized equations of P C C motion. They become, i„@S C j¤I J i j (S j CS i ), i /@t D  i where  i D 1 for down spins and  i D 1 for up spins. This alters the dynamical equations for diluted antiferromagnets, and requires a separate calculation of the eigenvalue spectrum. Neutron diffraction experiments exhibit both conventional magnon and fracton excitations. In the case of the site-diluted antiferromagnet Mnx Zn1x F2 both excitations are observed simultaneously for length scales in the vicinity of (p) [43]. Relaxation Dynamics on Fractal Structures Electron spin resonance and non-radiative decay experiments can be performed in fractal materials. It is of interest to compare these dynamical properties with those found in translationally ordered structures. At first sight, for vibrational relaxation, it might be thought that the only difference would originate with the differences in the densities of vibrational states D(!). However, the localized nature of the vibrational states (fractons) and their random positions in the fractal network, introduces properties that are specific to fractal lattices. There are two cases of interest: one-fracton relaxation, and two-fracton inelastic scattering. The former is referred to as the “direct” relaxation process; the latter the “Raman” relaxation process. In both cases, the main effect of vibrational localization is on the relaxation time profile. Different spatial sites can have very different relaxation rates because of the randomness of their distance from the appropriate fracton sites. The result is a strongly non-exponential time decay. For one-fracton relaxation [1], a straightforward generalization of the direct relaxation process rate, W(!0 ; L), for relaxation energy ! 0 arising from interaction with a fracton of the same energy centered a distance L away, is proportional to 2q1

W(!0 ; L) !0

f[(!0 )](D f ) g

 [coth(ˇ!0 /2)](1/ıL ) expf[L/(!0 )]d  g :

Here, ˇ D 1/kB T and the factor ı L represents the energy width of the fracton state. In the homogeneous limit, where phonons represent the extended vibrational states, an energy conserving delta function would replace ı L . There are two limits for ı L , case (a) when the fracton relaxation rate ı caused by anharmonicity dominates; and case (b) when the electronic relaxation rate caused by the electron-fracton interaction is greater than ı. In the former case, ıL D ı; while in the latter, ıL D W(!0 ; L) itself, leading to a self-consistent determination of the relaxation rate. The latter case requires explicit consideration of the L dependence of ı L , further complicating the calculation of W(!0 ; L). The calculation of the time profile of the electronic state population and the average relaxation rate are complex, and the reader is referred to [32] for details. For case (a), the population of the initial electronic state is found to vary as h i [(D f /d  )1] g : (ln t)[(D f d  )/2d  ] t fc 1 (ln t) Here, c1 is a constant independent of time, but dependent upon ! 0 and ı. The population of the initial electronic state decays is thus found to be faster than a power law, but slower than exponential or stretched exponential. For case (b), the population of the initial electronic state is found to vary as n o (1/t) (ln t)[(D f /d  )1] : The decay is slower than in case (a), and is closer to a power law. The average relaxation rates for cases (a) and (b) have the same functional form: hWi D(!0 )[(!0 )(2q1) ] coth(ˇ!0 /2) where q D d˜s (d /Df ). The temperature dependence is the usual one found for the direct process, but the energy dependence differs, depending on the values of the dimensions and parameters for fractal networks. Two fracton relaxation is considerably more complex [3,4]. In summary, the time profile of the initial electronic state in the long time regime begins as a stretched exponential, then crosses over into a form similar to case (a) for the one fracton decay process. In the presence of rapid electronic cross relaxation, the time profile is exponential, with a low-temperature relaxation time h1/T1 i ˜ proportional to T f2d s [1C2(d  /D f )]1g for Kramers transitions (between half-integer time-reversed spin states), and ˜ T f2d s [1C2(d  /D f )]3g for non-Kramers transitions (between integer time-reversed spin states). The complexity of random systems adds an interesting consideration: the nature of the relaxation rate at a specific site as compared to the average relaxation rate calculated above. The decay profile of single-site spin-lattice

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relaxation is always exponential. However, the temperature dependence does not simply follow that of the average relaxation rate. Instead, it exhibits irregular statistical fluctuations reflecting the environment of the chosen site. As the temperature is raised, new relaxation channels, involving higher-frequency localized vibrations, will become activated, adding to the relaxation processes responsible for the relaxation at low temperatures. The temperature dependence of the relaxation rate W(T) at different sites will fluctuate with strong correlations in the temperature dependence between them, underlying large accumulative fluctuations. Using a step function for the vibrational Bose functions, the relaxation rate has a T 2 temperature dependence between randomly spaced steps that occur when new relaxation channels are opened: a “devil’s staircase” structure. The full Bose function smoothes out these steps, but the qualitative features remain. All of these features point to the complexity and richness of electronic relaxation associated with localized vibrational spectra. Their remarkable complexity is worthy of investigation. Transport on Fractal Structures The localization of excitations discussed in Sect. “Fractal and Spectral Dimensions” has profound influence on the transport properties of and on fractal structures. For example, in the case of the silica aerogels, the “conventional” form for glasses or amorphous materials for the thermal conductivity (T) is found [12]: a rapid rise at low temperatures leveling off to a “plateau” as a function of temperature T, and then a rise in (T) roughly as T 2 . At first sight, this is to be “expected” as the aerogels are random, and one could make an analogy with glassy systems. However, there is a fundamental inconsistency in such an analysis. The fracton excitations are localized, and therefore cannot contribute to transport. Thus, one would expect instead a rapid rise in (T) at low temperatures where the long length scale excitations are phonon-like (and hence delocalized), flattening off to a constant value when all the extended state density of states have reached their duLong– Petit value. The specific heat for these states is then a constant, and (T) would be a constant independent of temperature T. So far, this behavior is consistent with experiment. But what happens for temperatures T greater than the plateau temperature region? How can (T) increase when the only excitations are fractons and therefore localized? The solution to this conundrum lies in the anharmonic forces connecting the silica clusters in the aerogel network. The

presence of the anharmonicity allows for a process of fracton hopping [2,20,26], in precisely the same fashion as in Mott’s [22,23] variable range-rate hopping for localized electronic impurity states in semiconductors. The fracton can absorb/emit a lower frequency extended state phonon, and hop to another fracton location, thereby transmitting excitation energy spatially. This possibility is amusing, for it means that for fractal networks the anharmonicity contributes to thermal transport, whereas in translationally invariant systems the anharmonicity inhibits thermal transport. In fact, were there no anharmonicity in a fractal network, the thermal conductivity (T) would stay at the plateau value for all T greater than the temperature T p , the onset of the plateau, for all T > Tp . In this sense, anharmonicity stands on its head. It is essential for transport in the fracton regime, defined by T  Tp  „!c /kB . The localized vibrational excitation hopping contribution to the thermal conductivity is hop (T) D (kB /2V)˙0 R2 (!0 )/sl (!0 ; T). Here sl means strongly localized modes, R(!0 ) is the hopping distance associated with the sl mode, 0 ; sl (!0 ; T) the hopping lifetime of the sl mode 0 caused by anharmonic interactions at temperature T, and V the volume of the system. We have already derived the ensemble averaged wave function for fractons, h (r)i, in Sect. “Fractal and Spectral Dimensions”. We adopt the sl notation [26] because the hopping transport arguments apply to strongly localized states, independent of whether or not they are fractons. The dynamical process for hot (T) is an sl mode at one site coupling with a low energy extended state to hop to an sl mode at another site, thereby transferring excitation energy spatially. The evaluation of sl (!0 ; T) will depend upon the hopping distance for transport of the sl modes, R(!0 ). The argument of Mott [22,23], derived by him for localized electronic states, can be taken over for sl vibrational modes [2]. The volume that contains at least one sl mode is given by (4/3)D(!0 )!sl [R(!0 )]3 D 1, where D(!0 ) is the density of sl states per unit energy. This condition assures that, for an sl mode at the origin, a second sl mode can be found within a hopping distance R(!0 ). We find R(!0 )  (!0 /!sl )(!0 ). Thus, the most probable hopping distance is at least the sl localization length. The hopping rate 1/sl (!0 ; T) caused by anharmonicity, with anharmonic coupling constant Ceff can be found from the Golden Rule, leading to the hopping contribution to the thermal conductivity, with ( M )3 the volume for finding a single sl mode, hop (T) D [43 3 (Ceff )2 (kB )2 T]/[ 3 (vs )5 3 ( M )3 h(! )i2 ]. Here h(! )i is the average sl length scale, vs the velocity of sound, and the density. There are very few undetermined parameters in this expression for hop (T). The anharmonic coupling

Dynamics on Fractals

constant can be found from the shift of the “boson peak” in the Raman spectra with pressure [48,49], typically a factor of 25 times larger than the long length scale third-order elastic coupling constant for extended modes. Making use of experiments on amorphous a-GeS2 one finds [26,27] hop (T)  0:0065T (W/mK) using the appropriate values for and vs , and  M  h(! )i D 15 Å. This contribution is comparable to that observed for other networkforming glasses above the plateau value. There is a quantum phenomenon associated with sl mode hopping. The hopping vertex is associated with a low energy extended mode coupling with an sl mode to hop to another (spatially separated) sl mode. As the temperature increases, the lifetime of the sl mode will become shorter than 1/! of the low energy extended mode. This quantum effect leads to a breakdown of the Golden Rule expression, causing a leveling off of the linear-in-T thermal conductivity above the plateau value. The sl mode hopping contribution to (T) above the plateau temperature for glasses and amorphous materials appears to provide a quantitative explanation of the observed phenomena. It is a specific example of how dynamics on a fractal network can be used to determine the properties of materials that, while certainly not fractal, have properties that map onto the behavior of fractal structures. Future Directions As stated in the Introduction, studying the dynamics on fractal networks is a convenient structure for analyzing the properties of localized states, with all of the frequency and temperature-dependent physical properties determined without arbitrary parameters. For example, the frequency dependence of the localization length scale, (!), is known precisely on fractal networks, and is used for precise calculation of the hopping transport in Sect. “Relaxation Dynamics on Fractal Structures”, without adjustable parameters. If one extrapolates the properties of fractal networks to those random systems that are certainly not fractal, but which exhibit localization, the transport and relaxation properties can be understood. Examples are the thermal transport of glasses and amorphous materials above the plateau temperature, and the lifetime of high energy vibrational states in a-Si. Measurements of these properties may offer opportunities for practical devices in frequency regimes beyond extended state frequencies. For example, the lifetime of high energy vibrational states in a-Si has been shown to increase with increasing vibrational energies [36,37]. Such behavior is opposite to that expected from extended vibrational states interacting anharmoni-

cally. But it is entirely consistent if these states are localized and behave as shown for fractal networks [32,33]. The thesis presented here points to a much broader application of the consequences of fractal geometry than simply of those physical systems which exhibit such structures. The claim is made that using what has been learned from fractal dynamics can point to explication of physical phenomena in disordered systems that are certainly not fractal, but display properties analogous to fractal structures. In that sense, the future lies in application of the ideas contained in the study of fractal dynamics to that of random systems in general. This very broad class of materials may well have properties of great practical importance, predicted from the dynamics on fractals.

Bibliography Primary Literature 1. Alexander S, Entin-Wohlman O, Orbach R (1985) Relaxation and nonradiative decay in disordered systems, I. One-fracton emission. Phys Rev B 32:6447–6455 2. Alexander S, Entin-Wohlman O, Orbach R (1986) Phonon-Fracton anharmonic interaction: the thermal conductivity of amorphous materials. Phys Rev B 34:2726–2734 3. Alexander S, Entin-Wohlman O, Orbach R (1986) Relaxation and non-radiative decay in disordered systems, II. Two-fracton inelastic scattering. Phys Rev B 33:3935–3946 4. Alexander S, Entin-Wohlman O, Orbach R (1987) Relaxation and non-radiative decay in disordered systems, III. Statistical character of Raman (two-quanta) spin-lattice relaxation. Phys Rev B 35:1166–1173 5. Alexander S, Orbach R (1982) Density of states on fractals, “fractons”. J Phys Lett (Paris) 43:L625–L631 6. Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109:1492–1505 7. Avnir D (ed) (1989) The Fractal Approach to Heterogeneous Chemistry. Wiley, Chichester 8. Barabaási A-L, Stanley HE (1995) Fractal Concepts in Crystal Growth. Cambridge University Press, Cambridge 9. Bunde A, Havlin S (eds) (1991) Fractals and Disordered Systems. Springer, Berlin 10. Bunde A, Havlin S (eds) (1994) Fractals in Science. Springer-Verlag, Berlin 11. Bunde A, Roman HE (1992) Vibrations and random walks on random fractals: anomalous behavior and multifractality. Philos Magazine B 65:191–211 12. Calemczuk R, de Goer AM, Salce B, Maynard R, Zarembowitch A (1987) Low-temperature properties of silica aerogels. Europhys Lett 3:1205–1211 13. Courtens E, Pelous J, Phalippou J, Vacher R, Woignier T (1987) Brillouin-scattering measurements of phonon-fracton crossover in silica aerogels. Phys Rev Lett 58:128–131 14. de Gennes PG (1976) La percolation: un concept unificateur. Recherche 7:919–927 15. Feder J (1988) Fractals. Plenum Press, New York

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16. Freltoft T, Kjems J, Richter D (1987) Density of states in fractal silica smoke-particle aggregates. Phys Rev Lett 59:1212–1215 17. Gefen Y, Aharony A, Alexander S (1983) Anomalous diffusion on percolating clusters. Phys Rev Lett 50:77–80 18. Ikeda H, Fernandez-Baca JA, Nicklow RM, Takahashi M, Iwasa I (1994) Fracton excitations in a diluted Heisenberg antiferromagnet near the percolation threshold: RbMn0.39 Mg0.61 F3 . J Phys: Condens Matter 6:10543–10549 19. Ikeda H, Itoh S, Adams MA, Fernandez-Baca JA (1998) Crossover from Homogeneous to Fractal Excitations in the Near-Percolating Heisenberg Antiferromagnet RbMn0.39 Mg0.61 F3 . J Phys Soc Japan 67:3376–3379 20. Jagannathan A, Orbach R, Entin-Wohlman O (1989) Thermal conductivity of amorphous materials above the plateau. Phys Rev B 39:13465–13477 21. Kistler SS (1932) Coherent Expanded–Aerogels. J Phys Chem 36:52–64 22. Mott NF (1967) Electrons in disordered structures. Adv Phys 16:49–144 23. Mott NF (1969) Conduction in non-crystalline materials III. Localized states in a pseudogap and near extremities of conduction and valence bands. Philos Mag 19:835–852 24. Nakayama T (1992) Dynamics of random fractals: large-scale simulations. Physica A 191:386–393 25. Nakayama T (1995) Elastic vibrations of fractal networks. Japan J Appl Phys 34:2519–2524 26. Nakayama T, Orbach R (1999) Anharmonicity and thermal transport in network glasses. Europhys Lett 47:468–473 27. Nakayama T, Orbach RL (1999) On the increase of thermal conductivity in glasses above the plateau region. Physica B 263– 264:261–263 28. Nakayama T, Yakubo K (2001) The forced oscillator method: eigenvalue analysis and computing linear response functions. Phys Rep 349:239–299 29. Nakayama T, Yakubo K (2003) Fractal Concepts in Condensed Matter Physics. Solid-State Sciences. Springer, Berlin 30. Nakayama T, Yakubo K, Orbach R (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 31. Orbach R (1986) Dynamics of Fractal Networks. Science 231:814–819 32. Orbach R (1996) Transport and vibrational lifetimes in amorphous structures. Physica B 219–220:231–234 33. Orbach R, Jagannathan A (1994) High energy vibrational lifetimes in a-Si:H. J Phys Chem 98:7411–7413 34. Rammal R (1984) Spectrum of harmonic excitations on fractals. J Phys (Paris) 45:191–206 35. Rammal R, Toulouse G (1983) Random walks on fractal structures and percolation clusters. J Phys Lett (Paris) 44:L13–L22

36. Scholten AJ, Dijkhuis JI (1996) Decay of high-frequency phonons in amorphous silicon. Phys Rev B 53:3837–3840 37. Scholten AJ, Verleg PAWE, Dijkjuis JI, Akimov AV, Meltzer RS, Orbach R (1995) The lifetimes of high-frequency phonons in amorphous silicon: evidence for phonon localization. Solid State Phenom 44–46:289–296 38. Stanley HE (1977) Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents. J Phys A 10:L211–L220 39. Stauffer D, Aharony A (1992) Introduction to Percolation Theory, 2nd edn. Taylor and Francis, London/Philadelphia 40. Stauffer D, Stanley HE (1995) From Newton to Mandelbrot: A primer in Theoretical Physics, 2nd edn. Springer, Berlin 41. Takayasu H (1990) Fractals in the Physical Sciences. Manchester University Press, Manchester 42. Tsujimi Y, Courtens E, Pelous J, Vacher R (1988) Raman-scattering measurements of acoustic superlocalization in silica aerogels. Phys Rev Lett 60:2757–2760 43. Uemura YJ, Birgeneau RJ (1987) Magnons and fractons in the diluted antiferromagnet Mnx Zn1x F2 . Phys Rev B 36:7024– 7035 44. Vacher R, Courtens E, Coddens G, Pelous J, Woignier T (1989) Neutron-spectroscopy measurement of a fracton density of states. Phys Rev B 39:7384–7387 45. Vacher R, Woignier T, Pelous J, Courtens E (1988) Structure and self-similarity of silica aerogels. Phys Rev B 37:6500–6503 46. Vicsek T (1993) Fractal Growth Phenomena. World Scientific, Singapore 47. Yakubo K, Nakayama T (1989) Direct observation of localized fractons excited on percolating nets. J Phys Soc Japan 58:1504–1507 48. Yamaguchi M, Nakayama T, Yagi T (1999) Effects of high pressure on the Bose peak in a-GeS2 studied by light scattering. Physica B 263–264:258–260 49. Yamaguchi M, Yagi T (1999) Anharmonicity of low-frequency vibrations in a-GeS2 studied by light scattering. Europhys Lett 47:462–467

Recommended Reading While many of the original articles are somewhat difficult to access, the reader is nevertheless recommended to read them as there are subtle issues that tend to disappear as references are quoted and re-quoted over time. For a very helpful introduction to percolation theory, Ref. 39 is highly recommended. A thorough review of dynamics of/on fractal structures can be found in Ref. 30. Finally, a comprehensive treatment of localization and multifractals, and their relevance to dynamics of/on fractal networks, can be found in Ref. 29.

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation ALAN CHAMPNEYS Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom Article Outline Glossary Definition of the Subject Introduction Linear Resonance or Nonlinear Instability? Multibody Systems Continuous Systems Future Directions Bibliography Glossary Parametric excitation Explicit time-dependent variation of a parameter of a dynamical system. Parametric resonance An instability that is caused by a rational relationship between the frequency of parametric excitation and the natural frequency of free oscillation in the absence of the excitation. If ! is the excitation frequency, and ! 0 the natural frequency, then parametric resonance occurs when ! D (n/2)!0 for any positive integer n. The case n D 1 is usually the most prominent form of parametric resonance, and is sometimes called the principle subharmonic resonance. Autoparametric resonance A virtual parametric resonance that occurs due to the coupling between two independent degrees of freedom within a system. The output of one degree of freedom acts like the parametric excitation of the other. Ince–Strutt diagram A two-parameter bifurcation diagram indicating stable and unstable regions, specifically plotting the instability boundaries as the required amplitude of parametric excitation against the square of the ratio of natural to excitation frequency. Floquet theory The determination of the eigenvalue spectrum that governs the stability of periodic solutions to systems of ordinary differential equations. Bifurcation A qualitative change in a system’s dynamics as a parameter is varied. One-parameter bifurcation diagrams often depict invariant sets of the dynamics against a single parameter, indicating stability and any bifurcation points. Two-parameter bifurcation diagrams depict curves in a parameter plane on which one-parameter bifurcations occur.

Modal analysis The study of continuum models by decomposing their spatial parts into eigenmodes of the dynamic operator. The projection of the full system onto each mode gives an infinite system of differential equations in time, one for each mode. Monodromy matrix The matrix used in Floquet theory, whose eigenvalues (also known as Floquet multipliers) determine stability of a periodic solution. Definition of the Subject Parametric excitation of a system differs from direct forcing in that fluctuations appear as temporal modulation of a parameter rather than as a direct additive term. A common paradigm is that of a pendulum hanging under gravity whose support is subjected to a vertical sinusoidal displacement. In the absence of any dissipative effects, instabilities occur to the trivial equilibrium whenever the natural frequency is a multiple of half the excitation frequency. At amplitude levels beyond the instability, further bifurcations (dynamical transitions) can lead to more complex quasi-periodic or chaotic dynamics. In multibody mechanical systems, one mode of vibration can effectively act as the parametric excitation of another mode through the presence of multiplicative nonlinearity. Thus autoparametric resonance occurs when one mode’s frequency is a multiple of half of the other. Other effects include combination resonance, where the excitation is at a sum or difference of two modal frequencies. Parametric excitation is important in continuous systems too and can lead to counterintuitive effects such as stabilization “upside-down” of slender structures, complex wave patterns on a fluid free surface, stable pulses of light that overcome optical loss, and fluid-induced motion from parallel rather than transverse flow. Introduction Have you observed a child on a playground swing? In either a standing or a sitting position, generations of children have learned how to “pump” with their legs to set the swing in motion without making contact with the ground, and without external forcing. Note that pumping occurs at twice the natural frequency of swing of the pendulum, since the child extends her legs maximally at the two extremities of the swing’s cycle, that is, two times per period. What is happening here? The simplest case to analyze is where the child is standing. Then, it has been argued by Curry [17] that the child plus swing is effectively a simple pendulum system, where the child’s pumping has the effect of periodic variation of the position of the center of the mass along the arm of the pendulum (see Fig. 1a,b).

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Dynamics of Parametric Excitation, Figure 1 a A child standing on a swing modeled as b a parametrically excited pendulum c a directly forced pendulum

We shall return to the self-propelled swing problem in Sect. “Future Directions” below, where we shall find out that all is not what it seems with this motivating example, but for the time being lets make the simple hypothesis that the child’s pumping causes the center of mass to move up and down. Using the standard model of a pendulum that consists of a lumped mass suspended by a rigid massless rod of length l from a pivot, the equations of motion for the angular displacement (t) of a pendulum hanging under gravity can be written in the form g ¨ (t) C sin[ (t)] D 0 : l Here g is the acceleration due to gravity, a dot denotes differentiation with respect to time t and, for the time being, we have ignored any damping that must inevitably be present. Now, the moving center of mass means that the effective length of the rigid rod oscillates, and let us suppose for simplicity that this oscillation is sinusoidal about some mean length l0 : l(t) D l0 [1  " cos(! t)] : Here " > 0 assumed to be small and as yet we haven’t specified the frequency !. Then, upon Taylor expansion in ", we find to leading order ¨ C !02 [1 C " cos(! t)] sin D 0 ; p where !0 D (g/l0 ) is the natural frequency of small amplitude swings in the absence of the pump. Finally, upon rescaling time ˜t D ! t and dropping the tildes we find ¨ C [˛ C ˇ cos t] sin D 0 ;

(1)

where ˛D

!02 !2

and ˇ D "

!02 !2

(2)

are dimensionless parameters that respectively describe the square of the ratio of natural frequency to excitation frequency, and the amplitude of the excitation. Equation (1) is a canonical example of a parametrically excited system. Note that (1) also describes, at least to leading order, the case of a pendulum hanging under gravity that is excited by a vertical force of size "!02 cos t which we can think of as a periodic modulation of the gravity. This would be in contrast to a directly forced simple pendulum where the forcing would act in the direction of swing for small amplitude motion; see Fig. 1c. For the forced pendulum, the equivalent equations of motion would look more like ¨ C ˛ sin D ˇ cos t ;

(3)

where the periodic forcing occurs as an additive input into the differential equation. In contrast, for (1) the periodic excitation occurs as a modulation of a parameter (which you can think of as gravity), hence the name ‘parametric’. Another feature of (1) that makes it a good paradigm for this article is that it is nonlinear and, as we shall see, a full appreciation of the dynamics resulting from parametric excitation requires treatment of nonlinear terms. An interesting contrast between parametrically excited and directly forced systems is in the nature of the trivial response to a non-zero input. In particular, for any ˛ and ˇ, Eq. (1) admits the trivial solution (t)  0, whereas there is no such solution to (3) for non-zero ˇ. The simplest “steady” solution to the directly forced system for small ˇ is a small-amplitude periodic oscillation of period 2 (in our dimensionless time units). So any analysis of a parametrically excited system should start with an analysis of the steady state D 0, which corresponds to the pendulum just hanging under gravity. To analyze stability of this equilibrium state let D 0C x where x is small, then x sat-

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 2 Possible behavior of Floquet multipliers  (eigenvalues of the monodromy matrix M) for a a conservative Hill equation and b in the presence of small damping

isfies linearization (1) x¨ C (˛ C ˇ cos t)x D 0 :

(4)

The equilibrium position is stable if all solutions x(t) of this equation, remain bounded as t ! 1 and it is unstable if there are solutions x(t) ! 1 as t ! 1.

Equation (4) is known as the Mathieu equation [42], and is a specific example of Hill’s equation [29,40] x¨ C [˛ C V(t)]x D 0 ;

(5)

where V(t) is any periodic function with period 2. Hill’s equation can be solved using classical methods for linear

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time-dependent ordinary differential equations (ODEs), see e. g. [31,33]. In particular, by writing y1 D x and y2 D x˙ , Hill’s equation (5) is just a two-dimensional example of a linear first-order time-periodic equation y˙ D P(t)y ;

y 2 Rn :

(6)

The general solutions to (6) for any initial condition can be expressed in terms of a fundamental matrix ˚ (t), with ˚(0) D I n (the identity matrix of dimension n): y(t) D ˚(t)y(0) : Note that just because P(t) has period 2 it does not follow that ˚(t) does. Also ˚(t) cannot be computed exactly, except in highly specialized cases, and one has to rely on approximate methods as described in the next Sect. “Floquet Analysis” and in the companion article to this one  Perturbation Analysis of Parametric Resonance. However, in order to study stability we don’t actually need to construct the full solution ˚ (t), it is sufficient to consider M D ˚(2), which is called the monodromy matrix associated with Hill’s equation. Stability is determined by

studying the eigenvalues of M: solutions to (5) with small initial conditions remain small for all time if all eigenvalues of M lie on or within the unit circle in the complex plane; whereas a general initial condition will lead to unbounded motion if there are eigenvalues outside this circle. See Fig. 2a. Eigenvalues of the Monodromy matrix are also known as Floquet multipliers of the system, and the general study of the stability of periodic systems is called Floquet theory. In the case of the Mathieu equation, where V (t) D ˇ cos(t), we have

 0 1 P(t) D : (7) 0 ˛ C ˇ cos(t) Figure 3 depicts graphically the behavior of Floquet multipliers for this case. Here, in what is known as an Ince– Strutt diagram [31,63], the shaded regions represent the values of parameters at which the origin x D 0 is stable (with Floquet multipliers on the unit circle) and the white regions are where the origin is unstable (at least one multiplier outside the unit circle). We describe how to calculate such a diagram in the next section. For the time be-

Dynamics of Parametric Excitation, Figure 3 Ince–Strutt diagram showing resonance tongues of the Mathieu equation (4). Regions of instability of the trivial solution x D 0 are shaded and a distinction is drawn between instability curves corresponding to a period-T (T D 2) and to a period-2T orbit. After (2nd edn. in [33]), reproduced with permission from Oxford University Press

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 4 Similar to Fig. 3 but for the damped Mathieu equation (8) with ı D 0:1. After http://monet.unibas.ch/~elmer/pendulum/parres.htm, reproduced with permission

ing we note that there are resonance tongues that emanate from ˇ D 0 whenever ˛ is the square of a half-integer: ˛ D 1/4, 1, 9/4, 4, 25/4, etc. The width (in ˛) of the tongue for small ˇ turns out to be proportional to ˇ to the power of twice the half-integer that is being squared. That is, the ˛ D 1/4 tongue has width O(ˇ), the ˛ D 1 tongue has width O(ˇ)2 , the ˛ D 9/4 tongue has width O(ˇ 3 ) etc. On the tongue boundaries (between the shaded and the unshaded regions of Fig. 3), the Mathieu equation admits non-trivial periodic solutions. For the tongues that are rooted at square integers, ˛ D 1, 9, 16, etc., these solutions have period 2 (the same period as that of the excitation) and the instability that occurs upon crossing such a tongue boundary is referred to as a harmonic instability. On the other boundaries, those of tongues rooted at ˛ D 1/4, 9/4, 25/4 etc., the periodic solution has period 4 (twice that of the excitation). Hence such instabilities are characterized by frequencies that are half that of the excitation and are hence referred to as subharmonic instabilities. So far, we have ignored any dissipative effects, so that Eq. (5) can be regarded as a conservative or Hamiltonian system. Damping can be introduced via a dissipative force that is proportional to velocity, hence (4) becomes x¨ C ı x˙ C (˛ C ˇ cos t)x D 0 ;

(8)

where ı is a dimensionless damping coefficient. One can also apply Floquet theory to this damped Mathieu equation; see Fig. 2b. The results for Eq. (8) are plotted in Fig. 4, where we can see the intuitive effect that damping

increases the areas of stability. In particular, in comparison with Fig. 3, note that the resonance tongues have been lifted from the ˛-axis, with the amount of the raise being inversely proportional to the width of the undamped tongue. Thus, the resonance tongue corresponding to ˛ D 1/4 is easily the most prominent. Not only for fixed forcing amplitude ˇ does this tongue occupy for the greatest interval of square frequency ratio ˛, but it is accessible with the least forcing amplitude ˇ. Thus, practicing engineers often think of this particular subharmonic instability close to ˛ D 1/4 (i. e. where the forcing frequency is approximately half the natural frequency) as being the hallmark of parametric resonance. Such an instability is sometimes called the principal parametric resonance. Nevertheless, we shall see that other instability tongues, in particular the fundamental or first harmonic parametric resonance near to ˛ D 1 can also sometimes be of significance. Returning briefly to the problem of how children swing, it would seem that the choice ˛  1/4 is in some sense preferable. This is the largest instability tongue for small ˇ. That is, the smallest effort is required in order to cause an exponential instability of the downwards hanging solution, and set the swing in motion. And, according to (2), ˛ D 1/4 corresponds to ! D 2!0 . Thus generations of children seem to have hit upon an optimal strategy [48], by pumping their legs at precisely twice the frequency of the swing. Finally, looking at Fig. 4, note the curious feature that we have continued the diagrams into negative ˛, despite the fact that ˛ was defined as a square term. In the case of

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a pendulum though, ˛ D l/(g! 2 ), and so we can think of negative ˛ as representing negative g. That is, a pendulum with negative ˛ is defying gravity by ‘hanging upwards’. Clearly without any excitation, such a situation would be violently unstable. The big surprise then is that there is a thin wedge of stability (which is present either with or without damping) for ˛ < 0. Note though that the wedge is thickest for small ˇ and small j˛j, which limit corresponds to small amplitude, high-frequency oscillation. This remarkable stabilization of an upside-down pendulum by tiny, rapid vertical vibration was first described by Stephenson [53] in 1908, re-discovered by Kapitza [34] in 1951 (and hence often called the Kapitza pendulum problem) and has been experimentally verified many times; see e. g. [3]. We shall return to this upside down stability in Sect. “Upside-Down Stability”. The rest of this article is outlined as follows. In Sect. “Linear Resonance or Nonlinear Instability?” we shall examine how the Ince–Strutt diagram is constructed and see that this is just an example of the linearized analysis around any periodic state of a nonlinear system. Further quasi-periodic or chaotic motion can ensue as we push beyond this initial instability. Section “Multibody Systems” concerns mechanical systems with multiple degrees of freedom. We consider autoparametric resonance where the motion of one degree of freedom acts as the parametric excitation of another. We also look at conditions in multibody systems where a combination of two natural frequencies can be excited parametrically, focusing in particular on the somewhat illusive phenomenon of difference combination resonances. Section “Continuous Systems” then looks at parametric excitation in continuous systems, looking at phenomena as diverse as pattern formation on the surface of a liquid, flow-induced oscillation of pipes, a clever way of overcoming loss in optical fibers and the oscillation-induced stiffening of structures. Finally, Sect. “Future Directions” draws conclusions by looking at future and emerging ideas. Linear Resonance or Nonlinear Instability? Let us now consider a more detailed analysis of parametric resonance of the Mathieu equation and of the implications for a nonlinear system, using the parametrically excited pendulum as a canonical example.

behave as parameters are varied. Essentially, we explain the behavior of Fig. 2a. Our treatment follows the elementary account in [33], to which the interested reader is referred for more details. For a general linear system of the form (6), a lot of insight can be gained by considering the so-called Wronskian W(t) D det(˚(t)). From the elementary theory of differential equations (e. g. [33]) we have Z t  trace[P(s)] ds ; (9) W(t) D W(t0 ) exp t0

where the trace of a matrix is the sum of its diagonal elements. Note that the specific form of P(t) for the Mathieu equation (7) has trace[P(t)] D 0, hence, from (9) that that det(M) D det[W(2)] D 1. But from elementary algebra we know that the trace of the matrix is the product of its eigenvalues. So we know that the two Floquet multipliers 1;2 satisfy 1 2 D 1 and must therefore solve a characteristic equation of the form 2  2(˛; ˇ) C 1 D 0 ; for some unknown real function . There are two generic cases jj < 1 and jj > 1. If  > 1, then the Floquet multipliers are complex and lie on the unit circle. This corresponds to stability of the zero-solution of the Mathieu equation. Conversely if jj < 1, then the Floquet multipliers are real and satisfy j1 j < 1 and j2 j D 1/j1 j > 1. The larger multiplier represents an exponentially growing solution and hence corresponds to an instability. The two boundary cases are:  D 1, in which case we have a double Floquet multiplier at  D 1; or  D 1, which corresponds to a double multiplier at  D 1. These represent respectively the harmonic and the subharmonic stability boundaries in the Ince–Strutt diagram. But, how do we compute ? The answer is that we don’t. We simply recognize that  D 1 corresponds to the existence of a pure 2-periodic solution of the fundamental matrix M D ˚ (2), and that  D 1 represents a periodic solution of ˚(2)2 D ˚(4) and hence a 4periodic solution of the fundamental matrix. To look for a 2-periodic solution of the Mathieu equation, it is convenient to use Fourier series. That is, we seek a solution (4) in the form x(t) D

nDC1 X

c n e i nt :

nD1

Floquet Analysis We begin by giving a reasoning as to why the Ince–Strutt diagram Fig. 3 for the Mathieu equation (ignoring damping for the time being) looks the way it does, by making a more careful consideration of how Floquet multipliers

Substitution into (4), using cos(t) D (1/2)(e i t Cei t ) leads to infinitely many equations of the form (1/2)ˇc nC1 C (˛  n2 )c n C (1/2)ˇc n1 D 0 ; n D 1; : : : ; 1 ;

with cn D c n ;

(10)

Dynamics of Parametric Excitation

where an asterisk represents complex conjugation. Note that the final condition implies that x is real. Clearly, for ˇ D 0, a nontrivial solution can be found for any integer p which has c n D 0 for all n ¤ p, provided ˛ 2 D p2 . That is, solution curves bifurcate from ˇ D 0 for each ˛-value that is the square of a whole integer. Furthermore, nonlinear analysis can be used to show that for small positive ˇ, there are two solution branches that emerge from each such point (provided p ¤ 0); one corresponding to cp being real, which corresponds to motion x(t) cos(pt) in phase with the excitation; and another corresponding to cp imaginary, which corresponds to motion x(t) sin(pt) out of phase with the excitation. Similarly, if we look for solutions which are 4-periodic by making the substitution x(t) D

mDC1 X

d m e i mt/2 ;

(11)

mD1

we arrive at an infinite system of equations (1/2)ˇd mC2 C (˛  (1/4)m2 )d m C (1/2)ˇd m1 D 0 ; m D 1; : : : ; 1 ;

 with dm D d m :

(12)

Note that the equations for m even and m odd in (12) are decoupled. The equations for even m become, on setting m D 2n, precisely the system (10) with d2m D c n . For m D 2n C 1, however, we arrive at new points of bifurcation, by exactly the same logic as we used to analyze (10). Namely whenever ˛ 2 D (p/2)2 for any odd integer p we have the bifurcation from ˇ D 0 of two 4-periodic solutions (one like cos(pt/2) and one like sin(pt/2)). Perturbation analysis can be used to obtain asymptotic expressions for the transition curves that bifurcate from ˇ D 0 at these special ˛-values; see the companion article  Perturbation Analysis of Parametric Resonance for the details. In particular it can be shown that the width of the stability tongue (separation in ˛-values from the sine and the cosine instability curves) originating at ˛ D (p/2)2 (for p being odd or even) scales like tongue width p ˇ p C h.o.t : Hence the principal subharmonic instability arising from ˛ D 1/4 has width O(ˇ) (actually the transition curves are given by ˛ D 1/4 ˙ (1/2)ˇ C O(ˇ 2 )), and the fundamental harmonic instability arising from ˛ D 1 has width O(ˇ 2 ), etc. Hence the principal subharmonic instability is by far the most prevalent as it occupies the greatest frequency width for small ˇ, and is often thought to be the hallmark of parametric resonance in applications.

Note that we have not as yet mentioned the transition curve emerging from ˛ D ˇ D 0, which corresponds to n D 0 in the above analysis. This case can also be analyzed asymptotically, and the single transition curve can be shown to scale like ˛ D (1/2)ˇ 2 C O(ˇ 3 ) for small ˇ. It is this bending back into ˛ < 0 that leads to the Kapitza pendulum effect, which we shall return to in Sect. “Upside-Down Stability” below. Finally, the addition of small damping can be shown to narrow the tongues and lift them off from the ˇ D 0 axis as shown in Fig. 4. The amount of lift is in proportion to the width of the tongue. This property further enhances the pre-eminence of the principal subharmonic instability for parametrically excited systems in practice, as they thus require the least amplitude of vibration in order to overcome the damping forces. Bifurcation Theory We have just analyzed the Mathieu equation, which is purely linear. But, for example as shown in the Introduction, such an equation is often derived in the course of analyzing the stability of a trivial solution to a nonlinear parametrically excited system. It is worthwhile therefore to consider the nonlinear implications of the harmonic and subharmonic instabilities. The time-periodic Mathieu equation can be written as an example of a three-dimensional autonomous system of ODEs x˙ D y ;

y˙ D f (x; y; s) ;

s˙ D 1 (mod 2) ;

for (x; y; s) 2 R2  S 1 , where f (x; y; s) D ı y  (˛ C ˇ cos(s))x. As such, the trivial solution x D 0 should actually be seen as the periodic solution x D y D 0, s D t (mod 2). Hence we can understand instability of the trivial solution as points of bifurcation of periodic solutions, for which there is a mature theory, see for example [27,37,65,67]. In particular, the subharmonic instability with Floquet multiplier  1 is a period doubling bifurcation, and the harmonic bifurcation corresponding to multiplier + 1 is a pitchfork bifurcation. Note that both these bifurcations come in super- and sub-critical forms. Hence, depending on the nature of the nonlinearity we may find either stable bounded motion close to the instability inside the resonant tongue close to the boundary edge, or motion that is not of small amplitude. See Fig. 5. The case of the Mathieu or Hill equations without damping have received special attention in the literature. This is because of the extra mathematical structure they posses, namely that they can be expressed as reversible and Hamiltonian systems. Specifically, Hill’s equation of the form (5) can be shown to conserve the energy function

189

190

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 5 Contrasting a super-critical and b sub-critical bifurcations close to the resonance tongue boundary of a subharmonic (period-doubling) instability, here represented by the abbreviation P.D. Solid lines represent stable solutions, dashed lines represent unstable solutions. c A one-parameter bifurcation diagram close to the super/sub-critical boundary. d A two parameter unfolding of the codimension-two point P where the super/sub-critical transition occurs. Shading is used to represent where a stable period-two cycle exists. Note in this case that for higher ˇ-values the right-hand boundary of this existence region is therefore the fold curve and not the period-doubling point. Between the period-doubling and the fold there is hysteresis (coexistence) between the stable trivial solution and the stable period-two cycle

H(x; y; t) D (1/2)y 2 C (1/2)(˛ C V(t))x 2 ; y D x˙ , and to be expressible in Hamiltonian form x˙ D

@H ; @y

y˙ D 

@H : @x

Reversibility of Hill’s equation is expressed in the fact that the model is invariant under the transformation t ! t, y ! y. Broer and Levi [7] used Hamiltonian and reversible theory to look at Hill’s equations for general periodic functions V(t). They showed that under certain conditions, the resonance tongues may become of zero width for nonzero ˇ. Effectively, the left and right-hand boundaries of the tongue pass through each other. This is a generic phenomenon, in that a small perturbation within the class of Hill’s equations will lead not destroy such an intersection. However, the Mathieu equation for which V(t) D cos(t) does not posses this property. Many nonlinear models that undergo parametric resonance also preserve the properties of being Hamiltonian and reversible. Special tools from reversible or Hamiltonian systems theory can sometimes be used to analyze the bifurcations that occur under parametric resonance in nonlinear systems, see for example [8,9]. Points outside

the resonance tongues correspond to where the trivial solution is an elliptic (weakly stable) fixed point of the time2 map, and those inside the resonance tongue to where it is a hyperbolic (unstable) fixed point. One therefore does not generally see asymptotically stable non-trivial solutions being born at a resonance tongue boundary, rather one sees elliptic nontrivial dynamics. In fact, for a general periodic solution in a Hamiltonian system, pitchfork and period-doubling bifurcations are not the only kind of bifurcations that produce non-trivial elliptic periodic orbits. If the Floquet multipliers pass through any nth root of unity, there is a period multiplying (subharmonic) bifurcation at which a periodic orbit of period 2n is born (see, e. g. [62]). A period-doubling is just the case n D 2. Such bifurcations can also occur for the non-trivial elliptic periodic orbits within a resonance tongue, but they are all destroyed by the presence of even a small amount of damping. Beyond the Initial Instability For a nonlinear system, the pitchfork or period-doubling bifurcation that occurs upon crossing a resonance tongue might just be the first in a sequence of bifurcations that is

Dynamics of Parametric Excitation

encountered as a parameter is varied. As a general rule, the more ˇ is increased for fixed ˛, the more irregular the stable dynamics becomes. For example, the period-doubling bifurcation that is encountered upon increasing ˇ for ˛ close to 1/4 is typically just the first in a sequence of perioddoublings, that accumulate in the creation of a chaotic attractor, obeying the famous Feigenbaum scaling of this route to chaos (see e. g. [18]). Van Noort [64] produced a comprehensive numerical and analytical study into the nonlinear dynamics of the parametrically excited pendulum (1) without damping. Of interest in that study was to complete the nonlinear dynamics of one of the simplest non-integrable Hamiltonian and reversible dynamical systems. The parameter ˇ acts like a perturbation to integrability. Certain structures that can be argued to exist in the integrable system, like closed curves of quasiperiodic motion in a Poincaré section around an elliptic fixed point, serve as approximate organizers of the dynamics for non-zero ˇ. However, KAM theory (see e. g. [55]) shows that these closed curves become invariant tori that typically breakup as ˇ is increased. At the heart of the regions of chaotic dynamics lie the resonance tongues associated with a period-multiplying bifurcation. Inside the resonance tongues are nontrivial fixed points, whose surrounding invariant tori break up into heteroclinic chains, which necessarily imply chaos. Acheson [2] studied the dynamics of the parametrically forced pendulum in the presence of damping (i. e. Eq. (1) with an additional term x˙ on the right-hand side) via computer simulation. He found parameter regions inside the principal resonance tongue of the pendulum where it undergoes what he referred to as multiple nodding oscillations. The subharmonic motion inside the tongue should be such that during one cycle of the excitation, the pendulum swings to the left and back, and during the next cycle it swings to the right and back. That is, one complete oscillation per two cycles. Multiple nodding motion by contrast is asymmetric and involves two swings to the left, say, (one swing per cycle of the excitation), followed by one swing to the right, with the motion repeating not every two but every three cycles. He also found similar four-cycle and five-cycle asymmetric motion. This behavior can be explained as pockets of stable behavior that survives from the period-multiplying bifurcations that are only present in the case of zero damping [21]. Clearly, beyond the initial instability the universality of the Ince–Strutt diagram for the Mathieu equation, or for any other Hill equation for that matter, disappears and the precise features of the nonlinear dynamics depends crucially on the details of the particular nonlinear system being investigated.

Multibody Systems Of course, there is little in the real world that can be accurately described by deterministic, parametrically excited single degree-of-freedom ODEs. In this section we shall consider slightly more realistic situations that are governed by higher-dimensional systems, albeit still deterministic, and still described by finitely many dynamic variables. For simplicity, we shall restrict attention to rigidbody mechanisms that can be described by a system of (finitely many) coupled modes of vibration. Furthermore, we shall adopt a Lagrangian framework where the dynamics of mode i can be represented by a generalized coordinate qi and its associated generalized velocity q˙i , such that we arrive at a system of (generally nonlinear) coupled second-order ODEs; one equation for each acceleration q¨i . Autoparametric Resonance Autoparametric resonance occurs when there is asymmetric nonlinear multiplicative coupling between modes of a multibody system. Specifically, suppose that a mode (q1 ; q˙1 ) is coupled to a mode (q2 ; q˙2 ) through a term proportional to q1 q2 in the equation of motion for the 2nd mode (i. e. the q¨2 equation), yet there is no corresponding q2 q1 term in the q¨1 equation. For example, consider q¨1 C ı1 q˙1 C ˛1 q1 D f1 (q1 ; t)

(13)

q¨2 C ı2 q˙2 C ˛2 q2 C ˇq1 q2 D f2 (q2 ) ;

(14)

where f 1 and f 2 are nonlinear functions of their arguments and the dependence of f 1 on t is periodic. Furthermore suppose that mode 1 undergoes a periodic motion (either as a result of direct forcing, its own parametric resonance, or self-excitation). For example, if f2 D 0 and f 1 were simply cos(! t) for some ! 2 sufficiently detuned from ˛ 1 , then the solution q1 (t) to (13) would, after allowing sufficient time for transients to decay, be simply proportional to cos(! t C ) for some phase . Then, the q1 q2 term in (14) would be effectively act like ˇ cos(! t C )q2 and so (14) would be precisely the Mathieu equation. Values of ! 2 D 4˛2 /n for any positive integer n correspond to the root points of the resonance tongues of the Ince–Strutt diagram. Such points are referred to as internal or autoparametric resonances. A canonical example of an autoparametrically resonant system is the simple spring-mass-pendulum device depicted in Fig. 6, which we shall use to discuss the some key generic features of autoparametric resonance; see the book by Tondl et al. [58] for more details. In the figure, a mass M is mounted on a linear spring with stiffness k,

191

192

Dynamics of Parametric Excitation

x(t) D R0 cos(t C where R0 D q and tan

0

D

0)

; A

ı12 C (˛1  1)2

ı1 : ˛1  1

(17)

Note that the solution (17) exists for all values of the physical parameters, in particular for all positive ˛ 1 provided ı1 > 0. Substitution of the semi-trivial solution (17) into the pendulum Eq. (16), leads to ¨ C ı2 ˙ C [˛2 C R0 cos(t C

0 )] sin 

D0;

which, after a time shift t ! t  0 is precisely the Eq. (1) for a parametrically excited pendulum in the presence of damping. Thus, the Ince–Struttqdiagram of the damped

Dynamics of Parametric Excitation, Figure 6 A mass-spring-pendulum system

is forced by an applied displacement P cos(! ) and is attached to the pivot of a pendulum of length l and lumped mass m that hangs under gravity. The motion of the mass and the pendulum are assumed to be subject to proportional viscous damping with coefficients b and c respectively. Using standard techniques, the equations of motion can of such a device can be written in the dimensionless form x¨ C ı1 x˙ C ˛1 C (¨ sin() C ˙ 2 cos ) D A cos t (15) ¨ C ı2 ˙ C ˛2 sin  C x¨ sin  D 0 ;

(16)

where x D y/l, time t has been scaled so that t D !, and the dimensionless parameters are b k m ; ˛1 D 2 ; D ; !(M C m) ! (M C m) MCm c g P ; ˛2 D : ; ı2 D AD 2 l! (M C m) !ml 2 l! 2

ı1 D

The simplest form of output of such a model is the socalled semi-trivial solution in which the pendulum is at rest, ˙ D   0, and the mass-spring system alone satisfies the simple equation x¨ C ı1 x˙ C 1 x D A cos t : After transients have died out, this single equation has the attracting solution

Mathieu equation, with ˇ D A/( ı12 C (˛1  1)2 ), ı D ı2 and ˛ D ˛2 tell us precisely the parameter values for which the semi-trivial solution is stable. Inside the resonance tongues we have an instability of the semi-trivial solution, which leads to stable non-trivial motion with the pendulum swinging. To determine precisely what happens away from the boundaries of the stability tongue, one again has to perform fully nonlinear analysis, as outlined in Sect. “Linear Resonance or Nonlinear Instability?”. Note now though that for nonzero pendulum motion,  6 0, the Eqs. (15) and (16) become a fully coupled two-degreeof-freedom system, whose dynamics can be even more complex than that of the one-degree-of-freedom parametrically excited pendulum. Indeed in [58], a further destabilizing secondary-Hopf (Neimark–Sacker) bifurcation is identified within the principal (˛2 D 1/4) subharmonic resonance tongue, and numerical evidence is found for several different kinds of quasiperiodic motion. There is one difference though between autoparametric resonance and pure parametric resonance. The paradigm for parametric resonance is that the external parametric excitation is held constant (its amplitude and frequency are parameters of the system). Here, however, for nonzero , the system is fully coupled, so that the parametric term (17) can no longer be thought of as being independent of , effectively the excitation becomes a dynamic variable itself. Thus, motion of the pendulum can affect the motion of the mass-spring component of the system. In particular if there is an autoparametric instability where the pendulum is set in motion, then energy must be transferred from the mass-spring component to the pendulum, a phenomenon known as quenching. But the pendulum is only set in motion close to certain resonant frequency ratios ˛ 2 . Thus, we have a tuned damper that is designed to

Dynamics of Parametric Excitation

take energy out of the motion of the mass at certain input frequencies ! frequencies. To see how such a damper might work in practice, suppose we build a civil engineering structure such as a bridge or a building, that has a natural frequency ! 1 in some mode, whose amplitude we call x(t). Suppose the system is subjected to an external dynamic loading (perhaps from traffic, wind or earthquakes) that is rich in a frequency ! which is close to ! 1 . Then, if the damping is small (as is typical in large structures) using (15) with  D 0, we find that the response given by (17) is large since ˛ 1 is close to 1 and ı 1 is small. This is the fundamental resonant response of the structure close to its natural frequency. Now suppose we add the pendulum as a tuned damper. In particular we design the pendulum so that ˛2  ˛1 . The pendulum now sees a large amplitude parametric excitation within its main subharmonic resonance tongue. It becomes violently unstable, thus sucking energy from the structure. To see just how effective this quenching effect can be, consider the following asymptotic calculation [58]. Supˆ for some pose the external forcing is very small, A D "2 A, small parameter ", and that the damping and frequency detunings scale with ": ı1 D "ıˆ1 , ı2 D "ıˆ2 , ˛1 D 1 C "ˆ 1 , ˛2 D (1/4) C "ˆ 2 , where the hatted quantities are all assumed to be O(1) as " ! 0. Upon making these substitutions, and dropping all hats, Eqs. (15) and (16) become 1 x¨ C x C "[ı1 x˙ C 1   2 C ˙ 2 ] 4 D "A cos(t) C O("2 ) ; 1 1 ¨ C  C "[ı2 ˙ C 2   x] D O("2 ) : 4 2

(18) (19)

We then look for solutions of the form  x D R1 (t) cos t C 1 (t) ;

 t  D R2 (t) cos C 2 (t) : 2 Making this substitution into (18), (19), and keeping only O(") terms we find

 1 2R˙ 1 D " ı1 R1  R22 sin( 1  2 2 ) A sin( 1 ) ; 4

 1 2R1 ˙ 1 D " 1 R1  R22 cos( 1  2 2 )  A cos 1 ; 4  2R˙ 2 D "  ı2 R2 C R1 R2 sin( 1  2 2 ) ;  2R2 ˙ 2 D " 2 R2  R1 R2 cos( 1  2 2 ) : Equilibrium solutions of these equations correspond to 4-periodic solutions that are valid close to the instability

tongue. Upon setting the left-hand sides to zero, a straightforward combination of the final two equations yields q R1 D 22 C ı22 ; (20) and a slightly longer calculation reveals that this leads to a unique (stable) solution for R2 provided A2 > (ı12 C 12 )(ı22 C 22 ). Equation (20) gives the amplitude of the quenched solution for the structure. Compare this with the amplitude R0 of the semi-trivial solution (17), which is the forced response of the structure without the pendulum-tuneddamper added. Written in the rescaled coordinates, this solution is R0 D

A2 : 12 C ı12

Here we make the important observation that, unlike the simple forced response R0 , the quenched amplitude R1 is independent of the forcing amplitude A, and does not blow up at the fundamental resonance 1 ! 0. In fact, at least to leading order in this asymptotic approximation, the amplitude of the quenched solution is independent of the frequency detuning and damping constant of the structure, and depends only on the frequency detuning and proportional damping of the pendulum! Tondl et al. [58] give a number of applications of autoparametric resonance. First, the above quenching effect means that analogues of the mass-spring-pendulum system can be used as tuned dampers, that are capable of suppressing certain undesirable response frequencies from structures or machines. Another important example of parametric resonance they cite is in flow-induced oscillation, where periodic flow features such as vortex shedding can cause a parametric response of a structure. The galloping of cables in strong winds is an example of just such an effect. Cable-stayed bridges are engineering structures that are particularly susceptible to autoparametric effects. Not only is there potential for resonant interaction between the cables and the wind, but, since the cables are all of different lengths, there is a good chance that there is at least one cable whose natural frequency is either half or twice that of a global vibration mode of the entire bridge. Models of deck-cable interaction have shown the propensity for undesirable effects, see e. g. [23,24]. For example, vibrations caused by traffic could cause particular cables to undergo large-amplitude vibration leading to potential problems with wear. Alternatively, large-amplitude cable vibrations (caused for example by fluid-structure interaction) could cause significant deck vibration leading to an uncomfortable ride for those crossing the bridge.

193

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Dynamics of Parametric Excitation

Combination Parametric Resonances Mailybayev and Seyranian [41] consider general systems of second-order equations of the form M y¨ C ıD y˙ C (A C ˇB(˝ t))y D 0 ;

y 2 Rn ;

(21)

where M, D and A are positive definite symmetric matrices, and B is a periodic matrix with period 2/˝. They derive conditions under which parametric resonance may occur. Suppose that when ı D 0 D ˇ D 0, the linear system has non-degenerate natural frequencies, !1 ; !2 ; : : : ! n . Then they showed that the only kinds of parametric resonance that are possible are when ˝D

2! j ; k

j D 1; : : : m ; k D 1; 2; : : : ;

(simple resonance) ;

!i ˙ ! j or ˝ D ; i; j D 1; : : : m ; i > j ; k k D 1; 2; : : : ; (combination resonance) :(22)

that the system with ˇ D 0 is written in diagonal form, and finally that time has been rescaled so that ˝ D 1. Hence we obtain a system of the form        x¨1 ˛1 0 b11 b12 C Cˇ cos(t) 0 ˛2 x¨2 b21 b22   x1 D 0 ; (23) x2 where ˛1 D !12 /˝ 2 , ˛2 D !22 /˝ 2 and the matrix Bˆ D fb i j g is the original constant matrix within B0 written in the transformed coordinates. Recalling, how in Subsect. “Floquet Analysis” we found the transition curves for the Mathieu equation using Floquet theory, we can look for solutions to (23) in the form x1 D c n

1 X

p 2

e i nt/

nD1 1 X

x2 D ˙c n

e

;

p si nt/ 2

(24) ;

nD1

The sign ‘+’ or ‘’ in (22) gives what are called sum or difference combination resonances respectively. There are many examples of such combination resonances in the literature. For example, the book by Nayfeh [45] considers many such cases, especially of mechanical systems where the two modes are derived from a Galerkin reduction of a continuum system such as a plate or beam. In all these examples, though, it is a sum combination resonance that is excited. That is, by forcing the system at the sum of the two individual frequencies ˝ D !1 C !2 , individual responses are found at frequency ! 1 or ! 2 . However, there do not seem to be any concrete examples of difference resonances in the literature. In fact, such a mechanical device might be somewhat strange. If we had !1  !2 , then ˝ D !1  !2 would be small, perhaps several orders of magnitude smaller. So a difference combination resonance would give a response at a high frequency from low frequency excitation. This would be like making a drum vibrate by sending it up and down in an elevator! In fact, for the case that B D B0 (t) is a constant matrix times a function of time, then it is shown in (Theorem 1 in [41]) that a given system can only excite either sum or difference combination resonances, but not both. To explain this result, it is sufficient to consider a simplification of (21) to the case n D 2. Consider (21) in the case ı D 0 and where B(t) is a constant matrix times a pure function of time with period ˝. Without loss of generality, let us suppose this function to be a pure cosine B D B0 cos(˝ t). Suppose that we change coordinates so

where s D ˙1. We find an infinite system of algebraic equations analogous to (10). It is straightforward to see that in the case ˇ D 0, there is a non-trivial solution with c n D 0 for all n ¤ k and c k ¤ 0, whenever ˛1 C s˛2 D k 2 . This would suggest that both sum and difference combination resonances are possible. However, when looking at the conditions for the bifurcation equations to have a real solution for nonzero ˇ, one finds the following condition sign(b12 b21 ) D s :

(25)

That is, to excite a sum resonance, the diagonal entries of B0 must be of the same sign, and to excite a difference resonance these diagonal entries must be of opposite sign. In particular, if B0 is a symmetric matrix (as is the case in many mechanical applications) then only sum combinations can be excited. In fact, it is argued in [68] that pure Hamiltonian systems cannot ever excite difference combination resonance; in other words, if difference parametric resonance is possible at all, then the matrix B(˝ t) must contain dissipative terms. One might wonder whether difference combination resonance can ever be excited in any physically derived system. To show that they can, consider the following example with a non-conservative parametric excitation force, a detailed analysis of which will appear elsewhere [59]. Figure 7 depicts a double pendulum (which we think of as lying in a horizontal plane so that gravity does not affect it) with stiff, damped joints and is loaded by an end

Dynamics of Parametric Excitation

Now, (26) is not in diagonal form, but a simple coordinate transformation can it in the form (21) with M the identity matrix,  DDAD  and

B0 D

0:1492 0 0:0466 0:1288

0 3:3508



0:3788 1:0466

 :

Hence the system has two natural frequencies !1 D p p 0:1492 D 0:3863 and !2 D 3:3508 D 1:8305, and the quantities b12 b21 D 0:3788  0:1288 D 0:0488 < 0. Hence, according to (25), a difference combination resonance should be possible by exciting the system at ˝ D !2  !1 D 1:4442, as verified in [59]. Upside-Down Stability Dynamics of Parametric Excitation, Figure 7 The partially follower-loaded elastic double pendulum, depicting a definition of the various physical quantities. In this study we take m1 D m2 D m and ˛ D 1

force – a so-called follower load – that is maintained in the direction of the axis of the second pendulum. Note that such a load is by its nature non-conservative because work (either positive or negative) has to be done to maintain the direction of the follower. In practice, such a force might be produced by a jet of fluid emerging from the end of the outer pendulum, for example if the pendula were actually pipes, although the equations of motion (and the consequent nonlinear dynamics) are rather different for that case [5,12,13]. Non-dimensional equations of motion for this system are derived in [56], where it was shown that the system can undergo quite violent chaotic dynamics for large deflections. We shall restrict ourselves here to small deflection theory and allow the follower load to be a harmonic function of time P D A cos ˝ t. After p nondimensionalization using ˇ D jAjl/k;pı D C/ kml 2 , and rescaling time according to tnew D k/m2 l 2 ) told , we obtain the equations  ¨  1 ¨2     ˙  2 1 1 2 1 1 Cı 1 1 2 1 1 ˙2    1 1 1 C p cos ˝ t D 0 : (26) 2 0 0



2 1

1 1



C

As mentioned in the Introduction, it has been known for 100 years [53] that the ‘negative gravity’ (˛ < 0) region of the Ince–Strutt diagram for the Mathieu equation implies that a simple pendulum can be stabilized in the inverted position by application of parametric excitation. More recently Acheson and Mullin [3] demonstrated experimentally that a double and even a triple pendulum can be similarly stabilized when started in the inverted position. Moreover, the stabilized equilibrium is quite robust, such that the system relaxes asymptotically back to its inverted state when given a moderate tap. At first sight, these observations seem remarkable. A multiple pendulum has several independent normal modes of oscillation and naively one might expect that single frequency excitation could at best stabilize only one of these modes at a time. However, Acheson [1] (see also the earlier analysis of Otterbein [46]) provided a simple one-line proof that, in fact, for sufficiently high frequency and sufficiently small amplitude sinusoidal excitation, in theory any ideal finite chain of N pendulum can be stabilized in the inverted position. To see why this is true, consider the multiple pendulum system represented in Fig. 8 but without the springs at the joints. Then, upon performing a modal analysis, we find that the equations of motion reduce to the study of N uncoupled Mathieu equations each with different parameters ˛ i > 0 and ˇ i > 0, for i D 1; : : : ; N. Hence the frequency can be chosen to be sufficiently high and the amplitude chosen to be sufficiently small for each ˛ i , ˇ i to lie in the thin wedge of stability for ˛ < 0 in Fig. 4. Thus, each mode is stable, and so the whole finite chain has been stabilized by parametric excitation.

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Dynamics of Parametric Excitation, Figure 8 a A multiple pendulum with stiff joints. b,c Ince–Strutt diagrams of dimensionless amplitude of vibration " against scaled elastic stiffness B for eight identical jointed pendulums with weak damping. b For dimensionless applied frequency ! D 10; c ! D 20. Shading represents the stability region of the upright equilibrium and digits in other regions give the number of unstable Floquet multipliers. After [22], to which the reader is referred to for the precise details; reproduced with permission from Elsevier

It has been suggested that the limit of such a multiple pendulum, in which the total length and mass of the system stays constant while the number of pendulums in the chain becomes infinite, could be used as a possible explanation of the so called ‘Indian rope trick’, a classic conjuring trick in which a rope is ‘charmed’ by a magician to appear vertically out of a basket like a snake. Unfortunately, as noted [1], in this limit, which is that of a piece of string, the stability region becomes vanishingly small and so this potential scientific explanation of the magic trick fails. However, a further experiment by Mullin announced in [4], demonstrated that a piece of ‘bendy curtain wire’, clamped at the bottom and free at the top, that is just too long to support its own weight can be stabilized by parametric excitation (see Fig. 9). In order to explain this result, Galán et al. [22] introduced the model depicted in Fig. 8 and studied the stability of the vertical equilibrium

position in the presence of both damping and stiffness in each of the joints. The normal modes of this problem lead to fully coupled equations and so Acheson’s argument does not apply. Using numerical bifurcation analysis Galán et al. were able to find stability boundaries and also to find an appropriate continuum limit; providing a good motivation for us to turn our attention to parametric resonance in continuous systems. Continuous Systems Parametric resonance is also of importance in continuum systems. Canonical effects we shall discuss here include the wire stiffening we have just introduced, the excitation of patterns at fluid interfaces, more general theories of flowinduced oscillations of structures, and the stabilizing effects of periodic modulation of optical fibers.

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 9 Experimental results on a piece of ‘bendy curtain wire’ which is just long enough to buckle under its own weight (a). For tiny vertical oscillations between about 15 and 35 Hz, the wire can be made to stand upright (b), and to be stable against even quite large perturbations (c). Beyond the upper frequency threshold, a violent dynamic stability sets in which involves the third vibration mode excited harmonically (d). After [44], reproduced with permission from the Royal Society

Structural Stiffening; the ‘Indian Rod Trick’ Thomsen [57] has shown that parametric excitation generally speaking has a stiffening effect on a structure. This is essentially because of the same effect that causes the upside down stability of the simple pendulum for high-enough frequencies and small enough amplitude. However, detailed experiments by Mullin and co-workers [44], reproduced in Fig. 9, showed that there can also be a destabilizing effect of parametric resonance. In particular, a piece of curtain wire (a thin helical spring surrounded by plastic) that was just too long (about 50 cm) to support its own weight, was clamped vertically upright and subjected to rapid, small-amplitude oscillations of about 1 cm peakto-peak. At around 15 Hz the rod was found to be stabilized, such that when released it remains erect (see Fig. 9b). This effect continues for higher frequencies, until about 30 Hz at which point a violent dynamic instability sets in that is commensurate with the driving frequency (harmonic, rather than subharmonic). The experimentally determined parameter values in which the upside-down stabilization can be achieved are represented in terms of dimensionless parameters in Fig. 10b. Champneys and Fraser [14,15,20], introduced a continuum theory to explain these experimental results, which produced the shaded region in Fig. 10b, that we now explain. Consider an intrinsically straight column of length `, with a uniform circular cross-section of radius a  ` and mass linear density m per unit length (see Fig. 10a). The column is assumed to be inextensible, unshearable and linearly elastic with bending stiffness Bˆ and material damping coefficient ˆ (where hatted variables are

dimensional). The lower end is clamped upright and is to vertical displacement  cos ! ˆt , whereas the upper end is free. In [15], it is shown that the stability of the vertical equilibrium position can be studied by analyzing solutions to the PDE Bussss C [(1  s)us ]s C  (u t t C usssst  "[(1  s)us ]s cos t) D 0 ;

(27)

for the scaled lateral displacement u(s; t) assumed without loss of generality to be in a fixed coordinate direction, where the scaled arc length s 2 (0; 1), the scaled time variable t D ! ˆt, and the boundary conditions are u D us D 0 ;

at s D 0 ;

(28)

uss D ussss D 0 ; at s D 1 : The dimensionless parameters appearing in (27) are

D

 `! ; mg

BD

Bˆ ; mg`3

D

! 2` ; g

"D

 ; (29) `

which represent respectively material damping, bending stiffness, the ratio of forcing frequency to the equivalent pendulum natural frequency, and forcing amplitude. If we momentarily consider the problem with no forcing or damping, " D D 0, then the peigenmodes  n and corresponding natural frequencies n / of free vibration of the rod satisfy B( n )ssss C [(1  s)( n )s ]s  n  n  n D 0 ; and form a complete basis of functions that satisfy the boundary conditions (28). Now the eigenmodes  n (s; B)

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Dynamics of Parametric Excitation, Figure 10 a Definition sketch of the parametrically excited flexible rod model. b Comparison between theory (solid line and shading) and the experimental results of [44] (dashed line), for the stability region as a function of the dimensionless parameters B and  using the experimental value " D 0:02, and the representative damping value 1 D 0:01 (which was not measured in the experiment) The theory is based on calculations close to the resonance tongue interaction between when B0;1 D B1;3 when  D c D 460:7. After [15], reproduced with permission from SIAM

are in general not expressible in closed form except at the special values of B at which n (B) D 0, in which case [20] there is a solution in terms of the Bessel function J1/3 . The same analysis shows that there are infinitely many such B-values, B0;n , for n D 1; 2; 3; 4; : : :, accumulating at B D 0, the first few values of which are B0;1 D 0:127594, B0;2 D 0:017864, B0;3 D 0:0067336 and B0;4 D 0:0003503. These correspond respectively to where each eigenvalue locus n (B), for n D 1; 2; 3; 4, crosses the B-axis, with the corresponding eigenmodes there having n  1 internal zeros. The lowering of B through each B0;n -value implies that the nth mode becomes linearly unstable. Hence for B > B c :D B0;1 the unforced rod is stable to self-weight buckling (a result known to Greenhill [26]). Moreover, it can be shown [14,15] that each eigenmode  n (s) retains a qualitatively similar mode shape for B > B0;n with the nth mode having n D 1 internal zeros, and that the corresponding loci n (B) are approximately straight lines as shown in the upper part of Fig. 11. The lower part of Fig. 11 shows the results of a parametric analysis of the Eq. (27) for D 0. Basically, each time one of the eigenvalue loci passes through  times the square of a half-integer ( j/2)2 we reach a B-value for which there is the root point of an instability tongue. Essentially we have a Mathieu-equation-like Ince–Strutt diagram in the (B; ") plane for each mode n. These diagrams are overlaid on top of each other to produce the overall stability

plot. However, note from Fig. 11 that the slope of each eigenvalue locus varies with the frequency ratio . Hence as  increases, the values of all the B j/2;n , j D 0; : : : ; 1 for a given n increase with the same fixed speed. But this speed varies with the mode number n. Hence the instability tongues slide over each other as the frequency varies. Note the schematic diagram in the lower panel of Fig. 11 indicates a region of the stability for B < B0;1 for small positive ". Such a region would indicate the existence of rod of (marginally) lower bending stiffness than that required for static stability, that can be stabilized by the introduction of the parametric excitation. Essentially, the critical bending stiffness for the buckling instability is reduced (or, equivalently since B / `3 , the critical length for buckling is increased). This effect has been dubbed the ‘Indian rod trick’ because the need for bending stiffness in the wire means that mathematically this is a rod, not a rope (or string); see, e. g. the account in [4]. To determine in theory the precise parameter values at which the rod trick works turns out to be an involved process. That the instability boundary emerging from B0;1 in the lower panel of Fig. 11 should usually bend back to the left can be argued geometrically [7]. However, for -values at which B0;1  B1;n for some n, it has been shown that there is actually a singularity in the coefficient of the quadratic part of this buckling curve causing the instability boundary to bend to the right [20]. A complete un-

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 11 Summarizing schematically the results of the parametric resonance analysis of the parametric excited vertical column in the absence of damping. The upper part shows eigenvalue loci n (B) and the definition of the points Bj/2;n , j D 0; 1; 2; 3; : : : In the lower part, instability tongues in the (B; ")-plane are shown to occur with root points Bj/2;n and to have width "j . The shaded regions correspond to the where the vertical solution is stable. Solid lines represent neutral stability curves with Floquet multiplier + 1 (where ˛ is an integer) and dashed lines to multipliers  1 (where ˛ is half an odd integer). Reproduced from [14] with permission from the Royal Society

folding of this so-called resonance tongue interaction was performed in [15], also allowing for the presence of material damping. The resulting asymptotic analysis needs to be carried out in the presence of four small parameters; ",

, B  B0;1 and  c where c is the frequency-ratio value at which the resonance-tongue interaction occurs. The results produce the shaded region shown in Fig. 10b, which can be seen to agree well with the experimental results. Faraday Waves and Pattern Formation When a vessel containing a liquid is vibrated vertically, it is well known that certain standing wave patterns form on the free surface. It was Faraday in 1831 [19] who first noted that the frequency of the liquid motion is half that of the vessel. These results were confirmed by Rayleigh [50] who suggested that what we now know as parametric resonance was the cause. This explanation was shown to be

correct by Benjamin and Ursell [6] who derived a system of uncoupled Mathieu equations from the Navier–Stokes equations describing the motion of an ideal (non-viscous) fluid. There is one Mathieu equation (4), for each wave number i that fits into the domain, each with its own parameters ˛ i and ˇ i . They argue that viscous effects would effectively introduce a linear damping term, as in (8) and they found good agreement between the broad principal (subharmonic) instability tongue and the results of a simple experiment. Later, Kumer and Tuckerman [36] produced a more accurate Ince–Strutt diagram (also known as a dispersion relation in this context) for the instabilities of free surfaces between two viscous fluids. Through numerical solution of appropriate coupled Mathieu equations that correctly take into account the boundary conditions, they showed that viscosity acts in a more complicated way than the simple velocity proportional damping term in (8). They produced the diagram shown in Fig. 12b, which showed a good match with the experimental results. Note that, compared with the idealized theory, Fig. 12a, the correct treatment of viscosity causes a small shift in the critical wavelength of each instability, a broadening of the higher-harmonic resonance tongues and a lowering of the amplitude required for the onset of the principal subharmonic resonance if damping is introduced to the idealized theory (see the inset to Fig. 12b). More interest comes when Faraday waves are excited in a vessel where two different wave modes have their subharmonic resonance at nearly the same applied frequency, or when the excitation contains two (or more) frequencies each being close to twice the natural frequency of a standing wave. See for example the book by Hoyle [30] and references therein, for a glimpse at the amazing complexity of the patterns of vibration that can result and for their explanation in terms of dynamical systems with symmetry. Recently a new phenomenon has also been established, namely that of localized spots of pattern that can form under parametric resonance, so-called oscillons see e. g. [61] for a review and [39] for the first steps at an explanation. Fluid-Structure Interaction Faraday waves are parametric effects induced by the gross mechanical excitation of a fluid. In Subsect. “Autoparametric Resonance” we briefly mentioned that fluid-structure interaction can go the other way, namely that external fluid flow effects such as vortex shedding can excite (auto)parametric resonances in simple mechanical systems. Such flow-induced parametric resonance can also occur for continuous structures, in which waves can be excited in the structure due to external parametric loading of

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Dynamics of Parametric Excitation, Figure 12 Stability boundaries for Faraday waves from: a Benjamin and Ursell’s theory for ideal fluids; b the results of full hydrodynamic stability analysis by Kumar and Tuckerman. The horizontal axis is the frequency and the vertical scale the amplitude of the excitation. The inset to panel b shows the difference between the true instability boundary for the principal subharmonic resonance (lower curve) and the idealized boundary in the presence of simple proportional damping. After [36], to which the reader is referred for the relevant parameter values; reproduced with permission from Cambridge University Press

the fluid; see for example the extensive review by Païdousis and Li [47]. They also treat cases where the flow is internal to the structure: we are all familiar with the noisy ducts and pipes, for example in central heating or rumbling digestive systems, and it could well be that many of these effects are a result of parametric resonance. Semler and Païdousis [52] consider in detail parametric resonance effects in the simple paradigm of a cantilevered (free at one end, clamped at the other) flexible pipe conveying fluid. Think of this as a hosepipe resting on a garden lawn. Many authors have considered such an apparatus as a paradigm for internal flow-induced oscillation

problems (see the extensive references in [47,52]), starting from Benjamin’s [5] pioneering derivation of equations of motion from the limit of a segmented rigid pipe. The equations of motion for such a situation are similar to those of the rod (27), but with significant nonlinear terms arising from inertial and fluid-structure interaction effects. In the absence of periodic fluctuations in the flow, then it is well known that there is a critical flow rate beyond which the pipe will become oscillatory, through a Hopf bifurcation (flutter instability). This effect can be seen in the way that hosepipes writhe around on a lawn, and indeed forms the basis for a popular garden toy. Another form of instability can occur when the mean flow rate is beneath the critical value for flutter, but the flow rate is subject to a periodic variation, see [52] and references therein. There, a parametric instability can occur when the pulse rate is twice the natural frequency of small amplitude vibrations of the pipe. In [52] this effect was observed in theory, in numerical simulation and in a careful experimental study, with good agreement between the three. Parametric resonance could also be found for super-critical flow rates. In this case, the effect was that the periodic oscillation born at the Hopf bifurcation was excited into a quasi-periodic response. Another form of fluid structure interaction can occur when the properties of an elastic body immersed in a fluid are subject to a temporal periodic variation. With a view to potential biological applications such as active hearing processes and heart muscles immersed in blood, Cortez et al. [16] consider a two-dimensional patch of viscous fluid containing a ring filament that is excited via periodic variation of its elastic modulus. Using a mixture of immersed boundary numerical methods and analytic reductions, they are able to find an effective Ince–Strutt diagram for this system and demonstrate the existence of parametric instability in which the excitation of the structure causes oscillations in the fluid. Taking a fixed wave number p of the ring in space, they find a sequence of harmonic and subharmonic instabilities as a parameter effectively like ˛ in the Mathieu equation is increased. Interestingly though, even for low viscosity, the principal subharmonic resonance tongue (corresponding to ˛ D 1/4) does not occur for low amplitude forcing, and the most easily excited instability is actually the fundamental, harmonic resonance (corresponding to ˛ D 1). Dispersion Managed Optical Solitons The stabilizing effect of high-frequency parametric excitation has in recent years seen a completely different application, in an area of high technological importance, namely

Dynamics of Parametric Excitation

optical communications. Pulses of light sent along optical fibers are a crucial part of global communication systems. A pulse represents a packet of linear waves, with a continuum of different frequencies. However, optical fibers are fundamentally dispersive, which means that waves with different wavelengths travel at different group velocities. In addition, nonlinearity, such as the Kerr effect of intensity-dependent refractive index change, and linear losses mean that there is a limit to how far a single optical signal can be sent without the need for some kind of amplifier. Depending on the construction of the fiber, dispersion can be either normal which means that lower frequencies travel faster, or anomalous in which case higher frequencies travel faster. One idea to overcome dispersive effects, to enable pulses to be travel over longer distances without breaking up, is to periodically vary the medium with which the fiber is constructed so that the dispersion is alternately normally and anomalously dispersive [38]. Another idea, first proposed by Hasegawa and Tappert [28], is to balance dispersion with nonlinearity. This works because the complex wave envelope A of the electric field of the pulse can be shown to leading order to satisfy the nonlinear Schrödinger (NLS) equation i

dA 1 d2 A C jAj2 A D 0 ; C dz 2 dt 2

(30)

in which time and space have been rescaled so that the anomalous dispersion coefficient (multiplying the secondderivative term) and the coefficient of the Kerr nonlinearity are both unity. Note the peculiarity in optics compared to other wave-bearing system in that time t plays the role of the spatial coordinate, and the propagation distance z is the evolution variable. Thus we think of pulses as being time-traces that propagate along the fiber. Now, the NLS equation is completely integrable and has the well-known soliton solution 2

A(z; t) D  sech(t  cz)e i z ;

(31)

which represents a smooth pulse with amplitude , which propagates at ‘speed’ c. Moreover, Eq. (30) is completely integrable [69] and so arbitrary initial conditions break up into solitons of different speed and amplitude. Optical solitons were first realized experimentally by Mollenauer et al. [43]. However, true optical fibers still suffer linear losses, and the necessary periodic amplification of the optical solitons to enable them to survive over long distances can cause them to jitter [25] and to interact with each other as they move about in time. The idea of combining nonlinearity with dispersion management in order to stabilize soliton propagation was

first introduced by Knox, Forysiak and Doran [35], was demonstrated experimentally by Suzuki et al. [54] and has shown surprisingly advantageous performance that is now being exploited technologically (see [60] for a review). Essentially, dispersion management introduces a periodic variation of the dispersion coefficient d(z), that can be used to compensate the effects of periodic amplification, which with the advent of modern erbium-doped optical amplifiers can be modeled as a periodic variation of the nonlinearity. As an envelope equation we arrive at the modified form of (30), i

dA d(z) d2 A C  (z)jAj2 A D 0 ; C dz 2 dt 2

(32)

where the period L of the dispersion map d(z) and the amplifier spacing (the period of  (z)) Z may not necessarily be the same. Two particular regimes are of interest. One, of application to long-haul, sub-sea transmission systems where the cable is already laid with fixed amplifiers, is to have L  Z. Here, one modifies an existing cable by applying bits of extra fiber with large dispersion of the opposite sign only occasionally. In this case, one can effectively look only at the long scale and average out the effect of the variation of  . Then the parametric term only multiplies the second derivative. Another interesting regime, of relevance perhaps when designing new terrestrial communication lines, is to have L D Z so that one adds loops of extra opposite-dispersion fiber at each amplification site. In both of these limits we have an evolution equation whose coefficients are periodic functions of the evolution variable z with period L. In either regime, the periodic (in z) parameter variation has a profound stabilizing influence, such that the effects of linear losses and jitter are substantially reduced. A particular observation is that soliton solutions appear to (32) breathe during each period; that is, their amplitude and phase undergo significant periodic modulation. Also, with increase in the amplitude of the oscillatory part of d, so the average shape of the breathing soliton becomes less like a sech-pulse and more like a Gaussian. An advantage is that the Gaussian has much greater energy the regular NLS soliton (31) for the same average values of d and  , hence there is less of a requirement for amplification. Surprisingly, it has been found that the dispersion managed solitons can propagate stably even if the average of d is slightly positive, i. e. in the normal dispersion region where there is no corresponding NLS soliton. This has a hugely advantageous consequence. The fact that the local dispersion can be chosen to be high, but the average dispersion can be chosen to be close to zero, is the key to enabling the jitter introduced by the amplifiers to be greatly suppressed

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compared with the regular solitons. Finally the large local modulation of phase that is a consequence of the breathing, means that neighboring solitons are less likely to interact with each other. A mathematical analysis of why these surprising effects arise is beyond the scope of this article, see e. g. [32,60] for reviews, but broadly-speaking the stable propagation of dispersion-managed solitons is due to the stabilizing effects of high frequency parametric excitation. Future Directions This article has represented a whistle-stop tour of a wide areas of science. One of the main simplifications we have taken is that we have assumed the applied excitation to the system to be periodic; in fact, in most examples it has been pure sinusoidal. There is a growing literature on quasiperiodic forcing, see e. g. [51], and it would be interesting to look at those effects that are unique to parametrically excited systems that contain two or more frequencies. The inclusion of noise in the excitation also needs to be investigated. What effect does small additive or multiplicative noise have on the shape of the resonance tongues? And what is the connection with the phenomenon of stochastic resonance [66]? There is a mature theory of nonlinear resonance in systems with two degrees of freedom which, when in an integrable limit, undergoes quasiperiodic motion with two independent frequencies. Delicate results, such as the KAM theorem, see e. g. [55], determine precisely which quasiperiodic motions (invariant tori) survive as a function of two parameters, the size of the perturbation from integrability and the frequency detuning between the two uncoupled periodic motions. For the tori that break up (inside so-called Arnol’d tongues) we get heteroclinic tangles and bands of chaotic motion. Parametric resonance tends to be slightly different though, since in the uncoupled limit, one of the two modes typically has zero amplitude (what we referred to earlier as a semi-trivial solution). It would seem that there is scope for a deeper connection to be established between the Arnol’d tongues of KAM theory and the resonance tongues that occur under parametric excitation. In terms of applications, a large area that remains to be explored further is the potential exploitation of parametric effects in fluid-structure interaction problems. As I write, the heating pipes in my office are buzzing at an extremely annoying frequency that seems to arise on warm days when many radiators have been switched off. Is this a parametric effect? It would also seem likely that nature uses parametric resonance in motion. Muscles contract

longitudinally, but motion can be in a transverse direction. Could phenomena such as the swishing of bacterial flagella, fish swimming mechanisms and the pumping of fluids through vessels, be understood in terms of exploitation of natural parametric resonances? How about motion of vocal chords, the Basilar membrane in the inner ear, or the global scale pumping of the heart? Does nature naturally try to tune tissue to find the primary subharmonic resonance tongue? Perhaps other effects of parametric excitation that we have uncovered here, such as the quenching of resonant vibrations, structural stiffening, and frequency tuned motion, are already being usefully exploited in nature. If so, there could be interesting consequences for nature inspired design in the engineered world. Finally, let us return to our opening paradigm of how children effectively excite a parametric instability in order to induce motion in a playground swing. In fact, it was carefully shown by Case and Swanson [11] that the mechanism the child uses when in the sitting position is far more accurately described as an example of direct forcing rather than parametric excitation. By pushing on the ropes of the swing during the backwards flight, and pulling on them during the forward flight, the child is predominately shifting his center of gravity to and fro, rather than up and down; effectively as in Fig. 1c rather than Fig. 1b. Later, Case [10] argued that, even in the standing position, the energy gained by direct forcing from leaning backwards and forwards is likely to greatly outweigh any gained from parametric up-and-down motion of the child. The popular myth that children swinging is a suitable playground science demonstrator of parametric resonance was finally put to bed by Post et al. [49]. They carried out experiments on human patients and analyzed the mechanisms they use to move, reaching the conclusion that even in the standing position, typically 95% of the energy comes from direct forcing, although parametric effects do make a slightly more significant contribution for larger amplitudes of swing.

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31. Ince E (1956) Ordinary Differential Equations. Dover, New York 32. Jones C (2003) Creating stability from instability. In: Nonlinear dynamics and chaos; where do we go from here? IOP Publishing, Bristol, chap 4, pp 73–90 33. Jordan DW, Smith P (1998) Nonlinear Ordinary Differential Equations, 3rd edn. Oxford University Press, Oxford 34. Kapitza P (1951) Dinamicheskaya ustoichivost mayatnika pri koleblyushcheisya tochke podvesa. Z Eksperimentalnoi I Teoreticheskoi Fiziki 21:588–597, in Russian 35. Knoz F, Forysiak W, Doran N (1995) 10 gbit/s soliton communication systems over standard fiber at 1.55m and the use of dispersion compensation. IEEE J Lightwave Technol 13:1960–1995 36. Kumar K, Tuckerman L (1994) Parametric instability of the interface between two fluids. J Fluid Mech 279:49–68 37. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York 38. Lin C, Kogelnik H, Cohen L (1980) Optical-pulse equilization of low-dispersion transmission in single fibers in the 1.3–1.7m spectral region. Opt Lett 5:476–480 39. Lloyd D, Sandstede B, Avitabilie D, Champneys A (2008) Localized hexagon patterns in the planar swift-hohenberg equation. To appear in SIAM Appl Dyn Sys 40. Magnus W, Winkler S (1979) Hill’s Equation. Dover, New York 41. Mailybayev A, Seyranian AP (2001) Parametric resonance in systems with small dissipation. PMM J App Math Mech 65:755–767 42. Mathieu E (1868) Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. J Math Pure Appl 13:137–203 43. Mollenauer L, Stolen R, Gordon J (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45:1095–1097 44. Mullin A, Champneys T, Fraser W, Galan J, Acheson D (2003) The ‘Indian wire trick’ via parametric excitation: a comparison between theory and experiment. Proc Roy Soc Lond A 459:539–546 45. Nayfeh A (2000) Nonlinear Interactions: Analytical, Computational and Experimental Methods. Wiley Interscience, New York 46. Otterbein S (1982) Stabilization of the N-pendulum and the Indian link trick. Arch Rat Mech Anal 78:381–393 47. Païdoussis MP, Li G (1993) Pipes conveying fluid: a model dynamical problem. J Fluids Structures 7:137–204 48. Piccoli B, Kulkarni J (2005) Pumping a swing by standing and squatting: Do children pump time optimally? IEEE Control Systems Magazine 25:48–56 49. Post A, de Groot G, Daffertshofer A, Beek P (2007) Pumping a playground swing. Motor Control 11:136–150 50. Rayliegh L (1883) On the crispations of fluid resting upon a vibrating support. Phil Mag 16:50 51. Romeiras F, Romeiras F, Bondeson A, Ott E, Antonsen TM, Grebogi C (1989) Quasiperiodic forcing and the observability of strange nonchaotic attractors. Phys Scr 40:442–444 52. Semler C, Païdoussis M (1996) Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe. J Fluids Structures 10:787–825 53. Stephenson A (1908) On a new type of dynamical stability. Mem Proc Manch Lit Phil Soc 52:1–10 54. Suzuki M, Morita N, Edagawa I, Yamamoto S, Taga H, Akiba S (1995) Reduction of gordon-haas timing jitter by periodic

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Entropy in Ergodic Theory

Entropy in Ergodic Theory JONATHAN L. F. KING University of Florida, Gainesville, USA Article Outline Glossary Definition of the Subject Entropy example: How many questions? Distribution Entropy A Gander at Shannon’s Noisy Channel Theorem The Information Function Entropy of a Process Entropy of a Transformation Determinism and Zero-Entropy The Pinsker–Field and K-Automorphisms Ornstein Theory Topological Entropy Three Recent Results Exodos Bibliography Glossary Some of the following definitions refer to the “Notation” paragraph immediately below. Use mpt for ‘measure-preserving transformation’. Measure space A measure space (X; X ; ) is a set X, a field (that is, a -algebra) X of subsets of X, and a countably-additive measure  : X ! [0; 1]. (We often just write (X; ), with the field implicit.) For a collection C X , use Fld(C)  C for the smallest field including C. The number (B) is the “-mass of B”. Measure-preserving map A measure-preserving map : (X; X ; ) ! (Y; Y; ) is a map : X ! Y such that the inverse image of each B 2 Y is in X , and ( 1 (B)) D (B). A (measure-preserving) transformation is a measure-preserving map T : (X; X ; ) ! (X; X ; ). Condense this notation to (T : X; X ; ) or (T : X; ). Probability space A probability space is a measure space (X; ) with (X) D 1; this  is a probability measure. All our maps/transformations in this article are on probability spaces. Factor map A factor map : (T : X; X ; ) ! (S : Y; Y; ) is a measure-preserving map : X ! Y which intertwines the transformations, ıT D S ı . And is an

isomorphism if – after deleting a nullset (a mass-zero set) in each space – this is a bijection and 1 is also a factor map. Almost everywhere (a.e.) A measure-theoretic statement holds almost everywhere, abbreviated a.e., if it holds off of a nullset. (Eugene Gutkin once remarked to me that the problem with Measure Theory is . . . that you have to say “almost everywhere”, almost everywhere.) a:e: For example, B  A means that (B X A) is zero. The a.e. will usually be implicit. Probability vector A probability vector vE D (v1 ; v2 ; : : : ) is a list of non-negative reals whose sum is 1. We generally assume that probability vectors and partitions (see below) have finitely many components. Write “countable probability vector/partition”, when finitely or denumerably many components are considered. Partition A partition P D (A1 ; A2 ; : : : ) splits X into pairwise disjoint subsets A i 2 X so that the disjoint union F # i A i is all of X. Each Ai is an atom of P. Use jPj or P for the number of atoms. When P partitions a probability space, then it yields a probability vector vE, where v j :D (A j ). Lastly, use Phxi to denote the P-atom that owns x. Fonts We use the font H ; E ; I for distribution-entropy, entropy and the information function. In contrast, the script font ABC : : : will be used for collections of sets; usually subfields of X . Use E() for the (conditional) expectation operator. Notation Z D integers, ZC D positive integers, and N D natural numbers D 0; 1; 2; : : :. (Some wellmeaning folk use N for ZC , saying ‘Nothing could be more natural than the positive integers’. And this is why 0 2 N. Use de and bc for the ceiling and floor functions; bc is also called the “greatest-integer function”. For an interval J :D [a; b) [1; C1], let [a : : : b) denote the interval of integers J \ Z (with a similar convention for closed and open intervals). E. g., (e : : : ] D (e : : : ) D f3g. For subsets A and B of the same space, ˝, use A B for inclusion and A ¦ B for proper inclusion. The difference set B X A is f! 2 B j ! … Ag. Employ Ac for the complement ˝ X A. Since we work in a probability space, if we let x :D (A), then a convenient convention is to have xc

denote 1  x ;

since then (Ac ) equals xc . Use A4B for the symmetric difference [AXB][[BXA]. For a collection C D fE j g j of sets in ˝, let the disjoint F F S union j E j or (C) represent the union j E j and also assert that the sets are pairwise disjoint.

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Use “8large n” to mean: “9n0 such that 8n > n0 ”. To refer to left hand side of an Eq. (20), use LhS(20); do analogously for RhS(20), the right hand side.

Definition of the Subject The word ‘entropy’ (originally German, Entropie) was coined by Rudolf Julius Emanuel Clausius circa 1865 [2,3], taken from the Greek  o ˛, ‘a turning towards’. This article thus begins (Prolegomenon, “introduction”) and ends (Exodos1 , “the path out”) in Greek. Clausius, in his coinage, was referring to the thermodynamic notion in physics. Our focus in this article, however, will be the concept in measurable and topological dynamics. (Entropy in differentiable dynamics2 would require an article by itself.) Shannon’s 1948 paper [6] on Information Theory, then Kolmogorov’s [4] and Sinai’s [7] generalization to dynamical systems, will be our starting point. Our discussion will be of the one-dimensional case, where the acting-group is Z. “My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. John von Neumann had a better idea, he told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function goes by that name in statistical mechanics. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.’ ” (Shannon as quoted in [59]) Entropy example: How many questions? Imagine a dartboard, Fig. 1, split in five regions A; : : : ; E with known probabilities. Blindfolded, you throw a dart 1 This is

the Greek spelling. 2 For instance, see [15,18,24,25].

Entropy in Ergodic Theory, Figure 1 This dartboard is a probability space with a 5-set partition. The atoms have probabilities 12 ; 18 ; 18 ; 18 ; 18 . This probability distribution will be used later in Meshalkin’s example

at the board. What is the expected number V of Yes/No questions needed to ascertain the region in which the dart landed? Solve this by always dividing the remaining probability in half. ‘Is it A?’ – if Yes, then V D 1. Else: ‘Is it B or C?’ – if Yes, then ‘Is it B?’ – if No, then the dart landed in C, and V D 3 was the number of questions. Evidently V D 3 also for regions B; D; E. Using “log” to denote base-2 logarithm3 , the expected number of questions4 is thus 1 1 1 1 1 1C 3C 3C 3C 3 2 8 8 8 8   4 X 1 note D p j log D 2: pj

E(V ) D

(1)

jD0

  Letting vE :D 12 ; 18 ; 18 ; 18 ; 18 be the probability vector, we can write this expectation as X (x) : E(V ) D x2E v

Here,  : [0; 1] ! [0; 1) is the important function5 (x) :D x log(1/x) ;

so extending by continuity gives

(0) D 0 : (2) An interpretation of “(x)” is the number of questions needed to winnow down to an event of probability x. Distribution Entropy Given a probability vector vE, define its distribution entropy as X H (E v ) :D (x) : (3) x2E v

This article will use the term distropy for ‘distribution entropy’, reserving the word entropy for the corresponding 3 In this paper, unmarked logs will be to base-2. In entropy theory, it does not matter much what base is used, but base-2 is convenient for computing entropy for messages described in bits. When using the natural logarithm, some people refer to the unit of information as a nat. In this paper, I have picked bits rather than nats. 4 This is holds when each probability p is a reciprocal power of two. For general probabilities, the “expected number of questions” interpretation holds in a weaker sense: Throw N darts independently at N copies of the dartboard. Efficiently ask Yes/No questions to determine where all N darts  landed. Dividing by N, then sending N ! 1, will be the p log 1p sum of Eq. (1). 5 There does not seem to be a standard name for this function. I use , since an uppercase  looks like an H, which is the letter that Shannon used to denote what I am calling distribution-entropy.

Entropy in Ergodic Theory

dynamical concept, when there is a notion of time involved. Getting ahead of ourselves, the entropy of a stationary process is the asymptotic average value that its distropy decays to, as we look at larger and larger finite portions of the process.   An equi-probable vector vE :D K1 ; : K: :; K1 evidently has H (E v ) D log(K). On a probability space, the “distropy of partition P”, written H (P) or H (A1 ; A2 ; : : :) shall mean the distropy of probability vector j 7! (A j ). A (finite) partition necessarily has finite distropy. A countable partition can have finite distropy, e. g. H (1/2; 1/4; 1/8; : : : ) D 2. One could also have infinite distropy: Consider a piece B X of mass 1/2 N . Splitting B into 2 k many equal-mass atoms gives an -sum of 2 k (k C N)/ (2 k 2 N ). Setting k D k N :D 2 N  N makes this -sum F1 equal 1; so splitting the pieces of X D ND1 B N , with 1 (B N ) D 2N , yields an 1-distropy partition.

Entropy in Ergodic Theory, Figure 2 Using natural log, here are the graphs of: (x) in solid red, H (x; x c ) in dashed green, 1  x in dotted blue. Both (x) and H (x; x c ) are strictly convex-down. The 1  x line is tangent to (x) at x D 1

Function  The (x) D x log(1/x) function6 has vertical tangent at x D 0, maximum at 1/e and, when graphed in nats slope 1 at x D 1. Consider partitions P and Q on the same space (X; ). Their join, written P _ Q, has atoms A \ B, for each pair A 2 P and B 2 Q. They are independent, written P?Q if (A\ B) D (A)(B) for each A; B pair. We write P < Q, and say that “P refines Q”, if each P-atom is a subset of some Q-atom. Consequently, each Q-atom is a union of Patoms. Recall, for ı a real number, our convention that ı c means 1  ı, in analogy with (B c ) equaling 1  (B) on a probability space. Distropy Fact

Entropy in Ergodic Theory, Figure 3 Using natural log: The graph of H (x1 ; x2 ; x3 ) in barycentric coordinates; a slice has been removed, between z D 0:745 and z D 0:821. The three arches are copies of the distropy curve from Fig. 2.

For partitions P; Q; R on probability space (X; ): (a) H (P)  log(# P), with equality IFF P is an equi-mass partition. (b) H (Q _ R)   H (Q) C H (R), with equality IFF Q?R. (c) For ı 2 0; 12 , the function ı 7! H (ı; ı c ) is strictly increasing. a:e: (d) R 4 P implies H (R)  H (P), with equality IFF R D P. 6 Curiosity:

Just in this paragraph we compute distropy in nats, that is, using natural logarithm. Given a small probability p 2 [0; 1] and setting x :D 1/p, note that (p) D log(x)/x 1/(x), where (x) denotes the number of prime numbers less-equal x. (This approximation is a weak form of the Prime Number Theorem.) Is there any actual connection between the ‘approximate distropy’ function P H (E p) :D p2Ep 1/(1/p) and Number Theory, other than a coincidence of growth rate?

Proof Use the strict concavity of (), together with Jensen’s inequality.  Remark 1 Although we will not discuss it in this paper, most distropy statements remain true with ‘partition’ replaced by ‘countable partition of finite distropy’. Binomial Coefficients The dartboard gave an example where distropy arises in a natural way. Here is a second example. For a small   ı > 0, one might guess that the binomial coefficient ınn grows asymptotically (as n ! 1) like 2"n , for some small ". But what is the correct relation between " and ı?

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Well, Stirling’s formula n!  [n/e]n gives nn

n!  c [ın]![ı c n]! [ın][ı n] [ı c n][ı n] 1 (recall ı c D 1  ı) : D ın c ıc n [ı [ı ] ]   Thus n1 log ınn  H (ı; ı c ). But by means of the above distropy inequalities, we get an inequality true for all n, not just asymptotically.  Lemma 2 (Binomial Lemma) Fix a ı 2 0; 12 and let H :D H (ı; ı c ). Then for each n 2 ZC : ! X n  2Hn : (4) j j2[0:::ı n]

Proof Let X f0; 1gn be the set of xE with #

Noisy Channel Shannon’s theorem says that a noisy channel has a channel capacity. Transmitting above this speed, there is a minimum error-rate (depending how much “above”) that no error-correcting code can fix. Conversely, one can transmit below – but arbitrarily close to – the channel capacity, and encode the data so as to make the error-rate less than any given ". We use Corollary 3 to show the existence of such codes, in the simplest case where the noise7 is a binary independent-process (a “Bernoulli” process, in the language later in this article). We have a channel which can pass one bit per second. Alas, there is a fixed noise-probability  2 [0; 12 ) so that a bit in the channel is perturbed into the other value. Each perturbation is independent of all others. Let H :D H (;  c ). The value [1  H] bits-per-second is the channel capacity of this noise-afflicted channel.

fi 2 [1 : : : n] j x i D 1g  ın :

On X, let P1 ; P2 ; : : : be the coordinate partitions; e. g. P7 D (A7 ; Ac7 ), where A7 :D fE x j x7 D 1g. Weighting each point 1 , the uniform distribution on X, gives that (A7 )  by jXj ı. So H (P7 )  H, by (c) in Sect. “Distropy Fact”. Finally, the join P1 _    _ Pn separates the points of X. So log(# X) D H (P1 _    _ Pn )  H (P1 ) C : : : C H (P n )  Hn ; making use of (a),(b) in “Distropy Fact”. And # X equals LhS (4).  A Gander at Shannon’s Noisy Channel Theorem We can restate the Binomial lemma using the Hamming metric on f0; 1gn , Dist(E x ; yE) :D # fi 2 [1 : : : n] j x i ¤ y i g :

Encoding/Decoding Encode using an “k; n–blockcode”; an injective map F : f0; 1g k ! f0; 1gn . The source text is split into consecutive k-bit blocks. A block xE 2 f0; 1g k is encoded to F(E x ) 2 f0; 1gn and then sent through the channel, where it comes out perturbed to ˛E 2 f0; 1gn . The transmission rate is thus k/n bits per second. For this example, we fix a radius r > 0 to determine the decoding map, Dr : f0; 1gn ! fOopsg t f0; 1g k : We set Dr (E ˛) to zE if there is a unique zE with F(Ez) 2 Bal(E ˛; r); else, set Dr (E ˛) :D Oops. One can think of the noise as a f0; 1g-independentprocess, with Prob(1) D , which is added mod-2 to the signal-process. Suppose we can arrange that the set fF(E x )) j xE 2 f0; 1g k g of codewords, is a strongly r-separated-set. Then

Use Bal(E x ; r) for the open radius-r ball centered at xE, and Bal(E x ; r) :D f yE j Dist (E x ; yE)  rg for the closed ball. The above lemma can be interpreted as saying that x ; ın)j  2H (ı;ı jBal(E

c )n

for each xE 2 f0; 1gn : (5)  Corollary 3 Fix n 2 ZC and ı 2 0; 12 , and let ;

H :D H (ı; ı c ) : Then there is a set C f0; 1gn , with # C  2[1H]n , that is strongly ın-separated. I. e., Dist (E x ; yE) > ın for each distinct pair xE; yE 2 C.

The probability that a block is mis-decoded is the probability, flipping a -coin n times that we get more than r many Heads:

(6)  1 Theorem 4 (Shannon) Fix a noise-probability  2 0; 2 and let H :D H (;  c ). Consider a rate R < [1  H] and an " > 0. Then 8large n there exists a k and a code F : f0; 1g k ! f0; 1gn so that: The F-code transmits bits at faster than R bits-per-second, and with error-rate < ". 7 The noise-process is assumed to be independent of the signalprocess. In contrast, when the perturbation is highly dependent on the signal, then it is sometimes called distortion.

Entropy in Ergodic Theory

Proof Let H0 :D H (ı; ı c ), where ı >  was chosen so close to  that ı<

1 2

and 1  H0 > R :

(7)

The information function has been defined so that its expectation is the distropy of P, E(IP ) D

where

k :D b[1  H0 ]nc :

(8)

By Corollary 3, there is a strongly ın-separated-set C

0 f0; 1gn with # C  2[1H ]n . So C is big enough to permit an injection F : f0; 1g k ! C. Courtesy Eq. (6), the probability of a decoding error is that of getting more than ın many Heads in flipping a -coin n times. Since ı > , the Weak Law of Large Numbers guarantees – once n is large enough – that this probability is less than the given ".  The Information Function Agree to use P D (A1 ; : : :), Q D (B1 ; : : :), R D (C1 ; : : :) for partitions, and F ; G for fields. With C a (finite or infinite) family of subfields of X , W their join G2C G is the smallest field F such that G F , for each G 2 C. A partition Q can be interpreted also as a field; namely, the field of unions of its atoms. A join of denumerably many partitions will be interpreted as a field, but a join of finitely many, P1 _    _ PN , will be viewed as a partition or as a field, depending on context. Conditioning a partition P on a positive-mass set B, let PjB be the probability vector A 7! (A\B) (B) . Its distropy is   X 1 (A \ B) H (PjB) D log : (A \ B)/(B) (B) A2P So conditioning P on a partition Q gives conditional distropy X H (PjQ) D H (PjB)(B) B2Q

D

X A2P;B2Q

log

1 (A\B) (B)

Conditioning on a Field For a subfield F X , recall that each function g 2 L1 () has a conditional expectation E(gjF ) 2 L1 (). It is the unique F -measurable function with Z

8B 2 F :

E(gjF )d D B

Z gd : B

Returning to distropy ideas, use (AjF ) for the conditional probability function; it is the conditional expectation E(1A jF ). So the conditional information function is IPjF (x) :D

X A2P

 log

 1 1A (x) : (AjF (x))

(11)

Its integral H (PjF ) :D

Z

IPjF d ;

is the conditional distropy of P on F . When F is the field of unions of atoms from some partition Q, then the number H (PjF ) equals the H (PjQ) from Eq. (9). Write G j % F to indicate  fields G1 G2 : : : S1 that are nested, and that Fld 1 G j D F , a.e. The Martingale Convergence Theorem (p. 103 in [20]) gives (c) below. Conditional-Distropy Fact Consider partitions P; Q; R and fields F and G j . Then a:e:

! (A \ B) :

(9)

A “dartboard” interpretation of H (PjQ) is The expected number of questions to ascertain the P-atom that a random dart x 2 X fell in, given that we are told its Q-atom. For a set A X, use 1A : X ! f0; 1g for its indicator function; 1A (x) D 1 IFF x 2 A. The information function of partition P, a map IP : X ! [0; 1), is   X 1 IP :D log (10) 1A () : (A) A2P

IP ()d D H (P) : X

Pick a large n for which k >R; n

Z

(a) 0  H (PjF )  H (P), with equality IFF P F , respectively, P?F . (b) H (Q _ RjF )  H (QjF ) C H (RjF ). (c) Suppose G j % F . Then H (PjG j ) & H (PjF ). (d) H (Q _ R) D H (QjR) C H (R). (d’) H (Q _ R1 jR0 ) D H (QjR1 _ R0 ) C H (R1 jR0 ). Imagining our dartboard (Fig. 1) divided by superimposed partitions Q and R, equality (d) can interpreted as saying: ‘You can efficiently discover where the dart landed in both partitions, by first asking efficient questions about R, then – based on where it landed in R – asking intelligent questions about Q.’

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Entropy in Ergodic Theory

Entropy of a Process Consider a transformation (T : X; ) and partition P D (A1 ; A2 ; : : :). Each “time” n determines a partition Pn :D T n P, whose jth-atom is T n (A j ). The process T; P refers W to how T acts on the subfield 1 0 P n X . (An alternative view of a process is as a stationary sequence V0 ; V1 ; : : : of random variables Vn : X ! ZC , where Vn (x) :D j because x is in the jth-atom of Pn .) Write E (T; P) or E T (P) for the “entropy” of the T; P process”. It is the limit of the conditional-distropy-numbers c n :D H (P0 jP1 _ P2 _    _ Pn1 ) : This limit exists since H (P) D c1  c2      0. Define the average-distropy-number n1 h n , where h n :D H (P0 _ P1 _    _ Pn1 ) : Certainly h n D c n C H (P1 _    _ Pn1 ) D c n C h n1 , since T is measure preserving. Induction gives h n D Pn 1 jD1 c j . So the Cesàro averages n h n converge to the entropy. Theorem 5 The entropy of process (T; P : X; X ; ) equals

T(x) :D [n 7! xnC1 ] : The time-zero partition P separates points, under the action of the shift. This L-atom time-zero partition has Phxi D Phyi IFF x0 D y0 . So no matter what shift-invariant measure is put on X, the time-zero partition will generate under the action of T. Time Reversibility A transformation need not be isomorphic to its inverse. Nonetheless, the average-distropy-numbers show that E(T 1 ; P) D E(T; P); although this is not obvious from the conditioning-definition of entropy. Alternatively, ! ˇ_ ˇ n H P0 ˇˇ Pj 1

D H (P0 _    _ Pn )  H (P1 _    _ Pn ) D H (Pn _    _ P0 )  H (Pn _    _ P1 ) ! ˇ_ ˇ n (12) D H P0 ˇˇ P j :

1 H (P0 _    _ Pn1 ) n!1 n ! ˇ_ ˇ n ˇ D lim H P0 ˇ P j n!1 1 ˇ1 ! ˇ_ D H P0 ˇˇ P j : lim

1

Bernoulli Processes

1

Both limits are non-increasing. The entropy E T (P)  0, W T with equality IFF P 1 1 P j . And E (P)  H (P), with equality IFF T; P is an independent process. Generators We henceforth only discuss invertible mpts, that is, when T 1 is itself an mpt. Viewing the atoms of P as “letters”, then, each x 2 X has a T; P-name : : : x2 x1 x0 x1 x2 x3 : : : ; PhT n (x)i,

IFF P separates points. That is, after deleting a (T-invariant) nullset, distinct points of X have distinct T; P-names. A finite set [1 : : : L] of integers, interpreted as an alphabet, yields the shift space X :D [1 : : : L]Z of doubly-infinite sequences x D (: : : x1 x0 x1 : : : ). The shift T : X ! X acts on X by

T n (x).

where xn is the P-letter owning A partition P generates (the whole field) under (T : W n 8 X; ), if 1 1 T P D X . It turns out that P generates 8 I am now at liberty to reveal that our X has always been a Lebesgue space, that is, measure-isomorphic to an interval of R together with countably many point-atoms (points with positive mass).

A probability vector vE :D (v1 ; : : : ; v L ) can be viewed as a measure on alphabet [1 : : : L]. Let vE be the resulting product measure on X :D [1 : : : L]Z , with T the shift on X and P the time-zero partition. The independent process (T; P : X; vE ) is called, by ergodic theorists, a Bernoulli process. Not necessarily consistently, we tend to refer to the underlying  transformation as a Bernoulli  shift.  The 12 ; 12 -Bernoulli and the 13 ; 13 ; 13 -Bernoulli have different process-entropies, but perhaps their underlying transformations are isomorphic? Prior to the Kolmogorov–Sinai definition of entropy9 of a transformation, this question remained unanswered. The equivalence of generating and separating is a technical theorem, due to Rokhlin. Assuming  to be Lebesgue is not much of a limitation. For instance, if  is a finite measure on any Polish space, then  extends to a Lebesgue measure on the -completion of the Borel sets. To not mince words: All spaces are Lebesgue spaces unless you are actively looking for trouble. 9 This is sometimes called measure(-theoretic) entropy or (perhaps unfortunately) metric entropy, to distinguish it from topological

Entropy in Ergodic Theory

Entropy of a Transformation The Kolmogorov–Sinai definition of the entropy of an mpt is E(T) :D supfE T (Q) j Q a partition on Xg :

Certainly entropy is an isomorphism invariant – but is it useful? After all, the supremum of distropies of partitions is always infinite (on non-atomic spaces) and one might fear that the same holds for entropies. The key observation (restated in Lemma 8c and proved below) was this, from [4] and [7]. Theorem 6 (Kolmogorov–Sinai Theorem) If P generates under T, then E(T) D E(T; P).     Thereupon the 12 ; 12 and 13 ; 13 ; 13 Bernoulli-shifts are not isomorphic, since their respective entropies are log(2) ¤ log(3). Wolfgang Krieger later proved a converse to the Kolmogorov–Sinai theorem. Theorem 7 (Krieger Generator Theorem, 1970) Suppose T ergodic. If E(T) < 1, then T has a generating partition. Indeed, letting K be the smallest integer K > E(T), there is a K-atom generator.10 Proof See Rudolph [21], or § 5.1 in  Joinings in Ergodic Theory, where Krieger’s theorem is stated in terms of joinings.  Entropy Is Continuous

Then Lemma 8b says that process-entropy varies continuously with varying the partition. Lemma 8 Fix a mpt (T : X; ). For partitions P; Q; Q0 , define R :D Q4Q0 and let ı :D H (R). Then (a) jH (Q)  H (Q0 )j  ı. (Distropy varies continuously with the partition.) (b) jE T (Q)  E T (Q0 )j  ı. (Process-entropy varies continuously with the partition.) (c) For all partitions Q Fld(T; P) : E T (Q)  E T (P). Proof (of (a)) Evidently Q0 _ R D Q0 _ Q D Q _ R. So H (Q0 )  H (Q _ R)  H (Q) C ı.  Proof (of (b)) As above, H

N _

! Q0j

N _

H

1

! Qj C H

1

N _

! :

Rj

1

Sending N ! 1 gives E T (Q0 )  E T (Q) C E T (R). Finally, E T (R)  H (R) and so E T (Q0 )  E T (Q) C ı.  Proof (of (c)) Let K :D jQj. Then there is a sequence of WL P` . By above, K-set partitions Q(L) ! Q with Q(L) 4 L E T (Q(L) ) ! E T (Q), so showing that E

T

L _

!

?

P`  E T (P)

L

will suffice. Note that 0

(B01 ; : : :), 0

Given ordered partitions Q D (B1 ; : : :) and Q D extend the shorter by null-atoms until jQj D jQ j. Let F Fat :D j [B j \ B0j ]; this set should have mass close to 1 if Q and Q0 are almost the same partition. Define a new partition Q4Q0 :D fFatg t fB i \ B0j j with i ¤ jg : (In other words, take Q _ Q0 and coalesce, into a single atom, all the B k \ B0k sets.) Topologize the space of parti  tions by saying11 that Q(L) ! Q when H Q4Q(L) ! 0. entropy. Tools known prior to entropy, such as spectral properties, did not distinguish the two Bernoulli-shifts; see  Spectral Theory of Dynamical Systems for the definitions. 10 It is an easier result, undoubtedly known much earlier, that every ergodic T has a countable generating partition – possibly of 1-distropy. 11 On the set of ordered K-set partitions (with K fixed) this conver   gence is the same as: Q(L) ! Q when  Fat Q(L) ; Q ! 1. An alternative approach is the Rokhlin metric, Dist(P; Q) :D H (PjQ) C H (QjP), which has the advantage of working for unordered partitions.

h N :D H

N1 _

L _

Tn

1 NH

W

N1 Pj 0

0 D H@

P`

L

nD0

So N1 h N  N ! 1.

!!

N1CL _

1 PjA :

jDL



C

1 N 2LH (P).

Now send 

Entropy Is Not Continuous The most common topology placed on the space ˝ of mpts is the coarse topology12 that Halmos discusses in his “little red book” [14]. The Rokhlin lemma (see p. 33 in [21]) implies that the isomorphism-class of each ergodic mpt is dense in ˝, (e. g., see p. 77 in [14]) disclosing that the S 7! E(S) map is exorbitantly discontinuous. 12 i. e, S ! T IFF 8A 2 X : (S 1 (A)4T 1 (A)) ! 0; this is n n a metric-topology, since our probability space is countably generated. This can be restated in terms of the unitary operator U T on L2 (), where U T ( f ) :D f ı T. Namely, S n ! T in the coarse topology IFF U S n ! U T in the strong operator topology.

211

212

Entropy in Ergodic Theory

Indeed, the failure happens already for process-entropy with respect to a fixed partition. A Bernoulli process T; P has positive entropy. Take mpts S n ! T, each isomorphic to an irrational rotation. Then each E(S n ; P) is zero, as shown in the later section "Determinism and ZeroEntropy". Further Results When F is a T-invariant subfield, agree to use T F for “T restricted to F ”, which is a factor (see Glossary) of T. Transformations T and S are weakly isomorphic if each is isomorphic to a factor of the other. The foregoing entropy tools make short shrift of the following. Lemma 9 (Entropy Lemma) Consider T-invariant subfields G j and F . (a) Suppose G j % F . Then E T G j % E(T F ). In particular, G F implies that E(T G )  E(T F ), so entropy is an invariant of weak-isomorphism. P (b) E(T G1 _G2 _::: )  j E(T G j ). P And E(T; Q1 _ Q2 _ : : :)  j E(T; Q j ). P (c) For mpts (S j : Yj ;  j ): E(S1  S2    ) D j E(S j ). (d) E(T 1 ) D E(T). More generally, E(T n ) D jnj  E(T). Meshalkin’s Map In the wake of Kolmogorov’s 1958 entropy paper, for two Bernoulli-shifts to be isomorphic one now knew that they had to have equal entropies. Meshalkin provided the first non-trivial example in 1959 [45]. Let S : Y ! Y be the Bernoulli-shift over the “letter”  alphabet  fE; D; P; Ng, with probability distribution 14 ; 14 ; 14 ; 14 . The letters E; D; P; N stand for Even, oDd, Positive, Negative, and will be used to describe the code (isomorphism) between the processes. Use T : X ! X for the Bernoulli-shift over “digit” alphabet 1;  1 1f0;1 C1;  C2; 2g, with probability  1 1 1 distribu 1 1 1 tion ; ; ; ; . Both distributions 4 ; 4 ; 4 ; 4 and  1 1 12 18 18  8 8 2 ; 8 ; 8 ; 8 ; 8 have distropy log(4). After deleting invariant nullsets from X and Y, the construction will produce a measure-preserving isomorphism : X ! Y so that T ı D ı S. The Code

The leftmost 0 is linked to the rightmost +1, as indicated by the longest-overbar. The left/right-parentheses form a 12 ; 12 -random-walk. Since this random walk is recurrent, every position in x will be linked (except for a nullset of points x). Below each 0, write “P” or “N” as the 0 is linked to a positive or negative digit. And below the other digits, write “E” or “D” as the digit is even or odd. So the upper name in X is mapped to the lower name, a point y 2 Y. map : X !  Y carries  1 This   the upstairs 1 1 1 1 1 1 1 1 distribution to ; ; ; ; ; ; ; 2 8 8 8 8 4 4 4 4 , downstairs. It takes some arguing to show that independence is preserved. The inverse map 1 views D and E as right-parentheses, and P and N as left. Above D, write the odd digit C1 or 1, as this D is linked to Positive or Negative. Markov Shifts A Bernoulli process T; P has independence P(1:::0] ?P1 whereas a Markov process is a bit less aloof: The infinite Past P(1:::0] doesn’t provide any more information about Tomorrow than Today did. That is, the conditional distribution P1 jP(1:::0] equals P1 jP0 . Equivalently,   note H (P1 jP0 ) D H P1 jP(1:::0] D E(T; P) : (13) The simplest non-trivial Markov process (T; P : X; ) is over a two-letter alphabet fa; bg, and has transition graph Fig. 4, for some choice of transition probabilities s and c. The graph’s Markov matrix is

 s c ; M D [m i; j ] i; j D 1 0 where c D 1  s, and m i; j denotes the probability of going from state i to state j. If Today’s distribution on the two states is the probability-vector vE :D [pa pb ], then Tomorrow’s is the product vEM. So a stationary process needs vEM D vE. This equa1 c tion has the unique solution pa D 1Cc and pb D 1Cc .

In X, consider this point x:

: : : 0 0 0 1 0 0 +1 +2 1 +1 0 : : : Regard each 0 as a left-parenthesis, and each non-zero as a right-parenthesis. Link them according to the legal way of matching parentheses, as shown in the top row, below: 0 0 0 1 0 0 +1 +2 1 +1 0 P N N D P P D

E

D

D ?

Entropy in Ergodic Theory, Figure 4 Call the transition probabilities s :D Prob(a!a) for stay, and c :D Prob a!b for change. These are non-negative reals, and sCcD1

Entropy in Ergodic Theory

An example of computing the probability of a word or cylinder set; (see Sect. “The Carathéodory Construction” in  Measure Preserving Systems) in the process, is

approximately 2E(T;P)n ; this, after discarding small mass from the space. But the growth of n 7! 2n is sub-exponential and so, for our rotation, E(T; P) must be zero. Theorem 10 (Shannon–McMillan–Breiman Theorem (SMB-Theorem)) Set E :D E(T; P), where the tuple (T; P : X; ) is an ergodic process. Then the average information function

s (baaaba) D pb mba maa maa mab mba c D 1ssc1: 1Cc The subscript on s indicates the dependence on the transition probabilities; let’s also mark the mpt and call it T s . Using Eq. (13), the entropy of our Markov map is

1 n!1 IP (x) ! E ; n [0:::n)

for a.e.

x2X:

(15)

The functions f n :D IP[0:::n) converge to the constant function E both in the L1 -norm and in probability. 14

E(Ts ) D pa  H (s; c) C pb  H (1; 0)

„ ƒ‚ … D0

1 D  [s log(s) C c log(c)] : 1Cc

(14)

Determinism and Zero-Entropy Irrational rotations have zero-entropy; let’s reveal this in two different ways. Equip X :D [0; 1) with “length” (Lebesgue) measure and wrap it into a circle. With “˚” denoting addition mod-1, have T : X ! X be the rotation T(x) :D x ˚ ˛, where the rotation number ˛ is irrational. Pick distinct points y0 ; z0 2 X, and let P be the partition whose two atoms are the intervals [y0 ; z0 ) and [z0 ; y0 ), wrapping around the circle. The T-orbit of each point x is dense13 in X. In particular, y0 has dense orbit, so P separates points – hence ?

generates – under T. Our goal, thus, is E(T; P) D 0.

Proof See the texts of Karl Petersen (p. 261 in [20]), or Dan Rudolph (p. 77 in [21]).  Consequences Recall that P[0:::n) means P0 _ P1 _    _ Pn1 , where P j :D T j P. As usual, P[0:::n) hxi denotes the P[0:::n) -atom owning x. Having deleted a nullset, we can restate Eq. (15) to now say that 8"; 8x; 8large n:   1/2[EC"]n   P[0:::n) hxi  1/2[E"]n :

(16)

This has the following consequence. Fixing a number ı > 0, we consider any set with (B)  ı and count the number of n-names of points in B. The SMB-Thm implies 

8"; 8large n; 8B  ı : ˇ ˇ ˇfn-names in Bgˇ  2[E"]n :

(17)

Rotations are Deterministic

Rank-1 Has Zero-Entropy

The forward T-orbit of each point is dense. This is true for y0 , and so the backward T; P-name of each x actually W n tells us which point x is. I. e., P 1 1 T P, which is our definition of “process T; P is deterministic”. Our P being finite, this determinism implies that E(T; P) is zero, by Theorem 5.

There are several equivalent definitions for “rank-1 transformation”, several of which are discussed in the introduction of [28]. (See Chap. 6 in [13] as well as [51] and [27] for examples of stacking constructions.) A rank-1 transformation (T : X; ) admits a generating partition P and a sequence of Rokhlin stacks S n X, with heights going to 1, and with (S n ) ! 1. Moreover, each of these Rokhlin stacks is P-monochromatic, that is, each level of the stack lies entirely in some atom of P. Taking a stack of some height 2n, let B D B n be the union of the bottom n levels of the stack. There are at most n many length-n names starting in Bn , by monochromaticity. Finally, (B n ) is almost 12 , so is certainly larger than ı :D 13 . Thus Eq. (17) shows that our rank-1 T has zero entropy.

Counting Names in a Rotation The P0 _    _ Pn1 partition places n translates of points y0 and of z0 , cutting the circle into at most 2n intervals. Thus H (P0 _ : : : _ Pn1 )  log(2n). And n1 log(2n) ! 0. Alternatively, the below SMB-theorem implies, for an ergodic process T; P, that the number of length-n names is an " > 0 and an N > 1/". Points x; T(x); : : : ; T N (x) have some two at distance less than N1 ; say, Dist(T i (x); T j (x)) < ", for some 0  i < j  N. Since T is an isometry, " > Dist(x; T k (x)) > 0, where k :D j  i. So the T k -orbit of x is "-dense. 13 Fix

14 In engineering circles, this is called the Almost-everywhere equipartition theorem.

213

214

Entropy in Ergodic Theory

Cautions on Determinism’s Relation to Zero-Entropy

K-Processes

A finite-valued process T; P has zero-entropy iff P

W1 1 P j . Iterating gives

Kolmogorov introduced the notion of a K-process or Kolmogorov process, in which the present becomes asymptotically independent of the distant past. The asymptotic past of the T; P process is called its tail field, where

1 _ 0

Pj

1 _

Pj ;

1

i. e., the future is measurable with respect to the past. This was the case with the rotation, where a point’s past uniquely identified the point, thus telling us its future. While determinism and zero-entropy mean the same thing for finite-valued processes, this fails catastrophically for real-valued (i. e., continuum-valued) processes, as shown by an example of the author’s. A stationary realvalued process V D : : : V1 V0 V1 V2 : : : is constructed in [40] which is simultaneously strongly deterministic: The two values V0 ; V1 determine all of V, future and past. and non-consecutively independent. This latter means that for each bi-infinite increasing integer sequence fn j g1 jD1 with no consecutive pair (always 1 C n j < n jC1 ), then the list of random variables : : : Vn 1 Vn 0 Vn 1 Vn 2 : : : is an independent process. Restricting the random variables to be countablyvalued, how much of the example survives? Joint work with Kalikow [39] produced a countably-valued stationary V which is non-consecutively independent as well as deterministic. (Strong determinism is ruled out, due to cardinality considerations.) A side-effect of the construction is that V’s time-reversal n 7! Vn is not deterministic. The Pinsker–Field and K-Automorphisms Consider the collection of zero-entropy sets, Z D Z T :D fA 2 X j E(T; (A; Ac )) D 0g :

(18)

Courtesy of Lemma 9b, Z is a T-invariant field, and 8Q Z : E(T; Q) D 0 :

(19)

The Pinsker field15 of T is this Z. It is maximal with respect to Eq. (19). Unsurprisingly, the Pinsker factor T Z has zero entropy, that is, E(T Z ) D 0. A transformation T is said to have completely-positive entropy if it has no (nontrivial) zero-entropy factors. That is, its Pinsker field Z T is the trivial field, ; :D f¿; Xg. 15 Traditionally, this called the Pinsker algebra where, in this context, “algebra” is understood to mean “ -algebra”.

Tail(T; P) :D

1 \

M _

Pj :

MD1 jD1

This T; P is a K-process if Tail(T; P) D ;. This turns out to be equivalent to what we might a call a strong form of “sensitive dependence on initial conditions”: For each fixed length L, the distant future _

Pj

j2(G:::GCL]

becomes more and more independent of _

Pj ;

j2(1:::0]

as the gap G ! 1. A transformation T is a Kolmogorov automorphism if it possesses a generating partition P for which T; P is a K-process. Here is a theorem that relates the “asymptotic forgetfulness” of Tail(T; P) D ;, to the lack of determinism implied by having no zero-entropy factors (See Walters, [23], p. 113. Related results appear in Berg [44]). Theorem 11 (Pinsker-Algebra Theorem) Suppose P is a generating partition for an ergodic T. Then Tail(T; P) equals Z T . Since Z T does not depend on P, this means that all generating partitions have the same tail field, and therefore K-ness of T can be detected from any generator. Another non-evident fact follows from the above. The future field of T; P is defined to be Tail (T 1 P). It is not obvious that if the present is independent of the distant past, then it is automatically independent of the distant future. (Indeed, the precise definitions are important; witness the Cautions on determinism section.) But since the entropy of a process, E(T; (A; Ac )), equals the entropy of the time-reversed process E(T 1 ; (A; Ac )), it follows that Z T equals Z T 1 . Ornstein Theory In 1970, Don Ornstein [46] solved the long-standing problem of showing that entropy was a complete isomorphism-

Entropy in Ergodic Theory

invariant of Bernoulli transformations; that is, that two independent processes with same entropy necessarily have the same underlying transformation. (Earlier, Sinai [52]) had shown that two such Bernoulli maps were weakly isomorphic, that is, each isomorphic to a factor of the other.) Ornstein introduced the notion of a process being finitely determined, see [46] for a definition, proved that a transformation T was Bernoulli IFF it had a finitely-determined generator IFF every partition was finitely-determined with respect to T, and showed that entropy completely classified the finitely-determined processes up to isomorphism. This seminal result led to a vast machinery for proving transformations to be Bernoulli, as well as classification and structure theorems [47,51,53]. Showing that the class of K-automorphisms far exceeds the Bernoulli maps, Ornstein and Shields produced in [48] an uncountable family of non-isomorphic K-automorphisms all with the same entropy.

And the topological entropy of T is Etop (T) :D supV Etop (T; V ) ;

(21)

taken over all open covers V. Thus Etop counts, in some sense, the growth rate in the number of T-orbits of length n. Evidently, topological entropy is an isomorphism invariant. Two continuous maps T : X ! X and S : Y ! Y are topologically conjugate (as isomorphism is called in this category) if there exists a homeomorphism : X ! Y with T D S . Lemma 12 (Subadditive Lemma) Consider a sequence s D (s l )1 1 [1; 1] satisfying s kCl  s k C s l , for all k; l 2 Z. Then the following limit exists in [1; 1], and lim n!1 snn D inf n snn . Topological entropy, or “top ent” for short, satisfies many of the relations of measure-entropy. Lemma 13

Topological Entropy

(a) V 4 W implies

Adler, Konheim and McAndrew, in 1965, published the first definition of topological entropy in the eponymous article [33]. Here, T : X ! X is a continuous self-map of a compact topological space. The role of atoms is played by open sets. Instead of a finite partition, one uses a finite16 open-cover V D fU j gLjD1 , i. e. each patch U j is open, and S their union (V) D X. (Henceforth, ‘cover’ means “open cover”.) Let Card(V) be the minimum cardinality over all subcovers. [ ˚ (V0 ) D X ; Card(V) :D Min ] V0 j V0 V and and let H (V ) D Htop (V ) :D log(Card(V )) :

Analogous to the definitions for partitions, define V _ W :D fV \ W j V 2 V and W 2 W g ;

T V :D fT 1 (U) j U 2 Vg and V[0:::n) :D V0 _ V1 _    _ V n1 ; W < V ; if each W -patch is a subset of some V -patch:

The T; V-entropy is E T (V ) D E(T; V ) D Etop (T; V )

  1 :D lim sup Htop V[0:::n) : n!1 n

(20)

16 Because we only work on a compact space, we can omit “finite”. Some generalizations of topological entropy to non-compact spaces require that only finite open-covers be used [37].

H (V )  H (W )

and E(T; V )  E(T; W ) :

(b) H (V _ W )  H (V) C H (W ). (c) H (T(V))  H (V), with equality if T is surjective. Also, E(T; V )  H (V).   (d) In Eq. (20), the limn!1 n1 H V[0:::n) exists. (e) Suppose T is a homeomorphism. Then E(T 1 ; V ) D E(T; V ) ;

for each cover V. Consequently, Etop (T 1 ) D Etop (T). (f) Suppose C is a collection of covers such that: For each cover W , there exists a V 2 C with V < W . Then Etop (T) equals the supremum of Etop (T; V ), just taken over those V 2 C. (g) For all ` 2 N:   Etop T ` D `Etop (T) : Proof (of (c)) Let C 4 V be a min-cardinality subcover. Then T C is a subcover of T V. So Card T V  jT Cj D jCj. As for entropy, inequality (b) and the foregoing give H (V [0:::n) )  H (V )n.  Proof (of (d)) Set s n :D H (V[0:::n) ). Then    s kCl  s k C H T k V[0:::l )  s k C s l ; by (b) and (c), and so the Subadditive Lemma 12, applies. 

215

216

Entropy in Ergodic Theory

Proof (of (g)) WLOG, ` D 3. Given V a cover, triple it to b V :D V \ T V \ T 2 V ; so _  j   _ T3 b V D T i (V) : j2[0:::N)

i2[0:::3N)

  Thus H T 3 ; b V ; N D H (T; V ; 3N), extending notation. Part (d) and sending N ! 1, gives E(T 3 ; b V) D 3H (T; V). Lastly, take covers such that 









E T 3 ; C(k) ! Etop T 3



and

E T; D(k) ! Etop (T) ;

as k!1. Define V(k) :D C(k) _ D(k) . Apply the above to V (k) , then send k!1. 

You Take the High Road and I’ll Take the Low Road There are several routes to computing top-ent, some via maximization, others, minimization. Our foregoing discussion computed Etop (T) by a family of sizes f k (n) D f kT (n), depending on a parameter k which specifies the fineness of scale. (In Sect. “Metric Preliminaries”, this k is an integer; in the original definition, an open cover.) Define two numbers: 1 b L f (k) :D lim sup log f k (n) and n!1 n 1 L f (k) :D lim inf log f k (n) : (22) n!1 n L f (k). If the limit exists Finally, let E f (T) :D sup k b f in Eq. (22) then agree to write L (k) for the common value. The A-K-M definition used the size fV (n) :D Card(V[0:::n) ), where Card(W ) :DMinimum cardinality of a subcover from W :

Using a Metric From now on, our space is a compact metric space (X; d). Dinaburg [36] and Bowen [34,35], gave alternative, equivalent, definitions of topological entropy, in the compact metric-space case, that are often easier to work with than covers. Bowen gave a definition also when X is not compact17 (see [35] and Chap. 7 in [23]). Metric Preliminaries An "-ball-cover comprises finitely many balls, all of radius ". Since our space is compact, every cover V has a Lebesgue number " > 0. I. e., for each z 2 X, the Bal(z; ") lies entirely inside at least one V-patch. (In particular, there is an "-ball-cover which refines V.) Let LEB(V) be the supremum of the Lebesgue numbers. Courtesy of Lemma 13f we can Fix a “universal” list V(1) 4 V(2) 4 : : : , with V(k) a 1k -ball-cover. For every T : X ! X, then, the lim k E(T; V(k) ) computes Etop (T). An "-Microscope Three notions are useful in examining a metric space (X; m) at scale ". Subset A X is an "separated-set, if m(z; z0 )  " for all distinct z; z0 2 A. Subset F X is "-spanning if 8x 2 X; 9z 2 F with m(x; z) < ". Lastly, a cover V is "-small if Diam(U) < ", for each U 2 V. 17 When X is not compact, the definitions need not coincide; e. g. [37]. And topologically-equivalent metrics, but which are not uniformly equivalent, may give the same T different entropies (see p. 171 in [23]).

Here are three metric-space sizes f" (n): Sep(n; ") :DMaximum cardinality of a dn –"-separated set: Spn(n; ") :DMinimum cardinality of a dn –"-spanning set: Cov(n; ") :DMinimum cardinality of a dn –"-small cover: These use a list (d n )1 nD1 of progressively finer metrics on X, where   d N (x; y) :D Max j2[0:::N) d T j (x); T j (y) : Theorem 14 (All-Roads-Lead-to-Rome Theorem) Fix " and let W be any d-"-small cover. Then 8n : Cov(n; 2")  Spn(n; ")  Sep(n; ")  Card(W [0:::n) ) : (ii) Take a cover V and a ı < LEB(V). Then 8n: Card(V[0:::n) )  Cov(n; ı) : (iii) The limit LCov (") D lim n n1 log(Cov(n; ")) exists in [0 : : : 1) : (iv)

(i)

ESep (T) D ESpn (T) D ECov (T) D ECard (T) by defn

D Etop (T) :

Proof (of (i)) Take F X, a min-cardinality dn -"-spanS ning set. So z2F Dz D X, where note

Dz :D dn -Bal(z; ") D

n1 \ jD0

   T  j Bal T j z; " :

Entropy in Ergodic Theory

This D :D fDz gz is a cover, and it is dn –2"–small. Thus Cov(n; 2")  jDj D jFj. For any metric, a maximal "-separated-set is automatically "-spanning; adjoin a putative unspanned point to get a larger separated set. Let A be a max-cardinality dn -"-separated set. Take C, a min-cardinality subcover of W [0:::n) . For each z 2 A, pick a C-patch Cz 3 z. Could some pair x; y 2 A pick T j the same C? Well, write C D n1 jD0 T (Wj ), with each Wj 2 W . For every j 2 [0 : : : n), then, d(T j (x); T j (y))  Diam(Wj ) < " :

Proof (of (ii)) Choose a min-cardinality d n -ı-small cover C. For each C 2 C and j 2 [0 : : : n), the d-Diam(T j C) < ı. So there is a V-patch VC; j  T j (C). Hence note

n1 \

  T  j VC; j  C :

jD0

Thus V[0:::n) 4 C. So   Card V[0:::n)  Card(C)  jCj D Cov(n; ı) :



Proof (of (iii)) To upper-bound Cov(k C l; ") let V and W be min-cardinality "-small covers, respectively, for metrics d k and d l . Then V \ T l (W ) is a "-small for d kCl . Consequently Cov(kCl; ")  Cov(k; ")  Cov(l; "). Thus n 7! log(Cov(n; ")) is subadditive.  Proof (of (iv)) Pick a V from the list in Sect. “Metric Preliminaries”, choose some 2" < LEB(V) followed by an "-small W from Sect. “Metric Preliminaries”. Pushing n ! 1 gives LCard (V)  LCov (2") 

b LSep (") LSpn (")  b LSpn (")  LSep (")

1 log(Sep(n; ")) (24) n limit exists, in arguments that subsequently send "&0. Ditto for LSpn ("). LSep (") D lim

n!1

This will be used during the proof of the Variational Principle. But first, here are two entropy computations which illustrate the efficacy in having several characterizations of topological entropy. Etop (Isometry) D 0

Hence d n (x; y) < "; so x D y. Accordingly, the z 7! Cz map is injective, whence jAj  jCj. 

V [0:::n) 3

zero as "&0. Consequently, we can pretend that the

 LCard (W ) : (23)

Now send V and W along the list in Sect. “Metric Preliminaries”.  Pretension Topological entropy takes its values in [0; 1]. A useful corollary of Eq. (23) can be stated in terms of any Distance(; ) which topologizes [0; 1] as a compact interval. For each continuous T : X  ! X on a compact metric-space, the Distance b LSep ("); LSep (") goes to

Suppose (T : X; d) is a distancepreserving map of a compact metric-space. Fixing ", a set is dn -"-separated IFF it is d-"-separated. Thus Sep(n; ") does not grow with n. So each b LSep (") is zero. Topological Markov Shifts Imagine ourselves back in the days when computer data is stored on large reels of fast-moving magnetic tape. One strategy to maximize the density of binary data stored is to not put timing-marks (which take up space) on the tape. This has the defect that when the tape-writer writes, say, 577 consecutive 1-bits, then the tape-reader may erroneously count 578 copies of 1. We sidestep this flaw by first encoding our data so : :1 word, then writing to tape. as to avoid the 11:577 Generalize this to a finite alphabet Q and a finite list F of disallowed Q-words. Extend each word to a common length K C 1; now F QKC1 . The resulting “K-step TMS (topological Markov shift)” is the shift on the set of doubly-1 Q-names having no substring in F . In the above magnetic-tape example, K D 576. Making it more realistic, suppose that some string of zeros, say 00:574 : :0, is also forbidden18 . Extending to length 577, we get 23 D8 new disallowed words of form 00:574 : :0b1 b2 b3 . We recode to a 1-step TMS (just called a TMS or a subshift of finite type) over the alphabet P :D QK . Each outlawed Q-word w0 w1    w K engenders a length-2 forbidden P-word (w0 ; : : : ; w K1 )(w1 ; : : : ; w K ). The resulting TMS is topologically conjugate to the original K-step. The allowed length-2 words can be viewed as the edges in a directed-graph and the set of points x 2 X is the set of doubly-1 paths through the graph. Once trivialities removed, this X is a Cantor set and the shift T : X ! X is a homeomorphism. The Golden Shift As the simplest example, suppose our magnetic-tape is constrained by the Markov graph, Fig. 5 that we studied measure-theoretically in Fig. 4. 18 Perhaps the ;-bad-length, 574, is shorter than the 1-bad-length because, say, ;s take less tape-space than 1s and so – being written more densely – cause ambiguity sooner.

217

218

Entropy in Ergodic Theory

Entropy in Ergodic Theory, Figure 5 Ignoring the labels on the edges, for the moment, the Golden shift, T, acts on the space of doubly-infinite paths through this graph. The space can be represented as a subset XGold  fa; bgZ , namely, the set of sequences with no two consecutive b letters.

We want to store the text of The Declaration of Independence on our magnetic tape. Imagining that English is a stationary process, we’d like to encode English into this Golden TMS as efficiently as possible. We seek a shift-invariant measure  on XGold of maximum entropy, should such exist. View PDfa; bg as the time-zero partition on X Gold ; that is, name xD: : : x1 x0 x1 x2 : : :, is in atom b IFF letter x0 is “b”. Any measure  gives conditional probabilities (aja) D: s ; note

note

1 H (s; 1  s) 2s 1  slog(s) C (1  s)log(1  s) : D 2s

f (s) :D

(25)

Certainly f (0) D f (1) D 0, so f ’s maximum occurs at 0 s) the (it turns out) unique pointb ps where the derivative f (b equals zero. Thisb s D (1 C 5)/2. Plugging in, the maximum entropy supportable by the Golden Shift is p



 p 1 C 5 p  # 2 3 5 C log p : (26) 2 3 5 1 C 2

5

log

2

Exponentiating, the number of -typical n-names grows like Gn , where 2 G :D 4

2 1 C

p

p 1C p 5 5 5

5

1 ; 1 C jmj

(28)

0 . for the smallest jmj with x m ¤ x m

Lemma 15 Consider a subshift X. Then the 1 log(Names X (n)) n

exists in [0; 1], and equals Etop (X).

But recall, E(T) D H (P1 jP[1:::0) )  H (P1 jP0 ). So among all measures that make the conditional distribution Pja equal (s; c), the unique one maximizing entropy is the (s; c)-Markov-process. Its entropy, derived in Eq. (14), is

"

d(x; x 0 ) :D

n!1

(bjb) D 0 :

2 p MaxEnt D 5 5

Top-ent of the Golden Shift For a moment, let’s work more generally on an arbitrary subshift (a closed, shiftinvariant subset) X QZ , where Q is a finite alphabet. Here, the transformation is always the shift – but the space is varying – so agree to refer to the top-ent as Etop (X). Let ˚ Names X (n) be the number of distinct words in the set x [0:::n) j x 2 X . Note that a metric inducing the product-topology on QZ is

lim

(bja) D: c ;

(ajb) D 1 ;

This expression19 looks unpleasant to simplify – it isn’t even obviously an algebraic number – and yet topological entropy will reveal its familiar nature. This, because the Variational Principle (proved in the next section) says that the top-ent of a system is the supremum of measure-entropies supportable by the system.

32

p 3 3p5 2 5 5 54 5: p 3 5

(27)

Proof With " 2 (0; 1) fixed, two n-names are d n -"-separated IFF they are not the same name. Hence Sep(n; ") D  Names X (n). To compute Etop (X Gold ), declare that a word is “golden” if it appears in some x 2 XGold . Each [n C 1]-golden word ending in a has form wa, where w is n-golden. An [n C 1]golden word ending in b, must end in ab and so has form wab, where w is [n  1]-golden. Summing up, NamesXGold (n C 1) D NamesXGold (n) C Names XGold (n  1) : This is the Fibonacci recurrence, and indeed, these are the Fibonacci numbers, since NamesXGold (0) D 1 and Names XGold (1) D 2. Consequently, we have that NamesXGold (n) Const  n ; p

where  D 1C2 5 is the Golden Ratio. So the sesquipedalian number G from Eq. (27) is simply , and Etop (X Gold ) D log(). Since log()  0:694, each thousand bits written on tape (subject to the “no bb substrings” constraint) can carry at most 694 bits of information. 19 A popular computer-algebra-system was not, at least under my inexpert tutelage, able to simplify this. However, once top-ent gave the correct answer, the software was able to detect the equality.

Entropy in Ergodic Theory

Top-ent of a General TMS A (finite) digraph G engenders a TMS T : XG ! XG , as well as a f0; 1g-valued adjacency matrix ADAG , where a i; j is the number of directed-edges from state i to j. (Here, each a i; j is 0 or 1.) The (i; j)-entry in power An is automatically the number of length-n paths from i to j. Employing the matrix-norm P kMk :D i; j jm i; j j, then, kAkn D NamesX (n) : Happily Gelfand’s formula (see 10.13 in [58] or Spectral_ radius in [60]) applies: For an arbitrary (square) complex matrix, 1

lim kAn k n D SpecRad(A) :

n!1

(29)

This right hand side, the spectral radius of A, means the maximum of the absolute values of A’s eigenvalues. So the top-ent of a TMS is thus the Etop (X G ) D SpecRad(AG )

˚ ˇ :D Max jej ˇ e is an eigenvalue of AG : (30)

The (a; b)-adjacency matrix of Fig. 5 is

 1 1 ; 1 0 whose eigenvalues are  and

1  .

Labeling Edges Interpret (s; c; 1) simply as edge-labels in Fig. 5. The set of doubly-1 paths can also be viewed as a subset YGold fs; c; 1gZ , and it too is a TMS. The shift on YGold is conjugate (topologically isomorphic) to the shift on XGold , so they a fortiori have the same topent, log(). The (s; c; 1)-adjacency matrix is 2 3 1 1 0 40 0 15 : 1 1 0 Its j  j-largest eigenvalue is still , as it must. Now we make a new graph. We modify Fig. 5 by manufacturing a total of two s-edges, seven c-edges, and three edges 11 ; 12 ; 13 . Give these 2 C 7 C 3 edges twelve distinct labels. We could compute the resulting TMS-entropy from the corresponding 12 12 adjacency matrix. Alternatively, look at the (a; b)-adjacency matrix

 2 7 A :D : 3 0 p The roots of its characteristic polynomial arep1 ˙ 22. Hence Etop of this 12-symbol TMS is log(1 C 22).

The Variational Principle Let M :D M (X; d) be the set of Borel probability measures, and M (T) :D M (T : X; d) the set of T-invariant  2 M . Assign EntSup(T) :D sup fE (T) j  2 M (T)g : Theorem 16 (Variational EntSup(T) D Etop (T).

principle

(Goodson))

This says that top-ent is the top entropy – if there is a measure  which realizes the supremum. There doesn’t have to be. Choose a sequence of metric-systems (S k : Yk ; m k ) whose entropies strictly increase Etop (S k ) % L to some limit in (0; 1]. Let (S1 : Y1 ; m1 ) be the identity-map on a 1-point space. Define a new system (T : X; d), where F X :D k2[1:::1] Yk . Have T(x) :D S k (x), for the unique k with Yk 3 x. As for the metric, on Yk let d be a scaled version of m k , so that the d-Diam(Yk ) is less than 1/2 k . Finally, for points in distinct ˇcomponents, ˇ x 2 Yk and z 2 Y` , decree that d(x; z) :D ˇ2k  2` ˇ. Our T is continuous, and is a homeomorphism if each of the S k is. Certainly Etop (T) D L > Etop (S k ), for every k 2 [1 : : : 1]. If L is finite then there is no measure  of maximal entropy; for  must give mass to some Yk ; this pulls the entropy below L, since there are no compensatory components with entropy exceeding L. In contrast, when L D 1 then there is a maximalentropy measure (put mass 1/2 j on some component Yk j , where k j %1 swiftly); indeed, there are continuum-many maximal-entropy measures. But there is no 20 ergodic measure of maximal entropy. For a concrete L D 1 example, let S k be the shift on [1 : : : k]Z . Topology on M Let’s arrange our tools for establishing the Variational Principle. The argument will follow Misiurewicz’s proof, adapted from the presentations in [23] and [11]. Equip M with the weak- topology. 21 An A X is nice if its topological boundary @(A) is -null. And a partition is -nice if each atom is. 20 The ergodic measures are the extreme points of M(T); call them MErg (T). This M(T) is the set of barycenters obtained from Borel probability measures on MErg (T) (see Krein-Milman_ theorem, Choquet_theory in [60]). In this instance, what explains the failure to have an ergodic maximal-entropy measure? Let k be an invariant ergodic measure on Yk . These measures do converge to the one-point (ergodic) probability measure 1 on Y1 . But the map  7! E (T) is not continuousR at 1 . R 21 Measures ˛ !  IFF f d˛L ! f d, for each continuL ous f : X ! R. This metrizable topology makes M compact. Always, M(T) is a non-void compact subset (see  Measure Preserving Systems.)

219

220

Entropy in Ergodic Theory

Proposition 17 If ˛L !  and A X is -nice, then ˛L (A) ! (A). Proof Define operator U (D) :D lim supL ˛L (D). It suffices to show that U (A)  (A). For since Ac is -nice too, then U (Ac )  (Ac ). Thus limL ˛L (A) exists, and equals (A). Because C :D A is closed, the continuous functions f N & 1C pointwise, where f N (x) :D 1  Min(N  d(x; C); 1) : By the Monotone Convergence theorem, then, Z N f N d ! (C) : And (C) D (A), since AR is nice. Fixing N, then, it suffices to establish U (A)  f N d. But f N is continuous, so Z Z f N d D lim sup f N d˛L L!1 Z  lim sup 1A d˛L D U (A) :  L!1

Corollary 18 Suppose ˛L ! , and partition P is -nice. Then H˛L (P) ! H (P). The diameter of partition P is MaxA2P Diam(A). Proposition 19 Take  2 M and " > 0. Then there exists a -nice partition with Diam(P) < ". Proof Centered at an x, the uncountably many balls fBal(x; r) j r 2 (0; ")g have disjoint boundaries. So all but countably many are -nice; pick one and call it B x . Compactness gives a finite nice cover, say, fB1 ; : : : ; B7 g, at different centers. Then the partition P :D (A1 ; : : : ; A7 ) is S  nice, 22 where A k :D B k X k1 jD1 B j . Here is a consequence of Jensen’s inequality. Lemma 20 (Distropy-Averaging Lemma) For ;  2 M , a partition R, and a number t 2 [0; 1], t H (R) C t c H (R)  H tCt c  (R) : Strategy for EntSup(T)  Etop (T). Choose an " > 0. For L D 1; 2; 3; : : : , take a maximal (L; ")–separated–set FL X, then define 1 F D F" :D lim sup log(jFL j) : L!1 L 22 For any two sets B; B 0 X, the union @B [ @B 0 is a superset of the three boundaries @(B [ B0 ); @(B \ B0 ); @(B X B0 ).

Let 'L () be the equi-probable measure on FL ; each point has weight 1/jFL j. The desired invariant measure  will come from the Cesàro averages 1 X T ` 'L ; ˛L :D L `2[0:::L)

which get more and more invariant. Lemma 21 Let  be any weak- accumulation point of the above f˛L g1 1 . (Automatically,  is T-invariant.) Then E (T)  F. Indeed, if Q is any -nice partition with Diam(Q) < ", then E (T; Q)  F. Tactics As usual, Q[0:::N) means Q0 _Q1 _: : :_Q N1 . Our goal is 8N :

?

F

  1 H Q[0:::N) : N

(31)

Fix N and P :D Q[0:::N) , and a ı > 0. It suffices to verify: 8large L  N, ? 1 1 log(jFL j)  ı C H˛L (P) ; L N

(32)

since this and Corollary 17 will prove Eq. (31): Pushing L ! 1 along the sequence that produced  essentially sends LhS(32) to F, courtesy Eq. (24). And RhS(32) goes to ı C N1 H (P), by Corollary 17, since P is -nice. Descending ı & 0, hands us the needed Eq. (31). Remark 22 The idea in the following proof is to mostly fill interval [0::L) with N-blocks, starting with a offset K 2 [0::N). Averaging over the offset will create a Cesàro average over each N-block. Averaging over the N-blocks will allow us to compute distropy with respect to the averaged measure, ˛L . Proof (of Eq. (32)) Since L is frozen, agree to use ' for the ' L probability measure. Our dL -"-separated set F L has at most one point in any given atom of Q[0:::L) , thereupon   log(jFL j) D H' Q[0:::L) : Regardless of the “offset” K 2 [0 : : : N), we can always fit C :D b LN N c many N-blocks into [0 : : : L). Denote by G(K) :D [K : : : K C CN), this union of N-blocks, the good set of indices. Unsurprisingly, B(K) :D [0 : : : L) X G(K) is the bad index-set. Therefore, Bad(K)





‚  …„ _

H' Q[0:::L)  H'

j2B(K)

ƒ 

Good(K)

‚  …„ _

Q j C H'

ƒ  Qj

:

j2G(K)

(33)

Entropy in Ergodic Theory

this last inequality, when n is large. The upshot: E (T)  2 C Etop (T). Applied to a power T ` , this asserts that E (T ` )  2 C Etop (T ` ). Thus

Certainly Bad(K)  3Nlog(jQj). So 1 NL

X

Bad(K) 

K2[0:::N)

3N log(jQj) : L

This is less than ı, since L is large. Applying to Eq. (28) now produces 1 1 X Good(K) : log(jFL j)  ı C L NL

1 NL

P

K2[0:::N)

(34)

2 C Etop (T) ; ` using Lemma 9d and using Lemma 13g. Now coax ` ! 1.  E (T) 

K

Note _

j

T (Q) D

j2G(K)

Three Recent Results

_

:

c2[0:::C)T KCcN (P)

So Good(K) 

X

Having given an survey of older results in measure-theoretic entropy and in topological entropy, let us end this survey with a brief discussion of a few recent results, chosen from many.

H' (T KCc N P) :

c

P

This latter, by definition, equals c H T KCcN(') (P). We conclude that 1 X 1 XX Good(K)  H T KCcN ' (P) NL NL c K K 1 X H T ` ' (P) ;  NL `2[0:::L)

by adjoining a few translates of P; 1  H˛L (P) ; N by the Distropy-averaging lemma 18; P since ˛L is the average L1 ` T ` '. Thus Eq. (34) implies Eq. (32), our goal.  Proof (of EntSup(T)  Etop (T)) Fix a T-invariant . For partition Q D (B1 ; : : : ; B K ), choose a compact set A k B k with (B k X A k ) small. (This can be done, since F is automatically a regular measure [58].) Letting c and P :D (D; A1 ; : : : ; A K ), we can have D :D i Ai made H (PjQ) as small as desired. Courtesy of Lemma 8b, then, we only need consider partitions of the form that P has. Open-cover V D (U1 ; : : : ; U K ) has patches U k :D D[ A k . What atoms of, say, P[0:::3) , can the intersection U9 \ T 1 (U2 ) \ T 2 (U5 ) touch? Only the eight atoms (D or A9 ) \ T 1 (D or A2 ) \ T 2 (D or A5 ) : Thus ] P[0:::n)  2n  ] V[0:::n) . (Here, ] () counts the number of non-void atoms/patches.) So     1 1 H P[0:::n)  1 C log ] V [0:::n) n n  1 C 1 C Etop (T) ;

Ornstein–Weiss: Finitely-Observable Invariant In a landmark paper [10], Ornstein and Weiss show that all “finitely observable” properties of ergodic processes are secretly entropy; indeed, they are continuous functions of entropy. This was generalized by Gutman and Hochman [9]; some of the notation below is from their paper. Here is the setting. Consider an ergodic process, on a non-atomic space, taking on only finitely many values in N; let C be some family of such processes. An observation scheme is a metric space (˝; d) and a sequence of n functions S D (S n )1 1 , where Sn maps N  : : :  N into ˝. 1 On a point xE 2 N , the scheme converges if n 7! S n (x1 ; x2 ; : : : x n )

(35)

converges in ˝. And on a particular process X , say that S converges, if S converges on a.e. xE in X . A function J : C ! ˝ is isomorphism invariant if, whenever the underlying transformations of two processes X ; X 0 2 C are isomorphic, then J(X ) D J(X 0 ). Lastly, say that S “converges to J”, if for each X 2 C, scheme S converges to the value J(X ). The work of David Bailey [38], a student of Ornstein, produced an observation scheme for entropy. The Lempel-Ziv algorithm [43] was another entropy observer, with practical application. Ornstein and Weiss provided entropy schemes in [41] and [42]. Their recent paper “Entropy is the only finitelyobservable invariant” [10], gives a converse, a uniqueness result. Theorem 23 (Ornstein, Weiss) Suppose J is a finitely observable function, defined on all ergodic finite-valued processes. If J is an isomorphism invariant, then J is a continuous function of the entropy.

221

222

Entropy in Ergodic Theory

Gutman–Hochman: Finitely-Observable Extension Extending the Ornstein–Weiss result, Yonatan Gutman and Michael Hochman, in [9] proved that it holds even when the isomorphism invariant, J, is well-defined only on certain subclasses of the set of all ergodic processes. In particular they obtain the following result on three classes of zero-entropy transformations. Theorem 24 (Gutman, Hochman) Suppose J() is a finitely observable invariant on one of the following classes: (i)

The Kronecker systems; the class of systems with pure point spectrum. (ii) The zero-entropy mild mixing processes. (iii) The zero-entropy strong mixing processes. Then J() is constant.

Consider (G; G), a topological group and its Borel field (sigma-algebra). Let G  X be the field on G  X generated by the two coordinate-subfields. A map (36)

is measurable if 1

The paper introduces a new isomorphism invariant, the “f invariant”, and shows that, for Bernoulli actions, the f invariant agrees with entropy, that is, with the distropy of the independent generating partition. Exodos Ever since the pioneering work of Shannon, and of Kolmogorov and Sinai, entropy has been front and center as a major tool in Ergodic Theory. Simply mentioning all the substantial results in entropy theory would dwarf the length of this encyclopedia article many times over. And, as the above three results (cherry-picked out of many) show, Entropy shows no sign of fading away. . . Bibliography

Entropy of Actions of Free Groups

 : GxX ! X

Theorem 25 (Lewis Bowen) Let G be a finite-rank free group. Then two Bernoulli G-actions are isomorphic IFF they have the same entropy.

(X ) G  X :

Use g (x) for (g; x). This map in Eq. (36) is a (measure-preserving) group action if 8g; h 2 G: g ı h D g h , and each g : X ! X is measure preserving. This encyclopedia article has only discussed entropy for Z-actions, i. e., when G D Z. The ergodic theorem, our definition of entropy, and large parts of ergodic theory, involve taking averages (of some quantity of interest) over larger and larger “pieces of time”. In Z, we typically use the intervals I n :D [0 : : : n). When G is Z  Z, we might average over squares I n  I n . The amenable groups are those which possess, in a certain sense, larger and larger averaging sets. Parts of ergodic theory have been carried over to actions of amenable groups, e. g. [49] and [55]. Indeed, much of the Bernoulli theory was extended to certain amenable groups by Ornstein and Weiss, [50]. The stereotypical example of a non-amenable group, is a free group (on more than one generator). But recently, Lewis Bowen [8] succeeded in extending the definition of entropy to actions of finite-rank free groups.

Historical 1. Adler R, Weiss B (1967) Entropy, a complete metric invariant for automorphisms of the torus. Proc Natl Acad Sci USA 57:1573– 1576 2. Clausius R (1864) Abhandlungen ueber die mechanische Wärmetheorie, vol 1. Vieweg, Braunschweig 3. Clausius R (1867) Abhandlungen ueber die mechanische Wärmetheorie, vol 2. Vieweg, Braunschweig 4. Kolmogorov AN (1958) A new metric invariant of transitive automorphisms of Lebesgue spaces. Dokl Akad Nauk SSSR 119(5):861–864 5. McMillan B (1953) The basic theorems of information theory. Ann Math Stat 24:196–219 6. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423,623–656 7. Sinai Y (1959) On the Concept of Entropy of a Dynamical System, Dokl Akad Nauk SSSR 124:768–771

Recent Results 8. Bowen L (2008) A new measure-conjugacy invariant for actions of free groups. http://www.math.hawaii.edu/%7Elpbowen/ notes11.pdf 9. Gutman Y, Hochman M (2006) On processes which cannot be distinguished by finitary observation. http://arxiv.org/pdf/ math/0608310 10. Ornstein DS, Weiss B (2007) Entropy is the only finitely-observable invariant. J Mod Dyn 1:93–105; http://www.math.psu. edu/jmd

Ergodic Theory Books 11. Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge 12. Cornfeld I, Fomin S, Sinai Y (1982) Ergodic theory. Grundlehren der Mathematischen Wissenschaften, vol 245. Springer, New York

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13. Friedman NA (1970) Introduction to ergodic theory. Van Nostrand Reinhold, New York 14. Halmos PR (1956) Lectures on ergodic theory. The Mathematical Society of Japan, Tokyo 15. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. (With a supplementary chapter by Katok and Leonardo Mendoza). Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge 16. Keller G, Greven A, Warnecke G (eds) (2003) Entropy. Princeton Series in Applied Mathematics. Princeton University Press, Princeton 17. Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 18. Mañé R (1987) Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete, ser 3, vol 8. Springer, Berlin 19. Parry W (1969) Entropy and generators in ergodic theory. Benjamin, New York 20. Petersen K (1983) Ergodic theory. Cambridge University Press, Cambridge 21. Rudolph DJ (1990) Fundamentals of measurable dynamics. Clarendon Press, Oxford 22. Sinai Y (1994) Topics in ergodic theory. Princeton Mathematical Series, vol 44. Princeton University Press, Princeton 23. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, New York

Differentiable Entropy 24. Ledrappier F, Young L-S (1985) The metric entropy of diffeomorphisms. Ann Math 122:509–574 25. Pesin YB (1977) Characteristic Lyapunov exponents and smooth ergodic theory. Russ Math Surv 32:55–114 26. Young L-S (1982) Dimension, entropy and Lyapunov exponents. Ergod Theory Dyn Syst 2(1):109–124

Finite Rank 27. Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1):35– 65 28. King JLF (1988) Joining-rank and the structure of finite rank mixing transformations. J Anal Math 51:182–227

Maximal-Entropy Measures 29. Buzzi J, Ruette S (2006) Large entropy implies existence of a maximal entropy measure for interval maps. Discret Contin Dyn Syst 14(4):673–688 30. Denker M (1976) Measures with maximal entropy. In: Conze J-P, Keane MS (eds) Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974. Lecture Notes in Mathematics, vol 532. Springer, Berlin, pp 70–112 31. Misiurewicz M (1973) Diffeomorphism without any measure with maximal entropy. Bull Acad Polon Sci Sér Sci Math Astron Phys 21:903–910

Topological Entropy 32. Adler R, Marcus B (1979) Topological entropy and equivalence of dynamical systems. Mem Amer Math Soc 20(219)

33. Adler RL, Konheim AG, McAndrew MH (1965) Topological entropy. Trans Am Math Soc 114(2):309–319 34. Bowen R (1971) Entropy for group endomorphisms and homogeneous spaces. Trans Am Math Soc 153:401–414. Errata 181:509–510 (1973) 35. Bowen R (1973) Topological entropy for noncompact sets. Trans Am Math Soc 184:125–136 36. Dinaburg EI (1970) The relation between topological entropy and metric entropy. Sov Math Dokl 11:13–16 37. Hasselblatt B, Nitecki Z, Propp J (2005) Topological entropy for non-uniformly continuous maps. http://www.citebase.org/ abstract?id=oai:arXiv.org:math/0511495

Determinism and Zero-Entropy, and Entropy Observation 38. Bailey D (1976) Sequential schemes for classifying and predicting ergodic processes. Ph D Dissertation, Stanford University 39. Kalikow S, King JLF (1994) A countably-valued sleeping stockbroker process. J Theor Probab 7(4):703–708 40. King JLF (1992) Dilemma of the sleeping stockbroker. Am Math Monthly 99(4):335–338 41. Ornstein DS, Weiss B (1990) How sampling reveals a process. Ann Probab 18(3):905–930 42. Ornstein DS, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inf Theory 39(1):78–83 43. Ziv J, Lempel A (1977) A universal algorithm for sequential data compression. IEEE Trans Inf Theory 23(3):337–343

Bernoulli Transformations, K-Automorphisms, Amenable Groups 44. Berg KR (1975) Independence and additive entropy. Proc Am Math Soc 51(2):366–370; http://www.jstor.org/stable/2040323 45. Meshalkin LD (1959) A case of isomorphism of Bernoulli schemes. Dokl Akad Nauk SSSR 128:41–44 46. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 5:337–352 47. Ornstein DS (1974) Ergodic theory randomness and dynamical systems, Yale Math Monographs, vol 5. Yale University Press, New Haven 48. Ornstein DS, Shields P (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88 49. Ornstein DS, Weiss B (1983) The Shannon–McMillan–Briman theorem for a class of amenable groups. Isr J Math 44(3):53– 60 50. Ornstein DS, Weiss B (1987) Entropy and isomorphism theorems for actions of amenable groups. J Anal Math 48:1–141 51. Shields P (1973) The theory of Bernoulli shifts. University of Chicago Press, Chicago 52. Sinai YG (1962) A weak isomorphism of transformations having an invariant measure. Dokl Akad Nauk SSSR 147:797– 800 53. Thouvenot J-P (1975) Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli. Isr J Math 21:177– 207 54. Thouvenot J-P (1977) On the stability of the weak Pinsker property. Isr J Math 27:150–162

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Abramov Formula 55. Ward T, Zhang Q (1992) The Abramov–Rohlin entropy addition formula for amenable group actions. Monatshefte Math 114:317–329

Miscellaneous 56. Newhouse SE (1989) Continuity properties of entropy. Ann Math 129:215–235 57. http://www.isib.cnr.it/control/entropy/ 58. Rudin W (1973) Functional analysis. McGraw-Hill, New York 59. Tribus M, McIrvine EC (1971) Energy and information. Sci Am 224:178–184 60. Wikipedia, http://en.wikipedia.org/wiki/. pages: http://en.

wikipedia.org/wiki/Spectral_radius, wiki/Information_entropy

http://en.wikipedia.org/

Books and Reviews Boyle M, Downarowicz T (2004) The entropy theory of symbolic extensions. Invent Math 156(1):119–161 Downarowicz T, Serafin J (2003) Possible entropy functions. Isr J Math 135:221–250 Hassner M (1980) A non-probabilistic source and channel coding theory. Ph D Dissertation, UCLA Katok A, Sinai YG, Stepin AM (1977) Theory of dynamical systems and general transformation groups with an invariant measure. J Sov Math 7(6):974–1065

Ergodicity and Mixing Properties

Ergodicity and Mixing Properties ANTHONY QUAS Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Article Outline Glossary Definition of the Subject Introduction Basics and Examples Ergodicity Ergodic Decomposition Mixing Hyperbolicity and Decay of Correlations Future Directions Bibliography Glossary Bernoulli shift Mathematical abstraction of the scenario in statistics or probability in which one performs repeated independent identical experiments. Markov chain A probability model describing a sequence of observations made at regularly spaced time intervals such that at each time, the probability distribution of the subsequent observation depends only on the current observation and not on prior observations. Measure-preserving transformation A map from a measure space to itself such that for each measurable subset of the space, it has the same measure as its inverse image under the map. Measure-theoretic entropy A non-negative (possibly infinite) real number describing the complexity of a measure-preserving transformation. Product transformation Given a pair of measure-preserving transformations: T of X and S of Y, the product transformation is the map of X  Y given by (T  S)(x; y) D (T(x); S(y)). Definition of the Subject Many physical phenomena in equilibrium can be modeled as measure-preserving transformations. Ergodic theory is the abstract study of these transformations, dealing in particular with their long term average behavior. One of the basic steps in analyzing a measure-preserving transformation is to break it down into its simplest possible components. These simplest components are its ergodic components, and on each of these components, the system enjoys the ergodic property: the long-term time

average of any measurement as the system evolves is equal to the average over the component. Ergodic decomposition gives a precise description of the manner in which a system can be split into ergodic components. A related (stronger) property of a measure-preserving transformation is mixing. Here one is investigating the correlation between the state of the system at different times. The system is mixing if the states are asymptotically independent: as the times between the measurements increase to infinity, the observed values of the measurements at those times become independent. Introduction The term ergodic was introduced by Boltzmann [8,9] in his work on statistical mechanics, where he was studying Hamiltonian systems with large numbers of particles. The system is described at any time by a point of phase space, a subset of R6N where N is the number of particles. The configuration describes the 3-dimensional position and velocity of each of the N particles. It has long been known that the Hamiltonian (i. e. the overall energy of the system) is invariant over time in these systems. Thus, given a starting configuration, all future configurations as the system evolves lie on the same energy surface as the initial one. Boltzmann’s ergodic hypothesis was that the trajectory of the configuration in phase space would fill out the entire energy surface. The term ergodic is thus an amalgamation of the Greek words for work and path. This hypothesis then allowed Boltzmann to conclude that the long-term average of a quantity as the system evolves would be equal to its average value over the phase space. Subsequently, it was realized that this hypothesis is rarely satisfied. The ergodic hypothesis was replaced in 1911 by the quasi-ergodic hypothesis of the Ehrenfests [17] which stated instead that each trajectory is dense in the energy surface, rather than filling out the entire energy surface. The modern notion of ergodicity (to be defined below) is due to Birkhoff and Smith [7]. Koopman [44] suggested studying a measure-preserving transformation by means of the associated isometry on Hilbert space, U T : L2 (X) ! L2 (X) defined by U T ( f ) D f ı T. This point of view was used by von Neumann [91] in his proof of the mean ergodic theorem. This was followed closely by Birkhoff [6] proving the pointwise ergodic theorem. An ergodic measure-preserving transformation enjoys the property that Boltzmann first intended to deduce from his hypothesis: that long-term averages of an observable quantity coincide with the integral of that quantity over the phase space.

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These theorems allow one to deduce a form of independence on the average: given two sets of configurations A and B, one can consider the volume of the phase space consisting of points that are in A at time 0 and in B at time t. In an ergodic measure-preserving transformation, if one computes the average of the volumes of these regions over time, the ergodic theorems mentioned above allow one to deduce that the limit is simply the product of the volume of A and the volume of B. This is the weakest mixing-type property. In this article, we will outline a rather full range of mixing properties with ergodicity at the weakest end and the Bernoulli property at the strongest end. We will set out in some detail the various mixing properties, basing our study on a number of concrete examples sitting at various points of this hierarchy. Many of the mixing properties may be characterized in terms of the Koopman operators operators mentioned above (i. e. they are spectral properties), but we will see that the strongest mixing properties are not spectral in nature. We shall also see that there are connections between the range of mixing properties that we discuss and measure-theoretic entropy. In measure-preserving transformations that arise in practice, there is a correlation between strong mixing properties and positive entropy, although many of these properties are logically independent. One important issue for which many questions remain open is that of higher-order mixing. Here, the issue is if instead of asking that the observations at two times separated by a large time T be approximately independent, one asks whether if one makes observations at more times, each pair suitably separated, the results can be expected to be approximately independent. This issue has an analogue in probability theory, where it is well-known that it is possible to have a collection of random variables that are pairwise independent, but not mutually independent. Basics and Examples In this article, except where otherwise stated, the measurepreserving transformations that we consider will be defined on probability spaces. More specifically, given a measurable space (X; B) and a probability measure  defined on B, a measure-preserving transformation of (X; B; ) is a B-measurable map T : X ! X such that (T 1 B) D (B) for all B 2 B. While this definition makes sense for arbitrary measures, not simply probability measures, most of the results and definitions below only make sense in the probability measure case. Sometimes it will be helpful to make the assumption that the underlying probability space is

a Lebesgue space (that is, the space together with its completed -algebra agrees up to a measure-preserving bijection with the unit interval with Lebesgue measure and the usual  -algebra of Lebesgue measurable sets). Although this sounds like a strong restriction, in practice it is barely a restriction at all, as almost all of the spaces that appear in the theory (and all of those that appear in this article) turn out to be Lebesgue spaces. For a detailed treatment of the theory of Lebesgue spaces, the reader is referred to Rudolph’s book [76]. The reader is referred also to the chapter on  Measure Preserving Systems. While many of the definitions that we shall present are valid for both invertible and non-invertible measure-preserving transformations, the strongest mixing conditions are most useful in the case of invertible transformations. It will be helpful to present a selection of simple examples, relative to which we will be able to explore ergodicity and the various notions of mixing. These examples and the lemmas necessary to show that they are measure-preserving transformations as claimed may be found in the books of Petersen [64], Rudolph [76] and Walters [92]. More details on these examples can also be found in the chapter on  Ergodic Theory: Basic Examples and Constructions. Example 1 (Rotation on the circle) Let ˛ 2 R. Let R˛ : [0; 1) ! [0; 1) be defined by R˛ (x) D x C ˛ mod 1. It is straightforward to verify that R˛ preserves the restriction of Lebesgue measure  to [0; 1) (it is sufficient to check that (R˛1 (J)) D (J) for an interval J). Example 2 (Doubling Map) Let M2 : [0; 1) ! [0; 1) be defined by M2 (x) D 2x mod 1. Again, Lebesgue measure is invariant under M 2 (to see this, one observes that for an interval J, M21 (J) consists of two intervals, each of half the length of J). This may be generalized in the obvious way to a map M k for any integer k  2. Example 3 (Interval Exchange Transformation) The class of interval exchange transformations was introduced by Sinai [85]. An interval exchange transformation is the map obtained by cutting the interval into a finite number of pieces and permuting them in such a way that the resulting map is invertible, and restricted to each interval is an order-preserving isometry. More formally, one takes a sequence of positive lengths `1 ; `2 ; : : : ; ` k summing to 1 and a permutation  P of f1; : : : ; kg and defines a i D j
Ergodicity and Mixing Properties

Example 4 (Bernoulli Shift) Let A be a finite set and fix a vector (p i ) i2A of positive numbers that sum to 1. Let AN denote the set of sequences of the form x0 x1 x2 : : : , where x n 2 A for each n 2 N and let AZ denote the set of biinfinite sequences of the form : : : x2 x1  x0 x1 x2 : : : (the  is a placeholder that allows us to distinguish (for example) between the sequences : : : 01010  10101 : : : and : : : 10101  01010 : : : ). We define a map (the shift map) S on AN by (S(x))n D x nC1 and define S on AZ by the same formula. Note that S is invertible as a transformation on AZ but non-invertible as a transformation on AN . We need to equip AN and AZ with measures. This is done by defining the measure of a preferred class of sets, checking certain consistency conditions and appealing to the Kolmogorov extension theorem. Here the preferred sets are the cylinder sets. Given m  n in the invertible case and a sequence a m : : : a n , we let [a m : : : a n ]nm denote fx 2 AZ : x m D a m ; : : : ; x n D a n g and define ([a m : : : a n ]nm ) D p a m p a mC1 : : : p a n . This is then shown to uniquely define a measure  on the -algebra of AZ generated by the cylinder sets. It is immediate to see that for any cylinder set C, (S 1 C) D (C), and it follows that S is a measure-preserving transformation of (AZ ; B; ). The construction is exactly analogous in the non-invertible case. See the chapter on  Measure Preserving Systems or the books of Walters [92] or Rudolph [76] for more details of defining measures in these systems.

of identical balls which move at constant velocity until two balls collide, whereupon they elastically swap momentum along the direction of contact. The phase space for this system is a region of R6N (with N 3-dimensional position vectors and N 3-dimensional velocity vectors). More abstractly, the system is equivalent to the motion of a single point particle in a region of R M  R M (with the first M-vector representing position and the second representing velocity). The system is constrained in that its position is required to lie in a bounded region S of R M with a piecewise smooth boundary. The system evolves by moving the position at a constant rate in the direction of the velocity vector until the point reaches @S, at which time the component of the velocity parallel to the normal to @S is reversed. This then defines a flow (i. e. a family of maps (Tt ) t2R satisfying TtCs D Tt ı Ts ) on the phase space. Since the magnitude of the velocity is conserved, it is convenient to restrict to flows with speed 1. This system is clearly the closest of the examples that we consider to the situation envisaged by Boltzmann. Perhaps not surprisingly, proofs of even the most basic properties for this system are much harder than those for the other examples that we consider.

The class of Bernoulli shifts will play a distinguished role in what follows.

1. g ı h and h ı g agree with the respective identity maps almost everywhere; 2. (g 1 F) D (F) and (h1 B) D (B) for all F 2 F and B 2 B; and 3. Sıg(x) D gıT(x) for -almost every x (or equivalently T ı h(y) D h ı S(y) for -almost every y).

Example 5 (Markov Shift) The spaces AN and AZ are exactly as above, as is the shift map. All that changes is the measure. To define a Markov shift, we need a stochastic matrix P (i. e. a matrix with non-negative entries whose rows sum to 1) with rows and columns indexed by A and a left eigenvector  for P with eigenvalue 1 with the property that the entries of  are non-negative and sum to 1. The existence of such an eigenvector is a consequence of the Perron–Frobenius theory of positive matrices. Provided that the matrix P is irreducible (for each a and a0 in A, there is n an n > 0 such that Pa;a 0 > 0), the eigenvector  is unique. Given the pair (P; ), one defines the measure of a cylinder set by ([a m : : : a n ]nm ) D  a m Pa m a mC1 : : : Pa n1 a n and extends  as before to a probability measure on AN or AZ . Example 6 (Hard Sphere Gases and Billiards) We wish to model the behavior of a gas in a bounded region. We make the assumption that the gas consists of a large number N

We will need to make use of the concept of measure-theoretic isomorphism. Two measure-preserving transformations T of (X; B; ) and S of (Y; F ; ) are measure-theoretically isomorphic (or just isomorphic) if there exist measurable maps g : X ! Y and h : Y ! X such that

Measure-theoretic isomorphism is the basic notion of ‘sameness’ in ergodic theory. It is in some sense quite weak, so that systems may be isomorphic that feel very different (for example, as we discuss later, the time one map of a geodesic flow is isomorphic to a Bernoulli shift). For comparison, the notion of sameness in topological dynamical systems (topological conjugacy) is far stronger. As an example of measure-theoretic isomorphism, it may be seen that the doubling map is isomorphic to the one-sided Bernoulli shift on f0; 1g with p0 D p1 D 1/2 (the map g takes an x 2 [0; 1) to the sequence of 0’s and 1’s in its binary expansion (choosing the sequence ending with 0’s, for example, if x is of the form p/2n ) and the inverse map h takes a sequence of 0’s and 1’s to the point in [0; 1) with that binary expansion.)

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Given a measure-preserving transformation T of a probability space (X; B; ), T is associated to an isometry of L2 (X; B; ) by U T ( f ) D f ı T. This operator is known as the Koopman Operator. In the case where T is invertible, the operator U T is unitary. Two measure-preserving transformations T and S of (X; B; ) and (Y; F ; ) are spectrally isomorphic if there is a Hilbert space isomorphism ‚ from L2 (X; B; ) to L2 (Y; F ; ) such that ‚ ı U T D U S ı ‚. As we shall see below, spectral isomorphism is a strictly weaker property than measure-theoretic isomorphism. Since in ergodic theory, measure-theoretic isomorphism is the basic notion of sameness, all properties that are used to describe measure-preserving systems are required to be invariant under measure-theoretic isomorphism (i. e. if two measure-preserving transformations are measure-theoretically isomorphic, the first has a given property if and only if the second does). On the other hand, we shall see that some mixing-type properties are invariant under spectral isomorphism, while others are not. If a property is invariant under spectral isomorphism, we say that it is a spectral property. There are a number of mixing type properties that occur in the probability literature (˛-mixing, ˇ-mixing, -mixing, -mixing etc.) (see Bradley’s survey [12] for a description of these conditions). Many of these are stronger than the Bernoulli property, and are therefore not preserved under measure-theoretic isomorphism. For this reason, these properties are not widely used in ergodic theory, although ˇ-mixing turns out to be equivalent to the so-called weak Bernoulli property (which turns out to be stronger than the Bernoulli property that we discuss in this article – see Smorodinsky’s paper [87]) and ˛-mixing is equivalent to strong-mixing. A basic construction (see the article on  Ergodic Theory: Basic Examples and Constructions) that we shall require in what follows is the product of a pair of measurepreserving transformations: given transformations T of (X; B; ) and S of (Y; F ; ), we define the product transformation T  S : (X  Y; B ˝ F ; ) by (T  S)(x; y) D (Tx; Sy). One issue that we face on occasion is that it is sometimes convenient to deal with invertible measure-preserving transformations. It turns out that given a non-invertible measure-preserving transformation, there is a natural way to uniquely associate an invertible measurepreserving transformation transformation sharing almost all of the ergodic properties of the original transformation. Specifically, given a non-invertible measure-preserving transformation T of (X; B; ), one lets X D f(x0 ; x1 ; : : : ) : x n 2 X and T(x n ) D x n1 for all ng, B

be the  -algebra generated by sets of the form A¯ n D ¯ A¯ n ) D (A) and T(x0 ; x1 ; : : : ) D fx¯ 2 X : x n 2 Ag, ( (T(x0 ); x0 ; x1 ; : : : ). The transformation T of (X; B; ) is called the natural extension of the transformation T of (X; B; ) (see the chapter on  Ergodic Theory: Basic Examples and Constructions for more details). In situations where one wants to use invertibility, it is often possible to pass to the natural extension, work there and then derive conclusions about the original non-invertible transformation. Ergodicity Given a measure-preserving transformation T : X ! X, if T 1 A D A, then T 1 Ac D Ac also. This allows us to decompose the transformation X into two pieces A and Ac and study the transformation T separately on each. In fact the same situation holds if T 1 A and A agree up to a set of measure 0. For this reason, we call a set A invariant if (T 1 AA) D 0. Returning to Boltzmann’s ergodic hypothesis, existence of an invariant set of measure between 0 and 1 would be a bad situation as his essential idea was that the orbit of a single point would ‘see’ all of X, whereas if X were decomposed in this way, the most that a point in A could see would be all of A, and similarly the most that a point in Ac could see would be all of Ac . A measure-preserving transformation will be called ergodic if it has no non-trivial decomposition of this form. More formally, let T be a measure-preserving transformation of a probability space (X; B; ). The transformation T is said to be ergodic if for all invariant sets, either the set or its complement has measure 0. Unlike the remaining concepts that we discuss in this article, this definition of ergodicity applies also to infinite measure-preserving transformations and even to certain non-measure-preserving transformations. See Aaronson’s book [1] for more information. The following lemma is often useful: Lemma 1 Let (X; B; ) be a probability space and let T : X ! X be a measure-preserving transformation. Then T is ergodic if and only if the only measurable functions f satisfying f ı T D f (up to sets of measure 0) are constant almost everywhere. For the straightforward proof, we notice that if the condition in the lemma holds and A is an invariant set, then 1A ı T D 1A almost everywhere, so that 1A is an a. e. constant function and so A or Ac is of measure 0. Conversely, if f is an invariant function, we see that for each ˛, fx : f (x) < ˛g is an invariant set and hence of measure 0

Ergodicity and Mixing Properties

or 1. It follows that f is constant almost everywhere. We remark for future use that it is sufficient to check that the bounded measurable invariant functions are constant. The following corollary of the lemma shows that ergodicity is a spectral property. Corollary 2 Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if 1 is a simple eigenvalue of U T . The ergodic theorems mentioned earlier due to von Neumann and Birkhoff are the following (see also the chapter on  Ergodic Theorems). Theorem 3 (von Neumann Mean Ergodic Theorem [91]) Let T be a measure-preserving transformation of the probability space (X; B; ). For f 2 L2 (X; B; ), let A N f D 1/N( f C f ı T C    C f ı T N1 ). Then for all f 2 L2 (X; B; ), A N f converges in L2 to an invariant function f . Theorem 4 (Birkhoff Pointwise Ergodic Theorem [6]) Let T be a measure-preserving transformation of the probability space (X; B; ). Let f 2 L1 (X; B; ). Let A N f be as above. Then for -almost every x 2 X, (A N f (x)) is a convergent sequence. Of these two theorems, the pointwise ergodic theorem is the deeper result, and it is straightforward to deduce the mean ergodic theorem from the pointwise ergodic theorem. The mean ergodic theorem was reproved very concisely by Riesz [71] and it is this proof that is widely known now. Riesz’s proof is reproduced in Parry’s book [63]. There have been many different proofs given of the pointwise ergodic theorem. Notable amongst these are the argument due to Garsia [23] and a proof due to Katznelson and Weiss [40] based on work of Kamae [35], which appears in a simplified form in work of Keane and Petersen [42]. If the measure-preserving transformation T is ergodic, then by virtue of Lemma 1, the limit functions appearing in the ergodic theorems are constant. One sees that the constant is simply the integral of f with respect to , so R that in this situation A N f (x) converges to f d in norm and pointwise almost everywhere, thereby providing a justification of Boltzmann’s original claim: for ergodic measure-preserving transformations, time averages agree with spatial averages. In the case where T is not ergodic, it is also possible to identify the limit in the ergodic theorems: we have f  D E( f jI ), where I is the -algebra of T-invariant sets. Note that the set on which the almost everywhere convergence in Birkhoff’s theorem takes place depends on the L1 function f that one is considering. Straightforward considerations show that there is no single full measure set

that works simultaneously for all L1 functions. In the case where X is a compact metric space, it is well known that C(X), the space of continuous functions on X with the uniform norm has a countable dense set, ( f n )n1 say. If the invariant measure  is ergodic, then for each n, there is aR set Bn of measure 1 such that for all x 2 B n , A N f n (x) ! T f n d. Letting B D n B n , one obtains a full Rmeasure set such that for all n and all x 2 B, A N f n (x) ! f n d. A simple approximation argument thenRshows that for all x 2 B and all f 2 C(X), A N f (x) ! f d. A point x with this property is said to be generic for . The observations above show that for an ergodic invariant measure , we have fx : x is generic for g D 1. If T is ergodic, but T n is not ergodic for some n, then one can show that the space X splits up as A1 ; : : : ; A d for some djn in such a way that T(A i ) D A iC1 for i < d and T(A d ) D A1 with T n acting ergodically on each Ai . The transformation T is totally ergodic if T n is ergodic for all n 2 N. One can check that a non-invertible transformation T is ergodic if and only if its natural extension is ergodic. The following lemma gives an alternative characterization of ergodicity, which in particular relates it to mixing. Lemma 5 (Ergodicity as a Mixing Property) Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if for all f and g in L2 , N 1 X h f ; g ı T n i ! h f ; 1ih1; gi: N nD0

P N1 In particular, if T is ergodic, then (1/N) nD0 (A \ T n B) ! (A)(B) for all measurable sets A and B. Proof Suppose that T is ergodic. Then the left-hand side P N1 of the equality is equal to h f ; (1/N) nD0 g ı T n i. The mean ergodic theorem shows that the second term conR verges in L2 to the constant function with value g d D hg; 1i, and the equality follows. Conversely, if the equation holds for all f and g in L2 , suppose that A is an invariant set. Let f D g D 1A . Then since g ı T n D 1A for all n, the left-hand side is h1A ; 1A i D (A). On the other hand, the right-hand side is (A)2 , so that the equation yields (A) D (A)2 , and (A) is either 0 or 1 as required. Taking f D 1A and g D 1B for measurable sets A and B gives the final statement.  We now examine the ergodicity of the examples presented above. Firstly, for the rotation of the circle, we claim that

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the transformation is ergodic if and only if the ‘angle’ ˛ is irrational. To see this, we argue as follows. If ˛ D p/q, then we see that f (x) D e 2 i qx is a non-constant R˛ invariant function, and hence R˛ is not ergodic. On the other hand, if ˛ is irrational, suppose f is a bounded measurable invariant function. Since f is bounded, it is an L2 function, and so f may be expressed in L2 as a Fourier seP 2 i nx . We then ries: f D n e n where e n (x) D e n2Z cP see that f ı R˛ D n2Z e 2 i n˛ c n e n . In order for f to be equal in L2 to f ı R˛ , they must have the same Fourier coefficients, so that c n D e 2 i n˛ c n for each n. Since ˛ is irrational, this forces c n D 0 for all n ¤ 0, so that f is constant as required. The doubling map and the Bernoulli shift are both ergodic, although we defer proof of this for the time being, since they in fact have the strong-mixing property. A Markov chain with matrix P and vector  is ergodic if and only if for all i and j in A with  i > 0 and  j > 0, there exists an n  0 with Pinj > 0. This follows from the ergodic theorem for Markov chains (which is derived from the Strong Law of Large Numbers) (see [18] for details). In particular, if the underlying Markov chain is irreducible, then the measure is ergodic. In the case of interval exchange transformations, there is a simple necessary condition on the permutation for irreducibility, namely for 1  j < k, we do not have f1; : : : ; jg D f1; : : : ; jg. Under this condition, Masur [49] and Veech [88] independently showed that for almost all values of the sequence of lengths (` i )1ik , the interval exchange transformation is ergodic. (In fact they showed the stronger condition of unique ergodicity: that the transformation has no other invariant measure than Lebesgue measure. This implies that Lebesgue measure is ergodic, because if there were a non-trivial invariant set, then the restriction of Lebesgue measure to that set would be another invariant measure). For the hard sphere systems, there are no results on ergodicity in full generality. Important special cases have been studied by Sinai [84], Sinai and Chernov [86], Krámli, Simányi and Szász [45], Simányi and Szász [81], Simnyi [79,80] and Young [95]. Ergodic Decomposition We already observed that if a transformation is not ergodic, then it may be decomposed into parts. Clearly if these parts are not ergodic, they may be further decomposed. It is natural to ask whether the transformation can be decomposed into ergodic parts, and if so what form does the decomposition take? In fact such a decomposition does exist, but rather than decompose the transforma-

tion, it is necessary to decompose the measure into ergodic pieces. This is known as ergodic decomposition. The set of invariant measures for a measurable map T of a measurable space (X; B) to itself forms a simplex. General functional analytic considerations (due to Choquet [14,15] – see also Phelps’ account [66] of this theory) mean that it is possible to write any member of the simplex as an integral-convex combination of the extreme points. Further, the extreme points of the simplex may be identified as precisely the ergodic invariant measures for T. It follows that any invariant probability measure  for T may be uniquely expressed in the form Z (A) D (A) dm() ; M erg (X;T)

where Merg (X; T) denotes the set of ergodic T-invariant measures on X and m is a measure on Merg (X; T). We will give a proof of this theorem in the special case of a continuous transformation of a compact space. Our proof is based on the Birkhoff ergodic theorem and the Riesz Representation Theorem identifying the dual space of the space of continuous functions on a compact space as the set of bounded signed measures on the space (see Rudin’s book [75] for details). We include it here because this special case covers many cases that arise in practice, and because few of the standard ergodic theory references include a proof of ergodic decomposition. An exception to this is Rudolph’s book [76] which gives a full proof in the case that X is a Lebesgue space. This is based on a detailed development of the theory of these spaces and builds measures using conditional exceptions. Kalikow’s notes [32] give a brief outline of a proof similar to that which follows. Oxtoby [62] also wrote a survey article containing much of the following (and much more besides). Theorem 6 Let X be a compact metric space, B be the Borel  -algebra,  be an invariant Borel probability measure and T be a continuous measure-preserving transformation of (X; B; ). Then for each x 2 X, there exists an invariant Borel measure x such that: R R R  1. For f 2 L1 (X; B; ), f d D f dx d(x); 2. Given f 2 L1 (X; R B; ), for -almost every x 2 X, one has A N f (x) ! f dx ; 3. The measure x is ergodic for -almost every x 2 X. Notice that conclusion (2) shows that x can be understood as the distribution on the phase space “seen” if one starts the system in an initial condition of x. This interpretation of the measures x corresponds closely with the ideas of Boltzmann and the Ehrenfests in the formulation

Ergodicity and Mixing Properties

of the ergodic and quasi-ergodic hypotheses, which can be seen as demanding that x is equal to  for (almost) all x. Proof The proof will be divided into 3 main steps: defining the measures x , proving measurability with respect to x and proving ergodicity of the measures. Step 1: Definition of x Given a function f 2 L1 (X; B; ), Birkhoff’s theorem states that for -almost every x 2 X, (A N f (x)) is convergent. It will be convenient to denote the limit by f˜(x). Let f1 ; f2 ; : : : be a sequence of continuous functions that is dense in C(X). For each k, there is a set Bk of x’s measure 1 for which (A n f k (x))1 nD1 is a convergent sequence. Intersecting these gives a set B of full measure such that for x 2 B, for each k  1, A n f k (x) is convergent. A simple approximation argument shows that for x 2 B and f an arbitrary continuous function, A n f (x) is convergent. Given x 2 B, define a map L x : C(X) ! R by L x ( f ) D f˜(x). This is a continuous linear functional on C(X), and hence by the Riesz Representation Theorem there exists a Borel R measure x such that f˜(x) D f dx for each f 2 C(X) and x 2 B. Since L x ( f )  0 when f is a non-negative function and L x (1) D 1, the measure x is a probability measure. Since L x ( f ı T) D L x ( f ) for f 2 C(X), one can check that x must be an invariant probability measure. For x 62 B, simply define x D . Since Bc is a set of measure 0, this will not affect any of the statements that we are trying to prove. Now for f continuous, we have A N f is a bounded seR to f d quence of functions withR A N f (x) converging x R almost everywhere and A N f d D f d since T is measure-preserving. It follows from the bounded convergence theorem that for f 2 C(X),  Z Z Z f d D f dx d(x) : (1) Step 2: Measurability of x 7! x (A) Lemma 7 Let C 2 B satisfy (C) D 0. Then x (C) D 0 for -almost every x 2 X. Proof Using regularity of Borel probability measures (see Rudin’s book [75] for details), there exist open sets U1  U2      C with (U k ) < 1/k. There exist continuous functions g k;m with (g k;m (x))1 mD1 increasing to 1U k everywhere R(e.Rg. g k;m (x) D min(1; m  d(x; U kc ))). By (1), we Rhave ( g k;m dx ) d(x) < 1/k for all k; m. Note that g k;m dx D lim n!1 A n g k;m (x) is a measurable function of x, so that using the monotone converR gence theorem (taking the limit in m), x ! 7 1 U k d x D R R x (U k ) is measurable and ( 1U k dx ) d(x)  1/k. T We now see that x 7! lim k!1 x (U k ) D x ( U k )

is measurable, and by monotone convergence we see R alsoT T x ( U k ) d(x) D 0. It follows that x ( U k ) D 0 T for -almost every x. Since U k  C, the lemma follows.  Given a set A 2 B, let f k be a sequence of continuous functions (uniformly bounded by 1) satisfying k f k  1A kL 1 () < 2n , so that in particular R f k (x) ! 1A (x) for -almost every x. For each k, x 7! f k dx D lim n!1 A n f k (x) is a measurable function. By Lemma 7, for -almost every x, f k ! 1A x -almost everywhere, so R that by the bounded convergence theorem limk!1 f k dx D x (A) for -almost every x. Since the limit of measurable functions is measurable, it follows that x 7! x (A) is measurable for any measurable set A 2 B. R This allows us to define a measure  by (A) D x (A) R d(x). RForR a bounded measurable function f , we have f d D ( f dx ) d(x). Since this agrees with R f d for continuous functions by (1), it follows that  D . Conclusion (1) of the theorem now follows easily. Given f 2 L1 (X), we let ( f k ) be a sequence of continuous functions such that k f k  f kL 1 () is summable. This implies that k f k  f kL 1 ( R for -almost evR x ) is summable ery x and in particular, f k dx ! f dx for almost every x. On the other hand, by the remark following the statement of Birkhoff ’s theorem, we have f˜k D E( f k jI ) so that k f˜  f˜k kL 1 () is summable and f˜k (x) ! f˜(x) for -almost every x. Combining these two statements, we see that for -almost every x, we have f˜(x) D lim f˜k (x) D lim k!1

k!1

Z

Z f k dx D

f dx :

This establishes conclusion (2) of the theorem. Step 3: Ergodicity of x We have shown how to disintegrate the invariant measure  as an integral combination of x ’s, and we have interpreted the x ’s as describing the average behavior starting from x. It remains to show that the x ’s are ergodic measures. Fix for now a continuous function f and a number 0 < < 1. Since A n f (x) ! f˜(x) -almost everywhere, there exists an N such that fx : jA N f (x) f˜(x)j > /2g < 3 /8. We now claim the following: R ˚  x : x fy : j f˜(y)  f dx j > g > < :

(2)

R To see this, note that fy : j f˜(y)  f dx j > g

fy : j f˜(y)A N f (y)j > /2g[fy : jA N f (y) f˜(x)j > /2g, R ˜ so that if x fy : j f (y)  f dx j > g > , then either

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x fy : j f˜(y) A N f (y)j > /2g > /2 or x fy : jA N f (y) f˜(x)j > /2g > /2. We show that the set of x’s satisfying each condition is small. 3 ˜ Firstly, R we have /8 > fy : j f (y)  A N f (y)j > /2g D x fy : j f˜(y)  A N f (y)j > /2g d(x), so that fx : x fy : j f˜(y)  A N f (y)j > /2g > /2g < 2 /4 < /2. For the second term, given c 2 R, let Fc (x) D jA N f (x)  cj and G(x) D F f˜(x) (x). Note that Z F f˜(x) (y) dx (y) D lim A n F f˜(x) (x) D lim A n G(x) n!1

n!1

(using the facts that y 7! F f˜(x) (y) is a continuous function and that since f˜(x) is R an invariant function, F f˜(x) (T k x) D G(T k x)). Since G(x) d(x) < 3 /8, it R follows that F f˜(x) (y) dx (y)  2 /4 except on a set of x’s of measure less than /2. Outside this bad set, we have x fy : jA N f (y)  f˜(x)j > /2g < /2 so that fx : x fy : jA N f (y)  f˜(x)j > /2g > /2g < /2 as required. This establishes our claim (2) above. Since > 0 is arbitrary, it follows that for each f 2 C(X), for R -almost every x, x -almost every y satisfies f˜(y) D f dx . As usual, taking a countable dense sequence ( f k ) in C(X), it ˜ is R the case that for all k and -almost every x, f k (y) D f k dx x -almost everywhere. Let the set of x’s with this property be D. We claim that for x 2 D, x is ergodic. Suppose not. Then let x 2 D and let J be an invariant set of x measure between ı and 1  ı for some ı > 0. Then by density of C(X) in L1 (x ), there exists an f k with k f k  1 J kL 1 (x ) < ı. Since 1 J is an invariant function, we have 1˜ J D 1 J . On the other hand, f˜k is a constant function. It follows that k f˜k  1˜ J kL 1 (x )  ı > k f k  1 J kL 1 (x ) . This contradicts the identification of the limit as a conditional expectation and concludes the proof of the theorem.  Mixing As mentioned above, ergodicity may be seen as an independence on average property. More specifically, one wants to know whether in some sense (A \ T n B) converges to (A)(B) as n ! 1. Ergodicity is the property that there is convergence in the Césaro sense. Weak-mixing is the property that there is convergence in the strong Césaro sense. That is, a measure-preserving transformation T is weak-mixing if N1 1 X j(A\ T n B)(A)(B)j ! 0 N nD0

as

N ! 1:

In order for T to be strong-mixing, we require simply (A \ T N B) ! (A)(B) as N ! 1. It is clear that strong-mixing implies weak-mixing and weak-mixing implies ergodicity. If T d is not ergodic (so that T d A D A for some A of measure strictly between 0 and 1), then j(T nd A \ A)  (A)2 j D (A)(1  (A)), so that T is not weak-mixing. An alternative characterization of weak-mixing is as follows: Lemma 8 The measure-preserving transformation T is weak-mixing if and only if for every pair of measurable sets A and B, there exists a subset J of N of density 1 (i. e. #(J \ f1; : : : ; Ng)/N ! 1) such that lim

n!1 n62 J

(A \ T n B) D (A)(B) :

(3)

By taking a countable family of measurable sets that are dense (with respect to the metric d(A; B) D (AB)) and taking a suitable intersection of the corresponding J sets, one shows that for a given weak-mixing measure-preserving transformation, there is a single set J N such that (3) holds for all measurable sets A and B (see Petersen [64] or Walters [92] for a proof). We show that an irrational rotation of the circle is not weak-mixing as follows: let ˛ 2 R n Q and let A be the interval [ 14 ; 34 ). There is a positive proportion of n’s in the natural numbers (in fact proportion 1/3) with the property that jT n ( 12 )  12 j < 16 . For these n’s (A \ T n A) > 13 , 1 . so that in particular j(A \ T n A)  (A)(A)j > 12 Clearly this precludes the required convergence to 0 in the definition of weak-mixing, so that an irrational rotation is ergodic but not weak-mixing. Since R˛n D R n˛ , the earlier argument shows that R˛n is ergodic, so that R˛ is totally ergodic. On the other hand, we show that any Bernoulli shift is strong-mixing. To see this, let A and B be arbitrary measurable sets. By standard measure-theoretic arguments, A and B may each be approximated arbitrarily closely by a finite union of cylinder sets. Since if A0 and B0 are finite unions of cylinder sets, we have that (A0 \ T n B0 ) is equal to (A0 )(B0 ) for large n, it is easy to deduce that (A \ T n B) ! (A)(B) as required. Since the doubling map is measure-theoretically isomorphic to a onesided Bernoulli shift, it follows that the doubling map is also strong-mixing. Similarly, if a Markov Chain is irreducible (i. e. for any states i and j, there exists an n  0 such that Pinj > 0) and aperiodic (there is a state i such that gcdfn : Pini > 0g D 1), then given any pair of cylinder sets A0 and B0 we have by standard theorems of Markov chains (A0 \ T n B0 ) !

Ergodicity and Mixing Properties

(A0 )(B0 ). The same argument as above then shows that an aperiodic irreducible Markov Chain is strong-mixing. On the other hand, if a Markov chain is periodic (d D gcdfn : Pini > 0g > 0), then letting A D B D fx : x0 D ig, we have that (A \ T n B) D 0 whenever d − n. It follows T d is not ergodic, so that T is not weak-mixing. Both weak- and strong-mixing have formulations in terms of functions: Lemma 9 Let T be a measure-preserving transformation of the probability space (X; B; ). 1. T is weak-mixing if and only if for every f ; g 2 L2 one has N1 1 X jh f ; g ı T n i  h f ; 1ih1; gij ! 0 as N ! 1 : N nD0

2. T is strong-mixing if and only if for every f ; g 2 L2 , one has h f ; g ı T N i ! h f ; 1ih1; gi as N ! 1 : Using this, one can see that both mixing conditions are spectral properties. Lemma 10 Weak- and strong-mixing are spectral properties. Proof Suppose S is a weak-mixing transformation of (Y; F ; ) and the transformation T of (X; B; ) is spectrally isomorphic to S by the Hilbert space isomorphism ‚. Then for f ; g 2 L2 (X; B; ), h f ; g ı T n i X  h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY  h‚( f ); ‚(1)iY h‚(1); ‚(g)iY . Since 1 is an eigenfunction of U T with eigenvalue 1, ‚(1) is an eigenfunction of U S with an eigenvalue 1, so since S is ergodic, ‚(1) must be a constant function. Since ‚ preserves norms, ‚(1) must have a constant value of absolute value 1 and hence h f ; g ı T n i X  h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY  h‚( f ); 1iY h1; ‚(g)iY . It follows from Lemma 9 that T is weak-mixing. A similar proof shows that strong-mixing is a spectral property.  Both weak- and strong-mixing properties are preserved by taking natural extensions. Recent work of Avila and Forni [4] shows that for interval exchange transformations of k  3 intervals with the underlying permutation satisfying the non-degeneracy condition above, almost all divisions of the interval (with respect to Lebesgue measure on the k1-dimensional simplex) lead to weak-mixing transformations. On the other

hand, work of Katok [36] shows that no interval exchange transformation is strong-mixing. It is of interest to understand the behavior of the ‘typical’ measure-preserving transformation. There are a number of Baire category results addressing this. In order to state them, one needs a set of measure-preserving transformations and a topology on them. As mentioned earlier, it is effectively no restriction to assume that a transformation is a Lebesgue-measurable map on the unit interval preserving Lebesgue measure. The classical category results are then on the collection of invertible Lebesgue-measure preserving transformations of the unit interval. One topology on these is the ‘weak’ topology, where a sub-base is given by sets of the form N(T; A; ) D fS : (S(A)T(A)) < g. With respect to this topology, Halmos [26] showed that a residual set (i. e. a dense Gı set) of invertible measurepreserving transformations is weak-mixing (see also work of Alpern [3]), while Rokhlin [72] showed that the set of strong-mixing transformations is meagre (i. e. a nowhere dense F  set), allowing one to conclude that with respect to this topology, the typical transformation is weak- but not strong-mixing. As often happens in these cases, even when a certain kind of behavior is typical, it may not be simple to exhibit concrete examples. In this case, a well-known example of a transformation that is weak-mixing but not strong-mixing was given by Chacon [13]. While on the face of it the formulation of weak-mixing is considerably less natural than that of strong-mixing, the notion of weak-mixing turns out to be extremely natural from a spectral point of view. Given a measure-preserving transformation T, let U T be the Koopman operator described above. Since this operator is an isometry, any eigenvalue must lie on the unit circle. The constant function 1 is always an eigenfunction with eigenvalue 1. If T is ergodic and g and h are eigenfunctions of U T with eigenvalue , then g h¯ is an eigenfunction with eigenvalue 1, hence invariant, so that g D Kh for some constant K. We see that for ergodic transformations, up to rescaling, there is at most one eigenfunction with any given eigenvalue. If U T has a non-constant eigenfunction f , then one has jhU Tn f ; f ij D k f k2 for each n, whereas by Cauchy– Schwartz, jh f ; 1ij2 < k f k2 . It follows that jhU Tn f ; f i  h f ; 1ih1; f ij  c for some positive constant c, so that using Lemma 9, T is not weak-mixing. Using the spectral theorem, the converse is shown to hold. Theorem 11 The measure-preserving transformation T is weak-mixing if and only U T has no non-constant eigenfunctions.

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Ergodicity and Mixing Properties

Of course this also shows that weak-mixing is a spectral property. Equivalently, this says that the transformation T is weak-mixing if and only if the apart from the constant eigenfunction, the operator U T has only continuous spectrum (that is, the operator has no other eigenfunctions). For a very nice and concise development of the part of spectral theory relevant to ergodic theory, the reader is referred to the Appendix in Parry’s book [63]. Using this theory, one can establish the following: Theorem 12 1. T is weak-mixing if and only if T  T is ergodic; 2. If T and S are ergodic, then T  S is ergodic if and only if U S and U T have no common eigenvalues other than 1. Proof The main factor in the proof is that the eigenvalues of U TS are precisely the set of ˛ˇ, where ˛ is an eigenvalue of U T and ˇ is an eigenvalue of U S . Further, the eigenfunctions of U TS with eigenvalue are spanned by eigenfunctions of the form f ˝ g, where f is an eigenfunction of U T , g is an eigenfunction of U S , and the product of the eigenvalues is . Suppose that T is weak-mixing. Then the only eigenfunction is the constant function, so that the only eigenfunction of U TT is the constant function, proving that T  T is ergodic. Conversely, if U T has an eigenvalue (so that f ı T D ˛T for some non-constant f ) then f ˝ f¯ is a non-constant invariant function of T  T so that T  T is not ergodic. For the second part, if U S and U T have a common eigenvalue other than 1 (say f ı T D ˛ f and g ı T D ˛g), then f ˝ g¯ is a non-constant invariant function. Conversely, if T  S has a non-constant invariant function h, then h can be decomposed into functions of the form f ˝g, where f and g are eigenfunctions of U T and U S respectively with eigenvalues ˛ and ˇ satisfying ˛ˇ D 1. Since the eigenvalues of S are closed under complex conjugation, we see that U T and U S have a common eigenvalue other than 1 as required.  For a measure-preserving transformation T, we let K be the subspace of L2 spanned by the eigenfunctions of U T . It is a remarkable fact that K may be identified as L2 (X; B0 ; ) where B0 is a sub--algebra of B. The space K is called the Kronecker factor of T. The terminology comes from the fact that any sub--algebra F of B gives rise to a factor mapping  : (X; B; ) ! (X; F ; ) with (x) D x. By construction L2 (X; B0 ; ) is the closed linear span of the eigenfunctions of T considered as a measure-preserving transformation of (X; B0 ; ). By the Discrete Spectrum Theorem of Halmos and von Neumann [27], T act-

ing on (X; B0 ; ) is measure-theoretically isomorphic to a rotation on a compact group. This allows one to split L2 (X; B; ) as L2 (X; B0 ; ) ˚ L2c (X; B; ), where, as mentioned above the first part is the discrete spectrum part, spanned by eigenfunctions, and the second part is the continuous spectrum part, consisting of functions whose spectral measure is continuous. Since we have split L2 into a discrete part and a continuous part, it is natural to ask whether the underlying transformation T can be split up in some way into a weak-mixing part and a discrete spectrum (compact group rotation) part, somewhat analogously to the ergodic decomposition. Unfortunately, there is nosuch decomposition available. However for some applications, for example to multiple recurrence (starting with the work of Furstenberg [20,21]), the decomposition of L2 (possibly into more complicated parts) plays a crucial role (see the chapters on  Ergodic Theory: Recurrence and  Ergodic Theory: Interactions with Combinatorics and Number Theory). For non-invertible measure-preserving transformations, the transformation is weak- or strong-mixing if and only if its natural extension has that property. The understanding of weak-mixing in terms of the discrete part of the spectrum of the operator also extends to total ergodicity. T n is ergodic if and only if T has no eigenvalues of the form e 2 i p/n other than 1. From this it follows that an ergodic measure-preserving transformation T is totally ergodic if and only if it has no rational spectrum (i. e. no eigenvalues of the form e 2 i p/q other than the simple eigenvalue 1). An intermediate mixing condition between strongand weak- mixing is that a measure-preserving transformation is mild-mixing if whenever f ı T n i ! f for an L2 function f and a sequence n i ! 1, then f is a.e. constant. Clearly mild-mixing is a spectral property. If a transformation has an eigenfunction f , then it is straightforward to find a sequence ni such that f ı T n i ! f , so we see that mild-mixing implies weak-mixing. To see that strong-mixing implies mild-mixing, suppose that T is and that f ı T n i ! f . Then we have R strong-mixing n ¯ i f ı T f ! k f k2 . On R the other hand, the strong mixing property implies that f ı T n i f¯ ! jh f ; 1ij2. The equality of these implies that f is a. e. constant. Mild-mixing has a useful reformulation in terms of ergodicity of general (not necessarily probability) measure-preserving transformations: A transformation T is mild-mixing if and only if for every conservative ergodic measure-preserving transformation S, T  S is ergodic. See Furstenberg and Weiss’ article [22] for further information on mild-mixing. The strongest spectral property that we consider is that of having countable Lebesgue spectrum. While we

Ergodicity and Mixing Properties

will avoid a detailed discussion of spectral theory in this article, this is a special case that can be described simply. Specifically, let T be an invertible measure-preserving transformation. Then T has countable Lebesgue spectrum if there is a sequence of functions f1 ; f2 ; : : : such that f1g [ fU Tn f j : n 2 Z; j 2 Ng forms an orthonormal basis for L2 (X). To see that this property is stronger than strongmixing, we simply observe that it implies that hU Tt U Tn f j ; U Tm f k i ! 0 as t ! 1. Then by approximating f and g by their expansions with respect to a finite part of the basis, we deduce that hU Tn f ; gi ! h f ; 1ih1; gi as required. Since already strong-mixing is atypical from the topological point of view, it follows that countable Lebesgue spectrum has to be atypical. In fact, Yuzvinskii [96] showed that the typical invertible measure-preserving transformation has simple singular spectrum. The property of countable Lebesgue spectrum is by definition a spectral property. Since it completely describes the transformation up to spectral isomorphism, there can be no stronger spectral properties. The remaining properties that we shall examine are invariant under measuretheoretic isomorphisms only. An invertible measure-preserving transformation T of (X; B; ) is said to be K (for Kolmogorov) if there is a sub-algebra F of B such that T n F is the trivial -algebra up to sets of mea1. 1 nD1 T sure 0 (i. e. the intersection consists only of null sets and sets of full measure). W n 2. 1 nD1 T F D B (i. e. the smallest -algebra containn ing T F for all n > 0 is B). The K property has a useful reformulation in terms of entropy as follows: T is K if and only if for every non-trivial partition P of X, the entropy of T with respect to the partition P is positive: T has completely positive entropy. See the chapter on  Entropy in Ergodic Theory for the relevant definitions. The equivalence of the K property and completely positive entropy was shown by Rokhlin and Sinai [74]. For a general transformation T, one can consider the collection of all subsets B of X such that with respect to the partition PB D fB; B c g, h(PB ) D 0. One can show that this is a -algebra. This -algebra is known as the Pinsker -algebra. The above reformulation allows us to say that a transformation is K if and only if it has a trivial Pinsker -algebra. The K property implies countable Lebesgue spectrum (see Parry’s book [63] for a proof). To see that K is not implied by countable Lebesgue spectrum, we point out that certain measure-preserving transformations derived from Gaussian systems (see for example the paper of Parry and

Newton [52]) have countable Lebesgue spectrum but zero entropy. The fact that (two-sided) Bernoulli shifts have the K property follows from Kolmogorov’s 0–1 law by taking W1 F D nD0 T n P , where P is the partition into cylinder sets (see Williams’s book [93] for details of the 0–1 law). Although the K property is explicitly an invertible property, it has a non-invertible counterpart, namely exactness. A transformation T of (X; B; ) is exact if T1 n B consists entirely of null sets and sets of meanD0 T sure 1. It is not hard to see that a non-invertible transformation is exact if and only if its natural extension is K. The final and strongest property in our list is that of being measure-theoretically isomorphic to a Bernoulli shift. If T is measure-theoretically isomorphic to a Bernoulli shift, we say that T has the Bernoulli property. While in principle this could apply to both invertible and non-invertible transformations, in practice the definition applies to a large class of invertible transformations, but occurs comparatively seldom for non-invertible transformations. For this reason, we will restrict ourselves to a discussion of the Bernoulli property for invertible transformations (see however work of Hoffman and Rudolph [29] and Heicklen and Hoffman [28] for work on the one-sided Bernoulli property). In the case of invertible Bernoulli shifts, Ornstein [53,58] developed in the early 1970s a powerful isomorphism theory, showing that two Bernoulli shifts are measure-theoretically isomorphic if and only if they have the same entropy. Entropy had already been identified as an invariant by Kolmogorov and Sinai [43,82], so this established that it was a complete invariant for Bernoulli shifts. Keane and Smorodinsky [41] gave a proof which showed that two Bernoulli shifts of the same entropy are isomorphic using a conjugating map that is continuous almost everywhere. With other authors, this theory was extended to show that the property of being isomorphic to a Bernoulli shift applied to a surprisingly large class of measure-preserving transformations (e. g. geodesic flows on manifolds of constant negative curvature (Ornstein and Weiss [60]), aperiodic irreducible Markov chains (Friedman and Ornstein [19]), toral automorphisms (Katznelson [39]) and more generally many Gibbs measures for hyperbolic dynamical systems (see the book of Bowen [11])). Initially, it was conjectured that the properties of being K and Bernoulli were the same, but since then a number of measure-preserving transformations that are K but not Bernoulli have been identified. The earliest was due to Ornstein [55]. Ornstein and Shields [59] then provided an uncountable family of non-isomorphic K automorphisms.

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Katok [37] gave an example of a smooth diffeomorphism that is K but not Bernoulli; and Kalikow [33] gave a very natural probabilistic example of a transformation that has this property (the T, T 1 process). While in systems that one regularly encounters there is a correlation between positive entropy and the stronger mixing properties that we have discussed, these properties are logically independent (for example taking the product of a Bernoulli shift and the identity transformation gives a positive entropy transformation that fails to be ergodic; also, the zero entropy Gaussian systems with countable Lebesgue spectrum mentioned above have relatively strong mixing properties but zero entropy). In many of the mixing criteria discussed above we have considered a pair of sets A and B and asked for asymptotic independence of A and B (so that for large n, A and T n B become independent). It is natural to ask, given a finite collection of sets A0 ; A1 ; : : : ; A k , under what conditions Q (A0 \T n 1 A1 \  \T n k A k ) converges to kjD0 (A j ). A measure-preserving transformation is said to be mixing of order k + 1 if for all measurable sets A0 ; : : : ; A k , lim

n 1 !1;n jC1n j !1

   A0 \ T n 1 A1 \    \ T n k A k D

k Y

(A j ) :

jD0

An outstanding open question asked by Rokhlin [73] appearing already in Halmos’ 1956 book [27] is to determine whether mixing (i. e. mixing of order 2) implies mixing of all orders. Kalikow [34] showed that mixing implies mixing of all orders for rank 1 transformations (existence of rank one mixing transformations having been previously established by Ornstein in [54]). Later Ryzhikov [77] used joining methods to establish the result for transformations with finite rank, and Host [30] also used joining methods to establish the result for measure-preserving transformations with singular spectrum, but the general question remains open. It is not hard to show using martingale arguments that K automorphisms and hence all Bernoulli measure-preserving transformations are mixing of all orders. For weak-mixing transformations, Furstenberg [21] has established the following weak-mixing of all orders statement: if a measure-preserving transformation T is weak-mixing, then given sets A0 ; : : : ; A k , there is a subsequence J of the integers of density 0 such that

lim

n!1 n62 J

k  Y   A0 \ T n A1 \    \ T kn A k D (A i ) : iD0

Bergelson [5] generalized this by showing that    A0 \ T p 1 (n) A1 \    \ T p k (n) A k n!1;n62 J k Y D (A i ) lim

iD0

whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n)  p j (n) is unbounded for i ¤ j. The method of proof of both of these results was a Hilbert space version of the van der Corput inequality of analytic number theory. Furstenberg’s proof played a key role in his ergodic proof [20] of Szemerédi’s theorem on the existence of arbitrarily long arithmetic progressions in a subset of the integers of positive density (see the chapter on  Ergodic Theory: Interactions with Combinatorics and Number Theory for more information about this direction of study). The conclusions that one draws here are much weaker than the requirement for mixing of all orders. For mixing of all orders, it was required that provided the gaps between 0; n1 ; : : : ; n k diverge to infinity, one achieves asymptotic independence, whereas for these weak-mixing results, the gaps are increasing along prescribed sequences with regular growth properties. It is interesting to note that the analogous question of whether mixing implies mixing of all orders is known to fail in higher-dimensional actions. Here, rather than a Z action, in which there is a single measure-preserving transformation (so that the integer n acts on a point x 2 X by mapping it to T n x), one takes a Zd action. For such an action, one has d commuting transformations T1 ; : : : ; Td and a vector (n1 ; : : : ; nd ) acts on a point x by sending it to n T1n 1    Td d x. Ledrappier [46] studied the following two2 dimensional action. Let X D fx 2 f0; 1gZ : xv C xvCe1 C xvCe2 D 0 (mod 2)g and let Ti (x)v D xvCe i . Since X is a compact Abelian group, it has a natural measure  invariant under the group operations (the Haar measure). It is not hard to show that this system is mixing (i. e. given any measurable sets A and B, (A \ T1n 1 T2n 2 B) ! (A)(B) as k(n1 ; n2 )k ! 1). Ledrappier showed that the system fails to be 3-mixing. Subsequently Masser [48] established necessary and sufficient conditions for similar higher-dimensional algebraic actions to be mixing of order k but not order k C 1 for any given k. Hyperbolicity and Decay of Correlations One class of systems in which the stronger mixing properties are often found is the class of smooth systems possessing uniform hyperbolicity (i. e. the tangent space to the manifold at each point splits into stable and unstable

Ergodicity and Mixing Properties

subspaces E s (x) and E u (x) such that the kDTj E s (x) k  a < 1 for all x and kDT 1 j E u (x) k  a and DT(Es (x)) D Es (T(x)) and DT(Eu (x)) D Eu (T(x))). In some cases similar conclusions are found in systems possessing non-uniform hyperbolicity. See Katok and Hasselblatt’s book [38] for an overview of hyperbolic dynamical systems, as well as the chapter in this volume on  Smooth Ergodic Theory. In the simple case of expanding piecewise continuous maps of the interval (that is, maps for which the absolute value of the derivative is uniformly bounded below by a constant greater than 1), it is known that if they are totally ergodic and topologically transitive (i. e. the forward images of any interval cover the entire interval), then provided that the map has sufficient smoothness (e. g. the map is C1 and the derivative satisfies a certain additional summability condition), the map has a unique absolutely continuous invariant measure which is exact and whose natural extension is Bernoulli (see the paper of Góra [25] for results of this type proved under some of the mildest hypotheses). These results were originally established for maps that were twice continuously differentiable, and the hypotheses were progressively weakened, approaching, but never meeting, C1 . Subsequent work of Quas [68,69] provided examples of C1 expanding maps of the interval for which Lebesgue measure was invariant, but respectively not ergodic and not weak-mixing. Some of the key tools in controlling mixing in one-dimensional expanding maps that are absent in the C1 case are bounded distortion estimates. Here, there is a constant 1  C < 1 such that given any interval I on which some power T n of T acts injectively and any sub-interval J of I, one has 1/C  (jT n Jj/jT n Ij)/(jJj/jIj)  C. An early place in which bounded distortion estimates appear is the work of Rényi [70]. One important class of results for expanding maps establishes an exponential decay of correlations. Here, one starts with f and g and one estiR a pair of smooth R functions R mates f  g ı T n d f d g d, where  is an absolutely continuous invariant measure. If  is mixing, we expect this to converge to 0. In fact though, in good cases this converges to 0 at an exponential rate for each pair of functions f and g belonging to a sufficiently smooth class. In this case, the measure-preserving transformation T is said to have exponential decay of correlations. See Liverani’s article [47] for an introduction to a method of establishing this based on cones. Exponential decay of correlations implies in particular that the natural extension is Bernoulli. Hu [31] has studied the situation of maps of the interval for which the derivative is bigger than 1 everywhere except at a fixed point, where the local behavior is of the form x 7! x C x 1C˛ for 0 < ˛ < 1. In this case, rather

than exhibiting exponential decay of correlations, the map has polynomial decay of correlations with a rate depending on ˛. In Young’s survey [94], a variety of techniques are outlined for understanding the strong ergodic properties of non-uniformly hyperbolic diffeomorphisms. In her article [95], methods are introduced for studying many classes of non-uniformly hyperbolic systems by looking at suitably high powers of the map, for which the power has strong hyperbolic behavior. The article shows how to understand the ergodic behavior of these systems. These methods are applied (for example) to billiards, one-dimensional quadratic maps and Hénon maps. Future Directions Problem 1 (Mixing of all orders) Does mixing imply mixing of all orders? Can the results of Kalikow, Ryzhikov and Host be extended to larger classes of measure-preserving transformations? Thouvenot observed that it is sufficient to establish the result for measure-preserving transformations of entropy 0. This observation (whose proof is based on the Pinsker  -algebra) was stated in Kalikow’s paper [34] and is reproduced as Proposition 3.2 in recent work of de la Rue [16] on the mixing of all orders problem. Problem 2 (Multiple weak-mixing) As mentioned above, Bergelson [5] showed that if T is a weak-mixing transformation, then there is a subset J of the integers of density 0 such that lim

n!1;n62 J

   A0 \ T p 1 (n) A1 \    \ T p k (n) A k D

k Y

(A i )

iD0

whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n)  p j (n) is unbounded for i ¤ j. It is natural to ask what is the most general class of times that can replace the sequences (p1 (n)); : : : ; (p k (n)). In unpublished notes, Bergelson and Håland considered as times the values taken by a family of integer-valued generalized polynomials (those functions of an integer variable that can be obtained by the operations of addition, multiplication, addition of or multiplication by apreal constantp and taking integer parts (e. g. g(n) D b 2b nc C b 3nc2 c)). They conjectured necessary and sufficient conditions for the analogue of Bergelson’s weakmixing polynomial ergodic theorem to hold, and proved the conjecture in certain cases.

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In a recent paper of McCutcheon and Quas [50], the analogous question was addressed in the case where T is a mild-mixing transformation. Problem 3 (Pascal adic transformation) Vershik [89,90] introduced a family of transformations known as the adic transformations. The underlying spaces for these transformations are certain spaces of paths on infinite graphs, and the transformations act by taking a path to its lexicographic neighbor. Amongst the adic transformations, the so-called Pascal adic transformation (so-called because the underlying graph resembles Pascal’s triangle) has been singled out for attention in work of Petersen and others [2,10,51,65]. In particular, it is unresolved whether this transformation is weak-mixing with respect to any of its ergodic measures. Weak-mixing has been shown by Petersen and Schmidt to follow from a number-theoretic condition on the binomial coefficients [2,10]. Problem 4 (Weak Pinsker Conjecture) Pinsker [67] conjectured that in a measure-preserving transformation with positive entropy, one could express the transformation as a product of a Bernoulli shift with a system with zero entropy. This conjecture (now known as the Strong Pinsker Conjecture) was shown to be false by Ornstein [56,57]. Shields and Thouvenot [78] showed that the collection of transformations that can be written as a product of a zero entropy transformation with a Bernoulli shift ¯ is closed in the so-called d-metric that lies at the heart of Ornstein’s theory. It is, however, the case that if T : X ! X has entropy h > 0, then for all h0  h, T has a factor S with entropy h0 (this was originally proved by Sinai [83] and reproved using the Ornstein machinery by Ornstein and Weiss in [61]). The Weak Pinsker Conjecture states that if a measure-preserving transformation T has entropy h > 0, then for all > 0, T may be expressed as a product of a Bernoulli shift and a measure-preserving transformation with entropy less than . Bibliography 1. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence 2. Adams TM, Petersen K (1998) Binomial coefficient multiples of irrationals. Monatsh Math 125:269–278 3. Alpern S (1976) New proofs that weak mixing is generic. Invent Math 32:263–278 4. Avila A, Forni G (2007) Weak-mixing for interval exchange transformations and translation flows. Ann Math 165:637–664 5. Bergelson V (1987) Weakly mixing pet. Ergodic Theory Dynam Systems 7:337–349 6. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci 17:656–660

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57. Ornstein DS (1973) A mixing transformation for which Pinsker’s conjecture fails. Adv Math 10:103–123 58. Ornstein DS (1974) Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, Newhaven 59. Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:89–102 60. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Israel J Math 14:184–198 61. Ornstein DS, Weiss B (1975) Unilateral codings of Bernoulli systems. Israel J Math 21:159–166 62. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116–136 63. Parry W (1981) Topics in Ergodic Theory. Cambridge, Cambridge 64. Petersen K (1983) Ergodic Theory. Cambridge, Cambridge 65. Petersen K, Schmidt K (1997) Symmetric Gibbs measures. Trans Amer Math Soc 349:2775–2811 66. Phelps R (1966) Lectures on Choquet’s Theorem. Van Nostrand, New York 67. Pinsker MS (1960) Dynamical systems with completely positive or zero entropy. Soviet Math Dokl 1:937–938 68. Quas A (1996) A C 1 expanding map of the circle which is not weak-mixing. Israel J Math 93:359–372 69. Quas A (1996) Non-ergodicity for C 1 expanding maps and g-measures. Ergodic Theory Dynam Systems 16:531–543 70. Rényi A (1957) Representations for real numbers and their ergodic properties. Acta Math Acad Sci Hungar 8:477–493 71. Riesz F (1938) Some mean ergodic theorems. J Lond Math Soc 13:274–278 72. Rokhlin VA (1948) A ‘general’ measure-preserving transformation is not mixing. Dokl Akad Nauk SSSR Ser Mat 60:349–351 73. Rokhlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 74. Rokhlin VA, Sinai Y (1961) Construction and properties of invariant measurable partitions. Dokl Akad Nauk SSSR 141:1038– 1041 75. Rudin W (1966) Real and Complex Analysis. McGraw Hill, New York 76. Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford, Oxford University Press 77. Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank. Funct Anal Appl 27:128–140 78. Shields P, Thouvenot J-P (1975) Entropy zero × Bernoulli pro¯ cesses are closed in the d-metric. Ann Probab 3:732–736 79. Simányi N (2003) Proof of the Boltzmann–Sinai ergodic hypothesis for typical hard disk systems. Invent Math 154:123– 178 80. Simányi N (2004) Proof of the ergodic hypothesis for typical hard ball systems. Ann Henri Poincaré 5:203–233 81. Simányi N, Szász D (1999) Hard ball systems are completely hyperbolic. Ann Math 149:35–96 82. Sinai YG (1959) On the notion of entropy of a dynamical system. Dokl Russ Acad Sci 124:768–771 83. Sinai YG (1964) On a weak isomorphism of transformations with invariant measure. Mat Sb (NS) 63:23–42 84. Sinai YG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat Nauk 25:141–192 85. Sinai Y (1976) Introduction to Ergodic Theory. Princeton, Princeton, translation of the 1973 Russian original

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Ergodic Theorems

Ergodic Theorems ANDRÉS DEL JUNCO Department of Mathematics, University of Toronto, Toronto, Canada

Article Outline Glossary Definition of the Subject Introduction Ergodic Theorems for Measure-Preserving Maps Generalizations to Continuous Time and Higher–Dimensional Time Pointwise Ergodic Theorems for Operators Subadditive and Multiplicative Ergodic Theorems Entropy and the Shannon–McMillan–Breiman Theorem Amenable Groups Subsequence and Weighted Theorems Ergodic Theorems and Multiple Recurrence Rates of Convergence Ergodic Theorems for Non-amenable Groups Future Directions Bibliography

Glossary Dynamical system in its broadest sense, any set X, with a map T : X ! X. The classical example is: X is a set whose points are the states of some physical system and the state x is succeeded by the state Tx after one unit of time. Iteration repeated applications of the map T above to arrive at the state of the system after n units of time. Orbit of x the forward images x; Tx; T 2 X : : : of x 2 X under iteration of T. When T is invertible one may consider the forward, backward or two-sided orbit of x. Automorphism a dynamical system T : X ! X, where X is a measure space and T is an invertible map preserving measure. Ergodic average if f is a function on X let A n f (x) D P i n1 n1 iD0 f (T x); the average of the values of f over the first n points in the orbit of x. Ergodic theorem an assertion that ergodic averages converge in some sense. Mean ergodic theorem an assertion that ergodic averages converge with respect to some norm on a space of functions.

Pointwise ergodic theorem an assertion that ergodic averages A n f (x) converge for some or all x 2 X, usually for a.e. x. Stationary process a sequence (X1 ; X2 ; : : :) of random variables (real or complex-valued measurable functions) on a probability space whose joint distributions are invariant under shifting (X1 ; X2 ; : : :) to (X2 ; X3 ; : : :). Uniform distribution a sequence fx n g in [0; 1] is uniformly distributed if for each interval I [0; 1], the time it spends in I is asymptotically proportional to the length of I. Maximal inequality an inequality which allows one to bound the pointwise oscillation of a sequence of functions. An essential tool for proving pointwise ergodic theorems. Operator any linear operator U on a vector space of functions on X, for example one arising from a dynamical system T by setting U f (x) D f (Tx). More generally any linear transformation on a real or complex vector space. Positive contraction an operator T on a space of functions endowed with a norm k  k such that T maps positive functions to positive functions and kT f k  j f j. Definition of the Subject Ergodic theorems are assertions about the long-term statistical behavior of a dynamical system. The subject arose out of Boltzmann’s ergodic hypothesis which sought to equate the spatial average of a function over the set of states in a physical system having a fixed energy with the time average of the function observed by starting with a particular state and following its evolution over a long time period. Introduction Suppose that (X; B; ) is a measure space and T : (X; B; ) ! (X; B; ) is a measurable and measurepreserving transformation, that is (T 1 E) D (E) for all E 2 B. One important motivation for studying such maps is that a Hamiltonian physical system (see the article by Petersen in this collection) gives rise to a one-parameter group fTt : t 2 Rg of maps in the phase space of the system which preserve Lebesgue measure. The ergodic theorem of Birkhoff asserts that for f 2 L1 () n1 1X f (T i x) n

(1)

iD0

converges a.e. and that if RT is ergodic (to be defined shortly) then the limit is f d. This may be viewed

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as a justification for Boltzmann’s ergodic hypothesis that “space averages equal time averages”. See Zund [214] for some history of the ergodic hypothesis. For physicists, then, the problem is reduced to showing that a given physical system is ergodic, which can be very difficult. However ergodic systems arise in many natural ways in mathematics. One example is rotation z 7! z of the unit circle if  is a complex number of modulus one which is not a root of unity. Another is the shift transformation on a sequence of i.i.d. random variables, for example a coin-tossing sequence. Another is an automorphism of a compact Abelian group. Often a transformation possesses an invariant measure which is not obvious at first sight. Knowledge of such a measure can be a very useful tool. See Petersen’s article for more examples. If (X; B; ) is a probability space and T is ergodic then Birkhoff’s ergodic theorem implies that if A is a measurable subset of X then for almost every x, the frequency with which x visits A is asymptotically equal to (A), a very satisfying justification of intuition. For example, applying this to the coin-tossing sequence one obtains the strong law of large numbers which asserts that almost every infinite sequence of coin tosses has tails occurring with asymptotic frequency 12 . One also obtains Borel’s theorem on normal numbers which asserts that for almost all x 2 [0; 1] each digit 0; 1; 2; : : : ; 9 occurs with limiting 1 frequency 10 . The so-called continued fraction transformation x 7! x 1 mod 1 on (0; 1) has a finite invariant dx . (Throughout this article x mod 1 denotes measure 1Cx the fractional part of x.) Applying Birkhoff’s theorem then gives precise information about the frequency of occurrence of any n 2 N in the continued fraction expansion of x, for a.e. x. See for example Billingsley [34]. These are the classical roots of the subject of ergodic theorems. The subject has evolved from these simple origins into a vast field in its own right, quite independent of physics or probability theory. Nonetheless it still has close ties to both these areas and has also forged new links with many other areas of mathematics. Our purpose here is to give a broad overview of the subject in a historical perspective. There are several excellent references, notably the books of Krengel [135] and Tempelman [194] which give a good picture of the state of the subject at the time they appeared. There has been tremendous progress since then. The time is ripe for a much more comprehensive survey of the field than is possible here. Many topics are necessarily absent and many are only glimpsed. For example this article will not touch on random ergodic theorems. See the articles [75,143] for some references on this topic.

I thank Mustafa Akcoglu, Ulrich Krengel, Michael Lin, Dan Rudolph and particularly Joe Rosenblatt and Vitaly Bergelson for many helpful comments and suggestions. I would like to dedicate this article to Mustafa Akcoglu who has been such an important contributor to the development of ergodic theorems over the past 40 years. He has also played a vital role in my mathematical development as well as of many other mathematicians. He remains a source of inspiration to me, and a valued friend. Ergodic Theorems for Measure-Preserving Maps Suppose (X; B; ) is a measure space. A (measure-preserving) endomorphism of (X; B; ) is a measurable mapping T : X ! X such that (T 1 E) D (E) for any measurable subset E X. If T has a measurable inverse then one says that it is an automorphism. In this article the unqualified terms “endomorphism” or “automorphism” will always mean measure-preserving. T is called ergodic if for all measurable E one has T 1 E D E ) (E) D 0

or (E c ) D 0 :

A very general class of examples comes from the notion of a stationary stochastic process in probability theory. A stochastic process is a sequence of measurable functions f1 ; f2 ; : : : on a probability space (X; B; ) taking values in a measurable space (Y; C ). The distribution of the process, a measure  on Y N , is defined as the image of  under the map ( f1 ; f2 ; : : :) : X ! Y N .  captures all the essential information about the process f f i g. In effect one may view any process f f i g as a probability measure on Y N . The process is said to be stationary if  is invariant under the left shift transformation S on Y N , S(y)(i) D y(i C 1), that is S is an endomorphism of the probability space (Y N ; ). From a probabilistic point of view the most natural examples of stationary stochastic processes are independent identically distributed processes, namely the case when  is a product measure N for some measure  on (Y; C ). More generally one can consider a stationary Markov process defined by transition probabilities on the state space Y and an invariant probability on Y. See for example Chap. 7 in [50], also Sect. “Pointwise Ergodic Theorems for Operators” below. The first and most fundamental result about endomorphisms is the celebrated recurrence theorem of Poincaré [173]). Theorem 1 Suppose  is finite, A 2 B and (A) > 0. Then for a.e. x 2 A there is an n > 0 such that T n x 2 A, in fact there are infinitely many such n. It may be viewed as an ergodic theorem, in that it is a qualitative statement about how x behaves under iteration of T.

Ergodic Theorems

For a proof, observe that if E A is the measurable set of points which never return to A then for each n > 0 the set T n E is disjoint from E. Applying T m one gets also T (nCm) E \ T m E D ;. Thus E; T 1 E; T 2 E; : : : is an infinite sequence of disjoint sets all having measure (E) so (E) D 0 since (X) < 1. Much later Kac [120] formulated the following quantitative version of Poincaré’s theorem. Theorem 2 Suppose that (X) D 1, T is ergodic and let r A x denote the time of first return to A, that is r A (x) is the least n > 0 such that T n x 2 A. Then Z 1 1 r A d D ; (2) (A) A (A) that is, the expected value of the return time to A is (A)1 . Koopman [131] made the observation that associated to an automorphism T there is a unitary operator U D U T defined on the Hilbert space L2 () by the formula U f D f ı T. This led von Neumann [151] to prove his mean ergodic theorem. Theorem 3 Suppose H is a Hilbert space, U is a unitary operator on H and let P denote the orthogonal projection on the subspace of U-invariant vectors. Then for any x 2 H one has  n1  1X    U i x  Px  ! 0 as n ! 1 : (3)  n  iD0

Von Neumann’s theorem is usually quoted as above but to be historically accurate he dealt with unitary operators indexed by a continuous time parameter. Inspired by von Neumann’s theorem Birkhoff very soon proved his pointwise ergodic theorem [35]. In spite of the later publication by von Neumann his result did come first. See [29,214] for an interesting discussion of the history of the two theorems and of the interaction between Birkhoff and von Neumann. Theorem 4 Suppose (X; B; ) is a measure space and T is an endomorphism of (X; B; ). Then for any f 2 L1 D L1 () there is a T-invariant function g 2 L1 such that A n f (x) D

n1 1X f (T i x) ! g(x) n

a.e.

(4)

iD0

Moreover if  is finite then the convergence alsoRholds with R respect to the L1 norm and one has E g d D E f d for all T-invariant subsets E.

Again this formulation of Birkhoff’s theorem is not historically accurate as he dealt with a smooth flow on a manifold. It was soon observed that the theorem, and its proof, remain valid for an abstract automorphism of a measure space although the realization that T need not be invertible seems to have taken a little longer. The notation A n f D A n (T) f as above will occur often in the sequel. Whenever T is an endomorphism one uses the notation T f D f ı T and with this notation P i A n (T) D n1 n1 iD0 T . When the scalars (R or C) for an L1 space are not specified the notation should be understood as referring to either possibility. In most of the theorems in this article the complex case follows easily from the real and any indications about proofs will refer to the real case. Although von Neumann originally used spectral theory to prove his result, there is a quick proof, attributed to Riesz by Hopf in his 1937 book [100], which uses only elementary properties of Hilbert space. Let I denote the (closed) subspace of U-invariant vectors and I 0 the (usually not closed) subspace of vectors of the form f  U f . It is easy to check that any vector orthogonal to I 0 must be in I, whence the subspace I C I 0 is dense in H . For any vector of the form x D y C y 0 ; y 2 I; y 0 2 I 0 it is clear P i that A n x D n1 n1 iD0 U x converges to y D Px, since 0 A n y D y and if y D z  Uz then the telescoping sum A n y 0 D n1 (z  U n z) converges to 0. This establishes the desired convergence for x 2 I C I 0 and it is easy to extend it to the closure of I C I 0 since the operators An are contractions of H (kA n k  1). Lorch [147] used a soft argument in a similar spirit to extend non Neumann’s theorem from the case of a unitary operator on a Hilbert space to that of an arbitrary linear contraction on any reflexive Banach space. Sine [188] gave a necessary and sufficient condition for the strong convergence of the ergodic averages of a contraction on an arbitrary Banach space. Birkhoff’s theorem has the distinction of being one of the most reproved theorems of twentieth century mathematics. One approach to the pointwise convergence, parallel to the argument to argument just seen, is to find a dense subspace E of L1 so that the convergence holds for all f 2 E and then try to extend the convergence to all f 2 L1 by an approximation argument. The first step is not too hard. For simplicity assume that  is a finite measure. As in the proof of von Neumann’s theorem the subspace E spanned by the T-invariant L1 functions together with functions of the form g  T g; g 2 L1 , is dense in L1 (). This can be seen by using the Hahn–Banach theorem and the duality of L1 and L1 . (Here one needs finiteness of  to know that L1 L1 .) The pointwise convergence of A n f for invariant f is trivial and for f D g  T g it follows from telescoping of the sum and the fact that n1 T n g ! 0 a.e. This

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last can be shown by using the Borel–Cantelli lemma. The second step, extending pointwise convergence, as opposed to norm convergence, for f in a dense subspace to all f in L1 is a delicate matter, requiring a maximal inequality. Roughly speaking a maximal inequality is an inequality which bounds the pointwise oscillation of A n f in terms of the norm of f . The now standard maximal inequality in the context of Birkhoff’s theorem is the following, due to Kakutani and Yosida [209]. Birkhoff’s proof of his theorem includes a weaker version of this result. Let S n f D nA n f , the nth partial sum of the iterates of f . S Theorem 5 Given any real f 2 L1 let A D n1 fS n f  0g. Then Z f d  0 : (5) A

Moreover if one sets M f D supn1 A n f then for any ˛ > 0 fM f > ˛g 

1 k f k1 ˛

(6)

A distributional inequality such as (6) will be referred to as a weak L1 inequality. Note that (6) follows easily from (5) by applying (5) to f ˛, at least in the case when  is finite. With the maximal inequality in hand it is straightforward to complete the proof of Birkhoff’s theorem. For a realvalued function f let Osc f D lim sup A n f  lim inf A n f :

(7)

Osc f D 0 a.e. if and only if A n f converges a.e.(to a possibly infinite limit). One has lim sup A n f  M f  Mj f j and by symmetry lim inf A n f  Mj f j, so Osc f  2Mj f j. To establish the convergence of A n f for a real-valued f 2 L1 let > 0 and write f D g C h with g 2 E (the subspace where convergence has already been established), h 2 L1 and khk < . Then since Osc g D 0 one has Osc f D Osc h. Thus for any fixed ˛ > 0, using (6) n ˛o fOsc f > ˛g D  fOsc h > ˛g   M h > 2 2khk1 2  < : (8) ˛ ˛ Since > 0 was arbitrary one concludes that fOsc f > ˛g D 0 and since ˛ > 0 is arbitrary it follows that fOsc f > 0g D 0, establishing the a.e. convergence. Moreover a simple application of Fatou’s lemma shows that the limiting function is in L1 , hence finite a.e. There are many proofs of (5). Two of particular interest are Garsia’s [89], perhaps the shortest and most mysterious, and the proof via the filling scheme of Chacón

and Ornstein [61], perhaps the most intuitive, which goes like this. Given a function g 2 L1 write g C D max(g; 0), g  D g C  g and let Ug D T g C  g  . Interpretation: the region between the graph of g C and the X-axis is a sheaf of vertical spaghetti sticks, the intervals [0; g C (x)], x in X, and g  is a hole. Now move the spaghetti (horizontally) by T and then let it drop (vertically) into the hole leaving a new hole and a new sheaf which are the negative and posS itive parts of Ug. Now let E 0 D n1 fU n f  0g, the set of points x at which the hole is eventually filled after finitely many iterations of U. The key point is that E D E 0 . Indeed if S n f (x)  0 for some n, then the total linear height of sticks over x; Tx; : : : T n1 x is greater than the total linear depth of holes at these points. The only way that spaghetti can escape from these points is by first filling the hole at x, which shows x 2 E 0 . Similar thinking shows that if x 2 E 0 and the hole at x is filled for the first time at time n then S n f (x)  0, so x 2 E, and that all the spaghetti that goes into the hole at x comes from points T i x which belong to E 0 . This shows that E D E 0 and that the part of the hole lying beneath E is eventually filled by spaghetti coming from E 0 D E. Thus the amount of spaghettiRover E is no less than the size of the hole under E, that is E f d  0. Most proofs of Birkhoff’s theorem use a maximal inequality in some form but a few avoid it altogether, for example [126,186]. It is also straightforward to deduce Birkhoff’s theorem directly from a maximal inequality, as Birkhoff does, without first establishing convergence on a dense subspace. However the technique of proving a pointwise convergence theorem by finding an appropriate dense subspace and a suitable maximal inequality has proved extremely useful, not only in ergodic theory. Indeed, in some sense maximal inequalities are unavoidable: this is the content of the following principle proved already in 1926 by Banach. The principle has many formulations; the following one is a slight simplification of the one to be found in [135]. Suppose B is a Banach space, (X; B; ) is a finite measure space and let E denote the space of -equivalence classes of measurable real-valued functions on X. A linear map T : B ! E is said to be continuous in measure if for each > 0 kx n  xk ! 0 ) fjTx n  Txj > g ! 0 : Suppose that T n is a sequence of linear maps from B to E which are continuous in measure and let Mx D sup n jTn xj. Of course if Tn x converges a.e. to a finite limit then Mx < 1 a.e. Theorem 6 (Banach principle) Suppose Mx < 1 a.e. for each x 2 X. Then there is a function C() such that

Ergodic Theorems

C() ! 0 as  ! 1 such that for all x 2 B and  > 0 one has fMx  kxkg  C() :

(9)

Moreover the set of x for which Tn x converges a.e. is closed in B. The first chapter of Garsia’s book [89] contains a nice introduction to the Banach principle. It should be noted that, for general integrable f , M f need not be in L1 . However if f 2 L p for p > 1 then M f does belong to Lp and one has the estimate kM f k p 

p k f kp : p1

(10)

This can be derived from (6) using also the (obvious) fact that kM f k1  k f k1 . Any such estimate on the norm of a maximal function will be called a strong Lp inequality. It also follows from (6) that if (X) is finite and f 2 L log L, R that is j f j logC j f jd < 1, then M f 2 L1 . In fact Ornstein [160] has shown that the converse of this last statement holds provided T is ergodic. There is a special setting where one has uniform convergence in the ergodic theorem. Suppose T is a homeomorphism of a compact metric space X. By a theorem of Krylov and Bogoliouboff [138] there is at least one probability measure on the Borel -algebra of X which is invariant under T. T is said to be uniquely ergodic if there is only one Borel probability measure, say , invariant under T. It is easy to see that when this is the case then T is an ergodic automorphism of (X; B; ). As an example, if ˛ is an irrational number then the rotation z 7! e2 i˛ z is a uniquely ergodic transformation of the circle fjzj D 1g. Equivalently x 7! x C ˛ mod 1 is a uniquely ergodic map on [0; 1]. A quick way to see this is to show that the ˆ Fourier co-efficients (n) of any invariant probability  are zero for n ¤ 0. The Jewett–Krieger theorem (see Jewett [109] and Krieger [137]) guarantees that unique ergodicity is ubiquitous in the sense that any automorphism of a probability space is measure-theoretically isomorphic to a uniquely ergodic homeomorphism. The following important result is due to Oxtoby [165]). Theorem 7 If T is uniquely ergodic,  is its unique invariant probability measure and f 2 C(X) R then the ergodic averages A n ( f ) converge uniformly to f d. This result can be proved along the same lines as the proof given above of von Neumann’s theorem. In a nutshell, one uses the fact that the dual of CR (X) is the space of finite signed measures on X and the unique ergodicity to show

that functions of the form f  f ı T together with the invariant functions (which are just constant functions) span a dense subspace of C(X). A sequence fx n g in the interval [0; 1]R is said to be uniP 1 formly distributed if n1 niD1 f (x n ) ! 0 f (x)dx for any f 2 C[0; 1] (equivalently for any Riemann integrable f or for any f D 1I where I is any subinterval of [0; 1]). As a simple application of Theorem 7 one obtains the linear case of the following result of Weyl [202]. Theorem 8 If ˛ is irrational and p(x) is any non-constant polynomial with integer coefficients then the sequence fp(n)˛mod1g is uniformly distributed. Furstenberg [84], see also [86], has shown that Weyl’s result, in full generality, can be deduced from the unique ergodicity of certain affine transformations of higher dimensional tori. For polynomials of degree k > 1 Weyl’s result is usually proved by inductively reducing to the case k D 1, using the following important lemma of van der Corput (see for example [139]). Theorem 9 Suppose that for each fixed h > 0 the sequence fx nCh  x n

mod 1gn

is uniformly distributed. Then fx n g is uniformly distributed. When  is infinite and T is ergodic the limiting function in Birkhoff’s theorem is 0 a.e. In 1937 Hopf [100] proved a generalization of Birkhoff’s theorem which is more meaningful in the case of an infinite invariant measure. It is a special case of a later theorem of Hurewicz [105], which we will discuss first. Suppose that (X; B; ) is a  -finite measure space. If  is another  -finite measure on B write    ( is absolutely continuous relative to ) if (E) D 0 implies (E) D 0 and one writes   if    and   . Consider a non-singular automorphism  : X ! X, meaning that  is measurable with a measurable inverse and that (E) D 0 if and only if ( E) D 0. In other words  D  ı  . By the Radon–Nikodym theorem there Ris a function 2 L1 () such that > 0 a.e. and (E) D E d for all measurable E. In order to obtain an associated operator T on L1 which is an (invertible) isometry one defines T f (x) D (x) f ( x) :

(11)

The dual operator on L1 is then given by T  g D g ı  1 . If  is a  -finite measure equivalent to  which is invariant under  then the ergodic theory of  can be reduced to the measure-preserving case using  . The interesting case is when there is no such . It was an open

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Ergodic Theorems

problem for some time whether there is always an equivalent invariant measure. In 1960 Ornstein [163] gave an example of a  which does not have an equivalent invariant measure. It is curious that, with hindsight, such examples were already known in the fifties to people studying von Neumann algebras. Pn i For f 2 L1 let S n f D iD0 T f .  is said to be conservative if there is no set E with (E) > 0 such that  i E; i D 0; 1; : : : are pairwise disjoint, that is if the Poincaré recurrence theorem remains valid. For example, the shift on Z is not conservative. Hurewicz [105] proved the following ratio ergodic theorem. Theorem 10 Suppose  is conservative, f ; g 2 L1 and a.e. to a -invarig(x) > 0 a.e. Then S n f /S n g converges R f d ant limit h. If  is ergodic then h D R gd . In the case when  is -invariant one has T f D f ı . If  is invariant and finite, taking g D 1 one recovers Birkhoff’s theorem. If  is invariant and -finite then Hurewicz’s theorem becomes the theorem of Hopf alluded to earlier. Wiener and Wintner [204] proved the following variant of Birkhoff’s theorem. Theorem 11 (Wiener–Wintner) Suppose T is an automorphism of a probability space (X; B; ). Then for any f 2 L1 there is a subset X 0 X of measure one such that for each x 2 X 0 and  2 C of modulus 1 the sequence 1 P n1 i i iD0  T f (x) converges. n It is an easy consequence of the ergodic theorem that one has a.e. convergence for a given f and  but the point here is that the set on which the convergence occurs is independent of . Generalizations to Continuous Time and Higher-Dimensional Time A (measure-preserving) flow is a one-parameter group fTt ; t 2 Rg of automorphisms of (X; B; ), that is TtCs D Tt Ts , such that Tt x is measurable as a function of (t; x). It will always be implicitly assumed that the map (t; x) 7! Tt x from R  X to X is measurable. Theorem 4 generalizes to flows by replacing sums with integrals and this generalization follows without difficulty from Theorem 4. (As already observed this observation reverses the historical record.) Theorem 4 may be viewed as a theorem about the “discrete flow” fTn D T n : n 2 Zg. Wiener was the first to generalize Birkhoff’s theorem to families of automorphisms fTg g indexed by groups more general than R or Z. A measure-preserving flow is an action of R while a single automorphism corresponds to an action of Z.

A (measure-preserving) action of a group G is a homomorphism T : g 7! Tg from G into the group of automorphisms of a measure space (X; B; ) (satisfying the appropriate joint measurability condition in case G is not discrete). Suppose now that G D R k or Z k and T is an action of G on (X; B; ). In the case of Z k an action amounts to an arbitrary choice of commuting maps T1 ; : : : Tk , Ti D T(e i ) where ei is the standard basis of Z k . In the case of R k one must specify k commuting flows. Let m denote counting measure on G in case G D Z k and Lebesgue measure in case G D R k . For any subset E of G with m(E) < 1 let Z 1 f (Tg x)dm(g) : (12) A E f (x) D m(E) E One may then ask whether A E f converges to a limit, either in the mean or pointwise, as E varies through some sequence of sets which “grow large” or, in case G D R k , “shrink to 0”. The second case is referred to as a local ergodic theorem. In the case of ergodic theorems at infinity the continuous and discrete theories are rather similar and often the continuous analogue of a discrete result can be deduced from the discrete result. In Wiener [203] proved the following result for actions of G D R k and ergodic averages over Euclidean balls Br D fx 2 R k : kxk2  rg. Theorem 12 Suppose T is an action of R k on (X; B; ) and f 2 L1 (). (a) For f 2 L1 limr!1 A B r f D g exists a.e. If  is finite the convergence also holds with respect to the L1 -norm, R R g is T-invariant and I gd D I f d for every T-invariant set I. (b) limr!0 A B r f D f a.e. The local aspect of Wiener’s theorem is closely related to the Lebesgue differentiation theorem, see for example Proposition 3.5.4 of [132], which, in its simplest form, states Rthat for f 2 L1 (R k ; m) one has a.e. convergence of 1 m(B r ) B r f (x C t)dt to f (x) as r ! 0. The local ergodic theorem implies Lebesgue’s theorem, simply by considering the action of R k on itself by translation. In fact the local ergodic theorem can also be deduced from Lebesgue’s theorem by a simple application of Fubini’s theorem (see for example [135], Chap. 1, Theorem 2.4 in the case k D 1). The key point in Wiener’s proof is the following weak L1 maximal inequality, similar to (6). Theorem 13 Let M f D supr>0 jA B r f j. Then one has fM f > ˛g 

C k f k1 ; ˛

(13)

Ergodic Theorems

where C is a constant depending only on the dimension d. (In fact one may take C D 3d .) In the case when T is the action of Rk on itself by translation (13) is the well-known maximal inequality for the Hardy–Littlewood maximal function ([132], Lemma 3.5.3). Wiener proves (13) by way of the following covering lemma. If B is a ball in R k let B0 denote the concentric ball with three times the radius. Rk

Theorem 14 Suppose a compact subset K of is covered by a (finite) collection U of open balls. Then there exist pairS wise disjoint B i 2 U; i D 1; : : : k such that i B0i covers K. To find the Bi it suffices to let B1 be the largest ball in U, then B2 the largest ball which does not intersect B1 and in general Bn the largest ball which does not intersect Sn1 0 iD1 B i . Then it is not hard to argue that the B i cover K. In general, a covering lemma is, roughly speaking, an assertion that, given a cover U of a set K in some space (often a group), one may find a subcollection U0 U which covers a substantial part of K and is in some sense efficient P in that U 2U0 1U  C, C some absolute constant. Covering lemmas play an important role in the proofs of many maximal inequalities. See Sect. 3.5 of [132] for a discussion of several of the best-known classical covering lemmas. As Wiener was likely aware, the same kind of covering argument easily leads to maximal inequalities and ergodic theorems for averages over “sufficiently regular” sets. For example, in the case of G D Z k use the standard total order  on G and for n 2 N k let S n D fm 2 Z k : 0  m  ng. Let e(n) denote the maximum value of n i /n j . Then one has the following result.

vergence if f 2 L p for some p > 1, provided  is finite. Let T1 ; : : : ; Tk be any k (possibly non-commuting!) automorphisms of a finite measure space (X; B; ). For n n D (n1 ; n2 ; : : : ; n k ) let Tn D T1n 1 : : : Tk k . (This is not an action of Z k unless the T i commute with each other.) As before when F Z k is a finite subset write A F f D 1 P n2F Tn f . Finally let P j f D lim n A n (T j ) f . jFj Theorem 16 For f 2 L p lim A S n f D P1 : : : Pk f

n!1

both a.e. and in Lp . The proof of Theorem 16 uses repeated applications of Birkhoff’s theorem and hinges on (10). Somewhat surprisingly Hurewicz’s theorem was not generalized to higher dimensions until the very recent work of Feldman [79]. In fact the theorem fails if one considers averages over the cube [0; n  1]d in Zd . However Feldman was able to prove a suitably formulated generalization of Hurewicz’s theorem for averages over symmetric cubes. For f 2 L1 (R) the classical Hilbert transform is defined for a.e. t by Z f (t  s) 1 lim ds (16) H f (t) D  !0C jsj> s R f (ts) Let H  f (t) D 1 sup >0 j jsj> s dsj, the corresponding maximal function. The proof that the limit (16) exists a.e. is based on the maximal inequality mfH f > g 

Theorem 15 For any e > 0 and any integrable f lim

n!1;e(n)<e

AS n f

exists a.e. and the limit is characterized by the same properties as in Theorem 12. Moreover there is a maximal inequality ( ) C  sup jA S n f j > ˛  k f k1 ; (14) ˛ e(n)<e

(15)

C k f k1 ; 

(17)

where C is an absolute constant. See Loomis [146] (1946) for a proof of (17) using real-variable methods. Cotlar [65] proved the existence of the following ergodic analogue of the Hilbert transform (actually for the n-dimensional Hilbert transform). Theorem 17 Suppose fTt g is a measure-preserving flow on a probability space (X; B; ) and f 2 L1 . Then Z f (Tt x) dt (18) lim !0
where C is a constant depending only on e and the dimension k.

exists for a.a. x.

If one does not impose a bound on e(n) the a.e. convergence of A S n f may fail for f 2 L1 , (although it does hold for any increasing sequence of n’s by Theorem 30 below). Nonetheless Dunford [72] and independently Zygmund [215] showed that one does have unrestricted con-

Motivated by Cotlar’s result Calderón [54]proved a general transfer principle which allows one to transfer maximal inequalities and convergence theorems for functions of a real or integer variable to the ergodic setting. Although stated for functions on R it applies equally well to R k

247

248

Ergodic Theorems

or Z k . This principle subsumes Birkhoff’s theorem, Wiener’s theorem and Cotlar’s result. For simplicity only a special case of Calderón’s result, in the discrete case, will be stated here. Let T be an automorphism of a probability space (X; B; ), let m denote counting measure on Z and suppose  is a probability measure on Z. For g 2 L1 (m) define P  (g)(n) D i2Z (i)g(nC i). For f 2 L1 () let (T) f D P i i2Z (i)T f , which is easily seen to converge a.e. and in L1 -norm. Given a fixed sequence  n of probabilities define M g D supn n g and M(T) f D supn n (T) f . Theorem 18 (Calderón) Suppose there is a constant C such that mfM g > g <

C kgk1 

for all g 2 L1 (m) :

(19)

Then one also has fM(T) f > g <

C k f k1 

for all T

and g 2 L1 () : (20)

In other words, in order to prove a maximal inequality for general T it suffices to prove it in case T is the shift map on the integers. The idea of transference could already be seen in Wiener’s proof of the Rn ergodic theorem. Transfer principles in various forms of have become an important tool in the study of ergodic theorems. See Bellow [18] for a very readable overview.

If  is a R -finite measure on X one may define the measure P D Px d(x). P is also meaningful if  is a finite signed measure. The case when P D  is the stochastic analogue of measure-preserving dynamics and the case when P   is the analogue of non-singular dynamics. It is easy to see that given any -finite measure  there is always a  such that    and P  . Let L˜ 1 denote the space of finite signed measures  such that   , which is identified with L1 D L1 (; R) via the Radon– Nikodym theorem. If P   then P maps L˜ 1 () into itself so the restriction of P is an operator T on L1 (). T is a positive contraction, that is kTk  1 and T maps nonnegative functions to non-negative functions. As proved in, for example, [153], every positive contraction arises in this way from a substochastic kernel under the assumption that (X; B) is standard. This simply means that there is some complete metric on X for which B is the -algebra of Borel sets. Virtually all measurable spaces encountered in analysis are standard so this should be viewed as a technicality only. See [153], [80] and [135] for more about the relation between kernels and positive contractions. The case when X is finite, P is stochastic and  is a a probability measure is classical in probability theory. P and  determine a probability measure  on X N characterized by its values on cylinder sets, namely for all x1 ; : : : ; x n 2 X fx 2 X N : x(i) D x i ; 1  i  n  1g D (x1 )

n1 Y

Px i (x iC1 ) :

(21)

iD1

Pointwise Ergodic Theorems for Operators Early in the history of ergodic theorems there were attempts to generalize the ergodic theorem to more general linear operators on Lp spaces, that is, operators which do not arise by composition with a mapping of X . In the case p D 1 the main motivation for this comes from the theory of Markov processes. If (X; B) is a measurable space a sub-stochastic kernel on X is a non-negative function P on X  B such that (a) for each x 2 XP(x; ) D Px is a measure on B such that Px (X)  1 and (b) P(; A) is a measurable function for each A 2 B. It is most intuitive to think about the stochastic case, namely when each Px is a probability measure. One then views P(x; A) as the probability that the point x moves into the set A in one unit of time, so one has stochastic dynamics as opposed to deterministic dynamics, namely the case when Px D ıT x for a map T. In this case the measures Px are called transition probabilities.

The co-ordinate functions X i (x) D x(i) on the space X N endowed with the probability  form a Markov process, which will be stationary if and only if  is P-invariant. For a general X the analogous construction is possible provided X is standard. Hopf [101] initiated the systematic study of positive L1 contractions and proved the following ergodic theorem. Theorem 19 Suppose (X; B; ) is a probability space and T is a positive contraction on L1 () satisfying T1 D 1 and P k T  1 D 1. Then limn!1 n1 n1 kD0 T f exists a.e. The importance of Hopf’s article lies less in this convergence result than in the methods he developed. He proved that the maximal inequality (5) generalizes to all positive L1 contractions T and used this to obtain the decomposition of X into its conservative and dissipative parts C and D, characterized by the fact that for any p 2 L1 such that P k p > 0 a.e. 1 kD0 T p is infinite a.e. on C and finite a.e. on D. These results are the cornerstone of much of the subsequent work on L1 contractions.

Ergodic Theorems

Theorem 19 contains Birkhoff’s theorem for an automorphism  of (X; B; ), simply by defining T f D f ı . In fact one can also deduce Theorem 19 from Birkhoff’s theorem (if one assumes only that X is standard). Indeed the hypotheses of Hopf’s theorem imply that the kernel P associated to T is stochastic (T  1 D 1) and and that  is P-invariant (T1 D 1). Hopf’s theorem then follows by applying Birkhoff’s theorem to the shift on the stationary Markov process associated to P and . In fact Kakutani [123] (see also Doob [71]) had already made essentially the same observation, except that his result assumes the stationary Markov process to be already given. In 1955 Dunford and Schwartz [73] made essential use of Hopf’s work to prove the following result. Theorem 20 Suppose (X) D 1 and T is a (not necessarily positive) contraction with respect to both the L1 and L1 norms. Then the conclusion of Hopf’s theorem remains valid. Note that the assumption that T contracts the L1 -norm is meaningful, as L1 L1 . The proof of the result is reduced to the positive case by defining a positive contraction jTj analogous to the total variation of a complex measure. Then in 1960 Chacón and Ornstein [61] proved a definitive ratio ergodic theorem for all positive contractions of L1 which generalizes both Hopf’s theorem and the Hurewicz theorem. Theorem 21 Suppose T is a positive contraction of L1 (), where  is -finite, f ; g 2 L1 and g  0. Then Pn Ti f PiD0 n i iD0 T g converges a.e. on the set f

(22) P1

iD0

T i g > 0g.

In 1963 Chacón [58] proved a very general theorem for non-positive operators which includes the Chacón–Ornstein theorem as well as the Dunford–Schwartz theorem. Theorem 22 Suppose T is a contraction of L1 and p n  0 is a sequence of measurable functions with the property that g 2 L1 ; jgj  p n ) jT gj  p nC1 :

(23)

Then Pn Ti f PiD0 n iD0 p i converges a.e. to a finite limit on the set f

(24) P1

iD0

p i > 0g.

If T is an L1 ; L1 -contraction and p n D 1 for all n then the hypotheses of this theorem are satisfied, so Theorem 22

reduces to the result of Dunford and Schwartz. See [59] for a concise overview of all of the above theorems in this section and the relations between them. The identification of the limit in the Chacón–Ornstein theorem on the conservative part C of X is a difficult problem. It was solved by Neveu [152] in case C D X, and in general by Chacón [57]. Chacón [60] has shown that there is a non-singular automorphism  of (X; B; ) such that for the associated invertible isometry T of L1 () given by (11) there is an f 2 L1 () such that lim sup A n f D 1 and lim inf A n f D 1. In 1975 Akcoglu [3] solved a major open problem when he proved the following celebrated theorem. Theorem 23 Suppose T : L p 7! L p is a positive contracP i tion. Then A n f D n1 n1 iD0 T f converges a.e. Moreover one has the strong Lp inequality     sup jA n f j     n

p

p k f kp : p1

(25)

As usual, the maximal inequality (25) is the key to the convergence. Note that it is identical in form to (10) the classical strong Lp inequality for automorphisms. (25) was proved by A. Ionesco–Tulcea (now Bellow) [106] in the case of positive invertible isometries of Lp . It is a result of Banach [16], see also [140], that in this case T arises from a non-singular automorphism  of (X; B; ) in the form T f D 1/p f ı . By a series of reductions Bellow was able to show that in this case (25) can be deduced from (10). Akcoglu’s brilliant idea was to consider a dilation S of T which is a positive invertible isometry on a larger Lp space L˜ p D L p (Y; C ; ). What this means is that there is a positive isometric injection D : L p ! L˜ p and a positive projection P on L˜ p whose range is D(L p ) such that DT n D PS n D for all n  0. Given the existence of such an S it is not hard to deduce (25) for T from (25) for S. In fact Akcoglu constructs a dilation only in the case when Lp is finite dimensional and shows how to reduce the proof of (25) to this case. In the finite dimensional case the construction is very concrete and P is a conditional expectation operator. Later Akcoglu and Kopp [7] gave a construction in the general case. It is noteworthy that the proof of Akcoglu’s theorem consists ultimately of a long string of reductions to the classical strong Lp inequality (10), which in turn is a consequence of (5). Subadditive and Multiplicative Ergodic Theorems Consider a family fX n;m g of real-valued random variables on a probability space indexed by the set of pairs (n; m) 2

249

250

Ergodic Theorems

Z2 such that 0  n < m. fX(n;m) g is called a (stationary) subadditive process if (a) the joint distribution of fX n;m g is the same as that of fX nC1;mC1 g (b) X n;m  X n;l C X l ;m whenever n < l < m. Denoting the index set by fn < mg Z2 the distribution of the process is a measure  on Rfn<mg which is invariant under the shift Tx(n; m) D x(n C 1; m C 1). Thus there is no loss of generality in assuming that there is an underlying endomorphism T such that X n;m ı T D X nC1;mC1 . In 1968 Kingman [129] proved the following generalizaR tion of Birkhoff’s theorem. D inf n1 X 0;n d is called the time constant of the process. Theorem 24 If the X n;m are integrable and > 1 then 1 n X 0;n converges a.e. and R in L1 -norm to a T -invariant limit ¯ D . X¯ 2 L1 () satisfying Xd It is easy to deduce from the above that if one assumes only C that X 0;1 is integrable then n1 X0;n still converges a.e. to a T-invariant limit X¯ taking values in [1; 1). Subadditive processes first arose in the work of Hammersley and Welsh [99] on percolation theory. Here is an example. Let G be the graph with vertex set Z2 and with edges joining every pair of nearest neighbors. Let E denote the edge set and let fTe : e 2 Eg be non-negative integrable i.i.d. random variables. To each finite path P in G P associate the “travel time” T(P) D e2E Te . For integers m > n  0 let X n;m be the infimum of T(P) over all paths P joining (0; R n) to (0; m). This is a subadditive process with 0  < Te d and it is not hard to see that the underlying endomorphism is ergodic. Thus Kingman’s theorem yields the result that n1 X0;n ! a.e. Suppose now that T is an ergodic automorphism of a probability space (X; B; ) and P is a function on X taking values in the space of d  d real matrices. Define Pn;m D P(T m1 x)P(T m2 x) : : : P(T n x) and let Pn;m (i; j) denote the i; j entry of Pn;m . Then X n;m D log(kPn;m k) (use any matrix norm) is a subadditive process so one obtains the first part of the following result of Furstenberg and Kesten [87] (1960) originally proved by more elaborate methods. The second part can also be deduced from the subadditive theorem with a little more work. See Kingman [130] for details and for some other applications of subadditive processes. Theorem 25 R (a) Suppose logC (jPj)d < 1. Then kP0;n k1/n converges a.e. to a finite limit. (b) Suppose that for each i; j P(i; j) is a strictly positive function such that log P(i; j) is integrable. Then the

1

limit p D lim(P0;n (i; j)) n exists a.e. and is independent of i and j. Partial results generalizing Kingman’s theorem to the multiparameter case were obtained by Smythe [189] and Nguyen [157]. In 1981 Akcoglu and Krengel [8] obtained a definitive multi-parameter subadditive theorem. They d consider an action fTm g of the semigroup G D Z0 by endomorphisms of a measure space (X; B; ). Using the standard total ordering  of G an interval in G is any set of the form fk 2 G : m  k  ng for any m  n 2 G. Let I denote the set of non-empty intervals. Reversing the direction of the inequality, they define a superadditive process as a collection of integrable functions FI ; I 2 I , such that (a) FI ı Tm D FICm , (b) FI  FI 1 C : : : C FI k whenever I is the disjoint union of I1 ; : : : ; I k and R (c) D supI2I jIj1 FI d < 1. A sequence fI n g of sets in I is called regular if there is an increasing sequence I 0n such that I n I 0n and jI 0n j  CjI n j for some constant C. Theorem 26 (Akcoglu–Krengel) Suppose F I is a superadditive process and fI n g is regular. Then jI1n j FI n converges a.e. fFI g is additive if the inequality in (b) is replaced by equalP ity. In this case FI D n2I f ı Tn where f is an integrable function. Thus in the additive case the Akcoglu–Krengel result is a theorem about ordinary multi-dimensional ergodic averages, which is in fact a special case of an earlier result of Tempelman [196] (see Sect. “Amenable Groups” below). Kingman’s proof of Theorem 24 hinged on the existence of a certain (typically non-unique) decomposition for subadditive processes. Akcoglu and Krengel’s proof of the multi-parameter result does not depend on a Kingman-type decomposition, in fact they show that there is no such decomposition in general. They prove a weak maximal inequality fsup jI n j1 FI n > g <

C

; 

(26)

where C is a constant depending only on the dimension, and show that this is sufficient to prove their result. In the case d D 1 the Akcoglu–Krengel argument provides a new and more natural proof of Kingman’s theorem, similar in spirit to Wiener’s arguments. Akcoglu and Sucheston [9] have proved a ratio ergodic theorem for subadditive processes with respect to

Ergodic Theorems

a positive L1 contraction, generalizing both the Chacón– Ornstein theorem and Kingman’s theorem. In 1968 Oseledec [164] proved his celebrated multiplicative ergodic theorem, which gives very precise information about the random matrix products studied by Furstenberg and Kesten. His theorem is an important tool for the study of Lyapunov exponents in differentiable dynamics, see notably Pesin [169]. If A is a d  d matrix let kAk D supfkAxk : kxk D 1g where kxk is the Euclidean norm on Rn . Theorem 27 Suppose T is an endomorphism of the probability space (X; B; ). Suppose P is a measurable function on XR whose values are d  d real matrices such that logC kPkd < 1 and let Pn (x) D P(T n1 x)P(T n2 x) : : : P(x). Then there is a T-invariant subset X 0 of X with measure 1 such that for x 2 X 0 the following hold. 1

(a) lim n!1 (Pn (x)Pn (x)) 2n D A(x) exists. (b) Let 0  exp 1 (x)  exp 2 (x)  : : :  exp r (x) be the distinct eigenvalues of A(x) (r D r(x) may depend on x and 1 may be 1) with multiplicities m1 (x); : : : mr (x). Let E i (x) be the eigenspace corresponding to exp( i (x)) and set F i (x) D E1 (x) C : : : C E i (x) : Then for each u 2 F i (x)nF i1 (x) lim

1 log kPn (x)uk D  i (x) : n

Entropy and the Shannon–McMillan–Breiman Theorem The notion of entropy was introduced by Shannon in his landmark work [185] which laid the foundations for a mathematical theory of information. Suppose (X; B; ) is a probability space, P is a finite measurable partition of X and T is an automorphism of (X; B; ). P(x) denotes the atom of P containing x. The entropy of P is h(P) D 

X Z

D

log((P(x)) d(x)  0 :

(d) If T is ergodic, det P(x) D 1 a.e. and lim sup n

1 n

Z log kPn kd > 0

then the i are constants, 1 < 0 and r > 0. Raghunathan [174] gave a much shorter proof of Oseledec’s theorem, valid for matrices with entries in a locally compact normed field. He showed that it could be reduced to the Furstenberg–Kesten theorem by considering the exterior powers of P. Ruelle [180] extended Oseledec’s theorem to the case where P takes values in the set of bounded operators on a Hilbert space. Walters [199] has given a proof (under slightly stronger hypotheses) which avoids the matrix calculations and tools from multilinear algebra used in other proofs.

(27)

 log((A)) may be viewed as a quantitative measure of the amount of information contained in the statement that a randomly chosen x 2 X happens to belong to A. So h(P) is the expected information if one is about to observe which atom of P a randomly chosen point falls in. See Billingsley [34] for more motivation of this concept. See also the article in this collection on entropy by J. King or any introductory book on ergodic theory, e. g. Petersen [171]. W If P and Q are partitions P Q denotes the common refinement which consists of all sets p\q, p 2 P; q 2 Q. It W is intuitive and not hard to show that h(P Q)  h(P) C W n i h(Q). Now let P0n D n1 iD0 T P and h n D h(P0 ). The subadditivity of entropy implies that h nCm  h n C h m , so by a well-known elementary lemma the limit h(P; T) D lim n

(c) The functions m i and i are T-invariant.

(p) log((p))

p2P

hn hn D inf 0 n n n

(28)

exists. If one thinks of P(x) as a measurement performed on the space X and Tx as the state succeeding x after one second has elapsed then h(P; T) is the expected information per second obtained by repeating the experiment every second for a very long time. See [177] for an alternative and very useful approach to h(P; T) via name-counting. The following result, which is known as the Shannon– McMillan–Breiman theorem, has proved to be of fundamental importance in ergodic theory, notably, for example, in the proof of Ornstein’s celebrated isomorphism theorem for Bernoulli shifts [159]. Theorem 28 If T is ergodic then 1 lim  log((P0n1 (x)) D h(P; T) n

n!1

a.e. and in L1 -norm.

(29)

251

252

Ergodic Theorems

In other words, the actual information obtained per second by observing x over time converges to the constant h(P; T), namely the limiting expected information per second. Shannon [185] formulated Theorem 28 and proved convergence in probability. McMillan [149] proved L1 convergence and Breiman [49] obtained the a.e. convergence. The original proofs of a.e. convergence used the martingale convergence theorem, were not very intuitive and did not generalize to Z n -actions, where the martingale theorem is not available. Ornstein and Weiss [161] (1983) found a beautiful and more natural argument which bypasses the martingale theorem and allows generalization to a class of groups which includes Z n . Amenable Groups Let G be any countable group and T D fTg g an action of G by automorphisms of a probability space. Suppose  is a complex measure on G, that is, f(g)g g2G is an absolutely summable sequence. Let T g act on functions via Tg f D f ı Tg , so T g is an isometry of Lp for every P 1  p  1. Let (T) D g2G (g)Tg . A very general framework for formulating ergodic theorems is to consider a sequence fn g and ask whether n (T) f converges, a.e. or in mean, for f in some Lp space. When n (T) f converges for all actions T and all f in Lp in p-norm or a.e. then one says that  n is mean or pointwise good in Lp . When the  n are probability measures it is natural to call such results weighted ergodic theorems and this terminology is retained for complex  as well. Birkhoff’s theorem says that if G D Z and  n is the normalized counting measure on f0; 1; : : : n  1g then fn g is pointwise good in L1 . This section will be concerned only with sequences fn g such that  n is normalized counting measure on a finite subset Fn G so one speaks of mean or pointwise good sequences fFn g. A natural condition to require of fFn g, which will ensure that the limiting function is invariant, is that it be asymptotically (left) invariant in the sense that 

jgFn Fn j !0 jFn j

8g 2 G :

(30)

Such a sequence is called a Følner sequence and a group G is amenable if it has a Følner sequence. As in most of this article G is restricted to be a discrete countable group for simplicity but most of the results to be seen actually hold for a general locally compact group. Amenability of G is equivalent to the existence of a finitely additive left invariant probability measure on G. It is not hard to see that any Abelian, and more generally

any solvable, group is amenable. On the other hand the free group F 2 on two generators is not amenable. See Paterson [167] for more information on amenable groups. The Følner property by itself is enough to give a mean ergodic theorem. Theorem 29 Any Følner sequence is mean good in Lp for 1  p < 1. The proof of this result is rather similar to the proof of Theorem 3. In fact Theorem 29 is only a special case of quite general results concerning amenable semi-groups acting on abstract Banach spaces. See the book of Paterson [167] for more on this. Turning to pointwise theorems, the Følner condition alone does not yield a pointwise theorem, even when G D Z and the F n are intervals. For example Akcoglu and del Junco [6] have shown that when G D Z and p Fn D [n; n C n] \ Z the pointwise ergodic theorem fails for any aperiodic T and for some characteristic function f . See also del Junco and Rosenblatt [119]. The following pointwise result of Tempelman [196] is often quoted. A Følner sequence fFn g is called regular if there is a constant C such that jFn1 Fn j  CjFn j and there is an increasing sequence Fn0 such that Fn Fn0 and jFn0 j  CjFn j. Theorem 30 Any regular Følner sequence is pointwise good in L1 . In case the F n are intervals in Z n this result can be proved by a variant of Wiener’s covering argument and in the general case by an abstraction thereof. The condition jFn1 Fn j  CjFn j captures the property of rectangles which is needed for the covering argument. Emerson [78] independently proved a very similar result. The work on ergodic theorems for abstract locally compact groups was pioneered by Calderón [53] who built on Wiener’s methods. The main result in this paper is somewhat technical but it already contains the germ of Tempelman’s theorem. Other ergodic theorems for amenable groups, whose main interest lies in the case of continuous groups, include Tempelman [195], Renaud [175], Greenleaf [93] and Greenleaf and Emerson[94]. The discrete versions of these results are all rather close to Tempelman’s theorem. Among pointwise theorems for discrete groups Tempelman’s result was essentially the best available for a long time. It was not known whether every amenable group had a Følner sequence which is pointwise good for some Lp . In 1988 Shulman [187] introduced the notion of a tempered

Ergodic Theorems

Følner sequence fFn g, namely one for which ˇ ˇ ˇ[ ˇ ˇ 1 ˇ F i Fn ˇ < CjFn j ; ˇ ˇ ˇ

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i
for some constant C. The advantage of the tempered condition is that any Følner sequence has a tempered subsequence, and in particular any amenable group has a tempered Følner sequence. Shulman proved a maximal inequality in L2 for such F n which implies that fFn g is pointwise good in L2 . An account of this work may be found in Section 5.6 of Tempelman’s book [194]. Lindenstrauss [145] was able to extend the result to L1 . Theorem 31 Any tempered Følner sequence is pointwise good in L1 . The key new idea in his proof is to use a probabilistic argument to establish a covering lemma. In the discrete case Ornstein and Weiss [201] have given a non-probabilistic proof of Lindenstrauss’s covering lemma. Lindenstrauss also generalizes the a.e. convergence in the Shannon–McMillan–Breiman theorem to this setting. L1 convergence was already established by Kieffer [128] in 1975. Subsequence and Weighted Theorems In this section G and T are as in the previous section and  n is a sequence of complex measures on G. This section will be concerned with conditions on  n that ensure that it is mean or pointwise good. For the most part G will be Z. Hopf’s ergodic theorem gives a class of examples for free. Choose any probability measure  on G, define the operator T D (T) and observe that (T )n D Tn , where n denotes the convolution power. Since T  satisfies the hypotheses of Hopf’s theorem it follows that the P i sequence n D n1 n1 iD0  is pointwise good in L1 . Another sort of example is given by choosing a seP quence g n in G and letting n D n1 n1 iD0 ı g n . If one has convergence for such a sequence one speaks of a subsequence ergodic theorem. This section will mainly focus on the case G D Z and sequences which are the increasing enumeration of a subset S N. Given S N write S;n for the corresponding probabilities  n and say S is good if S;n is good. Perhaps the first subsequence ergodic theorem is due to Blum and Hanson [38] who proved that an automorphism T of a probability space is strongly mixing if and P m i f converges in L norm for every only if n1 n1 2 iD0 T f 2 L2 and every increasing fm i g. Strong mixing means

that (T n A \ B) ! (A)(B). In 1969 Brunel and Keane [51] proved the first pointwise subsequence ergodic theorem Theorem 32 Suppose that T is a translation on a compact Abelian group G with Haar measure , g 2 G, and E G is any Borel set with (E) > 0 and (@E) D 0. Let S D fi > 0 : T i g 2 Eg. Then S is pointwise good in L1 . Krengel [133] constructed the first example of a sequence S N which is pointwise universally bad, in the strong sense that for any aperiodic T the a.e. convergence of S;n (T) f fails for some characteristic function f . Bellow [17] proved that any lacunary sequence (meaning a nC1 > ca n for some c > 1) is pointwise universally bad in L1 . Later Akcoglu at al. [1] were able to show that for lacunary sequences fS;n g is even strongly sweeping out. A sequence fn g of probability measures on Z is said to be strongly sweeping out if for any ergodic TRand for all ı > 0 there is a characteristic function f with f d < ı such that lim sup n (T) f D 1 a.e. It is not difficult to show that if fn g is strongly sweeping out then there are characteristic functions f such that lim inf n (T) f D 0 and lim sup n (T) f D 1. Thus for lacunary sequences the ergodic theorem fails in the worst possible way. Bellow and Losert [21] gave the first example of a sequence S Z of density 0 which is universally good for pointwise convergence, answering a question posed by Furstenberg. They construct an S which is pointwise good in L1 . This paper also contains a good overview of the progress on weighted and subsequence ergodic theorems at that time. Weyl’s theorem on uniform distribution (Theorem 9) suggests the possibility of an ergodic theorem for the sequence fn2 g. It is not hard to see that fn2 g is mean good in L2 . In fact the spectral theorem and the dominated convergence theorem show that it is enough to prove that theL1 P n 2 on the unit bounded sequence of functions n1 n1 iD0 z circle converges at each point z of the unit circle. When z is not a root of unity the sequence converges to 0 by Weyl’s result and when z is a root of unity the convergence is 2 trivial because fz n g is periodic. In 1987 Bourgain [39,43] proved his celebrated pointwise ergodic theorem for polynomial subsequences. Theorem 33 If p is any polynomial with rational coefficients taking integer values on the integers then S D fp(n)g is pointwise good in L2 . The first step in Bourgain’s argument is to reduce the problem of proving a maximal inequality to the case of the shift map on the integers, via Calderón’s transfer princi-

253

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Ergodic Theorems

ple. Then the problem is transferred to the circle by using Fourier transforms. At this point the problem becomes a very delicate question about exponential sums and a whole arsenal of tools is brought to bear. See Rosenblatt and Wierdl [176] and Quas and Wierdl [28](Appendix B) for nice expositions of Bourgain’s methods. Bourgain subsequently improved this to all Lp , p > 1 and also extended it to sequences f[q(n)]g where now q is an arbitrary real polynomial and [] denotes the greatest integer function. He also announced that his methods can be used to show that the sequence of primes is pointwise p 1C 3 good in Lp for any p > 2 . Wierdl [205] (1988) soon extended the result for primes to all p > 1. Theorem 34 The primes are pointwise good in Lp for p >1. It has remained a major open question for quite some time whether any of these results hold for p D 1. In 2005 there appeared a preprint of Mauldin and Buczolich [148], which remains unpublished, showing that polynomial sequences are L1 -universally bad. Another major result of Bourgain’s is the so-called return times theorem [44]. A simplification of Bourgain’s original proof was published jointly with Furstenberg, Katznelson and Ornstein as an appendix to an article [47] of Bourgain. To state it let us agree to say that a a sequence of complex numbers fa(n)gn0 has property P if the seP quence of complex measures n D n1 n1 iD0 a(i)ı i has property P, where ı i denotes the point mass at i. Theorem 35 (Bourgain) Suppose T is an automorphism of a probability space (X; B; ), 1  p; q  1 are conjugate exponents and f 2 L p (). Then for almost all x the sequence f f (T n x)g is pointwise good in Lq . Applying this to characteristic functions f D 1E one sees that the return time sequence fi > 0 : S i x 2 Eg is good for pointwise convergence in L1 . Theorem 32 is a very special case. It is also easy to see that Theorem 35 contains the Wiener–Wintner theorem. In 1998 Rudolph [179] proved a far-reaching generalization of the return times theorem using the technique of joinings. For an introduction to joinings see the article by de la Rue in this collection and also Thouvenot [198], Glasner [90] (2003) and Rudolph’s book [177]. Rudolph’s result concerns the convergence of multiple averages N1 k 1 XY f j (T jn x) N nD0

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jD1

where each T j is an automorphism of a probability space (X j ; B j ;  j ) and the f j are L1 functions. The point is that the convergence occurs whenever each x j 2 X 0j , sets of

measure one which may be chosen sequentially for j D 1; : : : ; k without knowing what T i or f i are for any i > j. He actually proves something stronger, namely he identifies an intrinsic property of a sequence fa i g, which he calls fully generic, such that the following hold. (a) The constant sequence {1} is fully generic. (b) If fa i g is fully generic then for any T and f 2 L1 the sequence a i f (T i x) is fully generic for almost all x. (c) Fully generic implies pointwise good in L1 . The definition of fully generic will not be quoted here as it is somewhat technical. For a proof of the basic return times theorem using joinings see Rudolph [178]. Assani, Lesigne and Rudolph [13] took a first step towards the multiple theorem, a Wiener–Wintner version of the return times theorem. Also Assani [11] independently gave a proof of Rudolph’s result in the case when all the T j are weakly mixing. Ornstein and Weiss [162] have proved the following version of the return times theorem for abstract discrete groups. As with Z, let us say that a sequence fa g g g2G of complex numbers has property P for fFn g if the sequence P n D jF1 j g2Fn a(g)ı g of complex measures has propn erty P. Theorem 36 Suppose that the increasing Følner sequence fFn g satisfies the Tempelman condition supn jFn1 Fn j/ S jFn j < 1 and Fn D G. If b 2 L1 then for a.a. x the sequence fb(Tg x)g is pointwise good in L1 for fFn g. Recently Demeter, Lacey, Tao and Thiele [67] have proved that the return times theorem remains valid for any 1 < p  1 and q  2. On the other hand Assani, Buczolich and Mauldin [14] (2005) showed that it fails for p D q D 1. Bellow, Jones and Rosenblatt have a series of papers [22,23,24,25] studying general weighted averages associated to a sequence  n of probability measures on Z, and, in some cases, more general groups. The following are a few of their results. [23] is concerned with Z-actions and moving block averages given by n D m I n , where the I n are finite intervals and mI denotes normalized counting measure on I. They resolve the problem completely, obtaining a checkable necessary and sufficient condition for such a sequence to be pointwise good in L1 . [24] gives sufficient conditions on a sequence  n for it to be pointwise good in Lp , p > 1, via properties of the Fourier transforms ˆ n . A particular consequence is P that if lim n!1 k2Z jn (k)  n (k  1)j D 0 then fn g has a subsequence which is pointwise good in Lp ,

Ergodic Theorems

p > 1. In [25] they obtain convergence results for sequences n D  n , the convolution powers of a probability measure . A consequence of one of their main results P is that if the expectation k2Z k(k) is zero, the second P moment k2Z k 2 (k) is finite and  is aperiodic (its support is not contained in any proper coset in Z) then  n is pointwise good in Lp for p > 1. Bellow and Calderón [19] later showed that this last result is valid also for p D 1. This is a consequence of the following sufficient condition for a sequence T to satisfy a weak L1 inequality. Given an automorphism of a probability space (X; B; ) let M f D sup jn (T) f (x)j be the maximal operator associated to fn g. Theorem 37 (Bellow and Calderón) Suppose there is an ˛ 2 (0; 1] and C > 0 such that for each n > 1 one has jn (x C y)  n (x)j  C

jyj˛ jxj1C˛

for all x; y 2 Z

Theorem 39 Suppose T is an automorphism of a probability space (X; B; ), (B) > 0 and k  1. Then there is T an n > 0 such that ( kiD1 T in B) > 0. In 1977, Furstenberg [85] proved the following ergodic theorem which implies the multiple recurrence theorem. He also established a general correspondence principle which puts the shaky analogy between the multiple recurrence theorem and Szemerédi’s theorem on a firm footing and allows each to be deduced from the other. Thus he obtained an ergodic theoretic proof of Szemerédi’s combinatorial result. Theorem 40 Suppose T is an automorphism of a probabilR ity space (X; B; ), f 2 L1 , f  0, f d > 0 and k  1. Then lim inf N

N1 Z k 1 X Y in T f d > 0 : N nD0

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iD1

such that 0 < 2jyj  jxj Then there is a constant D such that fM f > g 

D k f k1 

f 2 L1 ()

for all T;

and  > 0 :

Ergodic Theorems and Multiple Recurrence Suppose S N. The upper density of S is ¯ D lim sup jS \ [1; n]j : d(S) n n

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and the density d(S) is the limit of the same quantity, if it exists. In 1975 Szemerédi [190] proved the following celebrated theorem, answering an old question of Erd˝os and Turán. Theorem 38 Any subset of N with positive upper density contains an arithmetic progression of length k for each k  1. This result has a distinctly ergodic-theoretic flavor. Letting T denote the shift map on Z, it says that for each k there T is an n such that S 0 D kiD1 T i n S is non-empty. In fact ¯ 0 ) > 0. the result gives more: there is an n for which d(S In this light Szemerédi’s theorem becomes a multiple recurrence theorem for the shift map on N, equipped with ¯ Of course d¯ is not the invariant “measure-like” quantity d. even finitely additive so it is not a measure. d, however, is at least finitely additive, when defined, and d(N) D 1. This point of view suggests the following multiple recurrence theorem.

Furstenberg’s result opened the door to the study of socalled ergodic Ramsey theory which has yielded a vast array of deep results in combinatorics, many of which have no non-ergodic proof as yet. The focus of this article is not on this direction but the reader is referred to Furstenberg’s book [86] for an excellent introduction and to Bergelson [27,28] for surveys of later developments. There is also the article by Frantzikinakis and McCutcheon in this collection. Furstenberg’s proof relies on a deep structure theorem for a general automorphism which was also developed independently by Zimmer [213], [212] in a more general context. A factor of T is any sub- -algebra F B such that T(F ) D F . (It is more accurate to think of the factor as the action of T on the measure space (X; F ; jF ).) The structure theorem asserts that there is a transfinite increasing sequence of factors fF˛ g of T such that the following conditions hold. (a) F˛C1 is compact relative to F˛ . W (b) F˛ D ˇ <˛ Fˇ whenever ˛ is a limit ordinal. W (c) B is weakly mixing relative to F˛ . W ( denotes the  -algebra generated by a collection of  -alW gebras.) ˛ F˛ is called the maximal distal factor of T and W if F˛ D B then T is called distal. The definitions of the relative properties in (a) and (c) above are somewhat technical so only their absolute versions (i. e. relative to the trivial -algebra f;; Xg) will be described here. T is said to be compact if for each f 2 L2 the orbit fT i f : i 2 Zg is pre-compact in the norm topology of L2 . This turns out to be equivalent to the statement that T is

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Ergodic Theorems

a translation on a compact Abelian group endowed with its Haar measure. The property of weak mixing is a fundamental notion in ergodic theory which has many equivalent definitions. The most appropriate for our purposes is that T is weakly mixing if it has no compact factors. This turns out to be equivalent to the ergodicity of T  T acting on the product measure space (X; B; )  (X; B; ). The verification of 37 in the case of compact T is rather easy. In this case it is not hard to prove that for any f 2 L1 and > 0 the set fn 2 Z : kT n f  f k2 < g has bounded gaps and (34) follows easily. In the case when T is weakly mixing (34) is a consequence of the following theorem which Furstenberg proves in [85] (as a warm-up for it’s much harder relative version). Theorem 41 If T is weakly mixing and f1 ; f2 ; : : : ; f k are L1 functions then lim N

N1 Z k k Z Y 1 X Y in fi T f i d D N nD0 iD1

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iD1

Later Bergelson [26] showed that the result can be obtained easily by an induction argument using the following Hilbert space generalization of van der Corput’s lemma. Theorem 42 (Bergelson) Suppose fx n g is a bounded sequence of vectors in Hilbert space such that for each h > 0 P N1 one has N1 nD0 hx nCh ; x n i ! 0 as N ! 1. Then P N1 1 k N nD0 x n k ! 0. Ryzhikov has also given a beautiful short proof of Theorem 41 using joinings ([182]). Bergelson’s van der Corput lemma and variants of it have been a key tool in subsequent developments in ergodic Ramsey theory and in the convergence results to be be discussed in this section. Bergelson [26] used it to prove the following mean ergodic theorem for weakly mixing automorphisms. Theorem 43 Suppose T is weakly mixing, f1 ; : : : ; f k are L1 functions and p1 ; : : : ; p k are polynomials with rational coefficients taking integer values on the integers such that no p i  p j is constant for i ¤ j. Then   N1 k k Z  1 XY Y   p i (n) f i d D 0 : (36) T fi  lim   N!1  N nD0 iD1

iD1

Theorems 40 and 41 immediately raise the question of P N1 Q k in convergence of the multiple averages N1 nD0 iD1 T f i for a general T. Several authors obtained partial results on the question of mean convergence. It was finally resolved only recently by Host and Kra [104], who proved the following landmark theorem.

Theorem 44 Suppose f1 ; f2 ; : : : ; f k 2 L1 . Then there is a g 2 L1 such that   N1 k  1 XY   T in f i  g  D 0 : lim   N nD0 iD1

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2

Independently and somewhat later Ziegler [211]obtained the same result by somewhat different methods. Furstenberg had already established Theorem 44 for k D 2 in [85]. It was proved for k D 3 in the case of a totally ergodic T by Conze and Lesigne [63]and in general by Host and Kra [102]. It can also be obtained using the methods developed by Furstenberg and Weiss [88]. In this paper Furstenberg and Weiss proved a result for polynomial powers of T in the case k D 2. They also formalized the key notion of a characteristic factor. A factor C of T is said to be characteristic for the averages (37) if, roughly speaking, the L2 limiting behavior of the averages is unchanged when any one of the f i ’s is replaced by its conditional expectation on C . This means that the question of convergence of these averages may be reduced to the case when f i are all C -measurable. So the problem is to find the right (smallest) characteristic factor and prove convergence for that factor. The importance of characteristic factors was already apparent in Furstenberg’s original paper [85], where he showed that the maximal distal factor is characteristic for the averages (37). In fact he showed that for a given k a k-step distal factor is characteristic. (An automorphism is k-step distal if it is the top rung in a k-step ladder of factors as in the Furstenberg–Zimmer structure-theorem.) It turns out, though, that the right characteristic factor for (37) is considerably smaller. In their seminal paper [63] Conze and Lesigne identified the characteristic factor for k D 3, now called the Conze–Lesigne factor. As shown in [104], and [211], the characteristic factor for a general k is (isomorphic to) an inverse limit of k-step nilflows. A k-step nilflow is a compact homogeneous space N/ of a k-step nilpotent Lie group N, endowed with its unique left-invariant probability measure, on which T acts via left translation by an element of N. Ergodic properties of nilflows have been studied for some time in ergodic theory, for example in Parry [166]. In this way the problem of L2 -convergence of (37) is reduced to the case when T is a nilflow. In this case one has more: the averages converge pointwise by a result of Leibman [141] (See also Ziegler [210]). There have already been a good many generalizations of (37). Host and Kra [103], Frantzikinakis and Kra [82, 83], and Leibman [141] have proved results which replace

Ergodic Theorems

linear powers of T by polynomial powers. In increasing degrees of generality Conze and Lesigne [63], Frankzikinakis and Kra [81] and Tao [192] have obtained results which replace the maps T; T 2 ; : : : ; T k in (37) by commuting maps T1 ; : : : ; Tk . Bergelson and Leibman [30,31] have obtained results, both positive and negative, in the case of two noncommuting maps. In the direction of pointwise convergence the only general result is the following theorem of Bourgain [48] which asserts pointwise convergence in the case k D 2. Theorem 45 Suppose S and T are powers of a single autoPN f (T n x)g(S n x) morphism R and f ; g 2 L1 . Then N1 nD1 converges a.e. When T is a K-automorphism Derrien and Lesigne [68] have proved that the averages (35) converge pointwise to the product of the integrals, even with polynomial powers of T replacing the linear powers. Gowers [91] has given a new proof of Szemerédi’s theorem by purely finite methods using only harmonic analysis on Z n . His results give better quantitative estimates in the statement of finite versions of Szemerédi’s theorem. Although his proof contains no ergodic theory it is to some extent guided by Furstenberg’s approach. This section would be incomplete without mentioning the spectacular recent result of Green and Tao [92] on primes in arithmetic progression and the subsequent extensions of the Green-Tao theorem due to Tao [191] and Tao and Ziegler [193]. Rates of Convergence There are many results which say in various ways that, in general, there are no estimates for the rate of convergence of the averages A n f in Birkhoff’s theorem. For example there is the following result of Krengel [134]. Theorem 46 Suppose limn!1 c n D 1 and T is any ergodic automorphism of a probability space. Then there R is a bounded measurable f with f d D 0 such that lim sup c n A n f D 1 a.e. See Part 1 of Derriennic [69] for a selection of other results in this direction. In spite of these negative results one can obtain quantitative estimates by reformulating the ergodic theorem in various ways. Bishop [37] proved the following result which is purely finite and constructive in nature and evidently implies the a.e. convergence in Birkhoff’s theorem. If y D (y1 ; : : : ; y n ) is a finite sequence of real numbers and a < b, an upcrossing of y over [a; b] is a minimal integer interval [k; l] [1; n] satisfying y k < a and y l > b.

Theorem 47 Suppose T is an automorphism of the probability space (X; B; ). Let U(n; a; b; f ; x) be the number of upcrossings of the sequence A0 f (x); : : : ; A n f (x) over [a; b]. Then for every n fx : U(n; a; b; f ; x) > kg 

k f k1 : (b  a)k

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Ivanov [107] has obtained the following stronger upcrossing inequality for an arbitrary positive measurable f , which also implies Birkhoff’s theorem. Theorem 48 For any positive measurable f and 0 < a kg 

 a k b

:

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Note the exponential decay and the remarkable fact that the estimate does not depend on f . Ivanov has also obtained the following result (Theorem 23 in [121]) about fluctuations of A n f . An -fluctuation of a real sequence y D (y1 : : : y n ) is a minimal integer interval [k; l] satisfying jy k  y l j  . If f 2 L1 (R) let F( ; f ; x) be the number of -fluctuations of the sequence fA n f (x)g1 nD0 . Theorem 49 1

fx : F( ; f ; x)  kg  C

(log k) 2 k

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where C is a constant depending only on k f k1 / . If f 2 L1 then fx : F( ; f ; x)  kg  AeBk ;

(41)

where A and B are constants depending only on k f k1 / . See Kachurovskii’s survey [121] of results on rates of convergence in the ergodic theorem and also the article of Jones et al. [115] for more results on oscillation type inequalities. Under special assumptions on f and T it is possible to give more precise results on speed of convergence. If the sequence T i f is independent then there is a vast literature in probability theory giving very precise results, for example the central limit theorem and the law of the iterated logarithm (see, for example, [50]). See the surveys of Derrienic [69] (2006) and of Merlevède, Peligrad and Utev [150] for results on the central limit theorem for dynamical systems.

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Ergodic Theorems

Ergodic Theorems for Non-amenable Groups Guivarch [95] was the first to prove an ergodic theorem for a general pair of non-commuting unitary operators. Much work has been done in the last 15 years on mean and pointwise theorems for non-amenable groups. See the extensive survey by Nevo [155]. Here is just one result of Nevo [154] as a sample. Suppose G is a discrete group and fn g is a sequence of probability measures on G. Say that fn g is mean ergodic if for any unitary representation  of G on a Hilbert space H and any x 2 H one P has g2G n (g)(g)x converges in norm to the projection of x on the subspace of -invariant vectors. Say that fn g is pointwise ergodic if for any measure preserving action T of G on a probability space and f 2 L2 one has P g2G  n (g)Tg f converges a.e. to the projection of f on the subspace of T-invariant functions. Theorem 50 Let G be the free group on k generators, k  1. Let  n be the normalized counting measure on the set of elements whose word length (in terms of the generators and their inverses) is n. Let n D (n C nC1 )/2 and P n0 D n1 niD1  i . Then  n and n0 are mean and pointwise ergodic but  n is not mean ergodic. Future Directions This section will be devoted to a few open problems and some questions. Many of these have been suggested by my colleagues acknowledged in the introduction. The topic of convergence of Furstenberg’s multiple averages has seen some remarkable achievements in the last few years and will likely continue to be vital for some time to come. The question of pointwise convergence of multiple averages is completely open, beyond Bourgain’s result (Theorem 45). Even extending Bourgain’s result to three different powers of R or to any two commuting automorphisms S and T would be a very significant achievement. Another natural question is whether one has pointwise convergence of the averages of the sequence 2 f (T n (x))g(T n (x)), which are mean convergent by the result of Furstenberg and Weiss [88] (which is now subsumed in much more general results). A long-standing open problem relating to Akcoglu’s ergodic theorem (Theorem 23) for positive contractions of Lp is whether it can be extended to power-bounded operators T (this means that kTkn is bounded). It is also an open question whether it extends to non-positive contractions, excepting the case p D 2 where Burkholder [52] has shown that it fails. There are many natural questions in the area of subsequence and weighted ergodic theorems. For example,

which of Bourgain’s several pointwise theorems can be extended to L1 ? Are there other natural subsequences of an arithmetic character which have density 0 and for which an ergodic theorem is valid, either mean or pointwise, and in what Lp spaces might such theorems be valid? Are there higher dimensional analogues? Since lacunary sequences are bad, to have any sort of an pointwise theorem l n D log a nC1 must get close to 0 and for simplicity let us assume that lim l n D 0. How fast must the convergence be in order to get an ergodic theorem? Jones and Wierdl [113] have shown that if l n > 1 (log n) 2 C then the pointwise ergodic theorem fails in L2 while Jones, Lacey and Wierdl [116] have shown that an only slightly faster rate permits a sequence which is pointwise good in L2 . How well or badly does the ergodic theorem succeed or fail depending on the rate of convergence of ln ? In particular is there a (slow) rate which still guarantees strong sweeping out? [116] contains some interesting conjectures in this direction. There are also interesting questions concerning the mean and pointwise ergodic theorems for subsequences which are chosen randomly in some sense. See Bourgain [42] and [116] for some results in this direction. Again [116] contains some interesting conjectures along these lines. In a recent paper [32] Bergelson and Leibman prove some very interesting and surprising results about the distribution of generalized polynomials. A generalized polynomial is any function which can be built starting with polynomials in R[x] using the operations of addition, multiplication and taking the greatest integer. As a consequence they derive a generalization of von Neumann’s mean ergodic theorem to averages along generalized polynomial sequences. The following is a special case. Theorem 51 Suppose p is a generalized polynomial taking integer values on the integers and U is a unitary operator P p(i) x is norm convergent for all x 2 on H . Then n1 n1 iD0 U H. This begs the question: does one have pointwise convergence? If so, this would be a far-reaching generalization of Bourgain’s polynomial ergodic theorem. There are also lots of questions concerning the nature of Følner sequences fFn g in an amenable group which give a pointwise theorem. For example Lindenstrauss [145] has shown that in the lamplighter group, a semi-direct product L of Z with i2Z Z/2Z on which Z acts by the shift, there is no sequence satisfying Tempelman’s condition and that any fFn g satisfying the Shulman condition must grow super-exponentially. So, it is natural to ask for slower rates of growth. In particular, in any amenable group is there al-

Ergodic Theorems

ways a sequence fFn g which is pointwise good and grows at most exponentially? Can one do better either in general or in particular groups? Lindenstrauss’s theorem at least guarantees the existence of Følner sequences which are pointwise good in L1 but in particular groups there are often natural sequences L which one hopes might be good. For example in 1 iD1 Z one may take F n to be a cube based at 0 of sidelength ln and dimension dn (that is, all but the first dn co-ordinates are zero), where both sequences increase to 1. What conditions on ln and dn will give a good sequence? Note that no such sequence is regular in Tempelman’s sense. If d n D n then fl n g must be superexponential to ensure Shulman’s condition. Can one do better? What about l n D d n D n? Bibliography 1. Akcoglu M, Bellow A, Jones RL, Losert V, Reinhold–Larsson K, Wierdl M (1996) The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergodic Theory Dynam Systems 16(2):207–253 2. Akcoglu M, Jones RL, Rosenblatt JM (2000) The worst sums in ergodic theory. Michigan Math J 47(2):265–285 3. Akcoglu MA (1975) A pointwise ergodic theorem in Lp-spaces. Canad J Math 27(5):1075–1082 4. Akcoglu MA, Chacon RV (1965) A convexity theorem for positive operators. Z Wahrsch Verw Gebiete 3:328–332 (1965) 5. Akcoglu MA, Chacon RV (1970) A local ratio theorem. Canad J Math 22:545–552 6. Akcoglu MA, del Junco A (1975) Convergence of averages of point transformations. Proc Amer Math Soc 49:265–266 7. Akcoglu MA, Kopp PE (1977) Construction of dilations of positive Lp -contractions. Math Z 155(2):119–127 8. Akcoglu MA, Krengel U (1981) Ergodic theorems for superadditive processes. J Reine Angew Math 323:53–67 9. Akcoglu MA, Sucheston L (1978) A ratio ergodic theorem for superadditive processes. Z Wahrsch Verw Gebiete 44(4):269– 278 10. Alaoglu L, Birkhoff G (1939) General ergodic theorems. Proc Nat Acad Sci USA 25:628–630 11. Assani I (2000) Multiple return times theorems for weakly mixing systems. Ann Inst H Poincaré Probab Statist 36(2):153–165 12. Assani I (2003) Wiener–Wintner ergodic theorems. World Scientific Publishing Co. Inc., River Edge, NJ 13. Assani I, Lesigne E, Rudolph D (1995) Wiener–Wintner returntimes ergodic theorem. Israel J Math 92(1–3):375–395 14. Assani I, Buczolich Z, Mauldin RD (2005) An L1 counting problem in ergodic theory. J Anal Math 95:221–241 15. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. With the assistance of Markus L, Massey W and an appendix by Greenberg L Annals of Mathematics Studies, No 53. Princeton University Press, Princeton, NJ 16. Banach S (1993) Théorie des opérations linéaires. Éditions Jacques Gabay, Sceaux, reprint of the 1932 original 17. Bellow A (1983) On “bad universal” sequences in ergodic theory II. In: Belley JM, Dubois J, Morales P (eds) Measure theory and its applications. Lecture Notes in Math, vol 1033. Springer, Berlin, pp 74–78

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Ergodic Theory: Basic Examples and Constructions MATTHEW N ICOL1 , KARL PETERSEN2 Department of Mathematics, University of Houston, Houston, USA 2 Department of Mathematics, University of North Carolina, Chapel Hill, USA

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Article Outline Glossary Definition of the Subject Introduction Examples Constructions Future Directions Bibliography Glossary  A transformation T of a measure space (X; B; ) is measure-preserving if (T 1 A) D (A) for all measurable A 2 B.  A measure-preserving transformation (X; B; ; T) is ergodic if T 1 (A) D A (mod ) implies (A) D 0 or (Ac ) D 0 for each measurable set A 2 B.  A measure-preserving transformation (X; B; ; T) of a probability space is weak-mixing if lim n!1 n1 P n1 i iD0 j(T A \ B)  (A)(B)j D 0 for all measurable sets A; B 2 B. A measure-preserving transformation (X; B; ; T) of a probability space is strong-mixing if limn!1  (T n A\B) D (A)(B) for all measurable sets A; B 2 B.  A continuous transformation T of a compact metric space X is uniquely ergodic if there is only one T-invariant Borel probability measure on X. A continous transformation of a topological space X is topologically mixing for any two open sets U; V X there exists N > 0 such that T n (U) \ V ¤ ;, for each n  N.  Suppose (X; B; ) is a probability space. A finite partition P of X is a finite collection of disjoint (mod , i. e., up to sets of measure 0) measurable sets fP1 ; : : : ; Pn g such that X D [Pi (mod ). The entropy of P with reP spect to  is H(P ) D  i (Pi ) ln (Pi ) (other bases are sometimes used for the logarithm).  The metric (or measure-theoretic) entropy of T with respect to P is h (T; P ) D lim n!1 n1 H(P _    _ T nC1 (P )), where P _    _ T nC1 (P ) is the partition of X into sets of points with the same coding with







 









respect to P under T i , i D 0; : : : ; n  1. That is x, y are in the same set of the partition P _    _ T nC1 (P ) if and only if T i (x) and T i (y) lie in the same set of the partition P for i D 0; : : : ; n  1. The metric entropy h (T) of (X; B; ; T) is the supremum of h (T; P ) over all finite measurable partitions P . If T is a continuous transformation of a compact metric space X, then the topological entropy of T is the supremum of the metric entropies h (T), where the supremum is taken over all T-invariant Borel probability measures. A system (X; B; ; T) is loosely Bernoulli if it is isomorphic to the first-return system to a subset of positive measure of an irrational rotation or a (positive or infinite entropy) Bernoulli system. Two systems are spectrally isomorphic if the unitary operators that they induce on their L2 spaces are unitarily equivalent. A smooth dynamical system consists of a differentiable manifold M and a differentiable map f : M ! M. The degree of differentiability may be specified. Two submanifolds S1 , S2 of a manifold M intersect transversely at p 2 M if Tp (S1 ) C Tp (S2 ) D Tp (M). An ( -) small Cr perturbation of a Cr map f of a manifold M is a map g such that dC r ( f ; g) < i. e. the distance between f and g is less than in the Cr topology. A map T of an interval I D [a; b] is piecewise smooth (Ck for k  1) if there is a finite set of points a D x1 < x2 <    < x n D b such that Tj(x i ; x iC1 ) is Ck for each i. The degree of differentiability may be specified. A measure  on a measure space (X; B) is absolutely continuous with respect to a measure  on (X; B) if (A) D 0 implies (A) D 0 for all measurable A 2 B. A Borel measure  on a Riemannian manifold M is absolutely continuous if it is absolutely continuous with respect to the Riemannian volume on M. A measure  on a measure space (X; B) is equivalent to a measure  on (X; B) if  is absolutely continuous with respect to  and  is absolutely continuous with respect to .

Definition of the Subject Measure-preserving systems are a common model of processes which evolve in time and for which the rules governing the time evolution don’t change. For example, in Newtonian mechanics the planets in a solar system undergo motion according to Newton’s laws of motion: the planets

Ergodic Theory: Basic Examples and Constructions

move but the underlying rule governing the planets’ motion remains constant. The model adopted here is to consider the time-evolution as a transformation (either a map in discrete time or a flow in continuous time) on a probability space or more generally a measure space. This is the setting of the subject called ergodic theory. Applications of this point of view include the areas of statistical physics, classical mechanics, number theory, population dynamics, statistics, information theory and economics. The purpose of this chapter is to present a flavor of the diverse range of examples of measure-preserving transformations which have played a role in the development and application of ergodic theory and smooth dynamical systems theory. We also present common constructions involving measurepreserving systems. Such constructions may be considered a way of putting ‘building-block’ dynamical systems together to construct examples or decomposing a complicated system into simple ‘building-blocks’ to understand it better. Introduction In this chapter we collect a brief list of some important examples of measure-preserving dynamical systems, which we denote typically by (X; B; ; T) or (T; X; B; ) or slight variations. These examples have played a formative role in the development of dynamical systems theory, either because they occur often in applications in one guise or another or because they have been useful simple models to understand certain features of dynamical systems. There is a fundamental difference in the dynamical properties of those systems which display hyperbolicity: roughly speaking there is some exponential divergence of nearby orbits under iteration of the transformation. In differentiable systems this is associated with the derivative of the transformation possessing eigenvalues of modulus greater than one on a ‘dynamically significant’ subset of phase space. Hyperbolicity leads to complex dynamical behavior such as positive topological entropy, exponential divergence of nearby orbits (“sensitivity to initial conditions”) often coexisting with a dense set of periodic orbits. If ; are sufficiently regular functions on the phase space X of a hyperbolic measure-preserving transformation (T; X; ), then typically we have fast decay of correlations in the sense that ˇZ ˇ Z Z ˇ ˇ ˇ (T n x) (x) d   d ˇ  Ca(n) d ˇ ˇ X

where a(n) ! 0. If a(n) ! 0 at an exponential rate we say that the system has exponential decay of correlations. A theme in dynamical systems is that the time se-

ries formed by sufficiently regular observations on systems with some degree of hyperbolicity often behave statistically like independent identically distributed random variables. At this point it is appropriate to point out two pervasive differences between the usual probabilistic setting of a stationary stochastic process fX n g and the (smooth) dynamical systems setting of a time series of observations on a measure-preserving system f ı T n g. The most crucial is that for deterministic dynamical systems the time series is usually not an independent process, which is a common assumption in the strictly probabilistic setting. Even if some weak-mixing is assumed in the probabilistic setting it is usually a mixing condition on the  -algebras Fn D  (X1 ; : : : ; X n ) generated by successive random variables, a condition which is not natural (and usually very difficult to check) for dynamical systems. Mixing conditions on dynamical systems are given more naturally in terms of the mixing of the sets of the  -algebra B of the probability space (X; B; ) under the action of T and not by mixing properties of the  -algebras generated by the random variables f ı T n g. The other difference is that in the probabilistic setting, although fX n g satisfy moment conditions, usually no regularity properties, such as the Hölder property or smoothness, are assumed. In contrast in dynamical systems theory the transformation T is often a smooth or piecewise smooth transformation of a Riemannian manifold X and the observation  : X ! R is often assumed continuous or Hölder. The regularity of the observation  turns out to play a crucial role in proving properties such as rates of decay of correlation, central limit theorems and so on. An example of a hyperbolic transformation is an expanding map of the unit interval T(x) D (2x) (where (x) is x modulo the integers). Here the derivative has modulus 2 at all points in phase space. This map preserves Lebesgue measure, has positive topological entropy, Lebesgue almost every point x has a dense orbit and periodic points for the map are dense in [0; 1). Non-hyperbolic systems are of course also an important class of examples, and in contrast to hyperbolic systems they tend to model systems of ‘low complexity’, for example systems displaying quasiperiodic behavior. The simplest non-trivial example is perhaps an irrational rotation of the unit interval [0; 1) given by a map T(x) D (x C ˛), ˛ 2 R n Q. T preserves Lebesgue measure, every point has a dense orbit (there are no periodic orbits), yet the topological entropy is zero and nearby points stay the same distance from each other under iteration under T. There is a natural notion of equivalence for measurepreserving systems. We say that measure-preserving systems (T; X; B; ) and (S; Y; C ; ) are isomorphic if (possi-

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bly after deleting sets of measure 0 from X and Y) there is a one-to-one onto measurable map  : X ! Y with measurable inverse  1 such that  ı T D S ı   a.e. and ( 1 (A)) D (A) for all A 2 C . If X, Y are compact topological spaces we say that T is topologically conjugate to S if there exists a homeomorphism  : X ! Y such that  ı T D S ı . In this case we call  a conjugacy. If  is Cr for some r  1 we will call  a Cr -conjugacy and similarly for other degrees of regularity. We will consider X D [0; 1)(mod 1) as a representation of the unit circle S 1 D fz 2 C : jzj D 1g (under the map x ! e2 i x ) and similarly represent the k-dimensional torus T k D S 1      S 1 (k-times). If the -algebra is clear from the context we will write (T; X; ) instead of (T; X; B; ) when denoting a measure-preserving system. Examples Rigid Rotation of a Compact Group If G is a compact group equipped with Haar measure and a 2 G, then the transformation T(x) D ax preserves Haar measure and is called a rigid rotation of G. If G is abelian and the transformation is ergodic (in this setting transitivity implies ergodicity), then the transformation is uniquely ergodic. Such systems always have zero topological entropy. The simplest example of such a system is a circle rotation. Take X D [0; 1)(mod 1), with T(x) D (x C ˛) where ˛ 2 R : Then T preserves Lebesgue (Haar) measure and is ergodic (in fact uniquely ergodic) if and only if ˛ is irrational. Similarly, the map T(x1 ; : : : ; x k ) D (x1 C ˛1 ; : : : ; x k C ˛ k );

defined by  (k1  1; k2  1; : : : ) D (0; 0; : : : ) if each entry in the Z k i component is k i  1 , while  (k1  1; k2  1; : : : ; k n  1; x1 ; x2 ; : : : ) n times

‚ …„ ƒ D (0; 0; : : : ; 0; x1 C 1; x2 ; x3 ; : : : ) when x1 6D k nC1  1. The map  may be thought of as “add one and carry” and also as mapping each point to its successor in a certain order. See Sect. “Adic Transformations” for generalizations. If each k i D 2 then the system is called the dyadic (or von Neumann-Kakutani) adding machine or 2-odometer. Adding machines give examples of naturally occurring minimal systems of low orbit complexity in the sense that the topological entropy of an adding machine is zero. In fact if f is a continuous map of an interval with zero topological entropy and S is a closed, topologically transitive invariant set without periodic orbits, then the restriction of f to S is topologically conjugate to the dyadic adding machine (Theorem 11.3.13 in [46]). We say a non-empty set  is an attractor for a map T if there is an open set U containing  such that  D \n0 T n (U) (other definitions are found in the literature). The dyadic adding machine is topologically conjugate to the Feigenbaum attractor at the limit point of period doubling bifurcations (see Sect. “Unimodal Maps”). Furthermore, attractors for continuous unimodal maps of the interval are either periodic orbits, transitive cycles of intervals, or Cantor sets on which the dynamics is topologically conjugate to an adding machine [32].

where ˛1 ; : : : ; ˛ k 2 R ; preserves k-dimensional Lebesgue (Haar) measure and is ergodic (uniquely ergodic) if and only if there are no integers m1 ; : : : ; m k , not all 0, which satisfy m1 ˛1 C    C m k ˛ k 2 Z. Adding Machines Let fk i g i2N be a sequence of integers with k i  2. Equip each cyclic group Z k i with the discrete topology and form Q the product space ˙ D 1 iD1 Z k i equipped with the product topology. An adding machine corresponding to the seQ quence fk i g i2N is the topological space ˙ D 1 iD1 Z k i together with the map : ˙ !˙

Interval Exchange Maps A map T : [0; 1] ! [0; 1] is an interval exchange transformation if it is defined in the following way. Suppose that  is a permutation of f1; : : : ; ng and l i > 0, i D 1; : : : ; n, is a sequence of subintervals of I (open or closed) with P i l i D 1. Define ti by l i D t i  t i1 with t0 D 0. Suppose also that  is an n-vector with entries ˙1. T is defined by sending the interval t i1  x < t i1 of length li to the interval X X l( j)  x < l( j) ( j)<(i)

( j)>(i)

with orientation preserved if the ith entry of  is C1 and orientation reversed if the ith entry of  is -1. Thus on each

Ergodic Theory: Basic Examples and Constructions

interval li , T has the form T(x) D  i x Ca i , where  i is ˙1. If  i D 1 for each i, the transformation is called orientation preserving. The transformation T has finitely many discontinuities (at the endpoints of each li ), and modulo this set of discontinuities is smooth. T is also invertible (neglecting the finite set of discontinuities) and preserves Lebesgue measure. These maps have zero topological entropy and arise naturally in studies of polygonal billiards and more generally area-preserving flows. There are examples of minimal but non-ergodic interval exchange maps [56,69]. Full Shifts and Shifts of Finite Type Given a finite set (or alphabet) A D f0; : : : ; d  1g, take X D ˝ C (A) D AN (or X D AZ ) the sets of one-sided (two-sided) sequences, respectively, with entries from A. For example sequences in AN have the form x D x0 x1 : : : x n : : : . A cylinder set C(y n 1 ; : : : ; y n k ), y n i 2 A, of length k is a subset of X defined by fixing k entries; for example, C(y n 1 ; : : : ; y n k ) D fx : x n 1 D y n 1 ; : : : ; x n k D y n k g: We define the set Ak to consist of all cylinders C(y1 ; : : : ; y k ) determined by fixing the first k entries, i. e. an element of Ak is specified by fixing the first k entries of a sequence x0    x k by requiring x i D y i , i D 0; : : : ; k. Let p D (p0 ; : : : ; p d1 ) be a probability vector: all P p i  0 and d1 iD0 p i D 1. For any cylinder B D C(b1 ; : : : ; b k ) 2 Ak , define g k (B) D p b 1 : : : p b k :

(1)

It can be shown that these functions on Ak extend to a shift-invariant measure  p on AN (or AZ ) called product measure. (See the article on  Measure Preserving Systems.) The space AN or AZ may be given a metric by defining ( 1 if x0 6D y0 ; d(x; y) D 1 if x n 6D y n and x i D y i for jij < n : 2jnj The shift (:x0 x1    x n    ) D :x1 x2    x n    is ergodic with respect to  p . The measure-preserving system (˝; B; ; ) (with B the -algebra of Borel subsets of ˝(A), or its completion), is denoted by B(p) and is called the Bernoulli shift determined by p. This system models an infinite number of independent repetitions of an experiment with finitely many outcomes, the ith of which has probability pi on each trial. These systems are mixing of all orders (i. e.  n is mixing for all n  1) and have countable Lebesgue spectrum (hence are all spectrally isomorphic). Kolmogorov

and Sinai showed that two of them cannot be isomorphic unless they have the same entropy; Ornstein [81] showed the converse. B(1/2; 1/2) is isomorphic to the Lebesguemeasure-preserving transformation x ! 2x mod 1 on [0; 1]; similarly, B(1/3; 1/3; 1/3) is isomorphic to x ! 3x mod 1. Furstenberg asked whether the only nonatomic measure invariant for both x ! 2x mod 1 and x ! 3x mod 1 on [0; 1] is Lebesgue measure. Lyons [68] showed that if one of the actions is K; then the measure must be Lebesgue, and Rudolph [100] showed the same thing under the weaker hypothesis that one of the actions has positive entropy. For further work on this question, see [50,87]. This construction can be generalized to model onestep finite-state Markov stochastic processes as dynamical systems. Again let A D f0; : : : ; d1g, and let p D (p0 ; : : : ; p d1 ) be a probability vector. Let P be a d  d stochastic matrix with rows and columns indexed by A. This means that all entries of P are nonnegative, and the sum of the entries in each row is 1. We regard P as giving the transition probabilities between pairs of elements of A. Now we define for any cylinder B D C(b1 ; : : : ; b k ) 2 Ak  p;P (B) D p b 1 Pb 1 b 2 Pb 2 b 3 : : : Pb k1 b k :

(2)

It can be shown that  p;P extends to a measure on the Borel  -algebra of ˝ C (A), and its completion. (See the article on  Measure Preserving Systems.) The resulting stochastic process is a (one-step, finite-state) Markov process. If p and P also satisfy pP D p ;

(3)

then the Markov process is stationary. In this case we call the (one or two-sided) measure-preserving system the Markov shift determined by p and P. Aperiodic and irreducible Markov chains (those for which a power of the transition matrix P has all entries positive) are strongly mixing, in fact are isomorphic to Bernoulli shifts (usually by means of a complicated measure-preserving recoding). More generally we say a set  AZ is a subshift if it is compact and invariant under  . A subshift  is said to be of finite type (SFT) if there exists an d  d matrix M D (a i j ) such that all entries are 0 or 1 and x 2  if and only if a x i x iC1 D 1 for all i 2 Z. Shifts of finite type are also called topological Markov chains. There are many invariant measures for a non-trivial shift of finite type. For example the orbit of each periodic point is the support of an invariant measure. An important role in the theory, derived from motivations of statistical mechanics, is played by equilibrium measures (or equilibrium states) for continuous functions  :  ! R, i. e. those measures 

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R which maximize fh () C  dg over all shift-invariant probability measures, where h () is the measure-theoretic entropy of  with respect to . The study of full shifts or shifts of finite type has played a prominent role in the development of the hyperbolic theory of dynamical systems as physical systems with ‘chaotic’ dynamics ‘typically’ possess an invariant set with induced dynamics topologically conjugate to a shift of finite type (see the discussion by Smale in p. 147 in [108]). Dynamical systems in which there are transverse homoclinic connections are a common example (Theorem 5.3.5 in [45]). Furthermore in certain settings positive metric entropy implies the existence of shifts of finite type. One result along these lines is a theorem of Katok [53]. Let hto p ( f ) denote the topological entropy of a map f and h ( f ) denote metric entropy with respect to an invariant measure . Theorem 1 (Katok) Suppose T : M ! M is a C 1C diffeomorphism of a closed manifold and  is an invariant measure with positive metric entropy (i. e. h (T) > 0). Then for any 0 < < h (T) there exists an invariant set  topologically conjugate to a transitive shift of finite type with hto p (Tj) > h (T)  . More Examples of Subshifts We consider some further examples of systems that are given by the shift transformation on a subset of the set of (usually doubly-infinite) sequences on a finite alphabet, usually f0; 1g: Associated with each subshift is its language, the set of all finite blocks seen in all sequences in the subshift. These languages are extractive (or factorial ) (every subword of a word in the language is also in the language) and insertive (or extendable) (every word in the language extends on both sides to longer words in the language). In fact these two properties characterize the languages (subsets of the set of finite-length words on an alphabet) associated with subshifts. Prouhet–Thue–Morse An interesting (and often rediscovered) element of f0; 1gZC is produced as follows. Start with 0 and at each stage write down the opposite (00 D 1; 10 D 0) or mirror image of what is available so far. Or, repeatedly apply the substitution 0 ! 01; 1 ! 10: 0 01 0 1 10 0 1 10 0110 :: :

The nth entry is the sum, mod 2, of the digits in the dyadic expansion of n: Using Keane’s block multiplication [55] according to which if B is a block, B  0 D B; B  1 D B0 ; and B  (!1    ! n ) D (B  !1 )    (B  !n ); we may also obtain this sequence as 0  01  01  01     : The orbit closure of this sequence is uniquely ergodic (there is a unique shift-invariant Borel probability measure, which is then necessarily ergodic). It is isomorphic to a skew product (see Sect. “Skew Products”) over the von Neumann-Kakutani adding machine, or odometer (see Sect. “Adding Machines”). Generalized Morse systems, that is, orbit closures of sequences like 0001001001   ; are also isomorphic to skew products over compact group rotations. Chacon System This is the orbit closure of the sequence generated by the substitution 0 ! 0010; 1 ! 1: It is uniquely ergodic and is one of the first systems shown to be weakly mixing but not strongly mixing. It is prime (has no nontrivial factors) [30], and in fact has minimal self joinings [31]. It also has a nice description by means of cutting up the unit interval and stacking the pieces, using spacers (see Sect. “Cutting and Stacking”). This system has singular spectrum. It is not known whether or not its Cartesian square is loosely Bernoulli. Sturmian Systems Take the orbit closure of the sequence ! n D [1˛;1) (n˛); where ˛ is irrational. This is a uniquely ergodic system that is isomorphic to rotation by ˛ on the unit interval. These systems have minimal complexity in the sense that the number of n-blocks grows as slowly as possible (n C 1) [29]. Toeplitz Systems A bi-infinite sequence (x i ) is a Toeplitz sequence if the set of integers can be decomposed into arithmetic progressions such that each xi is constant on each arithmetic progression. A shift space X is a Toeplitz shift if it is the closure of the orbit of a Toeplitz sequence. It is possible to construct Toeplitz shifts which are uniquely ergodic and isomorphic to a rotation on a compact abelian group [34]. Sofic Systems These are images of SFT’s under continuous factor maps (finite codes, or block maps). They correspond to regular languages—languages whose words are recognizable by finite automata. These are the same as the languages defined by regular expressions—finite expressions built up from ; (empty set), (empty word), + (union of two languages),  (all concatenations of words

Ergodic Theory: Basic Examples and Constructions

from two languages), and  (all finite concatenations of elements). They also have the characteristic property that the family of all follower sets of all blocks seen in the system is a finite family; similarly for predecessor sets. These are also generated by phase-structure grammars which are linear, in the sense that every production is either of the form A ! Bw or A ! w; where A and B are variables and w is a string of terminals (symbols in the alphabet of the language). (A phase-structure grammar consists of alphabets V of variables and A of terminals, a set of productions, which is finite set of pairs of words (˛; w); usually written ˛ ! w; of words on V [ A; and a start symbol S. The associated language consists of all words on the alphabet A of terminals which can be made by starting with S and applying a finite sequence of productions.) Sofic systems typically support many invariant measures (for example they have many periodic points) but topologically transitive ones (those with a dense orbit) have a unique measures of maximal entropy (see [65]). Context-free Systems These are generated by phasestructure grammars in which all productions are of the form A ! w; where A is a variable and w is a string of variables and terminals. Coded Systems These are systems all of whose blocks are concatenations of some (finite or infinite) list of blocks. These are the same as the closures of increasing sequences of SFT’s [60]. Alternatively, they are the closures of the images under finite edge-labelings of irreducible countable-state topological Markov chains. They need not be context-free. Squarefree languages are not coded, in fact do not contain any coded systems of positive entropy. See [13,14,15]. Smooth Expanding Interval Maps Take X D [0; 1)(mod 1), m 2 N, m > 1 and define

If m D 2 the system is isomorphic to a model of tossing a fair coin, which is a common example of randomness. To see this let P D fP0 D [0; 1/2); P1 D [1/2; 1]g be a partition of [0; 1] into two subintervals. We code the orbit under T of any point x 2 [0; 1) by 0’s and 1’s by letting x k D i if T k x 2 Pi ; k D 0; 1; 2; : : : . The map  : X ! f0; 1gN which associates a point x to its itinerary in this way is a measure-preserving map from (X; ) to f0; 1gN equipped with the Bernoulli measure from p0 D p1 D 12 . The map  satisfies  ı T D  ı ,  a.e. and is invertible a.e., hence is an isomorphism. Furthermore, reading the binary expansion of x is equivalent to following the orbit of x under T and noting which element of the partition P is entered at each time. Borel’s theorem on normal numbers (base m) may be seen as a special case of the Birkhoff Ergodic Theorem in this setting. Piecewise C2 Expanding Maps The main statistical features of the examples in Sect. “Smooth Expanding Interval Maps” generalize to a broader class of expanding maps of the interval. For example: Let X D [0; 1] and let P D fI1 ; : : : ; I n g (n  2) be a partition of X into intervals (closed, half-open or open) such that I i \ I j D ; if i 6D j. Let I oi denote the interior of I i . Suppose T : X ! X satisfies: (a) For each i D 1; : : : ; n, TjI i has a C2 extension to the 0 closure I¯i of I i and jT (x)j  ˛ > 1 for all x 2 I oi . (b) T(I j ) D [ i2P j I i Lebesgue a.e. for some non-empty subset Pj f1; : : : ; ng. (c) For each I j there exists nj such that T n j (I j ) D [0; 1] Lebesgue a.e. Then T has an invariant measure  which is absolutely continuous with respect to Lebesgue measure m, and there d  C. Furthermore T is exists C > 0 such that C1  dm ergodic with respect to  and displays the same statistical properties listed above for the C2 expanding maps [20]. (See the “Folklore Theorem” in the article on  Measure Preserving Systems.)

T(x) D (mx): Then T preserves Lebesgue measure  (recall that T preserves  if (T 1 A) D (A) for all A 2 B). Furthermore it can be shown that T is ergodic. This simple map exemplifies many of the characteristics of systems with some degree of hyperbolicity. It is isomorphic to a Bernoulli shift. The map has positive topological entropy and exponential divergence of nearby orbits, and Hölder functions have exponential decay of correlations and satisfy the central limit theorem and other strong statistical properties [20].

More Interval Maps Continued Fraction Map This is the map T : [0; 1] ! [0; 1] given by Tx D 1/x mod 1, and it corresponds to the shift [0; a1 ; a2 ; : : : ] ! [0; a2 ; a3 ; : : : ] on the continued fraction expansions of points in the unit interval (a map on N N ): It preserves a unique finite measure equivalent to Lebesgue measure, the Gauss measure dx/(log 2)(1 C x): It is Bernoulli with entropy  2 /6 log 2 (in fact the natural partition into intervals is a weak Bernoulli generator, for the definition and details see [91]). By using the Ergodic

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Theorem, Khintchine and Lévy showed that

(a1    a n )1/n !

1

Y kD1

1 1C 2 k C 2k

there are good summaries by Bertrand-Mathis [9] and Blanchard [12]. p

k  log log 2

a.e. as n ! 1 ;

pn 1 2 if [0; a1 ; : : : ; a n ] D ; then log q n ! a.e. ; qn n 12 log 2 ˇ ˇ ˇ 1 p n (x) ˇˇ 2 log ˇˇx  a.e. ; ! ˇ n q n (x) 6 log 2 and if m is Lebesgue measure (or any equivalent measure) and  is Gauss measure, then for each interval I; m(T n I) ! (I); in fact exponentially fast, with a best constant 0:30366 : : : see [10,75]. The Farey Map This is the map U : [0; 1] ! [0; 1] given by Ux D x/(1  x) if 0  x  1/2; Ux D (1  x)/x if 1/2  x  1: It is ergodic for the -finite infinite measure dx/x (Rényi and Parry). It is also ergodic for the Minkowski measure d, which is a measure of maximal entropy. This map corresponds to the shift on the Farey tree of rational numbers which provide the intermediate convergence (best one-sided) as well as the continued fraction (best two-sided) rational approximations to irrational numbers. See [61,62]. f -Expansions Generalizing the continued fraction map, let f : [0; 1] ! [0; 1] and let fI n g be a finite or infinite partition of [0; 1] into subintervals. We study the map f by coding itineraries with respect to the partition fI n g: For many examples, absolutely continuous (with respect to Lebesgue measure) invariant measures can be found and their dynamical properties determined. See [104]. ˇ-Shifts This is the special case of f -expansions when f (x) D ˇx mod 1 for some fixed ˇ > 1: This map of the interval is called the ˇ-transformation. With a proper choice of partition, it is represented by the shift on a certain subshift of the set of all sequences on the alphabet D D f0; 1; : : : ; bˇcg, called the ˇ-shift. A point x is expanded as an infinite series in negative powers of ˇ with coefficients from this set; dˇ (x)n D bˇ f n (x)c: (By convention terminating expansions are replaced by eventually periodic ones.) A one-sided sequence on the alphabet D is in the ˇ-shift if and only if all of its shifts are lexicographically less than or equal to the expansion dˇ (1) of 1 base ˇ. A one-sided sequence on the alphabet D is the valid expansion of 1 for some ˇ if and only if it lexicographically dominates all its shifts. These were first studied by Bissinger [11], Everett [35], Rényi [93] and Parry [84,85];

For ˇ D 1C2 5 ; dˇ (1) D 10101010 : : : . For ˇ D 32 ; dˇ (1) D 101000001 : : : (not eventually periodic). Every ˇ-shift is coded. The topological entropy of a ˇ-shift is log ˇ: There is a unique measure of maximal entropy log ˇ. A ˇ-shift is a shift of finite type if and only if the ˇ-expansion of 1 is finite. It is sofic if and only if the expansion of 1 is eventually periodic. If ˇ is a Pisot–Vijayaragavhan number (algebraic integer all of whose conjugates have modulus less than 1), then the ˇ-shift is sofic. If the ˇ-shift is sofic, then ˇ is a Perron number (algebraic integer of maximum modulus among its conjugates). Theorem 2 (Parry [86]) Every strongly transitive (for every nonempty open set U, [n>0 T n U D X) piecewise monotonic map on [0; 1] is topologically conjugate to a ˇ-transformation. Gaussian Systems Consider a real-valued stationary process f f k :  1 < k < 1g on a probability space (˝; F ; P): The process (and the associated measure-preserving system consisting of the shift and a shift-invariant measure on RZ ) is called Gaussian if for each d  1, any d of the f k form an Rd valued Gaussian random variable on ˝ : this means that with E( f k ) D m for all k and Z Ai j D

˝

( f k i  m)( f k j  m)dP D C(k i  k j ) for i; j D 1; : : : ; d ;

where C() is a function, for each Borel set B R, Pf! : ( f k 1 (!); : : : ; f k d (!)) 2 Bg

Z 1 1 D p exp  (x  (m; : : : ; m)) tr d/2 2 2 det A B A1 (x  (m; : : : ; m)) dx1    dx d : where A is a matrix with entries (A i j ). The function C(k) is positive semidefinite and hence has an associated measure  on [0; 2) such that Z 2 e i k t d (t) : C(k) D 0

Theorem 3 (de la Rue [33]) The Gaussian system is ergodic if and only if the “spectral measure”  is continuous (i. e., nonatomic), in which case it is also weakly mixing. It

Ergodic Theory: Basic Examples and Constructions

is mixing if and only if C(k) ! 0 as jkj ! 1: If  is singular with respect to Lebesgue measure, then the entropy is 0; otherwise the entropy is infinite. For more details see [28]. Hamiltonian Systems (This paragraph is from the article on  Measure Preserving Systems.) Many systems that model physical situations can be studied by means of Hamilton’s equations. The state of the entire system at any time is specified by a vector (q; p) 2 R2n , the phase space, with q listing the coordinates of the positions of all of the particles, and p listing the coordinates of their momenta. We assume there is a time-independent Hamiltonian function H(q, p) such that the time development of the system satisfies Hamilton’s equations: dq i @H ; D dt @p i

dp i @H ; D dt @q i

i D 1; : : : ; n :

(4)

Often in applications the Hamiltonian function is the sum of kinetic and potential energy: H(q; p) D K(p) C U(q) :

(5)

Solving these equations with initial state (q, p) for the system produces a flow (q; p) ! Tt (q; p) in phase space which moves (q, p) to its position Tt (q; p)t units of time later. According to Liouville’s formula Theorem 3.2 in [69], this flow preserves Lebesgue measure on R2n . Calculating dH/dt by means of the Chain Rule  X  @H dp i dH @H dq i D C dt @p i dt @q i dt i

and using Hamilton’s equations shows that H is constant on orbits of the flow, and thus each set of constant energy X(H0 ) D f(q; p) : H(q; p) D H0 g is an invariant set. There is a natural invariant measure on a constant energy set X(H0 ) for the restricted flow, namely the measure given by rescaling the volume element dS on X(H0 ) by the factor 1/jjOHjj. Billiard Systems These form an important class of examples in ergodic theory and dynamical systems, motivated by natural questions in physics, particularly the behavior of gas models. Consider the motion of a particle inside a bounded region D in Rd with piecewise smooth (C1 at least) boundaries. In the case of planar billiards we have d D 2. The particle moves in a straight line with

constant speed until it hits the boundary, at which point it undergoes a perfectly elastic collision with the angle of incidence equal to the angle of reflection and continues in a straight line until it next hits the boundary. It is usual to normalize and consider unit speed, as we do in this discussion for convenience. We take coordinates (x, v) given by the Euclidean coordinates in x 2 D together with a direction vector v 2 S d1 . A flow  t is defined with respect to Lebesgue almost every (x, v) by translating x a distance t defined by the direction vector v, taking account of reflections at boundaries.  t preserves a measure absolutely continuous with respect to Riemannian volume on (x, v) coordinates. The flow we have described is called a billiard flow. The corresponding billiard map is formed by taking the Poincaré map corresponding to the cross-section given by the boundary @D. We will describe the planar billiard map; the higher dimensional generalization is clear. The billiard map is a map T : @D ! @D, where @D is coordinatized by (s; ), s 2 [0; L), where L is the length of @D and 2 (0; ) measures the angle that inward pointing vectors make with the tangent line to @D at s. Given a point (s; ), the angle defines an oriented line l(s; ) which intersects @D in two points s and s0 . Reflecting l in the tangent line to @D at the point s0 gives another oriented line passing through s0 with angle 0 (measured with respect to the angular coordinate system based at s0 ). The billiard map is the map T(s; ) D (s 0 ; 0 ). T preserves a measure  D sin ds  d . The billiard flow may be modeled as a suspension flow over the billiard map (see Sect. “Suspension Flows”). If the region D is a polygon in the plane (or polyhedron in Rd ), then @D consists of the faces of the polyhedron. The dynamical behavior of the billiard map or flow in regions with only flat (non-curved) boundaries is quite different to that of billiard flows or maps in regions D with strictly convex or strictly concave boundaries. The topological entropy of a flat polygonal billiard is zero. Research interest focuses on the existence and density of periodic or transitive orbits. It is known that if all the angles between sides are rational multiples of  then there are periodic orbits [17,74,112] and they are dense in the phase space [16]. It is also known that a residual set of polygonal billiards are topologically transitive and ergodic [58,117]. On the other hand, billiard maps in which @D has strictly convex components are physical examples of nonuniformly hyperbolic systems (with singularities). The meaning of concave or convex varies in the literature. We will consider a billiard flow inside a circle to be a system with a strictly concave boundary, while a billiard flow on the torus from which a circle has been excised to be a billiard flow with strictly convex boundary.

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Ergodic Theory: Basic Examples and Constructions

The class of billiards with some strictly convex boundary components, sometimes called dispersing billiards or Sinai billiards, was introduced by Sinai [106] who proved many of their fundamental properties. Lazutkin [63] proved that planar billiards with generic strictly concave boundary are not ergodic. Nevertheless Bunimovich [22,23] produced a large of billiard systems, Bunimovich billiards, with strictly concave boundary segments (perhaps with some flat boundaries as well) which were ergodic and non-uniformly hyperbolic. For more details see [26,54,66,109]. We will discuss possibly the simplest example of a dispersing billiard, namely a toral billiard with a single convex obstacle. Take the torus T 2 and consider a single strictly convex subdomain S with C 1 boundary. The domain of the billiard map is [0; L)(0; ), where L is the length of @S. The measure sin( )ds  d is preserved. If the curvature of @S is everywhere non-zero, then the billiard map T has positive topological entropy, periodic points are dense, and in fact the system is isomorphic to a Bernoulli shift [41]. KAM-Systems and Stably Non-Ergodic Behavior A celebrated theorem of Kolmogorov, Arnold and Moser (the KAM theorem) implies that the set of ergodic areapreserving diffeomorphisms of a compact surface without boundary is not dense in the Cr topology for r  4. This has important implications, in that there are natural systems in which ergodicity is not generic. The constraint of perturbing in the class of area-preserving diffeomorphisms is an appropriate imposition in many physical models. We will take the version of the KAM theorem as given in Theorem 5.1 in [69] (original references include [3,59,79]). An elliptic fixed point for an area-preserving diffeomorphism T of a surface M is called a nondegenerate elliptic fixed point if there is a local Cr , r  4, change of coordinates h so that in polar coordinates hT h1 (r; ) D (r; C ˛0 C ˛1 r) C F(r; ) ; where all derivatives of F up to order 3 vanish, ˛1 6D 0 and ˙2 ˛0 6D 0; ˙ 2 ; ; 3 . A map of the form (r; ) D (r; C ˛0 C ˛1 r) ; where ˛1 6D 0, is called a twist map. Note that a twist map leaves invariant the circle r D k, for any constant k, and rotates each invariant curve by a rigid rotation ˛1 r, the magnitude of the rotation depending upon r. With respect to two-dimensional Lebesgue measure a twist map is certainly not ergodic. Theorem 4 Suppose T is a volume-preserving diffeomorphism of class Cr , r  4, of a surface M. If x is a non-degenerate elliptic fixed point, then for every > 0 there exists

a neighborhood U of x and a set U0; U with the properties: (a) U0; is a union of T-invariant simple closed curves of class Cr1 containing x in their interior. (b) The restriction of T to each such invariant curve is topologically conjugate to an irrational rotation. (c) m(U U0; )  m(U ), where m is Lebesgue measure on M. It is possible to prove the existence of a Cr volume preserving diffeomorphism (r  4) with a non-degenerate elliptic fixed point and also show that if T possesses a non-degenerate elliptic fixed point then there is a neighborhood V of T in the Cr topology on volume-preserving diffeomorphisms such that each T 0 2 V possesses a non-degenerate elliptic fixed point Chapter II, Sect. 6 in [69]. As a corollary we have Corollary 1 Let M be a compact surface without boundary and Diffr (M) the space of Cr area-preserving diffeomorphisms with the Cr topology. Then the set of T 2 Diffr (M) which are ergodic with respect to the probability measure determined by normalized area is not dense in Diffr (M) for r  4. Smooth Uniformly Hyperbolic Diffeomorphisms and Flows Time series of measurements on deterministic dynamical systems sometimes display limit laws exhibited by independent identically distributed random variables, such as the central limit theorem, and also various mixing properties. The models of hyperbolicity we discuss in this section have played a key role in showing how this phenomenon of ‘chaotic behavior’ arises in deterministic dynamical systems. Hyperbolic sets and their associated dynamics have also been pivotal in studies of structural stability. A smooth system is Cr structurally stable if a small perturbation in the Cr topology gives rise to a system which is topologically conjugate to the original. When modeling a physical system it is desirable that slight changes in the modeling parameters do not greatly affect the qualitative or quantitative behavior of the ensemble of orbits considered as a whole. The orbit of a point may change drastically under perturbation (especially if the system has sensitive dependence on initial conditions) but the collection of all orbits should ideally be ‘similar’ to the original unperturbed system. In the latter case one would hope that statistical properties also vary only slightly under perturbation. Structural stability is one, quite strong, notion of stability. The conclusion of a body of work on structural

Ergodic Theory: Basic Examples and Constructions

stability is that a system is C1 structurally stable if and only if it is uniformly hyperbolic and satisfies a technical assumption called strong transversality (see below for details). Suppose M is a C1 compact Riemannian manifold equipped with metric d and tangent space TM with norm kk. Suppose also that U M is a non-empty open subset and T : U ! T(U) is a C1 diffeomorphism. A compact T invariant set  U is called a hyperbolic set if there is a splitting of the tangent space Tp M at each point p 2  into two invariant subspaces, Tp M D E u (p) ˚ E s (p), and a number 0 <  < 1 such that for n  0 kD p T n vk  Cn kvk for v 2 E s (p) ; kD p T n vk < Cn kvk for v 2 E u (p) : The subspace Eu is called the unstable or expanding subspace and the subspace Es the stable or contracting subspace. The stable and unstable subspaces may be integrated to produce stable and unstable manifolds W s (p) D fy : d(T n p; T n y) ! 0g u

W (p) D fy : d(T

n

p; T

n

as n ! 1 ;

y) ! 0g

as n ! 1 :

The stable and unstable manifolds are immersions of Euclidean spaces of the same dimension as E s (p) and E u (p), respectively, and are of the same differentiability as T. Moreover, Tp (W s (p)) D E s (p) and Tp (W u (p)) D E u (p). It is also useful to define local stable manifolds and local unstable manifolds by W s (p) D fy 2 W s (p) : d(T n p; T n y) < g for all n  0 ; W u (p) D fy 2 W u (p) : d(T n p; T n y) < g for all n  0 : Finally we discuss the notion of strong transversality. We say a point x is non-wandering if for each open neighborhood U of x there exists an n > 0 such that T n (U) \ U 6D ;. The NW set of non-wandering points is called the non-wandering set. We say a dynamical system has the strong transversal property if W s (x) intersects W u (y) transversely for each pair of points x; y 2 NW. In the Cr , r  1 topology Robbin [94], de Melo [78] and Robinson [95,96] proved that dynamical systems with the strong transversal property are structurally stable, and Robinson [97] in addition showed that strong transversality was also necessary. Mañé [70] showed that a C1 structurally stable diffeomorphism must be uniformly hyperbolic and Hayashi [47] extended this to flows. Thus a C1 diffeomorphism or flow on a compact manifold is structurally stable if and only it is uniformly hyperbolic and satisfies the strong transversality condition.

Geodesic Flow on Manifold of Negative Curvature The study of the geodesic flow on manifolds of negative sectional curvature by Hedlund and Hopf was pivotal to the development of the ergodic theory of hyperbolic systems. Suppose that M is a geodesically complete Riemannian manifold. Let p;v (t) be the geodesic with p;v (0) D p and ˙ p;v (0) D v, where ˙ p;v denotes the derivative with respect to time t. The geodesic flow is a flow  t on the tangent bundle TM of M,  t : R  TM ! TM, defined by  t (p; v) D ( p;v (t); ˙p;v (t)) : where (p; v) 2 TM. Since geodesics have constant speed, if kvk D 1 then k p;v (t)k D 1 for all t, and thus the unit tangent bundle T 1 M D f(p; v) 2 TM : kvk D 1g is preserved under the geodesic flow. The geodesic flow and its restriction to the unit tangent bundle both preserve a volume form, Liouville measure. In 1934 Hedlund [48] proved that the geodesic flow on the unit tangent bundle of a surface of strictly negative constant sectional curvature is ergodic, and in 1939 Hopf [49] extended this result to manifolds of arbitrary dimension and strictly negative (not necessarily constant) curvature. Hopf’s technique of proof of ergodicity (Hopf argument) was extremely influential and used the foliation of the tangent space into stable and unstable manifolds. For a clear exposition of this technique, and the property of absolute continuity of the foliations into stable and unstable manifolds, see [66]. The geodesic flow on manifolds of constant negative sectional curvature is an Anosov flow (see Sect. “Anosov Systems”). We remark that for surfaces sectional curvature is the same as Gaussian curvature. Recently the time-one map of the geodesic flow on the unit tangent bundle of a surface with constant negative curvature, which is a partially hyperbolic system (see Sect. “Partially Hyperbolic Dynamical Systems”), was shown to be stably ergodic [44], so the geodesic flow is still playing a major role in the development of ergodic theory. Horocycle Flow All surfaces endowed with a Riemannian metric of constant negative curvature are quotients of the upper half-plane H C :D fx C iy 2 C : y > 0g with dx 2 Cdy 2

, whose sectional curvature is the metric ds 2 D y2 1. The orientation-preserving isometries of this metric are exactly the linear fractional (also known as Möbius) transformations:   az C b a b 2 H C] : 2 SL(2; R); [z 2 H C 7! c d cz C d Since each matrix ˙I corresponds to the identity transformation, we consider matrices in PSL(2; R) :D SL(2; R)/f˙Ig.

273

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Ergodic Theory: Basic Examples and Constructions

The unit tangent bundle, S H C , of the upper halfplane can be identified with PSL(2; R). Then the geodesic flow corresponds to the transformations  t  e 0 t 2 R 7! 0 et seen as acting on PSL(2; R). The unstable foliation of an element A 2 PSL(2; R) Š S H C is given by   1 t A; t 2 R ; 0 1 and the flow along this foliation, given by   1 t ; t 2 R 7! 0 1 is called the horocycle flow. Similarly for the flow induced on the unit tangent bundle of each quotient of the upper half-plane by a discrete group of linear fractional transformations. The geodesic and horocycle flows acting on a (finitevolume) surface of constant negative curvature form the fundamental example of a transverse pair of actions. The geodesic flow often has many periodic orbits and many invariant measures, has positive entropy, and is in fact Bernoulli with respect to the natural measure [82], while the horocycle flow is often uniquely ergodic [39,72] and of entropy zero, although mixing of all orders [73]. See [46] for more details. Markov Partitions and Coding If (X; T; B; ) is a dynamical system then a finite partition of X induces a coding of the orbits and a semi-conjugacy with a subshift on a symbol space (it may not of course be a full conjugacy). For hyperbolic systems a special class of partitions, Markov partitions, induce a conjugacy for the invariant dynamics to a subshift of finite type. A Markov partition P for an invariant subset  of a diffeomorphism T of a compact manifold M is a finite collection of sets Ri , 1  i  n called rectangles. The rectangles have the property, for some > 0, if x; y 2 R i then W s (x) \ W u (y) 2 R i . This is sometimes described as being closed under local product structure. We let W u (x; R i ) denote W u (x) \ R i and W s (x; R i ) denote W s (x) \ R i . Furthermore we require for all i, j: (1) (2) (3) (4)

Each Ri is the closure of its interior.  [i R i R i \ R j D @R i \ @R j if i 6D j if x 2 R oi and T(x) 2 R oj then W u (T(x); R j )

T(W u (x; R i )) and W s (x; R i ) T 1 (W u (T(x); R j ))

Anosov Systems An Anosov diffeomorphism [2] is a uniformly hyperbolic system in which the entire manifold is a hyperbolic set. Thus an Anosov diffeomorphism is a C1 diffeomorphism T of M with a DT-invariant splitting (which is a continuous splitting) of the tangent space TM(x) at each point p into a disjoint sum Tp M D E u (p) ˚ E s (p) and there exist constants 0 <  < 1, constant C > 0 such that kDT n vk < Cn kvk for all v 2 E s (p) and kDT n wk  Cn kwk for all w 2 E u (p). A similar definition holds for Anosov flows  : R  M ! M. A flow is Anosov if there is a splitting of the tangent bundle into flow-invariant subspaces E u ; E s ; E c so D p  t E sp D Es t (p) , D p  t E up D Eu t (p) and D p  t E cp D Ec t (p) , and at each point p 2 M Tp M D E sp ˚ E up ˚ E cp k(D p  t )vk < C t kvk for v 2 E s (p) k(D p t )vk < C t kvk for v 2 E u (p) for some 0 <  < 1. The tangent to the flow direction E c (p) is a neutral direction: k(D p  t )vk D kvk for v 2 E c (p) : Anosov proved that Anosov flows and diffeomorphisms which preserve a volume form are ergodic [2] and are also structurally stable. Sinai [105] constructed Markov partitions for Anosov diffeomorphisms and hence coded trajectories via a subshift of finite type. Using ideas from statistical physics in [107] Sinai constructed Gibbs measures for Anosov systems. An SRB measure (see Sect. “Physically Relevant Measures and Strange Attractors”) is a type of Gibbs measure corresponding to the potential  log j det(DTj E u )j and is characterized by the property of absolutely continuous conditional measures on unstable manifolds. The simplest examples of Anosov diffeomorphisms are perhaps the two-dimensional hyperbolic toral automorphisms (the n > 2 generalization is clear). Suppose A is a 2  2 matrix with integer entries   a b c d such that det(A) D 1 and A has no eigenvalues of modulus 1. Then A defines a transformation of the two-dimensional torus T 2 D S 1  S 1 such that if v 2 T 2 ,   v vD 1 ; v2

Ergodic Theory: Basic Examples and Constructions

then   (av1 C bv2 ) : Av D (cv1 C dv2 ) A preserves Lebesgue (or Haar) measure and is ergodic. A prominent example of such a matrix is  2 1

1 1

 ;

which is sometimes called the Arnold Cat Map. Each point with rational coordinates (p1 /q1 ; p2 /q2 ) is periodic. p There are two eigenvalues 1/ < 1 <  D (3 C 5)/2 with orthogonal eigenvectors, and the projections of the eigenspaces from R2 to T 2 are the stable and unstable subspaces. Axiom A Systems In the case of Anosov diffeomorphisms the splitting into contracting and expanding bundles holds on the entire phase space M. A system T : M ! M is an Axiom A system if the non-wandering set NW is a hyperbolic set and periodic points are dense in the nonwandering set. NW M may have Lebesgue measure zero. A set  M is locally maximal if there exists an open set U such that  D \n2Z T n (U). The solenoid and horseshoe discussed below are examples of locally maximal sets. Bowen [18] constructed Markov partitions for Axiom A diffeomorphisms . Ruelle imported ideas from statistical physics, in particular the idea of an equilibrium state and the variational principle, to the study of Axiom A systems (see [101,102]) This work extended the notion of Gibbs measure and other ideas from statistical mechanics, introduced by Sinai for Anosov systems [107], into Axiom A systems. One achievement of the Axiom A program was the Smale Decomposition Theorem, which breaks the dynamics of Axiom A systems into locally maximal sets and describes the dynamics on each [18,19,108]. Theorem 5 (Spectral Decomposition Theorem) If T is Axiom A then there is a unique decomposition of the nonwandering set NW of T NW D 1 [    [  k as a disjoint union of closed, invariant, locally maximal hyperbolic sets  i such that T is transitive on each  i . Furthermore each  i may be further decomposed into a disj joint union of closed sets  i , j D 1; : : : ; n i such that T n i is j

topologically mixing on each  i and T cyclically permutes j

the  i .

Horseshoe Maps This type of map was introduced by Steven Smale in the 1960’s and has played a pivotal role in the development of dynamical systems theory. It is perhaps the canonical example of an Axiom A system [108] and is conjugate to a full shift on 2 symbols. Let S be a unit square in R2 and let T be a diffeomorphism of S onto its image such that S \ T(S) consists of two disjoint horizontal strips S0 and S1 . Think of stretching S uniformly in the horizontal direction and contracting uniformly in the vertical direction to form a long thin rectangle, and then bending the rectangle into the shape of a horseshoe and laying the straight legs of the horseshoe back on the unit square S. This transformation may be realized by a diffeomorphism and we may also require that T restricted to T 1 S i , i D 0; 1, acts as a linear map. The restriction of T i to the maximal invariant set H D \1 iD1 T S is a Smale horseshoe map. H is a Cantor set, the product of a Cantor set in the horizontal direction and a Cantor set in the vertical direction. The conjugacy with the shift on two symbols is realized by mapping x 2 H to its itinerary with respect to the sets S0 and S1 under powers of T (positive and negative powers). Solenoids The solenoid is defined on the solid torus X in R3 which we coordinatize as a circle of two-dimensional solid disks, so that X D f( ; z) : 2 [0; 1) and jzj  1; z 2 Cg The transformation T : X ! X is given by   1 1 T( ; z) D 2 (mod 1) ; z C e2 i 4 2 Geometrically the transformation stretches the torus to twice its length, shrinks its diameter by a factor of 4, then twists it and doubles it over, placing the resultant object without self-intersection back inside the original solid torus. T(X) intersects each disk D c D f( ; z) : D cg in two smaller disks of 14 the diameter. The transformation T contracts volume by a factor of 2 upon each application, yet there is expansion in the direction ( ! 2 ). The solenoid A D \n0 T n (X) has zero Lebesgue measure, is T-invariant and is (locally) topologically a line segment cross a two-dimensional Cantor set (A intersects each disk Dc in a Cantor set). The set A is an attractor, in that all points inside X limit under iteration by T upon A. T : A ! A is an Axiom A system. Partially Hyperbolic Dynamical Systems Partially hyperbolic dynamical systems are a generalization of uniformly hyperbolic systems in that an invariant central direction is allowed but the contraction in the

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central direction is strictly weaker than the contraction in the contracting direction and the expansion in the central direction is weaker than the expansion in the expanding direction. More precisely, suppose M is a C1 compact (adapted) Riemannian manifold equipped with metric d and tangent space TM with norm kk. A C1 diffeomorphism T of M is a partially hyperbolic diffeomorphism if there is a nontrivial continuous DT invariant splitting of the tangent space Tp M at each point p into a disjoint sum Tp M D E u (p) ˚ E c (p) ˚ E s (p) and continuous positive functions m; M; ˜ ; such that 

Es

is contracted: kD p T n v s k  m(p) < 1; if v s 2 E s (x) n f0g then kv sk  Eu is expanded: kD Tv u k if v u 2 E u (x) n f0g then kvp u k  M(p) > 1;  Ec is uniformly dominated by Eu and Es : ˜

(p) if v c 2 E c (x) n f0g then there are numbers (p); kD p Tv c k ˜ such that m(p) < (p)  kv c k  (p) < M(p) .

The notion of partial hyperbolicity was introduced by Brin and Pesin [21] who proved existence and properties, including absolute continuity, of invariant foliations in this setting. There has been intense recent interest in partially hyperbolic systems primarily because significant progress has been made in establishing that certain volume-preserving partially hyperbolic systems are ‘stably ergodic’—that is, they are ergodic and under small (Cr topology) volume-preserving perturbations remain ergodic. This phenomenon had hitherto been restricted to uniformly hyperbolic systems. For recent developments, and precise statements, on stable ergodicity of partially hyperbolic systems see [24,92]. Compact Group Extensions of Uniformly Hyperbolic Systems A natural example of a partially hyperbolic system is given by a compact group extension of an Anosov diffeomorphism. If the following terms are not familiar see Sect. “Constructions” on standard constructions. Suppose that (T; M; ) is an Anosov diffeomorphism, G is a compact connected Lie group and h : M ! G is a differentiable map. The skew product F : M  G ! M  G given by F(x; g) D (Tx; h(x)g) has a central direction in its tangent space corresponding to the Lie algebra LG of G (as a group element h acts isometrically on G so there is no expansion or contraction)

and uniformly expanding and contracting bundles corresponding to those of the tangent space of T : M ! M. Thus T(M  G) D E u ˚ LG ˚ E s . Time-One Maps of Anosov Flows Another natural context in which partial hyperbolicity arises is in time-one maps of uniformly hyperbolic flows. Suppose  t : R  M ! M is an Anosov flow. The diffeomorphism 1 : M ! M is a partially hyperbolic diffeomorphism with central direction given by the flow direction. There is no expansion or contraction in the central direction. Non-Uniformly Hyperbolic Systems The assumption of uniform hyperbolicity is quite restrictive and few ‘chaotic systems’ found in applications are likely to exhibit uniform hyperbolicity. A natural weakening of this assumption, and one that is non-trivial and greatly extends the applicability of the theory, is to require the hyperbolic splitting (no longer uniform) to hold only at almost every point of phase space. A systematic theory was built by Pesin [89,90] on the assumption that the system has non-zero Lyapunov exponents  almost everywhere where  is a Lebesgue equivalent invariant probability measure . Recall that a number  is a Lyapunov exponent for p 2 M if kD p T n vk en for some unit vector v 2 Tp M. Oseledet’s theorem [83] (see also p. 232 in [113]), which is also called the Multiplicative Ergodic Theorem, implies that if T is a C1 diffeomorphism of M then for any T-invariant ergodic measure  almost every point has well-defined Lyapunov exponents. One of the highlights of Pesin theory is the following structure theorem: If T : M ! M is a C 1C diffeomorphism with a T-invariant Lebesgue equivalent Borel measure  such that T has non-zero Lyapunov exponents with respect to  then T has at most a countable number of ergodic components fC i g on each of which the restriction of T is either Bernoulli or Bernoulli times a rotation (by which we mean the support of  i D jC i consists of a finite number ni of sets fS1i ; : : : S ni i g cyclically permuted and T n i is Bernoulli when restricted to each S ij ) [90,114]. This structure theorem has been generalized to SRB measures with non-zero Lyapunov exponents [64,90]. Physically Relevant Measures and Strange Attractors (This paragraph is from the article on  Measure Preserving Systems.) For Hamiltonian systems and other volumepreserving systems it is natural to consider ergodicity (and other statistical properties) of the system with respect to Lebesgue measure. In dissipative systems a measure equiv-

Ergodic Theory: Basic Examples and Constructions

alent to Lebesgue may not be invariant (for example the solenoid). Nevertheless Lebesgue measure has a distinguished role since sampling by experimenters is done with respect to Lebesgue measure. The idea of a physically relevant measure  is that it determines the statistical behavior of a positive Lebesgue measure set of orbits, even though the support of  may have zero Lebesgue measure. An example of such a situation in the uniformly hyperbolic setting is the solenoid , where the attracting set  has Lebesgue measure zero and is (locally) topologically the product of a two-dimensional Cantor set and a line segment. Nevertheless  determines the behavior of all points in a solid torus in R3 . More generally, suppose that T : M ! M is a diffeomorphism on a compact Riemannian manifold and that m is a version of Lebesgue measure on M, given by a smooth volume form. Although Lebesgue measure m is a distinguished physically relevant measure, m may not be invariant under T, and the system may even be volume contracting in the sense that m(T n A) ! 0 for all measurable sets A. Nevertheless an experimenter might observe long-term “chaotic” behavior whenever the state of the system gets close to some compact invariant set X which attracts a positive m-measure of orbits in the sense that these orbits limit on X. Possibly m(X) D 0, so that X is effectively invisible to the observer except through its effects on orbits not contained in X. The dynamics of T restricted to X can in fact be quite complicated—maybe a full shift, or a shift of finite type, or some other complicated topological dynamical system. Suppose there is a T-invariant measure  supported on X such that for all continuous functions  : M ! R Z n1 1X k  ı T (x) !  d ; n X

(6)

kD0

for a positive m-measure of points x 2 M. Then the longterm equilibrium dynamics of an observable set of points x 2 M (i. e. a set of points of positive m measure) is described by (X; T; ). In this situation  is described as a physical measure. There has been a great deal of research on the properties of systems with attractors supporting physical measures. In the dissipative non-uniformly hyperbolic setting the theory of ‘physically relevant’ measures is best developed in the theory of SRB (for Sinai, Ruelle and Bowen) measures. These dynamically invariant measures may be supported on a set of Lebesgue measure zero yet determine the asymptotic behavior of points in a set of positive Lebesgue measure. If T is a diffeomorphism of M and  is a T-invariant Borel probability measure with positive Lyapunov expo-

nents which may be integrated to unstable manifolds, then we call  an SRB measure if the conditional measure  induces on the unstable manifolds is absolutely continuous with respect to the Riemannian volume element on these manifolds. The reason for this definition is technical but is motivated by the following observations. Suppose that the diffeomorphism has no zero Lyapunov exponents with respect to . Since T is a diffeomorphism, this implies T has negative Lyapunov exponents as well as positive Lyapunov exponents and corresponding local stable manifolds as well as local unstable manifolds. Suppose that a T-invariant set A consists of a union of unstable manifolds and is the support of an ergodic SRB measure  and that  : M ! R is a continuous function. Since  has absolutely continuous conditional measures on unstable manifolds with respect to conditional Lebesgue measure on the unstable manifolds, almost every point x in the union of unstable manifolds U satisfies Z n1 1X lim  ı T j (x) D  d : (7) n!1 n jD0

If y 2 W s (x) for such an x 2 U then d(T n x; T n y) ! 1 and hence (7) implies Z n1 1X  ı T j (y) D  d : lim n!1 n jD0

Furthermore, if the holonomy between unstable manifolds defined by sliding along stable manifolds is absolutely continuous (takes sets of zero Lebesgue measure on W u to sets of zero Lebesgue measure on W u ), there is a positive Lebesgue measure of points (namely an unstable manifold and the union of stable manifolds through it) satisfying (7). Thus an SRB measure with absolutely continuous holonomy maps along stable manifolds is a physically relevant measure. If the stable foliation possesses this property it is called absolutely continuous. An Axiom A attractor for a C2 diffeomorphism is an example of an SRB attractor [19,101,102,107]. The examples we have given of SRB measures and attractors and measures have been uniformly hyperbolic. Recently much progress has been made in understanding the statistical properties of non-uniformly hyperbolic systems by using a tower (see Sect. “Induced Transformations”) to construct SRB measures. We refer to Young’s original papers [115,116], the book by Baladi [5] and to [114] for a recent survey on SRB measures in the nonuniformly hyperbolic setting. Unimodal Maps Unimodal Maps of an interval are simple examples of non-uniformly hyperbolic dynamical sys-

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Ergodic Theory: Basic Examples and Constructions

tems that have played an important role in the development of dynamical systems theory. Suppose I R is an interval; for simplicity we take I D [0; 1]. A unimodal map is a map T : [0; 1] ! [0; 1] such that there exists a point 0 < c < 1 and  T is C2 ;  T 0 (x) > 0 for x < c, T 0 (x) < 0 for x > c;  T 0 (c) D 0. Such a map is clearly not uniformly expanding, as jT 0 (x)j < 1 for points in a neighborhood of c. The family of maps T (x) D x(1  x), 0 <   4, is a family of unimodal maps with c D 1/2 and T2 (1/2) D 1/2, T4 (1/2) D 1. We could have taken the interval I to be [1; 1] or indeed any interval with an obvious modification of the definition above. A well-studied family of unimodal maps in this setting is the logistic family f a : [1; 1] ! [1; 1], f a (x) D 1  ax 2 , a 2 (0; 2]. The families f a and T are equivalent under a smooth coordinate change, so statements about one family may be translated into statements about the other. Unimodal maps are studied because of the insights they offer into transitions from regular or periodic to chaotic behavior as a parameter (e. g.  or a) is varied, the existence of absolutely continuous measures, and rates of decay of correlations of regular observations for non-uniformly hyperbolic systems. Results of Jakobson [52] and Benedicks and Carleson [6] implies that in the case of the logistic family there is a positive Lebesgue measure set of a such that f a has an absolutely continuous ergodic invariant measure  a . It has been shown by Keller and Nowicki [57] (see alsoYoung [116]) that if f a is mixing with respect to  a then the decay of correlations for Lipshitz observations on I is exponential. It is also known that the set of a such that f a is mixing with respect to  a has positive Lebesgue measure. There is a well-developed theory concerning bifurcations the maps T undergo as  varies [27]. We briefly describe the period-doubling route to chaos in the family T (x) D x(1  x). For a nice account see [46].p We let c denote the fixed point 1 . For  3 <   1 C 6, all points in [0; 1] except for 0; c and their preimages are attracted to a unique periodic orbit O(p ) of period 2. There is a monotone sequence of parameter values n ( 1 D 3) such that for n <   nC1 , T has a unique attracting periodic orbit O(n ) of period 2n and for each k D 1; 2; : : : ; n  1 a unique repelling orbit of period 2k . All points in the interval [0; 1] except for the repelling periodic orbits and their preimages are attracted to the attracting periodic orbit of period 2n . At

 D n the periodic orbit O(n ) undergoes a period-doubling bifurcation. Feigenbaum [36] found that the limit n1 ı D n  4:699 : : : exists and that in a wide class n nC1 of unimodal maps this period-doubling cascade occurs and the differences between successive bifurcation parameters give the same limiting ratio, an example of universality. At the end of the period-doubling cascade at a parameter 1 3:569 : : :, T1 has an invariant Cantor set C (the Feigenbaum attractor) which is topologically conjugate to the dyadic adding machine coexisting with isolated repelling orbits of period 2n , n D 0; 1; 2; : : : . There is a unique repelling orbit of period 2n for n  1 along with two fixed points. The Cantor set C is the !-limit set for all points that are not periodic or preimages of periodic orbits. C is the set of accumulation points of periodic orbits. Despite this picture of incredible complexity the topological entropy is zero for   1 . For  > 1 the map T has positive topological entropy and infinitely many periodic orbits whose periods are not powers of 2. For each   1 , T possesses an invariant Cantor set which is repelling for  > 1 . We say that T is hyperbolic if there is only one attracting periodic orbit and the only recurrent sets are the attracting periodic orbit, repelling periodic orbits and possibly a repelling invariant Cantor set. It is known that the set of  2 [0; 4] for which T is hyperbolic is open and dense [43]. Remarkably, by Jakobson’s result [52] there is also a positive Lebesgue measure set of parameters  for which T has an absolutely continuous invariant measure  with a positive Lyapunov exponent. Intermittent Maps Maps of the unit interval T : [0; 1] ! [0; 1] which are expanding except at the point x D 0, where they are locally of form T(x) x C x 1C˛ , ˛ > 0, have been extensively studied both for the insights they give into rates of decay of correlations for nonuniformly hyperbolic systems (hyperbolicity is lost at the point x D 0, where the derivative is 1) and for their use as models of intermittent behavior in turbulence [71]. A fixed point where the derivative is 1 is sometimes called an indifferent fixed point. It is a model of intermittency in the sense that orbits close to 1 will stay close for many iterates (since the expansion is very weak there) and hence a time series of observations will be quite uniform for long periods of time before displaying chaotic type behavior after moving away from the indifferent fixed into that part of the domain where the map is uniformly expanding. A particularly simple model [67] is provided by ( x(1 C 2˛ x ˛ ) if x 2 [0; 1/2) ; T(x) D 2x  1 if x 2 [1/2; 1] :

Ergodic Theory: Basic Examples and Constructions

For ˛ D 0 the map is uniformly expanding and Lebesgue measure is invariant. In this case the rate of decay of correlations for Hölder observations is exponential. For 0 < ˛ < 1 the map has an SRB measure ˛ with support the unit interval. For ˛  1 there are no absolutely continuous invariant probability measures though there are -finite absolutely continuous measures. Upper and lower polynomial bounds on the rate of decay of observations on such maps have been given as a function of 0 < ˛ < 1 and the regularity of the observable. For details see [51,67,103]. Hénon Diffeomorphisms The Henón family of diffeomorphisms was introduced and studied as Poincaré maps for the Lorenz system of equations. It is a two-parameter two-dimensional family which shares many characteristics with the logistic family and for small b > 0 may be considered a two-dimensional ‘perturbation’ of the logistic family. The parametrized mapping is defined as Ta;b (x; y) D (1  ax 2 C y; bx) ; so Ta;b : R2 ! R2 with 0 < a < 2 and b > 0. Benedicks and Carleson [7] showed that for a positivemeasure set of parameters (a, b), T a;b has a topologically transitive attractor  a;b . Benedicks and Young [8] later proved that for a positive-measure set of parameters (a, b), T a;b has a topologically transitive SRB attractor  a;b with SRB measure  a;b and that (Ta;b ;  a;b ;  a;b ) is isomorphic to a Bernoulli shift. Complex Dynamics Complex dynamics is concerned with the behavior of rational maps ˛1 z d C ˛2 z d1 C    ˛dC1 ˇ1 z d C ˇ2 z d1 C    ˇdC1 ¯ to itself, in which the of the extended complex plane C domain is C completed with the point at infinity (called the Riemann sphere). Recall that a family F of meromorphic functions is called normal on a domain D if every sequence possesses a subsequence that converges uniformly ¯ S 2 ) on compact subsets of (in the spherical metric C ¯ if it is normal on D. A family is normal at a point z 2 C ¯ of a rational a neighborhood of z. The Fatou set F(R) C ¯ ¯ ¯ map R : C ! C is the set of points z 2 C such that the family of forward iterates fR n gn0 is normal at z. The Julia set J(R) is the complement of the Fatou set F(R). The Fatou set is open and hence the Julia set is a closed set. Another characterization in the case d > 1 is that J(R) is the clo¯ ! C. ¯ sure of the set of all repelling periodic orbits of R : C

Both F(R) and J(R) are invariant under R. The dynamics of greatest interest is the restriction R : J(R) ! J(R). The Julia set often has a complicated fractal structure. In the case that R a (z) D z2  a, a 2 C, the Mandelbrot set is defined as the set of a for which the orbit of the origin 0 is bounded. The topology of the Mandelbrot set has been the subject of intense research. The study of complex dynamics is important because of the fascinating and complicated dynamics displayed and also because techniques and results in complex dynamics have direct implications for the behavior of one-dimensional maps. For more details see [25]. Infinite Ergodic Theory We may also consider a measure-preserving transformation (T; X; ) of a measure space such that (X) D 1. For example X could be the real line equipped with Lebesgue measure. This setting also arises with compact X in applications. For example, suppose T : [0; 1] ! [0; 1] is the simple model of intermittency given in Sect. “Intermittent Maps” and 2 (1; 2). Then T possesses an absolutely continuous invariant measure  with support [0; 1], but ([0; 1]) D 1. The Radon–Nikodym derivative of  with respect to Lebesgue measure m exists but is not in L1 (m). In this setting we say a measurable set A is a wandering set for T if fT n Ag1 nD0 are disjoint. Let D(T) be the measurable union of the collection of wandering sets for T. The transformation T is conservative with respect to  if (X n D(T)) D X (mod ) (see the article on Measure Preserving Systems). It is usually necessary to assume T conservative with respect to  to say anything interesting about its behavior. For example if T(x) D x C ˛, ˛ > 0, is a translation of the real line then D(T) D X. The definition of ergodicity in this setting remains the same: T is ergodic if A 2 B and T 1 A D A mod  implies that (A) D 0 or (Ac ) D 0. However the equivalence of ergodicity of T with respect to  and the equality of time and space averages for L1 () functions no longer holds. Thus in general  ergodic does not imply that lim

n!1

Z n1 1X  ı T j (x) D  d  a.e. x 2 X n X iD0

for all  2 L1 (). In the example of the intermittent map with 2 (1; 2) the orbit of Lebesgue almost every x 2 X is dense in X, yet the fraction of time spent near the indifferent fixed point x D 0 tends to one for Lebesgue almost every x 2 X. In fact it may be shown Sect. 2.4 in [1] that when (x) D 1 there are no constants a n > 0 such that Z n1 1 X lim  ı T j (x) D  d  a.e. x 2 X n!1 a n X iD0

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Nevertheless it is sometimes possible to obtain distributional limits, rather than almost sure limits, of Birkhoff sums under suitable normalization. We refer the reader to Aaronson’s book [1] for more details. Constructions We give examples of some of the standard constructions in dynamical systems. Often these constructions appear in modeling situations (for example skew products are often used to model systems which react to inputs from other systems, continuous time systems are often modeled as suspension flows over discrete-time dynamics) or to reduce systems to simpler components (often a factor system or induced system is simpler to study). Unless stated otherwise, in the sequel we will be discussing measure-preserving transformations on Lebesgue spaces (see the article on  Measure Preserving Systems). Products Given measure-preserving systems (X; B; ; T) and (Y; C ; ; S), their product consists of their completed product measure space with the transformation T S : X  Y ! X  Y defined by (T  S)(x; y) D (Tx; Sy) for all (x; y) 2 X  Y. Neither ergodicity nor transitivity is in general preserved by taking products; for example the product of an irrational rotation on the unit circle with itself is not ergodic. For a list of which mixing properties are preserved in various settings by forming a product see [113]. Given any countable family of measure-preserving transformations on probability spaces, their direct product is defined similarly. Factors We say that a measure-preserving system (Y; C ; ; S) is a factor of a measure-preserving system (X; B; ; T) if (possibly after deleting a set of measure 0 from X) there is a measurable onto map  : X ! Y such that  1 C B; T D S; T

1

and

(8)

D :

For Lebesgue spaces, there is a correspondence of factors of (X; B; ; T) and T-invariant complete sub- -algebras of B. According to Rokhlin’s theory of Lebesgue spaces [98] factors also correspond to certain partitions of X (see the article on  Measure Preserving Systems). A factor map  : X ! Y between Lebesgue spaces is an isomorphism if and only if it has a measurable inverse, or equivalently  1 C D B up to sets of measure 0.

Skew Products If (X; B; ; T) is a measure-preserving system, (Y; C ; ) is a measure-space, and fS x : x 2 Xg is a family of measurepreserving maps Y ! Y such that the map that takes (x, y) to S x y is jointly measurable in the two variables x and y, then we may define a skew product system consisting of the product measure space of X and Y equipped with product measure    together with the measure-preserving map T Ë S : X  Y ! X  Y defined by (T Ë S)(x; y) D (Tx; S x y) :

(9)

The space Y is called the fiber of the skew product and the space X the base. Sometimes in the literature the word skew product has a more general meaning and refers to the structure (T Ë S)(x; y) D (Tx; S x y) (without any assumption of measure-preservation), where the action of the map on the fiber Y is determined or ‘driven’ by the map T : X ! X. Some common examples of skew products include: Random Dynamical Systems Suppose that X indexes a collection of mappings S x : Y ! Y. We may have a transformation T : X ! X which is a full shift. Then the sequence of mappings fS T n x g may be considered a (random) choice of a mapping Y ! Y from the set fS x : x 2 Xg. The projection onto Y of the orbits of (Tx; S x y) give the orbits of a point y 2 Y under a random composition of maps S T n x ı  ıS T x ıS x . More generally we could consider the choice of maps Sx that are composed to come from any ergodic dynamical system, (T; X; ) to model the effect of perturbations by a stationary ergodic ‘noise’ process. Group Extensions of Dynamical Systems Suppose Y is a group,  is a measure on Y invariant under a left group action, and S x y :D g(x)y is given by a groupvalued function g : X ! Y. In this setting g is often called a cocycle, since upon defining g (n) (x) by (T Ë S)(n) (x; y) D (T n x; g (n) (x)y) we have a cocycle relation, namely g (mCn) (x) D g (m) (T n x)g (n) (x). Group extensions arise often in modeling systems with symmetry [37]. Common examples are provided by a random composition of matrices from a group of matrices (or more generally from a set of matrices which may form a group or not).

Induced Transformations Since by the Poincaré Recurrence Theorem (see [113]) a measure-preserving transformation (T; X; ; B) on a probability space is recurrent, given any set B of positive

Ergodic Theory: Basic Examples and Constructions

measure, the return-time function n B (x) D inffn  1 : T n x 2 Bg

(10)

is finite  a.e. We may define the first-return map by TB x D T n B (x) x :

(11)

Then (after perhaps discarding as usual a set of measure 0) TB : B ! B is a measurable transformation which preserves the probability measure B D /(B). The system (B; B \ B; B ; TB ) is called an induced, first-return or derived transformation. If (T; X; ; B) is ergodic then (B; B \ B; B ; TB ) is ergodic, but the converse is not in general true. The construction of the transformation T B allows us to represent the forward orbit of points in B via a tower or skyscraper over B. For each n D 1; 2; : : : , let B n D fx 2 B : n B (x) D ng :

(12)

Then fB1 ; B2 ; : : : g form a partition of B, which we think of as the bottom floor or base of the tower. The next floor is made up of TB2 ; TB3 ; : : : , which form a partition of TB n B, and so on. All these sets are disjoint. A column is a part of the tower of the form B n [ TB n [    [ T n1 B n for some n D 1; 2; : : : . The action of T on the entire tower is pictured as mapping each x not at the top of its column straight up to the point Tx above it on the next level, and mapping each point on the top level to T n B x 2 B. An equivalent way to describe the transformation on the tower is to write for each n and j < n, T j B n as f(x; j) : x 2 B n g, and then the transformation F on the tower is ( (x; l C 1) if l < n B (x)  1 ; F(x; l) D n (x) B x; 0) if l D n B (x)  1 : (T If T preserves a measure , then F preserves   dl, where l is counting measure. Sometimes the process of inducing yields an induced map which is easier to analyze (perhaps it has stronger hyperbolicity properties) than the original system. Sometimes it is possible to ‘lift’ ergodic or statistical properties from an induced system to the original system, so the process of inducing plays an important role in the study of statistical properties of dynamical systems [77]. It is possible to generalize the tower construction and relax the condition that nB (x) is the first-return time function. We may take a measurable set B X of positive  measure and define for almost every point x 2 B a height or ceiling function R : B ! N and take a countable partition fX n g of B into the sets on which R is constant. We

define the tower as the set  :D f(x; l) : x 2 B; 0  l < R(x)g and the tower map F :  !  by ( (x; l C 1) if l < R(x)  1 ; F(x; l) D R(x) x; 0) if l D R(x)  1 : (T R In this setting, if B R(x) d < 1, we may define an F-in  dl, where dl variant probability measure on  as C(R;B) is counting measure and C(R, B) is the normalizing conR stant C(R; B) D (B) B R(x)d. This viewpoint is connected with the construction of systems by cutting and stacking—see Sect. “Cutting and Stacking”. Suspension Flows The tower construction has an analogue in which the height function R takes values in R rather than N. Such towers are commonly used to model dynamical systems with continuous time parameter. Let (T; X; ) be a measure-preserving system and R : X ! (0; 1) a measurable “ceiling” function on X. The set X R D f(x; t) : 0  R(x) < tg ;

(13)

with measure  given locally by the product of  on X with Lebesgue measure m on R, is a measure space in a natural way. If  is a finite measure and R is integrable with respect to  then  is a finite measure. We define an action of R on X R by letting each point x flow at unit speed up the vertical lines f(x; t) : 0  t < R(x)g under the graph of R until it hits the ceiling, then jump to Tx, and so on. More precisely, defining R n (x) D R(x) C    C R(T n x), 8 (x; s C t) ˆ ˆ ˆ ˆ ˆ if 0  s C t < R(x); ˆ ˆ ˆ ˆ ˆ ˆ <(Tx; s C t  R(x)) Ts (x; t) D if R(x)  s C t < R(x) C R(Tx) ˆ ˆ ˆ ˆ ::: ˆ ˆ ˆ ˆ ˆ (T n x; s C t  [R(x) C    C R(T n1 x)]) ˆ ˆ : if R n1 (x)  s C t < R n (x) : (14) Ergodicity of (T; X; ) implies the ergodicity of (Ts ; X R ; ). Cutting and Stacking Several of the most interesting examples in ergodic theory have been constructed by this method; in fact, because of Rokhlin’s Lemma (see Sect. “Rokhlin’s Lemma”) every ergodic measure-preserving transformation on a Lebesgue

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space is isomorphic to one constructed by cutting and stacking. For example, the von Neumann-Kakutani adding machine (or 2-odometer) (Sect. “Adding Machines”), the Chacon weakly mixing but not strongly mixing system (Sect. “Chacon System”), Ornstein’s mixing rank one examples (see e. g. p. 160 ff. in [80]), and many more. We construct a Lebesgue measure-preserving transformation T on an interval X (bounded or maybe unbounded) by defining it as a translation on each of a pairwise disjoint countable collection of subintervals. The construction proceeds by stages, at each stage defining T on an additional part of X, until eventually T is defined a.e. At each stage X is represented as a tower, which is defined to be a disjoint union of columns. A column is defined to be a finite disjoint union of intervals of equal length, which are numbered from 0, for the “floor”, to the last one, for the “roof”, and which we picture as lying each above the preceding-numbered interval. T is defined on each level of a column (i. e. each interval in the column) except the roof by mapping it by translation to the next higher interval in the column. At stage 0, we have just one column, consisting of all of X as the floor, and T is not defined anywhere. To pass from one stage to the next, the columns are cut and stacked. This means that each column is divided, by vertical cuts, into a disjoint union of subcolumns of equal height (but maybe not equal width), and then some of these subcolumns are stacked above others (of the same width) so as to form a new tower. This allows the definition of T to be extended to some parts of X that were previously tops of towers, since they now may have levels above them. (Sometimes columns of height 1 are thought of as forming a reservoir for “spacers” to be inserted between subcolumns that are being stacked.) If the measure of the union of the tops of the columns tends to 0, eventually T becomes defined a.e. This description in words can be made precise with cumbersome notation, but the process can also be given a neater graphical description, which we sketch in the next section. Adic Transformations A.M. Vershik has introduced a family of models, called adic or Bratteli–Vershik transformations, into ergodic theory and dynamical systems. One begins with a graph which is arranged in levels, finitely many vertices on each level, with connections only from each level to the adjacent ones. The space X consists of the set of all infinite paths in this graph; it is a compact metric space in a natural way. We are given an order on the set of edges into each vertex,

and then X is partially ordered as follows: x and y are comparable if they agree from some point on, in which case we say that x < y if at the last level n where they traverse different edges, the edge xn of x is smaller than the edge yn of y: A map T is defined by letting Tx be the smallest y that is larger than x; if there is one. In nice situations, T is a homeomorphism after defining it and its inverse on perhaps countably many maximal and minimal elements. Invariant measures can sometimes be defined by assigning weights to edges, which are then multiplied to define the measure of each cylinder set. This is a nice combinatorial way to present the cutting and stacking method of constructing m.p.t.’s, allows for more convenient analysis of questions such as orbit equivalence, and leads to the construction of many interesting examples, such as those based on the Pascal or Euler graphs [4,38,76]. Odometers and generalizations are natural examples of adic systems. Vershik showed that in fact every ergodic measure-preserving transformation on a Lebesgue space is isomorphic to a uniquely ergodic adic transformation. See [111]. Rokhlin’s Lemma The following result is the fundamental starting point for many constructions in ergodic theory, from representing arbitrary systems in terms of cutting and stacking or adic systems, to constructing useful partitions and symbolic codings of abstract systems, to connecting convergence theorems in abstract ergodic theory with those in harmonic analysis. It allows us to picture arbitrarily long stretches of the action of a measure-preserving transformation as a translation within the set of integers. In the ergodic nonatomic case the statement follows readily from the construction of induced transformations. Lemma 1 (Rokhlin’s Lemma) Let T : X ! X be a measure-preserving transformation on a probability space (X; B; ). Suppose that (X; B; ) is nonatomic and T : X ! X is ergodic, or, more generally, (T; X; B; ) is aperiodic: that is to say, the set fx 2 X : there is n 2 N such that T n x D xg of periodic points has measure 0. Then given n 2 N and > 0, there is a measurable set B X such that the sets B; TB; : : : ; T n1 B are pairwise k disjoint and ([n1 kD0 T B) > 1  . Inverse Limits Suppose that for each i D 1; 2; : : : we have a Lebesgue probability space (X i ; B i ;  i ) and a measure-preserving transformation Ti : X i ! X i . Suppose also that for each i  j there is a factor map  ji : (T j ; X j ; B j ;  j ) ! (Ti ; X i ; B i ;  i ; ), such that each  j j is the identity on X j

Ergodic Theory: Basic Examples and Constructions

and  ji  k j D  ki whenever k  j  i. Let 1 X i :  ji x j D x i for all j  ig : X D fx 2 ˘ iD1

(15)

For each j, let  j : X ! X j be the projection defined by j x D x j. Let B be the smallest  -algebra of subsets of X which 1 contains all the  1 j B j . Define  on each  j B j by ( 1 j B) D  j (B)

for all B 2 B j :

(16)

Because  ji  j D  i for all j  i, the  1 j B j are increasing, and so their union is an algebra. The set function  can, with some difficulty, be shown to be countably additive on this algebra: since we are dealing with Lebesgue spaces, by means of measure-theoretic isomorphisms it is possible to replace the entire situation by compact metric spaces and continuous maps, then use regularity of the measures involved—see p. 137 ff. in [88]. Thus by Carathéodory’s Theorem (see the article on  Measure Preserving Systems)  extends to all of B. Define T : X ! X by T(x j ) D (T j x j ). Then (T; X; B; ) is a measure-preserving system such that any system which has all the (T j ; X j ; B j ;  j ) as factors, also has (T; X; B; ) a factor. Natural Extension The natural extension is a way to produce an invertible system from a non-invertible system. The original system is a factor of its natural extension and its orbit structure and ergodic properties are captured by the natural extension, as will be seen from its construction. Let (T; X; B; ) be a measure-preserving transformation of a Lebesgue probability space. Define ˝ : D f(x0 ; x1 ; x2 ; : : : ) : x n D T(x nC1 ); x n 2 X; n D 0; 1; 2; : : : g with  : ˝ ! ˝ defined by ((x0 ; x1 ; x2 ; : : : )) D (T(x0 ); x0 ; x1 ; : : : ). The map  is invertible on ˝. Given the invariant measure  we define the invariant measure ˜ on ˝ by defining it first on for the natural extension  cylinder sets C(A0 ; A1 ; : : : ; A k ) by ˜ (C(A 0 ; A1 ; : : : ; A k )) D (T k (A0 )\T kC1 (A1 )   \T kCi (A i )\  \A k ) and then extending it to ˝ using Kolmogorov’s extension theorem. We think of (x0 ; x1 ; x2 ; : : : ) as being an inverse branch of x0 2 X under the mapping T : X ! X. The ˜ if maps ;  1 : ˝ ! ˝ are ergodic with respect to  (T; X; B; ) is ergodic [113]. If  : ˝ ! X is projection onto the first component i. e. (x0 ; : : : ; x n ; : : : ) D x0 then

 ı  n (x0 ; : : : ; x n ; : : : ) D T n (x0 ) for all x0 and thus the natural extension yields information about the orbits of X under T. The natural extension is an inverse limit. Let (X; B; ) be a Lebesgue probability space and T : X ! X a map such that T 1 B B and T 1 D . For each i D 1; 2; : : : let (Ti ; X i ; B i ;  i ) D (T; X; B; ), and  ji D ˆ X; ˆ Bˆ ; ) ˆ T ji for each j > i. Then the inverse limit (T; of this system is an invertible measure-preserving system which is the natural extension of (T; X; B; ). We have Tˆ 1 (x1 ; x2 ; : : : ) D (x2 ; x3 ; : : : ) :

(17)

ˆ X; ˆ Bˆ ; ) ˆ The original system (T; X; B; ) is a factor of (T; (using any  i as the factor map), and any factor mapping from an invertible system onto (T; X; B; ) consists ˆ X; ˆ Bˆ ; ) ˆ followed by projecof a factor mapping onto (T; tion onto the first coordinate. Joinings Given measure-preserving systems (T; X; B; ) and (S; Y; C ; ), a joining of the two systems is a T  S-invariant measure P on their product measurable space that projects to  and , respectively, under the projections of X Y to X and Y, respectively. That is, if 1 : X Y ! X is the projection onto the first component i. e. 1 (x; y) D x then P(11 (A)) D (A) for all A 2 B and similarly for 2 : X  Y ! Y. This concept is the ergodic-theoretic version of the notion in probability theory of a coupling. The product measure  is always a joining of the two systems. If product measure is the only joining of the two systems, then we say that they are disjoint and write X ? Y [40]. If D is any family of systems, we write D? for the family of all measure-preserving systems which are disjoint from every system in D. Extensive recent accounts of the use of joinings in ergodic theory are in [42,99,110]. Future Directions The basic examples and constructions presented here are idealized, and many of the underlying assumptions (such as uniform hyperbolicity) are seldom satisfied in applications, yet they have given important insights into the behavior of real-world physical systems. Recent developments have improved our understanding of the ergodic properties of non-uniformly and partially hyperbolic systems. The ergodic properties of deterministic systems will continue to be an active research area for the foreseeable future. The directions will include, among others: establishing statistical and ergodic properties under weakened dependence assumptions; the study of systems which dis-

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play ‘anomalous statistics’; the study of the stability and typicality of ergodic behavior and mixing in dynamical systems; the ergodic theory of infinite-dimensional systems; advances in number theory (see the sections on Szemerédi and Ramsey theory); research into models with non-singular rather than invariant measures; and infinitemeasure systems. Other chapters in this Encyclopedia discuss in more detail these and other topics. Bibliography Primary Literature 1. Aaronson J (1997) An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol 50. American Mathematical Society, Providence. MR 1450400 (99d:28025) 2. Anosov DV (1967) Geodesic flows on closed riemannian manifolds of negative curvature. Trudy Mat Inst Steklov 90:209. MR 0224110 (36 #7157) 3. Arnol0 d VI (1963) Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat Nauk 18(6 (114)):91–192. MR 0170705 (30 #943) 4. Bailey S, Keane M, Petersen K, Salama IA (2006) Ergodicity of the adic transformation on the euler graph. Math Proc Cambridge Philos Soc 141(2):231–238. MR 2265871 (2007m:37010) 5. Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol 16. World Scientific Publishing Co Inc, River Edge. MR 1793194 (2001k:37035) 6. Benedicks M, Carleson L (1985) On iterations of 1  ax(X; B; )2 on (1; 1). Ann Math (2) 122(1):1–25. MR 799250 (87c:58058) 7. Benedicks M, Carleson L (1991) The dynamics of the hénon map. Ann Math (2) 133(1):73–169. MR 1087346 (92d:58116) 8. Benedicks M, Young LS (1993) Sina˘ı-bowen-ruelle measures for certain hénon maps. Invent Math 112(3):541–576. MR 1218323 (94e:58074) 9. Bertrand-Mathis A (1986) Développement en base ; répartition modulo un de la suite (x (X; B; )n)n0 ; langages codés et -shift. Bull Soc Math France 114(3):271–323. MR 878240 (88e:11067) 10. Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, N.Y., reprint of the 1965 original. MR 524567 (80b:28017) 11. Bissinger BH (1944) A generalization of continued fractions. Bull Amer Math Soc 50:868–876. MR 0011338 (6,150h) 12. Blanchard F (1989) ˇ -expansions and symbolic dynamics. Theoret Comput Sci 65(2):131–141. MR 1020481 (90j:54039) 13. Blanchard F, Hansel G (1986) Systèmes codés. Theoret Comput Sci 44(1):17–49. MR 858689 (88m:68029) 14. Blanchard F, Hansel G (1986) Systèmes codés et limites de systèmes sofiques. C R Acad Sci Paris Sér I Math 303(10):475–477. MR 865864 (87m:94009) 15. Blanchard F, Hansel G (1991) Sofic constant-to-one extensions of subshifts of finite type. Proc Amer Math Soc 112(1):259–265. MR 1050016 (91m:54050) 16. Boshernitzan M, Galperin G, Krüger T, Troubetzkoy S (1998) Periodic billiard orbits are dense in rational polygons. Trans Amer Math Soc 350(9):3523–3535. MR 1458298 (98k:58179)

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94. Robbin JW (1971) A structural stability theorem. Ann Math (2) 94:447–493. MR 0287580 (44 #4783) 95. Robinson C (1975) Errata to: “structural stability of vector fields” (ann. of math. (2) 99:154–175 (1974)). Ann Math (2) 101:368. MR 0365630 (51 #1882) 96. Robinson C (1976) Structural stability of c(X; B; )1 diffeomorphisms. J Differential Equations 22(1):28–73. MR 0474411 (57 #14051) 97. Robinson RC (1973) c(X; B; )r structural stability implies Kupka-Smale. In: Dynamical systems, (Proc Sympos, Univ Bahia, Salvador, 1971). Academic Press, New York, pp 443– 449. MR 0334282 (48 #12601) 98. Rohlin VA (1952) On the fundamental ideas of measure theory. Amer Math Soc Translation 1952(71):55. MR 0047744 (13,924e) 99. Rudolph DJ (1990) Fundamentals of Measurable Dynamics: Ergodic theory on Lebesgue spaces. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1086631 (92e:28006) 100. Rudolph DJ (1990) 2 and 3 invariant measures and entropy. Ergodic Theory Dynam Systems 10(2):395–406. MR 1062766 (91g:28026) 101. Ruelle D (1976) A measure associated with axiom-a attractors. Amer J Math 98(3):619–654. MR 0415683 (54 #3763) 102. Ruelle D (1978) Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics. Encyclopedia of Mathematics and its Applications, vol 5. AddisonWesley Publishing Co, Reading, MA, with a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655 (80g:82017) 103. Sarig O (2002) Subexponential decay of correlations. Invent Math 150(3):629–653. MR 1946554 (2004e:37010) 104. Schweiger F (1995) Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1419320 (97h:11083) 105. Sina˘ı JG (1968) Markov partitions and u-diffeomorphisms. Funkcional Anal i Priložen 2(1):64–89. MR 0233038 (38 #1361) 106. Sina˘ı JG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat Nauk 25(2 (152)):141–192. MR 0274721 (43 #481) 107. Sina˘ı JG (1972) Gibbs measures in ergodic theory. Uspehi Mat Nauk 27(4(166)):21–64. MR 0399421 (53 #3265) 108. Smale S (1980) The mathematics of time. Essays on dynamical systems, economic processes, and related topics. Springer, New York. MR 607330 (83a:01068) 109. Tabachnikov S (2005) Geometry and billiards, Student Mathematical Library, vol 30. American Mathematical Society, Providence, RI. MR 2168892 (2006h:51001) 110. Thouvenot JP (1995) Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc Lecture Note Ser, vol 205. Cambridge Univ Press, Cambridge, pp 207–235. MR 1325699 (96d:28017) 111. Vershik AM, Livshits AN (1992) Adic models of ergodic transformations, spectral theory, substitutions, and related topics. In: Representation theory and dynamical systems. Adv Soviet Math, vol 9. Amer Math Soc, Providence, RI, pp 185–204. MR 1166202 (93i:46131) 112. Vorobets YB, Gal0 perin GA, Stëpin AM (1992) Periodic billiard trajectories in polygons: generation mechanisms. Uspekhi Mat Nauk 47(3(285)):9–74, 207, (Russian with Russian

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Books and Reviews Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol 16. World Scientific Publishing Co Inc, River Edge. MR 1793194 (2001k:37035) Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, N.Y., pp xiii+194, reprint of the 1965 original. MR 524567 (80b:28017) Billingsley P (1995) Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 3rd edn. Wiley, New York, pp xiv+593, A Wiley-Interscience Publication. MR 1324786 (95k:60001) Bonatti C, Díaz LJ, Viana M (2005) Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective; Mathematical Physics, III. In: Encyclopaedia of Mathematical Sciences,vol. 102. Springer, Berlin, pp xviii+384. MR 2105774 (2005g:37001) Boyarsky A, Góra P (1997) Laws of chaos: Invariant measures and dynamical systems in one dimension. Probability and its Applications. Birkhäuser, Boston. MR 1461536 (99a:58102) Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge. MR 1963683 (2003m:37001) Carleson L, Gamelin TW (1993) Complex dynamics. Universitext: Tracts in Mathematics. Springer, New York. MR 1230383 (94h:30033) Chernov N, Markarian R (2006) Chaotic billiards, Mathematical Surveys and Monographs, vol 127. American Mathematical Society, Providence, RI. MR 2229799 (2007f:37050) Collet P, Eckmann JP (1980) Iterated maps on the interval as dynamical systems, Progress in Physics, vol 1. Birkhäuser, Boston. MR 613981 (82j:58078) Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic Theory, Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol 245. Springer, New York, translated from the Russian by A. B. Sosinski˘ı. MR 832433 (87f:28019) Denker M, Grillenberger C, Sigmund K (1976) Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol 527. Springer, Berlin. MR 0457675 (56 #15879)

Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies, No 29. Van Nostrand Reinhold Co, New York. MR 0435350 (55 #8310) Glasner E (2003) Ergodic Theory via Joinings, Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence, RI. MR 1958753 (2004c:37011) Halmos PR (1960) Lectures on Ergodic Theory. Chelsea Publishing Co, New York. MR 0111817 (22 #2677) Hasselblatt B, Katok A (2003) A First Course in Dynamics: With a panorama of recent developments. Cambridge University Press, Cambridge. MR 1995704 (2004f:37001) Hopf E (1937) Ergodentheorie, 1st edn. Ergebnisse der Mathematik und ihrer Grenzgebiete; 5. Bd, 2, J. Springer, Berlin Jacobs K (1965) Einige neuere Ergebnisse der Ergodentheorie. Jber Deutsch Math-Verein 67(Abt 1):143–182. MR 0186789 (32 #4244) Keller G (1998) Equilibrium States in Ergodic Theory. In: London Mathematical Society Student Texts, vol 42. Cambridge University Press, Cambridge, pp x+178. MR 1618769 (99e:28022) Mañé R (1987) Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), vol 8. Springer, Berlin, translated from the Portuguese by Silvio Levy. MR 889254 (88c:58040) Nadkarni MG (1998) Spectral Theory of Dynamical Systems. Birkhäuser Advanced Texts: Basler Lehrbücher. (Birkhäuser Advanced Texts: Basel Textbooks), Birkhäuser, Basel. MR 1719722 (2001d:37001) Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge, With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374 (96c:58055) Parry W, Pollicott M (1990) Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque (187-188):268. MR 1085356 (92f:58141) Ornstein DS, Rudolph DJ, Weiss B (1982) Equivalence of measure preserving transformations. Mem Amer Math Soc 37(262):xii+116. MR 653094 (85e:28026) Petersen K (1989) Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge. corrected reprint of the 1983 original. MR 1073173 (92c:28010) Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York. MR 1013117 (90g:00004) Rudolph DJ (1990) Fundamentals of Measurable Dynamics: Ergodic theory on Lebesgue spaces. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1086631 (92e:28006) Schweiger F (1995) Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1419320 (97h:11083) Thouvenot JP (1995) Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc Lecture Note Ser, vol 205. Cambridge Univ Press, Cambridge, pp 207–235. MR 1325699 (96d:28017) Walters P (1982) An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol 79. Springer, New York. MR 648108 (84e:28017)

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Ergodic Theory: Fractal Geometry JÖRG SCHMELING Center for Mathematical Sciences, Lund University, Lund, Sweden Article Outline Glossary Definition of the Subject Introduction Preliminaries Brief Tour Through Some Examples Dimension Theory of Low-Dimensional Dynamical Systems – Young’s Dimension Formula Dimension Theory of Higher-Dimensional Dynamical Systems Hyperbolic Measures General Theory Endomorphisms Multifractal Analysis Future Directions Bibliography Glossary Dynamical system A (discrete time) dynamical system describes the time evolution of a point in phase space. More precisely a space X is given and the time evolution is given by a map T : X ! X. The main interest is to describe the asymptotic behavior of the trajectories (orbits) T n (x), i. e. the evolution of an initial point x 2 X under the iterates of the map T. More generally one is interested in obtaining information on the geometrically complicated invariant sets or measures which describe the asymptotic behavior. Fractal geometry Many objects of interest (invariant sets, invariant measures etc.) exhibit a complicated structure that is far from being smooth or regular. The aim of fractal geometry is to study those objects. One of the main tools is the fractal dimension theory that helps to extract important properties of geometrically “irregular” sets. Definition of the Subject The connection between fractal geometry and dynamical system theory is very diverse. There is no unified approach and many of the ideas arose from significant examples. Also the dynamical system theory has been shown to have a strong impact on classical fractal geometry. In this article there are first presented some examples showing nontrivial results coming from the application of dimension the-

ory. Some of these examples require a deeper knowledge of the theory of smooth dynamical systems then can be provided here. Nevertheless, the flavor of these examples can be understood. Then there is a brief overview of some of the most developed parts of the application of fractal geometry to dynamical system theory. Of course a rigorous and complete treatment of the theory cannot be given. The cautious reader may wish to check the original papers. Finally, there is an outlook over the most recent developments. This article is by no means meant to be complete. It is intended to give some of the ideas and results from this field. Introduction In this section some of the aspects of fractal geometry in dynamical systems are pointed out. Some notions that are used will be defined later on and I intend only to give a flavor of the applications. The nonfamiliar reader will find the definitions in the corresponding sections and can return to this section later. The geometry of many invariant sets or invariant measures of dynamical systems (including attractors, measures defining the statistical properties) look very complicated at all scales, and their geometry is impossible to describe using standard geometric tools. For some important classes of dynamical systems, these complicated structures are intensively studied using notions of dimension. In many cases it becomes possible to relate these notions of dimension to other fundamental dynamical characteristics, such as Lyapunov exponents, entropies, pressure, etc. On the other hand tools from dynamical systems, especially from ergodic theory and thermodynamic formalism, are extremely useful to explore the fractal properties of the objects in question. This includes dimensions of limit sets of geometric constructions (the standard Cantor set being the most famous example), which a priori, are not related to dynamical systems [46,100]. Many dimension formulas for asymptotic sets of dynamical systems are obtained by means of Bowen-type formulas, i. e. as roots of some functionals arising from thermodynamic formalism. The dimension of a set is a subtle characteristic which measures the geometric complexity of the set at arbitrarily fine scales. There are many notions of dimension, and most definitions involve a measurement of geometric complexity at scale " (which ignores the irregularities of the set at size less than ") and then considers the limiting measurement as " ! 0. A priori (and in general) these different notions can be different. An important result is the affirmative solution of the Eckmann–Ruelle conjecture by Barreira, Pesin and Schmeling [17], which says that for smooth nonuniformly hyperbolic systems, the pointwise

Ergodic Theory: Fractal Geometry

dimension is almost everywhere constant with respect to a hyperbolic measure. This result implies that many dimension characteristics for the measure coincide. The deep connection between dynamical systems and dimension theory seems to have been first discovered by Billingsley [21] through several problems in number theory. Another link between dynamical systems and dimension theory is through Pesin’s theory of dimension-like characteristics. This general theory is a unification of many notions of dimension along with many fundamental quantities in dynamical system such as entropies, pressure, etc. However, there are numerous examples of dynamical systems exhibiting pathological behavior with respect to fractal geometrical characteristics. In particular higher-dimensional systems seem to be as complicated as general objects considered in geometric measure theory. Therefore, a clean and unified theory is still not available. The study of characteristic notions like entropy, exponents or dimensions is an essential issue in the theory of dynamical systems. In many cases it helps to classify or to understand the dynamics. Most of these characteristics were introduced for different questions and concepts. For example, entropy was introduced to distinguish nonisomorphic systems and appeared to be a complete invariant for Bernoulli systems (Ornstein). Later, the thermodynamic formalism (see [107]) introduced new quantities like the pressure. Bowen [27] and also Ruelle discovered a remarkable connection between the thermodynamic formalism and the dimension theory for invariant sets. Since then many efforts were taken to find the relations between all these different quantities. It occurred that the dimension of invariant sets or measures carries lots of information about the system, combining its combinatorial complexity with its geometric complexity. Unfortunately it is extremely difficult to compute the dimension in general. The general flavor is that local divergence of orbits and global recurrence cause complicated global behavior (chaos). It is impossible to study the exact (infinite) trajectory of all orbits. One way out is to study the statistical properties of “typical” orbits by means of an invariant measure. Although the underlying system might be smooth the invariant measures may often be singular. Preliminaries Throughout the article the following situation is considered. Let M be a compact Riemannian manifold without boundary. On M is acting a dynamical system generated by a C 1C˛ diffeomorphism T : M ! M. The presence of a dynamical system provides several important additional tools and methods for the theory of fractal dimensions.

Also the theory of fractal dimensions allows one to draw deep conclusions about the dynamical system. The importance and relevance of the study of fractal dimension will be explained in later sections. In the next sections some of the most important tools in the fractal theory of dynamical systems are considered. The definitions given here are not necessarily the original definitions but rather the ones which are closer to contemporary use. More details can be found in [95]. Some Ergodic Theory Ergodic theory is a powerful method to analyze statistical properties of dynamical systems. All the following facts can be found in standard books on ergodic theory like [103,124]. The main idea in ergodic theory is to relate global quantities to observations along single orbits. Let us consider an invariant measure: ( f 1 A) D (A) for all measurable sets A. Such a measure “selects” typical trajectories. It is important to note that the properties vary with the invariant measures. Any such invariant measure can be decomposed into elementary parts (ergodic components). An invariant measure is called ergodic if for any invariant set A D T 1 A one has (A)(M n A) D 0 (with the agreement 0  1 D 0), i. e. from the measure-theoretic point of view there are no nontrivial invariant subsets. The importance of ergodic probability measures (i. e. (M) D 1) lies in the following theorem of Birkhoff Theorem 1 (Birkhoff) Let  be an ergodic probability measure and ' 2 L1 (). Then Z n1 1X k '(T x) D ' d   a.e : lim n!1 n M kD0

Hausdorff Dimension With Z R N and s  0 one defines

X diam(B i )s : mH (s; Z) D lim inf ı!0 fB i g

i

sup diam(B i ) < " and i

[

 Bi  Z :

i

Note that this limit exists. mH (s; Z) is called the s-dimensional outer Hausdorff measure of Z. It is immediate that there exists a unique value s , called the Hausdorff dimension of Z, at which mH (s; Z) jumps from 1 to 0. In general it is very hard to find optimal coverings and hence it is often impossible to compute the Hausdorff dimension of a set. Therefore a simpler notion – the lower

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and upper Box dimension – was introduced. The difference to the Hausdorff dimension is that the covering balls are assumed to have the same radius ". Since then the limit as " ! 0 does not have to exist one arrives at the notion of the upper and lower dimension. Dimension of a Measure Definition 1 Let Z R N and let  be a probability measure supported on Z. Define the Hausdorff dimension of the measure  by dimH () 

inf

K Z : (K)D1

dimH (K):

Pointwise Dimension Most invariant sets or measures are not strongly self-similar, i. e. the local geometry at arbitrarily fine scales might look different from point to point. Therefore, the notion of pointwise dimension with respect to a Borel probability measure is defined. Let  be a Borel probability measure. By B(x; ") the ball with center x and radius " is denoted. The pointwise dimension of the measure  at the point x is defined as d (x) :D lim

"!0

log (B(x; ")) log "

if the above limit exists. If d (x) D d, then for small " the measure of small balls scales as (B(x; "))  "d . Proposition 1 Suppose  is a probability measure supported on Z R N . Then d (x) D d for  – almost all x 2 Z implies dimH () D d. One should not take the existence of a local dimension (even for good measures) for granted. Later on it will be seen that the spectrum of the pointwise dimension (dimension spectrum) is a main object of study in classical multifractal analysis. The dimension of a measure or a set of measures is its geometric complexity. However, under the presence of a dynamical system one also wants to measure the dynamical (combinatorial) complexity of the system. This leads to the notion of entropy. Dimension-Like Characteristics and Topological Entropy Pesin’s theory of dimension-like characteristics provides a unified treatment of dimensions and important dynamical quantities like entropies and pressure. The topological entropy of a continuous map f with respect to a subset Z

in a metric space (X; ) (in particular X D M – a Riemannian manifold) can be defined as a dimension-like characteristic. For each n 2 N and " > 0, define the Bowen ball B n (x; ") D fy 2 X : (T i (x); T i (y))  " for 0  i  ng. Then let m h (Z; ˛; e; n) :D ( ) X [ ˛n i e : n i > n; B n i (x; e)  Z : lim inf n!1

i

i

This gives rise to an outer measure that jumps from 1 to 0 at some value ˛  . This threshold value ˛  is called the topological entropy of Z (at scale e). However, in many situations this value does not depend on e. The topological entropy is denoted by htop (TjZ). If Z is f – invariant and compact, this definition of topological entropy coincides with the usual definition of topological entropy [124]. ˚The entropy h of a measure  is defined as h D inf htop (TjZ) : (Z) D 1 . For ergodic measures this definition coincides with the Kolmogorov–Sinai entropy (see [95]). One has to note that in the definition of entropy metric (“round”) balls are substituted by Bowen balls and the metric diameter by the “depth” of the Bowen ball. Therefore, the relation between entropy and dimension is determined by the relation of “round” balls to “oval” dynamical Bowen balls. If one understands how metric balls can be efficiently used to cover dynamical balls one can use the dynamical and relatively easy relation to compute notion of entropy to determine the dimension. However, in higher dimensions this relation is by far nontrivial. A heuristic argument for comparing “round” balls with dynamical balls is given in Subsect. “The Kaplan–Yorke Conjecture”. The Pressure Functional A useful tool in the dimension analysis of dynamical systems is the pressure functional. It was originally defined by means of statistical physics (thermodynamic formalism) as the free energy (or pressure) of a potential ' (see for example [107]). However, in this article a dimension-like definition (see [95]) is more suitable. Again an outer measure using Bowen balls will be used. Let ' : M ! R be a continuous function and (  X exp  ˛n i m P (Z; ˛; "; n; ') D lim inf n!1

C

i

sup

n X

x2B n i (x;") kD0

) '(T x) : k

Ergodic Theory: Fractal Geometry

This defines an outer measure that jumps from 1 to 0 as ˛ increases.The threshold value ˛  is called the topological pressure of the potential ' denoted by P('). In many situations it does not depend on ". There is also a third way of defining the pressure in terms of a variational principle (see [124]):   Z ' d P(') D sup h C invariant

M

Brief Tour Through Some Examples Before describing the fractal theory of dynamical systems in more detail some ideas about its role are presented. The application of dimension theory has many different aspects. At this point some (but by far not all) important examples are considered that should give the reader some feeling about the importance and wide use of dimension theory in dynamical system theory. Dimension of Conformal Repellers: Ruelle’s Pressure Formula Computing or estimating a dimension via a pressure formula is a fundamental technique. Explicit properties of pressure help to analyze subtle characteristics. For example, the smooth dependence of the Hausdorff dimension of basic sets for Axiom- A surface diffeomorphisms on the derivative of the map follows from smoothness of pressure. Ruelle proved the following pressure formula for the Hausdorff dimension of a conformal repeller. A conformal repeller J is an invariant set T(J) D J D fx 2 M : f n x 2 V8n 2 N and some neighborhood V of J} such that for any x 2 J the differential D x T D a(x) Isox where a(x) is a scalar with ja(x)j > 1 and Iso x an isometry of the tangent space Tx M. Theorem 2 ([108]) Let T : J ! J be a conformal repeller, and consider the function t ! P(t log jD x Tj), where P denotes pressure. Then P(s log jD x Tj) D 0

()

s D dimH (J)

Iterated Function Systems In fractal geometry one of the most-studied objects are iterated function systems, where there are given n contractions F1 ;    ; Fn of Rd . The unique compact set J fulfilling S J D i F i (J) is called the attractor of the iterated function system (see [42]). One question is to evaluate its Hausdorff dimension. Often (for example under the open set condition) the attractor J can be represented as the repeller of a piecewise expanding map T : Rd ! Rd where the F i are

the inverse branches of the map T. In general it is by far not trivial to determine the dimension of J or even the dimension of a measure sitting on J. The following example explains some of those difficulties. Let 1/2 <  < 1 and consider the maps F i : [0; 1] ! [0; 1] given by F1 (x) D x and F2 (x) D x C (1  ). Then the images of F1 ; F2 have an essential overlap and J D [0; 1]. If one randomizes this construction in the way that one applies both maps each with probability 1/2 a probability measure is induced on J. This measure might be absolute continuous with respect to Lebesgue measure or not. Already Erdös realized that for some special values of  (for example for the inverse of the golden mean) the induced measure is singular. In a breakthrough paper B. Solomyak ([119]) proved that for a.e.  the induced measure is absolutely continuous. A main ingredient in the proof is a transversality condition in the parameter space: the images of arbitrary two random samples of the (infinite) applications of the maps F i have to cross with nonzero speed when the parameter  changes. This is a general mechanism which allows one to handle more general situations. Homoclinic Bifurcations for Dissipative Surface Diffeomorphisms Homoclinic tangencies and their bifurcations play a fundamental role in the theory of dynamical systems [87,88,89]. Systems with homoclinic tangencies have a complicated and subtle quasi-local behavior. Newhouse showed that homoclinic tangencies can persist under small perturbations, and that horseshoes may co-exist with infinitely many sinks in a neighborhood of the homoclinic orbit and hence the system is not hyperbolic (Newhouse phenomenon). Let T : M 2 ! M 2 be a smooth parameter family of surface diffeomorphisms that exhibits for  D 0 an invariant hyperbolic set 0 (horseshoe) and undergoes a homoclinic bifurcation. The Hausdorff dimension of the hyperbolic set 0 for T 0 determines whether hyperbolicity is the typical dynamical phenomenon near T 0 or not. If dimH 0 < 1, then hyperbolicity is the prevalent dynamical phenomenon near f 0 . This is not the case if dimH 0 > 1. More precisely, let N W  denote the set of nonwandering points of T in an open neighborhood of 0 after the homoclinic bifurcation. Let ` denote Lebesgue measure. Theorem 3 ([87,88]) If dimH 0 < 1, then lim

0 !0

`f 2 [0; 0 ] : N W  is hyperbolicg D1: 0

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The hyperbolicity co-exists with the Newhouse phenomena for a residual set of parameter values. Theorem 4 (Palis–Yoccoz) If dimH 0 > 1, then `f 2 [0; 0 ] : N W  is hyperbolicg < 1: 0 !0 0 lim

Some Applications to Number Theory Sometimes dimension problems in number theory can be transferred to dynamical systems and attacked using tools from dynamics. Example Diadic expansion of numbers: Consider a real number expanded in base 2, i. e. x D P1 n nD1 x n /2 . Let ( Xp D

) n1 1X x : lim xk D p : n!1 n kD0

1

xD

1

a1 C

1

a2 C a3 C

1 : a4 C : :

D [a1 ; a2 ; a3 ; a4 ;    ] ; and the approximants p n (x)/q n (x) D [a1 ; a2 ; : : : ; a n ] (i. e. the approximation given by the finite continued fraction after step n). The set of numbers which admit a faster approximation by rational numbers is defined as follows. For   2, let F ˇ ˇ

ˇ pˇ 1 F D x 2 [0; 1] : ˇˇ x  ˇˇ   q q  for infinitely many p/q :

Borel showed that `(X 1/2 ) D 1, where ` denotes Lebesgue measure. This result is an easy consequence of the Birkhoff ergodic theorem applied to the characteristic function of the digit 1 (which in this simple case is the Strong Law of Large Numbers for i.i.d. processes).

It is well known that this set has zero measure for each   2. Jarnik [58] computed the Hausdorff dimension of F and showed that dimH (F ) D 2/. However, nowadays there are methods from dynamical systems which not only allow unified proofs but also can handle more subtle subsets of the reals defined by some properties of their continued fraction expansion (see for example [3,105]).

One can ask how large is the set X p in general. Eggleston [39] discovered the following wonderful dimension formula, which Billingsley [21] interpreted in terms of dynamics and reproved using tools from ergodic theory.

Infinite Iterated Function Systems and Parabolic Systems

Theorem 5 ([39]) The Hausdorff dimension of X p is given by dimH (X p ) D

1   p log p  (1  p) log(1  p) : log 2

The underlying dynamical system is E2 (x) D 2x mod 1 and the dimension is the dimension of the Bernoulli p measure. Other cases included that by Rényi who proposed generalization of the base d expansion from integer base d to noninteger base ˇ. In this case the underlying dynamical system is the (in general non-Markovian) beta shift ˇ(x) D ˇx mod 1 studied in [110]. There are many investigations concerning the approximation of real numbers by rationals by using dynamical methods. The underlying dynamical system in this case is the Gauss map T(x) D (1/x) mod 1. This map is uniformly expanding, but has infinitely many branches. Example Continued fraction approximation of numbers Consider the continued fraction expansion of x 2 [0; 1], i. e.,

In the previous section a system with infinitely many branches appeared. This can be regarded as an iterated function system with infinitely many maps F i . This situation is quite general. If one considers a (one-dimensional) system with a parabolic (indifferent) fixed point, i. e. there is a fixed point where the derivative has absolute value equal to 1, one often uses an induced system. For this one chooses nearby the parabolic point a higher iterate of the map in order to achieve uniform expansion away from the parabolic point. This leads to infinitely many branches since the number of iterates has to be increased the closer the parabolic point is. The main difference to the finite iterated function system is that the setting is no longer compact and many properties of the pressure functional are lost. Mauldin and Urba´nski and others (see for example [3,75,76,78]) developed a thermodynamic formalism adapted to the pressure functional for infinite iterated function systems. Besides noncompactness one of the main problems is the loss of analyticity (phase transitions) and convexity of the pressure functional for infinite iterated function systems.

Ergodic Theory: Fractal Geometry

Complex Dynamics Let T : C ! C be a polynomial and J its Julia set (repeller of this system). If this set is hyperbolic, i. e. the derivative at each point has absolute value larger than 1 the study of the dimension can be related to the study of a finite iterated function system. However, in the presence of a parabolic fixed point this leads to an infinite iterated function system. If one considers the coefficients of the polynomial as parameters one often sees a qualitative change in the asymptotic behavior. For example, the classical Mandelbrodt set for polynomials z2 C c is the locus of values c 2 C for which the orbit of the origin stays bounded. This set is well known to be fractal. However, its complete description is still not available. Embedology and Computational Aspects of Dimension Tools from dynamical systems are becoming increasingly important to study the time evolution of deterministic systems in engineering and the physical and biological sciences. One of the main ideas is to model a “real world” system by a smooth dynamical system which possesses a strange attractor with a natural ergodic invariant measure. When studying a complicated real world system, one can measure only a very small number of variables. The challenge is to reconstruct the underlying attractor from the time measurement of a scaler quantity. An idealized measurement is considered as a function h : M n ! R. The main tool researchers currently use to reconstruct the model system is called attractor reconstruction (see papers and references in [86]). This method is based on embedding with time delays (see the influential paper [33], where the authors attribute the idea of delay coordinates to Ruelle), where one attempts to reconstruct the attractor for the model using a single long trajectory. Then one considers the points in R pC1 defined by (x k ; x kC ; x kC2 ; : : : ; x kCp ). Takens [122] showed that for a smooth T : M n ! M n and for typical smooth h, the mapping ' : M n ! R2nC1 defined by x ! (h(x); h( f  (x));    ; h( f 2n (x)) is an embedding. Since the box dimension of the attractor  may be much less than the dimension of the ambient manifold n, an interesting mathematical question is whether there exists p < 2n C 1 such that the mapping on the attractor ' :  ! R p defined by x ! (h(x); h( f (x)); : : : ; h( f p (x)) is 1  1? It is known that for a typical smooth h the mapping ' is 1  1 for p > 2 dimB () [29]. Denjoy Systems This section will give some ideas indicating the principle difficulties that arise in systems with low complexity.

Contrary to hyperbolic systems (each vector in the tangent space is either contracted or expanded) finer mechanisms determine the local behavior of the scaling of balls. While in hyperbolic systems the dynamical scaling of small balls is exponential in a low-complexity system this scaling is subexponential and hence the linearization error is of the same magnitude. Up to now there is no general dimension theory for low complexity systems. A specific example presented here is considered in [67]. Poincaré showed that to each orientation preserving homeomorphism of the circle S 1 D R/Z is associated a unique real parameter ˛ 2 [0; 1), called the rotation number, so that the ordered orbit structure of T is the same as that of the rigid rotation R˛ , where R˛ (t) D (t C ˛) mod 1, provided that ˛ is irrational. Half a century later, Denjoy [35] constructed examples of C1 diffeomorphisms that are not conjugate (via a homeomorphism) to rotations. This was improved later on by Herman [55]. In these examples, the minimal set of T is necessarily a Cantor set ˝. The arithmetic properties of the rotation number have a strong effect on the properties of T. One area that has been well understood is the relation between the differentiability of T, the differentiability of the conjugation and the arithmetic properties of the rotation number. (See, for example, Herman [55]) Without stating any precise theorem, the results differ sharply for Diophantine and for Liouville rotation numbers (definition follows). Roughly speaking the conjugating map is always regular for Diophantine rotation numbers while it might be not smooth at all for Liuoville rotation numbers. Definition 2 An irrational number ˛ is of Diophantine class  D (˛) 2 RC if kq˛k <

1 q

has infinitely many solutions in integers q for  <  and at most finitely many for  >  where k  k denotes the distance to the nearest integer. In [67] the effect of the rotation number on the dimension of ˝ is studied. There the main result is Theorem 6 Assume that 0 < ı < 1 and that ˛ 2 (0; 1) is of Diophantine class  2 (0; 1). Then an orientation preserving C 1Cı diffeomorphism of the circle with rotation number ˛ and minimal set ˝˛ı satisfies dimH ˝˛ı 

ı : 

Furthermore, these results are sharp, i. e. the standard Denjoy examples attain the minimum.

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Return Times and Dimension Recently an interesting connection between the pointwise dimensions, multifractal analysis, and recurrence behavior of trajectories was discovered [1,11,23]. Roughly speaking, given an ergodic probability measure  the return time asymptotics (as the neighborhood of the point shrinks) of -a.e point is determined by the pointwise dimension of  at this point. The deeper understanding of this relation would help to get a unified approach to dimensions, exponents, entropies, recurrence times and correlation decay. Dimension Theory of Low-Dimensional Dynamical Systems – Young’s Dimension Formula In this section a remarkable extension of Ruelle’s dimension formula by Young [128] for the dimension of a measure is discussed. Theorem 7 Let T : M 2 ! M 2 be a C2 surface diffeomorphism and let  be an ergodic measure. Then   1 1 dimH () D h ( f )  ; 1 2 where 1  2 are the two Lyapunov exponents for .

2) Number theoretic properties of some scaling rates [104, 106] enter into dimension calculations in ways they do not in low dimensions (see Subsect. “Iterated Function Systems”). 3) The dimension theory of sets is often reduced to the theory of invariant measures. However, there is no invariant measure of full dimension in general and measure-theoretic considerations do not apply [79]. 4) The stable and unstable foliations for higher dimensional systems are typically not C1 [51,109]. Hence, to split the system into an expanding and a contracting part is far more subtle. Dimension Theory of Higher-Dimensional Dynamical Systems Here an example of a hyperbolic attractor in dimension 3 is considered to highlight some of the difficulties. Let 4 denote the unit disc in R2 . Let f : S 1  4 ! S 1  4 be of the form T(t; x; y) D ('(t);

Theorem 8 Let  be a basic set for a C2 Axiom-A surface diffeomorphism T : M 2 ! M 2 . Then dimH () D s1 C s2 , where s1 and s2 satisfy P(s log kDTx jE xu k) D 0 P(s log kDTx jE xs k) D 0 : where Es and Eu are the stable and unstable directions, respectively. Some Remarks on Dimension Theory for Low-Dimensional versus High-Dimensional Dynamical Systems Unlike lower dimensions (one, two, or conformal repellers), for higher-dimensional dynamical systems there are no general dimension formulas (besides the Ledrappier–Young formula), and in general dimension theory is much more difficult. This is due to several problems: 1) The geometry of the Bowen balls differs in a substantial way from round balls.

(t; x);

2

(t; y)) ;

with 0 < max

S 1 4

In [79], Manning and McCluskey prove the following dimension formula for a basic set (horseshoe) of an Axiom-A surface diffeomorphism which is a set-version of Young’s formula.

1

@ @x

1

(t; x) < min

S 1 4

@ @y

2

(t; y) <  < 1 :

The limit set \ T n (S 1  4)  :D n2N

is called the attractor or the solenoid. It is an example of a structurally stable basic set and is one of the fundamental examples of a uniformly hyperbolic attractor. The following result can be proved. Theorem 9 ([24,52]) For all t, the thermodynamic pressure ˇ ˇ  ˇ@ 2 ˇ s ˇ (t; y)ˇˇ D 0 : P dimH  t log ˇ @y In particular, the stable dimension is independent of the stable section. In this particular case the invariant axes for strong and weak contraction split the system smoothly and the difficulty is to show that the strong contraction is dominated by the weaker. In particular one has to ensure that effects as described in Subsect. “Iterated Function Systems” do not appear. In the general situation this is not the case and one lacks a similar theorem in the general situation. In particular, the unstable foliation is not better than Hölder and does not provide a “nice” coordinate.

Ergodic Theory: Fractal Geometry

Hyperbolic Measures

General Theory

Given x 2 M and a vector v 2 Tx M the Lyapunov exponent is defined as

In this section the dimension theory of higher-dimensional dynamical systems is investigated. Most developed is this theory for invariant measures. There is an important connection between Lyapunov exponents and the measure theoretic entropy and dimensions of measures that will be presented here. Let  be an ergodic invariant measure. The Oseledec and Pesin Theory guarantee that local stable manifolds exist at -a.e. point. As for the Kaplan–Yorke formula the idea is to consider the contributions to the entropy and to the dimension in the directions of i . Historically the first connections between exponents and entropy were discovered by Margulis and Ruelle. They proved that

(x; v) D lim

n!1

1 log kD x T n vk : n

provided this limit exists. For fixed x, the numbers (x; ) attain only finitely many values. By ergodicity they are a.e. constant, and the corresponding values are denoted by 1       p ; where p D dim M. Denote by s (s stands for stable) the largest index such that s < 0. Definition 3 The invariant ergodic measure  is said to be hyperbolic if  i ¤ 0 for every i D 1, : : :, p. The Kaplan–Yorke Conjecture In this section a heuristic argument for the dimension of an invariant ergodic measure will be given. This argument uses a specific cover of a dynamical ball by “round” balls. These ideas are essentially developed by Kaplan and Yorke [61] for the estimation of the dimension of an invariant set. Their estimates always provide an upper bound for the dimension. Kaplan and Yorke conjectured that in typical situations these estimates also provide a lower bound for the dimension of an attractor. Ledrappier and Young [72,73] showed that in case this holds for an invariant measure, then this measure has to be very special: an SBR-measure (after Sinai, Ruelle, and Bowen) (an SRBmeasure is a measure that describes the asymptotic behavior for a set of initial points of positive Lebesgue measure and has absolutely continuous conditional measures on unstable manifolds). Consider a small ball B in the phase space. The image T n B is almost an ellipsoid with axes of length e1 n ;    ; e p n : For 1  i  s, cover T n B by balls of radius e i n . Then approximately exp[ p n] exp[ iC1 n] ::: : exp[ i n] exp[ i n] balls are needed for covering. The dimension can be estimated from above by P j>i  j (1) C (p  i) :D dimLi : dimB   j i j This is the Kaplan–Yorke formula.

h ( f )  

s X

i

iD1

for a C1 diffeomorphism T. Pesin [91] showed that this inequality is actually an equality if the measure  is essentially the Riemannian volume on unstable manifolds. Ledrappier and Young [72] showed that this is indeed a necessary condition. They also provided an exact formula: Theorem 10 (Ledrappier–Young [72,73]) With d 0 D 0 for a C2 diffeomorphism holds h ( f ) D 

s X

   i d i  d i1

iD1

where di are the dimensions of the (conditional) measure on the ith unstable leaves. The proof of this theorem is difficult and uses the theory of nonuniform hyperbolic systems (Pesin theory). In dimension 1 and 2 the above theorem resembles Ruelle’s and Young’s theorems. The reader should note that the above theorem includes also the existence of the pointwise dimension along the stable and unstable direction. Here the question arises whether this implies the existence of the pointwise dimension itself. The Existence of the Pointwise Dimension for Hyperbolic Measure – the Eckmann–Ruelle Conjecture In [17], Barreira, Pesin, and Schmeling prove that every hyperbolic measure has an almost local product structure, i. e., the measure of a small ball can be approximated by the product of the stable conditional measure of the stable component and the unstable conditional measure of

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the unstable component, up to small exponential error. This was used to prove the existence of the pointwise dimension of every hyperbolic measure almost everywhere. Moreover, the pointwise dimension is the sum of the contributions from its stable and unstable part. This implies that most dimension-type characteristics of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide, and provides a rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures. The existence of the pointwise dimension for hyperbolic measures had been conjectured long before by Eckmann and Ruelle. The hypotheses of this theorem are sharp. Ledrappier and Misiurewicz [70] constructed an example of a nonhyperbolic measure for which the pointwise dimension is not constant a.e. In [101], Pesin and Weiss present an example of a Hölder homeomorphism with Hölder constant arbitrarily close to one, where the pointwise dimension for the unique measure of maximal entropy does not exist a.e. There is also a one-dimensional example by Cutler [34].

Endomorphisms The previous section indicates that the dimensional properties of hyperbolic measures under invariant conditions for a diffeomorphism are understood. However, partial differential equations often generate only semi-flows and the corresponding dynamical system is noninvertible. Also, Poincaré sections sometimes introduce singularities. For such dynamical systems the theory of diffeomorphisms does not apply. However, the next theorem allows under some conditions application of this theory. It essentially rules out similar situations as considered in Subsect. “Iterated Function Systems”.

The Dynamical Characteristic View of Multifractal Analysis The aim of multifractal analysis is an attempt to understand the fine structure of the level sets of the fundamental asymptotic quantities in ergodic theory (e. g., Birkhoff averages, local entropy, Lyapunov exponents). For ergodic measures these quantities are a.e. constant, however may depend on the underlying ergodic measure. Important elements of multifractal analysis entail determining the range of values these characteristics attain, an analysis of the dimension of the level sets, and an understanding of the sets where the limits do not exist. A general concept of multifractal analysis was proposed by Barreira, Schmeling and Pesin [16]. An important field of applications of multifractal analysis is to describe sets of real numbers that have constraints on their digits or continued fraction expansion. General Multifractal Formalism In this section the abstract theory of multifractal analysis is described. Let X; Y be two measurable spaces and g : X n B ! Y be any measurable function where B is a measurable (possibly empty) subset of X (in the standard applications Y D R or Y D C). The associated multifractal decomposition of X is defined as [ g K˛ X DB[ ˛2Y

where g

K˛ :D fx 2 X : g(x) D ˛g For a given set function G : 2 X ! R the multifractal spectrum is defined by g

F (˛) :D G(K˛ ) :

Definition 4 A system (possibly with singularities) is almost surely invertible if it is invertible on a set of full measure. This implies that a full measure set of points has unique forward and backward trajectories.

At this point some classical and concrete examples of this general framework are considered.

Theorem 11 (Schmeling–Troubetzkoy [114]) A two-dimensional system with singularities is almost surely invertible (w.r.t. an SRB–measure) if and only if Young’s formula holds.

Let  be an ergodic invariant measure for T : X ! X. If one sets

The Entropy Spectrum

g E (x) :D h (x) 2 Y D R and

Multifractal Analysis A group of physicists [50] suggested the idea of a multifractal analysis.

G E (Z) D htop (TjZ) the associated multifractal spectrum fE (˛) is called the entropy spectrum.

Ergodic Theory: Fractal Geometry

The Dimension Spectrum This is the classical multifractal spectrum. Let  be an invariant ergodic measure on a complete separable metric space X. Set g D (x) :D d (x) 2 Y D R and G D (Z) D dimH Z : The associated multifractal spectrum fD (˛) is called the dimension spectrum. The Lyapunov Spectrum Let n1 1X g L (x) :D (x) D lim (T k x) 2 Y D R n!1 n kD0

where

(x) D log jD x Tj and

G D (Z) D dimH Z

or G E (Z) D htop (jZ) :

The associated multifractal spectra f LD (˛) and f LE (˛) are called Lyapunov spectra. It was observed by H. Weiss [126] and Barreira, Pesin and Schmeling [16] that for conformal repellers   log 2 f LD (˛) D fDh (2) ˛ where the dimension spectrum on the right-hand side is with respect to the measure of maximal entropy, and f LE (˛) D fED (dimH   ˛)

(3)

where the entropy spectrum on the right-hand side is with respect to the measure of maximal dimension. The following list summarizes the state of the art for the dynamical characteristic multifractal analysis of dynamical systems. The precise statements can be found in the original papers.  [102,126] For conformal repellers and Axiom-A surface diffeomorphisms, a complete multifractal analysis exists for the Lyapunov exponent.  [16,102,126] For mixing a subshift of finite type, a complete multifractal analysis exists for the Birkhoff average for a Hölder continuous potential and for the local entropy for a Gibbs measure with Hölder continuous potential.

 [12,97] There is a complete multifractal analysis for hyperbolic flows.  [123] There is a generalization of the multifractal analysis on subshifts with specification and continuous potentials.  [13,19] There is an analysis of “mixed” spectra like the dimension spectrum of local entropies and also an analysis of joint level sets determined by more than one (measurable) function.  [105] For the Gauss map (and a class of nonuniformly hyperbolic maps) a complete multifractal analysis exists for the Lyapunov exponent.  [57] A general approach to multifractal analysis for repellers with countably many branches is developed. It shows in contrary to finitely many branches features of nonanalytic behavior. In the first three statements the multifractal spectra are analytic concave functions that can be computed by means of the Legendre transform of the pressure functional with respect to a suitable chosen family of potentials. In the remaining items this is no longer the case. Analyticity and convexity properties of the pressure functional are lost. However, the authors succeeded to provide a satisfactory theory in these cases. Multifractal Analysis and Large Deviation Theory There are deep connections between large deviation theory and multifractal analysis. The variational formula for pressure is an important tool in the analysis, and can be viewed (and proven) as a large deviation result [41]. Some authors use large deviation theory as a tool to effect multifractal analysis. Future Directions The dimension theory is fast developing and of great importance in the theory of dynamical systems. In the most ideal situations (low dimensions and hyperbolicity) a generally far reaching and powerful theory has been developed. It uses ideas from statistical physics, fractal geometry, probability theory and other fields. Unfortunately, the richness of this theory does not carry over to higher-dimensional systems. However, recent developments have shown that it is possible to obtain a general theory for the dimension of measures. Part of this theory is the development of the analytic tools of nonuniformly hyperbolic systems. Therefore, the dimension theory of dynamical systems is far from complete. In particular, it is usually difficult to apply the general theory to concrete examples, for instance

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if one really wants to compute the dimension. The general theory does not provide a way to compute the dimension but gives rather connections to other characteristics. Moreover, in the presence of neutral directions (zero Lyapunov exponents) one encounters all the difficulties arising in low-complexity systems. Another important open problem is to understand the dimension theory of invariant sets in higher-dimensional spaces. One way would be to relate the dimension of sets to the dimension of measures. Such a connection is not clear. The reason is that most systems do not exhibit a measure whose dimension coincides with the dimension of its support (invariant set). But there are some reasons to conjecture that any compact invariant set of an expanding map in any dimension carries a measure of maximal dimension (see [48,63]). If this conjecture is true one obtains an invariant measure whose unstable dimension coincides with the unstable dimension of the invariant set. There is also a measure of maximal stable dimension. Combining these two measures one could establish an analogous theory for invariant sets as for invariant measures. Last but not least one has to mention the impact of the dimension theory of dynamical systems on other fields. This new point of view makes in many cases the posed problems more tractable. This is illustrated in examples from number theory, geometric limit constructions and others. The applications of the dimension theory of dynamical systems to other questions seem to be unlimited.

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79. McCluskey H, Manning A (1983) Hausdorff Dimension For Horseshoes. Erg Th Dyn Syst 3:251–260 80. Moran P (1946) Additive Functions Of Intervals and Hausdorff Dimension. Proceedings Of Cambridge Philosophical Society 42:15–23 81. Moreira C, Yoccoz J (2001) Stable Intersections of Cantor Sets with Large Hausdorff Dimension. Ann of Math (2) 154(1):45– 96 82. Mãné R (1981) On the Dimension of Compact Invariant Sets for Certain Nonlinear Maps. Lecture Notes in Mathematics, vol 898. Springer 83. Mãné R (1990) The Hausdorff Dimension of Horseshoes of Surfaces. Bol Soc Bras Math 20:1–24 84. Neunhäuserer J (1999) An analysis of dimensional theoretical properties of some affine dynamical systems. Thesis. Free University Berlin, Berlin 85. Oseledets V (1968) A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc 19:197–221 86. Ott E, Sauer T, Yorke J (1994) Part I Background. In: Coping with chaos. Wiley Ser Nonlinear Sci, Wiley, New York, pp 1–62 87. Palis J, Takens F (1987) Hyperbolicity And The Creation Of Homoclinic Orbits. Ann Math 125:337–374 88. Palis J, Takens F (1993) Hyperbolicity And Sensitive Chaotic Dynamics At Homoclinic Bifurcations. Cambridge U Press, Cambridge 89. Palis J, Takens F (1994) Homoclinic Tangencies For Hyperbolic Sets Of Large Hausdorff Dimension. Acta Math 172:91–136 90. Palis J, Viana M (1988) On the continuity of Hausdorff dimension and limit capacity for horseshoes. Lecture Notes in Math, vol 1331. Springer 91. Pesin Y (1977) Characteristic exponents and smooth ergodic theory. Russian Math Surveys 32(4):55–114 92. Pesin Y (1992) Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Erg Th Dyn Syst 12:123–151 93. Pesin Y (1993) On Rigorous Mathematical Definition of Correlation Dimension and Generalized Spectrum for Dimensions. J Statist Phys 71(3–4):529–547 94. Pesin Y (1997) Dimension Theory In Dynamical Systems: Rigorous Results And Applications. Cambridge U Press, Cambridge 95. Pesin Y (1997) Dimension theory in dynamical systems: contemporary views and applications. In: Chicago Lectures in Mathematics. Chicago University Press, Chicago 96. Pesin Y, Pitskel’ B (1984) Topological pressure and the variational principle for noncompact sets. Funct Anal Appl 18:307– 318 97. Pesin Y, Sadovskaya V (2001) Multifractal analysis of conformal axiom A flows. Comm Math Phys 216(2):277–312 98. Pesin Y, Tempelman A (1995) Correlation Dimension of Measures Invariant Under Group Actions. Random Comput Dyn 3(3):137–156 99. Pesin Y, Weiss H (1994) On the Dimension of Deterministic and Random Cantor-like Sets. Math Res Lett 1:519–529 100. Pesin Y, Weiss H (1996) On The Dimension Of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, And The Eckmann-Ruelle Conjecture. Comm Math Phys 182:105–153

101. Pesin Y, Weiss H (1997) A Multifractal Analysis of Equilibrium Measures For Conformal Expanding Maps and Moran-like Geometric Constructions. J Stat Phys 86:233–275 102. Pesin Y, Weiss H (1997) The Multifractal Analysis of Gibbs Measures: Motivation. Mathematical Foundation and Examples. Chaos 7:89–106 103. Petersen K (1983) Ergodic theory. Cambridge studies in advanced mathematics 2. Cambridge Univ Press, Cambridge 104. Pollicott M, Weiss H (1994) The Dimensions Of Some Self Affine Limit Sets In The Plane And Hyperbolic Sets. J Stat Phys 77:841–866 105. Pollicott M, Weiss H (1999) Multifractal Analysis for the Continued Fraction and Manneville-Pomeau Transformations and Applications to Diophantine Approximation. Comm Math Phys 207(1):145–171 ´ 106. Przytycki F, Urbanski M (1989) On Hausdorff Dimension of Some Fractal Sets. Studia Math 93:155–167 107. Ruelle D (1978) Thermodynamic Formalism. Addison-Wesley 108. Ruelle D (1982) Repellers For Real Analytic Maps. Erg Th Dyn Syst 2:99–107 109. Schmeling J (1994) Hölder Continuity of the Holonomy Maps for Hyperbolic basic Sets II. Math Nachr 170:211–225 110. Schmeling J (1997) Symbolic Dynamics for Beta-shifts and Self-Normal Numbers. Erg Th Dyn Syst 17:675–694 111. Schmeling J (1998) A dimension formula for endomorphisms – the Belykh family. Erg Th Dyn Syst 18:1283–1309 112. Schmeling J (1999) On the Completeness of Multifractal Spectra. Erg Th Dyn Syst 19:1–22 113. Schmeling J (2001) Entropy Preservation under Markov Codings. J Stat Phys 104(3–4):799–815 114. Schmeling J, Troubetzkoy S (1998) Dimension and invertibility of hyperbolic endomorphisms with singularities. Erg Th Dyn Syst 18:1257–1282 115. Schmeling J, Weiss H (2001) Dimension theory and dynamics. AMS Proceedings of Symposia in Pure Mathematics series 69:429–488 116. Series C (1985) The Modular Surface and Continued Fractions. J Lond Math Soc 31:69–80 117. Simon K (1997) The Hausdorff dimension of the general Smale–Williams solenoidwith different contraction coefficients. Proc Am Math Soc 125:1221–1228 118. Simpelaere D (1994) Dimension Spectrum of Axiom-A Diffeomorphisms, II. Gibbs Measures. J Stat Phys 76:1359–1375 P 119. Solomyak B (1995) On the random series ˙n (an Erdös problem). Ann of Math (2) 142(3):611–625 120. Solomyak B (2004) Notes on Bernoulli convolutions. In: Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Proc Sympos Pure Math, 72, Part 1. Amer Math Soc, Providence, pp 207–230 121. Stratmann B (1995) Fractal Dimensions for Jarnik Limit Sets of Geometrically Finite Kleinian Groups; The Semi-Classical Approach. Ark Mat 33:385–403 122. Takens F (1981) Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics, vol 898. Springer 123. Takens F, Verbitzki E (1999) Multifractal analysis of local entropies for expansive homeomorphisms with specification. Comm Math Phys 203:593–612 124. Walters P (1982) Introduction to Ergodic Theory. Springer 125. Weiss H (1992) Some Variational Formulas for Hausdorff Di-

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Books and Reviews Bowen R (1975) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol 470. Springer Eckmann JP, Ruelle D (1985) Ergodic Theory Of Chaos And Strange Attractors. Rev Mod Phys 57:617–656 Federer H (1969) Geometric measure theory. Springer Hasselblatt B, Katok A (2002) Handbook of Dynamical Systems, vol 1, Survey 1. Principal Structures. Elsevier Katok A (1980) Lyapunov exponents, entropy and periodic or-

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Ergodic Theory on Homogeneous Spaces and Metric Number Theory DMITRY KLEINBOCK Department of Mathematics, Brandeis University, Waltham, USA Article Outline Glossary Definition of the Subject Introduction Basic Facts Connection with Dynamics on the Space of Lattices Diophantine Approximation with Dependent Quantities: The Set-Up Further Results Future Directions Acknowledgment Bibliography Glossary Diophantine approximation Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector-valued) functions attain values close to integers. Metric number theory Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties. Homogeneous spaces A homogeneous space G/ of a group G by its subgroup  is the space of cosets fg g. When G is a Lie group and  is a discrete subgroup, the space G/ is a smooth manifold and locally looks like G itself. Lattice; unimodular lattice A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1. Ergodic theory The study of statistical properties of orbits in abstract models of dynamical systems. Hausdorff dimension A nonnegative number attached to a metric space and extending the notion of topological dimension of “sufficiently regular” sets, such as smooth submanifolds of real Euclidean spaces. Definition of the Subject The theory of Diophantine approximation, named after Diophantus of Alexandria, in its simplest set-up deals with the approximation of real numbers by rational num-

bers. Various higher-dimensional generalizations involve studying values of linear or polynomial maps at integer points. Often a certain “approximation property” is fixed, and one wants to characterize the set of numbers (vectors, matrices) which share this property, by means of certain measures (Lebesgue, or Hausdorff, or some other interesting measures). This is usually referred to as metric Diophantine approximation. The starting point for the theory is an elementary fact that Q, the set of rational numbers, is dense in R, the reals. In other words, every real number can be approximated by rationals: for any y 2 R and any " > 0 there exists p/q 2 Q with jy  p/qj < " :

(1)

To answer questions like “how well can various real numbers be approximated by rational numbers? i. e., how small can " in (1) be chosen for varying p/q 2 Q?”, a natural approach has been to compare the accuracy of the approximation of y by p/q to the “complexity” of the latter, which can be measured by the size of its denominator q in its reduced form. This seemingly simple set-up has led to introducing many important Diophantine approximation properties of numbers/vectors/matrices, which show up in various fields of mathematics and physics, such as differential equations, KAM theory, transcendental number theory. Introduction As the first example of refining the statement about the density of Q in R, consider a theorem by Kronecker stating that for any y 2 R and any c > 0, there exist infinitely many q 2 Z such that jy  p/qj < c/jqj

i. e. jqy  pj < c

(2)

for some p 2 Z. A comparison of (1) and (2) shows that it makes sense to multiply both sides of (1) by q, since in the right hand side of (2) one would still be able to get very small numbers. In other words, approximation of y by p/q translates into approximating integers by integer multiples of y. Also, if y is irrational, (p; q) can be chosen to be relatively prime, i. e. one gets infinitely many different rational numbers p/q satisfying (2). However if y 2 Q the latter is no longer true for small enough c. Thus it seems to be more convenient to talk about pairs (p; q) rather than p/q 2 Q, avoiding a necessity to consider the two cases separately. At this point it is convenient to introduce the following central definition: if is a function N ! RC and y 2 R, say that y is -approximable (notation: y 2 W ( )) if

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

there exist infinitely many q 2 N such that jqy  pj <

(q)

(3)

for some p 2 Z. Because of Kronecker’s Theorem, it is natural to assume that (x) ! 0 as x ! 1. Often will be assumed non-increasing, although many results do not require monotonicity of . One can similarly consider a higher-dimensional version of the above set-up. Note that y 2 R in the above formulas plays the role of a linear map from R to another copy of R, and one asks how close values of this map at integers are from integers. It is natural to generalize it by taking a linear operator Y from Rn to Rm for fixed m; n 2 N, that is, an m  n-matrix (interpreted as a system of m linear forms Y i on Rn ). We will denote by M m;n the space of m  n matrices with real coefficients. For as above, one says that Y 2 M m;n is -approximable (notation: Y 2 Wm;n ( )) if there are infinitely many q 2 Z n such that kYq C pk 

(kqk)

Recurrence by Frantzikinakis and McCutcheon and  Ergodic Theory: Interactions with Combinatorics and Number Theory by Ward, as well as in the survey papers [32,33, 50,58,66,70,71]. Here is a brief outline of the rest of the article. In the next section we survey basic results, some classical, some obtained relatively recently, in metric Diophantine approximation. Sect. “Connection with Dynamics on the Space of Lattices” is devoted to a description of the connection between Diophantine approximation and dynamics, specifically flows on the space of lattices. In Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” and Sect. “Further Results”, we specialize to the set-up of Diophantine approximation on manifolds, or, more generally, approximation properties of vectors with respect to measures satisfying some natural conditions, and show how applications of homogeneous dynamics contributed to important recent developments in the field. Sect. “Future Directions” mentions several open questions and directions for further investigation.

(4)

for some p 2 Z m . Here k  k is the supremum norm on R k given by kyk D max1ik jy i j. (This definition is slightly different from the one used in [58], where powers of norms were considered). Traditionally, one of the main goals of metric Diophantine approximation has been to understand how big the sets Wm;n ( ) are for fixed m; n and various functions . Of course, (4) is not the only interesting condition that can be studied; various modifications of the approximation properties can also be considered. For example the Oppenheim Conjecture, now a theorem of Margulis [69] and a basis for many important recent developments [22,34,35], states that indefinite irrational quadratic forms can take arbitrary small values at integer points; Littlewood’s conjecture, see (18) below, deals with a similar statement about products of linear forms. See the article  Ergodic Theory: Rigidity by Nitica and surveys [32,70] for details. We remark that the standard tool for studying Diophantine approximation properties of real numbers (m D n D 1) is the continued fraction expansion, or, equivalently, the Gauss map x 7! 1/x mod 1 of the unit interval, see [49]. However the emphasis of this survey lies in higher-dimensional theory, and the dynamical system described below can be thought of as a replacement for the continued fraction technique applicable in the one-dimensional case. Additional details about interactions between ergodic theory and number theory can be found in the article by Nitica mentioned above, in  Ergodic Theory:

Basic Facts General references for this section: [17,80]. The simplest choice for functions happens to be the following: let us denote c;v (x) D cx v . It was shown by Dirichlet in 1842 that with the choice c D 1 and v D n/m, all Y 2 M m;n are -approximable. Moreover, Dirichlet’s Theorem states that for any Y 2 M m;n and for any t > 0 there exist q D (q1 ; : : : ; q n ) 2 Z n n f0g and p D (p1 ; : : : ; p m ) 2 Z m satisfying the following system of inequalities: kYq  pk < et/m

and kqk  e t/n :

(5)

From this it easily follows that Wm;n ( 1;n/m ) D M m;n . In fact, it is this paper of Dirichlet which gave rise to his box principle. Later another proof of the same result was given by Minkowski. The constant c D 1 is not optimal: p the smallest value of c for which W1;1 ( c;1 ) D R is 1/ 5, and the optimal constants are not known in higher dimensions, although some estimates can be given [80]. Systems of linear forms which do not belong to W m;n ( c;n/m ) for some positive c are called badly approximable; that is, we set def

BA m;n D M m;n n [c>0 Wm;n (

c;n/m )

:

Their existence in arbitrary dimensions was shown by Perron. Note that a real number y (m D n D 1) is badly approximable if and only if its continued fraction coefficients are uniformly bounded. It was proved by Jarnik [46] in the

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case m D n D 1 and by Schmidt in the general case [78] that badly approximable matrices form a set of full Hausdorff dimension: that is, dim(BA m;n ) D mn. On the other hand, it can be shown that each of the sets W m;n ( c;n/m ) for any c > 0 has full Lebesgue measure, and hence the complement BA m;n to their intersection has measure zero. This is a special case of a theorem due to Khintchine [48] in the case n D 1 and to Groshev [42] in full generality, which gives the precise condition on the function under which the set of -approximable matrices has full measure. Namely, if is non-increasing (this assumption can be removed in higher dimensions but not for n D 1, see [29]), then -almost no (resp. -almost every) Y 2 M m;n is -approximable, provided the sum 1 X

k n1 (k)m

(6)

kD1

converges (resp. diverges). (Here and hereafter  stands for Lebesgue measure). This statement is usually referred to as the Khintchine–Groshev Theorem. The convergence case of this theorem follows in a straightforward manner from the Borel–Cantelli Lemma, but the divergence case is harder. It was reproved and sharpened in 1960 by Schmidt [76], who showed that if the sum (6) diverges, then for almost all Y the number of solutions to (4) with kqk  N is asymptotic to the partial sum of the series (6) (up to a constant), and also gave an estimate for the error term. A special case of the convergence part of the theorem shows that Wm;n ( 1;v ) has measure zero whenever v > n/m. Y is said to be very well approximable if it belongs to W m;n ( 1;v ) for some v > n/m. That is, def

VWAm;n D [v>n/m Wm;n (

1;v )

:

More specifically, let us define the Diophantine exponent !(Y) of Y (sometimes called “the exact order” of Y) to be the supremum of v > 0 for which Y 2 Wm;n ( 1;v ). Then !(Y) is always not less than n/m, and is equal to n/m for Lebesgue-a.e. Y; in fact, VWA m;n D fY 2 M m;n : !(Y) > n/mg. The Hausdorff dimension of the null sets Wm;n ( 1;v ) was computed independently by Besicovitch [14] and Jarnik [45] in the one-dimensional case and by Dodson [26] in general: when v > n/m, one has  dim Wm;n (



1;v )

D (n  1)m C

mCn : vC1

(7)

See [27] for a nice exposition of ideas involved in the proof of both the aforementioned formula and the Khintchine– Groshev Theorem.

Note that it follows from (7) that the null set VWA m;n has full Hausdorff dimension. Matrices contained in the intersection \ W m;n ( 1;v ) D fY 2 M m;n : !(Y) D 1g v

are called Liouville and form a set of Hausdorff dimension (n  1)m, that is, to the dimension of Y for which Yq 2 Z for some q 2 Zn n f0g (the latter belong to Wm;n ( ) for any positive ). Note also that the aforementioned properties behave nicely with respect to transposition; this is described by the so-called Khintchine’s Transference Principle (Chap. V in [17]). For example, Y 2 BA m;n if and only if Y T 2 BA n;m , and Y 2 VWA m;n if and only if Y T 2 VWA n;m . In particular, many problems related to approximation properties of vectors (n D 1) and linear forms (m D 1) reduce to one another. We refer the readers to [43] and [8] for very detailed and comprehensive recent accounts of various further aspects of the theory. Connection with Dynamics on the Space of Lattices General references for this section: [4,86]. Interactions between Diophantine approximation and the theory of dynamical systems has a long history. Already in Kronecker’s Theorem one can see a connection. Indeed, the statement of the theorem can be rephrased as follows: the points on the orbit of 0 under the rotation of the circle R/Z by y approach the initial point 0 arbitrarily closely. This is a special case of the Poincare Recurrence Theorem in measurable dynamics. And, likewise, all the aforementioned properties of Y 2 M m;n can be restated in terms of recurrence properties of the Z n -action on the m-dimensional torus Rm /Z m given by x 7! Yx mod Z m . In other words, fixing Y gives rise to a dynamical system in which approximation properties of Y show up. However the theme of this section is a different dynamical system, whose phase space is (essentially) the space of parameters Y, and which can be used to read the properties of Y from the behavior of the associated trajectory. It has been known for a long time (see [81] for a historical account) that Diophantine properties of real numbers can be coded by the behavior of geodesics on the quotient of the hyperbolic plane by SL2 (Z). In fact, the latter flow can be viewed as the suspension flow of the Gauss map mentioned at the end of Sect. “Introduction”. There have been many attempts to construct a higher-dimensional analogue of the Gauss map so that it captures all the fea-

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

tures of simultaneous approximation, see [47,63,65] and references therein. On the other hand, it seems to be more natural and efficient to generalize the suspension flow itself, and this is where one needs higher rank homogeneous dynamics. As was mentioned above, in the basic set-up of simultaneous Diophantine approximation one takes a system of m linear forms Y1 ; : : : ; Ym on Rn and looks at the values of jYi (q) C p i j; p i 2 Z, when q D (q1 ; : : : ; q n ) 2 Z n is far from 0. The trick is to put together Y1 (q) C p1 ; : : : ; Ym (q) C p m

and

q1 ; : : : ; q n ;

fg t LY Z k : t 2 RC g

and consider the collection of vectors

 ˇ  Yq C p ˇˇ m n D LY Z k ; q 2 Z p 2 Z ˇ q where k D m C n and   Y def I LY D m ; Y 2 M m;n : 0 In

(8)

This collection is a unimodular lattice in R k , that is, a discrete subgroup of R k with covolume 1. Our goal is to keep track of vectors in such a lattice having small projections onto the first m components of R k and big projections onto the last n components. This is where dynamics comes into the picture. Denote by g t the one-parameter subgroup of SL k (R) given by g t D diag(et/m ; : : : ; e t/m ; et/n ; : : : ; et/n ) : „ ƒ‚ … „ ƒ‚ … mtimes

(9)

ntimes

The vectors in the lattice LY Z k are moved by the action of g t , t > 0, and a special role is played by the moment t when the “small” and “big” projections equalize. That is, one is led to consider a new dynamical system. Its phase space is the space of unimodular lattices in R k , which can be naturally identified with the homogeneous space def

˝ k D G/; where G D SL k (R) and  D SL k (Z) ;

Z k nf0gg such that g i (v i ) ! 0 as i ! 1. Equivalently, for " > 0 consider a subset K " of ˝ k consisting of lattices with no nonzero vectors of norm less than "; then all the sets K " are compact, and every compact subset of ˝ k is contained in one of them. Moreover, one can choose a metric on ˝ k such that dist(; Z k ) is, up to a uniform multiplicative constant, equal to  log minv2 nf0g kvk (see [25]); then the length of the smallest nonzero vector in a lattice  will determine how far away is this lattice in the “cusp” of ˝ k . Using Mahler’s Criterion, it is not hard to show that Y 2 BA m;n if and only if the trajectory

(10)

and the action is given by left multiplication by elements of the subgroup (9) of G, or perhaps other subgroups H G. Study of such systems has a rich history; for example, they are known to be ergodic and mixing whenever H is unbounded [74]. What is important in this particular case is that the space ˝ k happens to be noncompact, and its structure at infinity is described via Mahler’s Compactness Criterion, see Chap. V in [4]: a sequence of lattices g i Z k goes to infinity in ˝ k () there exists a sequence fv i 2

(11)

is bounded in ˝ k . This was proved by Dani [20] in 1985, and later generalized in [57] to produce a criterion for Y to be -approximable for any non-increasing function . An important special case is a criterion for a system of linear forms to be very well approximable: Y 2 VWAm;n if and only if the trajectory (11) has linear growth, that is, there exists a positive such that dist(g t LY Z k ; Z k ) > t for an unbounded set of t > 0. This correspondence allows one to link various Diophantine and dynamical phenomena. For example, from the results of [55] on abundance of bounded orbits on homogeneous spaces one can deduce the aforementioned theorem of Schmidt [78]: the set BAm;n has full Hausdorff dimension. And a dynamical Borel–Cantelli Lemma established in [57] can be used for an alternative proof of the Khintchine–Groshev Theorem; see also [87] for an earlier geometric approach. Note that both proofs are based on the following two properties of the g t -action: mixing, which forces points to return to compact subsets and makes preimages of cusp neighborhoods quasi-independent, and hyperbolicity, which implies that the behavior of points on unstable leaves is generic. The latter is important since the orbits of the group fLY Z k : Y 2 M m;n g are precisely the unstable leaves with respect to the g t -action. We note that other types of Diophantine problems, such as conjectures of Oppenheim and Littlewood mentioned in the previous section, can be reduced to statements involving ˝ k by means of the same principle: Mahler’s Criterion is used to relate small values of some function at integer points to excursions to infinity in ˝ k of orbit of the stabilizer of this function. Other important and useful recent applications of homogeneous dynamics to metric Diophantine approximation are related to the circle of ideas roughly called “Diophantine approximation with dependent quantities” (terminology borrowed from [84]), to be surveyed in the next two sections.

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Diophantine Approximation with Dependent Quantities: The Set-Up General references for this section: [12,84]. Here we restrict ourselves to Diophantine properties of vectors in Rn . In particular, we will look more closely at the set of very well approximable vectors, which we will simply denote by VWA, dropping the subscripts. In many cases it does not matter whether one works with row or column vectors, in view of the duality remark made at the end of Sect. “Basic Facts”. We begin with a non-example of an application of dynamics to Diophantine approximation: a celebrated and difficult theorem which currently, to the best of the author’s knowledge, has no dynamical proof. Suppose that y D (y1 ; : : : ; y n ) 2 Rn is such that each yi is algebraic and 1; y1 ; : : : ; y n are linearly independent over Q. It was established by Roth for n D 1 [75] and then generalized to arbitrary n by Schmidt [79], that y as above necessarily belongs to the complement of VWA. In other words, vectors with very special algebraic properties happen to follow the behavior of a generic vector in Rn . We would like to view the above example as a special case of a general class of problems. Namely, suppose we are given a Radon measure  on Rn . Let us say that  is extremal [85] if -a.e. y 2 Rn is not very well approximable. Further, define the Diophantine exponent !() of  to be the -essential supremum of the function !(); in other words, ˚   def !() D sup vj W ( 1;v ) > 0 : Clearly it only depends on the measure class of . If  is naturally associated with a subset M of Rn supporting  (for example, if M is a smooth submanifold of Rn and  is the measure class of the Riemannian volume on M, or, equivalently, the pushforward f  of  by a smooth map f parametrizing M), one defines the Diophantine exponent !(M) of M to be equal to that of , and says that M is extremal if f(x) is not very well approximable for -a.e. x. Then !()  n for any , and !() D !(Rn ) is equal to n. The latter justifies the use of the word “extremal”:  is extremal if !() is equal to n, i. e. attains the smallest possible value. The aforementioned results of Roth and Schmidt then can be interpreted as the extremality of atomic measures supported on algebraic vectors without rational dependence relations. Historically, the first measure (other than ) to be considered in the set-up described above was the pushforward of  by the map f(x) D (x; x 2 ; : : : ; x n ) :

(12)

The extremality of f  for f as above was conjectured in 1932 by K. Mahler [67] and proved in 1964 by Sprindžuk [82,83]. It was important for Mahler’s study of transcendental numbers: this result, roughly speaking, says that almost all transcendental numbers are “not very algebraic”. At about the same time Schmidt [77] proved the extremality of f  when f : I ! R2 , I R, is C3 and satisfies ˇ 0 ˇ ˇ f (x) f 0 (x) ˇ 2 ˇ 1 ˇ ˇ ˇ ¤ 0 for -a.e. x 2 I ; ˇ f 00 (x) f 00 (x)ˇ 1

2

in other words, the curve parametrized by f has nonzero curvature at almost all points. Since then, a lot of attention has been devoted to showing that measures f  are extremal for other smooth maps f. To describe a broader class of examples, recall the following definition. Let x 2 Rd and let f D ( f1 ; : : : ; f n ) be a Ck map from a neighborhood of x to Rn . Say that f is nondegenerate at x if Rn is spanned by partial derivatives of f at x up to some order. Say that f is nondegenerate if it is nondegenerate at -a.e. x. It was conjectured by Sprindžuk [84] in 1980 that f  for real analytic nondegenerate f are extremal. Many special cases were established since then (see [12] for a detailed exposition of the theory and many related results), but the general case stood open until the mid-1990s [56], when Sprindžuk’s conjecture was proved using the dynamical approach (later Beresnevich [6] succeeded in establishing and extending this result without use of dynamics). The proof in [56] uses the correspondence outlined in the previous section plus a measure estimate for flows on the space of lattices which is described below. In the subsequent work the method of [56] was adapted to a much broader class of measures. To define it we need to introduce some more notation and definitions. If x 2 Rd and r > 0, denote by B(x; r) the open ball of radius r centered at x. If B D B(x; r) and c > 0, cB will denote the ball B(x; cr). For B Rd and a real-valued function f on B, let def

k f kB D sup j f (x)j : x2B

If  is a measure on Rd such that (B) > 0, define k f k;B def Dk f kB \ supp  ; this is the same as the L1 ()-norm of f j B if f is continuous and B is open. If D > 0 and U Rd is an open subset, let us say that  is D-Federer on U if for any ball B U centered at supp  one has (3B)/(B) < D whenever 3B U. This condition is often called “doubling” in the literature. See [54,72] for examples and references.  is called Federer if for -a.e. x 2 Rd there exist a

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

neighborhood U of x and D > 0 such that  is D-Federer on U. Given C; ˛ > 0, open U Rd and a measure  on U, a function f : U ! R is called (C; ˛)-good on U with respect to  if for any ball B U centered in supp  and any " > 0 one has  ˛   "  fx 2 B : j f (x)j < "g  C (B) : (13) k f k;B This condition was formally introduced in [56] for  being Lebesgue measure, and in [54] for arbitrary . A basic example is given by polynomials, and the upshot of the above definition is the formalization of a property needed for the proof of several basic facts [19,21,68] about polynomial maps into the space of lattices. In [54] a strengthening of this property was considered: f was called absolutely (C; ˛)-good on U with respect to  if for B and " as above one has  ˛   "  fx 2 B : j f (x)j < "g  C (B) : (14) k f kB There is no difference between (13) and (14) when  has full support, but it turns out to be useful for describing measures supported on proper (e. g. fractal) subsets of Rd . Now suppose that we are given a measure  on Rd , an open U Rd with (U) > 0 and a map f D ( f1 ; : : : ; f n ) : Rd ! Rn . Following [62], say that a pair (f; ) is (absolutely) good on U if any linear combination of 1; f1 ; : : : ; f n is (absolutely) (C; ˛)-good on U with respect to . If for -a.e. X there exists a neighborhood U of X and C; ˛ > 0 such that  is (absolutely) (C; ˛)-good on U, we will say that the pair (f; ) is (absolutely) good. Another relevant notion is the nonplanarity of (f; ). Namely, (f; ) is said to be nonplanar if whenever B is a ball with (B) > 0, the restrictions of 1; f1 ; : : : ; f n to B \ supp  are linearly independent over R; in other words, f(B \ supp ) is not contained in any proper affine subspace of Rn . Note that absolutely good implies both good and nonplanar, but the converse is in general not true. Many examples of (absolutely) good and nonplanar pairs (f; ) can be found in the literature. Already the case n D d and f D Id is very interesting. A measure  on Rn is said to be friendly (resp., absolutely friendly) if and only if it is Federer and the pair (Id; ) is good and nonplanar (resp., absolutely good). See [54,88,89] for many examples. An important class of measures is given by limit measures of irreducible system of self-similar or self-conformal contractions satisfying the Open Set Condition [44]; those are shown to be absolutely friendly in [54]. The prime example is the middle-third Cantor set on the real line. The term

“friendly” was cooked up as a loose abbreviation for “Federer, nonplanar and decaying”, and later proved to be particularly friendly in dealing with problems arising in metric number theory, see e. g. [36]. Also let us say that a pair (f; ) is nondegenerate if f is nondegenerate at -a.e. X. When  is Lebesgue measure on Rd , it is proved in Proposition 3.4 in [56], that a nondegenerate (f; ) is good and nonplanar. The same conclusion is derived in Proposition 7.3 in [54], assuming that  is absolutely friendly. Thus volume measures on smooth nondegenerate manifolds are friendly, but not absolutely friendly. It turns out that all the aforementioned examples of measures can be proved to be extremal by a generalization of the argument from [56]. Specifically, let  be a Federer measure on Rd , U an open subset of Rd , and f : U ! Rn a continuous map such that the pair (f; ) is good and nonplanar; then f  is extremal. This can be derived from the Borel–Cantelli Lemma, the correspondence described in the previous section, and the following measure estimate: if , U and f are as above, then for -a.e. x0 2 U there ex˜ ˛ > 0 such that for ists a ball B U centered at x0 and C; any t 2 RC and any " > 0,  ˚ ˜ ˛: (15)  x 2 B : g t Lf(x) Z nC1 … K" < C" Here g t is as in (9) with m D 1 (assuming that the row vector viewpoint is adopted). This is a quantitative way of saying that for fixed t, the “flow” x 7! g t Lf(x) Z nC1 , B ! ˝nC1 , cannot diverge, and in fact must spend a big (uniformly in t) proportion of time inside compact sets K " . The inequality (15) is derived from a general “quantitative non-divergence” estimate, which can be thought of a substantial generalization of theorems of Margulis and Dani [19,21,68] on non-divergence of unipotent flows on homogeneous spaces. One of its most general versions [54] deals with a measure  on Rd , a continuous map h: e B ! G, where e B is a ball in Rd centered at supp  and G is as in (10). To describe the assumptions on h, one needs to employ the combinatorial structure of lattices in R k , and it will be convenient to use the following notation: if V is a nonzero rational subspace of R k and g 2 G, define `V (g) to be the covolume of g(V \ Z k ) in gV . Then, given positive constants C; D; ˛, there exists C1 D C1 (d; k; C; ˛; D) > 0 with the following property. Suppose  is D-Federer on e B, 0 <  1, and h is such that for each rational V R k (i) `V ı h is (C; ˛)-good on B˜ with respect to , and ˜ Then (ii) k`V ı hk;B  , where B D 3(k1) B. (iii) for any positive "  , one has  ˚  x 2 B : h(x)Z k … K"  C1 (" )˛ (B) : (16)

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Taking h(x) D g t Lf(x) and unwinding the definitions of good and nonplanar pairs, one can show that (i) and (ii) can be verified for some balls B centered at -almost every point, and derive (15) from (16). Further Results The approach to metric Diophantine approximation using quantitative non-divergence, that is, the implication (i) + (ii) ) (iii), is not omnipotent. In particular, it is difficult to use when more precise results are needed, such as for example computing/estimating the Hausdorff dimension of the set of 1;v -approximable vectors on a manifold. See [9,10] for such results. On the other hand, the dynamical approach can often treat much more general objects that its classical counterpart, and also can be perturbed in a lot of directions, producing many generalizations and modifications of the main theorems from the preceding section. One of the most important of them is the so-called multiplicative version of the set-up of Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”. def Q def Namely, define functions ˘ (x) D i jx i j and ˘C (x) D Q : N ! RC , one i max(jx i j; 1) Then, given a function says that Y 2 M m;n is multiplicatively -approximable  ( )) if there are infinitely many (notation: Y 2 Wm;n q 2 Z n such that   ˘ (Yq C p)1/m  ˘C (q)1/n (17) for some p 2 Z m . Since ˘ (x)  ˘C (x)  kxk k for x 2 R k , any -approximable Y is multiplicatively -approximable; but the converse is in general not true, see e. g. [37]. However if one, as before, considers the family f 1;v g, the critical parameter for which the drop from full measure to measure zero occurs is again n/m. That is, if one defines the multiplicative Diophantine exponent def  ( !  (Y) of Y by !  (Y) D supfv : Y 2 Wm;n 1;v )g, then  clearly ! (Y)  !(Y) for all Y, and yet !  (Y) D n/m for -a.e. Y 2 M m;n . Now specialize to Rn (by the same duality principle as before, it does not matter whether to think in terms of row or column vectors, but we will adopt the row vector  set-up), and define the multiplicative˚ exponent   ! () of def n  a measure  on R by ! () D sup vj W ( 1;v ) > 0 ; then !  () D n. Following Sprindžuk [85], say that  is strongly extremal if !  () D n. It turns out that all the results mentioned in the previous section have their multiplicative analogues; that is, the measures described there happen to be strongly extremal. This was conjectured by A. Baker [1] for the curve (12), and then by Sprindžuk in 1980 [85] for analytic nondegenerate manifolds. (We re-

mark that only very few results in this set-up can be obtained by the standard methods, see e. g. [10]). The proof of this stronger statement is based on using the multi-parameter action of gt D diag(e t1 CCt n ; et1 ; : : : ; et n ) ; where t D (t1 ; : : : ; t n ) instead of g t considered in the previous section. One can show that the choice h(x) D gt Lf(x) allows one to verify n , and the proof is fin(i) and (ii) uniformly in t 2 RC ished by applying a multi-parameter version of the correspondence described in Sect. “Connection with Dynamics on the Space of Lattices”. Namely, one can show that k : y 2 VWA 1;n if and only if the trajectory fg t Ly Z n t 2 RC g grows linearly, that is, for some > 0 one has dist(gt LY Z nC1 ; Z nC1 ) > ktk for an unbounded n . A similar correspondence was recently set of t 2 RC used in [30] to prove that the set of exceptions to Littlewood’s Conjecture, which, using the terminology introduced above, can be called badly multiplicatively approximable vectors: def n [  BA W n;1 ( c;1/n ) n;1 D R n

D y:

c>0

 inf

q2Znf0g; p2Z n

jqj  ˘ (qy  p) > 0

; (18)

has Hausdorff dimension zero. This was done using a measure rigidity result for the action of the group of diagonal matrices on the space of lattices. See [18] for an implicit description of this correspondence and [32,66,70] for more detail. The dynamical approach also turned out to be fruitful in studying Diophantine properties of pairs (f; ) for which the nonplanarity condition fails. Note that obvious examples of non-extremal measures are provided by proper affine subspaces of Rn whose coefficients are rational or are well enough approximable by rational numbers. On the other hand, it is clear from a Fubini argument that almost all translates of any given subspace are extremal. In [51] the method of [56] was pushed further to produce criteria for the extremality, as well as the strong extremality, of arbitrary affine subspaces L of Rn . Further, it was shown that if L is extremal (resp. strongly extremal), then so is any smooth submanifold of L which is nondegenerate in L at a.e. point. (The latter property is a straightforward generalization of the definition of nondegeneracy in Rn : a map f is nondegenerate in L at x if the linear part of L is spanned by partial derivatives of f at x). In other words, extremality and strong extremality pass from affine subspaces to their nondegenerate submanifolds.

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

A more precise analysis makes it possible to study Diophantine exponents of measures with supports contained in arbitrary proper affine subspaces of Rn . Namely, in [53] it is shown how to compute !(L) for any L, and furthermore proved that if  is a Federer measure on Rd , U an open subset of Rd , and f : U ! Rn a continuous map such that the pair (f; ) is good and nonplanar in L, then !(f ) D !(L). Here we say, generalizing the definition from Sect.“Diophantine Approximation with Dependent Quantities: The Set-Up”, that (f; ) is nonplanar in L if for any ball B with (B) > 0, the f-image of B \ supp  is not contained in any proper affine subspace of L. (It is easy to see that for a smooth map f : U ! L, (f; ) is good and nonplanar in L whenever f is nondegenerate in L at a.e. point). It is worthwhile to point out that these new applications require a strengthening of the measure estimate described at the end of Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”: it was shown in [53] that (i) and (ii) would still imply (iii) if in (ii) is replaced by dim V . Another application concerns badly approximable vectors. Using the dynamical description of the set BA

Rn due to Dani [20], it turns out to be possible to find badly approximable vectors inside supports of certain measures on Rn . Namely, if a subset K of Rn supports an absolutely friendly measure, then BA \ K has Hausdorff dimension not less than the Hausdorff dimension of this measure. In particular, it proves that limit measures of irreducible system of self-similar/self-conformal contractions satisfying the Open Set Condition, such as e. g. the middle-third Cantor set on the real line, contain subsets of full Hausdorff dimension consisting of badly approximable vectors. This was established in [60] and later independently in [64] using a different approach. See also [36] for a stronger result. The proof in [60] uses quantitative nondivergence estimates and an iterative procedure, which requires the measure in question to be absolutely friendly and not just friendly. A similar question for even the simplest not-absolutely friendly measures is completely open. For example, it is not known whether there exist uncountably many badly approximable pairs of the form (x; x 2 ). An analogous problem for atomic measures supported on algebraic numbers, that is, a “badly approximable” version of Roth’s Theorem, is currently beyond reach as well — there are no known badly approximable (or, for that matter, well approximable) algebraic numbers of degree bigger than two. It has been recently understood that the quantitative nondivergence method can be applied to the question of improvement to Dirichlet’s Theorem (see the beginning of Sect. “Basic Facts”). Given a positive " < 1, let us say

that Dirichlet’s Theorem can be "-improved for Y 2 M m;n , writing Y 2 DIm;n ("), if for every sufficiently large t the system kYq  pk < "et/m

and kqk < "e t/n

(19)

(that is, (5) with the right hand side terms multiplied by ") has a nontrivial integer solution (p; q). of  It is a theorem  Davenport and Schmidt [24] that  DIm;n (") D 0 for any " < 1; in other words, Dirichlet’s Theorem cannot be improved for Lebesgue-generic systems of linear forms. By a modification of the correspondence between dynamics and approximation, (19) is easily seen to be equivalent to g t LY Z k 2 K" , and since the complement to K " has nonempty interior for any " < 1, the result of Davenport and Schmidt follows from the ergodicity of the g t -action on ˝ k . Similar questions with  replaced by f  for some specific smooth maps f were considered in [2,3,15,23]. For example, [15], Theorem 7, provides an explicitly computable constant "0 D "0 (n) such that for f as in (12),   f  DI1;n (") D 0 for " < "0 : This had been previously done in [23] for n D 2 and in [2] for n D 3. In [62] this is extended to a much broader class of measures using estimates described in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”. In particular, almost every point of any nondegenerate smooth manifold is proved not to lie in DI(") for small enough " depending only on the manifold. Earlier this was done in [61] for the set of singular vectors, defined as the intersection of DI(") over all positive "; those correspond to divergent g t -trajectories. As before, the advantage of the method is allowing a multiplicative generalization of the Dirichlet-improvement set-up; see [62] for more detail. It is also worthwhile to mention that a generalization of the measure estimate discussed in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” was used in [13] to estimate the measure of the set of points x in a ball B Rd for which the system 8 ˆ <jf(x)  q C pj < " jf0 (x)  qj < ı ˆ : jq i j < Q i ; i D 1; : : : ; n; where f is a smooth nondegenerate map B ! Rn , has a nonzero integer solution. For that, Lf(x) as in (15) has to be replaced by the matrix 1 0 1 0 f(x) @0 1 f0 (x)A ; 0 0 In

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and therefore (i) and (ii) turn into more complicated conditions, which nevertheless can be checked when f is smooth and nondegenerate and  is Lebesgue measure. This has resulted in proving the convergence case of Khintchine–Groshev Theorem for nondegenerate manifolds [13], in both standard and multiplicative versions. The aforementioned estimate was also used in [7] for the proof of the divergence case, and in [38,40] for establishing the convergence case of Khintchine–Groshev theorem for affine hyperplanes and their nondegenerate submanifolds. This generalized results obtained by standard methods for the curve (12) by Bernik and Beresnevich [6,11]. Finally, let us note that in many of the problems mentioned above, the ground field R can be replaced by Q p , and in fact several fields can be taken simultaneously, thus giving rise to the S-arithmetic setting where S D fp1 ; : : : ; p s g is a finite set of normalized valuations of Q, which may or may not include the infinite valuation (cf. [83,90]). The space of lattices in RnC1 is replaced there by the space of lattices in QnC1 , where QS is the prodS uct of the fields R and Q p 1 ; : : : ; Q p s . This is the subject of the paper [59], where S-arithmetic analogues of many results reviewed in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” have been established. Similarly one can consider versions of the above theorems over local fields of positive characteristic [39]. See also [52] where Sprindžuk’s solution [83] of the complex case of Mahler’s Conjecture has been generalized (the latter involves studying small values of linear forms with coefficients in C at real integer points), and [31] which establishes a p-adic analogue of the result of [30] on the set of exceptions to Littlewood’s Conjecture. Future Directions Interactions between ergodic theory and number theory have been rapidly expanding during the last two decades, and the author has no doubts that new applications of dynamics to Diophantine approximation will emerge in the near future. Specializing to the topics discussed in the present paper, it is fair to say that the list of “further results” contained in the previous section is by no means complete, and many even further results are currently in preparation. This includes: extending proofs of extremality and strong extremality of certain measures to the set-up of systems of linear forms (namely, with min(m; n) > 1; this was mentioned as work in progress in [56]); proving Khintchine-type theorems (both convergence and divergence parts) for p-adic and S-arithmetic nondegenerate manifolds, see [73] for results in this direction; extending [38,40] to establish Khintchine-type theo-

rems for submanifolds of arbitrary affine subspaces. The work of Dru¸tu [28], who used ergodic theory on homogeneous spaces to compute the Hausdorff dimension of the intersection of Wn;1 ( 1;v ), v > 1, with some rational quadratic hypersurfaces in Rn deserves a special mention; it is plausible that using this method one can treat more general situations. Several other interesting open directions are listed in [41], Section 9, in the final sections of papers [7,54], in the book [15], and in surveys by Frantzikinakis–McCutcheon, Nitica and Ward in this volume. Acknowledgment The work on this paper was supported in part by NSF Grant DMS-0239463. Bibliography 1. Baker A (1975) Transcendental number theory. Cambridge University Press, London 2. Baker RC (1976) Metric diophantine approximation on manifolds. J Lond Math Soc 14:43–48 3. Baker RC (1978) Dirichlet’s theorem on diophantine approximation. Math Proc Cambridge Phil Soc 83:37–59 4. Bekka M, Mayer M (2000) Ergodic theory and topological dynamics of group actions on homogeneous spaces. Cambridge University Press, Cambridge 5. Beresnevich V (1999) On approximation of real numbers by real algebraic numbers. Acta Arith 90:97–112 6. Beresnevich V (2002) A Groshev type theorem for convergence on manifolds. Acta Mathematica Hungarica 94:99–130 7. Beresnevich V, Bernik VI, Kleinbock D, Margulis GA (2002) Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds. Moscow Math J 2:203–225 8. Beresnevich V, Dickinson H, Velani S (2006) Measure theoretic laws for lim sup sets. Mem Amer Math Soc:179:1–91 9. Beresnevich V, Dickinson H, Velani S (2007) Diophantine approximation on planar curves and the distribution of rational points. Ann Math 166:367–426 10. Beresnevich V, Velani S (2007) A note on simultaneous Diophantine approximation on planar curves. Ann Math 337: 769–796 11. Bernik VI (1984) A proof of Baker’s conjecture in the metric theory of transcendental numbers. Dokl Akad Nauk SSSR 277:1036–1039 12. Bernik VI, Dodson MM (1999) Metric Diophantine approximation on manifolds. Cambridge University Press, Cambridge 13. Bernik VI, Kleinbock D, Margulis GA (2001) Khintchine-type theorems on manifolds: convergence case for standard and multiplicative versions. Int Math Res Notices 2001:453–486 14. Besicovitch AS (1929) On linear sets of points of fractional dimensions. Ann Math 101:161–193 15. Bugeaud Y (2002) Approximation by algebraic integers and Hausdorff dimension. J London Math Soc 65:547–559 16. Bugeaud Y (2004) Approximation by algebraic numbers. Cambridge University Press, Cambridge 17. Cassels JWS (1957) An introduction to Diophantine approximation. Cambridge University Press, New York

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41. Gorodnik A (2007) Open problems in dynamics and related fields. J Mod Dyn 1:1–35 42. Groshev AV (1938) Une théorème sur les systèmes des formes linéaires. Dokl Akad Nauk SSSR 9:151–152 43. Harman G (1998) Metric number theory. Clarendon Press, Oxford University Press, New York 44. Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30:713–747 45. Jarnik V (1928-9) Zur metrischen Theorie der diophantischen Approximationen. Prace Mat-Fiz 36:91–106 46. Jarnik V (1929) Diophantischen Approximationen und Hausdorffsches Mass. Mat Sb 36:371–382 47. Khanin K, Lopes-Dias L, Marklof J (2007) Multidimensional continued fractions, dynamical renormalization and KAM theory. Comm Math Phys 270:197–231 48. Khintchine A (1924) Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math Ann 92:115–125 49. Khintchine A (1963) Continued fractions. P Noordhoff Ltd, Groningen 50. Kleinbock D (2001) Some applications of homogeneous dynamics to number theory. In: Smooth ergodic theory and its applications. American Mathematical Society, Providence, pp 639–660 51. Kleinbock D (2003) Extremal subspaces and their submanifolds. Geom Funct Anal 13:437–466 52. Kleinbock D (2004) Baker–Sprindžuk conjectures for complex analytic manifolds. In: Algebraic groups and Arithmetic. Tata Inst Fund Res, Mumbai, pp 539–553 53. Kleinbock D (2008) An extension of quantitative nondivergence and applications to Diophantine exponents. Trans AMS, to appear 54. Kleinbock D, Lindenstrauss E, Weiss B (2004) On fractal measures and diophantine approximation. Selecta Math 10: 479–523 55. Kleinbock D, Margulis GA (1996) Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In: Sina˘ı’s Moscow Seminar on Dynamical Systems. American Mathematical Society, Providence, pp 141–172 56. Kleinbock D, Margulis GA (1998) Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann Math 148:339–360 57. Kleinbock D, Margulis GA (1999) Logarithm laws for flows on homogeneous spaces. Invent Math 138:451–494 58. Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook on Dynamical Systems, vol 1A. Elsevier Science, North Holland, pp 813–930 59. Kleinbock D, Tomanov G (2007) Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comm Math Helv 82:519–581 60. Kleinbock D, Weiss B (2005) Badly approximable vectors on fractals. Israel J Math 149:137–170 61. Kleinbock D, Weiss B (2005) Friendly measures, homogeneous flows and singular vectors. In: Algebraic and Topological Dynamics. American Mathematical Society, Providence, pp 281–292 62. Kleinbock D, Weiss B (2008) Dirichlet’s theorem on diophantine approximation and homogeneous flows. J Mod Dyn 2:43– 62

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63. Kontsevich M, Suhov Y (1999) Statistics of Klein polyhedra and multidimensional continued fractions. In: Pseudoperiodic topology. American Mathematical Society, Providence, pp 9–27 64. Kristensen S, Thorn R, Velani S (2006) Diophantine approximation and badly approximable sets. Adv Math 203:132–169 65. Lagarias JC (1994) Geodesic multidimensional continued fractions. Proc London Math Soc 69:464–488 66. Lindenstrauss E (2007) Some examples how to use measure classification in number theory. In: Equidistribution in number theory, an introduction. Springer, Dordrecht, pp 261– 303 67. Mahler K (1932) Über das Mass der Menge aller S-Zahlen. Math Ann 106:131–139 68. Margulis GA (1975) On the action of unipotent groups in the space of lattices. In: Lie groups and their representations (Budapest, 1971). Halsted, New York, pp 365–370 69. Margulis GA (1989) Discrete subgroups and ergodic theory. In: Number theory, trace formulas and discrete groups (Oslo, 1987). Academic Press, Boston, pp 377–398 70. Margulis GA (1997) Oppenheim conjecture. In: Fields Medalists’ lectures. World Sci Publishing, River Edge, pp 272– 327 71. Margulis GA (2002) Diophantine approximation, lattices and flows on homogeneous spaces. In: A panorama of number theory or the view from Baker’s garden. Cambridge University Press, Cambridge, pp 280–310 ´ 72. Mauldin D, Urbanski M (1996) Dimensions and measures in infinite iterated function systems. Proc London Math Soc 73:105–154 73. Mohammadi A, Salehi Golsefidy A (2008) S-Arithmetic Khintchine-Type Theorem. Preprint 74. Moore CC (1966) Ergodicity of flows on homogeneous spaces. Amer J Math 88:154–178 75. Roth KF (1955) Rational Approximations to Algebraic Numbers. Mathematika 2:1–20

76. Schmidt WM (1960) A metrical theorem in diophantine approximation. Canad J Math 12:619–631 77. Schmidt WM (1964) Metrische Sätze über simultane Approximation abhängiger Größen. Monatsh Math 63:154–166 78. Schmidt WM (1969) Badly approximable systems of linear forms. J Number Theory 1:139–154 79. Schmidt WM (1972) Norm form equations. Ann Math 96: 526–551 80. Schmidt WM (1980) Diophantine approximation. Springer, Berlin 81. Sheingorn M (1993) Continued fractions and congruence subgroup geodesics. In: Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991). Dekker, New York, pp 239–254 82. Sprindžuk VG (1964) More on Mahler’s conjecture. Dokl Akad Nauk SSSR 155:54–56 83. Sprindžuk VG (1969) Mahler’s problem in metric number theory. American Mathematical Society, Providence 84. Sprindžuk VG (1979) Metric theory of Diophantine approximations. VH Winston & Sons, Washington DC 85. Sprindžuk VG (1980) Achievements and problems of the theory of Diophantine approximations. Uspekhi Mat Nauk 35: 3–68 86. Starkov A (2000) Dynamical systems on homogeneous spaces. American Mathematical Society, Providence 87. Sullivan D (1982) Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math 149:215–237 88. Stratmann B, Urbanski M (2006) Diophantine extremality of the Patterson measure. Math Proc Cambridge Phil Soc 140:297– 304 89. Urbanski M (2005) Diophantine approximation of selfconformal measures. J Number Theory 110:219–235 90. Želudeviˇc F (1986) Simultane diophantische Approximationen abhängiger Grössen in mehreren Metriken. Acta Arith 46: 285–296

Ergodic Theory: Interactions with Combinatorics and Number Theory

Ergodic Theory: Interactions with Combinatorics and Number Theory TOM W ARD School of Mathematics, University of East Anglia, Norwich, UK Article Outline Glossary Definition of the Subject Introduction Ergodic Theory Frequency of Returns Ergodic Ramsey Theory and Recurrence Orbit-Counting as an Analogous Development Diophantine Analysis as a Toolbox Future Directions Bibliography Glossary Almost everywhere (abbreviated a. e.) A property that makes sense for each point x in a measure space (X; B; ) is said to hold almost everywhere (or a. e.) if the set N X on which it does not hold satisfies N 2 B and (N) D 0. ˇ Cech–Stone compactification of N, ˇN A compact Hausdorff space that contains N as a dense subset with the property that any map from N to a compact Hausdorff space K extends uniquely to a continuous map ˇN ! K. This property and the fact that ˇN is a compact Hausdorff space containing N characterizes ˇN up to homeomorphism. Curvature An intrinsic measure of the curvature of a Riemannian manifold depending only on the Riemannian metric; in the case of a surface it determines whether the surface is locally convex (positive curvature), locally saddle-shaped (negative) or locally flat (zero). Diophantine approximation Theory of the approximation of real numbers by rational numbers: how small can the distance from a given irrational real number to a rational number be made in terms of the denominator of the rational? Equidistributed A sequence is equidistributed if the asymptotic proportion of time it spends in an interval is proportional to the length of the interval. Ergodic A measure-preserving transformation is ergodic if the only invariant functions are equal to a constant a. e.; equivalently if the transformation exhibits the convergence in the quasi-ergodic hypothesis.

Ergodic theory The study of statistical properties of orbits in abstract models of dynamical systems; more generally properties of measure-preserving (semi-) group actions on measure spaces. Geodesic (flow) The shortest path between two points on a Riemannian manifold; such a geodesic path is uniquely determined by a starting point and the initial tangent vector to the path (that is, a point in the unit tangent bundle). The transformation on the unit tangent bundle defined by flowing along the geodesic defines the geodesic flow. Haar measure (on a compact group) If G is a compact topological group, the unique measure  defined on the Borel sets of G with the property that (A C g) D (A) for all g 2 G and (G) D 1. Measure-theoretic entropy A numerical invariant of measure-preserving systems that reflects the asymptotic growth in complexity of measurable partitions refined under iteration of the map. Mixing A measure-preserving system is mixing if measurable sets (events) become asymptotically independent as they are moved apart in time (under iteration). (Quasi) Ergodic hypothesis The assumption that, in a dynamical system evolving in time and preserving a natural measure, there are some reasonable conditions under which the ‘time average’ along orbits of an observable (that is, the average value of a function defined on the phase space) will converge to the ‘space average’ (that is, the integral of the function with respect to the preserved measure). Recurrence Return of an orbit in a dynamical system close to its starting point infinitely often. S-Unit theorems A circle of results stating that linear equations in fields of zero characteristic have only finitely many solutions taken from finitely-generated multiplicative subgroups of the multiplicative group of the field (apart from infinite families of solutions arising from vanishing sub-sums). Topological entropy A numerical invariant of topological dynamical systems that measures the asymptotic growth in the complexity of orbits under iteration. The variational principle states that the topological entropy of a topological dynamical system is the supremum over all invariant measures of the measure-theoretic entropies of the dynamical systems viewed as measurable dynamical systems. Definition of the Subject Number theory is a branch of pure mathematics concerned with the properties of numbers in general, and integers in particular. The areas of most relevance to this ar-

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ticle are Diophantine analysis (the study of how real numbers may be approximated by rational numbers, and the consequences for solutions of equations in integers); analytic number theory, and in particular asymptotic estimates for the number of primes smaller than X as a function of X; equidistribution, and questions about how the digits of real numbers are distributed. Combinatorics is concerned with identifying structures in discrete objects; of most interest here is that part of combinatorics connected with Ramsey theory, asserting that large subsets of highly structured objects must automatically contain large replicas of that structure. Ergodic theory is the study of asymptotic behavior of group actions preserving a probability measure; it has proved to be a powerful part of dynamical systems with wide applications. Introduction Ergodic theory, part of the mathematical study of dynamical systems, has pervasive connections with number theory and combinatorics. This article briefly surveys how these arise through a small sample of results. Unsurprisingly, many details are suppressed, and of course the selection of topics reflects the author’s interests far more than it does the full extent of the flow of ideas between ergodic theory and number theory. In addition the selection of topics has been chosen in part to be complementary to those in related articles in the Encyclopedia. A particularly enormous lacuna is the theory of arithmetic dynamical systems itself – the recent monograph by Silverman [94] gives a comprehensive overview. More sophisticated aspects of this connection – in particular the connections between ergodic theory on homogeneous spaces and Diophantine analysis – are covered in the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory and  Ergodic Theory: Rigidity; more sophisticated overviews of the connections with combinatorics may be found in the article  Ergodic Theory: Recurrence. Ergodic Theory While the early origins of ergodic theory lie in the quasiergodic hypothesis of classical Hamiltonian dynamics, the mathematical study of ergodic theory concerns various properties of group actions on measure spaces, including but not limited to several special branches: 1. The classical study of single measure-preserving transformations. 2. Measure-preserving actions of Zd ; more generally of countable amenable groups.

3. Measure-preserving actions of Rd and more general amenable groups, called flows. 4. Measure-preserving actions of lattices in Lie groups. 5. Measure-preserving actions of Lie groups. The ideas and conditions surrounding the quasi-ergodic hypothesis were eventually placed on a firm mathematical footing by developments starting in 1890. For a single measure-preserving transformation T : X ! X of a probability space (X; B; ), Poincaré [74] showed a recurrence theorem: if E 2 B is any measurable set, then for a. e. x 2 E there is an infinite set of return times, 0 < n1 < n2 <    with T n j (x) 2 E (of course Poincaré noted this in a specific setting, concerned with a natural invariant measure for the “three-body” problem in planetary motion). Poincaré’s qualitative result was made quantitative in the 1930s, when von Neumann [105] used the approach of Koopman [52] to show the mean ergodic theorem: if f 2 L2 () then there is some f 2 L2 () for which  N1  1 X    f ı T n  f  ! 0 as  N  nD0

N ! 1;

2

clearly f then R R has the property that k f  f ı Tk2 D 0 and f d D f d. Around the same time, Birkhoff [13] showed the more delicate pointwise ergodic theorem: for any g 2 L1 () there is some g 2 L1 () for which N1 1 X g(T n x) ! g(x) a. e. ; N nD0

R again it is then clear that g(Tx) D g(x) a. e. and g d D R g d. The map T is called ergodic if the invariance condition forces the function (f or g) to be equal to a constant a. e. Thus an ergodic map has the property that the time or erP N1 godic average (1/N) nD0 f ı T j converges to the space R average d. An overview of ergodic theorems and their many extensions may be found in the article  Ergodic Theorems. Thus ergodic theory at its most basic level makes strong statements about the asymptotic behavior of orbits of a dynamical system as seen by observables (measurable functions on the space X). Applying the ergodic theorem to the indicator function of a measurable set A shows that ergodicity guarantees that a. e. orbit spends an asymptotic proportion of time in A equal to the volume (A) of that set (as measured by the invariant measure). This points to the start of the pervasive connections between ergodic theory and number theory – but as this and other articles relate, the connections extend far beyond this.

Ergodic Theory: Interactions with Combinatorics and Number Theory

Frequency of Returns

First Digits

In this section we illustrate the way in which a dynamical point of view may unify, explain and extend quite disparate results from number theory.

The astronomer Newcomb [67] noted that the first digits of large collections of numerical data that are not dimensionless have a specific and non-uniform distribution:

Normal Numbers Borel [15] showed (as a consequence of what became the Borel–Cantelli Lemma in probability) that a. e. real number (with respect to Lebesgue measure) is normal to every base: that is, has the property that any block of k digits in the base-r expansion appears with asymptotic frequency rk . Continued Fraction Digits Analogs of normality results for the continued fraction expansion of real numbers were found by Khinchin, Kuz’min, Lévy and others. Any irrational x 2 [0; 1] has a unique expansion as a continued fraction 1

a1 (x) C a2 (x) C

1 a3 (x) C   

and, just as in the case of the familiar base-r expansion, it turns out that the digits (a n (x)) obey precise statistical rules for a. e. x. Gauss conjectured that the appearance of individual digits would obey the law 1 jfk : 1  k  N; a k (x) D jgj N 2 log(1 C j)  log j  log(2 C j) ! : log 2

(1)

lim (a1 (x)a2 (x) : : : a n (x))1/n D

log n/ log 2 1  Y (n C 1)2 nD1

n(n C 2)

D 2:68545 : : :

n!1

1 jfk : 1  k  N; fa k g 2 [a; b]gj ! (b  a) N as N ! 1 ;

Z 1 N 1 X f (a k ) ! f (t)dt N 0

as

N !1

kD1

for all continuous functions f . Weyl showed that the sequence fn˛g is equidistributed if and only if ˛ is irrational. This result was refined and extended in many directions; for example, Hlawka [44] and others found rates for the convergence in terms of the discrepancy of the sequence, Weyl [110] proved equidistribution for fn2 ˛g, and Vinogradov for fp n ˛g where pn is the nth prime.

for a. e. x :

Lévy [60] showed that the denominator q n (x) of the nth convergent (p n (x))/(q n (x)) (the rational obtained by truncating the continued fraction expansion of x at the nth term) grows at a specific exponential rate, lim

Weyl [109] (and, separately, Bohl [14] and Sierpi´nski [91]) found an important instance of equidistribution. Writing fg for the fractional part, a sequence (a n ) of real numbers is said to be equidistributed modulo 1 if, for any interval [a; b] [0; 1),

equivalently if

This was eventually proved by Kuz’min [54] and Lévy [59], and the probability distribution of the digits is the Gauss– Kuz’min law. Khinchin [50] developed this further, showing for example that n!1

This is now known as Benford’s Law, following his popularization and possible rediscovery of the phenomenon [6]. In both cases, this was an empirical observation eventually made rigorous by Hill [42]. Arnold (App. 12 in [3]), pointed out the dynamics behind this phenomena in certain cases, best illustrated by the statistical behavior of the sequence 1; 2; 4; 8; 1; 3; 6; 1; : : : of first digits of powers of 2. Empirically, the digit 1 appears about 30% of the time, while the digit 9 appears about 5% of the time. Equidistribution

1

xD

“The law of probability of the occurrence of numbers is such that all mantissæ of their logarithms are equally probable.”

1 2 log q n (x) D n 12 log 2

for a. e. x :

The Ergodic Context All the results of this section are manifestations of various kinds of convergence of ergodic averages. Borel’s theorem on normal numbers is an immediate consequence of the fact that Lebesgue measure on [0; 1) is invariant and ergodic for the map x 7! bx modulo 1 with b > 2. The asymptotic properties of continued fraction digits are all

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a consequence of the fact that the Gauss measure defined by Z dx 1 (A) D for A [0; 1] log 2 A 1 C x ˚ is invariant and ergodic for the Gauss map x 7! x1 , and the orbit of an irrational number under the Gauss map determine the digits appearing in the continued fraction expansion much as the orbit under the map x 7! bx (mod 1) determines the digits in the base b expansion. The results on equidistribution and the frequency of first digits are related to ergodic averaging of a different sort. For example, writing R˛ (t) D t C ˛ modulo 1 for the circle rotation by ˛, the first digit of 2n is the digit j if and only if log10 j  Rlog10 (2) (0) < log10 ( j C 1) : Thus the asymptotic frequency of appearance concerns the orbit of a specific point. In order to see what this means, consider a continuous map T : X ! X of a compact metric space (X; d). The space M(T) of Borel probability measures on the Borel -algebra of (X; d) is a non-empty compact convex set in the weak*-topology, each extreme point is an ergodic measure for T, and these ergodic measures are mutually singular. If M(T) is not a singleton and 1 ; 2 2 M(T) are distinct ergodic measures, then for R R a continuous function f with X f d1 ¤ X f d2 it is P N1 f (T n x) must clear thatRthe ergodic averages N1 nD0 R converge to X f d1 a. e. with respect to 1 and to X f d2 a. e. with respect to 2 . Thus the presence of many invariant measures for a continuous map means that ergodic averages along the orbits of specific points need not converge to the space average with respect to a chosen invariant measure. In the extreme situation of unique ergodicity (a single invariant measure, which is necessarily an extreme point of M(T) and hence ergodic) the convergence of ergodic averages is much more uniform. Indeed, if T is uniquely ergodic with M(T) D fg then, for any continuous function f : X ! R, Z N1 1 X n f (T x) ! f d uniformly in x N nD0 X (see Oxtoby [69]). The circle rotation R˛ is uniquely ergodic for irrational ˛, leading to the equidistribution results. The ergodic viewpoint on equidistribution also places equidistribution results in a wider context. Weyl’s result that fn2 ˛g (indeed, the fractional part of any polynomial

with at least one irrational coefficient) is equidistributed for irrational ˛ was given another proof by Furstenberg [29] using the notion of unique ergodicity. These methods were then used in the study of nilsystems (translations on quotients of nilpotent Lie groups) by Auslander, Green and Hahn [4] and Parry [70], and these nilsystems play an essential role for polynomial (and other non-conventional) ergodic averaging (see Host and Kra [45] and Leibman [58] in the polynomial case; Host and Kra [46] in the multiple linear case). Remarkably, nilsystems are starting to play a role within combinatorics – an example is the work on the asymptotic number of 4-step arithmetic progressions in the primes by Green and Tao [40]. Pointwise ergodic theorems have also been found along sequences other than N; notably for integer-valued polynomials and along the primes for L2 functions by Bourgain [16,17]. For more details, see the survey paper of del Junco on ergodic theorems. Ergodic Ramsey Theory and Recurrence In 1927 van der Waerden proved a conjecture attributed to Baudet: if the natural numbers are written as a disjoint union of finitely many sets, N D C1 t C2 t    t C r ;

(2)

then there must be one set Cj that contains arbitrarily long arithmetic progressions. That is, there is some j 2 f1; : : : ; rg such that for any k > 1 there are a > 1 and n > 1 with a; a C n; a C 2n; : : : ; a C (k  1)n 2 C j : The original proof appears in van der Waerden’s paper [103], and there is a discussion of how he found the proof in [104]. Work of Furstenberg and Weiss [34] and others placed the theorem of van der Waerden in the context of topological dynamics, giving alternative proofs. Specifically, van der Waerden’s theorem is a consequence of topological multiple recurrence: the return of points under iteration in a topological dynamical system close to their starting point along finite sequences of times. The same approach readily gives dynamical proofs of Rado’s extension [76] of van der Waerden’s theorem, and of Hindman’s theorem [43]. The theorems of Rado and Hindman introduce a new theme: given a set A D fn1 ; n2 ; : : : g of natural numbers, write FS(A) for the set of numbers obtained as finite sums n i 1 C    C n i j with i1 < i2 <    < i j . Rado showed that for any large n there is some Cs containing some FS(A) for a set A of cardinality n. Hindman showed that there is some Cs containing some FS(A) for an infinite set A.

Ergodic Theory: Interactions with Combinatorics and Number Theory

In the theorem of van der Waerden, it is clear that for any reasonable notion of “proportion” or “density” one of the sets Cj must occupy a positive proportion of N. A set A N is said to have positive upper density if there are sequences (M i ) and (N i ) with N i  M i ! 1 as i ! 1 such that lim

i!1

1 jfa 2 A: M i < a < N i gj > 0 : Ni  Mi

Erd˝os and Turán [26] conjectured the stronger statement that any subset of N with positive upper density must contain arbitrary long arithmetic progressions. This statement was shown for arithmetic progressions of length 3 by Roth [79] in 1952, then for length 4 by Szemerédi [96] in 1969. The general result was eventually proved by Szemerédi [97] in 1975 in a lengthy and extremely difficult argument. Furstenberg saw that Szemerédi’s Theorem would follow from a deep extension of the Poincaré recurrence phenomena described in Sect. “Ergodic Theory” and proved that extension [30] (see also the survey article by Furstenberg, Katznelson and Ornstein [33]). The multiple recurrence result of Furstenberg says that for any measure-preserving system (X; B; ; T) and set A 2 B with (A) > 0, and for any k 2 N, lim inf

NM!1



N X

1 NMC1

   A \ T n A \ T 2n A \    \ T kn A > 0 :

nDM

An immediate consequence is that in the same setting there must be some n > 1 for which   (3)  A \ T n A \ T 2n A \    \ T kn A > 0 : A general correspondence principle, due to Furstenberg, shows that statements in combinatorics like Szemerédi’s Theorem are equivalent to statements in ergodic theory like (3). This opened up a significant new field of ergodic Ramsey theory, in which methods from dynamical systems and ergodic theory are used to produce new results in infinite combinatorics. For an overview, see the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory,  Ergodic Theory: Rigidity,  Ergodic Theory: Recurrence and the survey articles of Bergelson [7,8,10]. The field is too large to give an overview here, but a few examples will give a flavor of some of the themes. Call a set R Z a set of recurrence if, for any finite measure-preserving invertible transformation T of a finite

measure space (X; B; ) and any set A 2 B with (A) > 0, there are infinitely many n 2 R for which (A \ T n A) > 0. Thus Poincaré recurrence is the statement that N is a set of recurrence. Furstenberg and Katznelson [31] showed that if T1 ; : : : ; Tk form a family of commuting measurepreserving transformations and A is a set of positive measure, then lim inf N!1

N1 1 X (T1n A \    \ Tkn A) > 0 : N nD0

This remarkable multiple recurrence implies a multi-dimensional form of Szemerédi’s theorem. Recently, Gowers has found a non-ergodic proof of this [38]. Furstenberg also gave an ergodic proof of Sárközy’s theorem [80]: if p 2 Q[t] is a polynomial with p(Z) Z and p(0) D 0, then fp(n)gn>0 is a set of recurrence. This was extended to multiple polynomial recurrence by Bergelson and Leibman [11]. Topology and Coloring Theorems ˇ The existence of idempotent ultrafilters in the Cech–Stone compactification ˇN gives rise to an algebraic approach to many questions in topological dynamics (this notion has its origins in the work of Ellis [24]). Using these methods, results like Hindman’s finite sums theorem find elegant proofs, and many new results in combinatorics have been found. For example, in the partition (2) there must be one set Cj containing a triple x; y; z solving x  y D z2 . A deeper application is to improve a strengthening of Kronecker’s theorem. To explain this, recall that a set S is called IP if there is a sequence (n i ) of natural numbers (which do not need to be distinct) with the property that S contains all the terms of the sequence and all finite sums of terms of the sequence with distinct indices. A set S is called IP if it has non-empty intersection with every IP set, and  if there is some t 2 Z for which S  t is a set S is called IPC  ) is an extreme form of ‘fatness’ IP . Thus being IP (or IPC for a set. Now let 1; ˛1 ; : : : ; ˛ k be numbers that are linearly independent over the rationals, and for any d 2 N and kd non-empty intervals I i j [0; 1] (1  i  d; 1  j  k), let D D fn 2 N : fn i ˛ j g 2 I i j for all i; jg : Kronecker showed that if d D 1 then D is non-empty; Hardy and Littlewood showed that D is infinite, and Weyl showed that D has positive density. Bergelson [9] uses these algebraic methods to improve the result by showing  set. that D is an IPC

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Polynomialization and IP-sets

Sets of Primes

As mentioned above, Bergelson and Leibman [11] extended multiple recurrence to a polynomial setting. For example, let

In a remarkable development, Szemerédi’s theorem and some of the ideas behind ergodic Ramsey theory joined results of Goldston and Yıldırım [37] in playing a part in Green and Tao’s proof [39] that the set of primes contains arbitrarily long arithmetic progressions. This profound result is surveyed from an ergodic point of view in the article of Kra [53]. As with Szemerédi’s theorem itself, this result has been extended to a polynomial setting by Tao and Ziegler [101]. Given integer-valued polynomials f1 ; : : : ; f k 2 Z[t] with

fp i; j : 1  i  k; 1  j  tg be a collection of polynomials with rational coefficients and p i; j (Z) Z, p i; j (0) D 0. Then if (A) > 0, we have 11 1 0 0 N k t X \ Y 1 p i; j (n) A AA > 0 : @ @ Tj lim inf N!1 N nD1 iD1

jD1

Using the Furstenberg correspondence principle, this gives a multi-dimensional polynomial Szemerédi theorem: If P : Zr ! Z` is a polynomial mapping with the property that P(0) D 0, and F Zr is a finite configuration, then any set S Z` of positive upper Banach density contains a set of the form u C P(nF) for some u 2 Z` and n 2 N. In a different direction, motivated in part by Hindman’s theorem, the multiple recurrence results generalize to IP-sets. Furstenberg and Katznelson [32] proved a linear IP-multiple recurrence theorem in which the recurrence is guaranteed to occur along an IP-set. A combinatorial proof of this result has been found by Nagle, Rödl and Schacht [66]. Bergelson and McCutcheon [12] extended these results by proving a polynomial IP-multiple recurrence theorem. To formulate this, make the following definitions. Write F for the family of non-empty finite subsets of N, so that a sequence indexed by F is an IP-set. More generally, an F -sequence (n˛ )˛2F taking values in an abelian group is called an IP-sequence if n˛[ˇ D n˛ C nˇ whenever ˛ \ ˇ D ;. An IP-ring is a set of the form S F (1) D f i2ˇ ˛ i : ˇ 2 F g where ˛1 < ˛2 <    is a sequence in F , and ˛ < ˇ means a < b for all a 2 ˛, b 2 ˇ; write F<m for the set of m-tuples (˛1 ; : : : ; ˛m ) from F m with ˛ i < ˛ j for i < j. Write PE(m; d) for the collection of all expressions of the form T(˛1 ; : : : ; ˛m ) D (b) p i ((n ˛ j )1bk; 1 jm ) Qr m, T ; (˛1 ; : : : ; ˛m ) 2 (F [ ;)< iD1 i where each pi is a polynomial in a k  m matrix of variables with integer coefficients and zero constant term with degree  d. Then for every m; t 2 N, there is an IP-ring F (1) , and an a D a(A; m; t; d) > 0, such that, for every set of polynomial expressions fS0 ; : : : ; S t g PE(m; d), ! t \ 1 S i (˛1 ; : : : ; ˛m ) A > 0 : IP  lim  m (˛1 ;:::;˛ m )2(F (1) )<

iD0

There are a large number of deep combinatorial consequences of this result, not all of which seem accessible by other means.

f1 (0) D    D f k (0) D 0 and any " > 0, Tao and Ziegler proved that there are infinitely many integers x; m with 1  m  x " for which x C f1 (m); : : : ; x C f k (m) are primes. Orbit-Counting as an Analogous Development Some of the connections between number theory and ergodic theory arise through developments that are analogous but not directly related. A remarkable instance of this concerns the long history of attempts to count prime numbers laid alongside the problem of counting closed orbits in dynamical systems. Counting Orbits and Geodesics Consider first the fundamental arithmetic function (X) D jfp  X : p is primegj. Tables of primes prepared by Felkel and Vega in the 18th century led Legendre to suggest that (X) is approximately x/(log(X)  1:08). Gauss, using both computational evidence and a heuristic argument, suggested that (X) is approximated by Z

X

li(X) D 2

dt : log t

Both of these suggestions imply the well-known asymptotic formula (X)

X : log X

(4)

Riemann brought the analytic ideas of Dirichlet and Chebyshev (who used the zeta function to find a weaker version of (4) with upper and lower bounds for the quan(X) log(X) ) to bear by proposing that the zeta function tity X (s) D

1 Y X 1 1 D ; 1  ps s n p nD1

(5)

Ergodic Theory: Interactions with Combinatorics and Number Theory

already studied by Euler, would connect properties of the primes to analytic methods. An essential step in these developments, due to Riemann, is the meromorphic extension of  from the region <(s) > 1 in (5) to the whole complex plane and a functional equation relating the value of the extension at s to the value at 1  s. Moreover, Riemann showed that the extension has readily understood real zeros, and that all the other zeros he could find were symmetric about <(s) D 12 . The Riemann hypothesis asserts that zeros in the region 0 < <(s) < 1 all lie on the line <(s) D 12 , and this remains open. Analytic properties of the Riemann zeta function were used by Hadamard and de la Vallée Poussin to prove (4), the Prime Number Theorem, in 1896. Tauberian methods developed by Wiener and Ikehara [111] later gave different approaches to the Prime Number Theorem. These ideas initiated the widespread use of zeta functions in several parts of mathematics, but it was not until the middle of the 20th century that Selberg [88] introduced a zeta function dealing directly with quantities arising in dynamical systems: the lengths of closed geodesics on surfaces of constant curvature 1. The geodesic flow acts on the unit tangent bundle to the surface by moving a point and unit tangent vector at that point along the unique geodesic they define at unit speed. Closed geodesics are then in one-to-one correspondence with periodic orbits of the associated geodesic flow on the unit tangent bundle, and it is in this sense that the quantities are dynamical. The function defined by Selberg takes the form Z(s) D

1  YY

 1  e(sCk)jj ;

 kD0

in which  runs over all the closed geodesics, and jj denotes the length of the geodesic. In a direct echo of the Riemann zeta function, Selberg found an analytic continuation to the complex plane, and showed that the zeros of Z lie on the real axis or on the line <(s) D 12 (the analogue of the Riemann hypothesis for Z; see also the paper of Hejhal [41]). The zeros of Z are closely connected to the eigenvalues for the Laplace–Beltrami operator, and thus give information about the lengths of closed geodesics via Selberg’s trace formula in the same paper. Huber [47] and others used this approach to give an analogue of the prime number theorem for closed geodesics – a prime orbit theorem. Sinai [92] considered closed geodesics on a manifold M with negative curvature bounded between R2 and r2 , and found the bounds (dim(M)  1) r  lim inf T!1

log (T) T

 lim sup T!1

log (T) T

 (dim(M)  1)R for the number (T) of closed geodesics of multiplicity one with length less than T, analogous to Chebyshev’s result. The essential dynamical feature behind the geodesic flow on a manifold of negative curvature is that it is an example of an Anosov flow [2]. These are smooth dynamical R-actions (equivalently, first-order differential equations on Riemannian manifolds) with the property that the tangent bundle has a continuously varying splitting into a direct sum E u ˚ E s ˚ E o and the action of the differential of the flow acts on Eu as an exponential expansion, on Es as an exponential contraction, Eo is the one-dimensional bundle of vectors that are tangent to orbits, and the expansion and contraction factors are bounded. In the setting of Anosov flows, the natural orbit counting function is (X) D jf :  a closed orbit of length jj  Xgj. Margulis [62,63] generalized the picture to weak-mixing Anosov flows by showing a prime orbit theorem of the form (X)

e htop X htop X

(6)

for the counting function (X) D jf :  a closed orbit of length jj  Xgj where as before htop denotes the topological entropy of the flow. Integral to Margulis’ work is a result on the spatial distribution of the closed geodesics reflected in a flow-invariant probability measure, now called the Margulis measure. Anosov also studied discrete dynamical systems with similar properties: diffeomorphisms of compact manifolds with a similar splitting of the tangent space (though in this setting Eo disappears). The archetypal Anosov diffeomorphism is a hyperbolic toral automorphism of the sort considered in Subsect. “Orbit Growth and Convergence”; for such automorphisms of the 2-torus Adler and Weiss [1] constructed Markov partitions, allowing the dynamics of the toral automorphism to be modeled by a topological Markov shift, and used this to determine when two such automorphisms are measurably isomorphic. Sinai [93], Ratner [77], Bowen [18,20] and others developed the construction of Markov partitions in general for Anosov diffeomorphisms and flows. Around the same time, Smale [95] introduced a more permissive hyperbolicity axiom for diffeomorphisms, Axiom A. Maps satisfying Axiom A are diffeomorphisms satisfying the same hypothesis as that of Anosov diffeomor-

319

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phisms, but only on the set of points that return arbitrarily close under the action of the flow (or iteration of the map). Thus Markov partitions, and with them associated transfer operators became a substitute for the geometrical Laplace–Beltrami operators of the setting considered by Selberg. Bowen [19] extended the uniform distribution result of Margulis to this setting and found an analogue of Chebychev’s theorem for closed orbits. Parry [71] (in a restricted case) and Parry and Pollicott [73] went on to prove the prime orbit theorem in this more general setting. The methods are an adaptation of the Ikehara–Wiener– Tauberian approach to the prime number theorem. Thus many facets of the prime number theorem story find their echoes in the study of closed orbits for hyperbolic flows: the role played by meromorphic extensions of suitable zeta functions, Tauberian methods, and so on. Moreover, related results from number theory have analogues in dynamics, for example Mertens’ theorem [65] in the work of Sharp [89] and Noorani [68] and Dirichlet’s theorem in work of Parry [72]. The “elementary” proof (not using analytic methods) of the prime number theorem by Erd˝os [25] and Selberg [87] (see the survey by Goldfeld [36] for the background to the results and the unfortunate priority dispute) has an echo in some approaches to orbit-counting problems from an elementary (non-Tauberian) perspective, including work of Lalley [56] on special flows and Everest, Miles, Stevens and the author [27] in the algebraic setting. In a different direction Lalley [55] found orbit asymptotics for closed orbits satisfying constraints in the Axiom A setting without using Tauberian theorems. His more direct approach is still analytic, using complex transfer operators (the same objects used to by Parry and Pollicott to study the dynamical zeta function at complex values) and indeed somewhat parallels a Tauberian argument. Further resonances with number theory arise here. For example, there are results on the distribution of closed orbits for group extensions (analogous to Chebotarev’s theorem) and for orbits with homological constraints (see Sharp [90], Katsuda and Sunada [49]). Of course the great diversity of dynamical systems subsumed in the phrase “prime orbit theorem” creates new problems and challenges, and in particular if there is not much geometry to work with then the reliance on Markov partitions and transfer operators makes it difficult to find higher-order asymptotics. Dolgopyat [22] has nonetheless managed to push the Markov methods to obtain uniform bounds on iterates of the associated transfer operators to the region <(s) > 0 with 0 < 1. This result has wide implications; an exam-

ple most relevant to the analogy with number theory is the work of Pollicott and Sharp [75] in which Dolgopyat’s result is used to show that for certain geodesic flows there is a two-term prime orbit theorem of the form     (X) D li e htop X C O e c X for some c < htop . For non-positive curvature manifolds less is known: Knieper [51] finds upper and lower bounds for the function counting closed geodesics on rank-1 manifolds of non-positive curvature of the form A

eh X  (X)  Be h X X

for constants A; B > 0. Counting Orbits for Group Endomorphisms A prism through which to view some of the deeper issues that arise in Subsect. “Counting Orbits and Geodesics” is provided by group endomorphisms. The price paid for having simple closed formulas for all the quantities involved is of course a severe loss of generality, but the diversity of examples illustrates many of the phenomena that may be expected in more general settings when hyperbolicity is lost. Consider an endomorphism T : X ! X of a compact group with the property that Fn (T) < 1 for all n > 1. The number of closed orbits of length n under T is then On (T) D

1X (n/d)Fd (T) : n

(7)

djn

In simple situations (hyperbolic toral automorphisms for example) it is straightforward to show that T (X) D jf :  a closed orbit under T of length  Xgj

e(XC1)htop (T) : X

(8)

Waddington [106] considered quasihyperbolic toral automorphisms, showing that the asymptotic (8) in this case is multiplied by an explicit almost-periodic function bounded away from zero and infinity. This result has been extended further into non-hyperbolic territory, which is most easily seen via the socalled connected S-integer dynamical systems introduced by Chothi, Everest and the author [21]. Fix an algebraic number field K with set of places P(K) and set of infinite places P1 (K), an element of infinite multiplicative order

Ergodic Theory: Interactions with Combinatorics and Number Theory

 2 K , and a finite set S P(K) n P1 (K) with the property that jjw  1 for all w … S [ P1 (K). The associated ring of S-integers is R S D fx 2 K : jxjw  1 for all w … S [ P1 (K)g : Let X be the compact character group of RS , and define the endomorphism T : X ! X to be the dual of the map x 7!  x on RS . Following Weil [108], write Kw for the completion at w, and for w finite, write rw for the maximal compact subring of Kw . Notice that if S D P then R S D K and Fn (T) D 1 for all n > 1 by the product formula for A-fields. As the set S shrinks, more and more periodic orbits come into being, and if S is as small as possible (given ) then the resulting system is (more or less) hyperbolic or quasi-hyperbolic. For S finite, it turns out that there are still sufficiently many periodic orbits to have the growth rate result (10), but the asymptotic (8) is modified in much the same way as Waddington observed for quasi-hyperbolic toral automorphisms: XT (X) >0 (9) e(XC1)htop (T) and there is an associated pair (X  ; a T ), where X  is a compact group and a T 2 X  , with the property that if Nj a T converges in X  as j ! 1, then there is convergence in (9). A simple special case will illustrate this. Taking K D Q,  D 2 and S D f3g gives a compact group endomorphism T with lim inf X!1

Fn (T) D (2n  1)j2n  1j3 : For this example the results of [21] are sharper: The expression in (9) converges along (X j ) if and only if 2 X j converges in the ring of 3-adic integers Z3 , the expression has uncountably many limit points, and the upper and lower limits are transcendental. Similarly, the dynamical analogue of Mertens’ theorem found by Sharp may be found for S-integer systems with S finite. Writing X 1 MT (N) D ; e h(T)jj

Diophantine Analysis as a Toolbox Many problems in ergodic theory and dynamical system exploit ideas and results from number theory in a direct way; we illustrate this by describing a selection of dynamical problems that call on particular parts of number theory in an essential way. The example of mixing in Subsect. “Mixing and Additive Relations in Fields” is particularly striking for two reasons: the results needed from number theory are relatively recent, and the ergodic application directly motivated a further development in number theory. Orbit Growth and Convergence The analysis of periodic orbits – how their number grows as the length grows and how they spread out through space – is of central importance in dynamics (see Katok [48] for example). An instance of this is that for many simple kinds of dynamical systems T : X ! X (where T is a continuous map of a compact metric space (X; d)) the logarithmic growth rate of the number of periodic points exists and coincides with the topological entropy h(T) (an invariant giving a quantitative measure of the average rate of growth in orbit complexity under T). That is, writing F n (T) D jfx 2 X : T n x D xgj ; we find 1 log Fn (T) ! htop (T) n

for many of the simplest dynamical systems. For example, if X D T r is the r-torus and T D TA is the automorphism of the torus corresponding to a matrix A in GLr (Z), then T A is ergodic with respect to Lebesgue measure if and only if no eigenvalue of A is a root of unity. Under this assumption, we have Fn (TA ) D

r Y

MT (N) D k T log N C C T C O (1/N) :

Without the restriction that K D Q, it is shown that there are constants k T 2 Q, CT and ı > 0 with MT (N) D k T log N C C T C O(N ı ) :

jni  1j

iD1

and

jjN

it is shown in [21] that for an ergodic S-integer map T with K D Q and S finite, there are constants k T 2 Q and CT such that

(10)

htop (TA ) D

r X

log maxf1; j i jg

(11)

iD1

where 1 ; : : : ; r are the eigenvalues of A. It follows that the convergence in (10) is clear under the assumption that T A is hyperbolic (that is, no eigenvalue has modulus one). Without this assumption the convergence is less clear: for r > 4 the automorphism T A may be ergodic without being hyperbolic. That is, while no eigenvalues are unit

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roots some may have unit modulus. As pointed out by Lind [61] in his study of these quasihyperbolic automorphisms, the convergence (10) does still hold for these systems, but this requires a significant Diophantine result (the theorem of Gel’fond [35] suffices; one may also use Baker’s theorem [5]). Even further from hyperbolicity lie the family of S-integer systems [21,107]; their orbit-growth properties are intimately tied up with Artin’s conjecture on primitive roots and prime divisors of linear recurrence sequences. Mixing and Additive Relations in Fields The problem of higher-order mixing for commuting group automorphisms provides a striking example of the dialogue between ergodic theory and number theory, in which deep results from number theory have been used to solve problems in ergodic theory, and questions arising in ergodic theory have motivated further developments in number theory. An action T of a countable group  on a probability space (X; B; ) is called k-fold mixing or mixing on (k C 1) sets if    A0 \ T g 1 A1 \    \ T g k A k ! (A0 )    (A k ) (12) as g i g 1 j ! 1 for i ¤ j with the convention that g0 D 1 , for any sets A0 ; : : : ; A k 2 B; g n ! 1 in  means that for any finite set F  there is an N with n > N H) g n … F. For k D 1 the property is called simply mixing. This notion for single transformations goes back to the foundational work of Rohlin [78], where he showed that ergodic group endomorphisms are mixing of all orders (and so the notion is not useful for distinguishing between group endomorphisms as measurable dynamical systems). He raised the (still open) question of whether any measure-preserving transformation can be mixing without being mixing of all orders. A class of group actions that are particularly easy to understand are the algebraic dynamical systems studied systematically by Schmidt [83]: here X is a compact abelian group, each T g is a continuous automorphism of X, and  is the Haar measure on X. Schmidt [82] related mixing properties of algebraic dynamical systems with  D Zd to statements in arithmetic, and showed that a mixing action on a connected group could only fail to mix in a certain way. Later Schmidt and the author [85] showed that for X

connected, mixing implies mixing of all orders. The proof proceeds by showing that the result is exactly equivalent to the following statement: if K is a field of characteristic zero, and G is a finitely generated subgroup of the multiplicative group K , then the equation a1 x1 C    C a n x n D 1

(13)

for fixed a1 ; : : : ; a n 2 K has a finite number of soluP tions x1 ; : : : ; x n 2 G for which no subsum i2I a i x i with I ¨ f1; : : : ; ng vanishes. The bound on the number of solutions to (13) follows from the profound extensions to W. Schmidt’s subspace theorem in Diophantine geometry [86] by Evertse and Schlickewei (see [28,81,102] for the details). The argument in [85] may be cast as follows: failure of k-fold mixing in a connected algebraic dynamical system implies (via duality) an infinite set of solutions to an equation of the shape (13) in some field of characteristic zero. The S-unit theorem means that this can only happen if there is some proper subsum that vanishes infinitely often. This infinite family of solutions to a homogeneous form of (13) with fewer terms can then be translated back via duality to show that the system fails to mix for some strictly lower order, proving that mixing implies mixing of all orders by induction. Mixing properties for algebraic dynamical systems without the assumption of connectedness are quite different, and in particular it is possible to have mixing actions that are not mixing of all orders. This is a simple consequence of the fact that the constituents of a disconnected algebraic dynamical system are associated with fields of positive characteristic, where the presence of the Frobenius automorphism can prevent higher-order mixing. Ledrappier [57] pointed this out via examples of the following shape. Let n 2 X D x 2 F2Z : x(aC1;b) C x(a;b) C x(a;bC1) D0

o (mod 2)

and define the Z2 -action T to be the natural shift action, (T (n;m) x)(a;b) D x(aCn;bCm) : It is readily seen that this action is mixing with respect to the Haar measure. The condition x(aC1;b) C x(a;b) C x(a;bC1) D 0 (mod 2) implies that, for any k > 1, x(0;2k )

! 2k X 2k D x( j;0) D x(0;0) Cx(2k ;0) j jD0

(mod 2) (14)

Ergodic Theory: Interactions with Combinatorics and Number Theory

since every entry in the 2 k th row of Pascal’s triangle is even apart from the first and the last. Now let A D fx 2 X : x(0;0) D 0g and let x 2 X be any element with x(0;0) D 1. Then X is the disjoint union of A and A C x , so (A) D (A C x ) D

1 2

stants b1 ; : : : ; b m 2 K and some (g1 ; : : : ; g m ) 2 G m with the following properties:  g j ¤ 1 for 1  j  m;  g i g 1 j ¤ 1 for 1  i < j  m;  there are infinitely many k for which

:

However, (14) shows that x 2 A \ T(2k ;0) A H) x 2 T(0;2k ) A ; so A \ T(2k ;0) A \ T(0;2k ) (A C x ) D ; for all k > 1, which shows that T cannot be mixing on three sets. The full picture of higher-order mixing properties on disconnected groups is rather involved; see Schmidt’s monograph [83]. A simple illustration is the construction by Einsiedler and the author [23] of systems with any prescribed order of mixing. When such systems fail to be mixing of all orders, they fail in a very specific way – along dilates of a specific shape (a finite subset of Zd ). In the example above, the shape that fails to mix is f(0; 0); (1; 0); (0; 1)g. This gives an order of mixing as detected by shapes; computing this is in principle an algebraic problem. On the other hand, there is a more natural definition of the order of mixing, namely the largest k for which (12) holds; computing this is in principle a Diophantine problem. A conjecture emerged (formulated explicitly by Schmidt [84]) that for any algebraic dynamical system, if every set of cardinality r > 2 is a mixing shape, then the system is mixing on r sets. This question motivated Masser [64] to prove an appropriate analogue of the S-unit theorem on the number of solutions to (13) in positive characteristic as follows. Let H be a multiplicative group and fix n 2 N. An infinite subset A H n is called broad if it has both of the following properties:  if h 2 H and 1  j  n, then there are at most finitely many (a1 ; : : : ; a n ) in A with a j D h;  if n > 2, h 2 H and 1  i < j  n then there are at most finitely many (a1 ; : : : ; a n ) 2 H with a i a1 j D h. Then Masser’s theorem says the following. Let K be a field of characteristic p > 0, let G be a finitely-generated subgroup of K and suppose that the equation a1 x1 C    C a n x n D 1 has a broad set of solutions (x1 ; : : : ; x n ) 2 G n for some constants a1 ; : : : ; a n 2 K . Then there is an m  n, con-

k b1 g1k C b2 g2k C    C b m g m D1:

The proof that shapes detect the order of mixing in algebraic dynamics then proceeds much as in the connected case. Future Directions The interaction between ergodic theory, number theory and combinatorics continues to expand rapidly, and many future directions of research are discussed in the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory,  Ergodic Theory: Rigidity and  Ergodic Theory: Recurrence. Some of the directions most relevant to the examples discussed in this article include the following. The recent developments mentioned in Subsect. “Sets of Primes” clearly open many exciting prospects involving finding new structures in arithmetically significant sets (like the primes). The original conjecture of Erd˝os and P 1 Turán [26] asked if a2A N a D 1 is sufficient to force the set A to contain arbitrary long arithmetic progressions, and remains open. This would of course imply both Szemerédi’s theorem [97] and the result of Green and Tao [39] on arithmetic progressions in the primes. More generally, it is clear that there is still much to come from the dialogue subsuming the four parallel proofs of Szemerédi’s: one by purely combinatorial methods, one by ergodic theory, one by hypergraph theory, and one by Fourier analysis and additive combinatorics. For an overview, see the survey papers of Tao [98,99,100]. In the context of the orbit-counting results in Sect. “Orbit-Counting as an Analogous Development”, a natural problem is to on the one hand obtain finer asymptotics with better control of the error terms, and on the other to extend the situations that can be handled. In particular, relaxing the hypotheses related to hyperbolicity (or negative curvature) is a constant challenge. Bibliography Primary Literature 1. Adler RL, Weiss B (1970) Similarity of automorphisms of the torus. Memoirs of the American Mathematical Society, No. 98. American Mathematical Society, Providence

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2. Anosov DV (1967) Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat Inst Steklov 90:209 3. Arnol’d VI, Avez A (1968) Ergodic problems of classical mechanics. Translated from the French by A Avez. Benjamin Inc, New York 4. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. Annals of Mathematics Studies, No 53. Princeton University Press, Princeton 5. Baker A (1990) Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2nd edn 6. Benford F (1938) The law of anomalous numbers. Proc Amer Philos Soc 78:551–572 7. Bergelson V (1996) Ergodic Ramsey theory – an update. In: Ergodic theory of Zd actions (Warwick, 1993–1994), vol 228. London Math Soc Lecture Note Ser. Cambridge University Press, Cambridge, pp 1–61 8. Bergelson V (2000) Ergodic theory and Diophantine problems. In: Topics in symbolic dynamics and applications (Temuco, 1997), vol 279. London Math Soc Lecture Note Ser. Cambridge University Press, Cambridge, pp 167–205 9. Bergelson V (2003) Minimal idempotents and ergodic Ramsey theory. In: Topics in dynamics and ergodic theory, vol 310. London Math Soc, Lecture Note Series, Cambridge University Press, Cambridge, pp 8–39 10. Bergelson V (2006) Combinatorial and Diophantine applications of ergodic theory. In: Handbook of dynamical systems, vol 1B. Elsevier, Amsterdam, pp 745–869 11. Bergelson V, Leibman A (1996) Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J Amer Math Soc 9(3):725–753 12. Bergelson V, McCutcheon R (2000) An ergodic IP polynomial Szemerédi theorem. Mem Amer Math Soc 146(695):viii–106 13. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Natl Acad Sci USA 17:656–660 14. Bohl P (1909) Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J Math 135:189–283 15. Borel E (1909) Les probabilités denombrables et leurs applications arithmetiques. Rend Circ Math Palermo 27:247–271 16. Bourgain J (1988) An approach to pointwise ergodic theorems. In: Geometric aspects of functional analysis (1986/87), vol 1317. Lecture Notes in Math, pp 204–223. Springer, Berlin 17. Bourgain J (1988) On the maximal ergodic theorem for certain subsets of the integers. Israel J Math 61(1):39–72 18. Bowen R (1970) Markov partitions for Axiom A diffeomorphisms. Amer J Math 92:725–747 19. Bowen R (1972) The equidistribution of closed geodesics. Amer J Math 94:413–423 20. Bowen R (1973) Symbolic dynamics for hyperbolic flows. Amer J Math 95:429–460 21. Chothi V, Everest G, Ward T (1997) S-integer dynamical systems: periodic points. J Reine Angew Math 489:99–132 22. Dolgopyat D (1998) On decay of correlations in Anosov flows. Ann Math 147(2):357–390 23. Einsiedler M, Ward T (2003) Asymptotic geometry of nonmixing sequences. Ergodic Theory Dyn Syst 23(1):75–85 24. Ellis R (1969) Lectures on topological dynamics. Benjamin Inc, New York

˝ P (1949) On a new method in elementary number the25. Erdos ory which leads to an elementary proof of the prime number theorem. Proc Natl Acad Sci USA 35:374–384 ˝ P, Turán P (1936) On some sequences of integers. J Lon26. Erdos don Math Soc 11:261–264 27. Everest G, Miles R, Stevens S, Ward T (2007) Orbit-counting in non-hyperbolic dynamical systems. J Reine Angew Math 608:155–182 28. Evertse JH, Schlickewei HP (1999) The absolute subspace theorem and linear equations with unknowns from a multiplicative group. In: Number theory in progress, vol 1 (ZakopaneKo´scielisko, 1997). de Gruyter, Berlin, pp 121–142 29. Furstenberg H (1961) Strict ergodicity and transformation of the torus. Amer J Math 83:573–601 30. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Analyse Math 31:204–256 31. Furstenberg H, Katznelson Y (1979) An ergodic Szemerédi theorem for commuting transformations. J Analyse Math 34:275–291 32. Furstenberg H, Katznelson Y (1985) An ergodic Szemerédi theorem for IP-systems and combinatorial theory. J Analyse Math 45:117–168 33. Furstenberg H, Katznelson Y, Ornstein D (1982) The ergodic theoretical proof of Szemerédi’s theorem. Bull Amer Math Soc 7(3):527–552 34. Furstenberg H, Weiss B (1979) Topological dynamics and combinatorial number theory. J Analyse Math 34:61–85 35. Gel0 fond AO (1960) Transcendental and algebraic numbers. Translated from the first Russian edition by Leo F Boron. Dover Publications, New York 36. Goldfeld D (2004) The elementary proof of the prime number theorem: an historical perspective. In: Number theory (2003). Springer, New York, pp 179–192 37. Goldston DA, Yıldırım CY (2005) Small gaps between primes I. arXiv:math.NT/0504336 38. Gowers WT (2007) Hypergraph regularity and the multidimensional Szemerédi Theorem. Ann of Math 166:897–946 39. Green B, Tao T (2004) The primes contain arbitrarily long arithmetic progressions. arXiv:math.NT/0404188 40. Green B, Tao T (2006) Linear equations in primes. arXiv: math.NT/0606088 41. Hejhal DA (1976) The Selberg trace formula and the Riemann zeta function. Duke Math J 43(3):441–482 42. Hill TP (1995) Base-invariance implies Benford’s law. Proc Amer Math Soc 123(3):887–895 43. Hindman N (1974) Finite sums from sequences within cells of a partition of N. J Comb Theory Ser A 17:1–11 44. Hlawka E (1964) Discrepancy and uniform distribution of sequences. Compositio Math 16:83–91 45. Host B, Kra B (2005) Convergence of polynomial ergodic averages. Israel J Math 149:1–19, Probability in mathematics 46. Host B, Kra B (2005) Nonconventional ergodic averages and nilmanifolds. Ann Math 161(1):397–488 47. Huber H (1959) Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. Math Ann 138:1–26 48. Katok A (1980) Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst Hautes Études Sci Publ Math (51):137–173 49. Katsuda A, Sunada T (1990) Closed orbits in homology classes. Inst Hautes Études Sci Publ Math (71):5–32

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50. Khinchin AI (1964) Continued fractions. The University of Chicago Press, Chicago 51. Knieper G (1997) On the asymptotic geometry of nonpositively curved manifolds. Geom Funct Anal 7(4):755–782 52. Koopman B (1931) Hamiltonian systems and transformations in Hilbert spaces. Proc Natl Acad Sci USA 17:315–318 53. Kra B (2006) The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view. Bull Amer Math Soc 43(1):3–23 54. Kuz’min RO (1928) A problem of Gauss. Dokl Akad Nauk, pp 375–380 55. Lalley SP (1987) Distribution of periodic orbits of symbolic and Axiom A flows. Adv Appl Math 8(2):154–193 56. Lalley SP (1988) The “prime number theorem” for the periodic orbits of a Bernoulli flow. Amer Math Monthly 95(5):385–398 57. Ledrappier F (1978) Un champ markovien peut étre d’entropie nulle et mélangeant. C R Acad Sci Paris Sér A-B 287(7):A561–A563 58. Leibman A (2005) Convergence of multiple ergodic averages along polynomials of several variables. Israel J Math 146:303– 315 59. Levy P (1929) Sur les lois de probabilité dont dependent les quotients complets et incomplets d’une fraction continue. Bull Soc Math France 57:178–194 60. Levy P (1936) Sur quelques points de la théorie des probabilités dénombrables. Ann Inst H Poincaré 6(2):153–184 61. Lind DA (1982) Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory Dyn Syst 2(1):49–68 62. Margulis GA (1969) Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funkcional Anal i Priložen 3(4):89–90 63. Margulis GA (2004) On some aspects of the theory of Anosov systems. Springer Monographs in Mathematics. Springer, Berlin. 64. Masser DW (2004) Mixing and linear equations over groups in positive characteristic. Israel J Math 142:189–204 65. Mertens F (1874) Ein Beitrag zur analytischen Zahlentheorie. J Reine Angew Math 78:46–62 66. Nagle B, Rödl V, Schacht M (2006) The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms 28(2):113–179 67. Newcomb S (1881) Note on the frequency of the use of digits in natural numbers. Amer J Math 4(1):39–40 68. Noorani MS (1999) Mertens’ theorem and closed orbits of ergodic toral automorphisms. Bull Malaysian Math Soc 22(2):127–133 69. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116– 136 70. Parry W (1969) Ergodic properties of affine transformations and flows on nilmanifolds. Amer J Math 91:757–771 71. Parry W (1983) An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Israel J Math 45(1):41–52 72. Parry W (1984) Bowen’s equidistribution theory and the Dirichlet density theorem. Ergodic Theory Dyn Syst 4(1):117–134 73. Parry W, Pollicott M (1983) An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann Math 118(3):573–591 74. Poincaré H (1890) Sur le probléme des trois corps et les équations de la Dynamique. Acta Math 13:1–270

75. Pollicott M, Sharp R (1998) Exponential error terms for growth functions on negatively curved surfaces. Amer J Math 120(5):1019–1042 76. Rado R (1933) Studien zur Kombinatorik. Math Z 36(1):424– 470 77. Ratner M (1973) Markov partitions for Anosov flows on n-dimensional manifolds. Israel J Math 15:92–114 78. Rohlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 79. Roth K (1952) Sur quelques ensembles d’entiers. C R Acad Sci Paris 234:388–390 80. Sárközy A (1978) On difference sets of sequences of integers. III. Acta Math Acad Sci Hungar 31(3-4):355–386 81. Schlickewei HP (1990) S-unit equations over number fields. Invent Math 102(1):95–107 82. Schmidt K (1989) Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pacific J Math 137(2):371–385 83. Schmidt K (1995) Dynamical systems of algebraic origin, vol 128. Progress in Mathematics. Birkhäuser, Basel 84. Schmidt K (2001) The dynamics of algebraic Zd -actions. In: European Congress of Mathematics, vol I (Barcelona, 2000), vol 201. Progress in Math. Birkhäuser, Basel, pp 543–553 85. Schmidt K, Ward T (1993) Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent Math 111(1):69– 76 86. Schmidt WM (1972) Norm form equations. Ann Math 96(2):526–551 87. Selberg A (1949) An elementary proof of the prime-number theorem. Ann Math 50(2):305–313 88. Selberg A (1956) Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J Indian Math Soc 20:47–87 89. Sharp R (1991) An analogue of Mertens’ theorem for closed orbits of Axiom A flows. Bol Soc Brasil Mat 21(2):205–229 90. Sharp R (1993) Closed orbits in homology classes for Anosov flows. Ergodic Theory Dyn Syst 13(2):387–408 ´ W (1910) Sur la valeur asymptotique d’une certaine 91. Sierpinski somme. Bull Intl Acad Polonmaise des Sci et des Lettres (Cracovie), pp 9–11 92. Sina˘ı JG (1966) Asymptotic behavior of closed geodesics on compact manifolds with negative curvature. Izv Akad Nauk SSSR Ser Mat 30:1275–1296 93. Sina˘ı JG (1968) Construction of Markov partitionings. Funkcional Anal i Priložen 2(3):70–80 94. Silverman JH (2007) The Arithmetic of Dynamical Systems, vol 241. Graduate Texts in Mathematics. Springer, New York 95. Smale S (1967) Differentiable dynamical systems. Bull Amer Math Soc 73:747–817 96. Szemerédi E (1969) On sets of integers containing no four elements in arithmetic progression. Acta Math Acad Sci Hungar 20:89–104 97. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:199–245 98. Tao T (2005) The dichotomy between structure and randomness, arithmetic progressions, and the primes. arXiv: math/0512114v2 99. Tao T (2006) Arithmetic progressions and the primes. Collect Math, vol extra, Barcelona, pp 37–88 100. Tao T (2007) What is good mathematics? arXiv:math/ 0702396v1

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101. Tao T, Ziegler T (2006) The primes contain arbitrarily long polynomial progressions. arXiv:math.NT/0610050 102. van der Poorten AJ, Schlickewei HP (1991) Additive relations in fields. J Austral Math Soc Ser A 51(1):154–170 103. van der Waerden BL (1927) Beweis einer Baudet’schen Vermutung. Nieuw Arch Wisk 15:212–216 104. van der Waerden BL (1971) How the proof of Baudet’s conjecture was found. In: Studies in Pure Mathematics (Presented to Richard Rado). Academic Press, London, pp 251–260 105. von Neumann J (1932) Proof of the quasi-ergodic hypothesis. Proc Natl Acad Sci USA 18:70–82 106. Waddington S (1991) The prime orbit theorem for quasihyperbolic toral automorphisms. Monatsh Math 112(3):235–248 107. Ward T (1998) Almost all S-integer dynamical systems have many periodic points. Ergodic Theory Dyn Syst 18(2):471–486 108. Weil A (1967) Basic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 144. Springer, New York 109. Weyl H (1910) Über die Gibbssche Erscheinung und verwandte Konvergenzphänomene. Rendiconti del Circolo Matematico di Palermo 30:377–407 110. Weyl H (1916) Uber die Gleichverteilung von Zahlen mod Eins. Math Ann 77:313–352 111. Wiener N (1932) Tauberian theorems. Ann Math 33(1):1–100

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Ergodic Theory, Introduction to

Ergodic Theory, Introduction to BRYNA KRA Northwestern University, Evanston, USA Ergodic theory lies at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Problems, techniques, and results are related to many other areas of mathematics, and ergodic theory has had applications both within mathematics and to numerous other branches of science. Ergodic theory has particularly strong overlap with other branches of dynamical systems; to clarify what distinguishes it from other areas of dynamics, we start with a quick overview of dynamical systems. Dynamical systems is the study of systems that evolve with time. The evolution of a dynamical system is given by some fixed rule that determines the states of the system a short time into the future, given only the present states. Reflecting the origins of the subject in celestial mechanics, the set of states through which the system evolves with time is called an orbit. Many important concepts in dynamical systems are related to understanding the orbits in the system: Do the orbits fill out the entire space? Do orbits collapse? Do orbits return to themselves? What are statistical properties of the orbits? Are orbits stable under perturbation? For simple dynamical systems, knowing the individual orbits is often sufficient to answer such questions. However, in most dynamical systems it is impossible to write down explicit formulae for orbits, and even when one can, many systems are too complicated to be understood just in terms of individual orbits. The orbits may only be known approximately, some orbits may appear to be random while others exhibit regular behavior, and varying the parameters defining the system may give rise to qualitatively different behaviors. The various branches of dynamical systems have been developed for understanding long term properties of the orbits. To make the notion of a dynamical system more precise, let X denote the collection of all states of the system. The evolution of these states is given by some fixed rule T : X ! X, dictating where each state x 2 X is mapped. An application of the transformation T : X ! X corresponds to the passage of a unit of time and for a positive integer n, the map T n D T ı T ı : : : ı T denotes the composition of T with itself taken n times. Given a state x 2 X, the orbit of the point x under the transformation T is the collection of iterates x; Tx; T 2 x; : : : of the state x. Thus the single transformation T generates a semigroup of transformations acting on X, by considering the

powers T n . More generally, one can consider a family of transformations fTt : t 2 Rg with each Tt : X ! X. Assuming that T0 (x) D x and that TtCs (x) D Tt (Ts (x)) for all states x 2 X and all real t and s, this models the evolution of continuous time in a system. Autonomous differential equations are examples of such continuous time systems. In almost all cases of interest, the space X has some underlying structure which is preserved by the transformation T. Different underlying structures X and different properties of the transformation T give rise to different branches of dynamical systems. When X is a smooth manifold and T : X ! X is a differentiable mapping, one is in the framework of differentiable dynamics. When X is a topological space and T : X ! X is a continuous map, one is in the framework of topological dynamics. When X is a measure space and T : X ! X is a measure preserving map, one is in the framework of ergodic theory. These categories are not mutually exclusive, and the relations among them are deep and interesting. Some of these relations are explored in the articles  Topological Dynamics,  Symbolic Dynamics, and  Smooth Ergodic Theory. To further explain the role of ergodic theory, a few definitions are needed. The state space X is assumed to be a measure space, endowed with a  -algebra B of measurable sets and a measure . The measure  assigns each set B 2 B a non-negative number (its measure), and usually one assumes that (X) D 1 (thus  is a probability measure). The transformation T : X ! X is assumed to be a measurable and measure preserving map: For all B 2 B, (T 1 B) D (B). The quadruple (X; B; ; T) is called a measure preserving system. For the precise definitions and background on measure preserving transformations, see the article  Measure Preserving Systems. An extensive discussion of examples of measure preserving systems and basic constructions used in ergodic theory is given in  Ergodic Theory: Basic Examples and Constructions. The origins of ergodic theory are in the nineteenth century work of Boltzmann on the foundations of statistical mechanics. Boltzmann hypothesized that for large systems of interacting particles in equilibrium, the “time average” is equal to the “space average”. This question can be reformulated in the context of modern terminology. Assume that (X; B; ; T) is a measure preserving system and that f : X ! R is some measurement taken on the system. Thus if x 2 X is a state, evaluating the sequence f (x) ; f (Tx) ; f (T 2 x); : : : can be viewed as successive values of this measurement. Boltzmann’s question can be phrased as: Under what conditions is the time mean equal to the space mean? In short,

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when does lim

N!1

N1 1 X f (T n x) D N nD0

Z f d ? X

Boltzmann hypothesized that if orbits went “everywhere” in the space, then such a conclusion would hold. The study of the equality of space and time averages has been a major direction of research in ergodic theory. The long term behavior of the average lim

N!1

N1 1 X f (T n x) ; N nD0

and especially the existence of this limit, is a basic question. Roughly speaking, the ergodic theorem states that starting at almost any initial point, the distribution of its iterates obeys some asymptotic law. This, and more general convergence questions, are addressed in the article  Ergodic Theorems. Perhaps the earliest result in ergodic theory is the Poincaré Recurrence Theorem: In a finite measure space, some iterate of any set with positive measure intersects the set itself in a set of positive measure. More generally, the qualitative behavior of orbits is used to understand conditions under which the time average is equal to the space average of the system (see the article  Ergodic Theory: Recurrence). If the time average is equal almost everywhere to the space average, then the system is said to be ergodic. Ergodicity is a key notion, giving a simple expression for the time average of an arbitrary function. Moreover, using the Ergodic Decomposition Theorem, the study of arbitrary measure preserving systems can be reduced to ergodic ones. Ergodicity and related properties of a system are discussed in the article  Ergodicity and Mixing Properties. Another central problem in ergodic theory is the classification of measure preserving systems. There are various notions of equivalence, and a classical approach to checking if systems are equivalent is finding invariants that are preserved under the equivalence. This subject, including an introduction to Ornstein Theory, is covered in the article  Isomorphism Theory in Ergodic Theory. A map T : X ! X determines an associated unitary operator U D U T defined on L2 (X) by U T f (x) D f (Tx) : There are also numerical invariants that can be assigned to

a system, for example entropy, and this is discussed in the article  Entropy in Ergodic Theory. When two systems are not equivalent, one would like to understand what properties they do have in common. An essential tool in such a classification is the notion of joinings (see the article  Joinings in Ergodic Theory). Roughly speaking, a joining is a way of embedding two systems in the same space. When this can be done in a nontrivial manner, one obtains information on properties shared by the systems. Some systems have predictable behavior and can be classified according to the behavior of individual points and their iterates. Others have behavior that is too complex or unpredictable to be understood on the level of orbits. Ergodic theory provides a statistical understanding of such systems and this is discussed in the article  Chaos and Ergodic Theory. A prominent role in chaotic dynamical systems is played by one dimensional Gibbs measure and by equilibrium states and (see the article  Pressure and Equilibrium States in Ergodic Theory). Rigidity theory addresses the opposite case, studying what kinds of properties in a system are obstructions to general chaotic behavior. The role of ergodic theory in this area is discussed in the article  Ergodic Theory: Rigidity. Another important class of systems arises when one relaxes the condition that the transformation T preserves the measure of sets on X, only requiring that the transformation preserve the negligible sets. Such systems are discussed in  Joinings in Ergodic Theory. Ergodic theory has seen a burst of recent activity, and most of this activity comes from interaction with other fields. Historically, ergodic theory has interacted with numerous fields, including other areas of dynamics, probability, statistical mechanics, and harmonic analysis. More recently, ergodic theory and its techniques have been imported into number theory and combinatorics, proving new results that have yet to be proved by other methods, and in turn, combinatorial problems have given rise to new areas of research within ergodic theory itself. Problems related to Diophantine approximation are discussed in  Ergodic Theory on Homogeneous Spaces and Metric Number Theory and  Ergodic Theory: Rigidity and ones related to combinatorial problems are addressed in  Ergodic Theory: Interactions with Combinatorics and Number Theory and in  Ergodic Theory: Recurrence. Interaction with problems that are geometric in nature, in particular dimension theory, is discussed in  Ergodic Theory: Fractal Geometry.

Ergodic Theory: Non-singular Transformations

Ergodic Theory: Non-singular Transformations ALEXANDRE I. DANILENKO1, CESAR E. SILVA 2 1 Institute for Low Temperature Physics & Engineering, Ukrainian National Academy of Sciences, Kharkov, Ukraine 2 Department of Mathematics, Williams College, Williamstown, USA Article Outline Glossary Definition of the Subject Basic Results Panorama of Examples Mixing Notions and multiple recurrence Topological Group Aut(X, ) Orbit Theory Smooth Nonsingular Transformations Spectral Theory for Nonsingular Systems Entropy and Other Invariants Nonsingular Joinings and Factors Applications. Connections with Other Fields Concluding Remarks Bibliography Glossary Nonsingular dynamical system Let (X; B; ) be a standard Borel space equipped with a -finite measure. A Borel map T : X ! X is a nonsingular transformation of X if for any N 2 B, (T 1 N) D 0 if and only if (N) D 0. In this case the measure  is called quasi-invariant for T; and the quadruple (X; B; ; T) is called a nonsingular dynamical system. If (A) D (T 1 A) for all A 2 B then  is said to be invariant under T or, equivalently, T is measure-preserving. Conservativeness T is conservative if for all sets A of positive measure there exists an integer n > 0 such that (A \ T n A) > 0. Ergodicity T is ergodic if every measurable subset A of X that is invariant under T (i. e., T 1 A D A) is either null or -conull. Equivalently, every Borel function f : X ! R such that f ı T D f is constant a. e. Types II, II1 , II1 and III Suppose that  is non-atomic and T ergodic (and hence conservative). If there exists a -finite measure  on B which is equivalent to  and invariant under T then T is said to be of type II. It is easy to see that  is unique up to scaling. If  is finite then T is of type II 1 . If  is infinite then T is of type II1 . If T is not of type II then T is said to be of type III.

Definition of the Subject An abstract measurable dynamical system consists of a set X (phase space) with a transformation T : X ! X (evolution law or time) and a finite or -finite measure  on X that specifies a class of negligible subsets. Nonsingular ergodic theory studies systems where T respects  in a weak sense: the transformation preserves only the class of negligible subsets but it may not preserve . This survey is about dynamics and invariants of nonsingular systems. Such systems model ‘non-equilibrium’ situations in which events that are impossible at some time remain impossible at any other time. Of course, the first question that arises is whether it is possible to find an equivalent invariant measure, i. e. pass to a hidden equilibrium without changing the negligible subsets? It turns out that there exist systems which do not admit an equivalent invariant finite or even  -finite measure. They are of our primary interest here. In a way (Baire category) most of systems are like that. Nonsingular dynamical systems arise naturally in various fields of mathematics: topological and smooth dynamics, probability theory, random walks, theory of numbers, von Neumann algebras, unitary representations of groups, mathematical physics and so on. They also can appear in the study of probability preserving systems: some criteria of mild mixing and distality, a problem of Furstenberg on disjointness, etc. We briefly discuss this in Sect. “Applications. Connections with Other Fields”. Nonsingular ergodic theory studies all of them from a general point of view:    

What is the qualitative nature of the dynamics? What are the orbits? Which properties are typical withing a class of systems? How do we find computable invariants to compare or distinguish various systems?

Typically there are two kinds of results: some are extensions to nonsingular systems of theorems for finite measure-preserving transformations (for instance, the entire Sect. “Basic Results”) and the other are about new properly ‘nonsingular’ phenomena (see Sect. “Mixing Notions and Multiple Recurrence” or Sect. “Orbit Theory”). Philosophically speaking, the dynamics of nonsingular systems is more diverse comparatively with their finite measurepreserving counterparts. That is why it is usually easier to construct counterexamples than to develop a general theory. Because of shortage of space we concentrate only on invertible transformations, and we have not included as many references as we had wished. Nonsingular endomorphisms and general group or semigroup actions are practically not considered here (with some exceptions in

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Sect. “Applications. Connections with Other Fields” devoted to applications). A number of open problems are scattered through the entire text. We thank J. Aaronson, J.R. Choksi, V.Ya. Golodets, M. Lema´nczyk, F. Parreau, E. Roy for useful remarks. Basic Results This section includes the basic results involving conservativeness and ergodicity as well as some direct nonsingular counterparts of the basic machinery from classic ergodic theory: mean and pointwise ergodic theorems, Rokhlin lemma, ergodic decomposition, generators, Glimm–Effros theorem and special representation of nonsingular flows. The historically first example of a transformation of type III (due to Ornstein) is also given here with full proof. Nonsingular Transformations In this paper we will consider only invertible nonsingular transformations, i. e. those which are bijections when restricted to an invariant Borel subset of full measure. Thus when we refer to a nonsingular dynamical system (X; B; ; T) we shall assume that T is an invertible nonsingular transformation. Of course, each measure  on B which is equivalent to , i. e.  and  have the same null sets, is also quasi-invariant under T. In particular, since  is -finite, T admits an equivalent quasi-invariant probability measure. For each i 2 Z, we denote by ! i or ! i the Radon–Nikodym derivative d( ı T i )/d 2 L1 (X; ). The derivatives satisfy the cocycle equation ! iC j (x) D ! i (x) ! j (T i x) for a. e. x and all i; j 2 Z. Basic Properties of Conservativeness and Ergodicity A measurable set W is said to be wandering if for all i; j  0 with i ¤ j, T i W \ T  j W D ;. Clearly, if T has a wandering set of positive measure then it cannot be conservative. A nonsingular transformation T is incompressible if whenever T 1 C C, then (C n T 1 C) D 0. A set W of positive measure is said to be weakly wandering if there is a sequence n i ! 1 such that T n i W \ T n j W D ; for all i ¤ j. Clearly, a finite measure-preserving transformation cannot have a weakly wandering set. Hajian and Kakutani [83] showed that a nonsingular transformation T admits an equivalent finite invariant measure if and only if T does not have a weakly wandering set. Proposition 1 (see e. g. [123]) Let (X; B; ; T) be a nonsingular dynamical system. The following are equivalent: (i) T is conservative. S n A) D 0. (ii) For every measurable set A, (A n 1 nD1 T

(iii) T is incompressible. (iv) Every wandering set for T is null. Since any finite measure-preserving transformation is incompressible, we deduce that it is conservative. This is the statement of the classical Poincaré recurrence lemma. If T is a conservative nonsingular transformation of (X; B; ) and A 2 B a subset of positive measure, we can define an induced transformation T A of the space (A; B \ A;   A) by setting TA x :D T n x if n D n(x) is the smallest natural number such that T n x 2 A. T A is also conservative. As shownRin [179], if (X) D 1 and T is conservative and erP !(x) d(x) D 1, which is a nonsingular godic, A n(x)1 iD0 version of the well-known Kaçs formula. Theorem 2 (Hopf Decomposition, see e. g. [3]) Let T be a nonsingular transformation. Then there exist disjoint invariant sets C; D 2 B such that X D C t D, T reF1 n stricted to C is conservative, and D D nD1 T W, 1 where W is a wandering set. If f 2 L (X; ), f > 0, Pn1 i then C D fx : iD0 f (T x) ! i (x) D 1 a. e.g and D D P n1 i fx : iD0 f (T x) ! i (x) < 1 a. e.g. The set C is called the conservative part of T and D is called the dissipative part of T. If T is ergodic and  is non-atomic then T is automatically conservative. The translation by 1 on the group Z furnished with the counting measure is an example of an ergodic non-conservative (infinite measure-preserving) transformation. Proposition 4 Let (X; B; ; T) be a nonsingular dynamical system. The following are equivalent: (i) T is conservative and ergodic. (ii) For every set A of positive measure, (X n S1 n A) D 0. (In this case we will say A sweeps nD1 T out.) (iii) For every measurable set A of positive measure and for a. e. x 2 X there exists an integer n > 0 such that T n x 2 A. (iv) For all sets A and B of positive measure there exists an integer n > 0 such that (T n A \ B) > 0. (v) If A is such that T 1 A A, then (A) D 0 or (Ac ) D 0. This survey is mainly about systems of type III. For some time it was not quite obvious whether such systems exist at all. The historically first example was constructed by Ornstein in 1960. Example 5 (Ornstein [149]) Let A n D f0; 1; : : : ; ng, n (0) D 0:5 and n (i) D 1/(2n) for 0 < i  n and all n 2 N. Denote by (X; ) the infinite product probability

Ergodic Theory: Non-singular Transformations

N space 1 nD1 (A n ;  n ). Of course,  is non-atomic. A point of X is an infinite sequence x D (x n )1 nD1 with x n 2 A n for all n. Given a1 2 A1 ; : : : ; a n 2 A n , we denote the cylinder fx D (x i )1 iD1 2 X : x1 D a1 ; : : : ; x n D a n g by [a1 ; : : : ; a n ]. Define a Borel map T : X ! X by setting 8 if i < l(x) ˆ <0; (1) (Tx) i D x i C 1; if i D l(x) ˆ : if i > l(x) ; xi ; where l(x) is the smallest number l such that x l ¤ l. It is easy to verify that T is a nonsingular transformation of (X; ) and 1 Y n ((Tx)n ) d ı T (x) D d n (x n ) nD1 8 < (l(x)  1)! ; if x l (x) D 0 l(x) D : (l(x)  1)! ; if x l (x) ¤ 0 :

We prove that T is of type III by contradiction. Suppose that there exists a T-invariant -finite measure  equivalent to . Let ' :D d/d. Then 

! i (x) D '(x) '(T i x)1 for a. a. x 2 X and all i 2 Z: (2) Fix a real C > 1 such that the set EC :D ' 1 ([C 1 ; C])

X is of positive measure. By a standard approximation argument, for each sufficiently large n, there is a cylinder [a1 ; : : : ;a n ] such that (EC \ [a1 ; : : : ; a n ]) > 0:9([a1 ; : : : ; a n ]). Since nC1 (0) D 0:5, it follows that (EC \ [a1 ; : : : ; a n ; 0]) > 0:8([a1 ; : : : ; a n ; 0]). Moreover, by the pigeon hole principle there is 0 < i  n C 1 with (EC \ [a1 ; : : : ; a n ; i]) > 0:8([a1 ; : : : ; a n ; i]). Find N n > 0 such that T N n [a1 ; : : : ; a n ; 0] D [a1 ; : : : ; a n ; i]. Since ! Nn is constant on [a1 ; : : : ; a n ; 0], there is a subset E0 EC \ [a1 ; : : : ; a n ; 0] of positive measure such that T N n E0 EC \[a1 ; : : : ; a n ; i]. Moreover, ! Nn (x) D nC1 (i)/nC1 (0) D (n C 1)1 for a. a. x 2 [a1 ; : : : ; a n ; 0]. On the other hand,  we deduce from (2) that ! N n (x)  C 2 for all x 2 E0 , a contradiction. Mean and Pointwise Ergodic Theorems. Rokhlin Lemma Let (X; B; ; T) be a nonsingular dynamical system. Define a unitary operator U T of L2 (X; ) by setting s d( ı T)  f ıT: (3) U T f :D d We note that U T preserves the cone of positive functions L2C (X; ). Conversely, every positive unitary operator in

L2 (X; ) that preserves L2C (X; ) equals U T for a -nonsingular transformation T. Theorem 6 (von Neumann mean Ergodic Theorem, see e. g. [3]) If T has no -absolutely continuous T-invariant P i probability, then n1 n1 iD0 U T ! 0 in the strong operator topology. Denote by I the sub--algebra of T-invariant sets. Let E [:jI ] stand for the conditional expectation with Rrespect to I . Note that if T is ergodic, then E [ f jI ] D f d. Now we state a nonsingular analogue of Birkhoff’s pointwise ergodic theorem, due to Hurewicz [105] and in the form stated by Halmos [84]. Theorem 7 (Hurewicz pointwise Ergodic Theorem) If T is conservative, (X) D 1, f ; g 2 L1 (X; ) and g > 0, then n1 X iD0 n1 X

f (T i x) ! i (x) ! g(T i x) ! i (x)

E [ f jI ] as n ! 1 for a. e. x : E [gjI ]

iD0

A transformation T is aperiodic if the T-orbit of a. e. point from X is infinite. The following classical statement can be deduced easily from Proposition 1. Lemma 8 (Rokhlin’s lemma [161]) Let T be an aperiodic nonsingular transformation. For each " > 0 and integer N > 1 there exists a measurable set A such that the sets A; TA; : : : ; T N1 A are disjoint and (A [ TA [    [ T N1 A) > 1  ". This lemma was refined later (for ergodic transformations) by Lehrer and Weiss as follows. Theorem 9 ("-free Rokhlin lemma [132]) Let T be ergodic and  non-atomic. Then for a subset B X and any N S k N (X n B) D X, there is a set A such for which 1 kD0 T that the sets A; TA; : : : ; T N1 A are disjoint and A [ TA [    [ T N1 A  B. S k N (X n B) D X holds of course The condition 1 kD0 T for each B ¤ X if T is totally ergodic, i. e. T p is ergodic for any p, or if N is prime. Ergodic Decomposition A proof of the following theorem may be found in [3]. Theorem 10 (Ergodic Decomposition Theorem) Let T be a conservative nonsingular transformation on a standard probability space (X; B; ). There there exists a standard probability space (Y; ; A) and a family of probability measures y on (X; B), for y 2 Y, such that

331

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Ergodic Theory: Non-singular Transformations

(i)

For each A 2 B the map y 7!  y (A) is Borel and for each A 2 B Z (A) D  y (A) d(y) :

(ii) For y; y 0 2 Y the measures y and  y 0 are mutually singular. (iii) For each y 2 Y the transformation T is nonsingular and conservative, ergodic on (X; B;  y ). (iv) For each y 2 Y d y ı T d ı T  y -a. e : D d d y (v) (Uniqueness) If there exists another probability space (Y 0 ;  0 ; A0 ) and a family of probability measures 0y 0 on (X; B), for y 0 2 Y 0 , satisfying (i)–(iv), then there exists a measure-preserving isomorphism : Y ! Y 0 such that  y D 0 y for -a. e. y. It follows that if T preserves an equivalent -finite measure then the system (X; B;  y ; T) is of type II for a. a. y. The space (Y; ; A) is called the space of T-ergodic components. Generators It was shown in [157,162] that a nonsingular transformation T on a standard probability space (X; B; ) has a countable generator, i. e. a countable partition P so that W1 n nD1 T P generates the measurable sets. It was refined by Krengel [126]: if T is of type II1 or III then there exists a generator P consisting of two sets only. Moreover, given a sub--algebra F B such that F

S T F and k>0 T k F D B, the set fA 2 F j (A; X n A) is a generator of Tg is dense in F . It follows, in particular, that T is isomorphic to the shift on f0; 1gZ equipped with a quasi-invariant probability measure. The Glimm–Effros Theorem The classical Bogoliouboff–Krylov theorem states that each homeomorphism of a compact space admits an ergodic invariant probability measure [33]. The following statement by Glimm [76] and Effros [61] is a “nonsingular” analogue of that theorem. (We consider here only a particular case of Z-actions.) Theorem 11 Let X be a Polish space and T : X ! X an aperiodic homeomorphism. Then the following are equivalent: T has a recurrent point x, i. e. x D lim n!1 T n i x for a sequence n1 < n2 <    : (ii) There is an orbit of T which is not locally closed.

(i)

(iii) There is no a Borel set which intersects each orbit of T exactly once. (iv) There is a continuous probability Borel measure  on X such that (X; ; T) is an ergodic nonsingular system. A natural question arises: under the conditions of the theorem how many such  can exists? It turns out that there is a wealth of such measures. To state a corresponding result we first write an important definition. Definition 12 Two nonsingular systems (X; B; ; T) and (X; B0 ; 0 ; T 0 ) are called orbit equivalent if there is a oneto-one bi-measurable map ' : X ! X with 0 ı '  and such that ' maps the T-orbit of x onto the T 0 -orbit of '(x) for a. a. x 2 X. The following theorem was proved in [116,128] and [174]. Theorem 13 Let (X; T) be as in Theorem 11. Then for each ergodic dynamical system (Y; C ; ; S) of type II1 or III, there exist uncountably many mutually disjoint Borel measures  on X such that (X; T; B; ) is orbit equivalent to (Y; C ; ; S). On the other hand, T may not have any finite invariant measure. Indeed, let T be an irrational rotation on the circle T and X a non-empty T-invariant Gı subset of T of full Lebesgue measure. Let (X; T) contain a recurrent point. Then the unique ergodicity of (T ; T) implies that (X; T) has no finite invariant measures. Let T be an aperiodic Borel transformation of a standard Borel space X. Denote by M(T) the set of all ergodic T-nonsingular continuous measures on X. Given  2 M(T), let N() denote the family of all Borel -null subsets. Shelah and Weiss showed [178] T that 2M(T) N() coincides with the collection of all Borel T-wandering sets. Special Representations of Ergodic Flows Nonsingular flows (D R-actions) appear naturally in the study of orbit equivalence for systems of type III (see Sect. “Orbit Theory”). Here we record some basic notions related to nonsingular flows. Let (X; B; ) be a standard Borel space with a  -finite measure  on B. A nonsingular flow on (X; ) is a Borel map S : XR 3 (x; t) 7! S t x 2 X such that S t Ss D S tCs for all s; t 2 R and each St is a nonsingular transformation of (X; ). Conservativeness and ergodicity for flows are defined in a similar way as for transformations. A very useful example of a flow is a flow built under a function. Let (X; B; ; T) be a nonsingular dynamical system and f a positive Borel function on X such that P1 P1 i i iD0 f (T x) D iD0 f (T x) D 1 for all x 2 X. Set

Ergodic Theory: Non-singular Transformations

X f :D f(x; s) : x 2 X; 0  s < f (x)g. Define f to be the restriction of the product measure   Leb on X  R to X f and define, for t  0, ! n1 X f n i S t (x; s) :D T x; s C t  f (T x) ; iD0

where n is the unique integer that satisfies n1 X

f (T i x) < s C t 

iD0

n X

If x ¤ (0; 0; : : : ), we let l(x) be the smallest number l such that the lth coordinate of x is not m l  1. We define a Borel map T : X ! X by (1) if x ¤ (m1 ; m2 ; : : : ) and put Tx :D (0; 0; : : : ) if x D (m1 ; m2 ; : : : ). Of course, T is isomorphic to a rotation on a compact monothetic totally disconnected Abelian group. It is easy to check that T is -nonsingular and 1 Y d ı T n ((Tx)n ) (x) D d n (x n ) nD1

f (T i x) :

iD0

A similar definition applies when t < 0. In particular, f when 0 < s C t < '(x), S t (x; s) D (x; s C t), so that the flow moves the point (x; s) up t units, and when it reaches (x; '(x)) it is sent to (Tx; 0). It can be shown that f S f D (S t ) t2R is a free f -nonsingular flow and that it pref serves  if and only if T preserves  [148]. It is called the flow built under the function ' with the base transformation T. Of course, S f is conservative or ergodic if and only if so is T. Two flows S D (S t ) t2R on (X; B; ) and V D (Vt ) t2R on (Y; C ; ) are said to be isomorphic if there exist invariant co-null sets X 0 X and Y 0 Y and an invertible nonsingular map : X 0 ! Y 0 that interwines the actions of the flows: ı S t D Vt ı on X 0 for all t. The following nonsingular version of Ambrose–Kakutani representation theorem was proved by Krengel [120] and Kubo [130]. Theorem 14 Let S be a free nonsingular flow. Then it is isomorphic to a flow built under a function. Rudolph showed that in the Ambrose–Kakutani theorem one can choose the function ' to take two values. Krengel [122] showed that this can also be assumed in the nonsingular case. Panorama of Examples This section is devoted entirely to examples of nonsingular systems. We describe here the most popular (and simple) constructions of nonsingular systems: odometers, nonsingular Markov odometers, tower transformations, rankone and finite rank systems and nonsingular Bernoulli shifts. Nonsingular Odometers Given a sequence mn of natural numbers, we let A n :Df0; 1;: : : ;m n  1g. Let  n be a probability on An and n (a) > 0 for all a 2 A n . Consider now the infinite product N1 probability space (X; ) :D nD1 (A n ;  n ). Assume that Q1 maxf (a) j a 2 A g D 0. Then  is non-atomic. n n nD1 Given a1 2 A1 ; : : : ; a n 2 A n , we denote by [a1 ; : : : ; a n ] the cylinder x D (x i ) i>0 j x1 D a1 ; : : : ; x n D a n .

D

l (x)1 n (0)  l (x) (x l (x) C 1) Y  l (x) (x l (x) ) n (m n  1) nD1

for a. a. x D (x n )n>0 2 X. It is also easy to verify that T is ergodic. It is called the nonsingular odometer associated to (m n ; n )1 nD1 . We note that Ornstein’s transformation (Example 5) is a nonsingular odometer. Markov Odometers We define Markov odometers as in [54]. An ordered Bratteli diagram B [102] consists of (i)

a vertex set V which is a disjoint union of finite sets V (n) , n  0, V 0 is a singleton; (ii) an edge set E which is a disjoint union of finite sets E (n) , n > 0; (iii) source mappings s n : E (n) ! V (n1) and range mappings r n : E (n) ! V (n) such that s 1 n (v) ¤ ; for all (v) ¤ ; for all v 2 V (n) , n > 0; v 2 V (n1) and r1 n 0 (iv) a partial order on E so that e; e 2 E are comparable if and only if e; e 0 2 E (n) for some n and r n (e) D r n (e 0 ).

A Bratteli compactum X B of the diagram B is the space of infinite paths ˚ x D (x n )n>0 j x n 2 E (n) and r(x n ) D s(x nC1 ) on B. X B is equipped with the natural topology induced Q by the product topology on n>0 E (n) . We will assume always that the diagram is essentially simple, i. e. there is only one infinite path xmax D (x n )n>0 with xn maximal for all n and only one xmin D (x n )n>0 with xn minimal for all n. The Bratteli–Vershik map TB : X B ! X B is defined as follows: Txmax D xmin . If x D (x n )n>0 ¤ xmax then let k be the smallest number such that xk is not maximal. Let yk be a successor of xk . Let (y1 ; : : : ; y k ) be the unique path such that y1 ; : : : ; y k1 are all minimal. Then we let TB x :D (y1 ; : : : ; y k ; x kC1 ; x kC2 ; : : : ). It is easy to see that T B is a homeomorphism of X B . Suppose that we are given (n) a sequence P(n) D (P(v;e)2V n1 E (n) ) of stochastic matrices, i. e. (n) (i) Pv;e > 0 if and only if v D s n (e) and P (n) (n1) . (ii) f e2E (n) js n (e)Dv g Pv;e D 1 for each v 2 V

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For e1 2 E (1) ; : : : ; e n 2 E (n) , let [e1 ; : : : ; e n ] denote the cylinder fx D (x j ) j>0 j x1 D e1 ; : : : ; x n D e n g. Then we define a Markov measure on X B by setting P ([e1 ; : : : ; e n ]) D Ps11 (e 1 );e 1 Ps22 (e 2 );e 2    Psnn (e n );e n for each cylinder [e1 ; : : : ; e n ]. The dynamical system (X B ; P ; TB ) is called a Markov odometer. It is easy to see that every nonsingular odometer is a Markov odometer where the corresponding V (n) are all singletons. Tower Transformations This construction is a discrete analogue of flow under a function. Given a nonsingular dynamical system (X; ; T) and a measurable map f : X ! N, we define a new dynamical system (X f ;  f ; T f ) by setting X f :D f(x; i) 2 X  ZC j 0  i < f (x)g ; d f (x; i) :D d(x) and ( (x; i C 1); if i C 1 < f (x) f T (x; i) :D (Tx; 0); otherwise : Then T f is f -nonsingular and (d f ı T f /d f )(x; i) D (d ı T/d)(x) for a. a. (x; i) 2 X f . This transformation is called the (Kakutani) tower over T with height function f . It is easy to check that T f is conservative if and only if T is conservative; T f is ergodic if and only if T is ergodic; T f is of type III if and only if T is of type III. Moreover, the induced transformation (T f ) Xf0g is isomorphic to T. Given a subset A X of positive measure, T is the tower over the induced transformation T A with the first return time to A as the height function. Rank-One Transformations. Chacón Maps. Finite Rank The definition uses the process of “cutting and stacking”. We construct by induction a sequence of columns Cn . A column Cn consists of a finite sequence of bounded intervals (left-closed, right-open) C n D fI n;0 ; : : : ; I n;h n 1 g of height hn . A column Cn determines a column map TC n that sends each interval I n;i to the interval above it I n;iC1 by the unique orientation-preserving affine map between the intervals. TC n remains undefined on the top interval I n;h n 1 . Set C0 D f[0; 1)g and let fr n > 2g be a sequence of positive integers, let fs n g be a sequence of functions s n : f0; : : : ; r n  1g ! N0 , and let fw n g be a sequence of probability vectors on f0; : : : ; r n  1g. If Cn has been defined, column C nC1 is defined as follows. First “cut” (i. e., subdivide) each interval I n;i in Cn into rn subintervals I n;i [ j]; j D 0; : : : ; r n  1, whose lengths are in the proportions w n (0) : w n (1) :    : w n (r n  1). Next place, for each

j D 0; : : : ; r n  1, s n ( j) new subintervals above I n;h n 1 [ j], all of the same length as I n;h n 1 [ j]. Denote these intervals, called spacers, by S n;0 [ j]; : : : ; S n;s n ( j)1 [ j]. This yields, for each j 2 f0; : : : ; r n  1g, rn subcolumns each consisting of the subintervals I n;0 [ j]; : : : ; I n;h n 1 [ j] followed by the spacers S n;0 [ j]; : : : ; S n;s n ( j)1 [ j] : Finally each subcolumn is stacked from left to right so that the top subinterval in subcolumn j is sent to the bottom subinterval in subcolumn j C 1, for j D 0; : : : ; r n  2 (by the unique orientation-preserving affine map between the intervals). For example, S n;s n (0)1 [0] is sent to I n;0 [1]. This defines a new column C nC1 and new column map TC nC1 , which remains undefined on its top subinterval. Let X be the union of all intervals in all columns and let  be Lebesgue measure restricted to X. We assume that as n ! 1 the maximal length of the intervals in Cn converges to 0, so we may define a transformation T of (X; ) by Tx :D lim n!1 TC n x. One can verify that T is welldefined a. e. and that it is nonsingular and ergodic. T is said to be the rank-one transformation associated with (r n ; w n ; s n )1 nD1 . If all the probability vectors wn are uniform the resulting transformation is measure-preserving. The measure is infinite (-finite) if and only if the total mass of the spacers is infinite. In the case r n D 3 and s n (0) D s n (2) D 0, s n (1) D 1 for all n  0, the associated rank-one transformation is called a nonsingular Chacón map. It is easy to see that every nonsingular odometer is of rank-one (the corresponding maps sn are all trivial). Each rank-one map T is a tower over a nonsingular odometer (to obtain such an odometer reduce T to a column Cn ). A rank N transformation is defined in a similar way. A nonsingular transformation T is said to be of rank N or less if at each stage of its construction there exits N disjoint columns, the levels of the columns generate the  -algebra and the Radon–Nikodym derivative of T is constant on each non-top level of every column. T is said to be of rank N if it is of rank N or less and not of rank N  1 or less. A rank N transformation, N  2, need not be ergodic. Nonsingular Bernoulli Transformations – Hamachi’s Example A nonsingular Bernoulli transformation is a transformation T such that there exists a countable generator P (see Subsect. “Generators”) such that the partitions T n P , n 2 Z, are mutually independent and such that the Radon– Nikodym derivative ! 1 is measurable with respect to the W sub--algebra 0nD1 T n P .

Ergodic Theory: Non-singular Transformations

In [87], Hamachi constructed examples of conservative nonsingular Bernoulli transformations, hence ergodic (see Subsect. “Weak Mixing, Mixing, K-Property”), with a 2-set generating partition that are of type III. Krengel [121] asked if there are of type II1 examples of nonsingular Bernoulli automorphisms and the question remains open. Hamachi’s construction is the left-shift on Q the space X D 1 nD1 f0; 1g. The measure is a product Q1  D nD1 n where n D (1/2; 1/2) for n  0 and for n < 0 n is chosen carefully alternating on large blocks between the uniform measure and different non-uniform measures. Kakutani’s criterion for equivalence of infinite product measures is used to verify that  is nonsingular. Mixing Notions and multiple recurrence The study of mixing and multiple recurrence are central topics in classical ergodic theory [33,70]. Unfortunately, these notions are considerably less ‘smooth’ in the world of nonsingular systems. The very concepts of any kind of mixing and multiple recurrence are not well understood in view of their ambiguity. Below we discuss nonsingular systems possessing a surprising diversity of such properties that seem equivalent but are different indeed. Weak Mixing, Mixing, K-Property Let T be an ergodic conservative nonsingular transformation. A number  2 C is an L1 -eigenvalue for T if there exists a nonzero f 2 L1 so that f ı T D  f a. e. It follows that jj D 1 and f has constant modulus, which we assume to be 1. Denote by e(T) the set of all L1 -eigenvalues of T. T is said to be weakly mixing if e(T) D f1g. We refer to Theorem 2.7.1 in [3] for proof of the following Keane’s ergodic multiplier theorem: given an ergodic probability preserving transformation S, the product transformation T  S is ergodic if and only if S (e(T)) D 0, where  S denotes the measure of (reduced) maximal spectral type of the unitary U S (see (3)). It follows that T is weakly mixing if and only T  S is ergodic for every ergodic probability preserving S. While in the finite measure-preserving case this implies that T  T is ergodic, it was shown in [5] that there exits a weakly mixing nonsingular T with T  T not conservative, hence not ergodic. In [11], a weakly mixing T was constructed with T  T conservative but not ergodic. A nonsingular transformation T is said to be doubly ergodic if for all sets of positive measure A and B there exists an integer n > 0 such that (A \ T n A) > 0 and (A \ T n B) > 0. Furstenberg [70] showed that for finite measure-preserving transformations double ergodicity is equivalent to weak mixing. In [20] it is shown that for nonsingular transformations weak mixing does not imply

double ergodicity and double ergodicity does not imply that T  T is ergodic. T is said to have ergodic index k if the Cartesian product of k copies of T is ergodic but the product of k C 1 copies of T is not ergodic. If all finite Cartesian products of T are ergodic then T is said to have infinite ergodic index. Parry and Kakutani [113] constructed for each k 2 N [ f1g, an infinite Markov shift of ergodic index k. A stronger property is power weak mixing, which requires that for all nonzero integers k1 ; : : : ; kr the product T k 1      T k r is ergodic [47]. The following examples were constructed in [12,36,38]: (i) power weakly mixing rank-one transformations, (ii) non-power weakly mixing rank-one transformations with infinite ergodic index, (iii) non-power weakly mixing rank-one transformations with infinite ergodic index and such that T k 1      T k r are all conservative, k1 ; : : : ; kr 2 Z, of types II1 and III (and various subtypes of III, see Sect. “Orbit Theory”). Thus we have the following scale of properties (equivalent to weak mixing in the probability preserving case), where every next property is strictly stronger than the previous ones: T is weakly mixing ( T is doubly ergodic ( T  T is ergodic ( T  T  T is ergodic (  ( T has infinite ergodic index ( T is power weakly mixing : We also mention a recent example of a power weakly mixing transformation of type II1 which embeds into a flow [46]. We now consider several attempts to generalize the notion of (strong) mixing. Given a sequence of measurable sets fA n g let  k (fA n g) denote the  -algebra generated by A k ; A kC1 ; : : : . A sequence fA n g is said to be remotely trivT ial if 1 kD0  k (fA n g) D f;; Xg mod , and it is semi-remotely trivial if every subsequence contains a subsequence that is remotely trivial. Krengel and Sucheston [124] define a nonsingular transformation T of a  -finite measure space to be mixing if for every set A of finite measure the sequence fT n Ag is semi-remotely trivial, and completely mixing if fT n Ag is semi-remotely trivial for all measurable sets A. They show that T is completely mixing if and only if it is type II1 and mixing for the equivalent finite invariant measure. Thus there are no type III and II1 completely mixing nonsingular transformations on probability spaces. We note that this definition of mixing in infinite

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measure spaces depends on the choice of measure inside the equivalence class (but it is independent if we replace the measure by an equivalent measure with the same collection of sets of finite measure). Hajian and Kakutani showed [83] that an ergodic infinite measure-preserving transformation T is either of zero type: limn!1 (T n A \ A) D 0 for all sets A of finite measure, or of positive type: lim supn!1 (T n A \ A) > 0 for all sets A of finite positive measure. T is mixing if and only if it is of zero type [124]. For 0  ˛  1 Kakutani suggested a related definition of ˛-type: an infinite measure preserving transformation is of ˛-type if lim supn!1 (A \ T n A) D ˛(A) for every subset A of finite measure. In [153] examples of ergodic transformations of any ˛-type and a transformation of not any type were constructed. It may seem that mixing is stronger than any kind of nonsingular weak mixing considered above. However, it is not the case: if T is a weakly mixing infinite measure preserving transformation of zero type and S is an ergodic probability preserving transformation then T  S is ergodic and of zero type. On the other hand, the L1 -spectrum e(T  S) is nontrivial, i. e. T  S is not weakly mixing, whenever S is not weakly mixing. We also note that there exist rank-one infinite measure-preserving transformations T of zero type such that T  T is not conservative (hence not ergodic) [11]. In contrast to that, if T is of positive type all of its finite Cartesian products are conservative [7]. Another result that suggests that there is no good definition of mixing in the nonsingular case was proved recently in [110]. It is shown there that while the mixing finite measure-preserving transformations are measurably sensitive, there exists no infinite measure-preserving system that is measurably sensitive. (Measurable sensitivity is a measurable version of the strong sensitive dependence on initial conditions – a concept from topological theory of chaos.) A nonsingular transformation T of (X; B; ) is called K-automorphism [180] if there exists a sub--algebra F

T WC1 B such that T 1 F F , k0 T k F D f;; Xg, kD0 T k F D B and the Radon–Nikodym derivative d ı T/ d is F -measurable (see also [156] for the case when T is of type II1 ; the authors in [180] required T to be conservative). Evidently, a nonsingular Bernoulli transformation (see Subsect. “Nonsingular Bernoulli Transformations – Hamachi’s Example”) is a K-automorphism. Parry [156] showed that a type II1 K-automorphism is either dissipative or ergodic. Krengel [121] proved the same for a class of Bernoulli nonsingular transformations, and finally Silva and Thieullen extended this result to nonsingular K-automorphisms [180]. It is also shown in [180] that if T is

a nonsingular K-automorphism, for any ergodic nonsingular transformation S, if S  T is conservative, then it is ergodic. It follows that a conservative nonsingular K-automorphism is weakly mixing. However, it does not necessarily have infinite ergodic index [113]. Krengel and Sucheston [124] showed that an infinite measure-preserving conservative K-automorphism is mixing. Multiple and Polynomial Recurrence Let p be a positive integer. A nonsingular transformation T is called p-recurrent if for every subset B of positive measure there exists a positive integer k such that (B \ T k B \    \ T k p B) > 0 : If T is p-recurrent for any p > 0, then it is called multiply recurrent. It is easy to see that T is 1-recurrent if and only if it is conservative. T is called rigid if T n k ! Id for a sequence n k ! 1. Clearly, if T is rigid then it is multiply recurrent. Furstenberg showed [70] that every finite measure-preserving transformation is multiply recurrent. In contrast to that Eigen, Hajian and Halverson [64] constructed for any p 2 N [ f1g, a nonsingular odometer of type II1 which is p-recurrent but not (p C 1)-recurrent. Aaronson and Nakada showed in [7] that an infinite measure preserving Markov shift T is p-recurrent if and only if the product T      T (p times) is conservative. It follows from this and [5] that in the class of ergodic Markov shifts infinite ergodic index implies multiple recurrence. However, in general this is not true. It was shown in [12,45] and [82] that for each p 2 N [ f1g there exist (i) power weakly mixing rank-one transformations and (ii) non-power weakly mixing rank-one transformations with infinite ergodic index which are p-recurrent but not (p C 1)-recurrent (the latter holds when p ¤ 1, of course). A subset A is called p-wandering if (A \ T k A \    pk \T A) D 0 for each k. Aaronson and Nakada established in [7] a p-analogue of Hopf decomposition (see Theorem 2). Proposition 15 If (X; B; ; T) is conservative aperiodic nonsingular dynamical system and p 2 N then X D C p [ D p , where Cp and Dp are T-invariant disjoint subsets, Dp is a countable union of p-wandering sets, T  C p is p-reP k B \   \T d k B) D 1 for current and 1 kD1 (B \ T every B C p . Let T be an infinite measure-preserving transformation and let F be a  -finite factor (i. e., invariant subalgebra)

Ergodic Theory: Non-singular Transformations

of T. Inoue [106] showed that for each p > 0, if T  F is p-recurrent then so is T provided that the extension T ! T  F is isometric. It is unknown yet whether the latter assumption can be dropped. However, partial progress was recently achieved in [140]: if T  F is multiply recurrent then so is T. Let P :D fq 2 Q[t] j q(Z) Z and q(0) D 0g. An ergodic conservative nonsingular transformation T is called p-polynomially recurrent if for every q1 ; : : : ; q p 2 P and every subset B of positive measure there exists k 2 N with (B \ T q 1 (k) B \    \ T q p (k) B) > 0 : If T is p-polynomially recurrent for every p 2 N then it is called polynomially recurrent. Furstenberg’s theorem on multiple recurrence was significantly strengthened in [17], where it was shown that every finite measure-preserving transformation is polynomially recurrent. However, Danilenko and Silva [45] constructed (i)

nonsingular transformations T which are p-polynomially recurrent but not (p C 1)-polynomially recurrent (for each fixed p 2 N), (ii) polynomially recurrent transformations T of type II1 , (iii) rigid (and hence multiply recurrent) transformations T which are not polynomially recurrent. Moreover, such T can be chosen inside the class of rankone transformations with infinite ergodic index. Topological Group Aut(X, ) Let (X; B; ) be a standard probability space and let Aut(X; ) denote the group of all nonsingular transformations of X. Let  be a finite or -finite measure equivalent to ; the subgroup of the -preserving transformations is denoted by Aut0 (X; ). Then Aut(X; ) is a simple group [62] and it has no outer automorphisms [63]. Ryzhikov showed [169] that every element of this group is a product of three involutions (i. e. transformations of order 2). Moreover, a nonsingular transformation is a product of two involutions if and only if it is conjugate to its inverse by an involution. Inspired by [85], Ionescu Tulcea [107] and Chacon and Friedman [21] introduced the weak and the uniform topologies respectively on Aut(X; ). The weak one – we denote it by dw – is induced from the weak operator topology on the group of unitary operators in L2 (X; ) by the embedding T 7! U T (see Subsect. “Mean and Pointwise Ergodic Theorems. Rokhlin Lemma”). Then (Aut(X; ); dw ) is a Polish topological group and Aut0 (X; ) is a closed subgroup of Aut(X; ).

This topology will not be affected if we replace  with any equivalent measure. We note that T n weakly converges  to T if and only if (Tn1 A T 1 A) ! 0 for each A 2 B and d( ı Tn )/d ! d( ı T)/d in L1 (X; ). Danilenko showed in [34] that (Aut(X; ); dw ) is contractible. It follows easily from the Rokhlin lemma that periodic transformations are dense in Aut(X; ). For each p  1, one can also embed Aut(X; ) into the isometry group of L p (X; ) via a formula similar to (3) but with another power of the Radon–Nikodym derivative in it. The strong operator topology on the isometry group induces the very same weak topology on Aut(X; ) for all p  1 [24]. It is natural to ask which properties of nonsingular transformations are typical in the sense of Baire category. The following technical lemma (see see [24,68]) is an indispensable tool when considering such problems. Lemma 16 The conjugacy class of each aperiodic transformation T is dense in Aut(X; ) endowed with the weak topology. Using this lemma and the Hurewicz ergodic theorem Choksi and Kakutani [24] proved that the ergodic transformations form a dense Gı in Aut(X; ). The same holds for the subgroup Aut0 (X; ) [24,170]. Combined with [107] the above implies that the ergodic transformations of type III is a dense Gı in Aut(X; ). For further refinement of this statement we refer to Sect. “Orbit Theory”. Since the map T 7! T      T (p times) from Aut(X; ) to Aut(X p ; ˝p ) is continuous for each p > 0, we deduce that the set E1 of transformations with infinite ergodic index is a Gı in Aut(X; ). It is non-empty by [113]. Since this E1 is invariant under conjugacy, it is dense in Aut(X; ) by Lemma 16. Thus we obtain that E1 is a dense Gı . In a similar way one can show that E1 \ Aut0 (X; ) is a dense Gı in Aut0 (X; ) (see also [24,26,170] for original proofs of these claims). The rigid transformations form a dense Gı in Aut(X; ). It follows that the set of multiply recurrent nonsingular transformations is residual [13]. A finer result was established in [45]: the set of polynomially recurrent transformations is also residual. Given T 2 Aut(X; ), we denote the centralizer fS 2 Aut(X; ) j ST D T Sg of T by C(T). Of course, C(T) is a closed subgroup of Aut(X; ) and C(T)  fT n j n 2 Zg. The following problems solved recently (by the efforts of many authors) for probability preserving systems are still open for the nonsingular case. Are the properties: (i) T has square root; (ii) T embeds into a flow;

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(iii) T has non-trivial invariant sub--algebra; (iv) C(T) contains a torus of arbitrary dimension typical (residual) in Aut(X; )? The uniform topology on Aut(X; ), finer than dw , is defined by the metric du (T; S) D (fx : Tx ¤ Sxg) C (fx : T

1

x¤S

1

xg) :

This topology is also complete metric. It depends only on the measure class of . However the uniform topology is not separable and that is why it is of less importance in ergodic theory. We refer to [21,24,27] and [68] for the properties of du . Orbit Theory Orbit theory is, in a sense, the most complete part of nonsingular ergodic theory. We present here the seminal Krieger’s theorem on orbit classification of ergodic nonsingular transformations in terms of ratio sets and associated flows. Examples of transformations of various types III , 0    1 are also given here. Next, we consider the outer conjugacy problem for automorphisms of the orbit equivalence relations. This problem is solved in terms of a simple complete system of invariants. We discuss also a general theory of cocycles (of nonsingular systems) taking values in locally compact Polish groups and present an important orbit classification theorem for cocycles. This theorem is an analogue of the aforementioned result of Krieger. We complete the section by considering ITPFIsystems and their relation to AT-flows. Full Groups. Ratio Set and Types III , 0    1 Let T be a nonsingular transformation of a standard probability space (X; B; ). Denote by Orb T (x) the T-orbit of x, i. e. OrbT (x) D fT n x j n 2 Zg. The full group [T] of T consists of all transformations S 2 Aut(X; ) such that Sx 2 OrbT (x) for a. a. x. If T is ergodic then [T] is topologically simple (or even algebraically simple if T is not of type II1 ) [62]. It is easy to see that [T] endowed with the uniform topology du is a Polish group. If T is ergodic then ([T]; du ) is contractible [34]. The ratio set r(T) of T was defined by Krieger [126] and as we shall see below it is the key concept in the orbit classification (see Definition 1). The ratio set is a subset of [0; C1) defined as follows: t 2 r(T) if and only if for every A 2 B of positive measure and each > 0 there is a subset B A of positive measure and an integer k ¤ 0 such that  T k B A and j! k (x)  tj < for all x 2 B. It is easy to verify that r(T) depends only on the equivalence class of  and not on  itself. A basic fact is that 1 2 r(T) if and

only if T is conservative. Assume now T to be conservative and ergodic. Then r(T) \ (0; C1) is a closed subgroup of the multiplicative group (0; C1). Hence r(T) is one of the following sets: (i) f1g; (ii) f0; 1g; in this case we say that T is of type III 0 , (iii) fn j n 2 Zg [ f0g for 0 <  < 1; then we say that T is of type III  , (iv) [0; C1); then we say that T is of type III 1 . Krieger showed that r(T) D f1g if and only if T is of type II. Hence we obtain a further subdivision of type III into subtypes III0 , III , or III1 . Example 17 (i) Fix  2 (0; 1). Let n (0) :D 1/(1 C ) and n (1) :D /(1 C ) for all n D 1; 2; : : : . Let T be the nonsingular odometer associated with the sequence (2; n )1 nD1 (see Subsect. “Nonsingular Odometers”). We claim that T is of type III . Indeed, the group ˙ of finite permutations of N acts on X by ( x)n D x 1 (n) , for all n 2 N,  2 ˙ and x D (x n )1 nD1 2 X. This action preserves . Moreover, it is ergodic by the Hewitt–Savage 0–1 law. It remains to notice that (d ı T/d)(x) D  on the cylinder [0] which is of positive measure. (ii) Fix positive reals 1 and 2 such that log 1 and log 2 are rationally independent. Let n (0) :D 1/(1 C 1 C 2 ), n (1) :D 1 /(1 C 1 C 2 ) and n (2) :D 2 /(1 C 1 C

2 ) for all n D 1; 2; : : : . Then the nonsingular odometer associated with the sequence (3; n )1 nD1 is of type III1 . This can be shown in a similar way as (i). Non-singular odometer of type III0 will be constructed in Example 19 below. Maharam Extension, Associated Flow and Orbit Classification of Type III Systems On X  R with the  -finite measure   , where d(y) D exp (y)dy, consider the transformation   d ı T e T(x; y) :D Tx; y  log (x) : d We call it the Maharam extension of T (see [136], where these transformations were introduced). It is measure-preserving and it commutes with the flow S t (x; y) :D (x; y C t), t 2 R. It is conservative if and only if T is conservative [136]. However e T is not necessarily ergodic. Let (Z; ) denote the space of e T-ergodic components. Then (S t ) t2R acts nonsingularly on this space. The restriction of (S t ) t2R to (Z; ) is called the associated flow of T. The associated flow is ergodic whenever T is ergodic. It is easy to verify

Ergodic Theory: Non-singular Transformations

that the isomorphism class of the associated flow is an invariant of the orbit equivalence of the underlying system. Proposition 18 ([90]) (i)

T is of type II if and only if its associated flow is the translation on R, i. e. x 7! x C t, x; t 2 R, (ii) T is of type III , 0   < 1 if and only if its associated flow is the periodic flow on the interval [0;  log ), i. e. x 7! x C t mod ( log ), (iii) T is of type III1 if and only if its associated flow is the trivial flow on a singleton or, equivalently, e T is ergodic, (iv) T is of type III0 if and only if its associated flow is nontransitive.

We also note that every nontransitive ergodic flow can be realized as the associated flow of a type III0 transformation. However it is somewhat easier to construct a Z2 -action of type III0 whose associated flow is the given one. For this, we take an ergodic nonsingular transformation Q on a probability space (Z; B; ) and a measure-preserving transformation R of an infinite  -finite measure space (Y; F ; ) such that there is a continuous homomorphism  : R ! C(R) with (d ı (t)/d)(y) D exp (t) for a. a. y (for instance, take a type III1 transformation T and put R :D e T and (t) :D S t ). Let ' : Z ! R be a Borel map with inf Z ' > 0. Define two transformations R0 and Q0 of (Z  Y;   ) by setting: R0 (x; y) :D (x; Ry) ;

n

Example 19 Let A n D f0; 1; : : : ; 22 g and n (0) D 0:5 n and n (i) D 0:5  22 for all 0 < i  2n . Let T be the n nonsingular odometer associated with (22 C 1; n )1 nD0 . It is straightforward that the associated flow of T is the flow built under the constant function 1 with the probability preserving 2-adic odometer (associated with (2; n )1 nD1 , n (0) D n (1) D 0:5) as the base transformation. In particular, T is of type III0 . A natural problem arises: to compute Krieger’s type (or the ratio set) for the nonsingular odometers – the simplest class of nonsingular systems. Some partial progress was achieved in [56,141,152], etc. However in the general setting this problem remains open. The map  : Aut(X; ) 3 T 7! e T 2 Aut(X  R;  ) is a continuous group homomorphism. Since the set E of ergodic transformations on X  R is a Gı in Aut(X  R;   ) (see Sect. “Topological Group Aut(X, )”), the subset  1 (E ) of type III1 ergodic transformations on X is also Gı . The latter subset is non-empty in view of Example 17(ii). Since it is invariant under conjugacy, we deduce from Lemma 16 that the set of ergodic transformations of type III1 is a dense Gı in (Aut(X; ); dw ) [23,159]. Now we state the main result of this section – Krieger’s theorem on orbit classification for ergodic transformations of type III. It is a far reaching generalization of the basic result by H. Dye: any two ergodic probability preserving transformations are orbit equivalent [60]. Theorem 20 (Orbit equivalence for type III systems [125]–[129]) Two ergodic transformations of type III are orbit equivalent if and only if their associated flows are isomorphic. In particular, for a fixed 0 <   1, any two ergodic transformations of type III are orbit equivalent. The original proof of this theorem is rather complicated. Simpler treatment of it can be found in [90] and [117].

Q0 (x; y) D (Qx; U x y) ;

where U x D ('(x)  log(d ı Q/d)(x)). Notice that R0 and Q0 commute. The corresponding Z2 -action generated by these transformations is ergodic. Take any transformation V 2 Aut(Z  Y;   ) whose orbits coincide with the orbits of the Z2 -action. (According to [29], any ergodic nonsingular action of any countable amenable group is orbit equivalent to a single transformation.) Then V is of type III0 . It is now easy to verify that the associated flow of V is the special flow built under ' ı Q 1 with the base transformation Q 1 . Since Q and ' are arbitrary, we deduce the following from Theorem 14. Theorem 21 Every nontransitive ergodic flow is an associated flow of an ergodic transformation of type III0 . In [129] Krieger introduced a map ˚ as follows. Let T be an ergodic transformation of type III0 . Then the associated flow of T is a flow built under function with a base transformation ˚(T). We note that the orbit equivalence class of ˚(T) is well defined by the orbit equivalent class of T. If ˚ n (T) fails to be of type III0 for some 1  n < 1 then T is said to belong to Krieger’s hierarchy. For instance, the transformation constructed in Example 19 belongs to Krieger’s hierarchy. Connes gave in [28] an example of T such that ˚(T) is orbit equivalent to T (see also [73] and [90]). Hence T is not in Krieger’s hierarchy. Normalizer of the Full Group. Outer Conjugacy Problem Let N[T] D fR 2 Aut(X; ) j R[T]R1 D [T]g ; i. e. N[T] is the normalizer of the full group [T] in Aut(X; ). We note that a transformation R belongs to N[T] if and only if R(Orb T (x)) D OrbT (Rx) for a. a. x.

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Ergodic Theory: Non-singular Transformations

To define a topology on N[T] consider the T-orbit equivalence relation RT RX  X and a -finite measure R on P RT given by R T D X y2OrbT (x) ı(x;y) d(x). For R 2 N[T], we define a transformation i(R) 2 Aut(RT ; RT ) by setting i(R)(x; y) :D (Rx; Ry). Then the map R 7! i(R) is an embedding of N[T] into Aut(RT ; RT ). Denote by  the topology on N[T] induced by the weak topology on Aut(RT ; RT ) via i [34]. Then (N[T]; ) is a Polish group. A sequence Rn converges to R in N[T] if R n ! R weakly (in Aut(X; )) and R n TR1 ! RTR1 uniformly (in n [T]). Given R 2 N[T], denote by e R the Maharam extension of R. Then e R 2 N[e T] and it commutes with (S t ) t2R . Hence it defines a nonsingular transformation mod R on the space (Z; ) of the associated flow W D (Wt ) t2R of T. Moreover, mod R belongs to the centralizer C(W) of W in Aut(Z; ). Note that C(W) is a closed subgroup of (Aut(Z; ); dw ). Let T be of type II1 and let 0 be the invariant  -finite measure equivalent to . If R 2 N[T] then it is easy to see that the Radon–Nikodym derivative d0 ı R/d0 is invariant under T. Hence it is constant, say c. Then mod R D log c. Theorem 22 ([86,90]) If T is of type III then the map mod : N[T] ! C(W) is a continuous onto homomorphism. The kernel of this homomorphism is the -closure of  [T]. Hence the quotient group N[T]/[T] is (topologically)  isomorphic to C(W). In particular, [T] is co-compact in N[T] if and only if W is a finite measure-preserving flow with a pure point spectrum. The following theorem describes the homotopical structure of normalizers. Theorem 23 ([34]) Let T be of type II or III , 0   <  1. The group [T] is contractible. N[T] is homotopically equivalent to C(W). In particular, N[T] is contractible if T is of type II. If T is of type III with 0 <  < 1 then 1 (N[T]) D Z. The outer period p(R) of R 2 N[T] is the smallest positive integer n such that R n 2 [T]. We write p(R) D 0 if no such n exists. Two transformations R and R0 in N[T] are called outer conjugate if there are transformations V 2 N[T] and S 2 [T] such that V RV 1 D R0 S. The following theorem provides convenient (for verification) necessary and sufficient conditions for the outer conjugacy. Theorem 24 ([30] for type II and [18] for type III) Transformations R; R0 2 N[T] are outer conjugate if and only if p(R) D p(R0 ) and mod R is conjugate to mod R0 in the centralizer of the associated flow for T.

We note that in the case T is of type II, the second condition in the theorem is just mod R D mod R0 . It is always satisfied when T is of type II1 . Cocycles of Dynamical Systems. Weak Equivalence of Cocycles Let G be a locally compact Polish group and G a left Haar measure on G. A Borel map ' : X ! G is called a cocycle of T. Two cocycles ' and ' 0 are cohomologous if there is a Borel map b : X ! G such that ' 0 (x) D b(Tx)1 '(x)b(x) for a. a. x 2 X. A cocycle cohomologous to the trivial one is called a coboundary. Given a dense subgroup G 0 G, then every cocycle is cohomologous to a cocycle with values in G 0 [81]. Each cocycle ' extends to a (unique) map ˛' : RT ! G such that ˛' (Tx; x) D '(x) for a. a. x and ˛' (x; y)˛' (y; z) D ˛' (x; z) for a. a. (x; y); (y; z) 2 RT . ˛' is called the cocycle of RT generated by '. Moreover, ' and ' 0 are cohomologous via b as above if and only if ˛' and ˛' 0 are cohomologous via b, i. e. ˛' (x; y) D b(x)1 ˛' 0 (x; y)b(y) for RT -a. a. (x; y) 2 RT . The following notion was introduced by Golodets and Sinelshchikov [78,81]: two cocycles ' and ' 0 are weakly equivalent if there is a transformation R 2 N[T] such that the cocycles ˛' and ˛'0 ı (R  R) of RT are cohomologous. Let M(X; G) denote the set of Borel maps from X to G. It is a Polish group when endowed with the topology of convergence in measure. Since T is ergodic, it is easy to deduce from Rokhlin’s lemma that the cohomology class of any cocycle is dense in M(X; G). Given ' 2 M(X; G), we define the '-skew product extension T' of T acting on (X  G;   G ) by setting T' (x; g) :D (Tx; '(x)g). Thus Maharam extension is (isomorphic to) the Radon– Nikodym cocycle-skew product extension. We now specify some basic classes of cocycles [19,35,81,173]: (i) ' is called transient if T' is of type I. (ii) ' is called recurrent if T' is conservative (equivalently, T' is not transient). (iii) ' has dense range in G if T' is ergodic. (iv) ' is called regular if ' cobounds with dense range into a closed subgroup H of G (then H is defined up to conjugacy). These properties are invariant under the cohomology and the weak equivalence. The Radon–Nikodym cocycle ! 1 is a coboundary if and only if T is of type II. It is regular if and only if T is of type II or III , 0 <   1. It has dense range (in the multiplicative group RC ) if and only if T is of type III1 . Notice that ! 1 is never transient (since T is conservative).

Ergodic Theory: Non-singular Transformations

Schmidt introduced in [176] an invariant R(') :D fg 2 G j '  g is recurrentg. He showed in particular that (i) (ii) (iii) (iv)

R(') is a cohomology invariant, R(') is a Borel set in G, R(log !1 ) D f0g for each aperiodic conservative T, there are cocycles ' such that R(') and G n R(') are dense in G, (v) if (X) D 1, Rı T D  and ' : X ! R is integrable then R(') D f ' dg. We note that (v) follows from Atkinson theorem [15]. A nonsingular version of this theorem was established in [183]: if T is ergodic and -nonsingular and f 2 L1 () then ˇ ˇ n1 ˇ ˇX ˇ ˇ j f (T x)! j (x)ˇ D 0 for a. a. x lim inf ˇ n!1 ˇ ˇ jD0 R if and only if f d D 0. Since T' commutes with the action of G on X  G by inverted right translations along the second coordinate, this action induces an ergodic G-action W' D (W' (g)) g2G on the space (Z; ) of T' -ergodic components. It is called the Mackey range (or Poincaré flow) of ' [66,135,173,188]. We note that ' is regular (and cobounds with dense range into H G) if and only if W' is transitive (and H is the stabilizer of a point z 2 Z, i. e. H D fg 2 G j W' (g)z D zg). Hence every cocycle taking values in a compact group is regular. It is often useful to consider the double cocycle '0 :D '  !1 instead of '. It takes values in the group G  RC . Since T'0 is exactly the Maharam extension of T' , it follows from [136] that ' 0 is transient or recurrent if and only if ' is transient or recurrent respectively. Theorem 25 (Orbit classification of cocycles [81]) Let '; ' 0 : X ! G be two recurrent cocycles of an ergodic transformation T. They are weakly equivalent if and only if their Mackey ranges W'0 and W'00 are isomorphic. Another proof of this theorem was presented in [65]. Theorem 26 Let T be an ergodic nonsingular transformation. Then there is a cocycle of T with dense range in G if and only if G is amenable. It follows that if G is amenable then the subset of cocycles of T with dense range in G is a dense Gı in M(X; G) (just adapt the argument following Example 19). The ‘only if ’ part of Theorem 26 was established in [187]. The ‘if ’ part was considered by many authors in particular cases: G is compact [186], G is solvable or amenable almost connected [79], G is amenable unimodular [108], etc. The general case was proved in [78] and [100] (see also a recent treatment in [9]).

Theorem 21 is a particular case of the following result. Theorem 27 ([10,65,80]) Let G be amenable. Let V be an ergodic nonsingular action of G  RC . Then there is an ergodic nonsingular transformation T and a recurrent cocycle ' of T with values in G such that V is isomorphic to the Mackey range of the double cocycle ' 0 . Given a cocycle ' 2 M(X; G) of T, we say that a transformation R 2 N[T] is compatible with ' if the cocycles ˛' and ˛' ı (R  R) of RT are cohomologous. Denote by D(T; ') the group of all such R. It has a natural Polish topology which is stronger than  [41]. Since [T] is a normal subgroup in D(T; '), one can consider the outer conjugacy equivalence relation inside D(T; '). It is called '-outer conjugacy. Suppose that G is Abelian. Then an analogue of Theorem 24 for the '-outer conjugacy is established in [41]. Also, the cocycles ' with D(T; ') D N[T] are described there. ITPFI Transformations and AT-Flows A nonsingular transformation T is called ITPFI 1 if it is orbit equivalent to a nonsingular odometer (associated to a sequence (m n ; n )1 nD1 , see Subsect. “Nonsingular Odometers”). If the sequence mn can be chosen bounded then T is called ITPFI of bounded type. If m n D 2 for all n then T is called ITPFI2 . By [74], every ITPFI-transformation of bounded type is ITPFI2 . A remarkable characterization of ITPFI transformations in terms of their associated flows was obtained by Connes and Woods [31]. We first single out a class of ergodic flows. A nonsingular flow V D (Vt ) t2R on a space (˝; ) is called approximate transitive (AT) if given > 0 and f1 ; : : : ; f n 2 L1C (X; ), there exists f 2 L1C (X; ) and 1 ; : : : ; n 2 L1C (R; dt) such that ˇˇ ˇˇ Z ˇˇ ˇˇ d ı Vt ˇˇ f j  ˇˇ < f ı V (t)dt  t j ˇˇ ˇˇ d R 1 for all 1  j  n. A flow built under a constant ceiling function with a funny rank-one [67] probability preserving base transformation is AT [31]. In particular, each ergodic finite measure-preserving flow with a pure point spectrum is AT. Theorem 28 ([31]) An ergodic nonsingular transformation is ITPFI if and only if its associated flow is AT. The original proof of this theorem was given in the framework of von Neumann algebras theory. A simpler, purely 1 This abbreviates ‘infinite tensor product of factors of type I’ (came from the theory of von Neumann algebras).

341

342

Ergodic Theory: Non-singular Transformations

measure theoretical proof was given later in [96] (the ‘only if ’ part) and [88] (the ‘if ’ part). It follows from Theorem 28 that every ergodic flow with pure point spectrum is the associated flow of an ITPFI transformation. If the point spectrum of V is  , where  is a subgroup of Q and 2 R, then V is the associated flow of an ITPFT2 transformation [91]. Theorem 29 ([54]) Each ergodic nonsingular transformation is orbit equivalent to a Markov odometer (see Subsect. “Markov Odometers”). The existence of non-ITPFI transformations and ITPFI transformations of unbounded type was shown in [127]. In [55], an explicit example of a non-ITPFI Markov odometer was constructed. Smooth Nonsingular Transformations Diffeomorphisms of smooth manifolds equipped with smooth measures are commonly considered as physically natural examples of dynamical systems. Therefore the construction of smooth models for various dynamical properties is a well established problem of the modern (probability preserving) ergodic theory. Unfortunately, the corresponding ‘nonsingular’ counterpart of this problem is almost unexplored. We survey here several interesting facts related to the topic. r For r 2 N [ f1g, denote by DiffC (T ) the group of orientation preserving Cr -diffeomorphisms of the circle T . Endow this set with the natural Polish topology. Fix r T 2 DiffC (T ). Since T D R/Z, there exists a C1 -function f : R ! R such that T(x C Z) D f (x) C Z for all x 2 R. The rotation number (T) of T is the limit lim ( f ı    ı f )(x)(mod 1) : „ ƒ‚ …

n!1

n times

The limit exists and does not depend on the choice of x and f . It is obvious that T is nonsingular with respect to r (T ) and Lebesgue measure T . Moreover, if T 2 DiffC

(T) is irrational then the dynamical system (T ; T ; T) is ergodic [33]. It is interesting to ask: which Krieger’s type can such systems have? Katznelson showed in [114] that the subset of type III C 1 -diffeomorphisms and the subset of type II1 C 1 -dif1 (T ). Hawkins and feomorphisms are dense in DiffC Schmidt refined the idea of Katznelson from [114] to construct, for every irrational number ˛ 2 [0; 1) which is not of constant type (i. e. in whose continued fraction expansion the denominators are not bounded) a transformation 2 T 2 DiffC (T ) which is of type III1 and (T) D ˛ [97]. It should be mentioned that class C2 in the construction

is essential, since it follows from a remarkable result of 3 Herman that if T 2 DiffC (T ) then under some condition on ˛ (which determines a set of full Lebesgue measure), T is measure theoretically (and topologically) conjugate to a rotation by (T) [101]. Hence T is of type II1 . In [94], Hawkins shows that every smooth paracompact manifold of dimension  3 admits a type III diffeomorphism for every  2 [0; 1]. This extends a result of Herman [100] on the existence of type III1 diffeomorphisms in the same circumstances. It is also of interest to ask: which free ergodic flows are associated with smooth dynamical systems of type III0 ? Hawkins proved that any free ergodic C 1 -flow on a smooth, connected, paracompact manifold is the associated flow for a C 1 -diffeomorphism on another manifold (of higher dimension) [95]. 2 (T ) A nice result was obtained in [115]: if T 2 DiffC and the rotation number of T has unbounded continued fraction coefficients then (T ; T ; T) is ITPFI. Moreover, a converse also holds: given a nonsingular odometer R, the set of orientation-preserving C 1 -diffeomorphisms of the circle which are orbit equivalent to R is C 1 -dense in the Polish set of all C 1 -orientation-preserving diffeomorphisms with irrational rotation numbers. In contrast to that, Hawkins constructs in [93] a type III0 C 1 -diffeomorphism of the 4-dimensional torus which is not ITPFI. Spectral Theory for Nonsingular Systems While the spectral theory for probability preserving systems is developed in depth, the spectral theory of nonsingular systems is still in its infancy. We discuss below some problems related to L1 -spectrum which may be regarded as an analogue of the discrete spectrum. We also include results on computation of the maximal spectral type of the ‘nonsingular’ Koopman operator for rank-one nonsingular transformations. L1 -Spectrum and Groups of Quasi-Invariance Let T be an ergodic nonsingular transformation of (X; B; ). A number  2 T belongs to the L1 -spectrum e(T) of T if there is a function f 2 L1 (X; ) with f ı T D  f . f is called an L1 -eigenfunction of T corresponding to . Denote by E (T) the group of all L1 -eigenfunctions of absolute value 1. It is a Polish group when endowed with the topology of converges in measure. If T is of type II1 then the L1 -eigenfunctions are L2 (0 )-eigenfunctions of T, where 0 is an equivalent invariant probability measure. Hence e(T) is countable. Osikawa constructed in [151] the first examples of ergodic nonsingular transformations with uncountable e(T).

Ergodic Theory: Non-singular Transformations

We state now a nonsingular version of the von Neumann–Halmos discrete spectrum theorem. Let Q T be a countable infinite subgroup. Let K be a compact dual of Qd , where Qd denotes Q with the discrete topology. Let k0 2 K be the element defined by k0 (q) D q for all q 2 Q. Let R : K ! K be defined by Rk D k C k0 . The system (K; R) is called a compact group rotation. The following theorem was proved in [6]. Theorem 30 Assume that the L1 -eigenfunctions of T generate the entire -algebra B. Then T is isomorphic to a compact group rotation equipped with an ergodic quasi-invariant measure. A natural question arises: which subgroups of T can appear as e(T) for an ergodic T? Theorem 31 ([1,143]) e(T) is a Borel subset of T and carries a unique Polish topology which is stronger than the usual topology on T . The Borel structure of e(T) under this topology agrees with the Borel structure inherited from T . There is a Borel map : e(T) 3  7!  2 E (T) such that  ıT D   for each . Moreover, e(T) is of Lebesgue measure 0 and it can have an arbitrary Hausdorff dimension. A proper Borel subgroup E of T is called (i) weak Dirichlet if lim supn!1 b (n) D 1 for each finite complex measure  supported on E; (n)j  j(E)j for each fi(ii) saturated if lim supn!1 jb nite complex measure  on T , where b (n) denote the nth Fourier coefficient of . Every countable subgroup of T is saturated. Theorem 32 e(T) is -compact in the usual topology on T [104] and saturated [104,139]. It follows that e(T) is weak Dirichlet (this fact was established earlier in [175]). It is not known if every Polish group continuously embedded in T as a -compact saturated group is the eigenvalue group of some ergodic nonsingular transformation. This is the case for the so-called H 2 -groups and the groups of quasi-invariance of measures on T (see below). Given a sequence nj of positive integers and a sequence a j  0, P nj 2 the set of all z 2 T such that 1 jD1 a j j1  z j < 1 is a group. It is called an H 2 -group. Every H 2 -group is Polish in an intrinsic topology stronger than the usual circle topology. Theorem 33 ([104]) (i)

Every H 2 -group is a saturated (and hence weak Dirichlet) -compact subset of T .

P (ii) If 1 jD0 a j D C1 then the corresponding H 2 -group is a proper subgroup of T . P 2 (iii) If 1 jD0 a j (n j /n jC1 ) < 1 then the corresponding H 2 -group is uncountable. (iv) Any H 2 -group is e(T) for an ergodic nonsingular compact group rotation T. It is an open problem whether every eigenvalue group e(T) is an H 2 -group. It is known however that e(T) is close ‘to be an H 2 -group’: if a compact subset L T is disjoint from e(T) then there is an H 2 -group containing e(T) and disjoint from L. Example 34 ([6], see also [151]) Let (X; ; T) be the nonsingular odometer associated to a sequence (2;  j )1 jD1 . Let nj be a sequence of positive integers such that n j > P n for all j. For x 2 X, we put h(x) :D n l (x)  P i< j i j 0 then either H() is countable or  is equivalent to T [137]. Theorem 35 ([6]) Let  be an ergodic with respect to the H()-action by translations on T . Then there is a compact group rotation (K; R) and a finite measure on K quasiinvariant and ergodic under R such that e(R) D H(). Moreover, there is a continuous one-to-one homomorphism : e(R) ! E(R) such that  ı R D   for all  2 e(R). It was shown by Aaronson and Nadkarni [6] that if n1 D 1 and n j D a j a j1    a1 for positive integers a j  2 P 1 with 1 jD1 a j < 1 then the transformation S from Example 34 does not admit a continuous homomorphism : e(S) ! E(S) with  ı T D   for all  2 e(S). Hence e(S) ¤ H() for any measure  satisfying the conditions of Theorem 35.

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Ergodic Theory: Non-singular Transformations

Assume that T is an ergodic nonsingular compact group rotation. Let B0 be the -algebra generated by a subcollection of eigenfunctions. Then B0 is invariant under T and hence a factor (see Sect. “Nonsingular Joinings and Factors”) of T. It is not known if every factor of T is of this form. It is not even known whether every factor of T must have non-trivial eigenvalues. Unitary Operator Associated with a Nonsingular System Let (X; B; ; T) be a nonsingular dynamical system. In this subsection we consider spectral properties of the unitary operator U T defined by (3). First, we note that the spectrum of T is the entire circle T [147]. Next, if U T has an eigenvector then T is of type II1 . Indeed, if there are  2 T and 0 ¤ f 2 L2 (X; ) with U T f D  f then the measure , d(x) :D j f (x)j2 d(x), is finite, T-invariant and equivalent to . Hence if T is of type III or II1 then the maximal spectral type  T of U T is continuous. Another ‘restriction’ on  T was recently found in [166]: no Foïa¸s-Str˘atil˘a measure is absolutely continuous with respect to  T if T is of type II1 . We recall that a symmetric measure on T possesses Foïa¸s-Str˘atil˘a property if for each ergodic probability preserving system (Y; ; S) and f 2 L2 (Y; ), if  is the spectral measure of f then f is a Gaussian random variable [134]. For instance, measures supported on Kronecker sets possess this property. Mixing is an L2 -spectral property for type II1 transformations: T is mixing Rif and only if  T is a Rajchman measure, i. e. b  T (n) :D z n dT (z) ! 0 as jnj ! 1. P ki Also, T is mixing if and only if n1 n1 iD0 U T ! 0 in the strong operator topology for each strictly increasing sequence k1 < k2 <    [124]. This generalizes a well known theorem of Blum and Hanson for probability preserving maps. For comparison, we note that ergodicity is not an L2 -spectral property of infinite measure preserving systems. Now let T be a rank-one nonsingular transformation associated with a sequence (r n ; w n ; s n )1 nD1 as in Subsect. “Rank-One Transformations. Chacón Maps. Finite Rank”. Theorem 36 ([25,104]) The spectral multiplicity of U T is 1 and the maximal spectral type  T of U T (up to a discrete measure in the case T is of type II1 ) is the weak limit of the measures k defined as follows: d k (z) D

k Y

w j (0)jPj (z)j2 dz ;

jD1

where P j (z)

:D

1 C

p w j (1)/w j (0)zR1; j C    C

p

R

w j (m j  1)/w j (0)z r j 1; j , z 2 T , R i; j :D ih j1 C s j (0) C    C s j (i), 1  i  r k  1 and hj is the height of the jth column.

Thus the maximal spectral type of U T is given by a socalled generalized Riesz product. We refer the reader to [25,103,104,148] for a detailed study of Riesz products: their convergence, mutual singularity, singularity to T , etc. It was shown in [6] that H(T )  e(T) for any ergodic nonsingular transformation T. Moreover,  T is ergodic under the action of e(T) by translations if T is isomorphic to an ergodic nonsingular compact group rotation. However it is not known: (i) Whether H(T ) D e(T) for all ergodic T. (ii) Whether ergodicity of  T under e(T) implies that T is an ergodic compact group rotation. The first claim of Theorem 36 extends to the rank N nonsingular systems as follows: if T is an ergodic nonsingular transformation of rank N then the spectral multiplicity of U T is bounded by N (as in the finite measure-preserving case). It is not known whether this claim is true for a more general class of transformations which are defined as rank N but without the assumption that the Radon– Nikodym cocycle is constant on the tower levels. Entropy and Other Invariants Let T be an ergodic conservative nonsingular transformation of a standard probability space (X; B; ). If P is a finite partition of X, we define the entropy H(P ) of P P as H(P ) D  P2P (P) log (P). In the study of measure-preserving systems the classical (Kolmogorov–Sinai) entropy proved to be a very useful invariant for isomorphism [33]. The key fact of the theory is that if  ı T D  W then the limit limn!1 n1 H( niD1 T i P ) exists for every P . However if T does not preserve , the limit may no longer exist. Some efforts have been made to extend the use of entropy and similar invariants to the nonsingular domain. These include Krengel’s entropy of conservative measure-preserving maps and its extension to nonsingular maps, Parry’s entropy and Parry’s nonsingular version of Shannon–McMillan–Breiman theorem, critical dimension by Mortiss and Dooley, etc. Unfortunately, these invariants are less informative than their classical counterparts and they are more difficult to compute. Krengel’s and Parry’s Entropies Let S be a conservative measure-preserving transformation of a -finite measure space (Y; E ; ). The Krengel en-

Ergodic Theory: Non-singular Transformations

tropy [119] of S is defined by hKr (S) D sup f(E)h(S E ) j 0 < (E) < C1g ; where h(S E ) is the finite measure-preserving entropy of SE . It follows from Abramov’s formula for the entropy of induced transformation that hKr (S) D (E)h(S E ) whenS ever E sweeps out, i. e. i0 S i E D X. A generic transformation from Aut0 (X; ) has entropy 0. Krengel raised a question in [119]: does there exist a zero entropy infinite measure-preserving S and a zero entropy finite measurepreserving R such that hKr (S  R) > 0? This problem was recently solved in [44] (a special case was announced by Silva and Thieullen in an October 1995 AMS conference (unpublished)): (i) if hKr (S) D 0 and R is distal then hKr (S  R) D 0; (ii) if R is not distal then there is a rank-one transformation S with hKr (S  R) D 1. We also note that if a conservative S 2 Aut0 (X; ) commutes with another transformation R such that  ı R D c for a constant c ¤ 1 then hKr (S) is either 0 or 1 [180]. Now let T be a type III ergodic transformation of (X; B; ). Silva and Thieullen define an entropy h (T) T), where e T is the Maharam of T by setting h (T) :D hKr (e extension of T (see Subsect. “Maharam Extension, Associated Flow and Orbit Classification of Type III Systems”). Since e T commutes with transformations which ‘multiply’ e T-invariant measure, it follows that h (T) is either 0 or 1. Let T be the standard III -odometer from Example 17(i). Then h (T) D 0. The same is true for a socalled ternary odometer associated with the sequence (3; n )1 nD1 , where  n (0) D  n (2) D /(1 C 2) and  n (1) D /(1 C ) [180]. It is not known however whether every ergodic nonsingular odometer has zero entropy. On the other hand, it was shown in [180] that h (T) D 1 for every K-automorphism. The Parry entropy [158] of S is defined by ˚ hPa (S) :D H(S 1 FjF) j F is a -finite subalgebra of B such that F S 1 F : Parry showed [158] that hPa (S)  hKr (S). It is still an open question whether the two entropies coincide. This is the case when S is of rank one (since hKr (S) D 0) and when S is quasi-finite [158]. The transformation S is called quasifinite if there exists a subset of finite measure A Y such that the first return time partition (A n )n>0 of A has finite entropy. We recall that x 2 A n () n is the smallest positive integer such that T n x 2 A. An example of nonquasi-finite ergodic infinite measure preserving transformation was constructed recently in [8].

Parry’s Generalization of Shannon–MacMillan–Breiman Theorem Let T be an ergodic transformation of a standard nonatomic probability space (X; B; ). Suppose that f ı T 2 L1 (X; ) if and only if f 2 L1 (X; ). This means that there is K > 0 such that K 1 < (d ı T)/(d) (x) < K for a. a. x. Let P be a finite partition of X. Denote by C n (x) the W atom of niD0 T i P which contains x. We put !1 D 0. Parry shows in [155] that n X

  log  C n j (T j x) (! j (x)  ! j1 (x))

jD0 n X

! ! j (x)

iD0

! Z ˇ _ ˇ 1 1 d ı T T P  log E H P ˇˇ d X iD1

! ˇ _ ˇ 1 i ˇ T P d ˇ iD0

for a. a. x. Parry also shows that under the aforementioned conditions on T, 0 1@ n

n X jD0

0 H@

j _

1 T  j PA 

iD0

n1 X jD0

0 H@

jC1 _

11 T  j P AA

iD1

! ˇ 1 ˇ _ i ˇ T P : !H Pˇ iD1

Critical Dimension The critical dimension introduced by Mortiss [146] measures the order of growth for sums of Radon–Nikodym derivatives. Let (X; B; ; T) be an ergodic nonsingular dynamical system. Given ı > 0, let 8 9 n1 X ˆ > ˆ > ˆ > ! i (x) ˆ > ˆ ˇ > < = ˇ iD0 and (4) X ı :D x 2 X ˇˇ lim inf > 0 ı n!1 ˆ > n ˆ > ˆ > ˆ > ˆ > : ; 8 9 n1 X ˆ > ˆ > ˆ > ! i (x) ˆ > ˆ ˇ > < = ˇ iD0 ı ˇ : (5) X :D x 2 X ˇ lim inf D 0 n!1 ˆ > nı ˆ > ˆ > ˆ > ˆ > : ; Then Xı and X ı are T-invariant subsets. Definition 37 ([57,146]) The lower critical dimension ˛(T) of T is sup fı j (Xı ) D 1g. The upper critical dimension ˇ(T) of T is inf fı j (X ı ) D 1g.

345

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Ergodic Theory: Non-singular Transformations

It was shown in [57] that the lower and upper critical dimensions are invariants for isomorphism of nonsingular systems. Notice also that ! n X log ! i (x) ˛(T) D lim inf n!1

iD1

log n ! n X log ! i (x)

ˇ(T) D lim sup n!1

iD1

and

Theorem 40 For any > 0, there exists a Hamachi shift S with ˛(S) < and ˇ(S) > 1  . :

log n

Nonsingular Restricted Orbit Equivalence

Moreover, 0  ˛(T)  ˇ(T)  1. If T is of type II1 then ˛(T) D ˇ(T) D 1. If T is the standard III -odometer from Example 17 then ˛(T) D ˇ(T) D log(1 C ) /(1 C ) log . Theorem 38 (i) For every  2 [0; 1] and every c 2 [0; 1] there exists a nonsingular odometer of type III with critical dimension equal to c [145]. (ii) For every c 2 [0; 1] there exists a nonsingular odometer of type II1 with critical dimension equal to c [57]. Let T be the nonsingular odometer associated with a sequence (m n ; n )1 nD1 . Let s(n) D m1    m n and let H(P n ) denote the entropy of the partition of the first n coordinates with respect to . We now state a nonsingular version of Shannon–MacMillan–Breiman theorem for T from [57]. Theorem 39 Let mi be bounded from above. Then (i)  ˛(T) D lim inf inf

entropy (average coordinate). It also follows from Theorem 39 that if T is an odometer of bounded type then ˛(T 1 ) D ˛(T) and ˇ(T 1 ) D ˇ(T). In [58], Theorem 39 was extended to a subclass of Markov odometers. The critical dimensions for Hamachi shifts (see Subsect. “Nonsingular Bernoulli Transformations – Hamachi’s Example”) were investigated in [59]:

n X

log m i (x i )

iD1

log s(n) H(Pn ) ; D lim inf n!1 log s(n)

In [144] Mortiss initiated study of a nonsingular version of Rudolph’s restricted orbit equivalence [167]. This work is still in its early stages and does not yet deal with any form of entropy. However she introduced nonsingular orderings of orbits, defined sizes and showed that much of the basic machinery still works in the nonsingular setting. Nonsingular Joinings and Factors The theory of joinings is a powerful tool to study probability preserving systems and to construct striking counterexamples. It is interesting to study what part of this machinery can be extended to the nonsingular case. However, the definition of nonsingular joining is far from being obvious. Some progress was achieved in understanding 2-fold joinings and constructing prime systems of any Krieger type. As far as we know the higher-fold nonsingular joinings have not been considered so far. It turned out however that an alternative coding technique, predating joinings in studying the centralizer and factors of the classical measure-preserving Chacón maps, can be used as well to classify factors of Cartesian products of some nonsingular Chacón maps.

n!1

and (ii)  ˇ(T) D lim sup inf

n X

log m i (x i )

iD1

n!1

log s(n)

H(Pn ) D lim sup log s(n) n!1 for a. a. x D (x i ) i1 2 X. It follows that in the case when ˛(T) D ˇ(T), the critical dimension coincides with limn!1 H(Pn )/(log s(n)). In [145] this expression (when it exists) was called AC-

Joinings, Nonsingular MSJ and Simplicity In this section all measures are probability measures. A nonsingular joining of two nonsingular systems ˆ on the (X1 ; B1 ; 1 ; T1 ) and (X 2 ; B2 ; 2 ; T2 ) is a measure  product B1  B2 that is nonsingular for T1  T2 and satisˆ ˆ 1 B) D 2 (B) for all A 2 fies: (AX 2 ) D 1 (A) and (X B1 and B 2 B2 . Clearly, the product 1 2 is a nonsingular joining. Given a transformation S 2 C(T), the measure S given by S (A  B) :D (A \ S 1 B) is a nonsingular joining of (X; ; T) and (X; ıS 1 ; T). It is called a graphjoining since it is supported on the graph of S. Another important kind of joinings that we are going to define now is related to factors of dynamical systems. Recall that given a nonsingular system (X; B; ; T), a sub- -algebra A of B such that T 1 (A) D A mod  is called a factor of T.

Ergodic Theory: Non-singular Transformations

There is another, equivalent, definition. A nonsingular dynamical system (Y; C ; ; S) is called a factor of T if there exists a measure-preserving map ' : X ! Y, called a factor map, with 'T D S' a. e. (If ' is only nonsingular,  may be replaced with the equivalent measure  ı ' 1 , for which ' is measure-preserving.) Indeed, the sub--algebra ' 1 (C ) B is T-invariant and, conversely, any T-invariant sub--algebra of B defines a factor map by immanent properties of standard probability spaces, see e. g. [3]. If ' is a factor map as above,Rthen  has a disintegration with respect to ', i. e.,  D  y d(y) for a measurable map y 7!  y from Y to the probability measures on X so that  y (' 1 (y)) D 1, the measure S'(x) ı T is equivalent to '(x) and dS'(x) ı T d ı T d ı S (x) (x) D ('(x)) d d d'(x)

(6)

ˆ D for a. e. x 2 X. Define now theR relative product  ˆ D  y   y d(y). Then  '  on X  X by setting  ˆ is a nonsingular selfit is easy to deduce from (6) that  joining of T. We note however that the above definition of joining is not satisfactory since it does not reduce to the classical definition when we consider probability preserving systems. Indeed, the following result was proved in [168]. Theorem 41 Let (X 1 ; B1 ; 1 ; T1 ) and (X2 ; B2 ; 2 ; T2 ) be two finite measure-preserving systems such that T1  T2 is ergodic. Then for every ; 0 <  < 1, there exists a nonˆ of 1 and 2 such that (T1  T2 ; ) ˆ is singular joining  ergodic and of type III . ˆ can It is not known however if the nonsingular joining  be chosen in every orbit equivalence class. In view of the above, Rudolph and Silva [168] isolate an important subclass of joining. It is used in the definition of a nonsingular version of minimal self-joinings. Definition 42 ˆ of (X 1 ; 1 ; T1 ) and (i) A nonsingular joining  (X 2 ; 2 ; T2 ) is rational if there exit measurable functions c 1 : X1 ! RC and c 2 : X2 ! RC such that    !ˆ 1 (x1 ; x2 ) D !1 1 (x1 )!1 2 (x2 )c 1 (x1 ) ˆ

  ˆ a. e. D !1 1 (x1 )!1 2 (x2 )c 2 (x2 ) 

(ii) A nonsingular dynamical system (X; B; ; T) has minimal self-joinings (MSJ) over a class M of probability measures equivalent to , if for every 1 ; 2 2 M, ˆ of 1 ; 2 , a. e. ergodic for every rational joining  ˆ is either the product of its marginals component of  or is the graph-joining supported on T j for some j 2 Z.

Clearly, product measure, graph-joinings and the relative products are all rational joinings. Moreover, a rational joining of finite measure-preserving systems is measure-preserving and a rational joining of type II1 ’s is of type II1 [168]. Thus we obtain the finite measure-preserving theory as a special case. As for the definition of MSJ, it depends on a class M of equivalent measures. In the finite measure-preserving case M D fg. However, in the nonsingular case no particular measure is distinguished. We note also that Definition 42(ii) involves some restrictions on all rational joinings and not only ergodic ones as in the finite measure-preserving case. The reason is that an ergodic component of a nonsingular joining needs not be a joining of measures equivalent to the original ones [2]. For finite measure-preserving transformations, MSJ over fg is the same as the usual 2-fold MSJ [49]. A nonsingular transformation T on (X; B; ) is called prime if its only factors are B and fX; ;g mod . A (nonempty) class M of probability measures equivalent to  is said to be centralizer stable if for each S 2 C(T) and 1 2 M, the measure 1 ı S is in M. Theorem 43 ([168]) Let (X; B; ; T) be a ergodic nonatomic dynamical system such that T has MSJ over a class M that is centralizer stable. Then T is prime and the centralizer of T consists of the powers of T. A question that arises is whether if such nonsingular dynamical system (not of type II1 ) exist. Expanding on Ornstein’s original construction from [150], Rudolph and Silva construct in [168], for each 0    1, a nonsingular rank-one transformation T that is of type III and that has MSJ over a class M that is centralizer stable. Type II1 examples with analogues properties were also constructed there. In this connection it is worth to mention the example by Aaronson and Nadkarni [6] of II1 ergodic transformations that have no factor algebras on which the invariant measure is  -finite (except for the trivial and the entire ones); however these transformations are not prime. A more general notion than MSJ called graph self-joinings (GSJ), was introduced [181]: just replace the the words “on T j for some j 2 Z” in Definition 3(ii) with “on S for some element S 2 C(T)”. For finite measure-preserving transformations, GSJ over fg is the same as the usual 2fold simplicity [49]. The famous Veech theorem on factors of 2-fold simple maps (see [49]) was extended to nonsingular systems in [181] as follows: if a system (X; B; ; T) has GSJ then for every non-trivial factor A of T there exists a locally compact subgroup H in C(T) (equipped with the weak topology) which acts smoothly (i. e. the partition into H-orbits is measurable) and such that A D fB 2 B j (hB4B) D 0 for all h 2 Hg. It follows that there is a co-

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cycle ' from (X; A;   A) to H such that T is isomorphic to the '-skew product extension (T  A)' (see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). Of course, the ergodic nonsingular odometers and, more generally, ergodic nonsingular compact group rotation (see Subsect. “L1 -Spectrum and Groups of Quasi-Invariance”) have GSJ. However, except for this trivial case (the Cartesian square is non-ergodic) plus the systems with MSJ from [168], no examples of type III systems with GSJ are known. In particular, no smooth examples have been constructed so far. This is in sharp contrast with the finite measure preserving case where abundance of simple (or close to simple) systems are known (see [39,40,49,182]). Nonsingular Coding and Factors of Cartesian Products of Nonsingular Maps As we have already noticed above, the nonsingular MSJ theory was developed in [168] only for 2-fold self-joinings. The reasons for this were technical problems with extending the notion of rational joinings form 2-fold to n-fold self-joinings. However while the 2-fold nonsingular MSJ or GSJ properties of T are sufficient to control the centralizer and the factors of T, it is not clear whether it implies anything about the factors or centralizer of T  T. Indeed, to control them one needs to know the 4-fold joinings of T. However even in the finite measure-preserving case it is a long standing open question whether 2-fold MSJ implies n-fold MSJ. That is why del Junco and Silva [51] apply an alternative – nonsingular coding – techniques to classify the factors of Cartesian products of nonsingular Chacón maps. The techniques were originally used in [48] to show that the classical Chacón map is prime and has trivial centralizer. They were extended to nonsingular systems in [50]. For each 0 <  < 1 we denote by T  the Chacón map (see Subsect. “Rank-One Transformations. Chacón Maps. Finite Rank”) corresponding the sequence of probability vectors w n D (/(1 C 2); 1/(1 C 2); /(1 C 2)) for all n > 0. One can verify that the maps T are of type III . (The classical Chacón map corresponds to  D 1.) All of these transformations are defined on the same standard Borel space (X; B). These transformations were shown to be power weakly mixing in [12]. The centralizer of any finite Cartesian product of nonsingular Chacón maps is computed in the following theorem. Theorem 44 ([51]) Let 0 < 1 < : : : <  k  1 and n1 ; : : : ; n k be positive integers. Then the centralizer of the ˝n 1 Cartesian product T˝n  : : :  T k is generated by maps 1 k of the form U1  : : :  U k , where each U i , acting on the

ni -dimensional product space X n i , is a Cartesian product of powers of T i or a co-ordinate permutation on X n i . Let  denote the permutation on X  X defined by (x; y) D (y; x) and let B2ˇ denote the symmetric factor, i. e. B2ˇ D fA 2 B ˝ B j (A) D Ag. The following theorem classifies the factors of the Cartesian product of any two nonsingular type III , 0 <  < 1, or type II1 Chacón maps. Theorem 45 ([51]) Let T1 and T2 be two nonsingular Chacón systems. Let F be a factor algebra of T1  T2 . If 1 ¤ 2 then F is equal mod 0 to one of the four algebras B ˝ B, B ˝ N , N ˝ B, or N ˝ N , where N D f;; Xg. (ii) If 1 D 2 then F is equal mod 0 to one of the following algebras B ˝ C , B ˝ N , N ˝ C , N ˝ N , or (T m  Id)B2ˇ for some integer m.

(i)

It is not hard to obtain type III1 examples of Chacón maps for which the previous two theorems hold. However the construction of type II1 and type III0 nonsingular Chacón transformations is more subtle as it needs the choice of ! n to vary with n. In [92], Hamachi and Silva construct type III0 and type II1 examples, however the only property proved for these maps is ergodicity of their Cartesian square. More recently, Danilenko [38] has shown that all of them (in fact, a wider class of nonsingular Chacón maps of all types) are power weakly mixing. In [22], Choksi, Eigen and Prasad asked whether there exists a zero entropy, finite measure-preserving mixing automorphism S, and a nonsingular type III automorphism T, such that T  S has no Bernoulli factors. Theorem 45 provides a partial answer (with a mildly mixing only instead of mixing) to this question: if S is the finite measure-preserving Chacón map and T is a nonsingular Chacón map as above, the factors of T  S are only the trivial ones, so T  S has no Bernoulli factors. Applications. Connections with Other Fields In this – final – section we shed light on numerous mathematical sources of nonsingular systems. They come from the theory of stochastic processes, random walks, locally compact Cantor systems, horocycle flows on hyperbolic surfaces, von Neumann algebras, statistical mechanics, representation theory for groups and anticommutation relations, etc. We also note that such systems sometimes appear in the context of probability preserving dynamics (see also a criterium of distality in Subsect. “Krengel’s and Parry’s Entropies”).

Ergodic Theory: Non-singular Transformations

Mild Mixing An ergodic finite measure-preserving dynamical system (X; B; ; T) is called mildly mixing if for each non-trivial factor algebra A B, the restriction T  A is not rigid. For equivalent definitions and extensions to actions of locally compact groups we refer to [3] and [177]. There is an interesting criterium of the mild mixing that involves nonsingular systems: T is mildly mixing if and only if for each ergodic nonsingular transformation S, the product T  S is ergodic [71]. Furthermore, T  S is then orbit equivalent to S [98]. Moreover, if R is a nonsingular transformation such that R  S is ergodic for any ergodic nonsingular S then R is of type II1 (and mildly mixing) [177]. Disjointness and Furstenberg’s Class W ? Two probability preserving systems (X; ; T) and (Y; ; S) are called disjoint if    is the only T  S-invariant probability measure on X  Y whose coordinate projections are  and  respectively. Furstenberg in [69] initiated studying the class W ? of transformations disjoint from all weakly mixing ones. Let D denote the class of distal transformations and M(W ? ) the class of multipliers of W ? (for the definitions see [75]). Then D M(W ? ) W ? . In [43] and [133] it was shown by constructing explicit examples that these inclusions are strict. We record this fact here because nonsingular ergodic theory was the key ingredient of the arguments in the two papers pertaining to the theory of probability preserving systems. The examples are of the form T';S (x; y) D (Tx; S'(x) y), where T is an ergodic rotation on (X; ), (S g ) g2G a mildly mixing action of a locally compact group G on Y and ' : X ! G a measurable map. Let W' denote the Mackey action of G associated with ' and let (Z; ) be the space of this action. The key observation is that there exists an affine isomorphism of the simplex of T';S -invariant probability measures whose pullback on X is  and the simplex of W'  S quasi-invariant probability measures whose pullback on Z is  and whose Radon–Nikodym cocycle is measurable with respect to Z. This is a far reaching generalization of Furstenberg theorem on relative unique ergodicity of ergodic compact group extensions. Symmetric Stable and Infinitely Divisible Stationary Processes Rosinsky in [163] established a remarkable connection between structural studies of stationary stochastic processes and ergodic theory of nonsingular transformations (and flows). For simplicity we consider only real processes in

discrete time. Let X D (X n )n2Z be a measurable stationary symmetric ˛-stable (S˛S) process, 0 < ˛ < 2. This P means that any linear combination nkD1 a k X j k , j k 2 Z, a k 2 R has an S˛S-distribution. (The case ˛ D 2 corresponds to Gaussian processes.) Then the process admits a spectral representation Z Xn D

f n (y) M(dy) ;

n2Z;

(7)

Y

where f n 2 L˛ (Y; ) for a standard  -finite measure space (Y;B;) and M is an independently scattered random measure on B such that E exp (iuM(A)) D exp (juj˛ (A)) for every A 2 B of finite measure. By [163], one can choose the kernel ( f n )n2Z in a special way: there are a -nonsingular transformation T and measurable maps ' : X ! f1; 1g and f 2 L˛ (Y; ) such that f n D U n f , n 2 Z, where U is the isometry of L˛ (X; ) given by Ug D '  (d ı T/d)1/˛  g ı T. If, in addition, the smallest T-invariant  -algebra containing f 1 (BR ) coincides with B and Supp f f ı T n : n 2 Zg D Y then the pair (T; ') is called minimal. It turns out that minimal pairs always exist. Moreover, two minimal pairs (T; ') and (T 0 ; ' 0 ) representing the same S˛S process are equivalent in some natural sense [163]. Then one can relate ergodic-theoretical properties of (T; ') to probabilistic properties of (X n )n2Z . For instance, let Y D C t D be the HopfR decomposition of Y (see Theorem 2). We R let X nD : D D f n (y) M(dy) and X nC :D C f n (y) M(dy). Then we obtain a unique (in distribution) decomposition of X into the sum X D C X C of two independent stationary S˛S-processes. Another kind of decomposition was considered in [171]. Let P be the largest invariant subset of Y such that T  P has a finite invariant measure. Partitioning Y into P and N :D Y n N and restricting the integration in (7) to P and N we obtain a unique (in distribution) decomposition of X into the sum X P C X N of two independent stationary S˛S-processes. Notice that the process X is ergodic if and only if (P) D 0. Recently, Roy considered a more general class of infinitely divisible (ID) stationary processes [165]. Using Maruyama’s representation of the characteristic function of an ID process X without Gaussian part he singled out the Lévy measure Q of X. Then Q is a shift invariant  -finite measure on RZ . Decomposing the dynamical system (RZ ; ; Q) in various natural ways (Hopf decomposition, 0-type and positive type, so-called ‘rigidity free’ part and its complement) he obtains corresponding decompositions for the process X. Here  stands for the shift on RZ .

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Ergodic Theory: Non-singular Transformations

Poisson Suspensions Poisson suspensions are widely used in statistical mechanics to model ideal gas, Lorentz gas, etc (see [33]). Let (X; B; ) be a standard -finite non-atomic measure space and (X) D 1. Denote by e X the space of unordered countable subsets of X. It is called the space of configurations. Fix t > 0. Let A 2 B have positive finite measure and let j 2 ZC . Denote by [A; j] the subset of all configurations e x2e X such that #(e x \ A) D j. Let e B be the -algebra generated by all [A; j]. We define a probability measure  et on e B by two conditions:

Boundaries of Random Walks

j

exp (t(A)); (i)  et ([A; j]) D (t(A)) j! (ii) if A1 ; : : : ; A p are pairwise disjoint Tp Qp et ([A k ; j k ]).  et ( kD1 [A k ; j k ]) D kD1 

singular transformation of a standard probability space (X; B; ) and let f : X ! Rn a measurable function. DeP n fine for m  1, Ym : X ! Rn by Ym :D m1 nD0 f ı T . In other words, (Ym )m1 is the random walk associated with the (non-stationary) process ( f ı T n )n0 . Let us call this random walk recurrent if the cocycle f of T is recurrent (see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). It was shown in [176] that in the case  ı T D , i. e. the process is stationary, this definition is equivalent to the standard one.

then

If T is a -preserving transformation of X and e x D (x1 ; x2 ; : : : ) is a configuration then we set e T! :D T is a e -preserving (Tx1 ; Tx2 ; : : : ). It is easy to verify that e transformation of e X. The dynamical system (e X; e B; e ; e T) is called the Poisson suspension above (X; B; ; T). It is ergodic if and only if T has no invariant sets of finite positive measure. There is a canonical representation of L2 (e X; e ) as the Fock space over L2 (X; ) such that the unitary operator Ue of U T . Thus, the T is the ‘exponent’ P 1  n , where  is (n!) maximal spectral type of Ue n0 T is a measure of the maximal spectral type of U T . It is easy to see that a -finite factor of T corresponds to a factor (called Poissonian) of e T. Moreover, a -finite measurepreserving joining (with -finite projections) of two infinite measure-preserving transformations T 1 and T 2 gene2 [52,164]. e1 and T erates a joining (called Poissonian) of T Thus we see a similarity with the well studied theory of Gaussian dynamical systems [134]. However, the Poissonian case is less understood. There was a recent progress in this field. Parreau and Roy constructed Poisson suspensions whose ergodic self-joinings are all Poissonian [154]. In [111] partial solutions of the following (still open) problems are found: (i) whether the Pinsker factor of e T is Poissonian, (ii) what is the relationship between Krengel’s entropy of T, Parry’s entropy of T and Kolmogorov–Sinai entropy of e T.

Recurrence of Random Walks with Non-stationary Increments Using nonsingular ergodic theory one can introduce the notion of recurrence for random walks obtained from certain non-stationary processes. Let T be an ergodic non-

Boundaries of random walks on groups retain valuable information on the underlying groups (amenability, entropy, etc.) and enable one to obtain integral representation for harmonic functions of the random walk [112,186,187]. Let G be a locally compact group and  a probability measure on G. Let T denote the (one-sided) shift on the probability space (X; B X ; ) :D (G; BG ; )ZC and ' : X ! G a measurable map defined by (y0 ; y1 ; : : : ) 7! y0 . Let T' be the '-skew product extension of T acting on the space (X  G;   G ) (for noninvertible transformations the skew product extension is defined in the very same way as for invertible ones, see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). Then T' is isomorphic to the homogeneous random walk on G with jump probability . Let I (T' ) denote the sub- -algebra of T' -invariant sets and T let F (T' ) :D n>0 T'n (B X ˝ BG ). The former is called the Poisson boundary of T' and the latter one is called the tail boundary of T' . Notice that a nonsingular action of G by inverted right translations along the second coordinate is well defined on each of the two boundaries. The two boundaries (or, more precisely, the G-actions on them) are ergodic. The Poisson boundary is the Mackey range of ' (as a cocycle of T). Hence the Poisson boundary is amenable [187]. If the support of  generates a dense subgroup of G then the corresponding Poisson boundary is weakly mixing [4]. As for the tail boundary, we first note that it can be defined for a wider family of non-homogeneous random walks. This means that the jump probability  is no longer fixed and a sequence (n )n>0 of probability measures on G is considered instead. Now let Q (X; B X ; ) :D n>0 (G; BG ; ). The one-sided shift on X may not be nonsingular now. Instead of it, we consider the tail equivalence relation R on X and a cocycle ˛ : R ! G given by ˛(x; y) D x1    x n y 1 n    y 1 , where x D (x i ) i>0 and y D (y i ) i>0 are R-equivalent and n in the smallest integer such that x i D y i for all i > n. The tail boundary of the random walk on G with time dependent jump

Ergodic Theory: Non-singular Transformations

probabilities (n )n>0 is the Mackey G-action associated with ˛. In the case of homogeneous random walks this definition is equivalent to the initial one. Connes and Woods showed [32] that the tail boundary is always amenable and AT. It is unknown whether the converse holds for general G. However it is true for G D R and G D Z: the class of AT-flows coincides with the class of tail boundaries of the random walks on R and a similar statement holds for Z [32]. Jaworsky showed [109] that if G is countable and a random walk is homogeneous then the tail boundary of the random walk possesses a so-called SAT-property (which is stronger than AT). Classifying  -Finite Ergodic Invariant Measures The description of ergodic finite invariant measures for topological (or, more generally, standard Borel) systems is a well established problem in the classical ergodic theory [33]. On the other hand, it seems impossible to obtain any useful information about the system by analyzing the set of all ergodic quasi-invariant (or just -finite invariant) measures because this set is wildly huge (see Subsect. “The Glimm–Effros Theorem”). The situation changes if we impose some restrictions on the measures. For instance, if the system under question is a homeomorphism (or a topological flow) defined on a locally compact Polish space then it is natural to consider the class of ( -finite) invariant Radon measures, i. e. measures taking finite values on the compact subsets. We give two examples. First, the seminal results of Giordano, Putnam and Skau on the topological orbit equivalence of compact Cantor minimal systems were extended to locally compact Cantor minimal (l.c.c.m.) systems in [37] and [138]. Given a l.c.c.m. system X, we denote by M(X) and M1 (X) the set of invariant Radon measures and the set of invariant probability measures on X. Notice that M1 (X) may be empty [37]. It was shown in [138] that two systems X and X 0 are topologically orbit equivalent if and only if there is a homeomorphism of X onto X 0 which maps bijectively M(X) onto M(X 0 ) and M1 (X) onto M1 (X 0 ). Thus M(X) retains an important information on the system – it is ‘responsible’ for the topological orbit equivalence of the underlying systems. Uniquely ergodic l.c.c.m. systems (with unique up to scaling infinite invariant Radon measure) were constructed in [37]. The second example is related to study of the smooth horocycle flows on tangent bundles of hyperbolic surfaces. Let D be the open disk equipped with the hyperbolic metric jdzj/(1  jzj2 ) and let Möb(D) denote the group of Möbius transformations of D. A hyperbolic surface can be written in the form M :D  nMöb(D), where  is a tor-

sion free discrete subgroup of Möb(D). Suppose that  is a nontrivial normal subgroup of a lattice  0 in Möb(D). Then M is a regular cover of the finite volume surface M0 : D 0 nMöb(D). The group of deck transformations G D 0 / is finitely generated. The horocycle flow (h t ) t2R and the geodesic flow (g t ) t2R defined on the unit tangent bundle T 1 (D) descend naturally to flows, say h and g, on T 1 (M). We consider the problem of classification of the h-invariant Radon measures on M. According to Ratner, h has no finite invariant measures on M if G is infinite (except for measures supported on closed orbits). However there are infinite invariant Radon measures, for instance the volume measure. In the case when G is free Abelian and  0 is co-compact, every homomorphism ' : G ! R determines a unique up to scaling ergodic invariant Radon measure (e.i.r.m.) m on T 1 (M) such that m ı dD D exp ('(D))m for all D 2 G [16] and every e.i.r.m. arises this way [172]. Moreover all these measures are quasi-invariant under g. In the general case, an interesting bijection is established in [131] between the e.i.r.m. which are quasi-invariant under g and the ‘nontrivial minimal’ positive eigenfunctions of the hyperbolic Laplacian on M. Von Neumann Algebras There is a fascinating and productive interplay between nonsingular ergodic theory and von Neumann algebras. The two theories alternately influenced development of each other. Let (X; B; ; T) be a nonsingular dynamical system. Given ' 2 L1 (X; ) and j 2 Z, we define operators A' and U j on the Hilbert space L2 (Z  Z;   ) by setting (A' f )(x; i) :D '(T i x) f (x; i) ; (U j f )(x; i) :D f (x; i  j) : Then U j A' U j D A'ıT j . Denote by M the von Neumann algebra (i. e. the weak closure of the -algebra) generated by A' , ' 2 L1 (X; ) and U j , j 2 Z. If T is ergodic and aperiodic then M is a factor, i. e. M \ M0 D C1, where M0 denotes the algebra of bounded operators commuting with M. It is called a Krieger’s factor. Murray–von Neumann–Connes’ type of M is exactly the Krieger’s type of T. The flow of weights of M is isomorphic to the associated flow of T. Two Krieger’s factors are isomorphic if and only if the underlying dynamical systems are orbit equivalent [129]. Moreover, a number of important problems in the theory of von Neumann algebras such as classification of subfactors, computation of the flow of weights and Connes’ invariants, outer conjugacy for

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automorphisms, etc. are intimately related to the corresponding problems in nonsingular orbit theory. We refer to [42,66,73,74,89,142] for details. Representations of CAR Representations of canonical anticommutation relations (CAR) is one of the most elegant and useful chapters of mathematical physics, providing a natural language for many body quantum physics and quantum field theory. By a representation of CAR we mean a sequence of bounded linear operators a1 ; a2 ; : : : in a separable Hilbert space K such that a j a k C a k a j D 0 and a j ak C a k a j D ı j;k . Consider f0; 1g as a group with addition mod 2. Then X D f0; 1gN is a compact Abelian group. Let  :D fx D (x1 ; x2 ; : : : ) : limn!1 x n D 0g. Then  is a dense countable subgroup of X. It is generated by elements k whose k-coordinate is 1 and the other ones are 0.  acts on X by translations. Let  be an ergodic  -quasi-invariant measure on X. Let (C k ) k1 be Borel maps from X to the group of unitary operators in a Hilbert space H satisfying C k (x) D C k (x C ı k ), C k (x)C l (x C ı l ) D C l (x)C k (x C ı k ), k ¤ l for a. a. x. In other words, (C k ) k1 f :D defines a cocycle of the  -action. We now put H 2 f L (X; ) ˝ H and define operators ak in H by setting (a k f )(x) D (1)x 1 CCx k1 (1  x k ) s d ı ı k (x) f (x C ı k ) ;  C k (x) d f and x D (x1 ; x2 ; where f : X ! H is an element of H : : : ) 2 X. It is easy to verify that a1 ; a2 ; : : : is a representation of CAR. The converse was established in [72] and [77]: every factor-representation (this means that the von Neumann algebra generated by all ak is a factor) of CAR can be represented as above for some ergodic measure , Hilbert space H and a  -cocycle (C k ) k1 . Moreover, using nonsingular ergodic theory Golodets [77] constructed for each k D 2; 3; : : : ; 1, an irreducible representation of CAR such that dim H D k. This answered a question of Gårding and Wightman [72] who considered only the case k D 1. Unitary Representations of Locally Compact Groups Nonsingular actions appear in a systematic way in the theory of unitary representations of groups. Let G be a locally compact second countable group and H a closed normal subgroup of G. Suppose that H is commutative (or, more generally, of type I, see [53]). Then the natural action of G by conjugation on H induces a Borel G-action, say ˛, on the dual space b H – the set of unitarily equivalent

classes of irreducible unitary representations of H. If now U D (U g ) g2G is a unitary representation of G in a separable Hilbert space then by applying Stone decomposition theorem to U  H one can deduce that ˛ is nonsingular with respect to a measure  of the ‘maximal spectral b Moreover, if U is irreducible then ˛ type’ for U  H on H. is ergodic. Whenever  is fixed, we obtain a one-to-one correspondence between the set of cohomology classes of irreducible cocycles for ˛ with values in the unitary group on a Hilbert space H and the subset of b G consisting of classes of those unitary representations V for which the measure associated to V  H is equivalent to . This correspondence is used in both directions. From information about cocycles we can deduce facts about representations and vise versa [53,118]. Concluding Remarks While some of the results that we have cited for nonsingular Z-actions extend to actions of locally compact Polish groups (or subclasses of Abelian or amenable ones), many natural questions remain open in the general setting. For instance: what is Rokhlin lemma, or the pointwise ergodic theorem, or the definition of entropy for nonsingular actions of general countable amenable groups? The theory of abstract nonsingular equivalence relations [66] or, more generally, nonsingular groupoids [160] and polymorphisms [184] is also a beautiful part of nonsingular ergodic theory that has nice applications: description of semifinite traces of AF-algebras, classification of factor representations of the infinite symmetric group [185], path groups [14], etc. Nonsingular ergodic theory is getting even more sophisticated when we pass from Z-actions to noninvertible endomorphisms or, more generally, semigroup actions (see [3] and references therein). However, due to restrictions of space we do not consider these issues in our survey. Bibliography 1. Aaronson J (1983) The eigenvalues of nonsingular transformations. Isr J Math 45:297–312 2. Aaronson J (1987) The intrinsic normalizing constants of transformations preserving infinite measures. J Analyse Math 49:239–270 3. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. Amer Math Soc, Providence ´ 4. Aaronson J, Lemanczyk M (2005) Exactness of Rokhlin endomorphisms and weak mixing of Poisson boundaries, Algebraic and Topological Dynamics. Contemporary Mathematics, vol 385. Amer Math Soc, Providence 77–88 5. Aaronson J, Lin M, Weiss B (1979) Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products. Isr J Math 33:198–224

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Ergodic Theory: Recurrence

Ergodic Theory: Recurrence N IKOS FRANTZIKINAKIS, RANDALL MCCUTCHEON Department of Mathematics, University of Memphis, Memphis, USA Article Outline Glossary Definition of the Subject Introduction Quantitative Poincaré Recurrence Subsequence Recurrence Multiple Recurrence Connections with Combinatorics and Number Theory Future Directions Bibliography Glossary Almost every, essentially Given a Lebesgue measure space (X; B; ), a property P(x) predicated of elements of X is said to hold for almost every x 2 X, if the set X n fx : P(x) holdsg has zero measure. Two sets A; B 2 B are essentially disjoint if (A \ B) D 0. Conservative system Is an infinite measure preserving system such that for no set A 2 B with positive measure are A; T 1 A; T 2 A; : : : pairwise essentially disjoint. (cn )-Conservative system If (c n )n2N is a decreasing sequence of positive real numbers, a conservative ergodic measure preserving transformation T is (c n )conservative if for some non-negative function f 2 P n L1 (), 1 nD1 c n f (T x) D 1 a. e. Doubling map If T is the interval [0; 1] with its endpoints identified and addition performed modulo 1, the (non-invertible) transformation T : T ! T , defined by Tx D 2x mod 1, preserves Lebesgue measure, hence induces a measure preserving system on T . Ergodic system Is a measure preserving system (X; B; ; T) (finite or infinite) such that every A 2 B that is T-invariant (i. e. T 1 A D A) satisfies either (A) D 0 or (X n A) D 0. (One can check that the rotation R˛ is ergodic if and only if ˛ is irrational, and that the doubling map is ergodic). Ergodic decomposition Every measure preserving system (X; X ; ; T) can be expressed as an integral of ergodic systems; for example, one can write  D R  t d(t), where  is a probability measure on [0; 1] and  t are T-invariant probability measures on (X; X ) such that the systems (X; X ;  t ; T) are ergodic for t 2 [0; 1].

Ergodic theorem States that if (X; B; ; T) is a measure 2 preserving system and  1 PN  f 2 L (), then lim N!1 nf P   T D 0, where P f denotes the f L 2 () nD1 N orthogonal projection of the function f onto the subspace f f 2 L2 () : T f D f g. Hausdorff a-measure Let (X; B; ; T) be a measure preserving system endowed with a -compatible metric d. The Hausdorff a-measure H a (X) of X is an outer measure defined for all subsets of X as follows: First, for P a A X and " > 0 let H a;"(A) D inff 1 iD1 r i g, where the infimum is taken over all countable coverings of A by sets U i X with diameter r i < ". Then define H a (A) D lim sup"!0 H a;"(A). Infinite measure preserving system Same as measure preserving system, but (X) D 1. Invertible system Is a measure preserving system (X; B; ; T) (finite or infinite), with the property that there exists X0 2 X, with (X n X 0 ) D 0, and such that the transformation T : X0 ! X0 is bijective, with T 1 measurable. Measure preserving system Is a quadruple (X; B; ; T), where X is a set, B is a  -algebra of subsets of X (i. e. B is closed under countable unions and complementation),  is a probability measure (i. e. a countably additive function from B to [0; 1] with (X) D 1), and T : X ! X is measurable (i. e. T 1 A D fx 2 X : Tx 2 Ag 2 B for A 2 B), and -preserving (i. e. (T 1 A) D (A)). Moreover, throughout the discussion we assume that the measure space (X; B; ) is Lebesgue (see Sect. 1.0 in [2]). -Compatible metric Is a separable metric on X, where (X; B; ) is a probability space, having the property that open sets are measurable. Positive definite sequence Is a complex-valued sequence (a n )n2Z such that for any n1 ; : : : ; n k 2 Z and z1 ; : : : ; P z k 2 C, ki; jD1 a n i n j z i z j  0. Rotations on T If T is the interval [0; 1] with its endpoints identified and addition performed modulo 1, then for every ˛ 2 R the transformation R˛ : T ! T , defined by R˛ x D x C ˛mod1, preserves Lebesgue measure on T and hence induces a measure preserving system on T . Syndetic set Is a subset E Z having bounded gaps. If G is a general discrete group, a set E G is syndetic if G D FE for some finite set F G. Upper density Is the number d() D lim sup N!1 (j \ fN; : : : ; Ngj)/(2N C 1), where  Z (assuming the limit to exist). Alternatively for measurable E

Rm , D(E) D lim sup l (S)!1 (m(S \ E))/(m(S)), where S ranges over all cubes in Rm , and l(S) denotes the length of the shortest edge of S.

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Notation The following notation will be used throughout the article: T f D f ı T, fxg D x  [x], D- limn!1 (a n ) D a $ d fn : ja n  aj > "g D 0 for every " > 0.

a finite (or conservative) measure preserving system, every set of positive measure (or almost every point) comes back to itself infinitely many times under iteration. Despite the profound importance of these results, their proofs are extremely simple.

Definition of the Subject

Theorem 1 (Poincaré Recurrence for Sets) Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then (A \ T n A) > 0 for infinitely many n 2 N.

The basic principle that lies behind several recurrence phenomena is that the typical trajectory of a system with finite volume comes back infinitely often to any neighborhood of its initial point. This principle was first exploited by Poincaré in his 1890 King Oscar prize-winning memoir that studied planetary motion. Using the prototype of an ergodic-theoretic argument, he showed that in any system of point masses having fixed total energy that restricts its dynamics to bounded subsets of its phase space, the typical state of motion (characterized by configurations and velocities) must recur to an arbitrary degree of approximation. Among the recurrence principle’s more spectacularly counterintuitive ramifications is that isolated ideal gas systems that do not lose energy will return arbitrarily closely to their initial states, even when such a return entails a decrease in entropy from equilibrium, in apparent contradiction to the second law of thermodynamics. Such concerns, previously canvassed by Poincaré himself, were more infamously expounded by Zermelo [74] in 1896. Subsequent clarifications by Boltzmann, Maxwell and others led to an improved understanding of the second law’s primarily statistical nature. (For an interesting historical/philosophical discussion, see [68]; also [10]. For a probabilistic analysis of the likelihood of observing second law violations in small systems over short time intervals, see [28]). These discoveries had a profound impact in dynamics, and the theory of measure preserving transformations (ergodic theory) evolved from these developments. Since then, the Poincaré recurrence principle has been applied to a variety of different fields in mathematics, physics, and information theory. In this article we survey the impact it has had in ergodic theory, especially as pertains to the field of ergodic Ramsey theory. (The heavy emphasis herein on the latter reflects authorial interest, and is not intended to transmit a proportionate image of the broader landscape of research relating to recurrence in ergodic theory.) Background information we assume in this article can be found in the books [35,63,71] ( Measure Preserving Systems). Introduction In this section we shall give several formulations of the Poincaré recurrence principle using the language of ergodic theory. Roughly speaking, the principle states that in

Proof Since T is measure preserving, the sets A; T 1 A; T 2 A; : : : have the same measure. These sets cannot be pairwise essentially disjoint, since then the union of finitely many of them would have measure greater than (X) D 1. Therefore, there exist m; n 2 N, with n > m, such that (T m A\T n A) > 0. Again since T is measure preserving, we conclude that (A \ T k A) > 0, where k D n  m > 0. Repeating this argument for the iterates A; T m A; T 2m A; : : :, for all m 2 N, we easily deduce  that (A \ T n A) > 0 for infinitely many n 2 N. We remark that the above argument actually shows that 1 (A \ T n A) > 0 for some n  [ (A) ] C 1. Theorem 2 (Poincaré Recurrence for Points) Let (X; B; ; T) be a measure preserving system and A 2 B. Then for almost every x 2 A we have that T n x 2 A for infinitely many n 2 N. Proof Let B be the set of x 2 A such that T n x … A for all S n 2 N. Notice that B D A n n2N T n A; in particular, B is measurable. Since the iterates B; T 1 B; T 2 B; : : : are pairwise essentially disjoint, we conclude (as in the proof of Theorem 1) that (B) D 0. This shows that for almost every x 2 A we have that T n x 2 A for some n 2 N. Repeating this argument for the transformation T m in place of T for all m 2 N, we easily deduce the advertised statement.  Next we give a variation of Poincaré recurrence for measure preserving systems endowed with a compatible metric: Theorem 3 (Poincaré Recurrence for Metric Systems) Let (X; B; ; T) be a measure preserving system, and suppose that X is endowed with a -compatible metric. Then for almost every x 2 X we have lim inf d(x; T n x) D 0 : n!1

The proof of this result is similar to the proof of Theorem 2 (see p. 61 in [35]). Applying this result to the doubling map Tx D 2x on T , we get that for almost every x 2 X, every string of zeros and ones in the dyadic expansion of x occurs infinitely often.

Ergodic Theory: Recurrence

n x 2 X the set R S x D fn 2 N : T x 2 Ag has well defined density and d(S x ) d(x) D (A). Furthermore, for ergodic measure preserving systems we have d(S x ) D (A) a. e.

We remark that all three formulations of the Poincaré Recurrence Theorem that we have given hold for conservative systems as well. See, e. g., [2] for details. This article is structured as follows. In Sect. “Quantitative Poincaré Recurrence” we give a few quantitative versions of the previously mentioned qualitative results. In Sects. “Subsequence Recurrence” and “Multiple Recurrence” we give several refinements of the Poincaré recurrence theorem, by restricting the scope of the return time n, and by considering multiple intersections (for simplicity we focus on Z-actions). In Sect. “Connections with Combinatorics and Number Theory” we give various implications of the recurrence results in combinatorics and number theory ( Ergodic Theory: Interactions with Combinatorics and Number Theory). Lastly, in Sect. “Future Directions” we give several open problems related to the material presented in Sects. “Subsequence Recurrence” to “Connections with Combinatorics and Number Theory”.

Theorem 6 (Kac [51]) Let (X; B; ; T) be an ergodic measure preserving system and A 2 B with (A) > 0. For x 2 X define R A (x) D minfn 2 N : T n x 2 Ag. RThen for x 2 A the expected value of R A (x) is 1/(A), i. e. A R A (x) d D 1.

Quantitative Poincaré Recurrence

More Recent Results

Early Results

As we mentioned in the previous section, if the space X is endowed with a -compatible metric d, then for almost every x 2 X we have that lim infn!1 d(x; T n x) D 0. A natural question is, how much iteration is needed to come back within a small distance of a given typical point? Under some additional hypothesis on the metric d we have the following answer:

For applications it is desirable to have quantitative versions of the results mentioned in the previous section. For example one would like to know how large (A \ T n A) can be made and for how many n. Theorem 4 (Khintchine [55]) Let (X; B; ; T) be a measure preserving system and A 2 B. Then for every " > 0 we have (A \ T n A) > (A)2  " for a set of n 2 N that has bounded gaps. By considering the doubling map Tx D 2x on T and letting A D 1[0;1/2) , it is easy to check that the lower bound of the previous result cannot be improved. We also remark that it is not possible to estimate the size of the gap by a function of (A) alone. One can see this by considering the rotations R k x D x C 1/k for k 2 N, defined on T , and letting A D 1[0;1/3] . Concerning the second version of the Poincaré recurrence theorem, it is natural to ask whether for almost every x 2 X the set of return times S x D fn 2 N : T n x 2 Ag has bounded gaps. This is not the case, as one can see by considering the doubling map Tx D 2x on T with the Lebesgue measure, and letting A D 1[0;1/2) . Since Lebesgue almost every x 2 T contains arbitrarily large blocks of ones in its dyadic expansion, the set S x has unbounded gaps. Nevertheless, as an easy consequence of the Birkhoff ergodic theorem [19], one has the following: Theorem 5 Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then for almost every

Another question that arises naturally is, given a set A with positive measure and an x 2 A, how long should one wait until some iterate T n x of x hits A? By considering an irrational rotation R˛ on T , where ˛ is very near to, but not 1 , and letting A D 1[0;1/2] , one can see that less than, 100 the first return time is a member of the set f1; 50; 51g. So it may come as a surprise that the average first return time does not depend on the system (as long as it is ergodic), but only on the measure of the set A.

Theorem 7 (Boshernitzan [20]) Let (X; B; ; T) be a measure preserving system endowed with a -compatible metric d. Assume that the Hausdorff a-measure H a (X) of X is  -finite (i. e., X is a countable union of sets X i with H a (X i ) < 1). Then for almost every x 2 X, ˚ 1 lim inf n a  d(x; T n x) < 1 : n!1

Furthermore, if H a (X) D 0, then for almost every x 2 X, ˚ 1 lim inf n a  d(x; T n x) D 0 : n!1

One can see from rotations by “badly approximable” vectors ˛ 2 T k that the exponent 1/a in the previous theorem cannot be improved. Several applications of Theorem 7 to billiard flows, dyadic transformations, symbolic flows and interval exchange transformations are given in [20]. For a related result dealing with mean values of the limits in Theorem 7 see [67]. An interesting connection between rates of recurrence and entropy of an ergodic measure preserving system was established by Ornstein and Weiss [62], following earlier work of Wyner and Ziv [73]:

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Theorem 8 (Ornstein and Weiss [62]) Let (X; B; ; T) be an ergodic measure preserving system and P be a finite partition of X. Let Pn (x) be the element of the partition Wn1 i Tn1 i (i) : P(i) 2 P ; 0  i < ng iD0 T P D f iD0 T P that contains x. Then for almost every x 2 X, the first return time R n (x) of x to Pn (x) is asymptotically equivalent to e h(T;P )n , where h(T; P ) denotes the entropy of the system with respect to the partition P . More precisely, lim

n!1

log R n (x) D h(T; P ) : n

An extension of the above result to some classes of infinite measure preserving systems was given in [42]. Another connection of recurrence rates, this time with the local dimension of an invariant measure, is given by the next result: Theorem 9 (Barreira [4]) Let (X; B; ; T) be an ergodic measure preserving system. Define the upper and lower recurrence rates log r (x) and  log r log r (x) R(x) D lim sup ;  log r r!0

R(x) D lim inf r!0

where r (x) is the first return time of T k x to B(x; r), and the upper and lower pointwise dimensions log (B(x; r)) and log r log (B(x; r)) d  (x) D lim sup : log r r!0

d  (x) D lim inf r!0

Then for almost every x 2 X, we have R(x)  d  (x) and

R(x)  d  (x) :

Roughly speaking, this theorem asserts that for typical x 2 X and for small r, the first return time of x to B(x; r) is at most rd (x) . Since d  (x)  H a (X) for almost every x 2 X, we can conclude the first part of Theorem 7 from Theorem 9. For related results the interested reader should consult the survey [5] and the bibliography therein. We also remark that the previous results and related concepts have been applied to estimate the dimension of certain strange attractors (see [49] and the references therein) and the entropy of some Gibbsian systems [25]. We end this section with a result that connects “wandering rates” of sets in infinite measure preserving systems with their “recurrence rates”. The next theorem follows easily from a result about lower bounds on ergodic aver-

ages for measure preserving systems due to Leibman [57]; a weaker form for conservative, ergodic systems can be found in Aaronson [1]. Theorem 10 Let (X; B; ; T) be an infinite measure preserving system, and A 2 B with (A) < 1. Then for all N 2 N,  0 S N1 1 n A N1  T X nD0 1 @ (A \ T n A)A  ((A))2 :  N 2 nD0

Subsequence Recurrence In this section we discuss what restrictions one can impose on the set of return times in the various versions of the Poincaré recurrence theorem. We start with: Definition 11 Let R Z. Then R is a set of: (a) Recurrence if for any invertible measure preserving system (X; B; ; T), and A 2 B with (A) > 0, there is some nonzero n 2 R such that (A \ T n A) > 0. (b) Topological recurrence if for every compact metric space (X; d), continuous transformation T : X ! X and every " > 0, there are x 2 X and nonzero n 2 R such that d(x; T n x) < ". It is easy to check that the existence of a single n 2 R satisfying the previous recurrence conditions actually guarantees the existence of infinitely many n 2 R satisfying the same conditions. Moreover, if R is a set of recurrence then one can see from existence of some T-invariant measure  that R is also a set of topological recurrence. A (complicated) example showing that the converse is not true was given by Kriz [56]. Before giving a list of examples of sets of (topological) recurrence, we discuss some necessary conditions: A set of topological recurrence must contain infinitely many multiples of every positive integer, as one can see by considering rotations on Zd , d 2 N . Hence, the sets f2n C 1; n 2 Ng, fn2 C 1; n 2 Ng, fp C 2; p primeg are not good for (topological) recurrence. If (s n )n2N is a lacunary sequence (meaning lim infn!1 (s nC1 /s n ) D > 1), then one can construct an irrational number ˛ such that fs n ˛g 2 [ı; 1  ı] for all large n 2 N, where ı > 0 depends on (see [54], for example). As a consequence, the sequence (s n )n2N is not good for (topological) recurrence. Lastly, we mention that by considering product systems, one can immediately show that any set of (topological) recurrence R is partition regular, meaning that if R is partitioned into finitely many pieces then at least one of these pieces must still be a set of (topological) recur-

Ergodic Theory: Recurrence

rence. Using this observation, one concludes for example that any union of finitely many lacunary sequences is not a set of recurrence. We present now some examples of sets of recurrence: Theorem 12 The following are sets of recurrence: S (i) Any set of the form n2N fa n ; 2a n ; : : : ; na n g where a n 2 N. (ii) Any IP-set, meaning a set that consists of all finite sums of the members of some infinite set. (iii) Any difference set S  S, meaning a set that consists of all possible differences of the members of some some infinite set S. (iv) The set fp(n); n 2 Ng where p is any nonconstant integer polynomial with p(0) D 0 [35,66] (In fact we only have to assume that the range of the polynomial contains multiples of an arbitrary positive integer [53]). (v) The set fp(n); n 2 Sg, where p is an integer polynomial with p(0) D 0 and S is any IP-set [12]. (vi) The set of values of an admissible generalized polynomial (this class contains in particular the smallest function algebra G containing all integer polynomials having zero constant term and such that if g1 ; : : : ; g k 2 G and c1 ; : : : ; c k 2 R then P [[ kiD1 c i g i ]] 2 G, where [[x]] D [x C 12 ] denotes the integer nearest to x) [13]. (vii) The set of shifted primes fp  1; p primeg, and the set fp C 1; p primeg [66]. (viii) The set of values of a random non-lacunary sequence. (Pick n 2 N independently with probability b n where 0  b n  1 and limn!1 nb n D 1. The resulting set is almost surely a set of recurrence. If lim supn!1 nb n < 1 then the resulting set is almost surely a finite union of sets, each of which is the range of some lacunary sequence, hence is not a set of recurrence). Follows from [22]. Showing that the first three sets are good for recurrence is a straightforward modification of the argument used to prove Theorem 1. Examples (iv)  (viii) require more work. A criterion of Kamae and Mendés-France [53] provides a powerful tool that may be used in many instances to establish that a set R is a set of recurrence. We mention a variation of their result: Theorem 13 (Kamae and Mendés-France [53]) Suppose that R D fa1 < a2 < : : :g is a subset of N such that: (i) The sequence fa n ˛gn2N is uniformly distributed in T for every irrational ˛.

(ii) The set R m D fn 2 N : mja n g has positive density for every m 2 N. Then R is a set of recurrence. We sketch a proof for this result. First, recall Herglotz’s theorem: if (a n )n2Z is a positive definite sequence, then there isR a unique measure  on the torus T such that 2 i nt Ra n D e n d (t). The case of interest to us is a n D T f (x)  f (T x) d, where T is measure preserving and f 2 L1 (); (a n ) is positive definite, and we call  D  f the spectral measure of f . Let now (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Putting f D 1A , one has lim

N!1

N Z 1 X f (x)  f (T a n x) d N nD1

Z D

lim

T N!1

N 1 X 2 i a n t e N nD1

! d f (t) :

(1)

For t irrational the limit inside the integral is zero (by condition (i)), so the last integral can be taken over the rational points in T . Since the spectral measure of a function orthogonal to the subspace H D f f 2 L2 () : there exists k 2 N with T k f D f g

(2) has no rational point masses, we can easily deduce that when computing the first limit in (1), we can replace the function f by its orthogonal projection g onto the subspace H (g is again nonnegative and g ¤ 0). To complete the argument, we approximate g by a function g 0 such that T m g 0 D g 0 for some appropriately chosen m, and use condition (ii) to deduce that the limit of the average (1) is positive. In order to apply Theorem 13, one uses the standard machinery of uniform distribution. Recall Weyl’s criterion: a real-valued sequence (x n )n2N is uniformly distributed mod 1 if for every non-zero k 2 Z, lim

N!1

N 1 X 2 i kx n e D0: N nD1

This criterion becomes especially useful when paired with van der Corput’s so-called third principal property: if, for every h 2 N, (x nCh  x n )n2N is uniformly distributed mod 1, then (x n )n2N is uniformly distributed mod 1. Using the foregoing criteria and some standard (albeit nontrivial) exponential sum estimates, one can verify for example that the sets (iv) and (vii) in Theorem 12 are good for recurrence.

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In light of the connection elucidated above between uniform distribution mod 1 and recurrence, it is not surprising that van der Corput’s method has been adapted by modern ergodic theorists for use in establishing recurrence properties directly.

pute

N 1 X hx nCm ; x n i N!1 N nD1

lim

D lim

N!1

Theorem 14 (Bergelson [7]) Let (x n )n2N be a bounded sequence in a Hilbert space. If D-lim

m!1

! N 1 X hx nCm ; x n i D 0 ; lim N!1 N nD1

then   N 1 X    xn  D 0 : lim   N!1  N nD1

Let us illustrate how one uses this “van der Corput trick” by showing that S D fn2 : n 2 Ng is a set of recurrence. We will actually establish the following stronger fact: If (X; B; ; T) is a measure preserving system and f 2 L1 () is nonnegative and f ¤ 0 then lim inf N!1

1 N

N Z X

2

f (x)  f (T n x) d > 0 :

(3)

nD1

Then our result follows by setting f D 1A for some A 2 B with (A) > 0. The main idea is one that occurs frequently in ergodic theory; split the function f into two components, one of which contributes zero to the limit appearing in (3), and the other one being much easier to handle than f . To do this consider the T-invariant subspace of L2 (X) defined by H D f f 2 L2 () : there exists k 2 N with T k f D f g :

(4) Write f D g C h where g 2 H and h?H , and expand the average in (3) into a sum of four averages involving the functions g and h. Two of these averages vanish because iterates of g are orthogonal to iterates of h. So in order to show that the only contribution comes from the average that involves the function g alone, it suffices to establish that   N  1 X  n2  lim  T h D0: (5)  2 N!1  N nD1 L ()

To show this we will apply the Hilbert space van der Cor2 put lemma. For given h 2 N, we let x n D T n h and com-

N Z 1 X 2 2 2 T n C2nmCm h  T n h d N nD1

N Z 1 X 2 D lim T 2nm (T m h)  h d : N!1 N nD1

Applying the ergodic theorem to the transformation T 2m and using the fact that h?H , we get that the last limit is 0. This implies (5). Thus far we have shown that in order to compute the limit in (3) we can assume that f D g 2 H (g is also nonnegative and g ¤ 0). By the definition of H , given any " > 0, there exists a function f 0 2 H such that T k f 0 D f 0 for some k 2 N and k f  f 0 kL 2 ()  ". Then the limit in (3) is at least 1/k times the limit N Z 1 X 2 f (x)  f (T (kn) x) d: lim inf N!1 N nD1 Applying the triangle inequality twice we get that this is greater or equal than N Z 1 X 2 lim f 0 (x)  f 0 (T (kn) x) d  c  " N!1 N nD1 Z D ( f 0 (x))2 d  2" Z 

2 f 0 (x) d  c  ";

for some constant c that does not depend on " (we used that T k f 0 D f 0 and the Cauchy–Schwartz inequality). Choosing " small enough we conclude that the last quantity is positive, completing the proof. Multiple Recurrence Simultaneous multiple returns of positive measure sets to themselves were first considered by H. Furstenberg [34], who gave a new proof of Szemerédi’s theorem [69] on arithmetic progressions by deriving it from the following theorem: Theorem 15 (Furstenberg [34]) Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then for every k 2 N, there is some n 2 N such that (A \ T n A \    \ T kn A) > 0 :

(6)

Furstenberg’s proof came by means of a new structure theorem allowing one to decompose an arbitrary measure preserving system into component elements exhibiting

Ergodic Theory: Recurrence

one of two extreme types of behavior: compactness, characterized by regular, “almost periodic” trajectories, and weak mixing, characterized by irregular, “quasi-random” trajectories. On T , these types of behavior are exemplified by rotations and by the doubling map, respectively. To see the point, imagine trying to predict the initial digit of the dyadic expansion of T n x given knowledge of the initial digits of T i x, 1  i < n. We use the case k D 2 to illustrate the basic idea. It suffices to show that if f 2 L1 () is nonnegative and f ¤ 0, one has N Z 1 X f (x)  f (T n x)  f (T 2n x) d > 0 : (7) lim inf N!1 N nD1 An ergodic decomposition argument enables us to assume that our system is ergodic. As in the earlier case of the squares, we split f into “almost periodic” and “quasi-random” components. Let K be the closure in L2 of the subspace spanned by the eigenfunctions of T, i. e. the functions f 2 L2 () that satisfy f (Tx) D e 2 i˛ f (x) for some ˛ 2 R. We write f D g C h, where g 2 K and h?K . It can be shown that g; h 2 L1 () and g is again nonnegative with g ¤ 0. We expand the average in (7) into a sum of eight averages involving the functions g and h. In order to show that the only non-zero contribution to the limit comes from the term involving g alone, it suffices to establish that   N  1 X  n 2n  lim  T g  T h D0; (8)  2 N!1  N nD1

L ()

(and similarly with h and g interchanged, and with g D h, which is similar). To establish (8), we use the Hilbert space van der Corput lemma on x n D T n g  T 2n h. Some routine computations and a use of the ergodic theorem reduce the task to showing that  Z D-lim h(x)  h(T 2m x) d D 0 : m!1

But this is well known for h?K (in virtue of the fact that for h?K the spectral measure  h is continuous, for example). We are left with the average (7) when f D g 2 K . In this case f can be approximated arbitrarily well by a linear combination of eigenfunctions, which easily implies that given " > 0 one has kT n f  f kL 2 ()  " for a set of n 2 N with bounded gaps. Using this fact and the triangle inequality, one finds that for a set of n 2 N with bounded gaps, Z 3 Z f (x)  f (T n x)  f (T 2n x) d  f d  c  "

for a constant c that is independent of ". Choosing " small enough, we get (7). The new techniques developed for the proof of Theorem 15 have led to a number of extensions, many of which have to date only ergodic proofs. To expedite discussion of some of these developments, we introduce a definition: Definition 16 Let R Z and k 2 N. Then R is a set of k-recurrence if for every invertible measure preserving system (X; B; ; T) and A 2 B with (A) > 0, there is some nonzero n 2 R such that (A \ T n A \    \ T kn A) > 0 : The notions of k-recurrence are distinct for different values of k. An example of a difference set that is a set of 1-recurrence but not a set of 2-recurrence was given in [34]; sets of k-recurrence that are not sets of (k C 1)recurrencepfor general k were given in [31] (R k D fn 2 N : fn kC1 2g 2 [1/4; 3/4]g is such). Aside from difference sets, the sets of (1-)recurrence given in Theorem 12 may well be sets of k-recurrence for every k 2 N, though this has not been verified in all cases. Let us summarize the current state of knowledge. The following are sets of k-recurrence for every k: Sets of S the form n2N fa n ; 2a n ; : : : ; na n g where a n 2 N (this follows from a uniform version of Theorem 15 that can be found in [15]). Every IP-set [37]. The set fp(n); n 2 Ng where p is any nonconstant integer polynomial with p(0) D 0 [16], and more generally, when the range of the polynomial contains multiples of an arbitrary integer [33]. The set fp(n); n 2 Sg where p is an integer polynomial with p(0) D 0 and S is any IP-set [17]. The set of values of an admissible generalized polynomial [60]. Moreover, the set of shifted primes fp  1; p primeg, and the set fp C 1; p primeg are sets of 2-recurrence [32]. More generally, one would like to know for which sequences of integers a1 (n); : : : ; a k (n) it is the case that for every invertible measure preserving system (X; B; ; T) and A 2 B with (A) > 0, there is some nonzero n 2 N such that (A \ T a 1 (n) A \    \ T a k (n) A) > 0 :

(9)

Unfortunately, a criterion analogous to the one given in Theorem 13 for 1-recurrence is not yet available for k-recurrence when k > 1. Nevertheless, there have been some notable positive results, such as the following: Theorem 17 (Bergelson and Leibman [16])

Let (X;

B; ; T) be an invertible measure preserving system and

p1 (n); : : : ; p k (n) be integer polynomials with zero constant term. Then for every A 2 B with (A) > 0, there is some

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n 2 N such that (A \ T p 1 (n) A \    \ T p k (n) A) > 0 :

(10)

Furthermore, it has been shown that the n in (10) can be chosen from any IP set [17], and the polynomials p1 ; : : : ; p k can be chosen to belong to the more general class of admissible generalized polynomials [60]. Very recently, a new boost in the area of multiple recurrence was given by a breakthrough of Host and Kra [50]. Building on work of Conze and Lesigne [26,27] and Furstenberg and Weiss [41] (see also the excellent survey [52], exploring close parallels with [45] and the seminal paper of Gowers [43]), they isolated the structured component (or factor) of a measure preserving system that one needs to analyze in order to prove various multiple recurrence and convergence results. This allowed them, in particular, to prove existence of L2 limits for the so-called PN Qk in “Furstenberg ergodic averages” N1 nD1 iD0 f (T x), which had been a major open problem since the original ergodic proof of Szemerédi’s theorem. Subsequently Ziegler in [75] gave a new proof of the aforementioned limit theorem and established minimality of the factor in question. It turns out that this minimal component admits of a purely algebraic characterization; it is a nilsystem, i. e. a rotation on a homogeneous space of a nilpotent Lie group. This fact, coupled with some recent results about nilsystems (see [58,59] for example), makes the analysis of some otherwise intractable multiple recurrence problems much more manageable. For example, these developments have made it possible to estimate the size of the multiple intersection in (6) for k D 2; 3 (the case k D 1 is Theorem 4): Theorem 18 (Bergelson, Host and Kra [14]) Let (X; B; ; T) be an ergodic measure preserving system and A 2 B. Then for k D 2; 3 and for every " > 0, (A \ T n A \    \ T kn A) >  kC1 (A)  "

(11)

for a set of n 2 N with bounded gaps. Based on work of Ruzsa that appears as an appendix to the paper, it is also shown in [14] that a similar estimate fails for ergodic systems (with any power of (A) on the right hand side) when k  4. Moreover, when the system is nonergodic it also fails for k D 2; 3, as can be seen with the help of an example in [6]. Again considering the doubling map Tx D 2x and the set A D [0; 1/2], one sees that the positive results for k  3 are sharp. When the polynomials n; 2n; : : : ; kn are replaced by linearly independent polynomials p1 ; p2 ; : : : ; p k with zero constant term, similar lower bounds hold for every k 2 N without assuming

ergodicity [30]. The case where the polynomials n; 2n; 3n are replaced with general polynomials p1 ; p2 ; p3 with zero constant term is treated in [33]. Connections with Combinatorics and Number Theory The combinatorial ramifications of ergodic-theoretic recurrence were first observed by Furstenberg, who perceived a correspondence between recurrence properties of measure preserving systems and the existence of structures in sets of integers having positive upper density. This gave rise to the field of ergodic Ramsey theory, in which problems in combinatorial number theory are treated using techniques from ergodic theory. The following formulation is from [8]. Theorem 19 Let  be a subset of the integers. There exists an invertible measure preserving system (X; B; ; T) and a set A 2 B with (A) D d() such that d( \ (  n1 ) \ : : : \ (  n k ))  (A \ T n 1 A \    \ T n k A) ;

(12)

for all k 2 N and n1 ; : : : ; n k 2 Z. Proof The space X will be taken to be the sequence space f0; 1gZ , B is the Borel  -algebra, while T is the shift map defined by (Tx)(n) D x(n C 1) for x 2 f0; 1gZ , and A is the set of sequences x with x(0) D 1. So the only thing that depends on  is the measure  which we now define. For m 2 N set 0 D Z n  and 1 D . Using a diagonal argument we can find an increasing sequence of integers (N m )m2N such that limm!1 j \ [1; N m ]j/N m D d() and such that ˇ ˇ ˇ ( i 1  n1 ) \ ( i 2  n2 ) \    ˇ ˇ ˇ ˇ ˇ \( i r  nr ) \ [1; N m ] (13) lim m!1 Nm exists for every n1 ; : : : ; nr 2 Z, and i1 ; : : : ; i r 2 f0; 1g. For n1 ; n2 ; : : : ; nr 2 Z, and i1 ; i2 ; : : : ; i r 2 f0; 1g, we define the measure  of the cylinder set fx(n1 ) D i1 ; x(n2 ) D i2 ; : : : ; x(nr ) D i r g to be the limit (13). Thus defined,  extends to a premeasure on the algebra of sets generated by cylinder sets and hence by Carathéodory’s extension theorem [24] to a probability measure on B. It is easy to check that (A) D d(), the shift transformation T preserves the measure  and (12) holds.  Using this principle for k D 1, one may check that any set of recurrence is intersective, that is intersects E  E for every set E of positive density. Using it for n1 D n; n2 D

Ergodic Theory: Recurrence

2n; : : : ; n k D kn, together with Theorem 15, one gets an ergodic proof of Szemerédi’s theorem [69], stating that every subset of the integers with positive upper density contains arbitrarily long arithmetic progressions (conversely, one can easily deduce Theorem 15 from Szemerédi’s theorem, and that intersective sets are sets of recurrence). Making the choice n1 D n2 and using part (iv) of Theorem 13, we get an ergodic proof of the surprising result of Sárközy [66] stating that every subset of the integers with positive upper density contains two elements whose difference is a perfect square. More generally, using Theorem 19, one can translate all of the recurrence results of the previous two sections to results in combinatorics. (This is not straightforward for Theorem 18 because of the ergodicity assumption made there. We refer the reader to [14] for the combinatorial consequence of this result). We mention explicitly only the combinatorial consequence of Theorem 17: Theorem 20 (Bergelson and Leibman [16]) Let  Z with d() > 0, and p1 ; : : : ; p k be integer polynomials with zero constant term. Then  contains infinitely many configurations of the form fx; x C p1 (n); : : : ; x C p k (n)g. The ergodic proof is the only one known for this result, even for patterns of the form fx; xCn2 ; xC2n2 g or fx; xC n; x C n2 g. Ergodic-theoretic contributions to the field of geometric Ramsey theory were made by Furstenberg, Katznelson, and Weiss [40], who showed that if E is a positive upper density subset of R2 then: (i) E contains points with any large enough distance (see also [21] and [29]), (ii) Every ı-neighborhood of E contains three points forming a triangle congruent to any given large enough dilation of a given triangle (in [21] it is shown that if the three points lie on a straight line one cannot always find three points with this property in E itself). Recently, a generalization of property (ii) to arbitrary finite configurations of Rm was obtained by Ziegler [76]. It is also worth mentioning some recent exciting connections of multiple recurrence with some structural properties of the set of prime numbers. The first one is in the work of Green and Tao [45], where the existence of arbitrarily long arithmetic progressions of primes was demonstrated, the authors, in addition to using Szemerédi’s theorem outright, use several ideas from its ergodic-theoretic proofs, as appearing in [34] and [39]. The second one is in the recent work of Tao and Ziegler [70], where a quantitative version of Theorem 17 was used to prove that the primes contain arbitrarily long polynomial progressions. Furthermore, several recent results in ergodic theory, related to the structure of the minimal characteristic fac-

tors of certain multiple ergodic averages, play an important role in the ongoing attempts of Green and Tao to get asymptotic formulas for the number of k-term arithmetic progressions of primes up to x (see for example [46] and [47]). This project has been completed for k D 3, thus verifying an interesting special case of the Hardy– Littlewood k-tuple conjecture predicting the asymptotic growth rate of N a 1 ;:::;a k (x) D the number of configurations of primes having the form fp; p C a1 ; : : : ; p C a k g with p  x. Finally, we remark that in this article we have restricted attention to multiple recurrence and Furstenberg correspondence for Z actions, while in fact there is a wealth of literature on extensions of these results to general commutative, amenable and even non-amenable groups. For an excellent exposition of these and other recent developments the reader is referred to the surveys [9] and [11]. Here, we give just one notable combinatorial corollary to some work of this kind, a density version of the classical Hales–Jewett coloring theorem [48]. Theorem 21 (Furstenberg and Katznelson [38]) Let Wn (A) denote the set of words of length n with letters in the alphabet A D fa1 ; : : : ; a k g. For every " > 0 there exists N0 D N0 ("; k) such that if n  N0 then any subset S of Wn (A) with jSj  "k n contains a combinatorial line, i. e., a set consisting of k n-letter words, having fixed letters in l positions, for some 0  l < n, the remaining n  l positions being occupied by a variable letter x, for x D a1 ; : : : ; a k . (For example, in W4 (A) the sets f(a1 ; x; a2 ; x) : x 2 Ag and f(x; x; x; x); : x 2 Ag are combinatorial lines). At first glance, the uninitiated reader may not appreciate the importance of this “master” density result, so it is instructive to derive at least one of its immediate consequences. Let A D f0; 1; : : : ; k  1g and interpret Wn (A) as integers in base k having at most n digits. Then a combinatorial line in Wn (A) is an arithmetic progression of length k – for example, the line f(a1 ; x; a2 ; x) : x 2 Ag corresponds to the progression fm; m C n; m C 2n; m C 3ng, where m D a1 C a2 d 2 and n D d C d 3 . This allows one to deduce Szemerédi’s theorem. Similarly, one can deduce from Theorem 21 multidimensional and IP extensions of Szemerédi’s theorem [36,37], and some related results about vector spaces over finite fields [37]. Again, the only known proof for the density version of the Hales– Jewett theorem relies heavily on ergodic theory. Future Directions In this section we formulate a few open problems relating to the material in the previous three sections. It should be

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noted that this selection reflects the authors’ interests, and does not strive for completeness. We start with an intriguing question of Katznelson [54] about sets of topological recurrence. A set S N is a set of Bohr recurrence if for every ˛1 ; : : : ; ˛ k 2 R and " > 0 there exists s 2 S such that fs˛ i g 2 [0; "] [ [1  "; 1) for i D 1; : : : ; k. Problem 1 Is every set of Bohr recurrence a set of topological recurrence? Background for this problem and evidence for a positive answer can be found in [54,72]. As we mentioned in Sect. “Subsequence Recurrence”, there exists a set of topological recurrence (and hence Bohr recurrence) that is not a set of recurrence. Problem 2 Is the set S D fl!2m 3n : l; m; n 2 Ng a set of recurrence? Is it a set of k-recurrence for every k 2 N? It can be shown that S is a set of Bohr recurrence. Theorem 13 cannot be applied since the uniform distribution condition fails for some irrational numbers ˛. A relevant question was asked by Bergelson in [9]: “Is the set S D f2m 3n : m; n 2 Ng good for single recurrence for weakly mixing systems?” As we mentioned in Sect. “Multiple Recurrence”, the set of primes shifted by 1 (or  1) is a set of 2-recurrence [32]. Problem 3 Show that the sets P  1 and P C 1, where P is the set of primes, are sets of k-recurrence for every k 2 N. As remarked in [32], a positive answer to this question will follow if some uniformity conjectures of Green and Tao [47] are verified. We mentioned in Sect. “Subsequence Recurrence” that random non-lacunary sequences (see definition there) are almost surely sets of recurrence. Problem 4 Show that random non-lacunary sequences are almost surely sets of k-recurrence for every k 2 N. The answer is not known even for k D 2, though, in unpublished work, Wierdl and Lesigne have shown that the answer is positive for random sequences with at most quadratic growth. We refer the reader to the survey [65] for a nice exposition of the argument used by Bourgain [22] to handle the case k D 1. It was shown in [31] that if S is a set of 2-recurrence then the set of its squares is a set of recurrence for circle rotations. The same method shows that it is actually a set of Bohr recurrence. Problem 5 If S Z is a set of 2-recurrence, is it true that S 2 D fs 2 : s 2 Sg is a set of recurrence?

A similar question was asked in [23]: “If S is a set of k-recurrence for every k, is the same true of S2 ?”. One would like to find a criterion that would allow one to deduce that a sequence is good for double (or higher order) recurrence from some uniform distribution properties of this sequence. Problem 6 Find necessary conditions for double recurrence similar to the one given in Theorem 13. It is now well understood that such a criterion should involve uniform distribution properties of some generalized polynomials or 2-step nilsequences. We mentioned in Sect. “Connections with Combinatorics and Number Theory” that every positive density subset of R2 contains points with any large enough distance. Bourgain [21] constructed a positive density subset E of R2 , a triangle T, and numbers t n ! 1, such that E does not contain congruent copies of all t n -dilations of T. But the triangle T used in this construction is degenerate, which leaves the following question open: Problem 7 Is it true that every positive density subset of R2 contains a triangle congruent to any large enough dilation of a given non-degenerate triangle? For further discussion on this question the reader can consult the survey [44]. The following question of Aaronson and Nakada [1] is related to a classical question of Erd˝os concerning whether P every K N such that n2K 1/n D 1 contains arbitrarily long arithmetic progressions: Problem 8 Suppose that (X; B; ; T) is a f1/ng-conservative ergodic measure preserving system. Is it true that for every A 2 B with (A) > 0 and k 2 N we have (A \ T n A \    \ T kn A) > 0 for some n 2 N? The answer is positive for the class of Markov shifts, and it is remarked in [1] that if the Erd˝os conjecture is true then the answer will be positive in general. The converse is not known to be true. For a related result showing that multiple recurrence is preserved by extensions of infinite measure preserving systems see [61]. Our next problem is motivated by the question whether Theorem 21 has a polynomial version (for a precise formulation of the general conjecture see [9]). Not even this most special consequence of it is known to hold. Problem 9 Let " > 0. Does there exist N D N(") having the property that every family P of subsets of f1; : : : ; Ng2 2 satisfying jPj  "2 N contains a configuration fA; A [ ( 

)g, where A f1; : : : ; Ng2 and f1; : : : ; Ng with A \ (  ) D ;?

Ergodic Theory: Recurrence

A measure preserving action of a general countably infinite group G is a function g ! Tg from G into the space of measure preserving transformations of a probability space X such that Tg h D Tg Th . It is easy to show that a version of Khintchine’s recurrence theorem holds for such actions: if (A) > 0 and " > 0 then fg : (A \ Tg A) > ((A))2  "g is syndetic. However it is unknown whether the following ergodic version of Roth’s theorem holds. Problem 10 Let (Tg ) and (S g ) be measure preserving G-actions of a probability space X that commute in the sense that Tg S h D S h Tg for all g; h 2 G. Is it true that for all positive measure sets A, the set of g such that (A \ Tg A \ S g A) > 0 is syndetic? We remark that for general (possibly amenable) groups G not containing arbitrarily large finite subgroups nor elements of infinite order, it is not known whether one can find a single such g ¤ e. On the other hand, the answer is known to be positive for general G in case (Tg1 S g ) is a G-action [18]; even under such strictures, however, it is unknown whether a triple recurrence theorem holds. Bibliography 1. Aaronson J (1981) The asymptotic distribution behavior of transformations preserving infinite measures. J Analyse Math 39:203–234 2. Aaronson J (1997) An introduction to infinite ergodic theory. Mathematical Surveys and Monographs 50. American Mathematical Society, Providence 3. Aaronson J, Nakada H (2000) Multiple recurrence of Markov shifts and other infinite measure preserving transformations. Israel J Math 117:285–310 4. Barreira L (2001) Hausdorff dimension of measures via Poincaré recurrence. Comm Math Phys 219:443–463 5. Barreira L (2005) Poincaré recurrence: old and new. XIVth International Congress on Mathematical Physics, World Sci Publ Hackensack, NJ, pp 415–422 6. Behrend F (1946) On sets of integers which contain no three in arithmetic progression. Proc Nat Acad Sci 23:331–332 7. Bergelson V (1987) Weakly mixing PET. Ergod Theory Dynam Syst 7:337–349 8. Bergelson V (1987) Ergodic Ramsey Theory. In: Simpson S (ed) Logic and Combinatorics. Contemporary Mathematics 65. American Math Soc, Providence, pp 63–87 9. Bergelson V (1996) Ergodic Ramsey Theory – an update. In: Pollicot M, Schmidt K (eds) Ergodic theory of Zd -actions. Lecture Note Series 228. London Math Soc, London, pp 1–61 10. Bergelson V (2000) The multifarious Poincaré recurrence theorem. In: Foreman M, Kechris A, Louveau A, Weiss B (eds) Descriptive set theory and dynamical systems. Lecture Note Series 277. London Math Soc, London, pp 31–57 11. Bergelson V (2005) Combinatorial and diophantine applications of ergodic theory. In: Hasselblatt B, Katok A (eds) Handbook of dynamical systems, vol 1B. Elsevier, pp 745–841 12. Bergelson V, Furstenberg H, McCutcheon R (1996) IP-sets and polynomial recurrence. Ergod Theory Dynam Syst 16:963–974

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57. Leibman A (2002) Lower bounds for ergodic averages. Ergod Theory Dynam. Syst 22:863–872 58. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold. Ergod Theory Dynam Syst 25:201–213 59. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial actions of Zd by translations on a nilmanifold. Ergod Theory Dynam Syst 25:215–225 60. McCutcheon R (2005) FVIP systems and multiple recurrence. Israel J Math 146:157–188 61. Meyerovitch T (2007) Extensions and multiple recurrence of infinite measure preserving systems. Preprint. Available at http://arxiv.org/abs/math/0703914 62. Ornstein D, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inform Theory 39:78–83 63. Petersen K (1989) Ergodic theory. Cambridge Studies in Advanced Mathematics 2. Cambridge University Press, Cambridge 64. Poincaré H (1890) Sur le problème des trois corps et les équations de la dynamique. Acta Math 13:1–270 65. Rosenblatt J, Wierdl M (1995) Pointwise ergodic theorems via harmonic analysis. Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc, Lecture Note Series 205, Cambridge University Press, Cambridge, pp 3–151 66. Sárközy A (1978) On difference sets of integers III. Acta Math Acad Sci Hungar 31:125–149 67. Shkredov I (2002) Recurrence in the mean. Mat Zametki 72(4):625–632; translation in Math Notes 72(3–4):576–582 68. Sklar L (2004) Philosophy of Statistical Mechanics. In: Zalta EN (ed) The Stanford Encyclopedia of Philosophy (Summer 2004 edn), http://plato.stanford.edu/archives/sum2004/ entries/statphys-statmech/ 69. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:299–345 70. Tao T, Ziegler T () The primes contain arbitrarily long polynomial progressions. Acta Math (to appear). Available at http:// www.arxiv.org/abs/math.DS/0610050 71. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, Berlin 72. Weiss B (2000) Single orbit dynamics. CBMS Regional Conference Series in Mathematics 95. American Mathematical Society, Providence 73. Wyner A, Ziv J (1989) Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression. IEEE Trans Inform Theory 35:1250–1258 74. Zermelo E (1896) Über einen Satz der Dynamik und die mechanische Wärmetheorie. Annalen der Physik 57:485–94; English translation, On a theorem of dynamics and the mechanical theory of heat. In: Brush SG (ed) Kinetic Theory. Oxford, 1966, II, pp 208–17 75. Ziegler T (2007) Universal characteristic factors and Furstenberg averages. J Amer Math Soc 20:53–97 76. Ziegler T (2006) Nilfactors of Rm -actions and configurations in sets of positive upper density in Rm . J Anal Math 99:249– 266

Ergodic Theory: Rigidity

Ergodic Theory: Rigidity ˘ 1,2 VIOREL N I TIC ¸ A 1 West Chester University, West Chester, USA 2 Institute of Mathematics, Bucharest, Romania

Article Outline Glossary Definition of the Subject Introduction Basic Definitions and Examples Differentiable Rigidity Local Rigidity Global Rigidity Measure Rigidity Future Directions Acknowledgment Bibliography Glossary Differentiable rigidity Differentiable rigidity refers to finding invariants to the differentiable conjugacy of dynamical systems, and, more general, group actions. Local rigidity Local rigidity refers to the study of perturbations of homomorphisms from discrete or continuous groups into diffeomorphism groups. Global rigidity Global rigidity refers to the classification of all group actions on manifolds satisfying certain conditions. Measure rigidity Measure rigidity refers to the study of invariant measures for actions of abelian groups and semigroups. Lattice A lattice in a Lie group is a discrete subgroup of finite covolume. Conjugacy Two elements g1 ; g2 in a group G are said to be conjugated if there exists an element h 2 G such that g1 D h1 g2 h. The element h is called conjugacy. Ck Conjugacy Two diffeomorphisms 1 ; 2 acting on the same manifold M are said to be Ck -conjugated if there exists a Ck diffeomorphism h of M such that 1 D h1 ı 2 ı h. The diffeomorphism h is called Ck conjugacy. Definition of the Subject As one can see from this volume, chaotic behavior of complex dynamical systems is prevalent in nature and in large classes of transformations. Rigidity theory can be viewed as the counterpart to the generic theory of dynamical systems which often investigates chaotic dynamics for a typical transformation belonging to a large class. In rigid-

ity one is interested in finding obstructions to chaotic, or generic, behavior. This often leads to rather unexpected classification results. As such, rigidity in dynamics and ergodic theory is difficult to define precisely and the best approach to this subject is to study various results and themes that developed so far. A classification is offered below in local, global, differentiable and measurable rigidity. One should note that all branches are strongly intertwined and, at this stage of the development of the subject, it is difficult to separate them. Rigidity is a well developed and prominent topic in modern mathematics. Historically, rigidity has two main origins, one coming from the study of lattices in semi-simple Lie groups, and one coming from the theory of hyperbolic dynamical systems. From the start, ergodic theory was an important tool used to prove rigidity results, and a strong interdependence developed between these fields. Many times a result in rigidity is obtained by combining techniques from the theory of lattices in Lie groups with techniques from hyperbolic dynamical systems and ergodic theory. Among other mathematical disciplines using results and contributing to this field one can mention representation theory, smooth, continuous and measurable dynamics, harmonic and spectral analysis, partial differential equations, differential geometry, and number theory. Additional details about the appearance of rigidity in ergodic theory as well as definitions for some terminology used in the sequel can be found in the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory by Kleinbock,  Ergodic Theory: Recurrence by Frantzikinakis, McCutcheon, and  Ergodic Theory: Interactions with Combinatorics and Number Theory by Ward. The theory of hyperbolic dynamics is presented in the article  Hyperbolic Dynamical Systems by Viana and in the article  Smooth Ergodic Theory by Wilkinson. Introduction The first results about classification of lattices in semi-simple Lie groups were local, aimed at trying to understand the space of small perturbations of a given linear representation. A major contributor was Weil [110,111,112], who proved local rigidity of linear representations for large classes of groups, in particular lattices. Another breakthrough was the contribution of Kazhdan [66], who introduced property (T), allowing to show that large classes of lattices are finitely generated. Rigidity theory matured due to the remarkable global rigidity results obtained by Mostow [90] and Margulis [82], leading to a complete classification of lattices in large classes of semi-simple Lie groups.

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Briefly, a hyperbolic (Anosov) dynamical system is one that exhibits strong expansion and contraction along complementary directions. An early contribution introducing this class of objects is the paper of Smale [106], in which basic examples and techniques are introduced. A breakthrough came with the results of Anosov [1], who proved structural stability of the Anosov systems and ergodicity of the geodesic flow on a manifold of negative curvature. Motivated by questions arising in mathematical physics, chaos theory and other areas, hyperbolic dynamics emerged as one of the major fields of contemporary mathematics. From the beginning, a major unsolved problem in the field was the classification of Anosov diffeomorphisms and flows. In the 80s a change in philosophy occurred, partially motivated by a program introduced by Zimmer [114]. The goal of the program was to classify the smooth actions of higher rank semi-simple Lie groups and of their (irreducible) lattices on compact manifolds. It was expected that any such lattice action that preserves a smooth volume form and is ergodic can be reduced to one of the following standard models: isometric actions, linear actions on infranilmanifolds, and left translations on compact homogeneous spaces. This original conjecture was disproved by Katok, Lewis (see [56]): by blowing up a linear nilmanifold-action at some fixed points they exhibit real-analytic, volume preserving, ergodic lattice actions on manifolds with complicated topology. Nevertheless, imposing extra assumptions on a higher rank action, for example the existence of some hyperbolicity, allows local and global classification results. The concept of Anosov action, that is, an action that contains at least one Anosov diffeomorphism, was introduced for general groups by Pugh, Shub [99]. The significant differences between the classical Z and R cases and those of higher rank lattices, or at a more basic level, of higher rank abelian groups, went unnoticed for a while. The surge of activity in the 80s allowed these differences to surface: for lattices in the work of Hurder, Katok, Lewis, Zimmer (see [43,55,56,63]); and for higher rank abelian groups in the work of Katok, Lewis [55] and Katok, Spatzier [59]. As observed in these papers, local and global rigidity are typical for such Anosov actions. This generated additional research which is summarized in Sects. “Local Rigidity” and “Global Rigidity”. Differentiable rigidity is covered in Sect. “Differentiable Rigidity”. An interesting problem is to find moduli for the Ck conjugacy, k  1, of Anosov diffeomorphisms and flows. This was tackled so far only for low dimensional cases (n D 2 for diffeomorphisms and n D 3 for flows). Another direction that can be included here refers to find-

ing obstructions for higher transverse regularity of the stable/unstable foliation of a hyperbolic system. A spin-off of the research done so far, which is of high interest by itself, and has applications to local and global rigidity, consists of results lifting the regularity of solutions of cohomological equations over hyperbolic systems. In turn, these results motivated a more careful study of analytic questions about lifting the regularity of real valued continuous functions that enjoy higher regularity along webs of foliations. We also include in this section rigidity results for cocycles over higher rank abelian actions. These are crucial to the proof of local rigidity of higher rank abelian group actions. A more detailed presentation of the material relevant to differentiable rigidity can be found in the forthcoming monograph [58]. Measure rigidity refers to the study of invariant measures under actions of abelian groups and semigroups. If the actions are hyperbolic, higher-rank, and satisfy natural algebraic and irreducibility assumptions, one expects the invariant measures to be rare. This direction was started by a question of Furstenberg, asking if any nonatomic probability measure on the circle, invariant and ergodic under multiplications by 2 and 3, is the Lebesgue measure. An early contribution is that of Rudolph [103], who answered positively if the action has an element of strictly positive entropy. Katok, Spatzier [61] extended the question to more general higher rank abelian actions, such as actions by linear automorphisms of tori and Weyl chamber flows. A related direction is the study of the invariant sets and measures under the action of horocycle flows, where important progress was made by Ratner [100,101] and earlier by Margulis [16,81,83]. An application of these results present in the last papers is the proof of the long standing Oppenheim’s conjecture, about the density of the values of the quadratic forms at integer points. Recent developments due to and Einsiedler, Katok, Lindenstrauss [20] give a partial answer to another outstanding conjecture in number theory, Littlewood’s conjecture, and emphasize measure rigidity as one of a more promising directions in rigidity. More details are shown in Sect. “Measure Rigidity”. Four other recent surveys of rigidity theory, each one with a fair amount of overlap but also complementary in part to the present one, that discuss various aspects of the field and its significance are written by Fisher [23], Lindenstrauss [68], Ni¸tic˘a, Török [95], and Spatzier [107]. Among these, [23] concentrates mostly on local and global rigidity, [95] on differentiable rigidity, [68] on measure rigidity, and [107] gives a general overview. Here is a word of caution for the reader. Many times, instead of the most general results, we present an example

Ergodic Theory: Rigidity

that contains the essence of what is available. Also, several important facts that should have been included, are left out. This is because stating complete results would require more space than allocated to this material. The limited knowledge of the author also plays a role here. He apologizes for any obvious omissions and hopes that the bibliography will help fill the gaps. Basic Definitions and Examples A detailed introduction to the theory of Anosov systems and hyperbolic dynamics is given in the monograph [51]. The proofs of the basic results for diffeomorphisms stated below can be found there. The proofs for flows are similar. Surveys about hyperbolic dynamics in this volume are the article  Hyperbolic Dynamical Systems by Viana and the article  Smooth Ergodic Theory by Wilkinson. Consider a compact differentiable manifold M and f : M ! M a C1 diffeomorphism. Let TM be the tangent bundle of M, and D f : TM ! TM be the derivative of f . The map f is said to be an Anosov diffeomorphism if there is a smooth Riemannian metric k  k on M, which induces a metric dM called adapted, a number  2 (0; 1), and a continuous Df invariant splitting TM D E s ˚ E u such that kD f vk  kvk ; v 2 E s ;

kD f 1 vk  kvk ; v 2 E u :

For each x 2 M there is a pair of embedded C1 discs s u (x), Wloc (x), called the local stable manifold and the Wloc local unstable manifold at x, respectively, such that: s (x) D E s (x) ; T W u (x) D E u (x); 1. Tx Wloc x loc s s ( f x) ; f 1 (W u (x)) W u ( f 1 x); 2. f (Wloc (x)) Wloc loc loc 3. For any  2 (; 1), there exists a constant C > 0 such that for all n 2 N,

d M ( f n x; f n y)  Cn d M (x; y) ;

for

s y 2 Wloc (x) ;

d M ( f n x; f n y)  Cn d M (x; y) ;

for

u y 2 Wloc (x) :

The local stable (unstable) manifolds can be extended to global stable (unstable) manifolds W s (x) and W u (x) which are well defined and smoothly injectively immersed. These global manifolds are the leaves of global foliations W s and W u of M. In general, these foliations are only continuous, but their leaves are differentiable. Let  : R  M ! M be a C1 flow. The flow  is said to be an Anosov flow if there is a Riemannian metric k  k on M, a constant 0 <  < 1, and a continuous Df invariant splitting TM D E s ˚ E 0 ˚ E u such that for all x 2 M and t > 0: 1.

d t dt j tD0 

2 E xc n f0g ; dim E xc D 1,

2. kD t vk   t kvk ; v 2 E s , 3. kD t vk   t kvk ; v 2 E u . For each x 2 M there is a pair of embedded C1 discs s (x) ; W u (x), called the local (strong) stable manifold Wloc loc and the local (strong) unstable manifold at x, respectively, such that: s (x) D E s (x) ; T W u (x) D E u (x); 1. Tx Wloc x loc s s (x)) Wloc ( t x) ; 2.  t (Wloc u (x)) W u ( t x) for t > 0;  t (Wloc loc 3. For any  2 (; 1), there exists a constant C > 0 such that for all n 2 N,

d M ( t x ;  t y)  C t d M (x; y) ; for d M (

t

x ;

t

s y 2 Wloc (x) ; t > 0 ;

t

y)  C d M (x; y) ; for

u y 2 Wloc (x) ; t > 0 :

The local stable (unstable) manifolds can be extended to global stable (unstable) manifolds W s (x) and W u (x). These global manifolds are the leaves of global foliations W s and W u of M. One can also define weak stable and weak unstable foliations with leaves given by W cs (x) D [ t2R (W s (x)) and W cu (x) D [ t2R (W u (x)), which have as tangent distributions E cs D E c ˚E s and E cu D E c ˚E s . In general, all these foliations are only continuous, but their leaves are differentiable. Any Anosov diffeomorphism is structurally stable, that is, any C1 diffeomorphism that is C1 close to an Anosov diffeomorphism is topologically conjugate to the unperturbed one via a Hölder homeomorphism. An Anosov flow is structurally stable in the orbit equivalence sense: any C1 small perturbation of an Anosov flow has the orbit foliation topologically conjugate via a Hölder homeomorphism to the orbit foliation of the unperturbed flow. Let SL(n; R) be the group of all n-dimensional square matrices with real valued entries of determinant 1. Let SL(n; Z) SL(n; R) be the subgroup with integer entries. Basic examples of Anosov diffeomorphisms are automorphisms of the n-torus T n D Rn /Z n induced by hyperbolic matrices in SL(n; Z). A hyperbolic matrix is one that has only nonzero eigenvalues, all away in absolute value from 1. A specific example of such matrix in SL(2; Z) is   2 1 : 1 1 Basic examples of Anosov flows are given by the geodesic flows of surfaces of constant negative curvature. The unitary bundle of such a surface can be realized as M D  n PSL(2; R), where PSL(2; R) D SL(2; R)/f˙1g and  is a cocompact lattice in PSL(2; R). The action of

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the geodesic flow on M is induced by right multiplication with elements in the diagonal one-parameter subgroup  

 t/2 0 e ; t 2 R : 0 et/2 A related transformation, which is not hyperbolic, but will be of interest in this presentation, is the horocycle flow induced by right multiplication on M by elements in the one parameter subgroup

   1 t ;t 2 R : 0 1 Of interest in this survey are also actions of more general groups than Z and R. Typical examples of higher rank Z k Anosov actions are constructed on tori using groups of units in number fields. See [65] for more details about this construction. A particular example of Anosov Z2 -action on T 3 is induced by the hyperbolic matrices: 0 1 0 1 0 1 0 2 1 0 A D @0 0 1A ; B D @0 2 1A : 1 8 2 1 8 4 One can check, by looking at the eigenvalues, that A and B are not multiples of the same matrix. Moreover, A and B commute. Typical examples of higher rank Anosov R k -actions are given by Weyl chamber flows, which we now describe using some notions from the theory of Lie groups. A good reference for the background in Lie group theory necessary here is the book of Helgason [40]. Note that for a hyperbolic element of such an action the center distribution is k-dimensional and coincides with the tangent distribution to the orbit foliation of R k . Let G be a semi-simple connected real Lie group of the noncompact type, with Lie algebra g. Let K G be a maximal compact subgroup that gives a Cartan decomposition g D k C p, where k is the Lie algebra of K and p is the orthogonal complement of k with respect to the Killing form of g. Let a p be a maximal abelian subalgebra and A D exp a be the corresponding subgroup. The simultaneous diagonalization of adg (a) gives the decomposition X gDgC g ; g0 D a C m ;

cones in a called Weyl chambers. The faces of the Weyl chambers are called Weyl chamber walls. Let M be the centralizer of A in K. Suppose  is an irreducible torsion-free cocompact lattice in G. Since A commutes with M, the action of A by right translations on  n G descends to an A-action on N :D  n G/M. This action is called a Weyl chamber flow. Any Weyl chamber flow is an Anosov action, that is, has an element that acts hyperbolically transversally to the orbit foliation of A. Note that all maximal connected R diagonalizable subgroups of G are conjugate and their common dimension is called the R-rank of G. If the R-rank k of G is higher than 2, then the Weyl chamber flow is a higher rank hyperbolic R k -action. An example of semi-simple Lie group is SL(n; R). Let A be the diagonal subgroup of matrices with positive entries in SL(n; R). An example of Weyl chamber flow that will be discussed in the sequel is the action of A by right translations on  nSL(n; R), where  is a cocompact lattice. In this case the centralizer M is trivial. The rank of this action is n  1. The picture of the Weyl chambers for n D 3 is shown in Fig. 1. The signs that appear in each chamber are the signs of half of the Lyapunov exponents of a regular element from the chamber with respect to a certain fixed basis. For this action, the Lyapunov exponents appear in pairs of opposite signs. An example of higher rank lattice Anosov action that will be discussed in the sequel is the standard action of SL(n; Z) on the torus T n , (A; x) 7! Ax; A 2 SL(n; Z); x 2 T n : SL(n; Z) is a (noncocompact!) lattice in SL(n; R). As shown in [55], this action is generated by Anosov diffeomorphisms. We describe now a class of dynamical systems more general than the hyperbolic one. A C1 diffeomorphism f of a compact differentiable manifold M is called partially

2

where  is the set of restricted roots. The spaces g are called root spaces. A point H 2 a is called regular if (H) ¤ 0 for all  2 . Otherwise it is called singular. The set of regular elements consists of the complement of a union of finitely many hyperplanes. Its components are

Ergodic Theory: Rigidity, Figure 1 Weyl chambers for SL(3; R)

Ergodic Theory: Rigidity

hyperbolic if there exists a continuous invariant splitting of the tangent bundle TM D E s ˚ E 0 ˚ E u such the the derivative of f expands Eu much more than E0 , and contracts Es much more than E0 . See [9,41] and [10] for the theory of partially hyperbolic diffeomorphisms. Es and Eu are called stable, respectively unstable distributions, and are integrable. E0 is called center distribution and, in general, it is not integrable. A structural stability result proved by Hirsch, Pugh, Shub [41], that is a frequently used tool in rigidity, shows that, if E0 is integrable to a smooth foliation, then any perturbation f¯ of f is partially hyperbolic and has an integrable center foliation. Moreover, the center foliations of f¯ of f are mapped one into the other by a homeomorphism that conjugates the maps induced on the factor spaces of the center foliations by f¯ of f respectively. We review now basic facts about cocycles. These basic definitions refer to several regularity classes: measurable, continuous, or differentiable. Let G be a group acting on a set M, and denote the action G  M ! M by (g; x) 7! gx; thus (g1 g2 )x D g1 (g2 x). Let  be a group with unit 1 . M is usually endowed with a measurable, continuous, or differentiable structure. A cocycle ˇ over the action is a function ˇ : G  M !  such that ˇ(g1 g2 ; x) D ˇ(g1 ; g2 x)ˇ(g2 ; x) ;

(1)

for all g1 ; g2 2 G; x 2 M. Note that any group representation  : G !  defines a cocycle called constant cocycle. The trivial representation defines the trivial cocycle. A natural equivalence relation for the set of cocycles is given by cohomology. Two cocycles ˇ1 ; ˇ2 : G  M !  are cohomologous if there exists a map P : M !  , called transfer map, such that ˇ1 (g; x) D P(gx)ˇ2 (g; x)P(x)1 ;

(2)

for all g 2 G; x 2 M. Differentiable Rigidity We start by reviewing cohomological results. Several basic notions are already defined in Sect. “Basic Definitions and Examples”. In this section we assume that the cocycles are at least continuous. A cocycle ˇ : G  M !  over an action (g; x) 7! gx; g 2 G; x 2 M, is said to satisfy closing conditions if for any g 2 G and x 2 M such that gx D x, one has ˇ(g; x) D 1 . Note that closing conditions are necessary in order for a cocycle to be cohomologous to the trivial one. Since a Z-cocycle is determined by a function ˇ : M ! ; ˇ(x) :D ˇ(1; x), the closing conditions become f nx D x

implies ˇ( f n1 x) : : : ˇ(x) D 1 ;

(3)

where f : M ! M is the function that implements the Zaction. The first cohomological results for hyperbolic systems were obtained by Livshits [70,71]. Let M be a compact Riemannian manifold with a Z-action implemented by a topologically transitive Anosov C1 diffeomorphism f . Then an ˛-Hölder function ˇ : M ! R determines a cocycle cohomologous to a trivial cocycle if and only if ˇ satisfies the closing conditions (3). The transfer map is ˛-Hölder. Moreover, for each Hölder class ˛ and each finite dimensional Lie group  , there is a neighborhood U of the identity in  such that an ˛-Hölder function ˇ : M !  determines a cocycle cohomologous to the trivial cocycle if and only if ˇ satisfies the closing conditions (3). The transfer map is again ˛-Hölder. Similar results are true for Anosov flows. Using Fourier analysis, Veech [108] extended Livshits’s result to real valued cocycles over Z-actions induced by ergodic endomorphisms of an n-dimensional torus, not necessarily hyperbolic. For cocycles with values in abelian groups, the question of two arbitrary cocycles being cohomologous reduces to the question of an arbitrary cocycle being cohomologous to a trivial one. This is not the case for cocycles with values in nonabelian groups. Parry [98] extended Livshits’s criteria to one for cohomology of two arbitrary cocycles with values in compact Lie groups. Parry’s result was generalized by Schmidt [104] to cocycles with values in Lie groups that, in addition, satisfy a center bunching condition. Ni¸tic˘a, Török [92] extended Livshits’s result to cocycles with values in the group Diff k (M) of Ck diffeomorphism of a compact manifold M with stably trivial bundle, k  3. Examples of such manifolds are the tori and the spheres. In this case, the transfer map takes values in Diff k3 (M), and it is Hölder with respect to a natural metric on Diff k3 (M). In [92] one can also find a generalization of Livshits’s result to generic Anosov actions, that is, actions generated by families of Anosov diffeomorphisms that do not interchange the stable and unstable directions of elements in the family. An example of such an action is the standard action of SL(n; Z) on the n-dimensional torus. A question of interest is the following: if two Ck cocycles, 1  k  !, over a hyperbolic action, are cohomologous through a continuous/measurable transfer map P, what can be said about the higher regularity of P? For real valued cocycles the question can be reduced to one about cohomologically trivial cocycles. Livshits showed that for a real valued C1 cocycle cohomologous to a constant the transfer map is C1 . He also obtained C 1 regularity results if the action is given by hyperbolic automorphisms of a torus. After preliminary results by Guillemin and Kazh-

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dan for geodesic flows on surfaces of negative curvature, for general hyperbolic systems the question was answered positively by de la Llave, Marco, Moriyon [76] in the C 1 case and by de la Llave [74] in the real analytic case. Ni¸tic˘a, Török [93] considered the lift of regularity for a transfer map between two cohomologous cocycles with values in a Lie group or a diffeomorphism group. In contrast to the case of cocycles cohomologous to trivial ones, here one needs to require for the transfer map a certain amount of Hölder regularity that depends on the ratio between the expansion/contraction that appears in the base and the expansion/contraction introduced by the cocycle in the fiber. This assumption is essential, as follows from a counterexample found by de la Llave’s [73]. Useful tools in this development have been results from analysis that lift the regularity of a continuous real valued function which is assumed to have higher regularity along pairs of transverse Hölder foliations. Many times the foliations are the stable and unstable ones associated to a hyperbolic system. Journé [46] proved the C n;˛ regularity of a continuous function that is C n;˛ along two transverse continuous foliations with C n;˛ leaves. If one is interested only in C 1 regularity, a convenient alternative is a result of Hurder, Katok [44]. This has a simpler proof and can be applied to the more general situation in which the function is regular along a web of transverse foliations. A real analytic regularity result along these lines belongs to de la Llave [74]. In certain problems, for example when working with Weyl chamber flows, it is difficult to control the regularity in enough directions to span the whole tangent space. Nevertheless, the tangent space can be generated if one consider higher brackets of good directions. A C 1 regularity result for this case belongs to Katok, Spatzier [60]. In order to apply this result, the foliations need to be C 1 not only along the leaves, but also transversally. An application of the above regularity results is to questions about transverse regularity of the stable and unstable foliations of the geodesic flow on certain C 1 surfaces of nonpositive curvature. For compact negatively curved C 1 surfaces, E. Hopf showed that these foliations are C1 , and it follows from the work of Anosov that individual leaves are C 1 . Hurder, Katok [44] showed that once the weak-stable and weak-unstable foliations of a volume-preserving Anosov flow on a compact 3-manifold are C2 , they are C 1 . Another application of the regularity results is to the study of invariants for Ck conjugacy of hyperbolic systems. By structural stability, a small C1 perturbation of a hyperbolic system is C0 conjugate to the unperturbed one. The conjugacy, in general, is only Hölder. If the conjugacy is

C1 then it preserves the eigenvalues of the derivative at the periodic orbit. The following two results describe the invariants of smooth and real analytic conjugacy of low dimensional hyperbolic systems. They are proved in a series of papers written in various combinations by de la Llave, Marco, Moryion [72,74,75,79,80]. Let X; Y be two C 1 (C ! ) transitive Anosov vector fields on a compact three-dimensional manifold. If they are C0 conjugate and the eigenvalues of the derivative at the corresponding periodic orbits are the same, then the conjugating homeomorphism is C 1 (C! ). In particular, any C1 conjugacy is C 1 (C! ). Assume now that f ; g are two C 1 (C! ) Anosov diffeomorphisms on a compact two dimensional manifold. If they are C0 conjugate and the eigenvalues of the derivative at the corresponding periodic orbits are the same, then the conjugating diffeomorphism is C 1 (C! ). In particular, any C1 conjugacy is C 1 (C! ). An important direction was initiated by Katok, Spatzier [59] who studied cohomological results over hyperbolic Z k or R k -actions, k  2. They show that real valued smooth/Hölder cocycles over typical classes of hyperbolic Z k or R k , k  2, actions are smoothly/Hölder cohomologous to constants. These results cover, in particular, actions by hyperbolic automorphisms of a torus, and Weyl chamber flows. The proofs rely on harmonic analysis techniques, such as Fourier transform and group representations for semi-simple Lie groups. A geometric method for cocycle rigidity was developed in [64]. One constructs a differentiable form using invariant structures along stable/unstable foliations, and the commutativity of the action. The form is exact if and only if the cocycle is cohomologous to a constant one. The method covers actions on nilmanifolds satisfying a condition called TNS (Totally Non-Symplectic). This condition means that the action is higher rank abelian hyperbolic, and that the tangent space is a direct sum of invariant distributions, with each pair of these included in the stable distribution of a hyperbolic element of the action. The method was also applied to small (i. e. close to identity on a set of generators) Lie group valued cocycles. A related paper is [96] which contains rigidity results for cocycles over (TNS) actions with values in compact Lie groups. In this situation the number of cohomology classes is finite. An example of (TNS) action is given by the action of a maximal diagonalizable subgroup of SL(n; Z) on T n . Recently Damjanovi´c, Katok [14] developed a new method that was applied to the action of the matrix diagonal group on  n SL(n; R). They use techniques from [54], where one finds cohomology invariants for cocycles over partially hyperbolic actions that satisfy accessibility prop-

Ergodic Theory: Rigidity

erty. Accessibility means that one can connect any two points from the manifold supporting the partially hyperbolic dynamical system by transverse piecewise smooth paths included in stable/unstable leaves. This notion was introduced by Brin, Pesin [9] and it is playing a crucial role in the recent surge of activity in the field of partially hyperbolic diffeomorphisms. See [10] for a recent survey of the subject. The cohomology invariants described in [54] are heights of the cocycle over cycles constructed in the base out of pieces inside stable/unstable leaves. They provide a complete set of obstructions for solving the cohomology equation. A new tool introduced in [14] is algebraic K-theory [88]. The method can be extended to cocycles with non-abelian range. In [57] one finds related results for small cocycles with values in a Lie group or the diffeomorphism group of a compact manifold. The equivalent of the Livshits theorem in the higherrank setting appears to be a description of the highest cohomology rather than the first cohomology. Indeed, for higher rank partially hyperbolic actions of the torus, the intermediate cohomologies are trivial, while for the highest one the closing conditions characterize the cohomology classes. This behavior provides a generalization of Veech cohomological result and of Katok, Spatzier cohomological result for toral automorphisms, and was discovered by A. Katok, S. Katok [52,53]. Flaminio, Forni [27] studied the cohomological equation over the horocycle flow. It is shown that there are infinitely many obstructions to the existence of a smooth solution. Moreover, if these obstructions vanish, then one can solve the cohomological equation. In [28] it is shown a similar result for cocycles over area preserving flows on compact higher-genus surfaces under certain assumptions that hold generically. Mieczkowski [87] extended these techniques and studied the cohomology of parabolic higher rank abelian actions. All these results rely on noncommutative Fourier analysis, more specifically representation theory of SL(2; R) and SL(2; C). Local Rigidity Let  be a finitely (or compactly) generated group, G a topological group, and  :  ! G a homomorphism. The target of local rigidity theory is to understand the space of perturbations of various homomorphisms . Trivial perturbations of a homomorphism arise from conjugation by an arbitrary element of G. In order to rule them out, one says that  is locally rigid if any nearby homomorphism  0 , (that is,  0 close to  on a finite or compact set of generators of  ), is conjugate to  by an element g 2 G, that is, ( ) D g 0 ( )g 1 for all 2  . If G

is path-wise connected, one can also consider deformation rigidity, meaning that any nearby continuous path of homomorphisms is conjugated to the initial one via a continuous path of elements in G that has an end in the identity. Initial results on local rigidity are about embeddings of lattices into semi-simple Lie groups. The main results belong to Weil [110,111,112]. He showed that if G is a semisimple Lie group that is not locally isomorphic to SL(2; R) and if  G is an irreducible cocompact lattice, then the natural embedding of  into G is locally rigid. Earlier results were obtained by Selberg [105], Calabi, Vesentini [12], and Calabi [11]. Selberg proved the local rigidity of the natural embedding of cocompact lattices into SL(n; R). His proof used the dynamics of iterates of matrices, in particular the existence of singular directions, or walls of Weyl chambers, in the maximal diagonalizable subgroups of SL(n; R). Selberg’s approach inspired Mostow [90] to use the boundaries at infinity in his proof of strong rigidity of lattices, which in turn was crucial to the development of superrigidity due to Margulis [82]. See Sect. “Global Rigidity” for more details. Recall that hyperbolic dynamical systems are structurally stable. Thus they are, in a certain sense, locally rigid. We introduce now a precise definition of local rigidity in the infinite-dimensional setup. The fact that for general group actions one needs to consider different regularities for the actions, perturbations and conjugacies is apparent from the description of structural stability for Anosov systems. A Ck action ˛ of a finitely generated discrete group  on a manifold M, that is, a homomorphism ˛ :  ! Diff k (M), is said to be C k;l ;p;r locally rigid if any Cl perturbation ˛˜ which is Cp close to ˛ on a family of generators, is Cr conjugate to ˛, i.e there exists a Cr diffeomorphism H : M ! M which conjugates ˛˜ to ˛, that is, ˜ H ı ˛(g) D ˛(g) ı H for all g 2  . Note that for Anosov Z-actions, C 1;1;1;0 rigidity is known as structural stability. One can also introduce the notion of deformation rigidity if the initial action and the perturbation are conjugate by a continuous path of diffeomorphisms that has an end coinciding to the identity. A weaker notion of local rigidity can be defined in the presence of invariant foliations for the initial group action and for the perturbation. The map H is now required to preserve the leaves of the foliations and to conjugate only after factorization by the invariant foliations. The importance of this notion is apparent from the leafwise conjugacy structural stability theorem of Hirsch, Pugh, Shub [41]. See Sect. “Basic Definitions and Examples”. Moreover, for Anosov flows this is the natural notion of structural stability, and appears by taking the invariant fo-

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liation to be the one-dimensional orbit foliation. For more general actions, of lattices or higher rank abelian groups, this property is often used in combination to cocycle rigidity in order to show local rigidity. We discuss more about this when we review local rigidity results for partially hyperbolic actions. We summarize now several developments in local rigidity that emerged in the 80s. Initial results [67,115] were about infinitesimal rigidity, that is, a weaker version of local rigidity suitable for discrete groups representations in infinite-dimensional spaces of smooth vector fields. Then Hurder [43] proved C 1;1;1;1 deformation rigidity and Katok, Lewis, Zimmer [55,56,63] proved C 1;1;1;1 local rigidity of the standard action of SL(n; Z); n  3, on the n-dimensional torus. In these results crucial use was made of the presence of an Anosov element in the action. Due to the uniqueness of the conjugacy coming from structural stability, one has a continuous candidate for the conjugacy between the actions. Margulis, Qian [85] used the existence of a spanning family of directions that are hyperbolic for certain elements of the action to show local rigidity of partially hyperbolic actions that are not hyperbolic. Another important tool present in many proofs is Margulis and Zimmer superrigidity. These results allow one to produce a measurable conjugacy for the perturbation. Then one shows that the conjugacy has higher regularity using the presence of hyperbolicity. Having enough directions to span the whole tangent space is essential to lift the regularity. A cocycle to which superrigidity can be applied is the derivative cocycle. The study of local rigidity of partially hyperbolic actions that contain a compact factor was initiated by Ni¸tic˘a, Török [92,94]. Let n  3 and d  1. Let be the action of SL(n; Z) on T nCd D T n  T d given by (A)(x; y) D (Ax; y) ; x 2 T n ; y 2 T d ; A 2 SL(n; Z). Then, for K  1, [92] shows that is C 1;1;5;K1 deformation rigid. The proof is based on three results in hyperbolic dynamics: the generalization of Livshits’s cohomological results to cocycles with values in diffeomorphism groups, the extension of Livshits’s result to general Anosov actions, and a version of the Hirsch, Pugh, Shub structural stability theorem improving the regularity of the conjugacy. Assume now n  3 and K  1. If is the action of SL(n; Z) on T nC1 D T n  T given by (A)(x; y) D (Ax; y) ; x 2 T n ; y 2 T ; A 2 SL(n; Z), [94] shows that the action is C 1;1;2;K1 locally rigid. Ingredients in the proof are two rigidity results, one about TNS actions, and one about actions of property (T) groups. A locally compact group has property (T) if the trivial representation is isolated in the Fell topology. This means that if G acts on a Hilbert space unitarily and it has almost invari-

ant vectors, then it has invariant vectors. Hirsch–Pugh– Shub theorem implies that perturbations of abelian partially hyperbolic actions of product type are conjugated to skew-products of abelian Anosov actions via cocycles with values in diffeomorphism groups. In addition, the TNS property implies that the sum of the stable and unstable distributions of any regular element of the perturbation is integrable. The leaves of the integral foliation are closed, covering the base simply. Thus one obtains a conjugacy between the perturbation and a product action. Property (T) is used to show that the conjugacy reduces the perturbed action to a family of perturbations of hyperbolic actions. But the last ones are already known to be conjugate to the hyperbolic action in the base. Recent important progress in the question of local rigidity of lattice actions was made by Fisher, Margulis [24, 25,26]. Their proofs are modeled along the proof of Weil’s local rigidity result [112] and use an analog of Hamilton’s [39] hard implicit function theorem. Let G be a connected semi-simple Lie group with all simple factors of rank at least two, and  G a lattice. The main result shows that a volume preserving affine action of G or  on a compact smooth manifold X is C 1;1;1;1 locally rigid. Lower regularity results are also available. A component of the proof shows that if  is a group with property (T), X a compact smooth manifold, and a smooth action of  on X by Riemannian isometries, then is C 1;1;1;1 locally rigid. An earlier local rigidity result for this type of actions by cocompact lattices was obtained by Benveniste [5]. Many lattices act naturally on “boundaries” of type G/P, where G is a semi-simple algebraic Lie group and P is a parabolic subgroup. An example is given by G D SL(2; R) and P the subgroup in G consisting of upper triangular matrices. Local rigidity results for this type of actions were found by Ghys [34], Kanai [50] and Katok, Spatzier [62]. Starting with the work of Katok and Lewis, a related direction was the study of local rigidity for higher rank abelian actions. They prove in [55] the C 1;1;1;1 local rigidity of the action of a Zn maximal diagonalizable (over R) subgroup of SL(n C 1; Z); n  2, acting on the torus T nC1 . These type of results were later pushed forward by Katok, Spatzier [62]. Using the theory of nonstationary normal forms developed in [38] and [37] by Katok, Guysinsky, they proved several local rigidity results. The first one assumes that G is a semisimple Lie group with all simple factors of rank at least two,  a lattice in G, N a nilpotent Lie group and  a lattice in N. Then any Anosov affine action of  on N/ is C 1;1;1;1 locally rigid. Second, let Zd be a group of affine transformations

Ergodic Theory: Rigidity

of N/ for which the derivatives are simultaneously diagonalizable over R with no eigenvalues on the unit circle. Then the Zd -action on N/ is C 1;1;1;1 locally rigid. A related result for continuous groups is the C 1;1;1;1 local rigidity (after factorization by the orbit foliation) of the action of a maximal abelian R-split subgroup in an R-split semi-simple Lie group of real rank at least two on G/, where  is a cocompact lattice in G. One can also study rigidity of higher rank abelian partially hyperbolic actions that are not hyperbolic. Natural examples appear as automorphisms of tori and as variants of Weyl chamber flows. For the case of ergodic actions by automorphisms of a torus, this was investigated using a version of the KAM (Kolmogorov, Arnold, Moser) method by Damianovi´c, Katok [15]. As usual in the KAM method, one starts with a linearization of the conjugacy equation. At each step of the iterative KAM scheme, some twisted cohomological equations are solved. The existence of the solutions is forced by the ergodicity of the action and the higher rank assumptions. Diophantine conditions present in this case allow to control the fixed loss of regularity which is necessary for the convergence of these solutions to a conjugacy. Global Rigidity The first remarkable result in global rigidity belongs to Mostow [90]. For G a connected non-compact semi-simple Lie group not locally isomorphic to SL(2; R), and two irreducible cocompact lattices 1 ; 2 G, Mostow showed that any isomorphism from  1 into  2 extends to an isomorphism of G into itself. G has an involution  whose fixed set is a maximal compact subgroup K. One constructs the symmetric Riemannian space X D G/K. To each chamber of X corresponds a parabolic group and these parabolic groups are endowed with a Tits geometry similar to the projective geometry of lines, planes etc. formed in the classical case when G D PGL(n; R). The proof of Mostow’s result starts by building a -equivariant pseudo-isometric map  : G/K1 ! G/K2 . The map  induces an incidence preserving -equivariant isomorphism  0 of the Tits geometries. By Tits’ generalized fundamental theorem of projective geometry,  0 is induced by an isomorphism of G. Finally, ( ) D 0   01 gives the desired conclusion. The next remarkable result in global rigidity is Margulis’ superrigidity theorem. An account of this development can be found in the monograph [82]. For large classes of irreducible lattices in semi-simple Lie groups, this result classifies all finite dimensional representations. Let G be a semi-simple simply connected Lie group of rank

higher than two and  < G an irreducible lattice. Then any linear representation  of  is almost the restriction of a linear representation of G. That is, there exists a linear representation  1 of G and a bounded image representation  2 of  such that  D 1 2 . The possible representations  2 are also classified by Margulis up to some facts concerning finite image representations. As in the case of Mostow’s result, the proof involved the analysis of a map defined on the boundary at infinity. In this case the map is studied using deep results from dynamics like the multiplicativity ergodic theorem of Oseledec [97] or the theory of random walks on groups developed by Furstenberg [31]. An important consequence of Margulis superrigidity result is the arithmeticity of irreducible lattices in connected semi-simple Lie groups of rank higher than two. A basic example of arithmetic lattice can be obtained by taking the integer points in a semi-simple Lie group that is a matrix group, like taking SL(n; Z) inside SL(n; R). Special cases of superrigidity theorems were proved by Corlette [13] and Gromov, Schoen [36] for the rank one groups Sp(1; n) and respectively F 4 using the theory of harmonic maps. A consequence is the arithmeticity of lattices in these groups. Some of these results are put into differential geometric setting in [89]. Margulis supperrigidity result was extended to cocycles by Zimmer. A detailed exposition, including a self contained presentation of several rigidity results of Margulis, can be found in the monograph [113]. We mention here a version of this result that can be found in [24]. Let M be a compact manifold, H a matrix group, P D M  H, and  a lattice in a simply connected, semi-simple Lie group with all factors of rank higher that two. Assume that  acts on M and H in a way that makes the projection from P to M equivariant. Moreover, the action of  on P is measure preserving and ergodic. Then there exists a measurable map s : M ! H, a representation  : G ! H, a compact subgroup K < H which commute with (G) and a measurable map k :   M ! K such that  s(m) D k( ; m)  ( )  s(  m). One can easily check from the last equation that k is a cocycle. So, up to a measurable change of coordinates given by the map s, the action of  on P is a compact extension via a cocycle of a linear representation of G. Developing further the method of Mostow for studying the Tits building associated to a symmetric space of non-positive curvature led Ballman, Brin, Eberlein, Spatzier [2,3] to a number of characterizations of symmetric spaces. In particular, they showed that if M is a complete Riemannian manifold of non-positive curvature, finite volume, with simply connected cover, irreducible and of rank at least two, then M is isometric to a symmetric

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space with the connected component of Isom(M) having no compact factors. A topological rigidity theorem has been proved by Farrell, Jones [21]. They showed that if N is a complete connected Riemannian manifold whose sectional curvature lies in a closed interval included in (1; 0], and M is a topological manifold of dimension greater than 5, then any proper homotopy equivalence f : M ! N is properly homotopic to a homeomorphism. In particular, if M and N are both compact connected negatively curved Riemannian manifolds with isomorphic fundamental groups, then M and N are homeomorphic. Likewise to the case of local rigidity, a source of inspiration for results in global rigidity was the theory of hyperbolic systems, in particular their classification. The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. Moreover, any Anosov diffeomorphism on an infranilmanifold is topologically conjugate to a hyperbolic automorphism [29,78]. It is conjectured that any Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism of an infranilmanifold. Partial results are obtained in [91], where the conjecture is proved for Anosov diffeomorphisms with codimension one stable/unstable foliation. The proof of the general conjecture eluded efforts done so far. It is not even known if any Anosov diffeomorphism is topologically transitive, that is, if it has a dense orbit. A few positive results are available. Let M be a C 1 compact manifold endowed with a C 1 affine connection. Let f be a topologically transitive Anosov diffeomorphism preserving the connection and such that the stable and unstable distributions are C 1 . Then Benoist, Labourie [4] proved that f is C 1 conjugate to a hyperbolic automorphism of an infranilmanifold. The situation for Anosov flows is somehow different. As shown in [30], there exist Anosov flows that are not topologically transitive, so a general analog of the conjecture is false. Nevertheless, for the case of codimension one stable or unstable foliation, it is conjectured in [109] that any Anosov flow on a manifold of dimension greater than three admits a global cross-section. This would imply that the flow is topologically conjugate to the suspension of a linear automorphism of a torus. For actions of groups larger than Z, or R, global classification results are more abundant. A useful strategy in these type of results, which are quite technical, is to start by obtaining a measurable description of the action, most of the time using Margulis–Zimmer superrigidity results, and then use extra assumptions on the action, such as the presence of a hyperbolic element, or the presence of an invariant geometric structure, or both, in order to show that

the measurable model is actually continuous or even differentiable. For actions of higher rank Lie groups and their lattices some representative papers are by Katok, Lewis, Zimmer [63] and Goetze, Spatzier [35]. For actions of higher rank abelian groups see Kalinin, Spatzier [49]. Measure Rigidity Measure rigidity is the study of invariants measures for actions of one parameter and multiparameter abelian groups and semigroups acting on manifolds. Typical situations when interesting rigidity phenomena appear are for one parameter unipotent actions and higher rank hyperbolic actions, discrete or continuous. A unipotent matrix is one all of whose eigenvalues are one. An important case where the action of a unipotent flow appears is that of the horocycle flow. The invariant measures for it were studied by Furstenberg [33], who showed that the horocycle flow on a compact surface is uniquely ergodic, that is, the ergodic measure is unique. Dani and Smillie [17] extended this result to the case of non-compact surfaces, with the only other ergodic measures appearing being those supported on compact horocycles. An important breakthrough is the work of Margulis [81], who solved a long standing question in number theory, Oppenheim’s conjecture. The conjecture is about the density properties of the values of an indefinite quadratic form in three or more variables, provided the form is not proportional to a rational form. The proof of the conjecture is based on the study of the orbits for unipotent flows acting by translations on the homogenous space SL(n; Z) n SL(n; R). All these results were special cases of the Raghunathan conjecture about the structure of the orbits of the actions of unipotent flows on homogenous spaces. Raghunathan’s conjecture was proved in full generality by Ratner [100,101]. Borel, Prasad [8] raised the question of an analog of Raghunathan’s conjecture for S-algebraic groups. S-algebraic groups are products of real and p-adic algebraic groups. This was answered independently by Margulis, Tomanov [86] and Ratner [102]. A basic example of higher rank abelian hyperbolic action is given by the action of S m;n , the multiplicative semigroup of endomorphisms generated by the multiplication by m and n, two nontrivial integers, on the one dimensional torus T 1 . In a pioneering paper [32] Furstenberg showed that for m; n that are not powers of the same integer the action of S m;n has a unique closed, infinite invariant set, namely T 1 itself. Since there are many closed, infinite invariant sets for multiplication by m, and by n, this result shows a remarkable rigidity property of the joint action. Furstenberg’s result was generalized later by Berend

Ergodic Theory: Rigidity

for other group actions on higher dimensional tori and on other compact abelian groups in [6] and [7]. Furstenberg further opened the field by raising the following question: Conjecture 1 Let  be a S m;n -invariant and ergodic probability measure on T 1 . Then  is either an atomic measure supported on a finite union of (rational) periodic orbits or  is the Lebesgue measure. While the statement appears to be simple, proving it has been elusive. The first partial result was given by Lyons [77] under the strong additional assumption that the measure makes one of the endomorphisms generating the action exact. Later Rudolph [103] and Johnson [45] weaken the exactness assumption and proved that  must be the Lebesgue measure provided that multiplication by m (or multiplication by n) has positive entropy with respect to . Their results have been proved again using slightly different methods by Feldman [22]. A further extension was given by Host [42]. Katok proposed another example for which measure rigidity can be fruitfully tested, the Z2 -action induced by two commuting hyperbolic automorphisms on the torus T 3 . An example of such action is shown in Sect. “Basic Definitions and Examples”. One can also consider the action induced by a Z n1 maximal abelian group of hyperbolic automorphisms acting on the torus T n . Katok, Spatzier [61] developed a more geometric technique allowing to prove measure rigidity for these actions if they have an element acting with positive entropy. Their technique can be applied in higher generality if the action is irreducible in a strong sense and, in addition, it has individual ergodic elements or it is TNS. See also [47]. This method is based on the study of conditional measures induced by the invariant measure on certain invariant foliations that appear naturally in the presence of a hyperbolic action. Besides stable and unstable foliations, one can also consider various intersections of them. Einsiedler, Lindenstrauss [19] were able to eliminate the ergodicity and TNS assumptions. Yet another interesting example of higher rank abelian action is given by Weyl chamber flows. These do not satisfy the TNS condition. Einsiedler, Katok [18] proved that if G is SL(n; R);  G is a lattice, H is the subgroup of positive diagonal matrices in G, and  a H-invariant and ergodic measure on G/ such that the entropy of  with respect to all one parameter subgroups of H is positive, then  is the G invariant measure on G/ . These results are useful in the investigation of several deep questions in number theory. Define X D SL(3; Z) n SL(3; R) the diagonal subgroup of matrices with posi-

tive entries in SL(3; R). This space is not compact but is endowed with a unique translation invariant probability measure. The diagonal subgroup 80 s 9 1 0 0 < e = H D @ 0 et 0 A : s; t 2 R : ; 0 0 est acts naturally on X by right translations. It was conjectured by Margulis [83] that any compact H-invariant subset of X is a union of compact H-orbits. A positive solution to this conjecture implies a long standing conjecture of Littlewood: Conjecture 2 Let kxk denote the distance between the real number x and the closest integer. Then lim inf nkn˛kknˇk D 0 n!1

(4)

for any real numbers ˛ and ˇ. A partial result was obtained by Einsiedler, Katok, Lindenstrauss [20] who proved that the set of pairs (˛; ˇ) 2 R2 for which (4) is not satisfied has Hausdorff dimension zero. Applications of these techniques to questions in quantum ergodicity were found by Lindenstrauss [69]. A current direction in measure rigidity is attempting to classify the invariant measures under rather weak assumptions about the higher rank abelian action, like the homotopical data for the action. Kalinin and Katok [48] proved that any Z k ; k  2, smooth action ˛ on a k C 1dimensional torus whose elements are homotopic to the corresponding elements of an action by hyperbolic automorphisms preserves an absolutely continuous measure. Future Directions An important open problem in differential rigidity is to find invariants for the Ck conjugacy of the perturbations of higher dimensional hyperbolic systems. For Anosov diffeomorphisms, de la Llave counterexample [73] shows that this extension is not possible for a four dimensional example that appears as a direct product of two dimensional Anosov diffeomorphisms. Indeed, there are C 1 perturbations of the product that are only Ck conjugate to the unperturbed system for any k  1. In the positive direction, Katok conjectured that generalizations are possible for the diffeomorphism induced by an irreducible hyperbolic automorphism of a torus. One can also investigate this question for Anosov flows. Examples of higher rank Lie groups can be obtained by taking products of rank one Lie groups. Many actions of irreducible lattices in these type of groups are believed

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to be locally rigid, but the techniques available so far cannot be applied. A related problem is to study local rigidity in low regularity classes, for example the local rigidity of homomorphisms from higher rank lattices into homeomorphism groups. More problems related to local rigidity can be found in [23]. An important problem in global rigidity, emphasized by Katok and Spatzier, is to show that, up to differentiable conjugacy, any higher rank Anosov or partially hyperbolic action is algebraic under the assumption that it is sufficiently irreducible. The irreducibility assumption is needed in order to exclude actions obtained by successive application of products, extensions, restrictions and time changes from basic ingredients which include some rank one actions. Another problem of current research in measure rigidity is to develop a counterpart of Ratner’s theory for the case of actions by hyperbolic higher rank abelian groups on homogenous spaces. It was conjectured by Katok, Spatzier [61] that the invariant measures for such actions given by toral automorphisms or Weyl chamber flows are essentially algebraic, that is, supported on closed orbits of connected subgroups. Margulis in [84] extended this conjecture to a rather general setup addressing both the topological and measurable aspects of the problem. More details about actions on homogenous spaces, as well as connections to diophantine approximation, can be found in the article  Ergodic Theory on Homogeneous Spaces and Metric Number Theory by Kleinbock. Acknowledgment This research was supported in part by NSF Grant DMS0500832. Bibliography Primary Literature 1. Anosov DV (1967) Geodesic flows on closed Riemannian manifolds with negative curvature. Proc Stek Inst 90:1–235 2. Ballman W, Brin M, Eberlein P (1985) Structure of manifolds of non-negative curvature. I. Ann Math 122:171–203 3. Ballman W, Brin M, Spatzier R (1985) Structure of manifolds of non-negative curvature. II. Ann Math 122:205–235 4. Benoist Y, Labourie F (1993) Flots d’Anosov á distribuitions stable et instable différentiables. Invent Math 111:285–308 5. Benveniste EJ (2000) Rigidity of isometric lattice actions on compact Riemannian manifolds. Geom Func Anal 10:516–542 6. Berend D (1983) Multi-invariant sets on tori. Trans AMS 280:509–532 7. Berend D (1984) Multi-invariant sets on compact abelian groups. Trans AMS 286:505–535 8. Borel A, Prasad G (1992) Values of isotropic quadratic forms at S-integral points. Compositio Math 83:347–372

9. Brin MI, Pesin YA (1974) Partially hyperbolic dynamical systems. Izvestia 38:170–212 10. Burns K, Pugh C, Shub M, Wilkinson A (2001) Recent results about stable ergodicity. In: Katok A, Pesin Y, de la Llave R, Weiss H (eds) Smooth ergodic theory and its applications, Seattle, 1999. Proc Symp Pure Math, vol 69. AMS, Providence, pp 327–366 11. Calabi E (1961) On compact Riemannian manifolds with constant curvature. I. Proc Symp Pure Math, vol 3. AMS, Providence, pp 155–180 12. Calabi E, Vesentini E (1960) On compact, locally symmetric Kähler manifolds. Ann Math 17:472–507 13. Corlette K (1992) Archimedian superrigidity and hyperbolic geometry. Ann Math 135:165–182 14. Damianovi´c D, Katok A (2005) Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions. Disc Cont Dynam Syst 13:985–1005 15. Damianovi´c D, Katok A (2007) Local rigidity of partially hyperbolic actions KAM, I.method and Zk -actions on the torus. Preprint available at http://www.math.psu.edu/katok_a 16. Dani SG, Margulis GA (1990) Values of quadratic forms at integer points: an elementary approach. Enseign Math 36:143– 174 17. Dani SG, Smillie J (1984) Uniform distributions of horocycle orbits for Fuchsian groups. Duke Math J 51:185–194 18. Einsiedler M, Katok A (2003) Invariant measures on G/ for split simple Lie groups. Comm Pure Appl Math 56:1184–1221 19. Einsiedler M, Lindenstrauss E (2003) Rigidity properties of Zd actions on tori and solenoids. Elec Res Ann AMS 9:99–110 20. Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the set of exceptions to Littlewood conjecture. Ann Math 164:513–560 21. Farrell FT, Jones LE (1989) A topological analog of Mostow’s rigidity theorem. J AMS 2:237–370 22. Feldman J (1993) A generalization of a result of R Lyons about measures on [0,1). Isr J Math 81:281–287 23. Fisher D (2006) Local rigidity of group actions: past, present, future. In: Dynamics, Ergodic Theory and Geometry (2007). Cambridge University Press 24. Fisher D, Margulis GA (2003) Local rigidity for cocycles. In: Surv Diff Geom VIII. International Press, Cambridge, pp 191– 234 25. Fisher D, Margulis GA (2004) Local rigidity of affine actions of higher rank Lie groups and their lattices. 2003 26. Fisher D, Margulis GA (2005) Almost isometric actions, property T, and local rigidity. Invent Math 162:19–80 27. Flaminio L, Forni G (2003) Invariant distributions and time averages for horocycle flows. Duke Math J 119:465–526 28. Forni G (1997) Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann Math 146:295–344 29. Franks J (1970) Anosov diffeomorphisms. In: Chern SS, Smale S (eds) Global Analysis (Proc Symp Pure Math, XIV, Berkeley 1968). AMS, Providence, pp 61–93 30. Franks J, Williams R (1980) Anomalous Anosov flows. In: Global theory of dynamical systems, Proc Inter Conf Evanston, 1979. Lecture Notes in Mathematics, vol 819. Springer, Berlin, pp 158–174 31. Furstenberg H (1963) A Poisson formula for semi-simple Lie groups. Ann Math 77:335–386

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32. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theor 1:1–49 33. Furstenberg H (1973) The unique ergodicity of the horocycle flow. In: Recent advances in topological dynamics, Proc Conf Yale Univ, New Haven 1972. Lecture Notes in Mathematics, vol 318. Springer, Berlin, pp 95–115 34. Ghys E (1985) Actions localement libres du groupe affine. Invent Math 82:479–526 35. Goetze E, Spatzier R (1999) Smooth classification of Cartan actions of higher rank semi-simple Lie groups and their lattices. Ann Math 150:743–773 36. Gromov M, Schoen R (1992) Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ Math IHES 76:165–246 37. Guysinsky M (2002) The theory of nonstationary normal forms. Erg Th Dyn Syst 22:845–862 38. Guysinsky M, Katok A (1998) Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations. Math Res Lett 5:149–163 39. Hamilton R (1982) The inverse function theorem of Nash and Moser. Bull AMS 7:65–222 40. Helgason S (1978) Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York 41. Hirsch M, Pugh C, Shub M (1977) Invariant Manifolds. Lecture Notes in Mathematics, vol 583. Springer, Berlin 42. Host B (1995) Nombres normaux, entropie, translations. Isr J Math 91:419–428 43. Hurder S (1992) Rigidity of Anosov actions of higher rank lattices. Ann Math 135:361–410 44. Hurder S, Katok A (1990) Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Publ Math IHES 72:5–61 45. Johnson AS (1992) Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of integers. Isr J Math 77:211–240 46. Journé JL (1988) A regularity lemma for functions of several variables. Rev Mat Iberoam 4:187–193 47. Kalinin B, Katok A (2002) Measurable rigidity and disjointness for Zk -actions by toral automorphisms. Erg Theor Dyn Syst 22:507–523 48. Kalinin B, Katok A (2007) Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori. J Modern Dyn 1:123–146 49. Kalinin B, Spatzier R (2007) On the classification of Cartan actions. Geom Func Anal 17:468–490 50. Kanai M (1996) A new approach to the rigidity of discrete group actions. Geom Func Anal 6:943–1056 51. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54. Cambridge University Press, Cambridge 52. Katok A, Katok S (1995) Higher cohomology for abelian groups of toral automorphisms. Erg Theor Dyn Syst 15:569– 592 53. Katok A, Katok S (2005) Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum. Erg Theor Dyn Syst 25:1909–1917 54. Katok A, Kononenko A (1996) Cocycles’ stability for partially hyperbolic systems. Math Res Lett 3:191–210 55. Katok A, Lewis J (1991) Local rigidity for certain groups of toral automorphisms. Isr J Math 75:203–241

56. Katok A, Lewis J (1996) Global rigidity results for lattice actions on tori and new examples of vol preserving actions. Isr J Math 93:253–280 57. Katok A, Ni¸tic˘a V (2007) Rigidity of higher rank abelian cocycles with values in diffeomorphism groups. Geometriae Dedicata 124:109–131 58. Katok A, Ni¸tic˘a V () Differentiable rigidity of abelian group actions. Cambridge University Press (to appear) 59. Katok A, Spatzier R (1994) First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ Math IHES 79:131–156 60. Katok A, Spatzier R (1994) Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math Res Lett 1:193–202 61. Katok A, Spatzier R (1996) Invariant measures for higherrank abelian actions. Erg Theor Dyn Syst 16:751–778; Katok A, Spatzier R (1998) Corrections to: Invariant measures for higher-rank abelian actions.; (1996) Erg Theor Dyn Syst 16:751–778; Erg Theor Dyn Syst 18:503–507 62. Katok A, Spatzier R (1997) Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Trudy Mat Inst Stek 216:292–319 63. Katok A, Lewis J, Zimmer R (1996) Cocycle superrigidity and rigidity for lattice actions on tori. Topology 35:27–38 64. Katok A, Ni¸tic˘a V, Török A (2000) Non-abelian cohomology of abelian Anosov actions. Erg Theor Dyn Syst 2:259–288 65. Katok A, Katok S, Schmidt K (2002) Rigidity of measurable structure for Zd -actions by automorphisms of a torus. Comm Math Helv 77:718–745 66. Kazhdan DA (1967) On the connection of a dual space of a group with the structure of its closed subgroups. Funkc Anal Prilozen 1:71–74 67. Lewis J (1991) Infinitezimal rigidity for the action of SLn (Z) on T n . Trans AMS 324:421–445 68. Lindenstrauss E (2005) Rigidity of multiparameter actions. Isr Math J 149:199–226 69. Lindenstrauss E (2006) Invariant measures and arithmetic quantum unique ergodicity. Ann Math 163:165–219 70. Livshits A (1971) Homology properties of Y-systems. Math Zametki 10:758–763 71. Livshits A (1972) Cohomology of dynamical systems. Izvestia 6:1278–1301 he Livšic cohomology equation. Ann Math 123:537–611 72. de la Llave R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm Math Phys 109:369–378 73. de la Llave R (1992) Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic dynamical systems. Comm Math Phys 150:289–320 74. de la Llave R (1997) Analytic regularity of solutions of Livshits’s cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems. Erg Theor Dyn Syst 17:649–662 75. de la Llave R, Moriyon R (1988) Invariants for smooth conjugacy of hyperbolic dynamical systems. IV. Comm Math Phys 116:185–192 76. de la Llave R, Marco JM, Moriyon R (1986) Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann Math 123:537–611 77. Lyons R (1988) On measures simultaneously 2- and 3-invariant. Isr J Math 61:219–224

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78. Manning A (1974) There are no new Anosov diffeomorphisms on tori. Amer J Math 96:422–429 79. Marco JM, Moriyon R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm Math Phys 109:681–689 80. Marco JM, Moriyon R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. III. Comm Math Phys 112:317–333 81. Margulis GA (1989) Discrete subgroups and ergodic theory. In: Number theory, trace formulas and discrete groups, Oslo, 1987. Academic Press, Boston, pp 277–298 82. Margulis GA (1991) Discrete subgroups of semi-simple Lie groups. Springer, Berlin 83. Margulis GA (1997) Oppenheim conjecture. In: Fields Medalists Lectures, vol 5. World Sci Ser 20th Century Math. World Sci Publ, River Edge, pp 272–327 84. Margulis GA (2000) Problems and conjectures in rigidity theory. In: Mathematics: frontiers and perspectives. AMS, Providence, pp 161–174 85. Margulis GA, Qian N (2001) Local rigidity of weakly hyperbolic actions of higher rank real Lie groups and their lattices. Erg Theor Dyn Syst 21:121–164 86. Margulis GA, Tomanov G (1994) Invariant measures for actions of unipotent groups over local fields of homogenous spaces. Invent Math 116:347–392 87. Mieczkowski D (2007) The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J Modern Dyn 1:61–92 88. Milnor J (1971) Introduction to algebraic K-theory. Princeton University Press, Princeton 89. Mok N, Siu YT, Yeung SK (1993) Geometric superrigidity. Invent Math 113:57–83 90. Mostow GD (1973) Strong rigidity of locally symmetric spaces. Ann Math Studies 78. Princeton University Press, Princeton 91. Newhouse SE (1970) On codimension one Anosov diffeomorphisms. Amer J Math 92:761–770 92. Ni¸tic˘a V, Török A (1995) Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher rank lattices. Duke Math J 79:751–810 93. Ni¸tic˘a V, Török A (1998) Regularity of the transfer map for cohomologous cocycles. Erg Theor Dyn Syst 18:1187– 1209 94. Ni¸tic˘a V, Török A (2001) Local rigidity of certain partially hyperbolic actions of product type. Erg Theor Dyn Syst 21:1213– 1237 95. Ni¸tic˘a V, Török A (2002) On the cohomology of Anosov actions. In: Rigidity in dynamics and geometry, Cambridge, 2000. Springer, Berlin, pp 345–361 96. Ni¸tic˘a V, Török A (2003) Cocycles over abelian TNS actions. Geom Ded 102:65–90 97. Oseledec VI (1968) A multiplicative ergodic theorem. Characteristic Lyapunov, exponents of dynamical systems. Trudy Mosk Mat Obsc 19:179–210 98. Parry W (1999) The Livšic periodic point theorem for nonabelian cocycles. Erg Theor Dyn Syst 19:687–701 99. Pugh C, Shub M (1972) Ergodicity of Anosov actions. Invent Math 15:1–23 100. Ratner M (1991) On Ragunathan’s measure conjecture. Ann Math 134:545–607

101. Ratner M (1991) Ragunathan’s topological conjecture and distributions of unipotent flows. Duke Math J 63:235–280 102. Ratner M (1995) Raghunathan’s conjecture for Cartesians products of real and p-adic Lie groups. Duke Math J 77:275–382 103. Rudolph D (1990) × 2 and × 3 invariant measures and entropy. Erg Theor Dyn Syst 10:395–406 104. Schmidt K (1999) Remarks on Livšic’ theory for nonabelian cocycles. Erg Theor Dyn Syst 19:703–721 105. Selberg A (1960) On discontinuous groups in higher-dimensional symmetric spaces. In: Contributions to function theory. Inter Colloq Function Theory, Bombay. Tata Institute of Fundamental Research, pp 147–164 106. Smale S (1967) Differentiable dynamical systems. Bull AMS 73:747–817 107. Spatzier R (1995) Harmonic analysis in rigidity theory. In: Ergodic theory and its connections with harmonic analysis. Alexandria, 1993. London Math Soc Lect Notes Ser, vol 205. Cambridge University Press, Cambridge, pp 153–205 108. Veech WA (1986) Periodic points and invariant pseudomeasures for toral endomorphisms. Erg Theor Dyn Syst 6:449–473 109. Verjovsky A (1974) Codimension one Anosov flows. Bul Soc Math Mex 19:49–77 110. Weil A (1960) On discrete subgroups of Lie groups. I. Ann Math 72:369–384 111. Weil A (1962) On discrete subgroups of Lie groups. II. Ann Math 75:578–602 112. Weil A (1964) Remarks on the cohomology of groups. Ann Math 80:149–157 113. Zimmer R (1984) Ergodic theory and semi-simple groups. Birhhäuser, Boston 114. Zimmer R (1987) Actions of semi-simple groups and discrete subgroups. Proc Inter Congress of Math (1986). AMS, Providence, pp 1247–1258 115. Zimmer R (1990) Infinitesimal rigidity of smooth actions of discrete subgroups of Lie groups. J Diff Geom 31:301–322

Books and Reviews de la Harpe P, Valette A (1989) La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 Feres R (1998) Dynamical systems and semi-simple groups: An introduction. Cambridge Tracts in Mathematics, vol 126. Cambridge University Press, Cambridge Feres R, Katok A (2002) Ergodic theory and dynamics of G-spaces. In: Handbook in Dynamical Systems, 1A. Elsevier, Amsterdam, pp 665–763 Gromov M (1988) Rigid transformation groups. In: Bernard D, Choquet-Bruhat Y (eds) Géométrie Différentielle (Paris, 1986). Hermann, Paris, pp 65–139; Travaux en Cours. 33 Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook in Dynamical Systems, 1A. Elsevier, Amsterdam, pp 813–930 Knapp A (2002) Lie groups beyond an introduction, 2nd edn. Progress in Mathematics, 140. Birkhäuser, Boston Raghunathan MS (1972) Discrete subgroups of Lie groups. Springer, Berlin Witte MD (2005) Ratner’s theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago

Existence and Uniqueness of Solutions of Initial Value Problems

Existence and Uniqueness of Solutions of Initial Value Problems GIANNE DERKS Department of Mathematics, University of Surrey, Guildford, UK Article Outline Glossary Definition of the Subject Introduction Existence Uniqueness Continuous Dependence on Initial Conditions Extended Concept of Differential Equation Further Directions Bibliography Glossary Ordinary differential equation An ordinary differential equation is a relation between a (vector) function u : I ! Rm , (I an interval in R, m 2 N) and its derivatives. A function u, which satisfies this relation, is called a solution. Initial value problem An initial value problem is an ordinary differential equation with a prescribed value for the solution at one instance of its variable, often called the initial time. The initial value is the pair consisting of the initial time and the prescribed value of the solution. Vector field For an ordinary differential equation of the m ! Rm form du dt D f (t; u), the function f : R  R is called the vector field. Thus, at a solution, the vector field gives the tangent to the solution. Flow map The flow map describes the solution of an ordinary differential equation for varying initial values. Hence, for an ordinary differential equation du dt D f (t; u), the flow map is a function ˚ : R  R  Rm ! Rm such that ˚(t; t0 ; u0 ) is a solution of the ordinary differential equation, starting at t D t0 in u D u0 . Functions: bounded, continuous, uniformly Lipschitz continuous Let D be a connected set in R k with k 2 N and let f : D ! Rm be a function on D:  The function f is bounded if there is some M > 0 such that j f (x)j  M for all x 2 D.  The function f is continuous on D if lim x!x 0 f (x) D f (x0 ), for every x0 2 D. A continuous function on a bounded and closed set D is bounded. The function f is equicontinuous or

uniform continuous if the convergence of the limits in the definition of continuity is uniform for all x0 2 D, i. e., for every " > 0, there is a ı > 0, such that for all x0 2 D and all x 2 D with jx  x0 j < ı it holds that j f (x)  f (x0 )j < ". A continuous function on a compact interval is equicontinuous.  Let D R  Rm . The function f : D ! Rm is uniformly Lipschitz continuous on D with respect to its second variable, if f is continuous on D and there exists some constant L > 0 such that j f (t; u)  f (t; u)j  Lju  vj ; for all (t; u); (t; v) 2 D : The constant L is called the Lipschitz constant. Pointwise and uniform convergence A sequence of functions fu n g with u n : I ! Rm is Pointwise convergent if lim n!1 u n (t) exists for every t 2 I. The sequence of functions fu n g is uniform convergent with limit function u : I ! Rm if lim n!1 supfju  u n j j t 2 Ig D 0. A sequence of pointwise convergent, equicontinuous functions is uniform convergent and the limit function is equicontinuous. Notation u˙ Derivative of u, i. e., u(k)

du dt

The kth derivative of u, i. e.,

dk u dt k

I a (t0 ) The closed interval [t0 ; t0 C a] Bb (u0 ) The closed ball with radius b about u0 in Rm , i. e., Bb (u0 ) :D fu 2 Rm j ju  u0 j  bg kuk1 The supremum norm for a bounded function u : I ! Rm , i. e., kuk1 D supfju(t)j j t 2 Ig Definition of the Subject Many problems in physics, engineering, biology, economics, etc., can be modeled as relations between observables or states and their derivatives, hence as differential equations. When only derivatives with respect to one variable play a role, the differential equation is called an ordinary differential equation. The field of differential equations has a long history, starting with Newton and Leibniz in the seventeenth century. In the beginning of the study of differential equations, the focus is on finding explicit solutions as the emphasis is on solving the underlying physical problems. But soon one starts to wonder: If a starting point for a solution of a differential equation is given, does the solution always exist? And if such a solution exists, how long does it exist and is there only one such solution? These are the questions of existence and uniqueness of solutions of initial value problems. The first existence

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result is given in the middle of the nineteenth century by Cauchy. At the end of the nineteenth century and the beginning of the twentieth century, substantial progress is made on the existence and uniqueness of solutions of initial value problems and currently the heart of the topic is quite well understood. But there are many open questions as soon as one considers delay equations, functional differential equations, partial differential equations or stochastic differential equations. Another area of intensive current research, which uses the existence and uniqueness of differential equations, is the area of finite- and infinitedimensional dynamical systems. Introduction As indicated before, an initial value problem is the problem of finding a solution of an ordinary differential equation with a given initial condition. To be precise, let D be an open, connected set in R  Rm . Given are a function f : D ! Rm and a point (t0 ; u0 ) 2 D. The initial value problem is the problem of finding an interval I R and a function u : I ! Rm such that t0 2 I, f(t; u(t)) j t 2 Ig D and du D f (t; u(t)) ; dt

t2I;

with u(t0 ) D u0 :

(1)

The function f is called the vector field and the point (t0 ; u0 ) the initial value. One often writes the deriva˙ tive as du dt D u. If f is continuous, then the initial value problem (1) is equivalent to finding a function u : I ! Rm such that Z

t

u(t) D u0 C

f (; u())d ;

for t 2 I :

(2)

t0

If f is continuous and a solution exists, then clearly this solution is continuously differentiable on I. It might seem restrictive to only consider first-order systems; however, most higher-order equations can be written as a system of first-order equations. Consider for example the nth order differential equation for v : I ! R   v (n) (t) D F t; v; v˙; : : : ; v (n1) ; with initial conditions v (k) (t0 ) D v k ;

k D 0; : : : ; n  1 :

This can be written as a first-order system by using the vector function u : I ! Rm defined as u D (v; v˙; : : : ; v (n1) ).

The equivalent initial value problem for u is 0 B B u˙ D f (t; u) D B @

u2 :: : un F(t; u1 ; u2 ; : : : ; u n )

1 C C C; A 0

B B with u(t0 ) D B @

v0 v1 :: :

1 C C C: A

v n1 Obviously, this system is not a unique representation, there are many other ways of obtaining first-order systems from the nth-order problem. In the beginning of the study of differential equations, the focus is on finding explicit solutions. The first existence theorem is by Cauchy [3] in the middle of the nineteenth century and an initial value problem is also called a Cauchy problem. At the end of the nineteenth century substantial progress is made on the existence of solutions of an initial value problem when Peano [20] shows that if f is continuous, then a solution exists near the initial value (t0 ; u0 ), i. e., there is local existence of solutions. Global existence of solutions of initial value problems needs more than smoothness, as is illustrated by the following example. Example 1 Consider the initial value problem on D D R  R given as u˙ D u2

and

u(0) D u0

for some u0 > 0. As can be verified easily, the solution is u(t) D u0 /(1  tu0 ), for t 2 (1; 1/u0 ). This solution cannot be extended for t  1/u0 , even though the vector field f (t; u) D u2 is infinitely differentiable on the full domain. As can be seen from later theorems, the lack of global existence is related to the fact that the vector field f is unbounded. Once existence of solutions of initial value problems is known, the next question is if such a solution is unique. As can be seen from the following example, continuity of f is not sufficient for uniqueness of solutions. Example 2 Consider the following initial value problem on D D R  R: u˙ D juj˛

and u(0) D 0 :

If 0 < ˛ < 1, then there is an infinite number of solutions. Two obvious solutions for t 2 [0; 1) are u(t) D 0 and

Existence and Uniqueness of Solutions of Initial Value Problems

1

ˆ D ((1  ˛)t) 1˛ . But these solutions are members of u(t) a large family of solutions. For any c  0, the following functions are solutions for I D R: ( 0; t
in more general differential equations and the use of existence and uniqueness of solutions of initial value problems in dynamical systems. Existence Cauchy [3] seems to have been the first one to publish an existence result for differential equations, using an approximation of solutions by joined line segments. This work was extended considerably by Peano [20] in 1890. Theorem 1 (Cauchy–Peano Theorem) Let a; b > 0, define I a (t0 ) :D [t0 ; t0 C a] and Bb (u0 ) :D fu 2 Rm j ju  u0 j  bg and assume that the cylinder S :D I a (t0 )  Bb (u0 ) D. Let the vector field f be a continuous function on the cylinder S. This implies that f is bounded on S, say, M D maxfj f (t; u)j j (t; u) 2 Sg.  

b . Then there exDefine the parameter D min a; M ists a solution u(t), with t0  t  t0 C , which solves the initial value problem (1).

The theorem can also be phrased for an interval [t0  ; t0 ] or [t0  ; t0 C ]. The parameter gives a time interval such that that the solution (t; u(t)) is guaranteed to be within S. From the integral formulation of the initial value problem (2), it follows that Z t ju(t)  u0 j  j f (; u())jd  M(t  t0 ) : t0

To guarantee that u(t) is within S, the condition t  t0  is sufficient as it gives M(t  t0 )  b and t  t0  a. In Fig. 1, this is illustrated in the case D R  R.

Existence and Uniqueness of Solutions of Initial Value Problems, Figure 1 The parameter D min( Mb ; a) guarantees that the solution is within the shaded area S, on the left in the case a < if a > Mb

b M

and on the right

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Existence and Uniqueness of Solutions of Initial Value Problems

To prove the Cauchy–Peano existence theorem, a construction which goes back to Cauchy is used. This so-called Cauchy–Euler construction of approximate solutions uses joined line segments which are such that the tangent of each line segment is given by the vector field evaluated at the start of the line segment. To be specific, for any N 2 N, the interval I (t0 ) D [t0 ; t0 C ] is divided in N equal parts with intermediate points t k :D t0 C Nk , k D 1; : : : ; N. The approximate solution is the function u N : I (t0 ) ! R, with u N (t0 ) D u0 ; u N (t) D u N (t k1 ) C f (t k1 ; u N (t k1 )) (t  t k1 ) ; (3) t k1 < t  t k ;

k D 1; : : : ; N ;

see also Fig. 2. One sees immediately that function uN is continuous and piecewise differentiable and, using the bound on the vector field f , ju N (t)  u N (s)j  Mjt  sj ;

for all t; s 2 I (t0 ) : (4)

This estimate implies that the sequence of functions fu N g is uniformly bounded and that the functions uN are equicontinuous. Indeed, the estimate above gives that for any t 2 I (t0 ) and any N 2 N, ju N (t)j  ju N (t0 )j C Mjt  t0 j  ju0 j C M : And equicontinuity follows immediately from (4) as M does not depend on N. Arzela–Ascoli’s lemma states that a sequence of functions which is uniformly bounded and equicontinuous on a compact set has a subsequence which is uniformly convergent on this compact set [5]. As the sequence fu N g is uniformly bounded and equicontinuous on the com-

pact set I (t0 ), it follows immediately that it has a convergent subsequence. This convergent subsequence is denoted by u Nk and its limit by u. To prove the Cauchy– Peano theorem, we will show that this limit function u satisfies the integral equation (2) and hence is a solution of the initial value problem (1). Proof First it will be shown that if N is sufficiently large, then the functions uN are close to solutions of the differential equation in the following sense: for all " > 0, there is some N0 such that for all N > N0 ju˙ N (t)  f (t; u N (t))j < " ; t 2 I (t0 ) ;

t ¤ tk ;

k D 0; : : : ; N :

(5)

Let " > 0. As f is a continuous function on the compact set S, it follows that f is uniform continuous. Thus, there is some ı" such that for all (t; u); (s; v) 2 S with jt  sj C ju  vj < ı" j f (t; u)  f (s; v)j < " :

(6)

m l and let N > N0 . Then for Now define N0 D (MC1)

ın any t 2 I (t0 ), t ¤ t k , k D 0; : : : ; N, we have u˙ N (t) D f (t l ; u N (t l )), where l is such that t l < t < t l C1 . Hence, (6) gives ju˙ N (t)  f (t; u N (t))j D j f (t l ; u N (t l ))  f (t; u N (t))j < " as (4) shows that jt l  tj C ju N (t l )  u N (t)j < (1 C M)jt l  tj < (1CM)

 (1CM)

 ı" . N N0 Next it will be shown that (5) implies that the functions uN almost satisfy the integral equation (2) for N sufficiently large. From (5), it follows that f (t; u N (t))  " < u˙ N (t) < f (t; u N (t)) C " for all N > N0 and all t 2 I (t0 ), except at the special points tk , k D 0; : : : ; N. Hence, for any k D 1; : : : ; N and all t k1 < t  t k , this gives Z

t

Z

t k1 t



u N (t) < u N (t k1 ) C  u N (t k1 ) C

 f (; u N ()) C " d

f (; u N ())d C t k1

" : N

Thus, R t k also for any k D" 1; : : : ; N, u N (t k )  u N (t k1 ) < t k1 f (; u N ())d C N and hence Existence and Uniqueness of Solutions of Initial Value Problems, Figure 2 The approximate solution uN , with N D 5, and the solution u(t). The dotted lines show the tangent given by the vector field f(t; u) at the points (ti ; u) for various values of u and i D 0; : : : ; 5

u N (t k )  u N (t0 ) D

k X  u N (t j )  u N (t j1 ) jD1

Z

tk

<

f (; u N ())d C " : t0

Existence and Uniqueness of Solutions of Initial Value Problems

Combination of the last two results gives that for any t 2 I (t0 ) Z t   u N (t) < u N (t0 ) C f (; u N ()) C " d C " : t0

In R t a similar way it can be shown that u N (t) > u N (t0 ) C t 0 f (; u N ())d  " and hence for all N > N0 , the following holds: ˇ ˇ Z t ˇ ˇ ˇ < " ; ˇu N (t)  u N (t0 )  f (; u ()d N ˇ ˇ t0

for all t 2 I (t0 ) :

(7)

Thus, the function uN satisfies the integral equation (2) up to an order " error if N is sufficiently large. As the subsequence u N k converges R t uniformly to u and f is continuous, this implies that t0 f (; u N k ()) ! Rt t 0 f (; u())d. Thus, from (7), it can be concluded that u satisfies the integral equation (2) exactly.  Note that the full sequence fu N g does not necessarily converge. A counterexample can be found in exercise 12 in Chap. 1 of [5]. It is based on the initial value problem 1 u˙ D juj 4 sgn(u)Ct sin(/t) with u(0) D 0, and shows that on a small interval near t0 D 0, the even approximations are bounded below by a strictly positive constant, while the odd approximations are bounded above by a strictly negative constant. However, if it is known that the solution of the initial value problem is unique, then the full sequence of approximate solutions fu N g must converge to the solution. This can be seen by a contradiction argument. If there is a unique solution, but the sequence fu N g is not convergent, then there is a convergent subsequence and a remaining set of functions which does not converge to the unique solution. But Arzela–Ascoli’s lemma can be applied to the remaining set as well; thus, there exists a convergent subsequence in the remaining set, which must converge to a different solution. This is not possible as the solution is unique. In case of the initial value problem as presented in Example 2, the sequence of Cauchy–Euler approximate solutions fu N g converges to the zero function; hence, no subsequence is required to obtain a solution. As there are many solutions of this initial value problem, this implies that not all solutions of the initial value problem in Example 2 solutions can be approximated by the Cauchy–Euler construction of approximate solutions as defined in (3). As indicated in exercise 2.2 in Chap. 2 of [13], it is possible to get any solution by using appropriate Lipschitz continuous approximations of the vector field f , instead of the

original vector field itself, in the Cauchy–Euler construction. In the example p below this is illustrated for the vector field f (t; u) D juj. p Example 3 Let f : R  R be defined as f (t; u) D juj. First we define the approximate functions fˆn : [0; 2]  [0; 1) ! [0; 1) as (p u; u > n1 ˆf n (t; u) D   p p1  1 n 1  u ; u  1 2 n n n and the definition of the Euler–Cauchy approximate functions is modified to u N (t) D u N (t k1 ) C fˆN (t k1 ; u N (t k1 )) (t  t k1 ); t k1 < t  t k ;

k D 1; : : : ; N :

This sequence converges to the solution u(t) D 4t 2 for t 2 [0; 2]. Next define the approximate functions f˜n : [0; 2]  [0; 1) ! [0; 1) as f˜n (t; u) 8p u; ˆ ˆ ˆ1 1  ˆ ˆ u ˆ < 2n  2n 1 2 D Ca2 2n  u ˆ 1 3 ˆ ˆ ˆ ˆ Ca3 n  u ; ˆ : u;

t 1; u >

t 1;

1 2n

1 n

;


t 1; u 

1 2n

1 n

;

;

p p with a2 D 4n( n1) and a3 D 12n3 (3 n2), f˜n (t; u) D fˆn (t; u), for t  1 C n1 and and a smooth connection between those two components of the approximation for a < t < 1 C n1 . Then the related approximate solutions will converge to the solution u(t) D 4(t  1)2 , 1  t  2 and u(t) D 0, 0  t < 1. In a similar way all other solutions presented in Example 2 can be obtained. Example 2 gives a connected and closed family of solutions to the initial value problem. The following theorem shows this is typical. Theorem 2 ([15]) Assume that the assumptions of Theorem 1 are satisfied. Let t0 < c  t0 C and define Ac to be the set of points that can be reached at time t D c by some solution of the initial value problem. Then Ac is closed, bounded and connected. If the initial value problem is one-dimensional, i. e., u 2 R, then this implies that the set of points reached by possible solutions at time t D c is the empty set, a point or a closed interval. A proof of this theorem, based on [19], can be found in Theorem 4.1 in Chap. 2 of [13].

387

388

Existence and Uniqueness of Solutions of Initial Value Problems

The existence theorems presented so far give existence of solutions near the initial value (t0 ; u0 ). But this local result can be extended to a global one. Theorem 3 (Extension Theorem) Let f be continuous on D. If u(t) is a solution of the initial value problem (1) on some interval, then the solution can be extended to a maximal interval of existence (t ; tC ) such that (t; u(t)) converges to the boundary of D as t " tC or t # t (where t˙ D ˙1 is possible). Proof Let Dn be open subsets of D such that [n2N D n D D, the closures D n are compact and D n D nC1 (e. g., D n D f(t; u) 2 R j j(t; u)j < n; dist((t; u); @D) > 1/ng, see [13]). First it will be shown that for all n 2 N, there is some n such that for each (t0 ; u0 ) 2 D n , the initial value problem has a solution on the interval [t0  n ; t0 C n ]. Indeed, for each n 2 N, the function f is continuous on the compact set D n . Hence, there is some M n > 0 such that j f (t; u)j  M n for all (t; u) 2 D n . Furthermore, for each n 2 N, the distance d n :D dist(D n ; D nC1 ) > 0. Thus, the Cauchy–Peano theorem implies that for each (t0 ; u0 ) 2 D n , the solution of the initial value problem exists on the interval [t0  n ; t0 C n ], with n D 1/2 min (d n /M nC1 ; d n ). If the solution u(t) is defined on an interval I, which is not a right maximal interval, then the argument above shows that the right endpoint of I can be included in the interval of existence. So it can be assumed that the interval I is of the form [a1 ; a2 ]. The continuity of the solution u gives that the set f(t; u(t)) j t 2 [a1 ; a2 ]g is a compact set in the open set D; hence, there is some n 2 N such that f(t; u(t)) j t 2 [a1 ; a2 ]g D n D n . But this implies that the solution can be extended to the interval [a1 ; 12 C n ]. And if f(t; u(t)) j t 2 [a1 ; a2 C n ]g D n , then this can be repeated. Since D n is compact, there is some k 2 N such that f(t; u(t)) j t 2 [a1 ; a2 C k n ]g 6 D n ; hence, there exists some (t n ; u(t n )) such that (t n ; u(t n )) 2 D nC1 nD n . This argument can be repeated for each n 2 N to give sequence (t n ; u(t n )) 2 D nC1 nD n , with t n < t nC1 . Thus, the sequence tn is monotone increasing. Either it is unbounded, which implies that the solution exists on an interval [a1 ; 1) and hence tC D 1 or it is bounded and hence is convergent to some limit tC . Similarly, the sequence (t n ; u(t n )) is either an unbounded sequence in D or has a cluster point on the boundary of D. In either case it is clear that the solution cannot be extended outside the interval [a1 ; tC ). A similar argument can be used for the left endpoint and it can be concluded that there exists a maximal interval of existence of the form (t ; tC ). Finally it will be shown that the solution converges to the boundary of D by a contradiction argument. Assume

that the solution does not converge to the boundary if t " tC . Then there is some "0 > 0 and a sequence fn g with tC  n < n1 and d(n ; u(n )); D) > "0 . This implies that there is some N 2 N such that the sequence f(n ; u(n ))g D N . With the arguments above, this implies that the solution can be extend to an interval including tC , which contradicts tC being the maximal right point.  Example 1 gives an example of a solution for which u(t) ! 1 if t ! u10 , the boundary of its interval of existence. Example 2 gives an example of a solution with an unbounded interval of existence. Example 4 shows that the endpoint can also be related to the failure of continuity of f . Example 4 Consider the initial value problem with D D R  (0; 1) and 1 u˙ D  ; and u(0) D 2 : u p Then the solution is u(t) D 4  2t for t 2 (1; 2). If t ! 2, then (t; u(t)) ! (2; 0), hence the boundary of D and the point where the continuity of f fails. If the initial value problem has a nonunique solution, then the maximal interval of extension in the extension theorem will in general depend on the initial solution u(t). Uniqueness Uniqueness follows if the vector field f is Lipschitz continuous with respect to its second variable u as shown first by Picard [21] and Lindelöf [18]. Their proof is based again on approximate solutions, but the approximate solutions are smoother than the Cauchy–Euler ones. The approximate solutions are obtained by successive iterations and are defined as u0 (t) :D u0

and Z (t) :D u C u nC1 0

t

f (; u n ())d ;

n2N; t2I:

t0

(8) Sometimes this iteration is called Picard iteration. Clearly, this iteration is based on the integral formula (2). The iteration process fails if u n () 62 D for some  2 I. If f is continuous, then there is some interval I for which the sequence is well-defined, as follows from the proof below. Theorem 4 (Picard–Lindelöf Theorem) Let a; b > 0 be such that the cylinder S :D I a (t0 )  Bb (u0 ) D. Let the vector field f be a uniformly Lipschitz continuous function on the cylinder S with respect to its second variable u and let M be the upper bound on f , i. e., M D maxf f (t; u) j (t; u) 2 Sg.

Existence and Uniqueness of Solutions of Initial Value Problems

  b Define the parameter D min a; M . For t 2 [t0 ; t0 C ], there exists a unique solution u(t) of the initial value problem (1). Successive iterations play an important role in proving existence for more general differential equations. Thus, although the existence of solutions already follows from the Cauchy–Peano theorem, we will prove again existence by using the successive iterations instead of the Cauchy–Euler approximations as the ideas in the proof can be extended to more general differential equations. Proof By using the bound on the vector field f , we will show that if (t; u n (t)) 2 S for all t 2 [t0 ; t0 C 0 ], then (t; u nC1 (t)) 2 S for all t 2 [t0 ; t0 C 0 ]. Indeed, for any t 2 [t0 ; t0 C 0 ], Z t ju nC1 (t)  u0 j  j f (; u n ())jd

Z

t

ju nC2 (t)  u nC1 (t)j 

L ju nC1 ()  u n ()jd t0

 L2 ku n  u n1 k1

Z

t

(  t0 )d t0

D

L2 ku n  u n1 k1 (t  t0 )2 : 2

Repeating this process, it follows for any k 2 N that Lk ku n  u n1 k1 (t  t0 ) k k! Lk k  ku n  u n1 k1 : k!

ju nCkC1 (t)  u nCk (t)j 

By using the triangle inequality repeatedly this gives for any n; k; 2 N and t 2 [t0 ; t0 C 0 ], ju nCkC1 (t)  u n (t)j 

t0 Z t

k X

ju nCiC1 (t)  u nCi (t)j

iD0

Md  M(t  t0 )  b ;



(9)



t0

k X L nCi nCi ku1  u0 k1 (n C i)! iD0

thus, ju nC1 (t)  u0 j  M  b and therefore u nC1 (t) 2 B b (u0 ). The boundedness of f also ensures that the functions fu n g are equicontinuous as for any n 2 N and any t0  t1 < t2  C t0 : ju n (t1 )  u n (t2 )j  Z t2 j f (; u n ())jd  Mjt2  t1 j : t1

Up to this point, we have only used the boundedness of f (which follows from the continuity of f ). But for the next part, in which it will be shown that for any t 2 [t0 ; t0 C

0 ] the sequence fu n (t)g is a Cauchy sequence in Rm , we will need the Lipschitz continuity. Let L be the Lipschitz constant of f on S, then for any n 2 N and t 2 [t0 ; t0 C 0 ], we have Z t ju nC1 (t)  u n (t)j  j f (; u n ())  f (; u n1 ())jd Z

t0 t

L ju n ()  u n1 ()jd

 t0

Z

t

 L ku n  u n1 k1

d t0

D L ku n  u n1 k1 (t  t0 ) ; where ku n  u n1 k1 D supfju n ()  u n1 ()j j t0    t0 C g. This implies that



k X Li i (L )n ku1  u0 k1 n! i! iD0

(L )n b eL :  n! Thus, for any t 2 [t0 ; t0 C 0 ], the sequence fu n (t)g is a Cauchy sequence in Rm ; hence, the sequence has a limit, which will be denoted by u(t). In other words, the sequence of functions fu n g is pointwise convergent in I (t0 ). We have already seen that the functions un are equicontinuous; hence, the convergence is uniform and the limit function u(t) is equicontinuous [23]. To see that u satisfies the integral equation (2), observe that for any t 2 [t0 ; t0 C 0 ] ˇ ˇ Z t ˇ ˇ ˇu(t)  u0  f (; u())d ˇˇ ˇ t0

 ju(t)  u nC1 (t)j ˇZ t ˇ ˇ ˇ ˇ Cˇ j f (; u n ())  f (; u())jd ˇˇ t0

 ju(t)  u nC1 (t)j C Lku  u n k1 (t  t0 )  (1 C L ) (ku  u nC1 k1 C ku  u n k1 ) for any n 2 N. As the sequence fu n g is uniformly convergent, this implies that u(t) satisfies the integral equation (2). Finally the uniqueness will be proved using techniques similar to those in “Existence.” Assume that there are two

389

390

Existence and Uniqueness of Solutions of Initial Value Problems

solutions u and v of the integral equation (2). Hence, for any t0  t  t0 C , Z t j f (; u())  f (; v())jd ju(t)  v(t)j  t0

 Lku  vk1 (t  t0 ) : Thus, for k D 1, it holds that ju(t)  v(t)j 

L k ku  vk1 (t  t0 ) k ; k! for any t0  t  t0 C :

(10)

For the induction step, assume that (10) holds for some k 2 N. Then for any t0  t  t0 C , Z t k L ku  vk1 (  t0 ) k d ju(t)  v(t)j  L k! t0 D

L kC1 ku  vk1 (t  t0 ) kC1 : (k C 1)!

By using the principle of induction, it follows that (10) holds for any k 2 N. This implies that for any   t0kC1 satisfy ju(t)  v(t)j  (L ) t  t0 C ,the solutions ju  vk1 / (k C 1)! for any k 2 N. Since the expression on the right converges to 0 for k ! 1, this implies that u(t) D v(t) for all t 2 [t0 ; t0 C ] and hence the solution is unique.  Implicitly, the proof shows that the integral equation gives rise to a contraction on the space of continuous functions with the supremum norm and this could also have been used to show existence and uniqueness; see [11,25], where Schauder’s fixed-point theorem is used to prove the Picard–Lindelöf theorem. With Schauder’s fixed-point theorem, existence and uniqueness can be shown for much more general differential equations than ordinary differential equations. The iteration process in (8) does not necessarily have to start with the initial condition. It can start with any continuously differentiable function u0 (t) and the interation process will still converge. In specific problems, there are often obvious choices for the initial function u0 which give a faster convergence or are more efficient, for instance, when proving unboundedness. Techniques similar to those in the proof can be used to derive a bound on the difference between the successive approximations and the solution, showing that ju n (t)  u(t)j 

ML n (t  t0 )n ; (n C 1)!

t 2 [t0 ; t0 C ] ;

see [13]. This illustrates that the convergence is fast for t near t0 , but might be slow further away: see Fig. 3 for an example.

Existence and Uniqueness of Solutions of Initial Value Problems, Figure 3 The first four iterations for initial value problem with the vector field f(t; u) D t cos u C sin t and the initial value u(0) D 0. Note that the convergence in the interval [0; 1] is very fast (two steps seem sufficient for graphical purposes), but near t D 2, the convergence has not yet been reached in three steps

Continuous Dependence on Initial Conditions In this section, the vector field f is uniformly Lipschitz continuous on D with respect to its second variable u and its Lipschitz constant is denoted by L. By combining the Picard–Lindelöf theorem (Theorem 4) and the extension theorem (Theorem 3), it follows that for every (t0 ; u0 ) 2 D, there exists a unique solution of (1) passing through (t0 ; u0 ) with a maximal interval of existence (t (t0 ; u0 ); tC (t0 ; u0 )). The trajectory through (t0 ; u0 ) is the set of points (t; u(t)), where u(t) solves (1) and t (t0 ; u0 ) < t < tC (t0 ; u0 ). Now define the set E RnC2 as E :D f (t; t0 ; u0 ) j t (t0 ; u0 ) < t < tC (t0 ; u0 ); (t0 ; u0 ) 2 D g : The flow map ˚ : E ! Rm is a mapping which describes the solution of the initial value problem with the initial condition varying through D, i. e., d ˚(t; t0 ; u0 ) D f (t; ˚ (t; t0 ; u0 )) and dt

Existence and Uniqueness of Solutions of Initial Value Problems

˚(t0 ; t0 ; u0 ) D u0 ;

for (t; t0 ; u0 ) 2 E :

From the integral equation (2), it follows that the flow map satisfies Z

Z

t

(t)  K C

()()d ;

for all a  t  b :

a

Then

t

˚(t; t0 ; u0 ) D u0 C

such that

f (; ˚ (; t0 ; u0 ))d ; t0

Z (t)  K exp

()d

Lemma 5 (Gronwall’s Lemma) Let : [a; b] ! R and  : [a; b] ! R be nonnegative continuous functions on an interval [a; b]. Let K  0 be some nonnegative constant

;

for all a  t  b :

a

for (t; t0 ; u0 ) 2 E : Furthermore, the uniqueness implies for the flow map that starting at the point (t0 ; u0 ), flowing to (t; ˚ (t; t0 ; u0 )) is the same as starting at the point (t0 ; u0 ) and flowing to the intermediate point (s; ˚ (s; t0 ; u0 )) and continuing to the point (t; ˚ (t; t0 ; u0 )). This implies that the flow map has a group structure: for any (t0 ; u0 ) 2 D and any s; t 2 (t (t0 ; u0 ); tC (t0 ; u0 )), the flow map satisfies ˚(t; s; ˚ (s; t0 ; u0 )) D ˚(t; t0 ; u0 ); the identity map is given by ˚(t0 ; t0 ; u0 ) D u0 ; and by combining the last two results, it follows that ˚ (t0 ; t; ˚ (t; t0 ; u0 )) D ˚ (t0 ; t0 ; u0 ) D u0 , and hence there is an inverse. See also Fig. 4 for a sketch of the flow map for D R  R. To show that the flow map is continuous in all its variables, Gronwall’s lemma [9] will be very useful. There are many versions of Gronwall’s lemma; the one presented here follows [1].



t

Rt Proof Define F(t) D K C a ()()d, for a  t  b. Then the assumption in the lemma gives (t)  F(t) for a  t  b. Furthermore, F is differentiable, with F˙ D  ; hence, for a  t  b  

Z t d ()d F(t) exp  dt a

Z t  D (t)(t) exp  ()d a

Z t   F(t) (t) exp  ()d  0 a

as (t)  F(t). Integrating the left-hand side gives 

Z t ()d  F(a) D K ; F(t) exp  a

which implies for  that (t)  F(t)  K exp

hR

t a

i ()d . 

Gronwall’s lemma will be used to show that small variations in the initial conditions give locally small variations in the solution, or Theorem 6 The flow map ˚ : E ! RnC2 is continuous in E.

Existence and Uniqueness of Solutions of Initial Value Problems, Figure 4 The flow map ˚(t; t0 ; u0 ). Note the group property ˚(t; s; ˚(s; t0 ; u0 )) D ˚(t; t0 ; u0 ). The lightly shaded area is the tube Uı1 and the darker shaded area is the tube Uı as used in the proof of Theorem 6. If a solution starts from within Uı , then it will stay within the tube Uı1 for t1  t  t2 , as the solution ˚(t; s0 ; v0 ) demonstrates

Proof Let (t; t0 ; u0 ) 2 E. As t; t0 2 (t (t0 ; u0 ); tC (t0 ; u0 )), there is some t1 < t2 such that t; t0 2 (t1 ; t2 ), [t1 ; t2 ]

(t (t0 ; u0 ); tC (t0 ; u0 )) and the solution ˚(s; t0 ; u0 ) exists for any s 2 [t1 ; t2 ]. First we will show that there is some tube around the solution curve f(s; ˚ (s; t0 ; u0 )) j t1  s  t2 g such that for any (s0 ; v0 ) in this tube, the solution ˚(s; s0 ; v0 ) exists for t1  s  t2 . As D is open, there is some ı1 > 0 such that the closed tube Uı1 :D f(s; v) j j˚(s; t0 ; u0 )vj  ı1 ; t1  s  t2 g D. As f is Lipschitz continuous, hence continuous, there is some M > 0 such that j f (s; v)j  M for all (s; v) 2 Uı1 . Recall that the Lipschitz constant of f on D is denoted by L. Now define ı :D ı1 eL(t2 t1 ) < ı1 and the open tube Uı D f(s; v) j j˚(s; t0 ; u0 )  vj < ı; t1 < s < t2 g ; thus, Uı Uı1 D. Thus, for every (s0 ; v0 ) 2 Uı , the solution ˚(s; s0 ; v0 ) exists for s in some closed interval in

391

392

Existence and Uniqueness of Solutions of Initial Value Problems

[t1 ; t2 ] around s0 and the solution in this interval satisfies Z s ˚(s; s0 ; v0 ) D v0 C f (; ˚ (; s0 ; v0 ))d ; s0

see also Fig. 4. Furthermore, for any s 2 [t1 ; t2 ], we have ˚ (s; t0 ; u0 ) D ˚(s; s0 ; ˚ (s0 ; t0 ; u0 )) Z s f (; ˚ (; s0 ; ˚ (s0 ; t0 ; u0 )))d D ˚(s0 ; t0 ; u0 ) C s Z 0s f (; ˚ (; t0 ; u0 ))d : D ˚ (s0 ; t0 ; u0 ) C s0

Subtracting these expressions gives j˚(s; s0 ; v0 )  ˚(s; t0 ; u0 )j  jv0  ˚(s0 ; t0 ; u0 )j Z s C j f (; ˚ (; s0 ; v0 ))  f (; ˚ (; t0 ; u0 ))j d s0 Z s L j˚(; s0 ; v0 )  ˚(; t0 ; u0 )jd ; ıC s0

so Gronwall’s lemma implies that j˚(s; s0 ; v0 )  ˚ (s; t0 ; u0 )j  ıe L(ss 0 )  ı1 . Hence, for any s, ˚ (s; s0 ; v0 ) 2 Uı1 D, and hence this solution can be extended to its maximal interval of existence, which contains the interval [t1 ; t2 ]. So it can be concluded that for any (s0 ; v0 ) 2 Uı , the solution ˚(s; s0 ; v0 ) exists for any s 2 [t1 ; t2 ]. Next we will show continuity of the flow map in its last two variables, i. e., in the initial value. For any (s0 ; v0 ) 2 Uı and t1  t  t2 , we have j˚(t; s0 ; v0 )  ˚(t; t0 ; u0 )j  jv0  u0 j ˇZ t ˇ Z t ˇ ˇ ˇ C ˇ f (; ˚ (; s0 ; v0 ))d  f (; ˚ (; t0 ; u0 ))dˇˇ s0 t0 ˇZ t 0 ˇ ˇ ˇ ˇ j f (; ˚ (; s0 ; v0 ))jd ˇˇ  jv0  u0 j C ˇ s0 ˇ ˇZ t ˇ ˇ ˇ j f (; ˚ (; t0 ; u0 ))  f (; ˚ (; s0 ; v0 ))jd ˇˇ Cˇ t0

 jv0  u0 j C Mjt0  s0 j ˇ ˇZ t ˇ ˇ ˇ Lj˚(; t0 ; u0 )  ˚(; s0 ; v0 )jd ˇˇ : Cˇ t0

Thus, Gronwall’s lemma implies that j˚(t; s0 ; v0 )  ˚(t; t0 ; u0 )j  (jv0  u0 j C Mjt0  s0 j) e Ljtt0 j  (jv0  u0 j C Mjt0  s0 j) e Ljt2 t1 j ;

and it follows that ˚ is continuous in its last two arguments. The continuity of ˚ in its first argument follows immediately from the fact that the solution of the initial value problem is continuous.  If the vector field f is smooth, then the flow map ˚ is smooth as well. Theorem 7 Let f 2 C 1 (D; R m ). Then the flow map ˚ 2 C 1 (E; R m ) and det(Du 0 ˚(t; t0 ; u0 )) Z t    D exp tr Du f (; ˚ (; t0 ; u0 )) d t0

for any (t; t0 ; u0 ) 2 E. In this theorem, tr(Du f (; ˚ (; t0 ; u0 ))) stands for the trace of the matrix Du f (; ˚ (; t0 ; u0 )). For second-order linear systems, this identity is known as Abel’s identity and det(Du 0 ˚(t; t0 ; u0 )) is the Wronskian. The proof of Theorem 7 uses the fact that Du 0 ˚(t; t0 ; u0 ) satisfies the linear differential equation d Du ˚(t; t0 ; u0 ) D Du f (t; ˚ (t; t0 ; u0 )) Du 0 ˚(t; t0 ; u0 ); dt 0 with the initial condition Du 0 ˚(t0 ; t0 ; u0 )) D I. This last fact follows immediately from ˚ (t0 ; t0 ; u0 ) D u0 . And the linear differential equation follows by differentiating the differential equation for ˚(t; t0 ; u0 ) with respect to u0 . The full details of the proof can be found, for example, in [5,11]. Extended Concept of Differential Equation Until now, we have looked for continuously differentiable functions u(t), which satisfy the initial value problem (1). But the initial value problem can be phrased for less smooth functions as well. For example, one can define a solution as an absolute continuous function u(t) which satisfies (1). A function u : I ! Rm is absolutely continuous on I if for every positive number ", no matter how small, there is a positive number ı small enough so that whenever a sequence of pairwise disjoint subintervals P [s k ; t k ] of I, k D 1; 2; : : : ; N satisfies N kD1 jt k  s k j < ı then n X

d (u(t k ); u(s k )) < " :

kD1

An absolute continuous function has a derivative almost everywhere and is uniformly continuous, thus continuous [24]. Thus, the initial value problem for an absolute continuous function can be stated almost everywhere.

Existence and Uniqueness of Solutions of Initial Value Problems

For the existence of absolute continuous solutions, the continuity condition for the vector field in the Cauchy– Peano theorem (Theorem 1) is replaced by the so-called Carathéodory conditions on the vector field. A function f : D ! Rm satisfies the Carathéodory conditions if [2]  f (t; u) is Lebesgue measurable in t for each fixed u;  f (t; u) is continuous in u for each fixed t;  for each compact set A D, there is a measurable function m A (t) such that j f (t; u)j  m A (t) for any (t; u) 2 A. Similar theorems to the ones in the previous sections about existence and uniqueness can be stated in this case. Theorem 8 (Carathéodory) If D is an open set in RnC1 and f : D ! Rm satisfies the Carathéodory conditions on D, then for every (t0 ; u0 ) 2 D there is a solution of the initial value problem (1) through (t0 ; u0 ). Theorem 9 If D is an open set in RnC1 , f : D ! Rm satisfies the Carathéodory conditions on D and u(t) satisfies the initial value problem (1) on some interval, then there exists a maximal open interval of existence. Furthermore, if (t ; tC ) denotes the maximal interval of existence, then the solution u(t) tends to the boundary of D if t # t or t " tC . Theorem 10 If D is an open set in RnC1 , f : D ! Rm satisfies the Carathéodory conditions on D and for every compact set U D there is an integrable function LU (t) such that j f (t; u) f (t; v)j  LU juvj;

for all (t; u); (t; v) 2 U ;

then for every (t0 ; u0 ) 2 D, there is a unique solution ˚ (t; t0 ; u0 ) of the initial value problem (1). The domain E of the flow map ˚ is open and ˚ is continuous in E. Proofs of those theorems can be found, for example, in Sect. I.5 in [11]. Many other generalizations are possible, for example, to vector fields which are discontinuous in u. Such vector fields play an important role in control theory. More details can be found in [6,7,17].

similar to the ones for ordinary differential equations as presented in the earlier sections and can be proved by using Schauder’s fixed-point theorem. But not all solutions can be extended as in the extension theorem (Theorem 3); see Chap. 2 in [12]. If one considers differential equations with more variables, i. e., partial differential equations, then there are no straightforward general existence and uniqueness theorems. For partial differential equations, the existence and uniqueness depends very much on the details of the partial differential equation. One possible theorem is the Cauchy– Kowaleskaya theorem, which applies to partial differential 2 equations of the form @m t u D F(t; u; @x u; @ t u; @ tx u; : : : ), where the total number of derivatives of u in F should be less than or equal to m [22]. But there are still many open questions as well. A famous open question is the existence and uniqueness of solutions of the three-dimensional Navier–Stokes equations, a system of equations which describes fluid motion. Existence and uniqueness is known for a fluid in a two-dimensional domain, but not in a threedimensional domain. The question is one of the seven “Millennium Problems,” stated as prize problems at the beginning of the third millennium by the Clay Mathematics Institute [26]. Existence and uniqueness results are also used in dynamical systems. An area of dynamical systems is bifurcation theory and for bifurcation theory the smooth dependence on parameters is crucial. The following theorem gives sufficient conditions for the smooth dependence parameters; see Theorem 3.3 in Chap. 1 of [11]. Theorem 11 If the vector field depends on parameters  2 R k , i. e., f : D  R k ! Rm and f 2 C 1 (D  R k ; Rm ), then the flow map is continuously differentiable with respect to its parameters . Furthermore, D ˚ satisfies an inhomogeneous linear equation d D ˚(t; t0 ; u0 ; ) dt D Du f (t; ˚ (t; t0 ; u0 ; ); ) D ˚(t; t0 ; u0 ; ) C D f (t; ˚ (t; t0 ; u0 ; ); ) with initial condition D ˚(t0 ; t0 ; u0 ; ) D 0.

Further Directions As follows from the previous sections, the theory of initial value problems for ordinary differential equations is quite well understood, at least for continuous vector fields. For more general differential equations, the situation is quite different. Consider for example a retarded differential equation, hence a differential equation with delay effects [12]. For retarded differential equations, there are local existence and uniqueness theorems which are quite

Results and overviews of bifurcation theory can be found, for example, in [4,8,10,14,16]. Another area of dynamical systems is stability theory. Roughly speaking, a solution is called stable if other solutions which start near this solution stay near it for all time. Note that the local continuity result in Theorem 6 is not a stability result as it only states that nearby solutions will stay nearby for a short time. For stability results, long-time existence of nearby solutions is a crucial property [10,14].

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Bibliography Primary Literature 1. Bellman R (1943) The stability of solutions of linear differential equations. Duke Math J 10:643–647 2. Carathéodory C (1918) Vorlesungen über Reelle Funktionen. Teubner, Leipzig (reprinted: (1948) Chelsea Publishing Company, New York) 3. Cauchy AL (1888) Oevres complètes (1) 6. Gauthiers-Villars, Paris 4. Chow S-N, Hale JK (1982) Methods of bifurcation theory. Springer, New York 5. Coddington EA, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York 6. Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, Dordrecht 7. Flugge-Lotz I (1953) Discontinuous automatic control. Princeton University Press, Princeton 8. Golubitsky M, Stewart I, Schaeffer DG (1985–1988) Singularities and groups in bifurcation theory, vol 1 and 2. Springer, New York 9. Gronwall TH (1919) Note on the derivative with respect to a parameter of the solutions of a system of differential equations. Ann Math 20:292–296 10. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York 11. Hale JK (1969) Ordinary Differential Equations. Wiley, New York 12. Hale JK, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations. Springer, New York 13. Hartman P (1973) Ordinary Differential Equations. Wiley, Baltimore 14. Iooss G, Joseph DD (1980) Elementary stability and bifurcation theory. Springer, New York 15. Kneser H (1923) Ueber die Lösungen eines Systems gewöhnlicher Differentialgleichungen das der Lipschitzschen Bedingung nicht genügt. S-B Preuss Akad Wiss Phys-Math Kl 171– 174 16. Kuznetsov YA (1995) Elements of applied bifurcation analysis. Springer, New York 17. Lee B, Markus L (1967) Optimal Control Theory. Wiley, New York

18. Lindelöf ME (1894) Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre. Comptes rendus hebdomadaires des séances de l’Académie des sciences 114:454–457 19. Müller M (1928) Beweis eines Satzes des Herrn H. Kneser über die Gesamtheit der Lösungen, die ein System gewöhnlicher Differentialgleichungen durch einen Punkt schickt. Math Zeit 28:349–355 20. Peano G (1890) Démonstration de l’integrabilité des équations differentielles ordinaires. Math Ann 37:182–228 21. Picard É (1890) Mémoire sur la théorie de équations aux dérivées partielles et la méthode des approximations successives. J Math, ser 4, 6:145–210 22. Rauch J (1991) Partial Differential Equations. Springer, New York 23. Rudin W (1976) Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York 24. Rudin W (1987) Real and Complex Analysis, 3rd edn. McGrawHill, New York 25. Zeidler E (1995) Applied functional analysis. Vol 1 Applications to mathematical physics. Springer, New York 26. Clay Mathematics Institute (2000) The Millenium Problems: Navier Stokes equation. http://www.claymath.org/ millennium/Navier-Stokes_Equations/

Books and Reviews Arnold VI (1992) Ordinary Differential Equations. Springer, Berlin Arrowsmith DK, Place CM (1990) An Introduction to Dynamical Systems. Cambridge University Press, Cambridge Braun M (1993) Differential Equations and their Applications. Springer, Berlin Brock WA, Malliaris AG (1989) Differential Equations, stability and chaos in Dynamic Economics. Elsevier, Amsterdam Grimshaw R (1993) Nonlinear Ordinary Differential Equations. CRC Press, Boca Raton Ince EL (1927) Ordinary differential equations. Longman, Green, New York Jordan DW, Smith P (1987) Nonlinear Differential Equations. Oxford University Press, Oxford Werner H, Arndt H (1986) Gewöhnliche Differentialgleichungen: eine Einführung in Theorie und Praxis. Springer, Berlin

Finite Dimensional Controllability

Finite Dimensional Controllability LIONEL ROSIER Institut Elie Cartan, Vandoeuvre-lès-Nancy, France Article Outline Glossary Definition of the Subject Introduction Control Systems Linear Systems Linearization Principle High Order Tests Controllability and Observability Controllability and Stabilizability Flatness Future Directions Bibliography Glossary Control system A control system is a dynamical system incorporating a control input designed to achieve a control objective. It is finite dimensional if the phase space (e. g. a vector space or a manifold) is of finite dimension. A continuous-time control system takes the form dx/dt D f (x; u), x 2 X, u 2 U and t 2 R denoting respectively the state, the input, and the continuous time. A discrete-time system assumes the form x kC1 D f (x k ; u k ), where k 2 Z is the discrete time. Open/closed loop A control system is said to be in open loop form when the input u is any function of time, and in closed loop form when the input u is a function of the state only, i. e., it takes the more restrictive form u D h(x(t)), where h : X ! U is a given function called a feedback law. Controllability A control system is controllable if any pair of states may be connected by a trajectory of the system corresponding to an appropriate choice of the control input. Stabilizability A control system is asymptotically stabilizable around an equilibrium point if there exists a feedback law such that the corresponding closed loop system is asymptotically stable at the equilibrium point. Output function An output function is any function of the state. Observability A control system given together with an output function is said to be observable if two different

states give rise to two different outputs for a convenient choice of the input function. Flatness An output function is said to be flat if the state and the input can be expressed as functions of the output and of a finite number of its derivatives. Definition of the Subject A control system is controllable if any state can be steered to another one in the phase space by an appropriate choice of the control input. While the stabilization issue has been addressed since the XIXth century (Watt’s steam engine governor providing a famous instance of a stabilization mechanism at the beginning of the English Industrial Revolution), the controllability issue has been addressed for the first time by Kalman in the 1960s [20]. The controllability is a basic mathematical property which characterizes the degrees of freedom available when we try to control a system. It is strongly connected to other control concepts: optimal control, observability, and stabilizability. While the controllability of linear finite-dimensional systems is well understood since Kalman’s seminal papers, the situation is more tricky for nonlinear systems. For the later, concepts borrowed from differential geometry (e. g. Lie brackets, holonomy, . . . ) come into play, and the study of their controllability is still a field of active research. The controllability of finite dimensional systems is a basic concept in control theory, as well as a notion involved in many applications, such as spatial dynamics (with e. g. spatial rendezvous), airplane autopilot, industrial robots, quantic chemistry. Introduction A very familiar example of a controllable finite dimensional system is given by a car that one attempts to park at some place in a parking. The phase space is roughly the three dimensional Euclidean space R3 , a state being composed of the two coordinates of the center of mass together with the angle formed by some axis linked to the car with the (fixed) abscissa axis. The driver may act on the angle of the wheels and on their velocity, which may thus be taken as control inputs. In general, the presence of obstacles (e. g. other cars) impose to change the phase space R3 to a subset of it. The controllability issue is roughly how to combine changes of direction and of velocity to drive the car from a position to another one. Note that the system is controllable, even if the number of control inputs (2) is less than the number of independent coordinates (3). This is an important property resting upon the many connections between the coordinates of the state. While in nature each motion is generally controlled by an input (think of

395

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Finite Dimensional Controllability

the muscles in an arm), the control theory focuses on the study of systems in which an input living in a space of low dimension (typically, one) is sufficient to control the coordinates of a state living in a space of high dimension. The article is outlined as follows. In Sect. “Control Systems”, we introduce the basic concepts (controllability, stabilizability) used thereafter. In the next section, we review the linear theory, recalling the Kalman and Hautus tests for the controllability of a time invariant system, and the Gramian test for a time dependent system. Section “Linearization Principle” is devoted to the linearization principle, which allows to deduce the controllability of a nonlinear system from the controllability of its linearization along a trajectory. The focus in Sect. “High Order Tests” is on nonlinear systems for which the linear test fails, i. e., the linearized system fails to be controllable. High order conditions based upon Lie brackets ensuring controllability will be given, first for systems without drift, and next for systems with a drift. Section “Controllability and Observability” explores the connections between controllability and observability, while Sect. “Controllability and Stabilizability” shows how to derive stabilization results from the controllability property. A final section on the flatness, a new theory used in many applications to design explicit control inputs, is followed by some thoughts on future directions. Control Systems A finite dimensional (continuous-time) control system is a differential equation of the form x˙ D f (x; u)

(1)

where x 2 X is the state, u 2 U is the input, f : X  U ! U is a smooth (typically real analytic) nonlinear function, x˙ D dx/dt, and X and U denote finite dimensional manifolds. For the sake of simplicity, we shall assume here that X Rn and U Rm are open sets. Sometimes, we impose U to be bounded (or to be a compact set) to force the control input to be bounded. Given some control input u 2 L1 (I; U), i. e. a measurable essentially bounded function u : I ! U, a solution of (1) is a locally Lipschitz continuous function x() : J ! X, where J I, such that x˙ (t) D f (x(t); u(t))

for almost every t 2 J :

(2)

Note that J I only, that is, x needs not exist on all I, as it may escape to infinity in finite time. In general, u is piecewise smooth, so that (2) holds actually for all t except for finitely many values. The basic problem of the controllability is the issue whether, given an initial state x0 2 X,

a terminal state x T 2 X, and a control time T > 0, one may design a control input u 2 L1 ([0; T]; U) such that the solution of the system (

x˙ (t) D f (x(t) ; u(t)) x(0) Dx0

(3)

satisfies x(T) D x T . An equilibrium position for (1) is a point x 2 X such that there exists a value u 2 U (typically, 0) such that f (x; u) D 0. An asymptotically stabilizing feedback law is a function k : X ! U with k(x) D u, such that the closed loop system x˙ D f (x; k(x))

(4)

obtained by plugging the control input u(t) :D k(x(t))

(5)

into (1), is locally asymptotically stable at x. Recall that it means that (i) the equilibrium point is stable: for any " > 0, there exists some ı > 0 such that any solution of (4) starting from a point x0 such that jx0  xj < ı at t D 0, is defined on RC and satisfies jx(t)j  " for all t  0; (ii) the equilibrium point is attractive. For some ı > 0 as in (i), we have also that x(t) ! 0 as t ! 1 whenever jx0 j < ı. The feedback laws considered here will be continuous, and we shall mean by a solution of (4) any function x(t) satisfying (4) for all t. Notice that the solutions of the Cauchy problems exist (locally in time) by virtue of Peano theorem. A control system is asymptotically stabilizable around an equilibrium position if an asymptotically stabilizing feedback law as above does exist. In the following, we shall also consider time-varying systems, i. e. systems of the form x˙ D f (x; t; u)

(6)

where f : X  I  U ! X is smooth and I R denotes some interval. The controllability and stabilizability concepts extend in a natural way to that setting. Time-varying feedback laws u(t) D k(x(t); t)

(7)

where k : X  R ! U is a smooth (generally time periodic) function, prove to be useful in situations where the classical static stabilization defined above fails.

Finite Dimensional Controllability

and let r D rank R(A; B) < n. It may be seen that the reachable space from the origin, that is the set (   R D x T 2 Rn ; 9T > 0; 9u 2 L2 [0; T] ; Rn ;

Linear Systems Time Invariant Linear Systems A time invariant linear system is a system of the form x˙ D Ax C Bu

(8)

B 2 denote some time invariwhere A 2 ant matrices. This corresponds to the situation where the function f in (1) is linear in both the state and the input. Here, x 2 Rn and u 2 Rm , and we consider square integrable inputs u 2 L2 ([0; T]; Rm ). We say that (8) is controllable in time T if for any pair of states x0 ; x T 2 Rn , one may construct an input u 2 L2 ([0; T]; Rm ) that steers (8) from x0 to xT . Recall that the solution of (8) emanating from x0 at t D 0 is given by Duhamel formula x(t) D e tA x0 C

Z

t

e(ts)A Bu(s) ds :

0

Let us introduce the nnm matrix R(A; B) D (BjABjA2 Bj    jAn1 B) obtained by gathering together the matrices B; AB; A2 B; : : : ; An1 B. Then we have the following rank condition due to Kalman [20] for the controllability of a linear system. Theorem 1 (Kalman) The linear system (8) is controllable in time T if and only if rank R(A; B) D n. We notice that the controllability of a linear system does not depend of the control time T, and that the control input may actually be chosen very smooth (e. g. in C 1 ([0; T]; Rn )). Example 1 Consider a pendulum to which is applied a torque as a control input. A simplified model is then given by the following linear system x˙1 D x2 x˙2 D x1 C u

Z

(9) (10)

where x1 ; x2 and u stand respectively for the angle with the vertical, the angular velocity, and the torque. Here, n D 2, m D 1, and     0 1 0 AD and B D : (11) 1 0 1 As rank(B; AB) D 2, we infer from Kalman rank test that (9)–(10) is controllable. When (8) fails to be controllable, it may be important (e. g. when studying the stabilizability) to identify the uncontrollable part of (8). Assume that (8) is not controllable,

)

T

(Ts)A

e

Rnm

Rnn ,

Bu(s) ds D x T

(12) ;

0

coincides with the space spanned by the columns of the matrix R(A; B): R D fR(A; B)V ; V 2 Rnm g

(13)

In particular, dim R D rank R(A; B) D r < n. Let e D fe1 ; : : : ; e n g be the canonical basis of Rn , and let f f1 ; : : : ; f r g be a basis of R, that we complete in a basis f D ˜ denotes the vector of the f f1 ; : : : ; f n g of Rn . If x (resp. x) coordinates of a point in the basis e (resp. f), then x D T x˜ where T D ( f1 ; f2 ; : : : ; f n ) 2 Rnn . In the new coordi˜ (8) may be written nates x, ˜x C Bu ˜ x˙˜ D A˜

(14)

where A˜ :D T 1 AT and B˜ :D T 1 B read     A˜1 A˜ 2 B˜ 1 ˜D A˜ D B 0 A˜ 3 0

(15)

with A˜ 1 2 Rrr , B˜ 1 2 Rrm . Writing x˜ D (x˜1 ; x˜2 ) 2 Rr  Rnr , we have x˙˜ 1 D A˜ 1 x˜1 C A˜ 2 x˜2 C B˜ 1 u

(16)

x˙˜ 2 D A˜ 3 x˜2

(17)

and it may be proved that rank R(A˜1 ; B˜ 1 ) D r :

(18)

This is the Kalman controllability decomposition. By (18), the dynamics of x˜1 is well controlled. Actually, a solution of (18) evaluated at t D T assumes the form Z T ˜ ˜ x˜1 (T) D e T A 1 x˜1 (0) C e(Ts)A1 A˜ 2 x˜2 (s) ds 0

Z

T

C ˜

˜

e(Ts)A1 B˜ 1 u(s)ds

0

D eT A1 x 1 C

Z

T

˜

e(Ts)A1 B˜ 1 u(s)ds

0

RT ˜ if we set x 1 D x˜1 (0) C 0 es A 1 A˜ 2 x˜2 (s)ds. Hence x˜1 (T) r may be given any value in R . On the other hand, no control input is present in (17). Thus x˜2 stands for the uncontrolled part of the dynamics of (8).

397

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Finite Dimensional Controllability

Another test based upon a spectral analysis has been furnished by Hautus in [16]. Theorem 2 (Hautus) The control system (8) is controllable in time T if and only if rank(I  A; B) D n for all  2 C. Notice that in Hautus test we may restrict ourselves to the complex numbers  which are eigenvalues of A, for otherwise rank(I  A) D n.

Note that the Gramian test provides a third criterion to test whether a time invariant linear system (8) is controllable or not. Corollary 4 (8) is controllable in time T if and only if the Gramian Z T  GD e(Tt)A BB e(Tt)A dt 0

is invertible. Time-Varying Linear Systems Let us now turn to the controllability issue for a time-varying linear system x˙ (t) D A(t)x(t) C B(t)u(t)

(19)

where A 2 L1 ([0; T] ; Rnn ), B 2 L1 ([0; T] ; Rnm ) denote time-varying matrices. Such a system arises in a natural way when linearizing a control system (1) along a trajectory. The input u(t) is any function in L1 ([0; T] ; Rm ) and a solution of (19) is any locally Lipschitz continuous function satisfying (19) almost everywhere. We define the fundamental solution  associated with A as follows. Pick any s 2 [0; T], and let M : [0; T] ! Rnn denote the solution of the system ˙ M(t) D A(t)M(t)

(20)

M(s) D I

Z

t

x(t) D (t; t0 )x0 C

(t; s)B(s)u(s)ds : t0

The controllability Gramian of (20) is the matrix T





(T; t)B(t)B (t) (T; t)dt

GD

u(t) D B (t)  (T; t)G 1 (x T  (T; 0)x0 ) :

(21)

A remarkable property of the control input u is that u minimizes the control cost Z T ju(t)j2 dt E(u) D 0

among all the control inputs u 2 L1 ([0; T] ; Rm ) (or u 2 L2 ([0; T]; Rm )) steering (19) from x0 to xT . Actually a little more can be said. Proposition 5 If u 2 L2 ([0; T] ; Rm ) is such that the solution x of (19) emanating from x0 at t D 0 reaches xT at time T, and if u ¤ u, then E(u) < E(u):

Then (t; s) :D M(t). Notice that (t; s) D e(ts)A when A is constant. The solution x of (20) starting from x0 at time t0 reads then

Z

As the value of the control time T plays no role according to Kalman test, it follows that the Gramian G is invertible for all T > 0 whenever it is invertible for one T > 0. If (19) is controllable, then an explicit control input steering (19) from x0 to xT is given by

0

where  denotes transpose. Note that G 2 Rnn , and that G is a nonnegative symmetric matrix. Then the following result holds. Theorem 3 (Gramian test) The system (19) is controllable on [0; T] if and only if the Gramian G is invertible.

The main drawback of the Gramian test is that the knowledge of the fundamental solution (t; s) and the computation of an integral term are both required. In the situation where A(t) and B(t) are smooth functions of time, a criterion based only upon derivatives in time is also available. Assume that A 2 C 1 ([0; T] ; Rnn ) and that B 2 C 1 ([0; T] ; Rnm ), and define a sequence of functions B i 2 C 1 ([0; T] ; Rnm ) by induction on i by B0 D B B i D AB i1 

dB i1 : dt

(22)

Then the following result holds (see, e. g., [11]). Theorem 6 Assume that there exists a time t 2 [0; T] such that span(B i (t)v ; v 2 Rm ; i  0) D Rn : Then (19) is controllable.

(23)

Finite Dimensional Controllability

The converse of Theorem 6 is true when A and B are real analytic functions of time. More precisely, we have the following result.

 Hautus test admits the following extension due to Liu [25]: (24) is (exactly) controllable in time T if and only if there exists some constant ı > 0 such that

Theorem 7 If A and B are real analytic on [0; T], then (20) is controllable if and only if for all t 2 [0; T] span(B i (t)v ; v 2 R ; i  0) D R : m

jj(A  I)zjj2 C jjB zjj2  ıjjzjj2 8z 2 D(A ) ; 8 2 C :

(25)

n

Clearly, Theorem 7 is not valid when A and B are merely of class C 1 . (Take n D m D 1, A(t) D 0, B(t) D exp(t 1 ) and t D 0.) Linear Control Systems in Infinite Dimension Let us end this section with some comments concerning the extensions of the above controllability tests to control systems in infinite dimension (see [30] for more details). Let us consider a control system of the form x˙ D Ax C Bu

(24)

where A: D(A) X ! X is an (unbounded) operator generating a strongly continuous semigroup (S(t)) t0 on a (complex) Hilbert space X, and B : U ! X is a bounded operator, U denoting another Hilbert space. Definition 8 We shall say that (24) is  Exactly controllable in time T if for any x0 ; x T 2 X there exists u 2 L2 ([0; T]; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies x(T) D x T ;  Null controllable in time T if for any x0 2 X there exists u 2 L2 ([0; T] ; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies x(T) D 0;  Approximatively controllable in time T if for any x0 ; x T 2 X and any " > 0 there exists u 2 L2 ([0; T]; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies jjx(T)  x T jj < " (jj  jj denoting the norm in X). This setting is convenient for a partial differential equation with an internal control Bu :D gu(t), where g D g(x) is such that gU X. Let us review the above controllability tests.  Kalman rank test, which is based on a computation of the dimension of the reachable space, possesses some extension giving the approximate (not exact!) controllability of (24) (See [13], Theorem 3.16). More interestingly, a Kalman-type condition has been introduced in [23] to investigate the null controllability of a system of coupled parabolic equations.

In (25), A (resp. B ) denotes the adjoint of the operator A (resp. B), and jj  jj denotes the norm in X.  The Gramian test admits the following extension, due to Dolecky–Russell [14] and J.L. Lions [24]: (24) is exactly controllable in time T if and only if there exists some constant ı > 0 such that Z T jjB S  (t)x0 jj2 dt  ıjjx0jj2 8x0 2 X : 0

Linearization Principle Assume given a smooth nonlinear control system x˙ D f (x; u)

(26)

where f : X  U ! Rn is a smooth map (i. e. of class C 1 ) and X Rn , U Rm denote some open sets. Assume also given a reference trajectory (x; u) : [0; T] ! X  U where u 2 L1 ([0; T] ; U) is the control input and x solves (26) for u  u. We introduce the linearized system along the reference trajectory defined as y˙ D A(t)y C B(t)v

(27)

where @f (x(t); u(t)) 2 Rnn ; @x @f (x(t); u(t)) 2 Rnm B(t) D @u

A(t) D

(28)

and y 2 Rn , v 2 Rm . (27) is formally derived from (26) by letting x D x C y, u D u C v and observing that y˙ D x˙  x˙ D f (x C y; u C v)  f (x; u) 

@f @f (x; u)y C (x; u)v : @x @u

Notice that if (x 0 ; u0 ) is an equilibrium point of f (i. e. f (x 0 ; u 0 ) D 0) and x(t)  x0 , u(t)  u0 , then A(t) D @f @f A D @x (x 0 ; u 0 ) and B(t) D B D @u (x 0 ; u 0 ) take constant values. Equation (27) is in that case the time invariant linear system y˙ D Ay C Bv :

(29)

399

400

Finite Dimensional Controllability

Let x0 ; x T 2 X. We seek for a trajectory x of (26) connecting x0 to xT when x0 (resp. xT ) is close to x(0) (resp. x(T)). In addition, we will impose that the trajectory (x, u) be uniformly close to the reference trajectory (x; u). We are led to the following Definition 9 The system (26) is said to be controllable along (x; u) if for each " > 0, there exists some ı > 0 such that for each x0 ; x T 2 X with jjx0  x(0)jj < ı, jjx T  x(T)jj < ı, there exists a control input u 2 L1 ([0; T] ; U) such that the solution of (26) starting from x0 at t D 0 satisfies x(T) D x T and   sup jx(t)  x(t)j C ju(t)  u(t)j  " :

tory such that the linearization of (30)–(32) along it is controllable. Pick any time T > 0 and let u 1 (t) D cos( t/T), u 2 (t) D 0 for t 2 [0; T] and x0 D 0. Let x denote the corresponding solution of (30)–(32). Notice that x 1 (t) D (T/) sin( t/T) (hence x 1 (T) D 0), and x 2 D x 3  0. The linearization of (30)–(32) along (x; u) is (27) with 0 1 0 0 0 A(t) D @ 0 0 0 A; 0  cos( t/T) 0 (33) 0 1 1 0 A: B(t) D @ 0 1 0 (T/) sin( t/T)

t2[0;T]

We are in a position to state the linearization principle. Theorem 10 Let (x; u) be a trajectory of (26). If the linearized system (27) along (x; u) is controllable, then the system (26) is controllable along the trajectory (x; u). When the reference trajectory is stationary, we obtain the following result. Corollary 11 Let (x0 ; u0 ) be such that f (x0 ; u0 ) D 0. If the linearized system (29) is controllable, then the system (26) is controllable along the stationary trajectory (x; u) D (x0 ; u0 ). Notice that the converse of Corollary 11 is not true. (Consider the system x˙ D u3 with n D m D 1, x0 D u0 D 0, which is controllable at the origin as it may be seen by performing the change of inputs v D u3 .) Often, the nonlinear part of f plays a crucial role in the controllability of (26). This will be explained in the next section using Lie algebraic techniques. Another way to use the nonlinear contribution in f is to consider a linearization of (26) along a convenient (not stationary) trajectory. We consider a system introduced by Brockett in [6] to exhibit an obstruction to stabilizability. Example 2 The Brockett’s system reads

Notice that A and B are real analytic, so that we may apply Theorem 8 to check whether (27) is controllable or not on [0; T]. Simple computations give 0 1 0 0 A: B1 (t) D @ 0 0 0 2 cos( t/T) Clearly,  span B0 (0)

1 0



 ; B0 (0)

0 1



 ; B1 (0)

0 1

!

D R3

hence (27) is controllable. We infer from Theorem 10 that (30)–(32) is controllable along (x; u). Notice that if we want to prove the controllability around an equilibrium point as above for Brockett’s system, we have to design a reference control input u so that x0 D x(0) D x(T) :

(34)

When f is odd with respect to the control, i. e. f (x; u) D  f (x; u) ;

(35)

then (34) is automatically satisfied whenever u fulfills

x˙1 D u1

(30)

x˙2 D u2

(31)

x˙3 D x1 u2  x2 u1 :

(32)

u(t) D u(T  t) 8t 2 [0; T] :

(36)

Indeed, it follows from (35)–(36) that the solution x to (26) starting from x0 at t D 0 satisfies

Its linearization along (x0 ; u0 ) D (0; 0), which reads y˙1 D v1 y˙2 D v2 y˙3 D 0 ; is not controllable, by virtue of the Kalman rank condition. We may however construct a smooth closed trajec-

x(t) D x(T  t) 8t 2 [0; T] :

(37)

In particular, x(T) D x(0) D x0 . Of course, the control inputs of interest are those for which the linearized system (27) is controllable, and the latter property is “generically” satisfied for a controllable system. The above construction of the reference trajectory is due to J.-M. Coron,

Finite Dimensional Controllability

and is referred to as the return method. A precursor to that method is [34]. Beside giving interesting results for the stabilization of finite dimensional systems (see below Sect. “Controllability and Stabilizability”), the return method has also been successfully applied for the control of some important partial differential equations arising in Fluid Mechanics (see [11]). High Order Tests In this section, we shall derive new controllability tests for systems for which the linearization principle is inconclusive; that is, the linearization at an equilibrium point fails to be controllable. To simplify the exposition, we shall limit ourselves to systems affine in the control, i. e. systems of the form x˙ D f0 (x) C u1 f1 (x) C    C u m f m (x) ; x 2 Rn ; juj1  ı

(38)

where juj1 :D sup1im ju i j and ı > 0 denotes a fixed number. We assume that f0 (0) D 0 and that f i 2 C 1 (Rn ; Rn ) for each i 2 f0; : : : ; mg. To state the results we need to introduce a few notations. Let v D (v1 ; : : : ; v n ) and w D (w1 ; : : : ; w n ) be two vector fields of class C 1 on Rn . The Lie bracket of v and w, denoted by [v; w], is the vector field [v; w] D

@v @w v w @x @x

where @w/@x is the Jacobian matrix (@w i /@x j ) i; jD1;:::;n , and the vector v(x) !D (v1 (x); : : : ; v n (x)) is identified v 1 (x)

to the column

: : : v n (x)

. As [v; w] is still a smooth vector

field, we may bracket it with v, or w, etc. Vector fields like [v; [v; w]], [[v; w]; [v; [v; w]]], etc. are termed iterated Lie brackets of v, w. The Lie bracketing of vector fields is an operation satisfying the two following properties (easy to check) 1. Anticommutativity: [w; v] D [v; w]; 2. Jacobi identity: [ f ; [g; h]]C[g; [h; f ]]C[h; [ f ; g]] D 0. The Lie algebra generated by f 1 , . . . ,f m , denoted Lie( f1 ; : : : ; f m ), is the smallest vector subspace v of C 1 (Rn ; Rn ) which contains f 1 , . . . ,f m and which is closed under Lie bracketing (i. e. v; w 2 V ) [v; w] 2 V ). It is easily to see that Lie( f1 ; : : : ; f m ) is the vector space spanned by all the iterated Lie brackets of f 1 ,. . . ,f m . (Actually, using the anticommutativity and Jacobi identity, we may restrict ourselves to iterated Lie brackets of the form

[ f i 1 ; [ f i 2 ; [: : : [ f i p1 ; f i p ] : : : ]]] to span Lie( f1 ; : : : ; f m ).) For any x 2 Rn , we set Lie( f1 ; : : : ; f m )(x) D fg(x); g 2 Lie( f1 ; : : : ; f m )g Rn : Example 3 Let us consider the system 0

1 0 1 1 0 x˙ D u1 f1 (x) C u2 f2 (x) D u1 @ 0 A C u2 @ 1 A (39) 0 x1 where x D (x1 ; x2 ; x3 ) 2 R3 . Clearly, the linearized system at (x0 ; u0 ) D (0; 0), which reduces to x˙ D u1 f1 (0) C u2 f2 (0), is not controllable. However, we shall see later that the system (39) is controllable. First, each point (˙"; 0; 0) (resp. (0; ˙"; 0)) may be reached from the origin by letting u(t) D (˙1; 0) (resp. u(t) D (0; ˙1)) on the time interval [0; "]. More interestingly, any point (0; 0; "2 ) may also be reached from the origin in a time T D O("), even though (0; 0; 1) 62 span( f1 (0); f2 (0)). To prove the last claim, let us introduce for i D 1; 2 the flow map  it defined by  it (x0 ) D x(t), where x() solves x˙ D f i (x) ;

x(0) D x0 :

Then, it is easy to see that 2" 1" 2" 1" (0) D (0; 0; "2 ) : It means that the control 8 (1; 0) if ˆ ˆ < (0; 1) if u(t) D (1; 0) if ˆ ˆ : (0; 1) if

0t<" "  t < 2" 2"  t < 3" 3"  t < 4"

steers the solution of (39) from x0 D (0; 0; 0) at t D 0 to x T D (0; 0; "2 ) at T D 4". (See Fig. 1.) More generally, if f 1 and f 2 denote arbitrary smooth vector fields and 1t , 2t denote the corresponding flow maps, we have that 2" 1" 2" 1" (0) D "2 [ f1 ; f2 ](0) C O("3 ) :

(40)

Thus, it is possible to reach points in the direction of the Lie bracket [ f1 ; f2 ](0). However, the process is not very efficient: in the direction of the Lie bracket [ f1 ; f2 ](0), only points located at a distance from the origin of order "2 may be reached in a time of order ". Let us proceed to the controllability properties of affine systems. We first review the results for systems without drift (i. e. f0 D 0), and next consider the general (only partially understood) situation where a drift exists.

401

402

Finite Dimensional Controllability

Example 2 continued Brockett’s system is also controllable, since [ f1 ; f2 ] D (0; 0; 2) gives (43). Notice that Theorem 13 is almost sharp. Indeed, it has been proved by Sussmann–Jurdjevic in [36] that (43) has to be satisfied for a controllable system (41) with real analytic vector fields (i. e. f i 2 C ! (Rn ; Rn ) for each i). Affine Systems with Drifts We consider now a control system with a drift x˙ D f0 (x) C u1 f1 (x) C    C u m f m (x)

where f i 2 C 1 (Rn ; Rn ) for any i 2 f0; : : : ; mg, and f0 (0) D 0 but f0 6 0. As no control is associated with the vector field f 0 , one expects that the latter will occupy a singular place in the controllability tests. Let > 0 be a given number. We assume that the control input u is a measurable function taking its values in the compact set U D [ ; ]m . To emphasize the dependence in u, we shall sometimes denote by x(t, u) the solution of (45) such that x(0; u) D 0. The attainable set at time T > 0 is the set

Finite Dimensional Controllability, Figure 1 Trajectory from x0 D (0; 0; 0) to xT D (0; 0; "2 )

Affine Systems Without Drift We assume that f0 D 0, so that (38) takes the form x˙ D u1 f1 (x) C    C u m f m (x) :

(41)

When m < n, the linearized system at any stationary trajectory (x; 0), which reads y˙ D ( f1 (x)    f m (x))v ;

(42)

cannot be controllable. High order tests are therefore very useful in this setting. We adopt the following Definition 12 The system (41) is said to be controllable if for any T > 0 and any x0 ; x T 2 Rn , there exists a control u 2 L1 ([0; T] ; Rm ) such that the solution x(t) of (41) emanating from x0 at t D 0 reaches xT at time t D T. Note that the controllability may be supplemented with the words: local, with small controls, in small time, in large time, etc. An important instance, the small-time local controllability, is introduced below in Definition 14. The following result has been obtained by Rashevski [28] and Chow [8]. Theorem 13 (Rashevski–Chow) Assume that f i 2 C 1 (Rn ; Rn ) for all i 2 f1; : : : ; mg. If Lie( f1 ; : : : ; f m )(x) D Rn

8x 2 Rn ;

(43)

then (41) is controllable. Example 3 continued (39) is controllable, since [ f1 ; f2 ] D (0; 0; 1) gives span( f1 (x); f2 (x); [ f1 ; f2 ](x)) D R3

(45)

8x 2 R3 : (44)

˚



A (T) D x(T; u); u(t) 2 U 8t 2 [0; T] :

Definition 14 We shall say that the control system (45) is  accessible from the origin if the interior of A (T) is nonempty for any T > 0;  small time locally controllable (STLC) at the origin if for any T > 0 there exists a number ı > 0 such that for any pair of states (x0 ; x T ) with jx0 j < ı; jx T j < ı, there exists a control input u 2 L1 ([0; T] ; U ) such that the solution x of x˙ D f0 (x)Cu1 f1 (x)C    Cu m f m (x);

x(0) D x0 (46)

satisfies x(T) D x T . (Notice that in the definition of the small time local controllability, certain authors assume x0 D 0 in above definition, or require the existence of ı > 0 for any T > 0 and any > 0.) The accessibility property turns out to be easy to characterize. As a STLC system is clearly accessible, the accessibility property is often considered as the first property to test before investigating the controllability of a given affine system. The following result provides an accessibility test based upon the rank at the origin of the Lie algebra generated by all the vectors fields involved in the control system.

Finite Dimensional Controllability

Theorem 15 (Hermann–Nagano) If dim Lie( f0 ; f1 ; : : : ; f m )(0) D n

(47)

then the system (45) is accessible from the origin. Conversely, if (45) is accessible from the origin and the vector fields f 0 , f 1 , . . . , f m are real analytic, then (47) has to be satisfied. Example 4 Pick any k 2 N  and consider as in [22] the system

x˙1 D u; juj  1 ; (48) x˙2 D (x1 ) k so that n D 2, m D 1, and f0 (x) D (0; (x1 ) k ), f1 (x) D (1; 0). Setting

h D [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f3 ]] : Finally, we set X  (h) D h : 2S m

With h ˚ as in (49), and S3 D idf1;2;3g ; (1 2 3); (1 3 2); (1 2); (1 3); (2 3) , we obtain  (h) D [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f2 ]] C [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f3 ]]

(ad f1 ; f0 ) D [ f1 ; f0 ] (ad iC1 f1 ; f0 ) D [ f1 ; (ad i f1 ; f0 )]

Let Sm denote the usual symmetric group, i. e. Sm is the group of all the permutations of the set f1; : : : ; mg. If 2 S m and h is an iterated Lie bracket of f0 ; f1 ; : : : ; f m , we denote by h the iterated Lie bracket obtained by replacing, in the definition of h, f i by f (i) for each i 2 f1; : : : ; mg. For instance, if D (1 2 3) and h is as in (49), then

C [[ f3 ; [ f3 ; f0 ]]; [ f3 ; f1 ]] C [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f1 ]]

8i  1

C [[ f3 ; [ f3 ; f0 ]]; [ f3 ; f2 ]] C [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f3 ]] :

we obtain at once that

We need the following

k

(ad f1 ; f0 )(x) D (0; k !)

Definition 16 Let 2 [0; 1]. We say that the system (45) satisfies the condition S( ) if, for any iterated Lie bracket h with ı0 (h) odd and ı i (h) even for all i 2 f1; : : : ; mg, the vector  (h)(0) belongs to the vector space spanned by the vectors g(0) where g is an iterated Lie bracket satisfying

hence Lie( f0 ; f1 )(0) D R2 : It follows from Theorem 15 that (48) is accessible from the origin. On the other hand, it is clear that (48) is not STLC at the origin when k is even, since x˙2 D (x1 ) k  0, hence x2 is nondecreasing. Using a controllability test due to Hermes [17], it may be shown that (48) is STLC at the origin if and only if k is odd. Note that the linearized system at the origin fails to be controllable whenever k  2. Let us consider some affine system (45) which is accessible from the origin. We exclude the trivial situation when the linearization principle may be applied, and seek for a Lie algebraic condition ensuring that (45) is STLC. If h is a given iterated Lie bracket of f0 ; f1 ; : : : ; f m , we let ı i (h) denote the number of occurrences of f i in the definition of h. For instance, if m D 3 and h D [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f2 ]]

(49)

then ı0 (h) D 1 ; ı1 (h) D 3 ; ı2 (h) D 1 ; ı3 (h) D 0 : Notice that the fields f0 ; f1 ; : : : ; f m are considered as indeterminates when computing the ı i (h)’s, their effective values as vector fields being ignored.

ı0 (g) C

m X

ı i (g) < ı0 (h) C

iD1

m X

ı i (h) :

iD1

Then we have the following result proved in [35]. Theorem 17 (Sussmann) If the condition S( ) is satisfied for some 2 [0; 1], then the system (45) is small time locally controllable at the origin. When D 0, Sussmann’s theorem is nothing else than Hermes’ theorem. Sussmann’s theorem, which is in itself very useful in Engineering (see, e. g., [5]), has been extended in [1,4,18]. Controllability and Observability Assume given a control system x˙ D f (x; u)

(50)

together with an output function y D h(x)

(51)

where f : X  U ! Rn and h : X ! Y are smooth functions, and X Rn , U Rm and Y R p denote some

403

404

Finite Dimensional Controllability

open sets. y typically stands for a (partial) measurement of the state, e. g. the p first coordinates, where p < n. Often, only y is available, and for the stabilization of (50) we should consider an output feedback law of the form u D k(y). We shall say that (50)–(51) is observable on the interval [0; T] if, for any pair x0 ; x˜0 of points in X, one may find a control input u 2 L1 ([0; T] ; U) such that if x (resp. x˜ ) denotes the solution of (50) emanating from x0 (resp. x˜0 ) at time t D 0, and y (resp. y˜) denotes the corresponding output function, we have y(t) ¤ y˜(t) for some t 2 [0; T] :

(52)

For a time invariant linear control system and a linear output function x˙ D Ax C Bu

(53)

y D Cx

(54)

with A 2 Rnn , B 2 Rnm , C 2 R pn , the output is found to be Z t y(t) D Cx(t) D Ce tA x0 C C e(ts)A Bu(s)ds : (55) 0

In particular, y(t)  y˜(t) D Ce tA (x0  x˜0 )

Notice that the observability of the adjoint system is easily shown to be equivalent to the existence of a constant ı > 0 such that Z T  jjB e tA x0 jj2 dt  ıjjx0 jj2 8x0 2 Rn : (59) 0

As it has been pointed out in Sect. “Linear Systems”, the equivalence between the controllability of a system and the observability of its adjoint, expressed in the form (59), is still true in infinite dimension, and provides a very useful way to investigate the controllability of a partial differential equation. Finally, using the duality principle, we see that any controllability test gives rise to an observability test, and vice-versa. Controllability and Stabilizability In this section we shall explore the connections between the controllability and the stabilizability of a system. Let us begin with a linear control system x˙ D Ax C Bu :

(56)

and B does not play any role. Therefore, (53)–(54) is observable in time T if and only if the only state x 2 Rn such that Ce tA x D 0 for any t 2 [0; T] is x D 0. Differentiating in time and applying Cayley–Hamilton theorem, we obtain the following result. Theorem 18 System (53)–(54) is observable in time T if and only if rank O(A; C) D n, where the observability matrix O(A; C) 2 Rn pn is defined by 1 0 C B CA C C B O(A; C) D B C :: A @ : n1 CA Noticing that R(A; B) D O(A ; B )

and introducing the adjoint system (57)

y˜ D B x˜

(58)

(60)

Performing a linear change of coordinates if needed, we may assume that (60) has the block structure given by the Kalman controllability decomposition        x˙1 A1 A2 x1 B1 u (61) D C x˙2 0 A3 x2 0 where A1 2 Rrr , B1 2 Rrm , and where rank R (A1 ; B1 ) D r. For any square matrix M 2 C nn we denote by  (M) its spectrum, i. e.  (M) D f 2 C ; det(M  I) D 0g : Let us note C :D f 2 C ; Re  < 0g. Then the asymptotic stabilizability of (60) may be characterized as follows. Theorem 20 There exists a (continuous) asymptotically stabilizing feedback law u D k(x) with k(0) D 0 for (60) if and only if  (A3 ) C . If it is the case, then for any family S D ( i )1ir of elements of C invariant by conjugation, there exists a linear asymptotically stabilizing feedback law u(x) D Kx, with K 2 Rmn , such that  (A C BK) D  (A3 ) [ S :

x˙˜ D A x˜

we arrive to the following duality principle.

Theorem 19 A time invariant linear system is controllable if and only if its adjoint system is observable.

(62)

The property (62), which shows to what extend the spectrum of the closed loop system can be assigned, is referred to as the pole shifting theorem.

Finite Dimensional Controllability

As a direct consequence of Theorem 20, we obtain the following result. Corollary 21 A time invariant linear system which is controllable is also asymptotically stabilizable. It is natural to ask whether Corollary 21 is still true for a nonlinear system, i. e. if a controllable system is necessarily asymptotically stabilizable. A general result cannot be obtained, as is shown by Brockett’s system. Example 2 continued Brockett’s system (30)–(32) may be written x˙ D u1 f1 (x) C u2 f2 (x) D f (x; u), and it follows from Theorem 13 that (30)–(32) is controllable around (x0 ; u0 ) D (0; 0). Let us now recall a necessary condition for stabilizability due to R. Brockett [6]. Brockett third condition for stabilizability. Let f 2 C(Rn  Rm ; Rn ) with f (x0 ; u0 ) D 0. If the control system x˙ D f (x; u) can be locally asymptotically stabilized at x0 by means of a continuous feedback law u satisfying u(x0 ) D u0 , then the image by f of any open neighborhood of (x0 ; u0 ) 2 Rn  Rm contains an open neighborhood of 0 2 Rn . Here, for any neighborhood V of (x0 ; u0 ) in R5 , f (V) does not cover an open neighborhood of 0 in R3 , since (0; 0; ") 62 f (V) for any " 2 R. According to Brockett’s condition, the system (30)–(32) is not asymptotically stabilizable at the origin. Thus a controllable system may fail to be asymptotically stabilizable by a continuous feedback law u D k(x) due to topological obstructions. It turns out that this phenomenon does not occur when the phase space is the plane, as it has been demonstrated by Kawski in [21]. Theorem 22 (Kawski) Let f0 ; f1 be real analytic vector fields on R2 with f0 (0) D 0 and f1 (0) ¤ 0. Assume that the system x˙ D f0 (x) C u f1 (x) ;

u2R

(63)

is small time locally controllable at the origin. Then it is also asymptotically stabilizable at the origin by a Hölder continuous feedback law u D k(x). In larger dimension (n  3), a way to go round the topological obstruction is to consider a time-varying feedback law u D k(x; t). It has been first observed by Sontag and Sussmann in [33] that for one-dimensional state and input (n D m D 1), the controllability implies the asymptotic stabilizability by means of a time-varying feedback law. This kind of stabilizability was later established by Samson in [31] for Brockett’s system (n D 3 and m D 2). Finally, using the return method, Coron proved that the implication

Controllability ) Asymptotic Stabilizability by TimeVarying Feedback was a principle verified in most cases. To state precise results, we consider affine systems x˙ D f0 (x) C

m X

u i f i (x) ;

x 2 Rn

iD1

and distinguish again two cases: (1) f0  0 (no drift); (2) f0 6 0 (a drift). (1) System without drift Theorem 23 (Coron [9]) Assume that (43) holds for the system (41). Pick any number T > 0. Then there exists a feedback law u D (u1 ; : : : ; u m ) 2 C 1 (Rn  R; Rm ) such that u(0; t) D 0

8t 2 R

u(x; t C T) D u(x; t)

(64) 8(x; t) 2 Rn  R

(65)

and such that 0 is globally asymptotically stable for the system x˙ D

m X

u i (x; t) f i (x) :

iD1

(2) System with a drift Theorem 24 (Coron [10]) Assume that the system (45) satisfies the condition S( ) for some 2 [0; 1]. Assume also that n 62 f2; 3g and that Lie( f0 ; f1 ; : : : ; f m )(0) D Rn : Pick any T > 0. Then there exists a feedback law u D (u1 ; : : : ; u m ) 2 C 0 (Rn R; Rm ) with u 2 C 1 ((Rn nf0g) R; Rm ) such that (64)–(65) hold and such that 0 is locally asymptotically stable for the system x˙ D f0 (x) C

m X

u i (x; t) f i (x) :

iD1

Flatness While powerful criterions enable us to decide whether a control system is controllable or not, most of them do not provide any indication on how to design an explicit control input steering the system from a point to another one. Fortunately, there exists a large class of systems, the so-called flat systems, for which explicit control inputs may easily be found. The flatness theory has been introduced by Fliess, Levine, Martin, and Rouchon in [15], and since then it has attracted the interest of many researchers

405

406

Finite Dimensional Controllability

thanks to its numerous applications in Engineering. Here, we only sketch the main ideas, referring the interested reader to [27] for a comprehensive introduction to the subject. Let us consider a smooth control system x 2 Rn ; u 2 Rm

x˙ D f (x; u) ;

given together with an output y 2 Rm depending on x, u, and a finite number of derivatives of u ˙ : : : ; u(r) ) : y D h(x; u; u;

Example 6 Consider now the nonlinear system x˙1 D u1

(72)

x˙2 D u2

(73)

x˙3 D x2 u1 :

(74)

Eliminating the input u1 in (72)–(74) yields x˙3 D x2 x˙1 , so that x2 may be expressed as a function of x1 ; x3 and their derivatives. The same is true for u2 , thanks to (73). Let us pick y D (y1 ; y2 ) D (x1 ; x3 ) :

Following [15], we shall say that y is a flat output if the components of y are differentially independent, and both x and u may be expressed as functions of y and of a finite number of its derivatives   (66) x D k y; y˙; : : : ; y (p)   u D l y; y˙; : : : ; y (q) : (67) In (66)–(67), p and q denote some nonnegative integers, and k and l denote some smooth functions. Since the state x and the input u are parameterized by the flat output y, to solve the controllability problem

We claim that y is a flat output. Indeed,   y˙2 (x1 ; x2 ; x3 ) D y1 ; ; y2 ; y˙1   y¨2 y˙1  y˙2 y¨1 : (u1 ; u2 ) D y˙1 ; ( y˙1 )2

(75) (76)

Pick x0 D (0; 0; 0) and x T D (0; 0; 1). Notice that, by the mean value theorem, y˙1 has to vanish somewhere, say at t D ¯t . We shall construct y1 in such a way that y˙1 vanishes only at t D ¯t . For x2 not to be singular at ¯t , we have to impose the condition y˙2 ( ¯t ) D 0 :

x˙ D f (x; u) x(0) D x0 ;

(68) x(T) D x T

(69)

it is sufficient to pick any function y 2 C max(p;q) ([0; T]; Rm ) such that k(y; y˙; : : : ; y (p) )(0) D x(0) D x0 k(y; y˙; : : : ; y

(p)

)(T) D x(T) D x T :

(70)

If y1 and y2 are both analytic near ¯t , we notice that x2 D y˙2 / y˙1 is analytic near ¯t , hence u2 D x˙2 is well-defined (and analytic) near t¯. To steer the solution of (72)–(74) from x0 D (0; 0; 0) to x T D (0; 0; 1), it is then sufficient to pick a function y D (y1 ; y2 ) 2 C ! ([0; T]; R2 ) such that y1 (0) D y2 (0) D 0 ; y1 (T) D 0 ;

The constraints (70)–(71) are generally very easy to satisfy. The desired control input u is then given by (67). Let us show how this program may be carried out on two simple examples.

y˙1

y2 (T) D 1 ;

  T D0; 2

Example 5 For the simple integrator x˙1 Dx2

and y˙1 (0) ¤ 0 ;

y˙2 (T) D 0 ;

and y˙1 (T) ¤ 0 :     T T y˙2 D 0 ; y¨1 ¤0; 2 2 T and y˙1 (t) ¤ 0 for t ¤ : 2

Clearly, 

x˙2 Du the output y D x1 is flat, for x2 DR y˙ and u D y¨. The output z D x2 is not flat, as x1 D z(t) dt. To steer the system from x0 D (0; 0) to x T D (1; 0) in time T, we only have to pick a function y 2 C 2 ([0; T]; R) such that y(0) D 0 ;

y˙2 (0) D 0 ;

(71)

y˙(0) D 0 ;

y(T) D 1 ; and y˙(T) D 0 :

Clearly, y(t) D t 2 (2T  t)2 /T 4 is convenient.

t 4 Tt 3 T 2 t2 C (y1 (t); y2 (t)) D t(T  t);  4 2 4



is convenient. Future Directions Lie algebraic techniques have been used to provide powerful controllability tests for affine systems. However, there is still an important gap between the known necessary con-

Finite Dimensional Controllability

ditions (e. g. the Legendre Clebsh condition or its extensions) and the sufficient conditions (e. g. the S( ) condition) for the small time local controllability. On the other hand, it has been noticed by Kawski in ([22], Example 6.1) that certain systems can be controlled on small time intervals [0; T] only by using faster switching control variations, the number of switchings tending to infinity as T tends to zero. As for switched systems [2], it is not clear whether purely algebraic conditions be sufficient to characterize the controllability of systems with drift. Of great interest for applications is the development of methods providing explicit control inputs, or control inputs computed in real time with the aid of some numerical schemes, both for the motion planning and the stabilization issue. The flatness theory seems to be a very promising method, and it has been successfully applied in Engineering. An active research is devoted to filling the gap between necessary and sufficient conditions for the existence of flat outputs, and to extending the theory to partial differential equations. The control of nonlinear partial differential equations may sometimes be reduced to the control of a family of finite-dimensional systems by means of the Galerkin method [3]. On the other hand, the spatial discretization of partial differential equations by means of finite differences, finite elements, or spectral methods leads in a natural way to the control of finite dimensional systems (see, e. g., [38]). Of great importance is the uniform boundedness with respect to the discretization parameter of the L2 (0; T)-norms of the control inputs associated with the finite dimensional approximations. Geometric ideas borrowed from the control of finite dimensional systems (e. g. the return method, the power series expansion, the quasistatic deformation) have been applied to prove the controllability of certain nonlinear partial differential equations whose linearization fails to be controllable. (See [11] for a survey of these techniques.) The Korteweg–de Vries equation provides an interesting example of a partial differential equation whose linearization fails to be controllable for certain lengths of the space domain [29]. However, for these critical lengths the reachable space proves to be of finite codimension, and it may be proved that the full equation is controllable by using the nonlinear term in order to reach the missing directions [7,12]. Bibliography Primary Literature 1. Agrachev AA, Gamkrelidze RV (1993) Local controllability and semigroups of diffeomorphisms. Acta Appl Math 32:1–57

2. Agrachev A, Liberzon D (2001) Lie-algebraic stability criteria for switched systems. SIAM J Control Optim 40(1):253–269 3. Agrachev A, Sarychev A (2005) Navier–Stokes equations: controllability by means of low modes forcing. J Math Fluid Mech 7(1):108–152 4. Bianchini RM, Stefani G (1986) Sufficient conditions of local controllability. In: Proc. 25th IEEE–Conf. Decision & Control. Athens 5. Bonnard B (1992) Contrôle de l’altitude d’un satellite rigide. RAIRO Autom Syst Anal Control 16:85–93 6. Brockett RW (1983) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control Theory. Progr. Math., vol 27. Birkhäuser, Basel, pp 181–191 7. Cerpa E, Crépeau E (2007) Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain 8. Chow WL (1940) Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math Ann 117:98–105 9. Coron JM (1992) Global asymptotic stabilization for controllable systems without drift. Math Control Signals Syst 5: 295–312 10. Coron JM (1995) Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws. SIAM J Control Optim 33:804–833 11. Coron JM (2007) Control and nonlinearity, mathematical surveys and monographs. American Mathematical Society, Providence 12. Coron JM, Crépeau E (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths. J Eur Math Soc 6(3):367–398 13. Curtain RF, Pritchard AJ (1978) Infinite Dimensional Linear Systems Theory. Springer, New York 14. Dolecky S, Russell DL (1977) A general theory of observation and control. SIAM J Control Optim 15:185–220 15. Fliess M, Lévine J, Martin PH, Rouchon P (1995) Flatness and defect of nonlinear systems: introductory theory and examples. Intern J Control 31:1327–1361 16. Hautus MLJ (1969) Controllability and observability conditions for linear autonomous systems. Ned Akad Wetenschappen Proc Ser A 72:443–448 17. Hermes H (1982) Control systems which generate decomposable Lie algebras. J Differ Equ 44:166–187 18. Hermes H, Kawski M (1986) Local controllability of a singleinput, affine system. In: Proc. 7th Int. Conf. Nonlinear Analysis. Dallas 19. Hirschorn R, Lewis AD (2004) High-order variations for families of vector fields. SIAM J Control Optim 43:301–324 20. Kalman RE (1960) Contribution to the theory of optimal control. Bol Soc Mat Mex 5:102–119 21. Kawski M (1989) Stabilization of nonlinear systems in the plane. Syst Control Lett 12:169–175 22. Kawski M (1990) High-order small time local controllability. In: Sussmann HJ (ed) Nonlinear controllability and optimal control. Textbooks Pure Appl. Math., vol 113. Dekker, New York, pp 431–467 23. Khodja FA, Benabdallah A, Dupaix C, Kostin I (2005) Nullcontrollability of some systems of parabolic type by one control force. ESAIM Control Optim Calc Var 11(3):426–448 24. Lions JL (1988) Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. In: Recherches en Mathématiques Appliquées, Tome 1, vol 8. Masson, Paris

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25. Liu K (1997) Locally distributed control and damping for the conservative systems. SIAM J Control Optim 35:1574– 1590 26. Lobry C (1970) Contrôlabilité des systèmes non linéaires. SIAM J Control 8:573–605 27. Martin PH, Murray RM, Rouchon P (2002) Flat systems. Mathematical Control Theory, Part 1,2 (Trieste 2001). In: ICTP Lect. Notes, vol VIII. Abdus Salam Int Cent Theoret Phys, Trieste 28. Rashevski PK (1938) About connecting two points of complete nonholonomic space by admissible curve. Uch Zapiski ped inst Libknexta 2:83–94 29. Rosier L (1997) Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM: Control Optim Calc Var 2:33–55 30. Rosier L (2007) A Survey of controllability and stabilization results for partial differential equations. J Eur Syst Autom (JESA) 41(3–4):365–412 31. Samson C (1991) Velocity and torque feedback control of a nonholonomic cart. In: de Witt C (ed) Proceedings of the International workshop on nonlinear and adaptative control: Issues in robotics, Grenoble, France nov. 1990. Lecture Notes in Control and Information Sciences, vol 162. Springer, Berlin, pp 125–151 32. Silverman LM, Meadows HE (1967) Controllability and observability in time-variable linear systems. SIAM J Control 5: 64–73 33. Sontag ED, Sussmannn H (1980) Remarks on continuous feedbacks. In: Proc. IEEE Conf. Decision and Control, (Albuquerque 1980). pp 916–921

34. Stefani G (1985) Local controllability of nonlinear systems: An example. Syst Control Lett 6:123–125 35. Sussmann H (1987) A general theorem on local controllability. SIAM J Control Optim 25:158–194 36. Sussmann H, Jurdjevic V (1972) Controllability of nonlinear systems. J Differ Equ 12:95–116 37. Tret’yak AI (1991) On odd-order necessary conditions for optimality in a time-optimal control problem for systems linear in the control. Math USSR Sbornik 79:47–63 38. Zuazua E (2005) Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev 47(2):197–243

Books and Reviews Agrachev AA, Sachkov YL (2004) Control theory from the geometric viewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87. Springer, Berlin Bacciotti A (1992) Local stabilization of nonlinear control systems. In: Ser. Adv. Math. Appl. Sci, vol 8. World Scientific, River Edge Bacciotti A, Rosier L (2005) Liapunov functions and stability in control theory. Springer, Berlin Isidori A (1989) Nonlinear control systems. Springer, Berlin Nijmeijer H, van der Schaft AJ (1990) Nonlinear dynamical control systems. Springer, New York Sastry S (1999) Nonlinear systems. Springer, New York Sontag E (1990) Mathematical control theory. Springer, New York Zabczyk J (1992) Mathematical control theory: An introduction. Birkhäuser, Boston

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to ARMIN BUNDE1 , SHLOMO HAVLIN2 1 Institut für Theoretische Physik, Gießen, Germany 2 Institute of Theoretical Physics, Bar-Ilan-University, Ramat Gan, Israel Article Outline Definition of the Subject Deterministic Fractals Random Fractal Models How to Measure the Fractal Dimension Self-Affine Fractals Long-Term Correlated Records Long-Term Correlations in Financial Markets and Seismic Activity Multifractal Records Acknowledgments Bibliography Definition of the Subject In this chapter we present some definitions related to the fractal concept as well as several methods for calculating the fractal dimension and other relevant exponents. The purpose is to introduce the reader to the basic properties of fractals and self-affine structures so that this book will be self contained. We do not give references to most of the original works, but, we refer mostly to books and reviews on fractal geometry where the original references can be found. Fractal geometry is a mathematical tool for dealing with complex systems that have no characteristic length scale. A well-known example is the shape of a coastline. When we see two pictures of a coastline on two different scales, with 1 cm corresponding for example to 0.1 km or 10 km, we cannot tell which scale belongs to which picture: both look the same, and this features characterizes also many other geographical patterns like rivers, cracks, mountains, and clouds. This means that the coastline is scale invariant or, equivalently, has no characteristic length scale. Another example are financial records. When looking at a daily, monthly or annual record, one cannot tell the difference. They all look the same. Scale-invariant systems are usually characterized by noninteger (“fractal”) dimensions. The notion of noninteger dimensions and several basic properties of fractal objects were studied as long ago as the last century by Georg Cantor, Giuseppe Peano, and David Hilbert, and in

Fractal Geometry, A Brief Introduction to, Figure 1 The Dürer pentagon after five iterations. For the generating rule, see Fig. 8. The Dürer pentagon is in blue, its external perimeter is in red, Courtesy of M. Meyer

the beginning of this century by Helge von Koch, Waclaw Sierpinski, Gaston Julia, and Felix Hausdorff. Even earlier traces of this concept can be found in the study of arithmetic-geometric averages by Carl Friedrich Gauss about 200 years ago and in the artwork of Albrecht Dürer (see Fig. 1) about 500 years ago. Georg Friedrich Lichtenberg discovered, about 230 years ago, fractal discharge patterns. He was the first to describe the observed self-similarity of the patterns: A part looks like the whole. Benoit Mandelbrot [1] showed the relevance of fractal geometry to many systems in nature and presented many important features of fractals. For further books and reviews on fractals see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Before introducing the concept of fractal dimension, we should like to remind the reader of the concept of dimension in regular systems. It is well known that in regular systems (with uniform density) such as long wires, large thin plates, or large filled cubes, the dimension d characterizes how the mass M(L) changes with the linear size L of the system. If we consider a smaller part of the system of linear size bL (b < 1), then M(bL) is decreased by a factor of bd , i. e., M(bL) D b d M(L) :

(1)

The solution of the functional equation (1) is simply M(L) D ALd . For the long wire the mass changes linearly with b, i. e., d D 1. For the thin plates we obtain d D 2, and for the cubes d D 3; see Fig. 2.

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Fractal Geometry, A Brief Introduction to, Figure 2 Examples of regular systems with dimensions d D 1, d D 2, and dD3

Next we consider fractal objects. Here we distinguish between deterministic and random fractals. Deterministic fractals are generated iteratively in a deterministic way, while random fractals are generated using a stochastic process. Although fractal structures in nature are random, it is useful to study deterministic fractals where the fractal properties can be determined exactly. By studying deterministic fractals one can gain also insight into the fractal properties of random fractals, which usually cannot be treated rigorously. Deterministic Fractals In this section, we describe several examples of deterministic fractals and use them to introduce useful fractal concepts such as fractal and chemical dimension, self similarity, ramification, and fractal substructures (minimum path, external perimeter, backbone, and red bonds).

One of the most common deterministic fractals is the Koch curve. Figure 3 shows the first n D 4 iterations of this fractal curve. By each iteration the length of the curve is increased by a factor of 4/3. The mathematical fractal is defined in the limit of infinite iterations, n ! 1, where the total length of the curve approaches infinity. The dimension of the curve can be obtained as for regular objects. From Fig. 3 we notice that, if we decrease the linear size by a factor of b D 1/3, the total length (mass) of the curve decreases by a factor of 1/4, i. e., 1  3 L D

1 4

M(L) :

d D log 4/ log 3. For such non-integer dimensions Mandelbrot coined the name “fractal dimension” and those objects described by a fractal dimension are called fractals. Thus, to include fractal structures, Eq. (1) is generalized by M(bL) D b d f M(L) ;

(2)

This feature is very different from regular curves, where the length of the object decreases proportional to the linear scale. In order to satisfy Eqs. (1) and (2) we are led to introduce a noninteger dimension, satisfying 1/4 D (1/3)d , i. e.,

(3)

and M(L) D ALd f ;

The Koch Curve

M

Fractal Geometry, A Brief Introduction to, Figure 3 The first iterations of the Koch curve. The fractal dimension of the Koch curve is df D log 4/ log 3

(4)

where df is the fractal dimension. When generating the Koch curve and calculating df , we observe the striking property of fractals – the property of self-similarity. If we examine the Koch curve, we notice that there is a central object in the figure that is reminiscent of a snowman. To the right and left of this central snowman there are two other snowmen, each being an exact reproduction, only smaller by a factor of 1/3. Each of the smaller snowmen has again still smaller copies (by 1/3) of itself to the right and to the left, etc. Now, if we take any such triplet of snowmen (consisting of 1/3m of the curve), for any m, and magnify it by 3m , we will obtain exactly the original Koch curve. This property of self-similarity or scale invariance is the basic feature of all deterministic and random fractals: if we take a part of a fractal and magnify it by the same magnification factor in all directions, the magnified picture cannot be distinguished from the original.

Fractal Geometry, A Brief Introduction to

For the Koch curve as well as for all deterministic fractals generated iteratively, Eqs. (3) and (4) are of course valid only for length scales L below the linear size L0 of the whole curve (see Fig. 3). If the number of iterations n is finite, then Eqs. (3) and (4) are valid only above a lower cut off length Lmin , Lmin D L0 /3n for the Koch curve. Hence, for a finite number of iterations, there exist two cut-off length scales in the system, an upper cut-off Lmax D L0 representing the total linear size of the fractal, and a lower cut-off Lmin . This feature of having two characteristic cutoff lengths is shared by all fractals in nature. An interesting modification of the Koch curve is shown in Fig. 4, which demonstrates that the chemical distance is an important concept for describing structural properties of fractals (for a review see, for example, [16] and Chap. 2 in [13]). The chemical distance ` is defined as shortest path on the fractal between two sites of the fractal. In analogy to the fractal dimension df that characterizes how the mass of a fractal scales with (air) distance L, we introduce the chemical dimension d` in order to characterize how the mass scales with the chemical distance `, M(b`) D b d ` M(`) ;

or M(`) D B`d ` :

(5)

From Fig. 4 we see that if we reduce ` by a factor of 5, the mass of the fractal within the reduced chemical distance is reduced by a factor of 7, i. e., M(1/5 `)D 1/7 M(`), yielding d` D log 7/ log 5 Š 1:209. Note that the chemical dimension is smaller than the fractal dimension df D log 7/ log 4 Š 1:404, which follows from M(1/4 L) D 1/7 M(L).

Fractal Geometry, A Brief Introduction to, Figure 4 The first iterations of a modified Koch curve, which has a nontrivial chemical distance metric

The structure of the shortest path between two sites represents an interesting fractal by itself. By definition, the length of the path is the chemical distance `, and the fractal dimension of the shortest path, dmin , characterizes how ` scales with (air) distance L. Using Eqs. (4) and (5), we obtain ` Ld f /d `  Ld min ;

(6)

from which follows dmin D df /d` . For our example we find that dmin D log 5/ log 4 Š 1:161. For the Koch curve, as well as for any linear fractal, one simply has d` D 1 and hence dmin D df . Since, by definition, dmin  1, it follows that d`  df for all fractals. The Sierpinski Gasket, Carpet, and Sponge Next we discuss the Sierpinski fractal family: the “gasket”, the “carpet”, and the “sponge”. The Sierpinski Gasket The Sierpinski gasket is generated by dividing a full triangle into four smaller triangles and removing the central triangle (see Fig. 5). In the following iterations, this procedure is repeated by dividing each of the remaining triangles into four smaller triangles and removing the central triangles. To obtain the fractal dimension, we consider the mass of the gasket within a linear size L and compare it with the mass within 1/2 L. Since M(1/2 L) D 1/3 M(L), we have df D log 3/ log 2 Š 1:585. It is easy to see that d` D df and dmin D 1.

Fractal Geometry, A Brief Introduction to, Figure 5 The Sierpinski gasket. The fractal dimension of the Sierpinski gasket is df D log 3/ log 2

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Fractal Geometry, A Brief Introduction to, Figure 6 A Sierpinski carpet with n D 5 and k D 9. The fractal dimension of this structure is df D log 16/ log 5

The Sierpinski Carpet The Sierpinski carpet is generated in close analogy to the Sierpinski gasket. Instead of starting with a full triangle, we start with a full square, which we divide into n2 equal squares. Out of these squares we choose k squares and remove them. In the next iteration, we repeat this procedure by dividing each of the small squares left into n2 smaller squares and removing those k squares that are located at the same positions as in the first iteration. This procedure is repeated again and again. Figure 6 shows the Sierpinski carpet for n D 5 and the specific choice of k D 9. It is clear that the k squares can be chosen in many different ways, and the fractal structures will all look very different. However, since M(1/n L) D 1/(n2  k) M(L) it follows that df D log(n2  k)/ log n, irrespective of the way the k squares are chosen. Similarly to the gasket, we have d` D df and hence dmin D 1. In contrast, the external perimeter (“hull”, see also Fig. 1) of the carpet and its fractal dimension dh depend strongly on the way the squares are chosen. The hull consists of those sites of the cluster, which are adjacent to empty sites and are connected with infinity via empty sites. In our example, see Fig. 6, the hull is a fractal with the fractal dimension dh D log 9/ log 5 Š 1:365. On the other hand, if a Sierpinski gasket is constructed with the k D 9 squares chosen from the center, the external perimeter stays smooth and dh D 1. Although the rules for generating the Sierpinski gasket and carpet are quite similar, the resulting fractal structures belong to two different classes, to finitely ramified and infinitely ramified fractals. A fractal is called finitely ramified if any bounded subset of the fractal can be isolated by cutting a finite number of bonds or sites. The Sierpinski

Fractal Geometry, A Brief Introduction to, Figure 7 The Sierpinski sponge (third iteration). The fractal dimension of the Sierpinski sponge is df D log 20/ log 3

gasket and the Koch curve are finitely ramified, while the Sierpinski carpet is infinitely ramified. For finitely ramified fractals like the Sierpinski gasket many physical properties, such as conductivity and vibrational excitations, can be calculated exactly. These exact solutions help to provide insight onto the anomalous behavior of physical properties on fractals, as was shown in Chap. 3 in [13]. The Sierpinski Sponge The Sierpinski sponge shown in Fig. 7 is constructed by starting from a cube, subdividing it into 3  3  3 D 27 smaller cubes, and taking out the central small cube and its six nearest neighbor cubes. Each of the remaining 20 small cubes is processed in the same way, and the whole procedure is iterated ad infinitum. After each iteration, the volume of the sponge is reduced by a factor of 20/27, while the total surface area increases. In the limit of infinite iterations, the surface area is infinite, while the volume vanishes. Since M(1/3 L) D 1/20 M(L), the fractal dimension is df D log 20/ log 3 Š 2:727. We leave it to the reader to prove that both the fractal dimension dh of the external surface and the chemical dimension d` is the same as the fractal dimension df . Modification of the Sierpinski sponge, in analogy to the modifications of the carpet can lead to fractals, where the fractal dimension of the hull, dh , differs from df . The Dürer Pentagon Five-hundred years ago the artist Albrecht Dürer designed a fractal based on regular pentagons, where in each iteration each pentagon is divided into six smaller pentagons and five isosceles triangles, and the triangles are removed (see Fig. 8). In each triangle, the ratio of the larger side to

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 8 The first iterations of the Dürer pentagon. The fractal dimension of the Dürer pentagon is df D log 6/ log(1 C g) Fractal Geometry, A Brief Introduction to, Figure 10 The first iterations of the triadic Cantor set. The fractal dimension of this Cantor set is df D log 2/ log 3

Fig. 10). We divide a unit interval [0; 1] into three equal intervals and remove the central one. In each following iteration, each of the remaining intervals is treated in this way. In the limit of n D 1 iterations one obtains a set of points. Since M(1/3 L) D 1/2 M(L), we have df D log 2/ log 3 Š 0:631, which is smaller than one. In chaotic systems, strange fractal attractors occur. The simplest strange attractor is the Cantor set. It occurs, for example, when considering the one-dimensional logistic map x tC1 D x t (1  x t ) :

Fractal Geometry, A Brief Introduction to, Figure 9 The first iterations of the David fractal. The fractal dimension of the David fractal is df D log 6/ log 3

the smaller side is the famous proportio divina or golden p ratio, g  1/(2 cos 72ı )  (1 C 5)/2. Hence, in each iteration the sides of the pentagons are reduced by 1 C g. Since M(L/(1 C g)) D 1/6 M(L), the fractal dimension of the Dürer pentagon is df D log 6/ log(1 C g) Š 1:862. The external perimeter of the fractal (see Fig. 1) forms a fractal curve with dh D log 4/ log(1 C g). A nice modification of the Dürer pentagon is a fractal based on regular hexagons, where in each iteration one hexagon is divided into six smaller hexagons, six equilateral triangles, and a David-star in the center, and the triangles and the David-star are removed (see Fig. 9). We leave it as an exercise to the reader to show that df D log 6/ log 3 and dh D log 4/ log 3. The Cantor Set Cantor sets are examples of disconnected fractals (fractal dust). The simplest set is the triadic Cantor set (see

(7)

The index t D 0; 1; 2; : : : represents a discrete time. For 0    4 and x0 between 0 and 1, the trajectories xt are bounded between 0 and 1. The dynamical behavior of xt for t ! 1 depends on the parameter . Below 1 D 3, only one stable fixed-point exists to which xt is attracted. At 1 , this fixed-point becomes unstable and bifurcates into two new stable fixed-points. At large times, the trajectories move alternately between both fixed-points, p and the motion is periodic with period 2. At 2 D 1C 6 Š 3:449 each of the two fixed-points bifurcates into two new stable fix points and the motion becomes periodic with period 4. As  is increased, further bifurcation points n occur, with periods of 2n between n and nC1 . For large n, the differences between nC1 and n become smaller and smaller, according to the law nC1  n D (n  n1 )/ı, where ı Š 4:6692 is the socalled Feigenbaum constant. The Feigenbaum constant is “universal”, since it applies to all nonlinear “single-hump” maps with a quadratic maximum [17]. At 1 Š 3:569 945 6, an infinite period occurs, where the trajectories xt move in a “chaotic” way between the infinite attractor points. These attractor points define the strange attractor, which forms a Cantor set with a fractal dimension df Š 0:538 [18]. For a further discussion of strange attractors and chaotic dynamics we refer to [3,8,9].

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Fractal Geometry, A Brief Introduction to, Figure 11 Three generations of the Mandelbrot–Given fractal. The fractal dimension of the Mandelbrot–Given fractal is df D log 8/ log 3

The Mandelbrot–Given Fractal This fractal was suggested as a model for percolation clusters and its substructures (see Sect. 3.4 and Chap. 2 in [13]). Figure 11 shows the first three generations of the Mandelbrot–Given fractal [19]. At each generation, each segment of length a is replaced by 8 segments of length a/3. Accordingly, the fractal dimension is df D log 8/ log 3 Š 1:893, which is very close to df D 91/46 Š 1:896 for percolation in two dimensions. It is easy to verify that d` D df , and therefore dmin D 1. The structure contains loops, branches, and dangling ends of all length scales. Imagine applying a voltage difference between two sites at opposite edges of a metallic Mandelbrot–Given fractal: the backbone of the fractal consists of those bonds which carry the electric current. The dangling ends are those parts of the cluster which carry no current and are connected to the backbone by a single bond only. The red bonds (or singly connected bonds) are those bonds that carry the total current; when they are cut the current flow stops. The blobs, finally, are those parts of the backbone that remain after the red bonds have been removed. The backbone of this fractal can be obtained easily by eliminating the dangling ends when generating the fractal (see Fig. 12). It is easy to see that the fractal dimension of the backbone is dB D log 6/ log 3 Š 1:63. The red bonds are all located along the x axis of the figure and form a Cantor set with the fractal dimension dred D log 2/ log 3 Š 0:63. Julia Sets and the Mandelbrot Set A complex version of the logistic map (7) is z tC1 D z2t C c ;

(8)

Fractal Geometry, A Brief Introduction to, Figure 12 The backbone of the Mandelbrot–Given fractal, with the red bonds shown in bold

where both the trajectories zt and the constant c are complex numbers. The question is: if a certain c-value is given, for example c D 1:5652  i1:03225, for which initial values z0 are the trajectories zt bounded? The set of those values forms the filled-in Julia set, and the boundary points of them form the Julia set. To clarify these definitions, consider the simple case c D 0. For jz0 j > 1, zt tends to infinity, while for jz0 j < 1, zt tends to zero. Accordingly, the filled-in Julia set is the set of all points jz0 j  1, the Julia set is the set of all points jz0 j D 1. In general, points on the Julia set form a chaotic motion on the set, while points outside the Julia set move away from the set. Accordingly, the Julia set can be regarded as a “repeller” with respect to Eq. (8). To generate the Julia set, it is thus practical to use the inverted transformation p z t D ˙ z tC1  c ; (9) start with an arbitrarily large value for t C 1, and go back-

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 13 Julia sets for a c D i and b c D 0:11031  i0:67037. After [9]

ward in time. By going backward in time, even points far away from the Julia set are attracted by the Julia set. For obtaining the Julia set for a given value of c, one starts with some arbitrary value for z tC1 , for example, z tC1 D 2. To obtain zt , we use Eq. (9), and determine the sign randomly. This procedure is continued to obtain z t1 , z t2 , etc. By disregarding the initial points, e. g., the first 1000 points, one obtains a good approximation of the Julia set. The Julia sets can be connected (Fig. 13a) or disconnected (Fig. 13b) like the Cantor sets. The self-similarity of the pictures is easy to see. The set of c values that yield connected Julia sets forms the famous Mandelbrot set. It has been shown by Douady and Hubbard [20] that the Mandelbrot set is identical to that set of c values for which zt converges starting from the initial point z0 D 0. For a detailed discussion with beautiful pictures see [10] and Chaps. 13 and 14 in [3]. Random Fractal Models In this section we present several random fractal models that are widely used to mimic fractal systems in nature. We begin with perhaps the simplest fractal model, the random walk. Random Walks Imagine a random walker on a square lattice or a simple cubic lattice. In one unit of time, the random walker advances one step of length a to a randomly chosen nearest neighbor site. Let us assume that the walker is unwinding a wire, which he connects to each site along his way. The length (mass) M of the wire that connects the random walker with his starting point is proportional to the number of steps n (Fig. 14) performed by the walker. Since for a random walk in any d-dimensional space the mean end-to-end distance R is proportional to n1/2 (for a simple derivation see e. g., Chap. 3 in [13], it follows that

Fractal Geometry, A Brief Introduction to, Figure 14 a A normal random walk with loops. b A random walk without loops

M R2 . Thus Eq. (4) implies that the fractal dimension of the structure formed by this wire is df D 2, for all lattices. The resulting structure has loops, since the walker can return to the same site. We expect the chemical dimension d` to be 2 in d D 2 and to decrease with increasing d, since. Loops become less relevant. For d  4 we have d` D 1. If we assume, however, that there is no contact between sections of the wire connected to the same site (Fig. 14b), the structure is by definition linear, i. e., d` D 1 for all d. For more details on random walks and its relation to Brownian motion, see Chap. 5 in [15] and [21]. Self-Avoiding Walks Self-avoiding walks (SAWs) are defined as the subset of all nonintersecting random walk configurations. An example is shown in Fig. 15a. As was found by Flory in 1944 [22], the end-to-end distance of SAWs scales with the number of steps n as R n ;

(10)

with  D 3/(d C 2) for d  4 and  D 1/2 for d > 4. Since n is proportional to the mass of the chain, it follows from Eq. (4) that df D 1/. Self-avoiding walks serve as models for polymers in solution, see [23]. Subsets of SAWs do not necessarily have the same fractal dimension. Examples are the kinetic growth walk (KGW) [24] and the smart growth walk (SGW) [25], sometimes also called the “true” or “intelligent” self-avoiding walk. In the KGW, a random walker can only step on those sites that have not been visited before. Asymptotically, after many steps n, the KGW has the same fractal dimension as SAWs. In d D 2, however, the asymptotic regime is difficult to reach numerically, since the random walker can be trapped with high probability (see Fig. 15b). A related structure is the hull of a random walk in d D 2. It has been conjectured by Mandelbrot [1] that the fractal dimension of the hull is dh D 4/3, see also [26].

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Fractal Geometry, A Brief Introduction to, Figure 16 a Generation of a DLA cluster. The inner release radius is usually a little larger than the maximum distance of a cluster site from the center, the outer absorbing radius is typically 10 times this distance. b A typical off-lattice DLA cluster of 10,000 particles

Fractal Geometry, A Brief Introduction to, Figure 15 a A typical self-avoiding walk. b A kinetic growth walk after 8 steps. The available sites are marked by crosses. c A smart growth walk after 19 steps. The only available site is marked by “yes”

In the SGW, the random walker avoids traps by stepping only at those sites from which he can reach infinity. The structure formed by the SGW is more compact and characterized by df D 7/4 in d D 2 [25]. Related structures with the same fractal dimension are the hull of percolation clusters (see also Sect. “Percolation”) and diffusion fronts (for a detailed discussion of both systems see also Chaps. 2 and 7 in [13]). Kinetic Aggregation The simplest model of a fractal generated by diffusion of particles is the diffusion-limited aggregation (DLA) model, which was introduced by Witten and Sander in 1981 [27]. In the lattice version of the model, a seed particle is fixed at the origin of a given lattice and a second particle is released from a circle around the origin. This particle performs a random walk on the lattice. When it comes to a nearest neighbor site of the seed, it sticks and a cluster (aggregate) of two particles is formed. Next, a third particle is released from the circle and performs a random walk. When it reaches a neighboring site of the aggregate, it sticks and becomes part of the cluster. This procedure is repeated many times until a cluster of the desired number of sites is generated. For saving computational time it is convenient to eliminate particles that have diffused too far away from the cluster (see Fig. 16).

In the continuum (off-lattice) version of the model, the particles have a certain radius a and are not restricted to diffusing on lattice sites. At each time step, the length ( a) and the direction of the step are chosen randomly. The diffusing particle sticks to the cluster, when its center comes within a distance a of the cluster perimeter. It was found numerically that for off-lattice DLA, df D 1:71 ˙ 0:01 in d D 2 and df D 2:5 ˙ 0:1 in d D 3 [28,29]. These results may be compared with the mean field result df D (d 2 C 1)/(d C 1) [30]. For a renormalization group approach, see [31] and references therein. The chemical dimension d` is found to be equal to df [32]. Diffusion-limited aggregation serves as an archetype for a large number of fractal realizations in nature, including viscous fingering, dielectric breakdown, chemical dissolution, electrodeposition, dendritic and snowflake growth, and the growth of bacterial colonies. For a detailed discussion of the applications of DLA we refer to [5,13], and [29]. Models for the complex structure of DLA have been developed by Mandelbrot [33] and Schwarzer et al. [34]. A somewhat related model for aggregation is the cluster-cluster aggregation (CCA) [35]. In CCA, one starts from a very low concentration of particles diffusing on a lattice. When two particles meet, they form a cluster of two, which can also diffuse. When the cluster meets another particle or another cluster, a larger cluster is formed. In this way, larger and larger aggregates are formed. The structures are less compact than DLA, with df Š 1:4 in d D 2 and df Š 1:8 in d D 3. CCA seems to be a good model for smoke aggregates in air and for gold colloids. For a discussion see Chap. 8 in [13]. Percolation Consider a square lattice, where each site is occupied randomly with probability p or empty with probability 1  p.

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 17 Square lattice of size 20  20. Sites have been randomly occupied with probability p (p D 0:20, 0.59, 0.80). Sites belonging to finite clusters are marked by full circles, while sites on the infinite cluster are marked by open circles

At low concentration p, the occupied sites are either isolated or form small clusters (Fig. 17a). Two occupied sites belong to the same cluster, if they are connected by a path of nearest neighbor occupied sites. When p is increased, the average size of the clusters increases. At a critical concentration pc (also called the percolation threshold) a large cluster appears which connects opposite edges of the lattice (Fig. 17b). This cluster is called the infinite cluster, since its size diverges when the size of the lattice is increased to infinity. When p is increased further, the density of the infinite cluster increases, since more and more sites become part of the infinite cluster, and the average size of the finite clusters decreases (Fig. 17c). The percolation transition is characterized by the geometrical properties of the clusters near pc . The probability P1 that a site belongs to the infinite cluster is zero below pc and increases above pc as P1 (p  pc )ˇ :

(11)

The linear size of the finite clusters, below and above pc , is characterized by the correlation length . The correlation length is defined as the mean distance between two sites on the same finite cluster and represents the characteristic length scale in percolation. When p approaches pc ,  increases as  jp  pc j ;

(12)

with the same exponent  below and above the threshold. While pc depends explicitly on the type of the lattice (e. g., pc Š 0:593 for the square lattice and 1/2 for the triangular lattice), the critical exponents ˇ and  are universal and depend only on the dimension d of the lattice, but not on the type of the lattice. Near pc , on length scales smaller than , both the infinite cluster and the finite clusters are self-similar. Above pc , on length scales larger than , the infinite cluster can be regarded as an homogeneous system which is composed of

Fractal Geometry, A Brief Introduction to, Figure 18 A large percolation cluster in d D 3. The colors mark the topological distance from an arbitrary center of the cluster in the middle of the page. Courtesy of M. Meyer

many unit cells of size . Mathematically, this can be summarized as ( r df ; r   ; M(r) d (13) r ; r: The fractal dimension df can be related to ˇ and : df D d 

ˇ : 

(14)

Since ˇ and  are universal exponents, df is also universal. One obtains df D 91/48 in d D 2 and df Š 2:5 in d D 3. The chemical dimension d` is smaller than df , d` Š 1:15 in d D 2 and d` Š 1:33 in d D 3. A large percolation cluster in d D 3 is shown in Fig. 18. Interestingly, a percolation cluster is composed of several fractal sub-structures such as the backbone, dangling ends, blobs, External perimeter, and the red bonds, which are all described by different fractal dimensions. The percolation model has found applications in physics, chemistry, and biology, where occupied and empty sites may represent very different physical, chemical, or biological properties. Examples are the physics of two component systems (the random resistor, magnetic or superconducting networks), the polymerization process in chemistry, and the spreading of epidemics and forest fires. For reviews with a comprehensive list of references, see Chaps. 2 and 3 of [13] and [36,37,38].

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How to Measure the Fractal Dimension One of the most important “practical” problems is to determine the fractal dimension df of either a computer generated fractal or a digitized fractal picture. Here we sketch the two most useful methods: the “sandbox” method and the “box counting” method. The Sandbox Method To determine df , we first choose one site (or one pixel) of the fractal as the origin for n circles of radii R1 < R2 <    < R n , where Rn is smaller than the radius R of the fractal, and count the number of points (pixels) M1 (R i ) within each circle i. (Sometimes, it is more convenient to choose n squares of side length L1 . . . Ln instead of the circles.) We repeat this procedure by choosing randomly many other (altogether m) pixels as origins for the n circles and determine the corresponding number of points M j (R i ), j D 2; 3; : : : ; m within each circle (see Fig. 19a). We obtain the mean number of points M(R i ) within each P circle by averaging, M(R i ) D 1/m m jD1 M j (R i ), and plot M(R i ) versus Ri in a double logarithmic plot. The slope of the curve, for large values of Ri , determines the fractal dimension. In order to avoid boundary effects, the radii must be smaller than the radius of the fractal, and the centers of the circles must be chosen well inside the fractal, so that the largest circles will be well within the fractal. In order to obtain good statistics, one has either to take a very large fractal cluster with many centers of circles or many realizations of the same fractal. The Box Counting Method We draw a grid on the fractal that consists of N12 squares, and determine the number of squares S(N1 ) that are

needed to cover the fractal (see Fig. 19b). Next we choose finer and finer grids with N12 < N22 < N32 <    < 2 squares and calculate the corresponding numbers of Nm squares S(N1 ) . . . S(N m ) needed to cover the fractal. Since S(N) scales as S(N) N d f ;

we obtain the fractal dimension by plotting S(N) versus 1/N in a double logarithmic plot. The asymptotic slope, for large N, gives df . Of course, the finest grid size must be larger than the pixel size, so that many pixels can fall into the smallest square. To improve statistics, one should average S(N) over many realizations of the fractal. For applying this method to identify self-similarity in real networks, see Song et al. [39]. Self-Affine Fractals The fractal structures we have discussed in the previous sections are self-similar: if we cut a small piece out of a fractal and magnify it isotropically to the size of the original, both the original and the magnification look the same. By magnifying isotropically, we have rescaled the x, y, and z axis by the same factor. There exist, however, systems that are invariant only under anisotropic magnifications. These systems are called self-affine [1]. A simple model for a self-affine fractal is shown in Fig. 20. The structure is invariant under the anisotropic magnification x ! 4x, y ! 2y. If we cut a small piece out of the original picture (in the limit of n ! 1 iterations), and rescale the x axis by a factor of four and the y axis by a factor of two, we will obtain exactly the original structure. In other words, if we describe the form of the curve in Fig. 20 by the function F(x), this function satisfies the equation F(4x) D 2F(x) D 41/2 F(x). In general, if a self-affine curve is scale invariant under the transformation x ! bx, y ! ay, we have F(bx) D aF(x)  b H F(x) ;

Fractal Geometry, A Brief Introduction to, Figure 19 Illustrations for determining the fractal dimension: a the sandbox method, b the box counting method

(15)

(16)

where the exponent H D log a/ log b is called the Hurst exponent [1]. The solution of the functional equation (16) is simply F(x) D Ax H . In the example of Fig. 20, H D 1/2. Next we consider random self-affine structures, which are used as models for random surfaces. The simplest structure is generated by a one-dimensional random walk, where the abscissa is the time axis and the ordinate is the P displacement Z(t) D tiD1 e i of the walker from its starting point. Here, e i D ˙1 is the unit step made by the random walker at time t. Since different steps of the random walker are uncorrelated, he i e j i D ı i j , it follows that the

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 20 A simple deterministic model of a self-affine fractal

root mean square displacement F(t)  hZ 2 (t)i1/2 D t 1/2 , and the Hurst exponent of the structure is H D 1/2. Next we assume that different steps i and j are correlated in such a way that he i e j i D bji  jj , 1 >  0. To see how the Hurst exponent depends on , we have to Pt evaluate again hZ 2 (t)i D i; j he i e j i. For calculating the double sum it is convenient to introduce the Fourier transP form of ei , e! D (1/˝)1/2 ˝ l D1 e l exp(i! l), where ˝ is the number of sites in the system. It is easy to verify that hZ 2 (t)i can be expressed in terms of the power spectrum he! e! i [40]: hZ 2 (t)i D

1 X he! e! ij f (!; t)j2 ; ˝ !

(17a)

where f (!; t) 

ei!(tC1)  1 : ei!  1

Since the power spectrum scales as he! e! i ! (1 ) ;

(17b)

the integration of (17a) yields, for large t, hZ 2 (t)i t 2 :

(17c)

Therefore, the Hurst exponent is H D (2 )/2. According to Eq. (17c), for 0 < < 1, hZ 2 (t)i increases faster in time than the uncorrelated random walk. The long-range correlated random walks were called fractional Brownian motion by Mandelbrot [1].

There exist several methods to generate correlated random surfaces. We shall describe the successive random additions method [41], which iteratively generates the selfaffine function Z(x) in the unit interval 0  x  1. An alternative method that is detailed in the chapter of Jan Kantelhardt is the Fourier-filtering technique and its variants. In the n D 0 iteration, we start at the edges x D 0 and x D 1 of the unit interval and choose the values of Z(0) and Z(1) from a distribution with zero mean and variance 02 D 1 (see Fig. 21). In the n D 1 iteration, we choose the midpoint x D 1/2 and determine Z(1/2) by linear interpolation, i. e., Z(1/2) D (Z(0) C Z(1))/2, and add to all so-far calculated Z values (Z(0), Z(1/2), and Z(1)) random displacements from the same distribution as before, but with a variance 1 D (1/2)H (see Fig. 21). In the n D 2 iteration we again first choose the midpoints (x D 1/4 and x D 3/4), determine their Z values by linear interpolation, and add to all so-far calculated Z values random displacements from the same distribution as before, but with a variance 2 D (1/2)2H . In general, in the nth iteration, one first interpolates the Z values of the midpoints and then adds random displacements to all existing Z values, with variance n D (1/2)nH . The procedure is repeated until the required resolution of the surface is obtained. Figure 22 shows the graphs of three random surfaces generated this way, with H D 0:2, H D 0:5, and H D 0:8. The generalization of the successive random addition method to two dimensions is straightforward (see Fig. 21). We consider a function Z(x; y) on the unit square 0 

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Fractal Geometry, A Brief Introduction to, Figure 21 Illustration of the successive random addition method in d D 1 and d D 2. The circles mark those points that have been considered already in the earlier iterations, the crosses mark the new midpoints added at the present iteration. At each iteration n, first the Z values of the midpoints are determined by linear interpolation from the neighboring points, and then random displacements of variance n are added to all Z values

x; y  1. In the n D 0 iteration, we start with the four corners (x; y) D (0; 0); (1; 0); (1; 1); (0; 1) of the unit square and choose their Z values from a distribution with zero mean and variance 02 D 1 (see Fig. 21). In the n D 1 iteration, we choose the midpoint at (x; y) D (1/2; 1/2) and determine Z(1/2; 1/2) by linear interpolation, i. e., Z(1/2; 1/2) D (Z(0; 0)CZ(0; 1)CZ(1; 1)CZ(1; 0))/4. Then we add to all so far calculated Z-values (Z(0; 0), Z(0; 1), Z(1; 0), Z(1; 1) and Z(1/2; 1/2)) random displacements from the same p distribution as before, but with a variance 1 D (1/ 2)H (see Fig. 21). In the n D 2 iteration we again choose the midpoints of the five sites (0; 1/2), (1/2; 0), (1/2; 1) and (1; 1/2), determine their Z value by linear interpolation, and add to all so far calculated Z values random displacements from the same p distribution as before, but with a variance 2 D (1/ 2)2H . This procedure is repeated again and again, until the required resolution of the surface is obtained. At the end of this section we like to note that selfsimilar or self-affine fractal structures with features similar to those fractal models discussed above can be found in nature on all, astronomic as well as microscopic, length scales. Examples include clusters of galaxies (the fractal dimension of the mass distribution is about 1.2 [42]), the crater landscape of the moon, the distribution of earthquakes (see Chap. 2 in [15]), and the structure of coastlines, rivers, mountains, and clouds. Fractal cracks (see, for example, Chap. 5 in [13]) occur on length scales ranging from 103 km (like the San Andreas fault) to micrometers (like fractures in solid materials) [44]. Many naturally growing plants show fractal structures, examples range from trees and the roots of trees to cauliflower and broccoli. The patterns of blood vessels in the human body, the kidney, the lung, and some types of nerve cells have fractal features (see Chap. 3 in [15]). In materials sciences, fractals appear in polymers, gels, ionic glasses, aggregates, electro-deposition, rough interfaces and surfaces (see [13] and Chaps. 4 and 6 in [15]), as well as in fine particle systems [43]. In all these structures there is no characteristic length scale in the system besides the physical upper and lower cut-offs. The occurrence of self-similar or self-affine fractals is not limited to structures in real space as we will discuss in the next section. Long-Term Correlated Records

Fractal Geometry, A Brief Introduction to, Figure 22 Correlated random walks with H D 0:2, 0.5, and 0.8, generated by the successive random addition method in d D 1

Long-range dependencies as described in the previous section do not only occur in surfaces. Of great interest is longterm memory in climate, physiology and financial markets, the examples range from river floods [45,46,47,48,

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 23 Comparison of an uncorrelated and a long-term correlated record with  D 0:4. The full line is the moving average over 30 data points

49], temperatures [50,51,52,53,54], and wind fields [55] to market volatilities [56], heart-beat intervals [57,58] and internet traffic [59]. Consider a record xi of discrete numbers, where the index i runs from 1 to N. xi may be daily or annual temperatures, daily or annual river flows, or any other set of data consisting of N successive data points. We are interested in the fluctuations of the data around their (sometimes seasonal) mean value. Without loss of generality, we assume that the mean of the data is zero and the variance equal to one. In analogy to the previous section, we call the data long-term correlated, when the corresponding autocorrelation function C x (s) decays by a power law, C x (s) D hx i x iCs i 

1 N s

Ns X

x1 x iCs s  ;

(18)

iD1

where denotes the correlation exponent, 0 < < 1. Such correlations are Rnamed ‘long-term’ since the mean 1 correlation time T D 0 C x (s) ds diverges in the limit of infinitely long series where N ! 1. If the xi are uncorrelated, C x (s) D 0 for s > 0. More generally, if correlations exist up to a certain correlation time sx , then C(s) > 0 for s < s x and C(s) D 0 for s > s x . Figure 23 shows parts of an uncorrelated (left) and a long-term correlated (right) record, with D 0:4; both series have been generated by the computer. The red line is the moving average over 30 data points. For the uncorrelated data, the moving average is close to zero, while for the long-term correlated data set, the moving average can have large deviations from the mean, forming some kind of mountain valley structure. This structure is a consequence of the power-law persistence. The mountains and valleys in Fig. 23b look as if they had been generated by external trends, and one might be inclined to draw a trend-

line and to extrapolate the line into the near future for some kind of prognosis. But since the data are trend-free, only a short-term prognosis utilizing the persistence can be made, and not a longer-term prognosis, which often is the aim of such a regression analysis. Alternatively, in analogy to what we described above for self-affine surfaces, one can divide the data set in K s equidistant windows of length s and determine in each window  the squared sum !2 s X 2 F (s) D xi (19a) iD1

and detect how the average of this quantity over all winPK s F2 (s), scales with the window dows, F 2 (s) D 1/K s D1 size s. For long-term correlated data one can show that F 2 (s) scales as hZ 2 (t)i in the previous section, i. e. F 2 (s) s 2˛ ;

(19b)

where ˛ D 1  /2. This relation represents an alternative way to determine the correlation exponent . Since trends resemble long-term correlations and vice versa, there is a general problem to distinguish between trends and long-term persistence. In recent years, several methods have been developed, mostly based on the hierarchical detrended fluctuation analysis (DFAn) where longterm correlations in the presence of smooth polynomial trends of order n  1 can be detected [57,58,60] (see also  Fractal and Multifractal Time Series). In DFAn, one considers the cumulated sum (“profile”) of the xi and divides the N data points of the profile into equidistant windows of fixed length s. Then one determines, in each window, the best fit of the profile by an nth order polynomial and determines the variance around the fit. Finally, one av2 and erages these variances to obtain the mean variance F(n)

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Fractal Geometry, A Brief Introduction to, Figure 24 DFAn analysis of six temperature records, one precipitation record and two run-off records. The black curves are the DFA0 results, while the upper red curves refer to DFA1 and the lower red curves to DFA2. The blue numbers denote the asymptotic slopes of the curves

the corresponding mean standard deviation (mean fluctuation) F(n) (s). One can show that for long-term correlated trend-free data F(n) (s) scales with the window size s as F(s) in Eq. (19b), i. e., F(n) (s) s ˛ , with ˛ D 1  /2, irrespective of the order of the polynomial n. For short-term correlated records (including the case  1), the exponent is 1/2 for s above sx . It is easy to verify that trends of order k  1 in the original data are eliminated in F(k) (s) but contribute to F(k1) ; F(k2) etc., and this allows one to determine the correlation exponent in the presence of trends. For example, in the case of a linear trend, DFA0 and DFA1 (where F(0) (s) and F(1) (s) are determined) are affected by the trend and will exaggerate the asymptotic exponents ˛, while DFA2, DFA3 etc. (where F(2) (s) and F(3) (s) etc. is determined) are not affected by the trend and will show, in a double logarithmic plot, the same value of ˛, which then gives immediately the correlation exponent . When is known this way, one can try to detect the trend, but there is no unique treatment available. In recent papers [61,62,63], different kinds of analysis have been elaborated and applied to estimate trends in the temperature records of the Northern Hemisphere and Siberian locations.

Climate Records Figure 24 shows representative results of the DFAn analysis, for temperature, precipitation and run-off data. For continental temperatures, the exponent ˛ is around 0.65, while for island stations and sea surface temperatures the exponent is considerably higher. There is no crossover towards uncorrelated behavior at larger time scales. For the precipitation data, the exponent is close to 0.55, not being significantly larger than for uncorrelated records. Figure 25 shows a summary of the exponent ˛ for a large number of climate records. It is interesting to note that while the distribution of ˛-values is quite broad for run-off, sea-surface temperature, and precipitation records, the distribution is quite narrow, located around ˛ D 0:65 for continental atmospheric temperature records. For the island records, the exponent is larger. The quite universal exponent ˛ D 0:65 for continental stations can be used as an efficient test bed for climate models [62, 64,65]. The time window accessible by DFAn is typically 1/4 of the length of the record. For instrumental records, the time window is thus restricted to about 50 years. For ex-

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 25 Distribution of fluctuation exponents ˛ for several kinds of climate records (from [53,66,67])

tending this limit, one has to take reconstructed records or model data, which range up to 2000y. Both have, of course, large uncertainties, but it is remarkable that exactly the same kind of long-term correlations can be found in these data, thus extending the time scale where long-term memory exists to at least 500y [61,62]. Clustering of Extreme Events Next we consider the consequences of long-term memory on the occurrence of rare events. Understanding (and predicting) the occurrence of extreme events is one of the major challenges in science (see, e. g., [68]). An important quantity here is the time interval between successive extreme events (see Fig. 26), and by understanding the statistics of these return intervals one aims to better understand the occurrence of extreme events. Since extreme events are, by definition, very rare and the statistics of their return intervals poor, one usually studies also the return intervals between less extreme events, where the data are above some threshold q and where the statistics is better, and hopes to find some general “scaling” relations between the return intervals at low and high thresholds, which then allows one to extrapolate the results to very large, extreme thresholds (see Fig. 26).

Fractal Geometry, A Brief Introduction to, Figure 26 Illustration of the return intervals for three equidistant threshold values q1 ; q2 ; q3 for the water levels of the Nile at Roda (near Cairo, Egypt). One return interval for each threshold (quantile) q is indicated by arrows

For uncorrelated data, the return intervals are independent of each other and their probability density function (pdf) is a simple exponential, Pq (r) D (1/R q )  exp(r/R q ). In this case, all relevant quantities can be derived from the knowledge of the mean return interval Rq . Since the return intervals are uncorrelated, a sequential ordering cannot occur. There are many cases, however, where some kind of ordering has been observed where the

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hazardous events cluster, for example in the floods in Central Europe during the middle ages or in the historic water levels of the Nile river which are shown in Fig. 26 for 663y. Even by eye one can realize that the events are not distributed randomly but are arranged in clusters. A similar clustering was observed for extreme floods, winter storms, and avalanches in Central Europe (see, e. g., Figs. 4.4, 4.7, 4.10, and 4.13 in [69], Fig. 66 in [70], and Fig. 2 in [71]). The reason for this clustering is the long-term memory. Figure 27 shows Pq (r) for long-term correlated records with ˛ D 0:4 (corresponding to D 0:8), for three values of the mean return interval Rq (which is easily obtained from the threshold q and independent of the correlations). The pdf is plotted in a scaled way, i. e., R q Pq (r) as a function of r/R q . The figure shows that all three curves collapse. Accordingly, when we know the functional form of the pdf for one value of Rq , we can easily deduce its functional form also for very large Rq values which due to its poor statistics cannot be obtained directly from the data. This scaling is a very important property, since it allows one to make predictions also for rare events which otherwise are not accessible with meaningful statistics. When the data are shuffled, the long-term correlations are destroyed and the pdf becomes a simple exponential. The functional form of the pdf is a quite natural extension of the uncorrelated case. The figure suggests that ln Pq (r) (r/R q )

(20)

i. e. simple stretched exponential behavior [72,73]. For

approaching 1, the long-term correlations tend to vanish

Fractal Geometry, A Brief Introduction to, Figure 27 Probability density function of the return intervals in long-term correlated data, for three different return periods Rq , plotted in a scaled way. The full line is a stretched exponential, with exponent  D 0:4 (after [73])

and we obtain the simple exponential behavior characteristic for uncorrelated processes. For r well below Rq , however, there are deviations from the pure stretched exponential behavior. Closer inspection of the data shows that for r/R q  the decay of the pdf is characterized by a power law, with the exponent  1. This overall behavior does not depend crucially on the way the original data are distributed. In the cases shown here, the data had a Gaussian distribution, but similar results have been obtained also for exponential, power-law and log-normal distributions [74]. Indeed, the characteristic stretched exponential behavior of the pdf can also be seen in long historic and reconstructed records [73]. The form of the pdf indicates that return intervals both well below and well above their average value are considerably more frequent for long-term correlated data than for uncorrelated data. The distribution does not quantify, however, if the return intervals themselves are arranged in a correlated or in an uncorrelated fashion, and if clustering of rare events may be induced by long-term correlations. To study this question, [73] and [74] have evaluated the autocorrelation function of the return intervals in synthetic long-term correlated records. They found that also the return intervals are arranged in a long-term correlated fashion, with the same exponent as the original data. Accordingly, a large return interval is more likely to be followed by a large one than by a short one, and a small return interval is more likely to be followed by a small one than by a large one, and this leads to clustering of events above some threshold q, including extreme events. As a consequence of the long-term memory, the probability of finding a certain return interval depends on the preceding interval. This effect can be easily seen in synthetic data sets generated numerically, but not so well in climate records where the statistics is comparatively poor. To improve the statistics, we now only distinguish between two kinds of return intervals, “small” ones (below the median) and “large” ones (above the median), and determine  the mean RC q and R q of those return intervals following a large (C) or a small () return interval. Due to scal ing, RC q /R q and R q /R q are independent of q. Figure 28 shows both quantities (calculated numerically for longterm correlated Gaussian data) as a function of the correlation exponent . The lower dashed line is R q /R q , the /R . In the limit of vanishing longupper dashed line is RC q q term memory, for D 1, both quantities coincide, as ex pected. Figure 28 also shows RC q /R q and R q /R q for five climate records with different values of . One can see that the data agree very well, within the error bars, with the theoretical curves.

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 28 Mean of the (conditional) return intervals that either follow a return interval below the median (lower dashed line) or above the median (upper dashed line), as a function of the correlation exponent , for five long reconstructed and natural climate records. The theoretical curves are compared with the corresponding values of the climate records (from right to left): The reconstructed run-offs of the Sacramento River, the reconstructed temperatures of Baffin Island, the reconstructed precipitation record of New Mexico, the historic water levels of the Nile and one of the reconstructed temperature records of the Northern hemisphere (Mann record) (after [73])

Long-Term Correlations in Financial Markets and Seismic Activity The characteristic behavior of the return intervals, i. e. long-term correlations and stretched exponential decay, can also be observed in financial markets and seismic activity. It is well known (see, e. g. [56]) that the volatility of stocks and exchange rates is long-term correlated. Figure 29 shows that, as expected from the foregoing, also the return intervals between daily volatilities are long-term

correlated, with roughly the same exponent as the original data [75]. It has further been shown [75] that also the pdfs show the characteristic behavior predicted above. A further example where long-term correlations seem to play an important role, are earthquakes in certain bounded areas (e. g. California) in time regimes where the seismic activity is (quasi) stationary. It has been discovered recently by [76] that the magnitudes of earthquakes in Northern and Southern California, from 1995 until 1998, are long-term correlated with an exponent around

D 0:4, and that also the return intervals between the earthquakes are long-term correlated with the same exponent. For the given exponential distribution of the earthquake magnitudes (following the Gutenberg–Richter law), the long-term correlations lead to a characteristic dependence on the scaled variable r/R q which can explain, without any fit parameter, the previous results on the pdf of the return intervals by [77]. Multifractal Records Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. In some cases, there exist crossover (time-) scales sx separating regimes with different scaling exponents, e. g. long-term correlations on small scales below sx and another type of correlations or uncorrelated behavior on larger scales above sx . In other cases, the scaling behavior is more complicated, and different scaling exponents are required for different parts of the series. In even more complicated cases, such different scaling behavior can be observed for many interwoven fractal subsets of the time series. In this case a multitude of scaling exponents is required for a full description of the scaling behavior, and

Fractal Geometry, A Brief Introduction to, Figure 29 Long-term correlation exponent  for the daily volatility (left) and the corresponding return intervals (right). The studied commodities are (from left to right), the S&P 500 index, six stocks (IBM, DuPont, AT&T, Kodak, General Electric, Coca-Cola) and seven currency exchange rates (US$ vs. Japanese Yen, British Pound vs. Swiss Franc, US$ vs. Swedish Krona, Danish Krone vs. Australian $, Danish Krone vs. Norwegian Krone, US$ vs. Canadian $ and US$ vs. South African $). Courtesy of Lev Muchnik

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a multifractal analysis must be applied (see, e. g., [78,79] and literature therein). To see this, it is meaningful to extend Eqs. (19a) and (19b) by considering the more general average F q (s) D

1 Ks

Ks X D1



q/2 F2 (s)

(21)

with q between 1 and C1. For q  1 the small fluctuations will dominate the sum, while for q  1 the large fluctuations are dominant. It is reasonable to assume that the q-dependent average scales with s as F q (s) s qˇ (q) ;

(22)

with ˇ(2) D ˛. Equation (22) generalizes Eq. (19b). If ˇ(q) is independent of q, we have (F q (s))1/q s ˛ , independent of q, and both large and small fluctuations scale the same. In this case, a single exponent is sufficient to characterize the record, which then is referred to as monofractal. If ˇ(q) is not identical to ˛, we have a multifractal [1,4,12]. In this case, the dependence of ˇ(q) on q characterizes the record. Instead of ˇ(q) one considers frequently the spectrum f (!) that one obtains by Legendre transform of qˇ(q), ! D d(qˇ(q))/dq, f (!) D q!  qˇ(q) C 1. In the monofractal limit we have f (!) D 1. For generating multifractal data sets, one considers mostly multiplicative random cascade processes, described, e. g., in [3,4]. In this process, the data set is obtained in an iterative way, where the length of the record doubles in each iteration. It is possible to generate random cascades with vanishing autocorrelation function (Cx (s) D 0 for s  1) or with algebraically decaying autocorrelation functions (C x (s) s ). Here we focus on a multiplicative random cascade with vanishing autocorrelation function, which is particularly interesting since it can be used as a model for the arithmetic returns (Pi  Pi1 )/Pi of daily stock closing prices Pi [80]. In the zeroth iteration n D 0, the data set (x i ) consists of one value, x1(nD0) D 1. In the nth iteration, the data x (n) i consist of 2n values that are obtained from (n) (n1) (n) x2l m2l 1 1 D x l

and (n) x2l D x (n1) m(n) l 2l ;

(23)

where the multipliers m are independent and identically distributed (i.i.d.) random numbers with zero mean and unit variance. The resulting pdf is symmetric with lognormal tails, with vanishing correlation function C x (s) for s  1.

It has been shown that in this case, the pdf of the return intervals decays by a power-law  Pq (r)

r Rq

ı(q) ;

(24)

where the exponent ı depends explicitly on Rq and seems to converge to a limiting curve for large data sets. Despite of the vanishing autocorrelation function of the original record, the autocorrelation function of the return intervals decays by a power law with a threshold-dependent exponent [80]. Obviously, these long-term correlations have been induced by the nonlinear correlations in the multifractal data set. Extracting the return interval sequence from a data set is a nonlinear operation, and thus the return intervals are influenced by the nonlinear correlations in the original data set. Accordingly, the return intervals in data sets without linear correlations are sensitive indicators for nonlinear correlations in the data records. The power-law dependence of Pq (r) can be used for an improved risk estimation. Both power-law dependencies can be observed in economic and physiological records that are known to be multifractal [81]. Acknowledgments We like to thank all our coworkers in this field, in particular Eva Koscielny-Bunde, Mikhail Bogachev, Jan Kantelhardt, Jan Eichner, Diego Rybski, Sabine Lennartz, Lev Muchnik, Kazuko Yamasaki, John Schellnhuber and Hans von Storch. Bibliography 1. Mandelbrot BB (1977) Fractals: Form, chance and dimension. Freeman, San Francisco; Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco 2. Jones H (1991) Part 1: 7 chapters on fractal geometry including applications to growth, image synthesis, and neutral net. In: Crilly T, Earschaw RA, Jones H (eds) Fractals and chaos. Springer, New York 3. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New York 4. Feder J (1988) Fractals. Plenum, New York 5. Vicsek T (1989) Fractal growth phenomena. World Scientific, Singapore 6. Avnir D (1992) The fractal approach to heterogeneous chemistry. Wiley, New York 7. Barnsley M (1988) Fractals everywhere. Academic Press, San Diego 8. Takayasu H (1990) Fractals in the physical sciences. Manchester University Press, Manchester 9. Schuster HG (1984) Deterministic chaos – An introduction. Physik Verlag, Weinheim 10. Peitgen H-O, Richter PH (1986) The beauty of fractals. Springer, Heidelberg

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32. Meakin P, Majid I, Havlin S, Stanley HE (1984) Topological properties of diffusion limited aggregation and cluster-cluster aggregation. Physica A 17:L975 33. Mandelbrot BB (1992) Plane DLA is not self-similar; is it a fractal that becomes increasingly compact as it grows? Physica A 191:95; see also: Mandelbrot BB, Vicsek T (1989) Directed recursive models for fractal growth. J Phys A 22:L377 34. Schwarzer S, Lee J, Bunde A, Havlin S, Roman HE, Stanley HE (1990) Minimum growth probability of diffusion-limited aggregates. Phys Rev Lett 65:603 35. Meakin P (1983) Formation of fractal clusters and networks by irreversible diffusion-limited aggregation. Phys Rev Lett 51:1119; Kolb M (1984) Unified description of static and dynamic scaling for kinetic cluster formation. Phys Rev Lett 53:1653 36. Stauffer D, Aharony A (1992) Introduction to percolation theory. Taylor and Francis, London 37. Kesten H (1982) Percolation theory for mathematicians. Birkhauser, Boston 38. Grimmet GR (1989) Percolation. Springer, New York 39. Song C, Havlin S, Makse H (2005) Self-similarity of complex networks. Nature 433:392 40. Havlin S, Blumberg-Selinger R, Schwartz M, Stanley HE, Bunde A (1988) Random multiplicative processes and transport in structures with correlated spatial disorder. Phys Rev Lett 61:1438 41. Voss RF (1985) In: Earshaw RA (ed) Fundamental algorithms in computer graphics. Springer, Berlin, p 805 42. Coleman PH, Pietronero L (1992) The fractal structure of the universe. Phys Rep 213:311 43. Kaye BH (1989) A random walk through fractal dimensions. Verlag Chemie, Weinheim 44. Turcotte DL (1997) Fractals and chaos in geology and geophysics. Cambridge University Press, Cambridge 45. Hurst HE, Black RP, Simaika YM (1965) Long-term storage: An experimental study. Constable, London 46. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Wat Resour Res 5:321–340 47. Koscielny-Bunde E, Kantelhardt JW, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of river runoff records: Detrended fluctuation studies. Hydrol J 322:120–137 48. Mudelsee M (2007) Long memory of rivers from spatial aggregation. Wat Resour Res 43:W01202 49. Livina VL, Ashkenazy Y, Braun P, Monetti A, Bunde A, Havlin S (2003) Nonlinear volatility of river flux fluctuations. Phys Rev E 67:042101 50. Koscielny-Bunde E, Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber H-J (1998) Indication of a universal persistence law governing athmospheric variability. Phys Rev Lett 81: 729–732 51. Pelletier JD, Turcotte DL (1999) Self-affine time series: Application and models. Adv Geophys 40:91 52. Talkner P, Weber RO (2000) Power spectrum and detrended fluctuation analysis: Application to daily temperatures. Phys Rev E 62:150–160 53. Eichner JF, Koscielny-Bunde E, Bunde A, Havlin S, Schellnhuber H-J (2003) Power-law persistence and trends in the atmosphere: A detailed study of long temperature records. Phys Rev E 68:046133 54. Király A, Bartos I, Jánosi IM (2006) Correlation properties of daily temperature anormalies over land. Tellus 58A(5):593–600

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55. Santhanam MS, Kantz H (2005) Long-range correlations and rare events in boundary layer wind fields. Physica A 345: 713–721 56. Liu YH, Cizeau P, Meyer M, Peng C-K, Stanley HE (1997) Correlations in economic time series. Physica A 245:437; Liu YH, Gopikrishnan P, Cizeau P, Meyer M, Peng C-K, Stanley HE (1999) Statistical properties of the volatility of price fluctuations. Phys Rev E 60:1390 57. Peng C-K, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anticorrelations and non-gaussian behavior of the heartbeat. Phys Rev Lett 70:1343–1346 58. Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter J-H, Voigt K (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736 59. Leland WE, Taqqu MS, Willinger W, Wilson DV (1994) On the self-similar nature of Ethernet traffic. IEEE/Transactions ACM Netw 2:1–15 60. Kantelhardt JW, Koscielny-Bunde E, Rego HA, Bunde A, Havlin S (2001) Detecting long-range correlations with detrended fluctuation analysis. Physica A 295:441 61. Rybski D, Bunde A, Havlin S, Von Storch H (2006) Long-term persistence in climate and the detection problem. Geophys Res Lett 33(6):L06718 62. Rybski D, Bunde A (2008) On the detection of trends in longterm correlated records. Physica A 63. Giese E, Mossig I, Rybski D, Bunde A (2007) Long-term analysis of air temperature trends in Central Asia. Erdkunde 61(2): 186–202 64. Govindan RB, Vjushin D, Brenner S, Bunde A, Havlin S, Schellnhuber H-J (2002) Global climate models violate scaling of the observed atmospheric variability. Phys Rev Lett 89:028501 65. Vjushin D, Zhidkov I, Brenner S, Havlin S, Bunde A (2004) Volcanic forcing improves atmosphere-ocean coupled general circulation model scaling performance. Geophys Res Lett 31:L10206 66. Monetti A, Havlin S, Bunde A (2003) Long-term persistence in the sea surface temperature fluctuations. Physica A 320: 581–589 67. Kantelhardt JW, Koscielny-Bunde E, Rybski D, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of precipitation and river runoff records. Geophys J Res Atmosph 111:1106

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Fractal Growth Processes

Fractal Growth Processes LEONARD M. SANDER Physics Department and Michigan Center for Theoretical Physics, The University of Michigan, Ann Arbor, USA Article Outline Glossary Definition of the Subject Introduction Fractals and Multifractals Aggregation Models Conformal Mapping Harmonic Measure Scaling Theories Future Directions Bibliography Glossary Fractal A fractal is a geometric object which is self-similar and characterized by an effective dimension which is not an integer. Multifractal A multifractal measure is a non-negative real function) defined on a support (a geometric region) which has a spectrum of scaling exponents. Diffusion-limited aggregation Diffusion-limited aggregation (DLA) is a discrete model for the irreversible growth of a cluster. The rules of the model involve a sequence of random walkers that are incorporated into a growing aggregate when they wander into contact with one of the previously aggregated walkers. Dielectric breakdown model The dielectric breakdown model (DBM) is a generalization of DLA where the probability to grow is proportional to a power of the diffusive flux onto the aggregate. If the power is unity, the model is equivalent to DLA: in this version it is called Laplacian growth. Harmonic measure If a geometric object is thought of as an isolated grounded conductor of fixed charge, the distribution of electric field on its surface is the harmonic measure. The harmonic measure of a DLA cluster is the distribution of growth probability on the surface. Definition of the Subject Fractal growth processes are a class of phenomena which produce self-similar, disordered objects in the course of development far from equilibrium. The most famous of these processes involve growth which can be modeled on

the large scale by the diffusion-limited aggregation algorithm of Witten and Sander [1]. DLA is a paradigm for pattern formation modeling which has been very influential. The algorithm describes growth limited by diffusion: many natural processes fall in this category, and the salient characteristics of clusters produced by the DLA algorithm are observed in a large number of systems such as electrodeposition clusters, viscous fingering patterns, colonies of bacteria, dielectric breakdown patterns, and patterns of blood vessels. At the same time the DLA algorithm poses a very rich problem in mathematical physics. A full “solution” of the DLA problem, in the sense of a scaling theory that can predict the important features of computer simulations is still lacking. However, the problem shows many features that resemble thermal continuous phase transitions, and a number of partially successful approaches have been given. There are deep connections between DLA in two dimensions and theories such as Loewner evolution that use conformal maps. Introduction In the 1970s Benoit Mandelbrot [2] developed the idea of fractal geometry to unify a number of earlier studies of irregular shapes and natural processes. Mandelbrot focused on a particular set of such objects and shapes, those that are self-similar, i. e., where a part of the object is identical (either in shape, or for an ensemble of shapes, in distribution) to a larger piece. He dubbed these fractals. He noted the surprising ubiquity of self-similar shapes in nature. Of particular interest to Mandelbrot and his collaborators were incipient percolation clusters [3]. These are the shapes generated when, for example, a lattice is diluted by cutting bonds at random until a cluster of connected bonds just reaches across a large lattice. This model has obvious applications in physical processes such as transport in random media. The model has a non-trivial mapping to a statistical model [4] and can be treated by the methods of equilibrium statistical physics. It is likely that percolation processes account for quite a few observations of fractals in nature. In 1981 Tom Witten and the present author observed that a completely different type of process surprisingly appears to make fractals [1,5]. These are unstable, irreversible, growth processes which we called diffusion-limited aggregation (DLA). DLA is a kinetic process which is not related to equilibrium statistical physics, but rather defined by growth rules. The rules idealize growth limited by diffusion: in the model there are random walking “par-

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ticles” which attach irreversibly to a single cluster made up of previously aggregated particles. As we will see, quite a few natural processes can be described by DLA rules, and DLA-like clusters are reasonably common in nature. The Witten–Sander paper and subsequent developments unleashed a large amount of activity – the original work has been cited more than 2700 times as of this writing. The literature in this area up to 1998 was reviewed in a very comprehensive manner by T. Vicsek [6] and P. Meakin [7]. See also the chapter by the present author in [8]. For non-technical reviews see [9,10,11]. There are three major areas where self-similar shapes arising from non-equilibrium processes have been studied. The first is the related to the original DLA algorithm. The model may be seen as an idealization of solidification of an amorphous substance. The study of this simple-seeming model is quite unexpectedly rich, and quite difficult to treat theoretically. It will be our focus in this article. We will review the early work, but emphasize developments since [7]. Meakin [12] and Kolb, et al. [13] generalized DLA to consider cluster-cluster or colloid aggregation. In this process particles aggregate when they come into contact, but the clusters so formed are mobile, and themselves aggregate by attaching to each other. This is an idealization of colloid or coagulated aerosol formation. This model also produces fractals but this is not really a surprise: at each stage, clusters of similar size are aggregating, and the result is an approximately hierarchical object. This model is quite important in applications in colloid science. The interested reader should consult [6,14]. A third line of work arose from studies of ballistic aggregation [15] and the Eden model [16]. In the former case particles attach to an aggregate after moving in straight paths, and in the latter, particles are simply added to the surface of an aggregate at any available site, with equal probability. These models give rise to non-fractal clusters with random rough surfaces. The surfaces have scaling properties which are often characterized at the continuum level by a stochastic partial differential equation proposed by Kardar, Parisi, and Zhang [17]. For accounts of this area the reader can consult [18,19,20]. The remainder of this article is organized as follows: we first briefly review fractals and multifractals. Then we give details about DLA and related models, numerical methods, and applications of the models. The major development that has fueled a remarkable revival of interest in this area was the work of Hastings and Levitov [21] who related two-dimensional DLA to a certain class of conformal maps. Developments of this theme is the subject of the subsequent section. Then we discuss the ques-

tion of the distribution of growth probabilities on the cluster surface, and finally turn to scaling theories of DLA. Fractals and Multifractals The kind of object that we deal with in this article is highly irregular; see Fig. 1. We think of the object as being made up of a large number, N, of units, and to be of overall linear dimension R. In growth processes the objects are formed by adding the units according to some dynamics. We will call the units “particles”, and take their size to be a. Such patterns need not be merely random, but can have well-defined scaling properties in the regime a  R. The picture is of a geometric object in two dimensions, but the concepts we introduce here are often applied to more abstract spaces. For example, a strange attractor in phase space often has the kind of scaling properties described here. In order to characterize the geometry of such complex objects, we first cover the points in question with a set of n(l) “boxes” of fixed size, l such that a  l  R. Clearly, for a smooth curve the product ln(l) approaches a limit (the length of the curve) as l ! 0. This is a number of order R. For a planar region with smooth boundaries l 2 n(l) approaches the area, of order R2 . The objects of ordinary geometry in d dimensions have measures given by the limit of l d n(l). For an object with many scales (a fractal), in general none of these relations hold. Rather, the product l D n(l) approaches a limit with D not necessarily an integer; D is called the (similarity) fractal dimension.

Fractal Growth Processes, Figure 1 A partial covering of a pattern with boxes. Smaller boxes reveal smaller scales of the pattern. For a pattern with many scales (like this one) there is a non-trivial scaling between the box size, l and the number of boxes

Fractal Growth Processes

Said another way, we define the fractal dimension by: D

n(l) / (R/l) :

(1)

For many simple examples of mathematical objects with non-integer values of D see [2,6]. For an infinite fractal there are no characteristic lengths. For a finite size object there is a characteristic length, the overall scale, R. This can be taken to be any measure of the size such as the radius of gyration or the extremal radius – all such lengths must be proportional. It is useful to generalize this definition in two ways. First we consider not only a geometric object, but also a measure, that is a non-negative R function  defined on the points of the object such that d D 1. For the geometry, we take the measure to be uniform on the points. However, for the case of growing fractals, we could also consider the growth probability at a point. As we will see, this is very non-uniform for DLA. Second, we define a sequence of generalized dimensions. If we are interested in geometry, we denote the mass of the object covered by box i by pi . For an arbitrary measure, we define: Z p i D d ; (2) where the integral is over the box labeled i. Then, following [22,23] we define a partition function for the pi : (q) D

n X

q

pi ;

(3)

points can be found by focusing on any point, and drawing a d-dimensional disk of radius r around it. The number of other points in the disk will scale as r D 2 . For DLA, it is common to simply count the number of points within radius r of the origin, or, alternatively, the dependence of some mean radius, R on the total number of aggregated particles, N, that is N / R D 2 This method is closely related to the Grassberger–Procaccia correlation integral [24]. For a simple fractal all of the Dq are the same, and we use the symbol D. If the generalized dimensions differ, then we have a multifractal. Multifractals were introduced by Mandelbrot [25] in the context of turbulence. In the context of fractal growth processes, the clusters themselves are usually simple fractals. They are the support for a multifractal measure, the growth probability. We need another bit of formalism. It is useful to look at the fractal measure and note how the pi scale with l/R. Suppose we assume a power-law form, p i / (l/R)˛ , where there are different values of ˛ for different parts of the measure. Also, suppose that we make a histogram of the ˛, and look at the parts of the support on which we have the same scaling. It is natural to adopt a form like Eq. (1) for the size of these sets, (l/R) f (˛) . (It is natural to think of f (˛) as the fractal dimension of the set on which the measure has exponent ˛, but this is not quite right because f can be negative due to ensemble averaging.) Then it is not hard to show [23] that ˛; f (˛) are related to q; (q) by a Legendre transform:

iD1

where q is a real number. For an object with well-defined scaling properties we often find that  scales with l in the following way as l/R ! 0: (q) / (R/l)(q)  (R/l)(q1)D q ; (q) D (q  1)D q :

(4)

Objects with this property are called fractals if all the Dq are the same. Otherwise they are multifractals. Some of the Dq have special significance. The similarity (or box-counting) dimension mentioned above is D0 since in this case  D n. If we take the limit q ! 1 we have the information dimension of dynamical systems theory: ˇ X ln p i d ˇˇ D pi : (5) D1 D ˇ dq qD1 ln l i

D2 is called the correlation dimension since p2i measures the probability that two points are close together, i. e. the number of pairs within distance l. This interpretation gives rise to a popular way to measure D2 . If we suppose that the structure is homogeneous, then the number of pairs of

f (˛) D q

d  ; dq

˛D

d : dq

(6)

Aggregation Models In this section we review aggregation models of the DLA type, and their relationship to certain continuum processes. We look at practical methods of simulation, and exhibit a few applications to physical and biological processes. DLA The original DLA algorithm [1,5] was quite simple: on a lattice, declare that the point at the origin is the first member of the cluster. Then launch a random walker from a distant point and allow it to wander until it arrives at a neighboring site to the origin, and attach it to the cluster, i. e., freeze its position. Then launch another walker and let it attach to one of the two previous points, and so on. The name, diffusion-limited aggregation, refers to the fact that random walkers, i. e., diffusing particles, control the growth. DLA is a simplified view of a common physical process, growth limited by diffusion.

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Fractal Growth Processes, Figure 2 a A DLA cluster of 50,000,000 particles produced with an offlattice algorithm by E. Somfai. b A three-dimensional off-lattice cluster. Figure courtesy of R. Ball

It became evident that for large clusters the overall shape is dependent on the lattice type [26,27], that is, DLA clusters are deformed by lattice anisotropy. This is an interesting subject [26,28,29,30,31] but most modern work is on DLA clusters without anisotropy, off-lattice clusters. The off-lattice algorithm is similar to the original one: instead of a random walk on a lattice the particle is considered to have radius a. For each step of the walk the particle moves its center from the current position to a random point on its perimeter. If it overlaps a particle of the current cluster, it is backed up until it just touches the cluster, and frozen at that point. Then another walker is launched. A reasonably large cluster grown in two dimensions is shown in Fig. 2, along with a smaller three-dimensional example. Most of the work on DLA has been for two dimensions, but dimensions up to 8 have been considered [32]. Patterns like the one in Fig. 2 have been analyzed for fractal properties. Careful measurements of both D0 and D2 (using the method, mentioned above, of fitting n(r) to r D 2 ) give the same fractal dimension, D=1.71 [32,33,34]; see Fig. 3. There is some disagreement about the next digit. There have been suggestions that DLA is a mass multifractal [35], but most authors now agree that all of the Dq are the same for the mass distribution. For three dimensions D  2:5 [31,32], and for four dimensions D  3:4 [32]. However, some authors [36,37] have claimed that plane DLA is not a self-similar fractal at all, and that the fractal dimension will drift towards 2 as the number of particles increases. More recent work based on conformal maps [38] has cast doubt on this. We will return to this point below. Plane DLA clusters can be grown in restricted geometries, see Fig. 4. The shape of such clusters is an interesting problem in pattern formation [39,40,41,42]. It was long thought that DLA grown in a channel had a different fractal dimension than in radial geometry [43,44,45].

Fractal Growth Processes, Figure 3 The number of particles inside of radius r for a large DLA cluster. This plot gives an estimate of D2

However, more careful work has shown that the dimensions are the same in the two cases [34]. Laplacian Growth and DBM Suppose we ask how the cluster of Fig. 2 gets to be rough. A simple answer is that if we already have a rough shape, it is quite difficult for a random walker to penetrate a narrow channel. (Just how difficult is treated in the Sect. “Harmonic Measure”, below) The channels don’t fill in, and the shape might be preserved. But we can also ask why a a smooth outline, e. g. a disk, does not continue to grow smoothly. In fact, it is easy to test that any initial condition is soon forgotten in the growth [5]. If we start with a smooth shape it roughens immediately because of a growth instability intrinsic to diffusion-limited growth. This instability was discovered by Mullins and Sekerka [46] who used a statement of the problem of diffusion-limited growth in continuum terms: this is known as the Stefan problem (see [47,48]), and is the standard way to idealize crystallization in the diffusion-limited case. The Stefan problem goes as follows: suppose that we have a density (r; t) of particles that diffuse until they reach the growing cluster where they deposit. Then we have: @ D r 2  @t

(7)

Fractal Growth Processes

Fractal Growth Processes, Figure 4 DLA grown in a channel (a) and a wedge (b). The boxes in a are the hierarchical maps of the Sect. “Numerical Methods”. Figures due to E. Somfai

@ / vn @n

(8)

That is,  should obey the diffusion equation;  is the diffusion constant. The normal growth velocity, vn , of the interface is proportional to the flux onto the surface, @/@n. However the term @/@t is of order v@/@x, where v is a typical growth velocity. Now jr 2 j  (v/D)j@/@nj. In the DLA case we launch one particle at a time, so that the velocity goes to zero. Hence Eq. (7) reduces to the Laplace equation, r 2 D 0

(9)

Since the cluster absorbs the particles, we should think of it as having  D 0 on the surface. We are to solve an electrostatics problem: the cluster is a grounded conductor with fixed electric flux far away. We grow by an amount proportional to the electric field at each point on the surface. This is called the quasi-static or Laplacian growth regime for deterministic growth. A linear stability analysis of these equations gives the Mullins–Sekerka instability [46]. The qualitative reason for the instability is that near the tips of the cluster the contours of  are compressed so that @/@n,

the growth rate, is large. Thus tips grow unstably. We expect DLA to have a growth instability. However, we can turn the argument, and use these observations to give a restatement of the DLA algorithm in continuum terms: we calculate the electric field on the surface of the aggregate, and interpret Eq. (8) as giving the distribution of the growth probability, p, at a point on the surface. We add a particle with this probability distribution, recalculate the potential using Eq. (9) and continue. This is called Laplacian growth. Simulations of Laplacian growth yield the same sort of clusters as the original discrete algorithm. (Some authors use the term Laplacian growth in a different way, to denote deterministic growth according to the Stefan model without surface tension [49].) DLA is thus closely related to one of the classic problems of mathematical physics, dendritic crystal growth in the quasistatic regime. However, it is not quite the same for several reasons: DLA is dominated by noise, whereas the Stefan problem is deterministic. Also, the boundary conditions are different [48]: for a crystal, if we interpret u as T  Tm , where T is the temperature, and T m the melting temperature, we have  D 0 only on a flat surface. On a curved surface we need  /  where is the surface stiffness, and  is the curvature. The surface tension acts

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Fractal Growth Processes

Fractal Growth Processes, Figure 5 DBM patterns for 1000 particles on a triangular lattice. a  D 0:5 b  D 1. This is a small DLA cluster. c  D 2. d  D 3

as a regularization which prevents the Mullins–Sekerka instability from producing sharp cusps [50]. In DLA the regularization is provided by the finite particle size. And, of course, crystals have anisotropy in the surface tension. There is another classic problem very similar to this, that of viscous fingering in Hele–Shaw flow [48]. This is the description of the displacement of an incompressible viscous fluid by an inviscid one: the “bubble” of inviscid fluid plays the role of the cluster,  is the difference of pressures in the viscous fluid and the bubble, and the Laplace equation is the direct result of incompressibility, r  v D 0 and D’Arcy’s law, v D kr where k is the permeability, and v the fluid velocity [51]. These considerations led Niemeyer, Pietronero, and Weismann [52] to a clever generalization of Laplacian growth. They were interested in dielectric breakdown with  representing a real electrostatic potential. This is known to be a threshold process so that we expect that the breakdown probability is non-linear in @/@n. To generalize they chose:  p/

@ @n

 ;

(10)

where  is a non-negative real number. There are some interesting special cases for this model. For  D 0 each growth site is equally likely to be used. This is the Eden model [16]. For  D 1 we have the Laplacian growth version of DLA, and for larger  we get a higher probability to grow at the tips so that the aggregates are more spread out (as in a real dielectric breakdown pattern like atmospheric lightning). There is a remarkable fact which was suggested numerically in [53] and confirmed more recently [54,55]: for  > 4 the aggregate prefers growth at tips so much that it becomes essentially linear and non-fractal. DBM patterns for small numbers of particles are shown in Fig. 5.

A large number of variants of the basic model have been proposed such as having the random walkers perform Lévy flights, having a variable particle size, imposing a drift on the random walkers, imposing anisotropy in attachment, etc. For references on these and other variants, see [7]. There are two qualitative results from these studies that are worth mentioning here: In the presence of drift, DLA clusters cross over to (non-fractal) Eden-like clusters [56,57]. And, anisotropy deforms the shape of the clusters on the large scale, as we mentioned above. Numerical Methods In the foregoing we have talked about the algorithm for DLA, but we have not described what is actually done to compute the positions of 50,000,000 particles. This is a daunting computational task, and considerable ingenuity has been devoted to making efficient algorithms. The techniques are quite interesting in their own right, and have been applied to other types of simulation. The first observation to make is that the algorithm is an idealization of growth due to particles wandering into contact with the aggregate from far away. However, it is not necessary to start the particles far away: they arrive at the aggregate with uniform probability on a circle which just circumscribes all of the presently aggregated particles; thus we can start the particles there. As the cluster grows, the starting circle grows. This was already done in the original simulations [1,5]. However, particles can wander in and then out of the starting circle without attaching. These walkers must be followed, and could take a long time to find the cluster again. However, since there is no matter outside, it is possible to speed up the algorithm by noting that if the random walker takes large steps it will still have the same probability distribution, provided that it cannot encounter any matter. For example, if it walks onto the circumference of

Fractal Growth Processes

Fractal Growth Processes, Figure 6 A cluster is surrounded by a starting circle (gray). A random walker outside the circle can safely take steps of length of the black circle. If the walker is at any point, it can walk on a circle equal in size to the distance to the nearest point of the cluster. However, finding that circle appears to require a time-consuming search

a circle that just reaches the starting circle, it will have the correct distribution. The radius of this circle is easy to find. This observation, due to P. Meakin, is the key to what follows: see Fig. 6. We should note that the most efficient way to deal with particles that wander away is to return the particle to the starting circle in one step using the Green’s function for a point charge outside an absorbing circle. A useful algorithm to do this is given in [58]. However, the idea of taking big steps is still a good one because there is a good deal of empty space inside the starting circle. If we could take steps in this empty space (see Fig. 6) we could again speed up the algorithm. The trick is to efficiently find the largest circle centered on the random walker that has no point of the aggregate within it. One could imagine simply doing a spiral search from the current walker position. This technique has actually been used in a completely different setting, that of Kinetic Monte Carlo simulations of surface growth in materials science [59]. For the case of DLA an even more efficient method, the method of hierarchical maps, was devised [26]. It was extended and applied to higher dimensions and off-lattice in [32], and is now the standard method. One version of the idea is illustrated in Fig. 4a for the case of growth in a channel. What is shown is an adaptively refined square mesh. The cluster is covered with a square – a map. The square is subdivided into four smaller squares, and each is further divided, but only if the cluster is closer to it than half of the side of the square. The subdivision continues only up to a predefined maximum depth so that the smallest maps are a few particle diameters. All particles of the cluster will be in one of the smallest maps: a list of the particles is attached to these maps.

As the cluster grows the maps are updated. Each time a particle is added to a previously empty smallest map, the neighboring maps (on all levels) are checked to see whether they satisfy the rule. If not, they are subdivided until they do. When a walker lands, we find the smallest map containing the point. If this map is not at the maximum depth, then the particle is far away from any matter, and half the side of the map is a lower estimate of the walker’s distance from the cluster. If, on the other hand, the particle lands in a map of maximum depth, then it is close to the cluster. The particle lists of the map and of the neighboring smallest size maps can be checked to calculate the exact distance from the cluster. Either way, the particle is enclosed in an empty circle of known radius, and can be brought to the perimeter of the circle in one step. Note that updating the map means that there is only a search for the cluster if we are in the smallest map. Empirically, the computational time, T for an N particle cluster obeys T N 1:1 , and the memory is linear in N. A more recent version of the algorithm for three-dimensional growth uses a covering with balls rather than cubes [31]. For simulations of the Laplacian growth version of the situation is quite different, and simulations are much slower. The reason is that a literal interpretation of the algorithm requires that the Laplace equation be solved each time a particle is added, for example, by the relaxation or boundary-integral method. This is how corresponding simulations for viscous fingering are done [60]. For DBM clusters the growth step requires taking a power of the electric field at the surface. It is possible to make DBM clusters using random walkers [61], but the method is rather subtle. It involves estimating the growth probability at a point by figuring out the “age” of a site, i. e., the time between attachments. This can be converted into an estimate of the growth probability. This algorithm is quite fast. There is another class of methods involving conformal maps. These are slow methods: the practical limit of the size of clusters that can be grown this way is about 50,000. However, this method calculates the growth probability as it goes along, and thus is of great interest. Conformal mapping is discussed in the Sect. “Loewner Evolution and the Hastings–Levitov Scheme”, below. Selected Applications Probably the most important impact of the DLA model has been the realization that diffusion-limited growth naturally gives rise to tenuous, branched objects. Of course, in the context of crystallization, this was understood in the centuries-long quest to understand the shape of

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Fractal Growth Processes, Figure 7 a A scanning tunneling microscope picture of Rh metal islands. The color scale indicates height, and the figure is about 500 Å across. Figure courtesy of R. Clarke. b A bacteria colony on a Petri dish. The figure is a few centimeters across. Figure courtesy of E. Ben-Jacob

snowflakes. However, applications of DLA gave rise to a unification of this with the study of many other physical applications. Broad surveys are given in [6,7,10]. Here we will concentrate on a few illustrative examples. Rapid crystallization in a random environment is the most direct application of DLA scheme. One particularly accessible example is the growth of islands on surfaces in molecular beam epitaxy experiments. In the proper growth conditions it is easy to see patterns dominated by the Mullins–Sekerka instability. These are often referred to with the unlovely phrase “fractal-like”. An example is given in Fig. 7a. There are many examples of this type, e. g. [62]. A related example is the electrodeposition of metal ions from solution. For overpotential situations DLA patterns are often observed [44,63,64,65]. There are also examples of Laplacian growth. A case of this type was discovered by Matsushita and collaborators [66], and exploited in a series of very interesting studies by the group of Ben-Jacob [67] and others. This is the growth of colonies of bacteria on hard agar plates in conditions of low nutrient supply. In this case, bacteria movement is suppressed (by the hard agar) and the limiting step in colony growth is the diffusion of nutrients to the colony. Thus we have an almost literal realization of Laplacian growth, and, indeed, colonies do look like DLA clusters in these conditions; see Fig. 7b. The detailed study of this system has led to very interesting insights into the biophysics of bacteria: these are far from our subject here, and the reader should consult [67]. We remarked above that noisy viscous fingering patterns are similar to DLA clusters, but not the same in detail: in viscous fingering the surface tension boundary con-

dition is different from that of DLA which involves discrete particles. We should note that for viscous fingering patterns in a channel, the asymptotic state is not a disorderly cluster, but rather a single finger that fills half the channel [48] because surface tension smooths the finger. In radial growth this is not true: the Mullins–Sekerka instability gives rise to a disorderly pattern [51] which looks very much like a DLA cluster, see Fig. 8. Even for growth in a channel, if there is sufficient noise, patterns look rather like those in Fig. 4a. In fact, Tang [68] used a version of DLA which allowed him to reduce the noise (see below) and introduce surface tension to do simulations of viscous fingering.

Fractal Growth Processes, Figure 8 A radial viscous fingering pattern. Courtesy of M. Moore and E. Sharon

Fractal Growth Processes

We should ask if this resemblance is more than a mere coincidence. It is the case that the measured fractal dimension of patterns like the one in Fig. 8 is close to 1.71, as for DLA. On this basis there are several claims in the literature that the large-scale structure of the patterns is identical [9,51,69]. Many authors have disagreed and given ingenious arguments about how to verify this. For example, some have claimed that viscous patterns become two-dimensional at large scales [70], or that viscous fingering patterns are most like DBM patterns with   1:2 [71]. Most recently a measurement of the growth probability of a large viscous fingering pattern was found to agree with that of DLA [72]. On this basis, these authors claim that DLA and viscous fingering are in the same universality class. In our opinion, this subject is still open. Conformal Mapping For pattern formation in two dimensions the use of analytic function theory and conformal mapping methods allows a new look at growth processes. The idea is to think of a pattern in the z plane as the image of a simple reference shape, e. g. the unit circle in the w plane, under a time-dependent analytic function, z D F t (w). More precisely, we think of the region outside of the pattern as the image of the region outside of the reference shape. By the Riemann mapping theorem the map exists and is unique if we set a boundary condition such as F(w) ! r0 w as w ! 1. We will also use the inverse map, w D G(z) D F 1 (z). For Laplacian growth processes this idea is particularly interesting since the growth rules depend on solving the Laplace equation outside the cluster. We recall that the Laplace equation is conformally invariant; that is, it retains its form under a conformal transformation. Thus we can solve in the w plane, and transform the solution: if r 2 (w) D 0, and  D 0 on the unit circle, and we take  to be the real part of a complex potential ˚ in the w plane, then Re ˚ (G(z)) solves the Laplace equation in the z plane with Re ˚ D 0 on the cluster boundary. Thus we can solve for the potential outside the unit circle in w-space (which is easy): ˚ D ln(w). Then if we map to z space we have the solution outside of the cluster: ˚(z) D ln G(z) :

(11)

Note that the constant r0 has the interpretation of a mean radius since ˚ ! ln(z/r0 ) as z ! 1. In fact, r0 is the radius of the disk that gives the same potential as the cluster far away. This has another consequence: the electric field (i. e. the growth probability) is uniform around the unit circle in the w plane. This means that equal intervals on the unit

circle in w space map into equal regions of growth probability in the z plane. The map contains not only the shape of the cluster (the image of jwj D 1) but also information about the growth probability: using Eq. (11) we have: jr˚j D jG 0 j D

1 : jF 0 j

(12)

The problem remains to construct the maps G or F for a given cluster. Somfai, et al. gave a direct method [38]: for a given cluster release a large number, M, of random walkers and record where they hit, say at points zj . We know from the previous paragraph that the images of these points are spaced roughly equally on the unit circle in the w plane. That is, if we start somewhere on the cluster and number the landing positions sequentially around the cluster surface, the images in the w-plane, w j D r j e i j , are given by j D 2 j/M; r j D 1. Thus we have the boundary values of the map and by analytic continuation we can construct the entire map. In fact, if we represent F by a Laurent series: F(w) D r0 w C A0 C

1 X Aj ; wj

(13)

jD1

it is easy to see that the Fourier coefficients of the function F( j ) are the Aj . Unfortunately, for DLA this method only gives a map of the highly probable points on the surface of the cluster. The inner, frozen, regions are very badly sampled by random walkers, so the generated map will not represent these regions. Loewner Evolution and the Hastings–Levitov Scheme A more useful approach to finding F is to impose a dynamics on the map which gives rise to the growth of the cluster. This is very closely related to Loewner evolution, where a curve in the z plane is generated by a map that obeys an equation of motion: 2 dG t (z) D : dt G t (z)  (t)

(14)

The map is to the upper half plane from the upper half plane minus the set of singularities, G D . (For an elementary discussion and references see [73].) If (t) is a stochastic process then many interesting statistical objects such as percolation clusters can be generated. For DLA a similar approach was presented by Hastings and Levitov [21,30]. In this case the evolution is discrete, and is designed to represent the addition of particles to a cluster. The process is iterative: suppose we know the

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Fractal Growth Processes

Fractal Growth Processes, Figure 9 The Hastings–Levitov scheme for fractal growth. At each stage a “bump” of the proper size is added to the unit circle. The composition of the bump map, , with the map at stage N gives the map at stage N C 1

map for N particles, F N . Then we want to add a “bump” corresponding to a new particle. This is accomplished by adding the bump of area  in the w-plane on the surface of the unit circle at angle . There are various explicit functions that generate bumps; for the most popular example see [21]. This is a function that depends on a parameter which gives the aspect ratio of the bump. Let us call the resulting transformation f; . If we use F N to transform the unit circle with a bump we get a cluster with an extra bump in the z-plane: that is, we have added a particle to the cluster and F NC1 D F N ı f . The scheme is represented in Fig. 9. There are two matters that need to be dealt with. First, we need to pick . Since the probability to grow is uniform on the circle in w-space, we take to be a random variable uniformly distributed between 0 and 2. Also, we need the bump in z-space to have a fixed area, 0 . That means that the bump in w-space needs to be adjusted because conformal transformations stretch lengths by jF 0 j. A first guess for the area  is:  NC1 D

0 0 jF N (e i NC1 ) j2

:

(15)

However, this is just a first guess. The stretching of lengths varies over the cluster, and there can be some regions where the approximation of Eq. (15) is not adequate. In this case an iterative procedure is necessary to get the area right [30,34]. The transformation itself is given by: F N D f1 ; 1 ı f2 ; 2 ı    ı fN ; N :

(16)

Fractal Growth Processes, Figure 10 A DLA cluster made by iterated conformal maps. Courtesy of E. Somfai

All of the information needed to specify the mapping is contained in the list  j ; j ; 1  j  N. An example of a cluster made this way is shown in Fig. 10. If we choose uniformly in w-space, we have chosen points with the harmonic measure, jF 0 j1 and we make DLA clusters. To simulate DBM clusters with  ¤ 1 we must choose the angles non-uniformly. Hastings [54] has shown how to do this: since a uniform distribution gives a distribution according to jF 0 j1 (see Eq. (12)) we have to pick angles with probability jF 0 j1 in order to grow with probability jrj . This can be done with a Metropolis algorithm with p( NC1 ) D jF N0 (e i )j1 playing the role of a Boltzmann factor. Applications of the Hastings–Levitov Method: Scaling and Crossovers The Hastings–Levitov method is a new numerical algorithm for DLA, but not a very efficient one. Constructing the composed map takes of order N steps so that the algorithm is of order N 2 . One may wonder what has been gained. The essential point is that new understanding arises from considering the objects that are produced in the course of the computation. For example, Davidovitch and collaborators [74] showed that averages of  over the cluster boundary are related to the generalized dimensions of the growth probability measure, cf. Eq. (4). Further, the Laurent coefficients of the map, Eq. (13), are meaningful in themselves and have interesting scaling properties. We

Fractal Growth Processes

have seen above that r0 is the radius of a disk with the same capacitance as the cluster. It scales as N 1/D , as we expect. The constant, A0 gives the wandering of the center of the cluster from its original position. Its scaling can be worked out in terms of D and the generalized dimension, D2 . The other coefficients, Aj , are related to the moments of the charge density on the cluster surface. We expect them to scale in the same way as r0 for the following reason: there is an elementary theorem in complex analysis [75] for any univalent map that says, in our notation:  r02 D S N C 

X

kjA k j2 :

(17)

kD1

Here SN is the area of the cluster. However, this is just the area of N particles, and is linear in N. Therefore, in leading order, the sum must cancel the N 2/D dependence of r02 . The simplest way this can occur is if every term in the sum goes as N 2/D . This seemed to be incorrect according to the data in [74]: they found that for for the first few k the scaling exponents of the hjA k j2 i were smaller than 2/D. However, later work [38] showed that the asymptotic scaling of all of the coefficients is the same. The apparent difference in the exponents is due to a slow crossover. This effect also seems to resolve a long-standing controversy about the asymptotic scaling behavior of DLA clusters, namely the anomalous behavior of the penetration depth of random walkers as the clusters grow. The anomaly was pointed out numerically by Plischke and Racz [76] soon after the DLA algorithm was introduced. These authors showed numerically that the width of the region where deposition occurred, , (a measure of penetration of random walkers into the cluster) seemed to grow more slowly with N than the mean radius of deposition, Rdep . However for a simple fractal all of the characteristic lengths must scale in the same way, R / N 1/D . Mandelbrot and collaborators [36,37,77] used this and other numerical evidence to suggest that DLA would not be a simple fractal for large N. However, Meakin and Sander [78] gave numerical evidence that the anomaly in the scaling of  is not due to a different exponent, but is a crossover. The controversy is resolved [38] by noting that the penetration depth can be estimated from the Laurent coefficients of f : 1X jA k j2 : (18) 2 D 2 kD1

It is easy to see [38] that this version of the penetration depth can be interpreted as the rms deviation of the position of the end of a field line from those for a disk of radius r0 . Thus an anomaly in the scaling of the Ak is related

to the effect discovered in [76]. Further, a slow crossover in Ak for small k is reasonable geometrically since this is a slow crossover in low moments of the charge. These low moments might be expected to have intrinsically slow dynamics. In [38,79,80] strong numerical evidence was given for the crossover and for universal asymptotic scaling of the Ak . Indeed, many quantities with the interpretation of a length fit an expression of the form:   C 1/D N 1C  ; (19) N where the subdominant correction to scaling is characterized by an exponent  D 0:33˙0:06. This observation verifies the crossover of . The asymptotic value of the penetration depth is measured to be /Rdep ! 0:12. The same value is found in three dimensions [81]. In fact, the crossover probably accounts for the anomalies reported in [36,37,77]. Further, a careful analysis of numerical data shows that the reported multiscaling [82] (an apparent dependence of D on the distance from the center of the cluster) is also a crossover effect. These insights lead to the view that DLA clusters are simple fractals up to a slow crossover. The origin of the crossover exponent,   1/3, is not understood. In three dimensions a similar crossover exponent was found by different techniques [81] with a value of 0:22 ˙ 0:03. Conformal Mapping for Other Discrete Growth Processes The Hastings–Levitov scheme depends on the conformal invariance of the Laplace equation. Bazant and collaborators [83,84] pointed out that the same techniques can be used for more general growth processes. For example, consider growth of an aggregate in a flowing fluid. Now there are two relevant fields, the particle concentration, c, and the velocity of the fluid, v D r (for potential flow). The current associated with c can be written j D cv  rc, where  is a mobility and  a diffusion coefficient. For steady incompressible flow we have, after rescaling: Per

 rc D r 2 c ;

r2

D0:

(20)

Here Pe is the Péclet number, U L/, where U is the value of the velocity far from the aggregate and L its initial size. These equations are conformally invariant, and the problem of flow past a disk in the w plane is easily solved. In [83] growth was accomplished by choosing bumps with probability distribution @c/@n using the method described above. This scheme solves the flow equation past the complex growing aggregate and adds particles according to

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Fractal Growth Processes, Figure 11 An aggregate grown by the advection-diffusion mechanism, Eq. (20) for Pe=10. Courtesy of M. Bazant

their flux at the surface. An example of a pattern of this type is given in Fig. 11. It has long been known that it is possible to treat quasi-static fracture as a growth process similar to DLA [85,86,87]. This is due to the fact that the Lamé equation of elasticity is similar in form to the Laplace equation and the condition for breaking a region of the order of a process zone is related to the stress at the current crack, i. e., boundary values of derivatives of the displacement field. Recent work has exploited this similarity to give yet another application of conformal mapping. For example, for Mode III fracture the quasi-static elastic equation reduces to the Laplace equation, and it is only necessary to replace the growth probability and the boundary conditions in order to use the method of iterated conformal maps [88]. For Mode I and Mode II fracture it is necessary to solve for two analytic functions, but this can also be done [89]. Harmonic Measure The distribution of the boundary values of the normal derivatives of a potential on a electrode of complex shape is called the problem of the harmonic measure. For DLA it is equivalent to the distribution of growth probabilities on the surface of the cluster, or, in other terms, the penetration of random walkers into the cluster. For other complex shapes the problem is still interesting. Its practical significance is that of the study of electrodes [90] or catalytic surfaces. In some cases the harmonic measure has deep relationships to conformal field theory. For the case of the harmonic measure we can interpret the variable ˛ of Eq. (6) as the singularity strength of the electric field near a sharp tip. This is seen as follows: it is well known [91] that near the apex of a wedge-shaped conductor the electric field diverges as r/ˇ 1 where r is the

distance from the tip and the exterior angle of the wedge is ˇ. For example, near a square corner with ˇ D 3/2 there is a r1/3 divergence. Now the quantity pi is the integral over a box of size l. Thus a sequence of boxes centered on the tip will give a power-law l /ˇ D l ˛ . Smaller ˛ means stronger divergence. For fractals that can be treated with the methods of conformal field theory, a good deal is known about the harmonic measure. For example, for a percolation cluster at pc the harmonic measure is completely understood [92]. The Dq are given by a formula: Dq D

1 5 ; Cp 2 24q C 1 C 5

q

1 : 24

(21)

The f (˛) spectrum is easy to compute from this. This formula in good accord with the numerical results of Meakin and collaborators [93] who sampled the measure by firing many random walkers at a cluster. There is an interesting feature of this formula: D0 D 4/3 is the dimension of the support of the measure. This is less than the dimension of a percolation hull, 7/4. There is a large part of the surface of a percolation cluster which is inaccessible to random walkers – the interior surfaces are cut off from the exterior by narrow necks whose widths vanish in the scaling regime. For DLA the harmonic measure is much less well understood. Some results are known. For example, Makarov [94] proved that D1 D 1 under very general circumstances for a two-dimensional harmonic measure. Halsey [95] used Green’s functions for electrostatics to prove the following for DBM models with parameter : ( C 2)  () D D ;

(22)

Here D is the fractal dimension of the cluster. For DLA  D 1, and (1) D 0. Thus (3) D D D 1:71. We introduced the function f (˛) as the Legendre transform of D(q). There is an argument due to Turkevich and Sher [96] which allows us to see a feature of f (˛) directly by giving an estimate of the singularity associated with the most active tip of the growing cluster. Note that the growth rate of the extremal radius of the cluster is related to the fractal dimension because Rext / N 1/D . Suppose we imagine adding one particle per unit time and recall that ptip / (l/Rext )˛tip . Then: dRext dRext dN ˛ D / Rext tip dt dN dt D D 1 C ˛tip  1 C ˛min :

(23)

Since the singularity at the tip is close to being the most active one, we have an estimate of the minimum value of ˛.

Fractal Growth Processes

Fractal Growth Processes, Figure 12 A sketch of the f(˛) curve. Some authors find a maximum slope at a position like that marked by the dot so that the curve ends there. The curve is extended to the real axis with a straight line. This is referred to as a phase transition

There have been a very large number of numerical investigations of D(q); f (˛) for two dimensional DLA; see [7] for a comprehensive list. They proceed either by launching many random walkers, e. g. [93] or by solving the Laplace equation [97], or, most recently, by using the Hastings–Levitov method [98]. The general features of the results are shown in Fig. 12. There is fairly good agreement about the left hand side of the curve which corresponds to large probabilities. The intercept for small ˛ is close to the Turkevich–Sher relation above. The maximum of the curve corresponds to d f /d˛ D q D 0; thus it is the dimension of the support of the measure. This seems to be quite close to D so that the whole surface is accessible to random walkers. There is very little agreement about the right hand side of the curve. It arises from regions with small probabilities which are very hard to estimate. The most reliable current method is that of [98] where conformal maps are manipulated to get at probabilities as small as 1070 , quite beyond the reach of other techniques. Unfortunately, these computations are for rather small clusters (N  50; 000) which are the largest ones that can be made by the Hastings–Levitov method. A particularly active controversy relates to the value of ˛max , if it exists at all; that is, the question is whether there is a maximum value of the slope d/dq. The authors of [98] find ˛max  20. Scaling Theories Our understanding of non-equilibrium fractal growth processes is not very satisfactory compared to that of equi-

librium processes. A long-term goal of many groups has been to find a “theory of DLA” which has the nice features of the renormalization theory of critical phenomena. The result of a good deal of work is that we have theories that give the general features of DLA, but they do not explain things in satisfactory detail. We should note that a mere estimate of the fractal dimension, D, is not what we have in mind. There are several ad hoc estimates in the literature that give reasonable values of D [41,99]. We seek, rather, a theory that allows a good understanding of the fixed point of fractal growth with a description of relevant and irrelevant operators, crossovers, etc. There have been an number of such attempts. In the first section below we describe a semi-numerical scheme which sheds light on the fixed point. Then we look at several attempts at ab initio theory. Scaling of Noise The first question one might ask about DLA growth is the role of noise. It might seem that for a very large cluster the noise of individual particle arrivals should average out. However, this is not the case. Clusters fluctuate on all scales for large N. Further, the noise-free case seems to be unstable. We can see this by asking about the related question of the growth of viscous fingers in two dimensions. As was remarked long ago [51] this sort of growth is always unstable. Numerical simulations of viscous fingering [69] show that any asymmetric initial condition develops into a pattern with a fractal dimension close to that of DLA. DLA organizes its noise to a fixed point exactly as turbulence does. Several authors [79,81,100] have looked at the idea that the noise (measured in some way) flows to a fixed value as clusters grow large. For example, we could characterize the shape fluctuations by measuring the scaled variance of the number of particles necessary to grow to a fixed extremal radius: ˇ p ıN ˇˇ D A : (24) ˇ N Rext This is easy to measure for large clusters. In two dimensions A D :0036 [79]. In [100] it was argued that DLA will be at its fixed point if if one unit of growth acts as a coarse graining of DLA on finer length scales. This amounts to a kind of real-space renormalization. For the original DLA model the scaled noise to grow one unit of length is of order unity. We can reduce it to the fixed point value by noise reduction. The slow approach to the fixed point governed by the exponent  can be in-

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Fractal Growth Processes

Fractal Growth Processes, Figure 13 A small noise-reduced cluster and a larger one with no noise reduction. Noise reduction of the proper size accelerates the approach to the fixed point. From [79]

terpreted as the drift of the noise to its proper value as N grows. Noise reduction was introduced into lattice models by Tang [68]. He kept a counter on each site and only grew there if a certain number of random walkers, m, hit that point. For off-lattice clusters a similar reduction is obtained if shallow bumps of height A are added to the cluster by letting the particles overlap as they stick. We must add m D 1/A particles to advance the growth by one particle diameter [79]. For the Hastings–Levitov method it is equivalent to use a bump map with a small aspect ratio [30]. In either case if we examine a number of sites p whose mean advance is one unit, we will find ıN/N D A. We should expect that if we tune the input value of A to A we will get to asymptotic behavior quickly. This is what is observed in two dimensions [79] and in three dimensions [81]. The amplitude of the crossover, the parameter C in Eq. (19), is smallest for A near the fixed point. In Fig. 13 we show two clusters, a small one grown with noise reduction, and a much larger one with A D 1. They both have the same value of /Rdep , near the asymptotic value of 0.12. Attempts at Theoretical Description The last section described a semi-empirical approach to DLA. The reader may wonder why the techniques of phase transition theory are not simply applicable here. One problem is that one of the favorite methods in equilibrium theory, the "-expansion, cannot be used because DLA has no upper critical dimension [5,101]. To be more precise, in other cluster theories such as percolation there is a dimension, dc such that if d > d c the fractal dimension of the cluster does not change. For example, percolation clusters are 4 dimensional for all di-

mensions above 6. For DLA this is not true because if d is much bigger than D then random walkers will penetrate and fill up the cluster so that its dimension would increase. This results [5] from the fact that for a random walker the number of sites visited in radius R is Nw / R2 . In the same region there are RD sites of the cluster, so the density of cluster sites goes as R Dd . The mean number of intersections is R Dd R2 . Therefore if D C 2 < d there are a vanishing number of intersections, and the cluster will fill up. Thus we must have D  d  2. A related argument [101] sharpens the bound to D  d  1: the fractal dimension increases without limit as d increases. Halsey and collaborators have given a theoretical description based on branch competition [102,103]. This method has been used to give an estimate of Dq for positive q [104]. The idea is to think of two branches that are born from a single site. Then the probability to stick to the first or second is called p1 ; p2 where p1 C p2 D p b is the total probability to stick to that branch. Similarly there are numbers of particles, n1 C n2 D nb . Define x D p1 /p b ; y D n1 /nb . The two variables x; y regulate the branch competition. On the average p1 D dn1 /dnb so we have: dy nb Dxy: (25) dnb The equation of motion for x is assumed to be of similar form: dx nb D g(x; y) : (26) dnb Thus the competition is reduced to a two-dimensional dynamical system with an unstable fixed point at x D y D 1/2 corresponding to the point when the two branches start with equal numbers. The fixed point is unstable because one branch will eventually screen the other. If g is known, then the unstable manifold of this fixed point describes the growth of the dominant branch, and it turns out, by counting the number of particles that attach to the dominant branch, that the eigenvalue of the unstable manifold is 1/D. The starting conditions for the growth are taken to result from microscopic processes that distribute points randomly near the fixed point. The problem remains to find g. This was done several ways: numerically, by doing a large number of simulations of branches that start equally, or in terms of a complicated self-consistent equation [103]. The result is a fractal dimension of 1.66, and a multifractal spectrum that agrees pretty well with direct simulations [104]. Another approach is due to Pietronero and collaborators [105]. It is called the method of fixed scale transformations. It is a real-space method where a small system

Fractal Growth Processes

at one scale is solved essentially exactly, and the behavior at the next coarse-grained scale estimated by assuming that there is a scale-invariant dynamics and estimating the parameters from the fixed-scale solution. The method is much more general than a theory of DLA: in [105] it is applied to directed percolation, the Eden model, sandpile models, and DBM. For DLA the fractal dimension calculated is about 1.6. The rescaled noise (cf. the previous section) comes out to be of order unity rather than the small value, 0.0036, quoted above [106] . The most recent attempt at a fundamental theory is due to Ball and Somfai [71,107]. The idea depends on a mapping from DLA to an instance of the DBM which has different boundary conditions on the growing tip. The scaling of the noise and the multifractal spectrum (for small ˛) are successfully predicted.

Future Directions The DLA model is 27 years old as of this writing. Every year (including last year) there have been about 100 references to the paper. Needless to say, this author has only read a small fraction of them. Space and time prevented presenting here even the interesting ones that I am familiar with. For example, there is a remarkable literature associated with the viscous-fingering problem without surface tension which seems, on the one hand, to describe some facets of experiments [108] and on the other to have deep relationships with the theory of 2d quantum gravity [109,110]. Where this line of work will lead is a fascinating question. There are other examples: I hope that those whose work I have not covered will not feel slighted. There is simply too much going on. A direction that should be pursued is to use the ingenious techniques that have been developed for the DLA problem for problems in different areas; [59,83] are examples of this. It is clear that this field is as lively as ever after 27 years, and will certainly hold more surprises.

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94. Makarov NG (1985) On the distortion of boundary sets under conformal-mappings. P Lond Math Soc 51:369–384 95. Halsey TC (1987) Some consequences of an equation of motion for diffusive growth. Phys Rev Lett 59:2067–2070 96. Turkevich LA, Scher H (1985) Occupancy-probability scaling in diffusion-limited aggregation. Phys Rev Lett 55(9):1026–1029 97. Ball RC, Spivack OR (1990) The interpretation and measurement of the f(alpha) spectrum of a multifractal measure. J Phys A 23:5295–5307 98. Jensen MH, Levermann A, Mathiesen J, Procaccia I (2002) Multifractal structure of the harmonic measure of diffusion-limited aggregates. Phys Rev E 65:046109 99. Ball RC (1986) Diffusion limited aggregation and its response to anisotropy. Physica A: Stat Theor Phys 140(1–2):62–69 100. Barker PW, Ball RC (1990) Real-space renormalization of diffusion-limited aggregation. Phys Rev A 42(10):6289–6292 101. Ball RC, Witten TA (1984) Causality bound on the density of aggregates. Phys Rev A 29(5):2966–2967 102. Halsey TC, Leibig M (1992) Theory of branched growth. Phys Rev A 46:7793–7809 103. Halsey TC (1994) Diffusion-limited aggregation as branched growth. Phys Rev Lett 72(8):1228–1231 104. Halsey TC, Duplantier B, Honda K (1997) Multifractal dimensions and their fluctuations in diffusion-limited aggregation. Phys Rev Lett 78(9):1719–1722 105. Erzan A, Pietronero L, Vespignani A (1995) The fixed-scale transformation approach to fractal growth. Rev Mod Phys 67(3):545–604 106. Cafiero R, Pietronero L, Vespignani A (1993) Persistence of screening and self-criticality in the scale-invariant dynamics of diffusion-limited aggregation. Phys Rev Lett 70(25):3939–3942 107. Ball RC, Somfai E (2003) Diffusion-controlled growth: Theory and closure approximations. Phys Rev E 67(2):021401, part 1 108. Ristroph L, Thrasher M, Mineev-Weinstein MB, Swinney HL (2006) Fjords in viscous fingering: Selection of width and opening angle. Phys Rev E 74(1):015201, part 2 109. Mineev-Weinstein MB, Wiegmann PB, Zabrodin A (2000) Integrable structure of interface dynamics. Phys Rev Lett 84(22):5106–5109 110. Abanov A, Mineev-Weinstein MB, Zabrodin A (2007) Self-similarity in Laplacian growth. Physica D – Nonlinear Phenomena 235(1–2):62–71

Books and Reviews Vicsek T (1992) Fractal Growth Phenomena, 2nd edn. World Scientific, Singapore Meakin P (1998) Fractals, scaling, and growth far from equilibrium. Cambridge University Press, Cambridge Godreche G (1991) Solids far from equilibrium. Cambridge, Cambridge, New York Sander LM (2000) Diffusion limited aggregation, a kinetic critical phenomenon? Contemporary Physics 41:203–218 Halsey TC (2000) Diffusion-limited aggregation: A model for pattern formation. Physics Today 53(4)

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks SIDNEY REDNER Center for Polymer Studies and Department of Physics, Boston University, Boston, USA Article Outline Glossary Definition of the Subject Introduction Solving Resistor Networks Conduction Near the Percolation Threshold Voltage Distribution in Random Networks Random Walks and Resistor Networks Future Directions Bibliography Glossary Conductance (G) The relation between the current I in an electrical network and the applied voltage V: I D GV . Conductance exponent (t) The relation between the conductance G and the resistor (or conductor) concentration p near the percolation threshold: G (p p c ) t . Effective medium theory (EMT) A theory to calculate the conductance of a heterogeneous system that is based on a homogenization procedure. Fractal A geometrical object that is invariant at any scale of magnification or reduction. Multifractal A generalization of a fractal in which different subsets of an object have different scaling behaviors. Percolation Connectivity of a random porous network. Percolation threshold pc The transition between a connected and disconnected network as the density of links is varied. Random resistor network A percolation network in which the connections consist of electrical resistors that are present with probability p and absent with probability 1  p. Definition of the Subject Consider an arbitrary network of nodes connected by links, each of which is a resistor with a specified electrical resistance. Suppose that this network is connected to the leads of a battery. Two natural scenarios are: (a) the “busbar geometry” (Fig. 1), in which the network is connected

to two parallel lines (in two dimensions), plates (in three dimensions), etc., and the battery is connected across the two plates, and (b) the “two-point geometry”, in which a battery is connected to two distinct nodes, so that a current I injected at a one node and the same current withdrawn from the other node. In both cases, a basic question is: what is the nature of the current flow through the network? There are many reasons why current flows in resistor networks have been the focus of more than a century of research. First, understanding currents in networks is one of the earliest subjects in electrical engineering. Second, the development of this topic has been characterized by beautiful mathematical advancements, such as Kirchhoff’s formal solution for current flows in networks in terms of tree matrices [52], symmetry arguments to determine the electrical conductance of continuous two-component media [10,29,48,69,74], clever geometrical methods to simplify networks [33,35,61,62], and the use of integral transform methods to solve node voltages on regular networks [6,16,94,95]. Third, the nodes voltages of a network through which a steady electrical current flows are harmonic [26]; that is, the voltage at a given node is a suitably-weighted average of the voltages at neighboring nodes. This same harmonicity also occurs in the probability distribution of random walks. Consequently, there are deep connections between the probability distribution of random walks on a given network and the node voltages on the same network [26]. Another important theme in the subject of resistor networks is the essential role played by randomness on current-carrying properties. When the randomness is weak, effective medium theory [10,53,54,55,57,74] is appropriate to characterize how the randomness affects the conductance. When the randomness is strong, as embodied by a network consisting of a random mixture of resistors and insulators, this random resistor network undergoes a transition between a conducting phase and an insulating phase when the resistor concentration passes through a percolation threshold [54]. The feature underlying this phase change is that for a small density of resistors, the network consists of disconnected clusters. However, when the resistor density passes through the percolation threshold, a macroscopic cluster of resistors spans the system through which current can flow. Percolation phenomenology has motivated theoretical developments, such as scaling, critical point exponents, and multifractals that have advanced our understanding of electrical conduction in random resistor networks. This article begins with an introduction to electrical current flows in networks. Next, we briefly discuss ana-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 1 Resistor networks in a the bus-bar geometry, and b the two-point geometry

lytical methods to solve the conductance of an arbitrary resistor network. We then turn to basic results related to percolation: namely, the conduction properties of a large random resistor network as the fraction of resistors is varied. We will focus on how the conductance of such a network vanishes as the percolation threshold is approached from above. Next, we investigate the more microscopic current distribution within each resistor of a large network. At the percolation threshold, this distribution is multifractal in that all moments of this distribution have independent scaling properties. We will discuss the meaning of multifractal scaling and its implications for current flows in networks, especially the largest current in the network. Finally, we discuss the relation between resistor networks and random walks and show how the classic phenomena of recurrence and transience of random walks are simply related to the conductance of a corresponding electrical network. The subject of current flows on resistor networks is a vast subject, with extensive literature in physics, mathematics, and engineering journals. This review has the modest goal of providing an overview, from my own myopic perspective, on some of the basic properties of random resistor networks near the percolation threshold. Thus many important topics are simply not mentioned and the reference list is incomplete because of space limitations. The reader is encouraged to consult the review articles listed in the reference list to obtain a more complete perspective. Introduction In an elementary electromagnetism course, the following classic problem has been assigned to many generations

of physics and engineering students: consider an infinite square lattice in which each bond is a 1 ohm resistor; equivalently, the conductance of each resistor (the inverse resistance) also equals 1. There are perfect electrical connections at all vertices where four resistors meet. A current I is injected at one point and the same current I is extracted at a nearest-neighbor lattice point. What is the electrical resistance between the input and output? A more challenging question is: what is the resistance between two diagonal points, or between two arbitrary points? As we shall discuss, the latter questions can be solved elegantly using Fourier transform methods. For the resistance between neighboring points, superposition provides a simple solution. Decompose the current source and sink into its two constituents. For a current source I, symmetry tells us that a current I/4 flows from the source along each resistor joined to this input. Similarly, for a current sink I, a current I/4 flows into the sink along each adjoining resistor. For the source/sink combination, superposition tells us that a current I/2 flows along the resistor directly between the source and sink. Since the total current is I, a current of I/2 flows indirectly from source to sink via the rest of the lattice. Because the direct and indirect currents between the input and output points are the same, the resistance of the direct resistor and the resistance of rest of the lattice are the same, and thus both equal to 1. Finally, since these two elements are connected in parallel, the resistance of the infinite lattice between the source and the sink equals 1/2 (conductance 2). As we shall see in Sect. “Effective Medium Theory”, this argument is the basis for constructing an effective medium theory for the conductance of a random network. More generally, suppose that currents I i are injected at each node of a lattice network (normally many of these

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currents are zero and there would be both positive and negative currents in the steady state). Let V i denote the voltage at node i. Then by Kirchhoff’s law, the currents and voltages are related by Ii D

X

P P Fi D j gi j F j/ j g i j ! We then exploit this connection to provide insights about random walks in terms of known results about resistor networks and vice versa. Solving Resistor Networks

g i j (Vi  Vj ) ;

(1)

j

where g ij is the conductance of link ij, and the sum runs over all links ij. This equation simply states that the current flowing into a node by an external current source equals the current flowing out of the node along the adjoining resistors. The right-hand side of Eq. (1) is a discrete Laplacian operator. Partly for this reason, Kirchhoff’s law has a natural connection to random walks. At nodes where the external current is zero, the node voltages in Eq. (1) satisfy P 1X j g i j Vj ! Vj : (2) Vi D P z j gi j j

The last step applies if all the conductances are identical; here z is the coordination number of the network. Thus for steady current flow, the voltage at each unforced node equals the weighted average of the voltages at the neighboring sites. This condition defines V i as a harmonic function with respect to the weight function g ij . An important general question is the role of spatial disorder on current flows in networks. One important example is the random resistor network, where the resistors of a lattice are either present with probability p or absent with probability 1  p [54]. Here the analysis tools for regular lattice networks are no longer applicable, and one must turn to qualitative and numerical approaches to understand the current-carrying properties of the system. A major goal of this article is to outline the essential role that spatial disorder has on the current-carrying properties of a resistor network by such approaches. A final issue that we will discuss is the deep relation between resistor networks and random walks [26,63]. Consider a resistor network in which the positive terminal of a battery (voltage V D 1) is connected to a set of boundary nodes, defined to be BC ), and where a disjoint set of boundary nodes B are at V D 0. Now suppose that a random walk hops between nodes of the same geometrical network in which the probability of hopping from P node i to node j in a single step is g i j / k g i k , where k is one of the neighbors of i and the boundary sets are absorbing. For this random walk, we can ask: what is the probability F i for a walk to eventually be absorbed on BC when it starts at node i? We shall show in Sect. “Random Walks and Resistor Networks” that F i satisfies Eq. (2):

Fourier Transform The translational invariance of an infinite lattice resistor network with identical bond conductances g i j D 1 cries out for applying Fourier transform methods to determine node voltages. Let’s study the problem mentioned previously: what is the voltage at any node of the network when a unit current enters at some point? Our discussion is specifically for the square lattice; the extension to other lattices is straightforward. For the square lattice, we label each site i by its x; y coordinates. When a unit current is injected at r0 D (x0 ; y0 ), Eq. (1) becomes  ıx;x 0 ı y;y0 D V(x C 1; y) C V(x  1; y) C V(x; y C 1) C V(x; y  1)  4V(x; y) ;

(3)

which clearly exposes the second difference operator of the discrete Laplacian. To find the node voltages, we deP fine V(k) D r V(r) e ikr and then we Fourier transform Eq. (3) to convert this infinite set of difference equations into the single algebraic equation V(k) D

e ikr0 : 4  2(cos k x C cos k y )

(4)

Now we calculate V(r) by inverting the Fourier transform 1 V(r) D (2)2

Z Z  

eik(rr0 ) dk : 4  2(cos k x C cos k y )

(5)

Formally, at least, the solution is trivial. However, the integral in the inverse Fourier transform, known as a Watson integral [96], is non-trivial, but considerable understanding has gradually been developed for evaluating this type of integral [6,16,94,95,96]. For a unit input current at the origin and a unit sink of current at r0 , the resistance between these two points is V (0)  V(r0 ), and Eq. (5) gives R D V(0)  V(r0 ) Z Z (1  e ikr0 ) 1 dk : D (2)2 4  2(cos k x C cos k y )

(6)

 

Tables for the values of R for a set of closely-separated input and output points are given in [6,94]. As some specific examples, for r0 D (1; 0), R D 1/2, thus reproducing

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

the symmetry argument result. For two points separated by a diagonal, r0 D (1; 1), R D 2/. For r0 D (2; 0), R D 24/. Finally, for two points separated by a knight’s move, r0 D (2; 1), R D 4/  1/2.

method is prohibitively inefficient for larger networks because the number of spanning trees grows exponentially with network size. Potts Model Connection

Direct Matrix Solution Another way to solve Eq. (1), is to recast Kirchhoff’s law as the matrix equation Ii D

N X

G i j Vj ;

i D 1; 2; : : : ; N

(7)

jD1

where the elements of the conductance matrix are: (P k¤i g i k ; i D j Gi j D i ¤ j: g i j ; The conductance matrix is an example of a tree matrix, as G has the property that the sum of any row or any column equals zero. An important consequence of this tree property is that all cofactors of G are identical and are equal to the spanning tree polynomial [43]. This polynomial is obtained by enumerating all possible tree graphs (graphs with no closed loops) on the original electrical network that includes each node of the network. The weight of each spanning tree is simply the product of the conductances for each bond in the tree. Inverting Eq. (7), one obtains the voltage V i at each node i in terms of the external currents I j ( j D 1; 2; : : : ; N) and the conductances gij . Thus the two-point resistance Rij between two arbitrary (not necessarily connected) nodes i and j is then given by R i j D (Vi  Vj ) / I, where the network is subject to a specified external current; for example, for the two-point geometry, I i D 1, I j D 1, and I k D 0 for k ¤ i; j. Formally, the two-point resistance can be written as [99] jG (i j) j ; Ri j D jG ( j) j

(8)

where jG ( j) j is the determinant of the conductance matrix with the jth row and column removed and jG (i j) j is the determinant with the ith and jth rows and columns removed. There is a simple geometric interpretation for this conductance matrix inversion. The numerator is just the spanning tree polynomial for the original network, while the denominator is the spanning tree polynomial for the network with the additional constraint that nodes i and j are identified as a single point. This result provides a concrete prescription to compute the conductance of an arbitrary network. While useful for small networks, this

The matrix solution of the resistance has an alternative and elegant formulation in terms of the spin correlation function of the q-state Potts model of ferromagnetism in the q ! 0 limit [88,99]. This connection between a statistical mechanical model in a seemingly unphysical limit and an enumerative geometrical problem is one of the unexpected charms of statistical physics. Another such example is the n-vector model, in which ferromagnetically interacting spins “live” in an n-dimensional spin space. In the limit n ! 0 [20], the spin correlation functions of this model are directly related to all self-avoiding walk configurations. In the q-state Potts model, each site i of a lattice is occupied by a spin si that can assume one of q discrete values. The Hamiltonian of the system is X H D J ıs i ;s j ; i; j

where the sum is over all nearest-neighbor interacting spin pairs, and ıs i ;s j is the Kronecker delta function (ıs i ;s j D 1 if s i D s j and ıs i ;s j D 0 otherwise). Neighboring aligned spin pairs have energy J, while spin pairs in different states have energy zero. One can view the spins as pointing from the center to a vertex of a q-simplex, and the interaction energy is proportional to the dot product of two interacting spins. The partition function of a system of N spins is X ˇ P Jı ZN D e i; j s i ;s j ; (9) fsg

where the sum is over all 2N spin states fsg. To make the connection to resistor networks, notice that: (i) the exponential factor associated with each link ij in the partition function takes the values 1 or eˇ J , and (ii) the exponential of the sum can be written as the product XY ZN D (1 C vıs i ;s j ) ; (10) fs i g i; j

with v D tanh ˇJ. We now make a high-temperature (small-v) expansion by multiplying out the product in (10) to generate all possible graphs on the lattice, in which each bond carries a weight vıs i ;s j . Summing over all states, the spins in each disjoint cluster must be in the same state, and the last sum over the common state of all spins leads to

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each cluster being weighted by a factor of q. The partition function then becomes X ZN D qN c v N b ; (11) graphs

where N c is the number of distinct clusters and N b is the total number of bonds in the graph. It was shown by Kasteleyn and Fortuin [47] that the limit q D 1 corresponds to the percolation problem when one chooses v D p / (1  p), where p is the bond occupation probability in percolation. Even more striking [34], if one chooses v D ˛q1/2 , where ˛ is a constant, then lim q!0 Z N /q(NC1)/2 D ˛ N1 TN , where T N is again the spanning tree polynomial; in the case where all interactions between neighboring spins have the same strength, then the polynomial reduces to the number of spanning trees on the lattice. It is because of this connection to spanning trees that the resistor network and Potts model are intimately connected [99]. In a similar vein, one can show that the correlation function between two spins at nodes i and j in the Potts model is simply related to the conductance between these same two nodes when the interactions J ij between the spins at nodes i and j are equal to the conductances g ij between these same two nodes in the corresponding resistor network [99]. -Y and Y- Transforms In elementary courses on circuit theory, one learns how to combine resistors in series and parallel to reduce the complexity of an electrical circuit. For two resistors with resistances R1 and R2 in series, the net resistance is R D R1 C R2 , while for resistors in parallel, the net resistance  1 is R D R11 C R22 . These rules provide the resistance of a network that contains only series and parallel connections. What happens if the network is more complicated? One useful way to simplify such a network is by the Y and Y- transforms that was apparently first discovered by Kennelly in 1899 [50] and applied extensively since then [33,35,61,62,81]. The basic idea of the -Y transform is illustrated in Fig. 2. Any triangular arrangement of resistors R12 , R13 , and R23 within a larger circuit can be replaced by a star, with resistances R1 , R2 , and R3 , such that all resistances between any two points among the three vertices in the triangle and the star are the same. The conditions that all two-point resistances are the same are: (R1 C R2 ) D



1 R12

1 1

C (R13 C R23 )  a12

C cyclic permutations :

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 2 Illustration of the -Y and Y- transforms

Solving for R1 ; R2 , and R3 gives R1 D 12 (a12  a23 C a13 ) + cyclic permutations; the explicit result in terms of the Rij is: R1 D

R12 R13 R12 C R13 C R23

C cyclic permutations; (12)

as well as the companion result for the conductances G i D R1 i : G1 D

G12 G13 C G12 G23 C G13 G23 Ccyclic permutations: G23

These relations allow one to replace any triangle by a star to reduce an electrical network. However, sometimes we need to replace a star by a triangle to simplify a network. To construct the inverse Y- transform, notice that the -Y transform gives the resistance ratios R1 /R2 D R13 /R23 + cyclic permutations, from which R13 D R12 (R3 /R2 ) and R23 D R12 (R3 /R1 ). Substituting these last two results in Eq. (12), we eliminate R13 and R23 and thus solve for R12 in terms of the Ri : R12 D

R1 R2 C R1 R3 C R2 R3 Ccyclic permutations; (13) R3

and similarly for G i j D R1 i j . To appreciate the utility of the -Y and Y- transforms, the reader is invited to apply them on the Wheatstone bridge. When employed judiciously and repeatedly, these transforms systematically reduce planar lattice circuits to a single bond, and thus provide a powerful approach to calculate the conductance of large networks near the percolation threshold. We will return to this aspect of the problem in Sect. “Conductance Exponent”. Effective Medium Theory Effective medium theory (EMT) determines the macroscopic conductance of a random resistor network by a homogenization procedure [10,53,54,55,57,74] that is reminiscent of the Curie–Weiss effective field theory of mag-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

netism. The basic idea in EMT is to replace the random network by an effective homogeneous medium in which the conductance of each resistor is determined self-consistently to optimally match the conductances of the original and homogenized systems. EMT is quite versatile and has been applied, for example, to estimate the dielectric constant of dielectric composites and the conductance of conducting composites. Here we focus on the conductance of random resistor networks, in which each resistor (with conductance g 0 ) is present with probability p and absent with probability 1  p. The goal is to determine the conductance as a function of p. To implement EMT, we first replace the random network by an effective homogeneous medium in which each bond has the same conductance g m (Fig. 3). If a voltage is applied across this effective medium, there will be a potential drop V m and a current I m D g m Vm across each bond. The next step in EMT is to assign one bond in the effective medium a conductance g and adjust the external voltage to maintain a fixed total current I passing through the network. Now an additional current ıi passes through the conductor g. Consequently, a current ıi must flow through one terminal of g to the other terminal via the remainder of the network (Fig. 3). This current perturbation leads to an additional voltage drop ıV across g. Thus the current-voltage relations for the marked bond and the remainder of the network are I m C ıi D g(Vm C ıV ) ıi D G ab ıV ;

(14)

where Gab is the conductance of the rest of the lattice between the terminals of the conductor g. The last step in EMT is to require that the mean value ıV averaged over the probability distribution of individual bond conductances is zero. Thus the effective medium “matches” the current-carrying properties of the original network. Solving Eq. (14) for ıV , and using the probability distribution P(g) D pı(gg0 )C(1p)ı(g) appropriate

for the random resistor network, we obtain

 (g m  g0 )p g m (1  p) C D0: hıVi D Vm (G ab C g0 ) G ab

It is now convenient to write Gab D ˛g m , where ˛ is a lattice-dependent constant of the order of one. With this definition, Eq. (15) simplifies to g m D go

p(1 C ˛)  1 : ˛

(16)

The value of ˛ – the proportionality constant for the conductance of the initial lattice with a single bond removed – can usually be determined by a symmetry argument of the type presented in Sect. “Introduction to Current Flows”. For example, for the triangular lattice (coordination number 6), the conductance Gab D 2g m and ˛ D 2. For the hypercubic lattice in d dimensions (coordination number z D 2d ), Gab D ((z  2)/2)g m . The main features of the effective conductance g m that arises from EMT are: (i) the conductance vanishes at a lattice-dependent percolation threshold p c D 1/(1 C ˛); for the hypercubic lattice ˛ D (z  2)/2 and the percolation threshold p c D 2/z D 21d (fortuitously reproducing the exact percolation threshold in two dimensions); (ii) the conductance varies linearly with p and vanishes linearly in p  p c as p approaches pc from above. The linearity of the effective conductance away from the percolation threshold accords with numerical and experimental results. However, EMT fails near the percolation threshold, where large fluctuations arise that invalidate the underlying assumptions of EMT. In this regime, alternative methods are needed to estimate the conductance. Conduction Near the Percolation Threshold Scaling Behavior EMT provides a qualitative but crude picture of the current-carrying properties of a random resistor network. While EMT accounts for the existence of a percolation transition, it also predicts a linear dependence of the conductance on p. However, near the percolation threshold it is well known that the conductance varies non-linearly in p  p c near pc [85]. This non-linearity defines the conductance exponent t by G (p  p c ) t

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 3 Illustration of EMT. (left) The homogenized network with conductances gm and one bond with conductance g. (right) The equivalent circuit to the lattice

(15)

p # pc ;

(17)

and much research on random resistor networks [85] has been performed to determine this exponent. The conductance exponent generically depends only on the spatial dimension of the network and not on any other details (a notable exception, however, is when link resistances are

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 4 (left) Realization of bond percolation on a 25  25 square lattice at p=0.505. (right) Schematic picture of the nodes (shaded circles), links and blobs picture of percolation for p & pc

broadly distributed, see [40,93]). This universality is one of the central tenets of the theory of critical phenomena [64, 83]. For percolation, the mechanism underlying universality is the absence of a characteristic length scale; as illustrated in Fig. 4, clusters on all length scales exist when a network is close to the percolation threshold. The scale of the largest cluster defines the correlation length  by  (p c  p) as p ! p c . The divergence in  also applies for p > p c by defining the correlation length as the typical size of finite clusters only (Fig. 4), thus eliminating the infinite percolating cluster from consideration. At the percolation threshold, clusters on all length scales exist, and the absence of a characteristic length implies that the singularity in the conductance should not depend on microscopic variables. The only parameter remaining upon which the conductance exponent t can depend upon is the spatial dimension d [64,83]. As typifies critical phenomena, the conductance exponent has a constant value in all spatial dimensions d > d c , where dc is the upper critical dimension which equals 6 for percolation [22]. Above this critical dimension, mean-field theory (not to be confused with EMT) gives the correct values of critical exponents. While there does not yet exist a complete theory for the dimension dependence of the conductance exponent below the critical dimension, a crude but useful nodes, links, and blobs picture of the infinite cluster [21,82,84] provides partial information. The basic idea of this picture is that for p & p c , a large system has an irregular network-like

topology that consists of quasi-linear chains that are separated by the correlation length  (Fig. 4). For a macroscopic sample of linear dimension L with a bus bar-geometry, the percolating cluster above pc then consists of (L/)d1 statistically identical chains in parallel, in which each chain consists of L/ macrolinks in series, and the macrolinks consists of nested blob-like structures. The conductance of a macrolink is expected to vanish as (p  p c ) , with  a new unknown exponent. Although a theory for the conductance of a single macrolink, and even a precise definition of a macrolink, is still lacking, the nodes, links, and blobs picture provides a starting point for understanding the dimension dependence of the conductance exponent. Using the rules for combining parallel and series conductances, the conductance of a large resistor network of linear dimension L is then G(p; L)

 d1 L (p  p c )  L/

Ld2 (p  p c )(d2)C :

(18)

In the limit of large spatial dimension, we expect that a macrolink is merely a random walk between nodes. Since the spatial separation between nodes is , the number of bonds in the macrolink, and hence its resistance, scales as  2 [92]. Using the mean-field result  (p  p c )1/2 , the resistance of the macrolink scales as (p  p c )1 and thus the exponent  D 1. Using the mean-field exponents

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

 D 1/2 and  D 1 at the upper critical dimension of d c D 6, we then infer the mean-field value of the conductance exponent t D 3 [22,91,92]. Scaling also determines the conductance of a finitesize system of linear dimension L exactly at the percolation threshold. Although the correlation length formally diverges when p  p c D 0,  is limited by L in a finite system of linear dimension L. Thus the only variable upon which the conductance can depend is L itself. Equivalently, deviations in p  p c that are smaller than L1/ cannot influence critical behavior because  can never exceed L. Thus to determine the dependence of a singular observable for a finite-size system at pc , we may replace (pp c ) by L1/ . By this prescription, the conductance at pc of a large finite-size system of linear dimension L becomes G(p c ; L) Ld2 (L1/ )(d2)C L/ :

(19)

In this finite-size scaling [85], we fix the occupation probability to be exactly at pc and study the dependence of an observable on L to determine percolation exponents. This approach provides a convenient and more accurate method to determine the conductance exponent compared to studying the dependence of the conductance of a large system as a function of p  p c . Conductance Exponent In percolation and in the random resistor network, much effort has been devoted to computing the exponents that characterize basic physical observables – such as the correlation length  and the conductance G – to high precision. There are several reasons for this focus on exponents. First, because of the universality hypothesis, exponents are a meaningful quantifier of phase transitions. Second, various observables near a phase transition can sometimes be related by a scaling argument that leads to a corresponding exponent relation. Such relations may provide a decisive test of a theory that can be checked numerically. Finally, there is the intellectual challenge of developing accurate numerical methods to determine critical exponents. The best such methods have become quite sophisticated in their execution. A seminal contribution was the “theorists’ experiment” of Last and Thouless [58] in which they punched holes at random in a conducting sheet of paper and measured the conductance of the sheet as a function of the area fraction of conducting material. They found that the conductance vanished faster than linearly with (p  p c ); here p corresponds to the area fraction of the conductor. Until this experiment, there was a sentiment that the conductance should be related to the fraction of material in

the percolating cluster [30] – the percolation probability P(p) – a quantity that vanished slower than linearly with (p  p c ). The reason for this disparity is that in a resistor network, much of the percolating cluster consists of dangling ends – bonds that carry no current – and thus make no contribution to the conductance. A natural geometrical quantity that ought to be related to the conductance is the fraction of bonds B(p) in the conducting backbone – the subset of the percolating cluster without dangling ends. However, a clear relation between the conductivity and a geometrical property of the backbone has not yet been established. Analytically, there are primary two methods that have been developed to compute the conductance exponent: the renormalization group [44,86,87,89] and low-density series expansions [1,2,32]. In the real-space version of the renormalization group, the evolution of conductance distribution under length rescaling is determined, while the momentum-space version involves a diagrammatic implementation of this length rescaling in momentum space. The latter is a perturbative approach away from mean-field theory in the variable 6  d that become exact as d ! 6. Considerable effort has been devoted to determining the conductance exponent by numerical and algorithmic methods. Typically, the conductance is computed for networks of various linear dimensions L at p D p c , and the conductance exponent is extracted from the L dependence of the conductance, which should vanish as L/ . An exact approach, but computationally impractical for large networks, is Gauss elimination to invert the conductance matrix [79]. A simple approximate method is Gauss relaxation [59,68,80,90,97] (and its more efficient variant of Gauss–Seidel relaxation [71]). This method uses Eq. (2) as the basis for an iteration scheme, in which the voltage V i at node i at the nth update step is computed from (2) using the values of V j at the (n  1)st update in the right-hand side of this equation. However, one can do much better by the conjugate gradient algorithm [27] and speeding up this method still further by Fourier acceleration methods [7]. Another computational approach is based on the node elimination method, in which the -Y and Y- transforms are used to successively eliminate bonds from the network and ultimately reduce a large network to a single bond [33,35,62]. In a different vein, the transfer matrix method has proved to be extremely accurate and efficient [24,25,70,100]. The method is based on building up the network one bond at a time and immediately calculating the conductance of the network after each bond addition. This method is most useful when applied to very long strips of transverse dimension L so that a single realization gives an accurate value for the conductance.

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

As a result of these investigations, as well as by series expansions for the conductance, the following exponents have been found. For d D 2, where most of the computational effort has been applied, the best estimate [70] for the exponent t (using  D t in d D 2 only) is t D 1:299˙0:002. One reason for the focus on two dimensions is that early estimates for t were tantalizingly close to the correlation length exponent  that is now known to exactly equal 4/3 [23]. Another such connection was the Alexander–Orbach conjecture [5], which predicted t D 91/72 D 1:2638 : : :, but again is incompatible with the best numerical estimate for t. In d D 3, the best available numerical estimate for t appears to be t D 2:003 ˙ 0:047 [12,

36], while the low concentration series method gives an equally precise result of t D 2:02 ˙ 0:05 [1,2]. These estimates are just compatible with the rigorous bound that t  2 in d D 3 [37,38]. In greater than three dimensions, these series expansions give t D 2:40 ˙ 0:03 for d D 4 and t D 2:74 ˙ 0:03 for d D 5, and the dimension dependence is consistent with t D 3 when d reaches 6. Voltage Distribution in Random Networks Multifractal Scaling While much research has been devoted to understanding the critical behavior of the conductance, it was re-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 5 The voltage distribution on an L  L square lattice random resistor network at the percolation threshold for a L D 4 (exact), b L D 10, and c L D 130. The latter two plots are based on simulation data. For L D 4, a number of peaks, that correspond to simple rational fractions of the unit potential drop, are indicated. Also shown are the average voltage over all realizations, Vav , the most probable voltage, Vmp , and the average of the minimum voltage in each realization, hVmin i. [Reprinted from Ref. [19]]

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

alized that the distribution of voltages across each resistor of the network was quite rich and exhibited multifractal scaling [17,19,72,73]. Multifractality is a generalization of fractal scaling in which the distribution of an observable is sufficiently broad that different moments of the distribution scale independently. Such multifractal scaling arises in phenomena as diverse as turbulence [45,66], localization [13], and diffusion-limited aggregation [41,42]. All these diverse examples showed scaling properties that were much richer than first anticipated. To make the discussion of multifractality concrete, consider the example of the Maxwell–Boltzmann velocity distribution of a one-dimensional ideal gas r m 2 1 2 2 P(v) D ev /2v th ; emv /2k B T  q 2 kB T 2 2v th where kB is Boltzmann’s constant,pm is the particle mass, T is the temperature, and v th D kB T/m is the characteristic thermal velocity. The even integer moments of the velocity distribution are  n   p(n) h(v 2 )n i / v th2  v th2 : Thus a single velocity scale, vth , characterizes all positive moments of the velocity distribution. Alternatively, the exponent p(n) is linear in n. This linear dependence of successive moment exponents characterizes single-parameter scaling. The new feature of multifractal scaling is that a wide range of scales characterizes the voltage distribution (Fig. 5). As a consequence, the moment exponent p(n) is a non-linear function of n. One motivation for studying the voltage distribution is its relation to basic aspects of electrical conduction. If a voltage V D 1 is applied across a resistor network, then the conductance G and the total current flow I are equal: I D G. Consider now the power dissipated through the network P D IV D GV 2 ! G. We may also compute the dissipated power by adding up these losses in each resistor to give X X X PDGD g i j Vi2j ! Vi2j D V 2 N(V) : (20) ij

ij

tribution, we define the family of exponents p(k) for the scaling dependence of the voltage distribution at p D p c by X M(k)  N(V)V k Lp(k)/ : (21) V

Since M(2) is just the network conductance, p(2) D . Other moments of the voltage distribution also have simple interpretations. For example, hV 4 i is related to the magnitude of the noise in the network [9,73], while hV k i for k ! 1 weights the bonds with the highest currents, or the “hottest” bonds of the network, most strongly, and they help understand the dynamics of fuse networks of failure [4,18]. On the other hand, negative moments weight low-current bonds more strongly and emphasize the low-voltage tail of the distribution. For example, M(1) characterizes hydrodynamic dispersion [56], in which passive tracer particles disperse in a network due to a multiplicity of network paths. In hydrodynamics dispersion, the transit time across each bond is proportional to the inverse of the current in the bond, while the probability for tracer to enter a bond is proportional to the entering current. As a result, the kth moment of the transit time distribution varies as M(k C 1), so that the quantity that quantifies dispersion, ht 2 i  hti2 , scales as M(1). A simple fractal model [3,11,67] of the conducting backbone (Fig. 6) illustrates the multifractal scaling of the voltage distribution near the percolation threshold [17]. To obtain the Nth-order structure, each bond in the (N  1)st iteration is replaced by the first-order structure. The resulting fractal has a hierarchical embedding of links and blobs that captures the basic geometry of the percolating backbone. Between successive generations, the length scale changes by a factor of 3, while the number of bonds

V

Here g i j D 1 is the conductance of resistor ij, and V ij is the corresponding voltage drop across this bond. In the last equality, N(V) is the number of resistors with a voltage drop V. Thus the conductance is just the second moment of the distribution of voltage drops across each bond in the network. From the statistical physics perspective it is natural to study other moments of the voltage distribution and the voltage distribution itself. Analogous to the velocity dis-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 6 The first few iterations of a hierarchical model

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

changes by a factor of 4. Defining the fractal dimension df as the scaling relation between mass (M D 4 N ) and the length scale (` D 3 N ) via M `d f , gives a fractal dimension d f D ln 4/ ln 3. Now let’s determine the distribution of voltage drops across the bonds. If a unit voltage is applied at the opposite ends of a first-order structure (N D 1) and each bond is a 1 ohm resistor, then the two resistors in the central bubble each have a voltage drop of 1/5, while the two resistors at the ends have a voltage drop 2/5. In an Nth-order hierarchy, the voltage of any resistor is the product of these two factors, with number of times each factor occurs dependent on the level of embedding of a resistor within the blobs. It is a simple exercise to show that the voltage distribution is [17] ! N N N(V( j)) D 2 ; (22) j where the voltage V(j) can take the values 2 j /5 N (with j D 0; 1; : : : ; N). Because j varies logarithmically in V, the voltage distribution is log binomial [75]. Using this distribution in Eq. (21), the moments of the voltage distribution are " #N 2(1 C 2 k ) M(k) D : (23) 5k In particular, the average voltage, M(1)/M(0)  Vav equals ((3/2)/5) N , which ispvery different from the most probable voltage, Vmp D ( 2/5) N as N ! 1. The underlying multiplicativity of the bond voltages is the ultimate source of the large disparity between the average and most probable values. To calculate the moment exponent p(k), we first need to relate the iteration index N to a physical length scale. For percolation, the appropriate relation is based on Coniglio’s theorem [15], which is a simple but profound statement about the structure of the percolating cluster. This theorem states that the number of singly-connected bonds in a system of linear dimension L, Ns , varies as L1/ . Singly-connected bonds are those that would disconnect the network if they were cut. An equivalent form of the theorem is Ns D @p0 /@p, where p0 is the probability that a spanning cluster exists in the system. This relation reflects the fact that when p is decreased slightly, p0 changes only if a singly-connected bond happens to be deleted. In the Nth-order hierarchy, the number of such singlyconnected links is simply 2N . Equating these two gives an effective linear dimension, L D 2 N . Using this relation in (23), the moment exponent p(k) is iı h p(k) D k  1 C k ln (5/4)  ln(1 C 2k ) ln 2 : (24)

Because each p(k) is independent, the moments of the voltage distribution are characterized by an infinite set of exponents. Equation (24) is in excellent agreement with numerical data for the voltage distribution in two-dimensional random resistor networks at the percolation threshold [19]. A similar multifractal behavior was also found for the voltage distribution of the resistor network at the percolation threshold in three dimensions [8]. Maximum Voltage An important aspect of the voltage distribution, both because of its peculiar scaling properties [27] and its application to breakdown problems [18,27], is the maximum voltage in a network. The salient features of this maximum voltage are: (i) logarithmic scaling as a function of system size [14,27,28,60,65], and (ii) non-monotonic dependence on the resistor concentration p [46]. The former property is a consequence of the expected size of the largest defect in the network that gives maximal local currents. Here, we use the terms maximum local voltage and maximum local current interchangeably because they are equivalent. To find the maximal current, we first need to identify the optimal defects that lead to large local currents. A natural candidate is an ellipse [27,28] with major and minor axes a and b (continuum), or its discrete analog of a linear crack (hyperplanar crack in greater than two dimensions) in which n resistors are missing (Fig. 7). Because current has to detour around the defect, the local current at the ends of the defect is magnified. For the continuum problem, the current at the tip of the ellipse is Itip D I0 (1Ca / b), where I 0 is the current in the unperturbed system [27]. For the maximum current in the lattice system, one must integrate the continuum current over a one lattice spacing and identify a/b with n [60]. This approach gives the maximal current at the tip of a crack Imax / (1 C n1/2 ) in two dimensions and as Imax / (1 C n1/2(d1) ) in d dimensions. Next, we need to find the size of the largest defect, which is an extreme-value statistics exercise [39]. For a linear crack, each broken bond occurs with probability 1  p, so that the probability for a crack of length n is (1  p)n  ean , with a D  ln(1  p). In a network of d , we estimate the size of the largest defect by volume R 1 Lan d L n max e dx D 1; that is, there exists of the order of one defect of size nmax or larger in the network [39]. This estimate gives nmax varying as ln L. Combining this result with the current at the tip of a crack of length n, the largest current in a system of linear dimension L scales as (ln L)1/2(d1) . A more thorough analysis shows, however, that a single crack is not quite optimal. For a continuum two-com-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 7 Defect configurations in two dimensions. a An ellipse and its square lattice counterpart, b a funnel, with the region of good conductor shown shaded, and a 2-slit configuration on the square lattice

ponent network with conductors of resistance 1 with probability p and with resistance r > 1 with probability 1  p, the configuration that maximizes the local current is a funnel [14,65]. For a funnel of linear dimension `, the maximum current at the apex of the funnel is proportional to `1 , where  D (4/) tan1 (r1/2 ) [14,65]. The probability to find a funnel of linear dimension ` now scales as 2 eb` (exponentially in its area), with b a constant. By the same extreme statistics reasoning given above, the size of the largest funnel in a system of linear dimension L then scales as (ln L)1/2 , and the largest expected current correspondingly scales as (ln L)(1)/2 . In the limit r ! 1, where one component is an insulator, the optimal discrete configuration in two dimensions becomes two parallel slits, each of length n, between which a single resistor remains [60]. For this two-slit configuration, the maximum current is proportional to n in two dimensions, rather than n1/2 for the single crack. Thus the maximal current in a system of linear dimension L scales as ln L rather than as a fractional power of ln L. The p dependence of the maximum voltage is intriguing because it is non-monotonic. As p, the fraction of occupied bonds, decreases from 1, less total current flows (for a fixed overall voltage drop) because the conductance is decreasing, while local current in a funnel is enhanced because such defects grow larger. The competition between these two effects leads to Vmax attaining its

peak at ppeak above the percolation threshold that only slowly approaches pc as L ! 1. An experimental manifestation of this non-monotonicity in Vmax occurred in a resistor-diode network [77], where the network reproducibly burned (solder connections melting and smoking) when p ' 0:77, compared to a percolation threshold of p c ' 0:58. Although the directionality constraint imposed by diodes enhances funneling, similar behavior should occur in a random resistor network. The non-monotonic p dependence of Vmax can be understood within the quasi-one-dimensional “bubble” model [46] that captures the interplay between local funneling and overall current reduction as p decreases (Fig. 8). Although this system looks one-dimensional, it can be engineered to reproduce the percolation properties of a system in greater than one dimension by choosing the length L to scale exponentially with the width w. The probability for a spanning path in this structure is L  p0 D 1  (1  p)w ! exp[L epw ] L; w ! 1 ;

(25)

which suddenly changes from 0 to 1 – indicative of percolation – at a threshold that lies strictly within (0,1) as L ! 1 and L ew . In what follows, we take L D 2w , which gives p c D 1/2. To determine the effect of bottlenecking, we appeal to the statement of Coniglio’s theorem [15], @p0 /@p equals the average number of singly-connected bonds in the system. Evaluating @p0 /@p in Eq. (25) at the percolation threshold of p c D 1/2 gives @p0 D w C O(ew ) ln L : @p

(26)

Thus at pc there are w ln L bottlenecks. However, current focusing due to bottlenecks is substantially diluted because the conductance, and hence the total current through the network, is small at pc . What is needed is a single bottleneck of width 1. One such bottleneck ensures the total current flow is still substantial, while the narrowing to width 1 endures that the focusing effect of the bottleneck is maximally effective.

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 8 The bubble model: a chain of L bubbles in series, each consisting of w bonds in parallel. Each bond is independently present with probability p

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Clearly, a single bottleneck of width 1 occurs above the percolation threshold. Thus let’s determine when a such an isolated bottleneck of width 1 first appears as a function of p. The probability that a single non-empty bubble contains at least two bonds is (1 qw w pqw1 ) / (1 qw ), where q D 1 p. Then the probability P1 (p) that the width of the narrowest bottleneck has width 1 in a chain of L bubbles is  L w pqw1 P1 (p) D 1  1  1  qw   p pw : (27) 1  exp Lw e q(1  qw ) The subtracted term is the probability that L non-empty bubbles contain at least two bonds, and then P1 (p) is the complement of this quantity. As p decreases from 1, P1 (p) sharply increases from 0 to 1 when the argument of the outer exponential becomes of the order of 1; this change occurs at pˆ p c C O(ln(ln L) / ln L). At this point, a bottleneck of width 1 first appears and therefore Vmax also occurs for this value of p. Random Walks and Resistor Networks The Basic Relation We now discuss how the voltages at each node in a resistor network and the resistance of the network are directly related to first-passage properties of random walks [31,63,76, 98]. To develop this connection, consider a random walk on a finite network that can hop between nearest-neighbor sites i to j with probability pij in a single step. We divide the boundary points of the network into two disjoint classes, BC and B , that we are free to choose; a typical situation is the geometry shown in Fig. 9. We now ask: starting at an arbitrary point i, what is the probability that the walk eventually reaches the boundary set BC without first reaching any node in B ? This quantity is termed the exit probability EC (i) (with an analogous definition for the exit probability E (i) D 1  EC (i) to B ). We obtain the exit probability EC (i) by summing the probabilities for all walk trajectories that start at i and reach a site in BC without touching any site in B (and similarly for E (i)). Thus X E˙ (i) D P p ˙ (i) ; (28)

E˙ (i) D

X

p i j E˙ ( j) :

(29)

j

Thus E˙ (i) is a harmonic function because it equals a weighted average of E˙ at neighboring points, with weighting function pij . This is exactly the same relation obeyed by the node voltages in Eq. (2) for the corresponding resistor network when we identify the single-step hopP ping probabilities pij with the conductances g i j / j g i j . We thus have the following equivalence:  Let the boundary sets BC and B in a resistor network be fixed at voltages 1 and 0 respectively, with g ij the conductance of the bond between sites i and j. Then the voltage at any interior site i coincides with the probability for a random walk, which starts at i, to reach BC before reaching B , when the hopping probability from P i to j is p i j D g i j / j g i j . If all the bond conductances are the same – corresponding to single - step hopping probabilities in the equivalent random walk being identical – then Eq. (29) is just the discrete Laplace equation. We can then exploit this correspondence between conductances and hopping probabilities to infer non-trivial results about random walks and about resistor networks from basic electrostatics. This correspondence can also be extended in a natural way to general random walks with a spatially-varying bias and diffusion coefficient, and to continuous media. The consequences of this equivalence between random walks and resistor networks is profound. As an example [76], consider a diffusing particle that is initially at distance r0 from the center of a sphere of radius a < r0 in otherwise empty d-dimensional space. By the correspondence with electrostatics, the probability that this particle eventually hits the sphere is simply the electrostatic potential at r0 ; E (r0 ) D (a/r0 )d2 !

p˙

where P p ˙ (i) denotes the probability of a path from i to B˙ that avoids B . The sum over all these restricted paths can be decomposed into the outcome after one step, when the walk reaches some intermediate site j, and the sum over all path remainders from j to B˙ . This decomposition gives

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 9 a Lattice network with boundary sites BC or B . b Corresponding resistor network in which each rectangle is a 1 ohm resistor. The sites in BC are all fixed at potential V D 1, and sites in B are all grounded

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

fact, Eq. (31) gives the fundamental result

Network Resistance and Pólya’s Theorem An important extension of the relation between exit probability and node voltages is to infinite resistor networks. This extension provides a simple connection between the classic recurrence/transience transition of random walks on a given network [31,63,76,98] and the electrical resistance of this same network [26]. Consider a symmetric random walk on a regular lattice in d spatial dimensions. Suppose that the walk starts at the origin at t D 0. What is the probability that the walk eventually returns to its starting point? The answer is strikingly simple:  For d  2, a random walk is certain to eventually return to the origin. This property is known as recurrence.  For d > 2, there is a non-zero probability that the random walk will never return to the origin. This property is known as transience. Let’s now derive the transience and recurrence properties of random walks in terms of the equivalent resistor network problem. Suppose that the voltage V at the boundary sites BC is set to one. Then by Kirchhoff’s law, the total current entering the network is ID

X X X (1  Vj )gC j D (1  Vj )pC j gCk : (30) j

j

k

Here g C j is the conductance of the resistor between BC P and a neighboring site j, and pC j D gC j / j gC j . Because the voltage V j also equals the probability for the corresponding random walk to reach BC without reaching B , the term Vj pC j is just the probability that a random walk starts at BC , makes a single step to one of the sites j adjacent to BC (with hopping probability pij ), and then returns to BC without reaching B . We therefore deduce that X ID (1  Vj )gC j j

D

X k

D

X

gCk

escape probability  Pescape D P

G : k g Ck

(32)

Suppose now that a current I is injected at a single point of an infinite network, with outflow at infinity (Fig. 10). Thus the probability for a random walk to never return to its starting point, is simply proportional to the conductance G from this starting point to infinity of the same network. Thus a subtle feature of random walks, namely, the escape probability, is directly related to currents and voltages in an equivalent resistor network. Part of the reason why this connection is so useful is that the conductance of the infinite network for various spatial dimensions can be easily determined, while a direct calculation of the return probability for a random walk is more difficult. In one dimension, the conductance of an infinitely long chain of identical resistors is clearly zero. Thus Pescape D 0 or, equivalently, Preturn D 1. Thus a random walk in one dimension is recurrent. As alluded to at the outset of Sect. “Introduction to Current Flows”, the conductance between one point and infinity in an infinite resistor lattice in general spatial dimension is somewhat challenging. However, to merely determine the recurrence or transience of a random walk, we only need to know if the return probability is zero or greater than zero. Such a simple question can be answered by a crude physical estimate of the network conductance. To estimate the conductance from one point to infinity, we replace the discrete lattice by a continuum medium

X (1  Vj )pC j j

gCk  (1  return probability )

k

D

X

gCk  escape probability :

(31)

k

Here “escape” means that the random walk reaches the set B without returning to a node in BC . On the other hand, the current and the voltage drop across the network are related to the conductance G between the two boundary sets by I D GV D G. From this

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 10 Decomposition of a conducting medium into concentric shells, each of which consists of fixed-conductance blocks. A current I is injected at the origin and flows radially outward through the medium

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

of constant conductance. We then estimate the conductance of the infinite medium by decomposing it into a series of concentric shells of fixed thickness dr. A shell at radius r can be regarded as a parallel array of r d1 volume elements, each of which has a fixed conductance. The conductance of one such shell is proportional to its surface area, and the overall resistance is the sum of these shell resistances. This reasoning gives Z1 R

Rshell (r)dr Z1



dr r d1

( D

1 (Pescape

P

for d  2 j

gC j

)1

for d > 2 :

(33)

The above estimate gives an easy solution to the recurrence/transience transition of random walks. For d  2, the conductance to infinity is zero because there are an insufficient number of independent paths from the origin to infinity. Correspondingly, the escape probability is zero and the random walk is recurrent. The case d D 2 is more delicate because the integral in Eq. (33) diverges only logarithmically at the upper limit. Nevertheless, the conductance to infinity is still zero and the corresponding random walk is recurrent (but just barely). For d > 2, the conductance between a single point and infinity in an infinite homogeneous resistor network is non zero and therefore the escape probability of the corresponding random walk is also non zero – the walk is now transient. There are many amusing ramifications of the recurrence of random walks and we mention two such properties. First, for d  2, even though a random walk eventually returns to its starting point, the mean time for this event is infinite! This divergence stems from a power-law tail in the time dependence of the first-passage probability [31,76], namely, the probability that a random walk returns to the origin for the first time. Another striking aspect of recurrence is that because a random walk returns to its starting point with certainty, it necessarily returns an infinite number of times. Future Directions There is a good general understanding of the conductance of resistor networks, both far from the percolation threshold, where effective medium theory applies, and close to percolation, where the conductance G vanishes as (p  p c ) t . Many advancements in numerical techniques have been developed to determine the conductance accurately and thereby obtain precise values for the conductance exponent, especially in two dimensions. In spite of this progress, we still do not yet have the right way, if it

exists at all, to link the geometry of the percolation cluster or the conducting backbone to the conductivity itself. Furthermore, many exponents of two-dimensional percolation are known exactly. Is it possible that the exact approaches developed to determine percolation exponents can be extended to give the exact conductance exponent? Finally, there are aspects about conduction in random networks that are worth highlighting. The first falls under the rubric of directed percolation [51]. Here each link in a network has an intrinsic directionality that allows current to flow in one direction only – a resistor and diode in series. Links are also globally oriented; on the square lattice for example, current can flow rightward and upward. A qualitative understanding of directed percolation and directed conduction has been achieved that parallels that of isotropic percolation. However, there is one facet of directed conduction that is barely explored. Namely, the state of the network (the bonds that are forward biased) must be determined self consistently from the current flows. This type of non-linearity is much more serious when the circuit elements are randomly oriented. These questions about the coupling between the state of the network and its conductance are central when the circuit elements are intrinsically non-linear [49,78]. This is a topic that seems ripe for new developments. Bibliography 1. Adler J (1985) Conductance Exponents From the Analysis of Series Expansions for Random Resistor Networks. J Phys A Math Gen 18:307–314 2. Adler J, Meir Y, Aharony A, Harris AB, Klein L (1990) Low-Concentration Series in General Dimension. J Stat Phys 58:511– 538 3. Aharony A, Feder J (eds) (1989) Fractals in Physics. Phys D 38:1–398 4. Alava MJ, Nukala PKVV, Zapperi S (2006) Statistical Models for Fracture. Adv Phys 55:349–476 5. Alexander S, Orbach R (1982) Density of States of Fractals: Fractons. J Phys Lett 43:L625–L631 6. Atkinson D, van Steenwijk FJ (1999) Infinite Resistive Lattice. Am J Phys 67:486–492 7. Batrouni GG, Hansen A, Nelkin M (1986) Fourier Acceleration of Relaxation Processes in Disordered Systems. Phys Rev Lett 57:1336–1339 8. Batrouni GG, Hansen A, Larson B (1996) Current Distribution in the Three-Dimensional Random Resistor Network at the Percolation Threshold. Phys Rev E 53:2292–2297 9. Blumenfeld R, Meir Y, Aharony A, Harris AB (1987) Resistance Fluctuations in Randomly Diluted Networks. Phys Rev B 35:3524–3535 10. Bruggeman DAG (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann Phys (Leipzig) 24:636–679. [Engl Trans: Computation of Different Physical Constants of Hetero-

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Fractal and Multifractal Time Series JAN W. KANTELHARDT Institute of Physics, Martin-Luther-University Halle-Wittenberg, Halle, Germany Article Outline Glossary Definition of the Subject Introduction Fractal and Multifractal Time Series Methods for Stationary Fractal Time Series Analysis Methods for Non-stationary Fractal Time Series Analysis Methods for Multifractal Time Series Analysis Statistics of Extreme Events in Fractal Time Series Simple Models for Fractal and Multifractal Time Series Future Directions Acknowledgment Bibliography Glossary Time series One dimensional array of numbers (x i ); i D 1; : : : ; N, representing values of an observable x usually measured equidistant (or nearly equidistant) in time. Complex system A system consisting of many non-linearly interacting components. It cannot be split into simpler sub-systems without tampering with the dynamical properties. Scaling law A power law with a scaling exponent (e. g. ˛) describing the behavior of a quantity F (e. g., fluctuation, spectral power) as function of a scale parameter s (e. g., time scale, frequency) at least asymptotically: F(s) s˛ . The power law should be valid for a large range of s values, e. g., at least for one order of magnitude. Fractal system A system characterized by a scaling law with a fractal, i. e., non-integer exponent. Fractal systems are self-similar, i. e., a magnification of a small part is statistically equivalent to the whole. Self-affine system Generalization of a fractal system, where different magnifications s and s 0 D s H have to be used for different directions in order to obtain a statistically equivalent magnification. The exponent H is called Hurst exponent. Self-affine time series and time series becoming self-affine upon integration are commonly denoted as fractal using a less strict terminology.

Multifractal system A system characterized by scaling laws with an infinite number of different fractal exponents. The scaling laws must be valid for the same range of the scale parameter. Crossover Change point in a scaling law, where one scaling exponent applies for small scale parameters and another scaling exponent applies for large scale parameters. The center of the crossover is denoted by its characteristic scale parameter s in this article. Persistence In a persistent time series, a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. A fractal scaling law holds at least for a limited range of scales. Short-term correlations Correlations that decay sufficiently fast that they can be described by a characteristic correlation time scale; e. g., exponentially decaying correlations. A crossover to uncorrelated behavior is observed on larger scales. Long-term correlations Correlations that decay sufficiently slow that a characteristic correlation time scale cannot be defined; e. g., power-law correlations with an exponent between 0 and 1. Power-law scaling is observed on large time scales and asymptotically. The term long-range correlations should be used if the data is not a time series. Non-stationarities If the mean or the standard deviation of the data values change with time, the weak definition of stationarity is violated. The strong definition of stationarity requires that all moments remain constant, i. e., the distribution density of the values does not change with time. Non-stationarities like monotonous, periodic, or step-like trends are often caused by external effects. In a more general sense, changes in the dynamics of the system also represent non-stationarities. Definition of the Subject Data series generated by complex systems exhibit fluctuations on a wide range of time scales and/or broad distributions of the values. In both equilibrium and nonequilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders of magnitude. Such scaling laws allow for a characterization of the data and the generating complex system by fractal (or multifractal) scaling exponents, which can serve as characteristic fingerprints of the systems in comparisons with other systems and with models. Fractal scaling behavior has been observed, e. g., in many data series from experimental physics, geophysics, medicine, physiology, and

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even social sciences. Although the underlying causes of the observed fractal scaling are often not known in detail, the fractal or multifractal characterization can be used for generating surrogate (test) data, modeling the time series, and deriving predictions regarding extreme events or future behavior. The main application, however, is still the characterization of different states or phases of the complex system based on the observed scaling behavior. For example, the health status and different physiological states of the human cardiovascular system are represented by the fractal scaling behavior of the time series of intervals between successive heartbeats, and the coarsening dynamics in metal alloys are represented by the fractal scaling of the time-dependent speckle intensities observed in coherent X-ray spectroscopy. In order to observe fractal and multifractal scaling behavior in time series, several tools have been developed. Besides older techniques assuming stationary data, there are more recently established methods differentiating truly fractal dynamics from fake scaling behavior caused by non-stationarities in the data. In addition, short-term and long-term correlations have to be clearly distinguished to show fractal scaling behavior unambiguously. This article describes several methods originating from statistical physics and applied mathematics, which have been used for fractal and multifractal time series analysis in stationary and non-stationary data. Introduction The characterization and understanding of complex systems is a difficult task, since they cannot be split into simpler subsystems without tampering with the dynamical properties. One approach in studying such systems is the recording of long time series of several selected variables (observables), which reflect the state of the system in a dimensionally reduced representation. Some systems are characterized by periodic or nearly periodic behavior, which might be caused by oscillatory components or closed-loop regulation chains. However, in truly complex systems such periodic components are usually not limited to one or two characteristic frequencies or frequency bands. They rather extend over a wide spectrum, and fluctuations on many time scales as well as broad distributions of the values are found. Often no specific lower frequency limit – or, equivalently, upper characteristic time scale – can be observed. In these cases, the dynamics can be characterized by scaling laws which are valid over a wide (possibly even unlimited) range of time scales or frequencies; at least over orders of magnitude. Such dynamics are usually denoted as fractal or multifractal, depending on the

question if they are characterized by one scaling exponent or by a multitude of scaling exponents. The first scientist who applied fractal analysis to natural time series was Benoit B. Mandelbrot [1,2,3], who included early approaches by H.E. Hurst regarding hydrological systems [4,5]. For extensive introductions describing fractal scaling in complex systems, we refer to [6,7,8, 9,10,11,12,13]. In the last decade, fractal and multifractal scaling behavior has been reported in many natural time series generated by complex systems, including  Geophysics time series (recordings of temperature, precipitation, water runoff, ozone levels, wind speed, seismic events, vegetational patterns, and climate dynamics),  Medical and physiological time series (recordings of heartbeat, respiration, blood pressure, blood flow, nerve spike intervals, human gait, glucose levels, and gene expression data),  DNA sequences (they are not actually time series),  Astrophysical time series (X-ray light sources and sunspot numbers),  Technical time series (internet traffic, highway traffic, and neutronic power from a reactor),  Social time series (finance and economy, language characteristics, fatalities in conflicts), as well as  Physics data (also going beyond time series), e. g., surface roughness, chaotic spectra of atoms, and photon correlation spectroscopy recordings. If one finds that a complex system is characterized by fractal (or multifractal) dynamics with particular scaling exponents, this finding will help in obtaining predictions on the future behavior of the system and on its reaction to external perturbations or changes in the boundary conditions. Phase transitions in the regulation behavior of a complex system are often associated with changes in their fractal dynamics, allowing for a detection of such transitions (or the corresponding states) by fractal analysis. One example for a successful application of this approach is the human cardiovascular system, where the fractality of heartbeat interval time series was shown to reflect certain cardiac impairments as well as sleep stages [14,15]. In addition, one can test and iteratively improve models of the system until they reproduce the observed scaling behavior. One example for such an approach is climate modeling, where the models were shown to need input from volcanos and solar radiation in order to reproduce the long-term correlated (fractal) scaling behavior [16] previously found in observational temperature data [17]. Fractal (or multifractal) scaling behavior certainly cannot be assumed a priori, but has to be established. Hence,

Fractal and Multifractal Time Series

there is a need for refined analysis techniques, which help to differentiate truly fractal dynamics from fake scaling behavior caused, e. g., by non-stationarities in the data. If conventional statistical methods are applied for the analysis of time series representing the dynamics of a complex system [18,19], there are two major problems. (i) The number of data series and their durations (lengths) are usually very limited, making it difficult to extract significant information on the dynamics of the system in a reliable way. (ii) If the length of the data is extended using computer-based recording techniques or historical (proxy) data, non-stationarities in the signals tend to be superimposed upon the intrinsic fluctuation properties and measurement noise. Non-stationarities are caused by external or internal effects that lead to either continuous or sudden changes in the average values, standard deviations or regulation mechanism. They are a major problem for the characterization of the dynamics, in particular for finding the scaling properties of given data. Fractal and Multifractal Time Series Fractality, Self-Affinity, and Scaling The topic of this article is the fractality (and/or multifractality) of time series. Since fractals and multifractals in general are discussed in many other articles of the encyclopedia, the concept is not thoroughly explained here. In particular, we refer to the articles  Fractal Geometry, A Brief Introduction to and  Fractals and Multifractals, Introduction to for the formalism describing fractal and multifractal structures, respectively. In a strict sense, most time series are one dimensional, since the values of the considered observable are measured in homogeneous time intervals. Hence, unless there are missing values, the fractal dimension of the support is D(0) D 1. However, there are rare cases where most of the values of a time series are very small or even zero, causing a dimension D(0) < 1 of the support. In these cases, one has to be very careful in selecting appropriate analysis techniques, since many of the methods presented in this article are not accurate for such data; the wavelet transform modulus maxima technique (see Subsect. “Wavelet Transform Modulus Maxima (WTMM) Method”) is the most advanced applicable method. Even if the fractal dimension of support is one, the information dimension D(1) and the correlation dimension D(2) can be studied. As we will see in Subsect. “The Structure Function Approach and Singularity Spectra”, D(2) is in fact explicitly related to all exponents studied in monofractal time series analysis. However, usually a slightly different approach is employed based on the no-

tion of self-affinity instead of (multi-) fractality. Here, one takes into account that the time axis and the axis of the measured values x(t) are not equivalent. Hence, a rescaling of time t by a factor a may require rescaling of the series values x(t) by a different factor aH in order to obtain a statistically similar (i. e., self-similar) picture. In this case the scaling relation x(t) ! a H x(at)

(1)

holds for an arbitrary factor a, describing the data as selfaffine (see, e. g., [6]). The Hurst exponent H (after the water engineer H.E. Hurst [4]) characterizes the type of self affinity. Figure 1a shows several examples of self-affine time series with different H. The trace of a random walk (Brownian motion, third line in Fig. 1a), for example, is characterized by H D 0:5, implying that the position axis must be rescaled by a factor of two if the time axis is rescaled by a factor of four. Note that self-affine series are often denoted as fractal even though they are not fractal in the strict sense. In this article the term “fractal” will be used in the more general sense including all data, where a Hurst exponent H can be reasonably defined. The scaling behavior of self-affine data can also be characterized by looking at their mean-square displacement. Since the mean-square displacement of a random walker is known to increase linearly in time, hx 2 (t)i t, deviations from this law will indicate the presence of selfaffine scaling. As we will see in Subsect. “Fluctuation Analysis (FA)”, one can thus retrieve the Hurst (or self-affinity) exponent H by studying the scaling behavior of the mean-square displacement, or the mean-square fluctuations hx 2 (t)i t 2H . Persistence, Long- and Short-Term Correlations Self-affine data are persistent in the sense that a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. For the trace of a random walk, persistence on all time scales is trivial, since a later position is just a former one plus some random increment(s). The persistence holds for all time scales, where the self-affinity relation (1) holds. However, the degree of persistence can also vary on different time scales. Weather is a typical example: while the weather tomorrow or in one week is probably similar to the weather today (due to a stable general weather condition), persistence is much harder to be seen on longer time scales. Considering the increments x i D x i  x i1 of a selfaffine series, (x i ), i D 1; : : : ; N with N values measured equidistant in time, one finds that the x i can be either

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Fractal and Multifractal Time Series, Figure 1 a Examples of self-affine series x i characterized by different Hurst exponents H = 0.9, 0.7, 0.5, 0.3 (from top to bottom). The data has been generated by Fourier filtering using the same seed for the random number generator. b Differentiated series xi of the data from a; the xi are characterized by positive long-term correlations (persistence) with  = 0.2 and 0.6 (first and second line), uncorrelated behavior (third line), and anti-correlations (bottom line), respectively

persistent, independent, or anti-persistent. Examples for all cases are shown in Fig. 1b. In our example of the random walk with H D 0:5 (third line in the figure), the increments (steps) are fully independent of each other. Persistent and anti-persistent increments, where a positive increment is likely to be followed by another positive or negative increment, respectively, are also leading to persistent P integrated series x i D ijD1 x j . For stationary data with constant mean and standard deviation the auto-covariance function of the increments, ˛ ˝ C(s) D x i x iCs D

Ns 1 X x i x iCs Ns

(2)

iD1

can be studied to determine the degree of persistence. If C(s) is divided by the variance h(x i )2 i, it becomes the auto-correlation function; both are identical if the data are normalized with unit variance. If the x i are uncorrelated (as for the random walk), C(s) is zero for s > 0. Shortrange correlations of the increments x i are usually described by C(s) declining exponentially, C(s) exp(s/t )

(3)

with a characteristic decay time t . Such behavior is typical for increments generated by an auto-regressive (AR) process x i D cx i1 C " i

(4)

with random uncorrelated offsets "i and c D exp(1/t ). Figure 2a shows the auto-correlation function for one configuration of an AR process with t D 48.

Fractal and Multifractal Time Series, Figure 2 Comparison of the autocorrelation functions C(s) (decreasing functions) and fluctuation functions F2 (s) (increasing functions) for short-term correlated data (top panel) and long-term correlated data ( D 0:4, bottom panel). The asymptotic slope H  ˛ D 0:5 of F2 (s) clearly indicates missing long-term correlations, while H  ˛ D 1  /2 > 0:5 indicates long-term correlations. The difference is much harder to observe in C(s), where statistical fluctuations and negative values start occurring above s  100. The data have been generated by an AR process Eq. (4) and Fourier filtering, respectively. The dashed lines indicate the theoretical curves

R1 For so-called long-range correlations 0 C(s)ds diverges in the limit of infinitely long series (N ! 1). In practice, this means that t cannot be defined because it increases with increasing N. For example, C(s) declines as a power-law

Fractal and Multifractal Time Series

C(s) / s 

(5)

with an exponent 0 < < 1. Figure 2b shows C(s) for one configuration with D 0:4. This type of behavior can be modeled by the Fourier filtering technique (see Subsect. “Fourier Filtering”). Long-term correlated, i. e. persistent, behavior of the x i leads to self-affine scaling behavior of the xi , characterized by H D 1  /2, as will be shown below. Crossovers and Non-Stationarities in Time Series Short-term correlated increments x i characterized by a finite characteristic correlation decay time t lead to a crossover in the scaling behavior of the integrated sePi ries x i D jD1 x j , see Fig. 2a for an example. Since the position of the crossover might be numerically different from t , we denote it by s here. Time series with a crossover are not self-affine and there is no unique Hurst exponent H characterizing them. While H > 0:5 is observed on small time scales (indicating correlations in the increments), the asymptotic behavior (for large time scales s  t and  s ) is always characterized by H D 0:5, since all correlations have decayed. Many natural recordings are characterized by pronounced short-term correlations in addition to scaling long-term correlations. For example, there are short-term correlations due to particular general weather situations in temperature data and due to respirational effects in heartbeat data. Crossovers in the scaling behavior of complex time series can also be caused by different regulation mechanisms on fast and slow time scales. Fluctuations of river runoff, for example, show different scaling behavior on time scales below and above approximately one year. Non-stationarities can also cause crossovers in the scaling behavior of data if they are not properly taken into account. In the most strict sense, non-stationarities are variations in the mean or the standard deviation of the data (violating weak stationarity) or the distribution of the data values (violating strong stationarity). Non-stationarities like monotonous, periodic or step-like trends are often caused by external effects, e. g., by the greenhouse warming and seasonal variations for temperature records, different levels of activity in long-term physiological data, or unstable light sources in photon correlation spectroscopy. Another example for non-stationary data is a record consisting of segments with strong fluctuations alternating with segments with weak fluctuations. Such behavior will cause a crossover in scaling at the time scale corresponding to the typical duration of the homogeneous segments. Different mechanisms of regulation during different time segments – like, e. g., different heartbeat reg-

ulation during different sleep stages at night – can also cause crossovers; they are regarded as non-stationarities here, too. Hence, if crossovers in the scaling behavior of data are observed, more detailed studies are needed to find out the cause of the crossovers. One can try to obtain homogenous data by splitting the original series and employing methods that are at least insensitive to monotonous (polynomially shaped) trends. To characterize a complex system based on time series, trends and fluctuations are usually studied separately (see, e. g., [20] for a discussion). Strong trends in data can lead to a false detection of long-range statistical persistence if only one (non-detrending) method is used or if the results are not carefully interpreted. Using several advanced techniques of scaling time series analysis (as described in Sect. “Methods for Non-stationary Fractal Time Series Analysis”) crossovers due to trends can be distinguished from crossovers due to different regulation mechanisms on fast and slow time scales. The techniques can thus assist in gaining insight into the scaling behavior of the natural variability as well as into the kind of trends of the considered time series. It has to be stressed that crossovers in scaling behavior must not be confused with multifractality. Even though several scaling exponents are needed, they are not applicable for the same regime (i. e., the same range of time scales). Real multifractality, on the other hand, is characterized by different scaling behavior of different moments over the full range of time scales (see next section). Multifractal Time Series Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. As discussed in the previous section, there might exist crossover (time-) scales s separating regimes with different scaling exponents. In other cases, the scaling behavior is more complicated, and different scaling exponents are required for different parts of the series. In even more complicated cases, such different scaling behavior can be observed for many interwoven fractal subsets of the time series. In this case a multitude of scaling exponents is required for a full description of the scaling behavior in the same range of time scales, and a multifractal analysis must be applied. Two general types of multifractality in time series can be distinguished: (i) Multifractality due to a broad probability distribution (density function) for the values of the time series, e. g. a Levy distribution. In this case the multifractality cannot be removed by shuffling the series. (ii) Multifractality due to different long-term correlations

467

468

Fractal and Multifractal Time Series

of the small and large fluctuations. In this case the probability density function of the values can be a regular distribution with finite moments, e. g., a Gaussian distribution. The corresponding shuffled series will exhibit nonmultifractal scaling, since all long-range correlations are destroyed by the shuffling procedure. Randomly shuffling the order of the values in the time series is the easiest way of generating surrogate data; however, there are more advanced alternatives (see Sect. “Simple Models for Fractal and Multifractal Time Series”). If both kinds of multifractality are present, the shuffled series will show weaker multifractality than the original series. A multifractal analysis of time series will also reveal higher order correlations. Multifractal scaling can be observed if, e. g., three or four-point correlations scale differently from the standard two-point correlations studied by classical autocorrelation analysis (Eq. (2)). In addition, multifractal scaling is observed if the scaling behavior of small and large fluctuations is different. For example, extreme events might be more or less correlated than typical events.

C(s) / s  with a correlation exponent 0 < < 1 (see Eqs. (3) and (5), respectively). As illustrated by the two examples shown in Fig. 2, a direct calculation of C(s) is usually not appropriate due to noise superimposed on the data x˜ i and due to underlying non-stationarities of unknown origin. Non-stationarities make the definition of C(s) problematic, because the average hxi is not welldefined. Furthermore, C(s) strongly fluctuates around zero on large scales s (see Fig. 2b), making it impossible to find the correct correlation exponent . Thus, one has to determine the value of indirectly.

Methods for Stationary Fractal Time Series Analysis

The spectral exponent ˇ and the correlation exponent

can thus be obtained by fitting a power-law to a double logarithmic plot of the power spectrum S( f ). An example is shown in Fig. 3. The relation (6) can be derived from the Wiener–Khinchin theorem (see, e. g., [25]). If, instead of x˜ i D x i the integrated runoff time series is Fourier P transformed, i. e., x˜ i D x i X D ijD1 x j , the resulting power spectrum scales as S( f ) f 2ˇ .

In this section we describe four traditional approaches for the fractal analysis of stationary time series, see [21,22,23] for comparative studies. The main focus is on the determination of the scaling exponents H or , defined in Eqs. (1) and (5), respectively, and linked by H D 1  /2 in long-term persistent data. Methods taking non-stationarities into account will be discussed in the next chapter.

Spectral Analysis If the time series is stationary, we can apply standard spectral analysis techniques (Fourier transform) and calculate the power spectrum S( f ) of the time series (x˜ i ) as a function of the frequency f to determine self-affine scaling behavior [24]. For long-term correlated data characterized by the correlation exponent , we have S( f ) f ˇ

with ˇ D 1  :

(6)

Autocorrelation Function Analysis We consider a record (x i ) of i D 1; : : : ; N equidistant measurements. In most applications, the index i will correspond to the time of the measurements. We are interested in the correlation of the values xi and x iCs for different time lags, i. e. correlations over different time scales s. In order to remove a constant offset in the data, the mean PN ˜ i  x i  hxi. Alhxi D N1 iD1 x i is usually subtracted, x ternatively, the correlation properties of increments x˜ i D x i D x i  x i1 of the original series can be studied (see also Subsect. “Persistence, Long- and Short-Term Correlations”). Quantitatively, correlations between x˜ -values separated by s steps are defined by the (auto-) covariance function C(s) D hx˜ i x˜ iCs i or the (auto-) correlation function C(s)/hx˜ 2i i, see also Eq. (2). As already mentioned in Subsect. “Persistence, Longand Short-Term Correlations”, the x˜ i are short-term correlated if C(s) declines exponentially, C(s) exp(s/t ), and long-term correlated if C(s) declines as a power-law

Fractal and Multifractal Time Series, Figure 3 Spectral analysis of a fractal time series characterized by longterm correlations with  D 0:4 (ˇ D 0:6). The expected scaling behavior (dashed line indicating the slope –ˇ) is observed only after binning of the spectrum (circles). The data has been generated by Fourier filtering

Fractal and Multifractal Time Series

Spectral analysis, however, does not yield more reliable results than auto-correlation analysis unless a logarithmic binning procedure is applied to the double logarithmic plot of S( f ) [21], see also Fig. 3. I. e., the average of log S( f ) is calculated in successive, logarithmically wide bands from a n f0 to a nC1 f0 , where f 0 is the minimum frequency, a > 1 is a factor (e. g., a D 1:1), and the index n is counting the bins. Spectral analysis also requires stationarity of the data. Hurst’s Rescaled-Range Analysis The first method for the analysis of long-term persistence in time series based on random walk theory has been proposed by the water construction engineer Harold Edwin Hurst (1880–1978), who developed it while working in Egypt. His so-called rescaled range analysis (R/S analysis) [1,2,4,5,6] begins with splitting of the time series (x˜ i ) into non-overlapping segments  of size (time scale) s (first step), yielding Ns D int(N/s) segments altogether. In the second step, the profile (integrated data) is calculated in each segment  D 0; : : : ; Ns  1, Y ( j) D

j X

(x˜sCi  hx˜sCi is )

iD1

D

j X

x˜sCi 

iD1

s jX x˜sCi : s

(7)

By the subtraction of the local averages, piecewise constant trends in the data are eliminated. In the third step, the differences between minimum and maximum value (ranges) R (s) and the standard deviations S (s) in each segment are calculated,

(8)

jD1

Finally, the rescaled range is averaged over all segments to obtain the fluctuation function F(s), FRS (s) D

N s 1 R (s) 1 X sH N s D0 S (s)

for s  1 ;

Fluctuation Analysis (FA) The standard fluctuation analysis (FA) [8,26] is also based on random walk theory. For a time series (x˜ i ), i D 1; : : : ; N, with zero mean, we consider the global profile, i. e., the cumulative sum (cf. Eq. (7)) Y( j) D

j X

x˜ i ;

j D 0; 1; 2; : : : ; N ;

(10)

iD1

iD1

R (s) D maxsjD1 Y ( j)  minsjD1 Y ( j); v u X u1 s Y2 ( j) : S (s) D t s

that H actually characterizes the self-affinity of the profile function (7), while ˇ and refer to the original data. The values of H, that can be obtained by Hurst’s rescaled range analysis, are limited to 0 < H < 2, and significant inaccuracies are to be expected close to the bounds. Since H can be increased or decreased by one if Pj ˜ i ) or differentiated the data is integrated (x˜ j ! iD1 x (x˜ i ! x˜ i  x˜ i1 ), respectively, one can always find a way to calculate H by rescaled range analysis provided the data is stationary. While values H < 1/2 indicate long-term anticorrelated behavior of the data x˜ i , H > 1/2 indicates longterm positively correlated behavior. For power-law correlations decaying faster than 1/s, we have H D 1/2 for large s values, like for uncorrelated data. Compared with spectral analysis, Hurst’s rescaled range analysis yields smoother curves with less effort (no binning procedure is necessary) and works also for data with piecewise constant trends.

(9)

where H is the Hurst exponent already introduced in Eq. (1). One can show [1,24] that H is related to ˇ and

by 2H  1 C ˇ D 2  (see also Eqs. (6) and (14)). Note that 0 < < 1, so that the right part of the equation does not hold unless 0:5 < H < 1. The relationship does not hold in general for multifractal data. Note also

and study how the fluctuations of the profile, in a given time window of size s, increase with s. The procedure is illustrated in Fig. 4 for two values of s. We can consider the profile Y( j) as the position of a random walker on a linear chain after j steps. The random walker starts at the origin and performs, in the ith step, a jump of length x˜ i to the bottom, if x˜ i is positive, and to the top, if x˜ i is negative. To find how the square-fluctuations of the profile scale with s, we first divide the record of N elements into N s D int(N/s) non-overlapping segments of size s starting from the beginning (see Fig. 4) and another N s non-overlapping segments of size s starting from the end of the considered series. This way neither data at the end nor at the beginning of the record is neglected. Then we determine the fluctuations in each segment . In the standard FA, we obtain the fluctuations just from the values of the profile at both endpoints of each segment  D 1; : : : ; Ns , 2 (; s) D [Y(s)  Y((1)s)]2 ; FFA

(11)

(see Fig. 4) and analogous for  D N sC1 ; : : : ; 2N s , 2 (; s) D [Y(N(N s )s)Y(N(1N s )s)]2 : (12) FFA

469

470

Fractal and Multifractal Time Series

differentiation of the data, the same rules apply as listed for H in the previous subsection. The results of FA become statistically unreliable for scales s larger than one tenth of the length of the data, i. e. the analysis should be limited by s < N/10. Methods for Non-stationary Fractal Time Series Analysis Wavelet Analysis

Fractal and Multifractal Time Series, Figure 4 Illustration of the fluctuation analysis (FA) and the detrended fluctuation analysis (DFA). For two segment durations (time scales) s D 100 (a) and 200 (b), the profiles Y( j) (blue lines; defined in Eq. (11), the values used for fluctuation analysis in Eq. (12) (green circles), and least-square quadratic fits to the profiles (red lines) are shown

The origins of wavelet analysis come from signal theory, where frequency decompositions of time series were studied [27,28]. Like the Fourier transform, the wavelet transform of a signal x(t) is a convolution integral to be replaced by a summation in case of a discrete time series (x˜ i ); i D 1; : : : ; N, Z 1 1 L (; s) D x(t) [(t  )/s] dt s 1 N

2 (; s) over all subsequences to obtain Then we average FFA the mean fluctuation F2 (s),

" F2 (s) D

#1/2 2N 1 Xs 2 F (; s) s˛ : 2N s D1 FA

(13)

By definition, F2 (s) can be viewed as the root-mean-square displacement of the random walker on the chain, after s steps (the reason for the index 2 will become clear later). For uncorrelated xi values, we obtain Fick’s diffusion law F2 (s) s1/2 . For the relevant case of long-term correlations, in which C(s) follows the power-law behavior of Eq. (5), F2 (s) increases by a power law, F2 (s) s ˛

with ˛  H ;

(14)

where the fluctuation exponent ˛ is identical with the Hurst exponent H for mono-fractal data and related to

and ˇ by 2˛ D 1 C ˇ D 2  :

(15)

The typical behavior of F2 (s) for short-term correlated and long-term correlated data is illustrated in Fig. 2. The relation (15) can be derived straightforwardly by inserting Eqs. (10), (2), and (5) into Eq. (11) and separating sums over products x˜ i x˜ j with identical and different i and j, respectively. The range of the ˛ values that can be studied by standard FA is limited to 0 < ˛ < 1, again with significant inaccuracies close to the bounds. Regarding integration or

D

1X x˜ i s

[(i  )/s] :

(16)

iD1

Here, (t) is a so-called mother wavelet, from which all daughter wavelets ;s (t) D ((t  )/s) evolve by shifting and stretching of the time axis. The wavelet coefficients L (; s) thus depend on both time position  and scale s. Hence, the local frequency decomposition of the signal is described with a time resolution appropriate for the considered frequency f D 1/s (i. e., inverse time scale). All wavelets (t) must have zero mean. They are often chosen to be orthogonal to polynomial trends, so that the analysis method becomes insensitive to possible trends in the data. Simple examples are derivatives of a Gaus(n) sian, Gauss (t) D dn /(dt n ) exp(x 2 /2), like the Mexican (2) (1) hat wavelet  Gauss and the Haar wavelet, Haar (t) D C1 if 0  t < 1, 1 if 1  t < 2, and 0 otherwise. It is straightforward to construct higher order Haar wavelets that are orthogonal to linear, quadratic and cubic trends, (2) e. g., Haar (t) D 1 for t 2 [0; 1) [ [2; 3), 2 for t 2 [1; 2), (3) (t) D 1 for t 2 [0; 1), 3 for and 0 otherwise, or Haar t 2 [1; 2), +3 for t 2 [2; 3), 1 for t 2 [3; 4), and 0 otherwise. Discrete Wavelet Transform (WT) Approach A detrending fractal analysis of time series can be easily implemented by considering Haar wavelet coefficients of the profile Y( j), Eq. (10) [17,30]. In this case the convolution (16) corresponds to the addition and subtraction of mean values of Y( j) within segments of size s. Hence, P defining Y¯ (s) D 1s sjD1 Y(s C j), the coefficients can

Fractal and Multifractal Time Series

be written as FWT1 (; s)  L FWT2 (; s)  L

(0) Haar

(s; s) D Y¯ (s)  Y¯C1 (s) ;

(1) Haar

(17)

(s; s)

D Y¯ (s)  2Y¯C1 (s) C Y¯C2 (s) ;

Detrended Fluctuation Analysis (DFA) (18)

and FWT3 (; s)  L

(2) Haar

(s; s)

D Y¯ (s)  3Y¯C1 (s) C 3Y¯C2 (s)  Y¯C3 (s)

results become statistically unreliable for scales s larger than one tenth of the length of the data, just as for FA.

(19)

for constant, linear and quadratic detrending, respectively. The generalization for higher orders of detrending is ob2 vious. The resulting mean-square fluctuations FWTn (; s) are averaged over all  to obtain the mean fluctuation F2 (s), see Eq. (13). Figure 5 shows typical results for WT analysis of long-term correlated, short-term correlated and uncorrelated data. Regarding trend-elimination, wavelet transform WT0 corresponds to standard FA (see Subsect. “Fluctuation Analysis (FA)”), and only constant trends in the profile are eliminated. WT1 is similar to Hurst’s rescaled range analysis (see Subsect. “Hurst’s Rescaled-Range Analysis”): linear trends in the profile and constant trends in the data are eliminated, and the range of the fluctuation exponent ˛  H is up to 2. In general, WTn determines the fluctuations from the nth derivative, this way eliminating trends described by (n  1)st-order polynomials in the data. The

In the last 14 years Detrended Fluctuation Analysis (DFA), originally introduced by Peng et al. [31], has been established as an important method to reliably detect longrange (auto-) correlations in non-stationary time series. The method is based on random walk theory and basically represents a linear detrending version of FA (see Subsect. “Fluctuation Analysis (FA)”). DFA was later generalized for higher order detrending [15], separate analysis of sign and magnitude series [32] (see Subsect. “Sign and Magnitude (Volatility) DFA”), multifractal analysis [33] (see Subsect. “Multifractal Detrended Fluctuation Analysis (MFDFA)”), and data with more than one dimension [34]. Its features have been studied in many articles [35,36,37,38,39,40]. In addition, several comparisons of DFA with other methods for stationary and nonstationary time-series analysis have been published, see, e. g., [21,23,41,42] and in particular [22], where DFA is compared with many other established methods for short data sets, and [43], where it is compared with recently suggested improved methods. Altogether, there are about 600 papers applying DFA (till September 2008). In most cases positive auto-correlations were reported leaving only a few exceptions with anti-correlations, see, e. g., [44,45,46]. Like in the FA method, one first calculates the global profile according to Eq. (10) and divides the profile into N s D int(N/s) non-overlapping segments of size s starting from the beginning and another N s segments starting from the end of the considered series. DFA explicitly deals with monotonous trends in a detrending procedure. This m ( j) within is done by estimating a polynomial trend y;s each segment  by least-square fitting and subtracting this trend from the original profile (‘detrending’), m ( j) : Y˜s ( j) D Y( j)  y;s

Fractal and Multifractal Time Series, Figure 5 Application of discrete wavelet transform (WT) analysis on uncorrelated data (black circles), long-term correlated data ( D 0:8; ˛ D 0:6, red squares), and short-term correlated data (summation of three AR processes, green diamonds). Averages of F2 (s) averaged over 20 series with N D 216 points and divided by s1/2 are shown, so that a horizontal line corresponds to uncorrelated behavior. The blue open triangles show the result for one selected extreme configuration, where it is hard to decide about the existence of long-term correlations (figure after [29])

(20)

The degree of the polynomial can be varied in order to eliminate constant (m D 0), linear (m D 1), quadratic (m D 2) or higher order trends of the profile function [15]. Conventionally the DFA is named after the order of the fitting polynomial (DFA0, DFA1, DFA2, . . . ). In DFAm, trends of order m in the profile Y( j) and of order m  1 in the original record x˜ i are eliminated. The variance of the detrended profile Y˜s ( j) in each segment  yields the meansquare fluctuations, s

2 (; s) D FDFAm

1 X ˜2 Ys ( j) : s jD1

(21)

471

472

Fractal and Multifractal Time Series

Fractal and Multifractal Time Series, Figure 6 Application of Detrended Fluctuation Analysis (DFA) on the data already studied in Fig. 5 (figure after [29])

2 As for FA and discrete wavelet analysis, the FDFAm (; s) are averaged over all segments  to obtain the mean fluctuations F2 (s), see Eq. (13). Calculating F2 (s) for many s, the fluctuation scaling exponent ˛ can be determined just as with FA, see Eq. (14). Figure 6 shows typical results for DFA of the same long-term correlated, short-term correlated and uncorrelated data studied already in Fig. 5. We note that in studies that include averaging over many records (or one record cut into many separate pieces by the elimination of some unreliable intermediate data points) the averaging procedure (13) must be performed for all data. Taking the square root is always the final step after all averaging is finished. It is not appropriate to calculate F2 (s) for parts of the data and then average the F2 (s) values, since such a procedure will bias the results towards smaller scaling exponents on large time scales. If F2 (s) increases for increasing s by F2 (s) s ˛ with 0:5 < ˛ < 1, one finds that the scaling exponent ˛  H is related to the correlation exponent by ˛ D 1  /2 (see Eq. (15)). A value of ˛ D 0:5 thus indicates that there are no (or only short-range) correlations. If ˛ > 0:5 for all scales s, the data are long-term correlated. The higher ˛, the stronger the correlations in the signal are. ˛ > 1 indicates a non-stationary local average of the data; in this case, FA fails and yields only ˛ D 1. The case ˛ < 0:5 corresponds to long-term anti-correlations, meaning that large values are most likely to be followed by small values and vice versa. ˛ values below 0 are not possible. Since the maximum value for ˛ in DFAm is mC1, higher detrending orders should be used for very non-stationary data with large ˛. Like in FA and Hurst’s analysis, ˛ will decrease or increase by one upon additional differentiation or integration of the data, respectively.

Small deviations from the scaling law (14), i. e. deviations from a straight line in a double logarithmic plot, occur for small scales s, in particular for DFAm with large detrending order m. These deviations are intrinsic to the usual DFA method, since the scaling behavior is only approached asymptotically. The deviations limit the capability of DFA to determine the correct correlation behavior in very short records and in the regime of small s. DFA6, e. g., is only defined for s  8, and significant deviations from the scaling law F2 (s) s ˛ occur even up to s  30. They will lead to an over-estimation of the fluctuation exponent ˛, if the regime of small s is used in a fitting procedure. An approach for correction of this systematic artefact in DFA is described in [35]. The number of independent segments of length s is larger in DFA than in WT, and the fluctuations in FA are larger than in DFA. Hence, the analysis has to be based on s values lower than smax D N/4 for DFA compared with smax D N/10 for FA and WT. The accuracy of scaling exponents ˛ determined by DFA was recently studied as a function of the length N of the data [43] (fitting range s 2 [10; N/2] was used). The results show that statistical standard errors of ˛ (one standard deviation) are approximately 0.1 for N D 500, 0.05 for N D 3000, and reach 0.03 for N D 10,000. Findings of long-term correlations with ˛ D 0:6 in data with only 500 points are thus not significant. A generalization of DFA for two-dimensional data (or even higher dimensions d) was recently suggested [34]. The generalization works well when tested with synthetic surfaces including fractional Brownian surfaces and multifractal surfaces. In the 2D procedure, a double cumulative sum (profile) is calculated by summing over both directional indices analogous with Eq. (10), Y(k; l) D Pk Pl ˜ i; j . This surface is partitioned into squares iD1 jD1 x of size s  s with indices  and , in which polynomials 2 like y;;s (i; j) D ai 2 C b j2 C ci j C di C e j C f are fitted. The fluctuation function F2 (s) is again obtained by calculating the variance of the profile from the fits. Detection of Trends and Crossovers with DFA Frequently, the correlations of recorded data do not follow the same scaling law for all time scales s, but one or sometimes even more crossovers between different scaling regimes are observed (see Subsect. “Crossovers and Non-stationarities in Time Series”). Time series with a well-defined crossover at s and vanishing correlations above s are most easily generated by Fourier filtering (see Subsect. “Fourier Filtering”). The power spectrum S( f ) of an uncorrelated random series is multiplied by ( f / f )ˇ

Fractal and Multifractal Time Series

with ˇ D 2˛  1 for frequencies f > f D 1/s only. The series obtained by inverse Fourier transform of this modified power spectrum exhibits power-law correlations on time scales s < s only, while the behavior becomes uncorrelated on larger time scales s > s . The crossover from F2 (s) s ˛ to F2 (s) s 1/2 is clearly visible in double logarithmic plots of the DFA fluctuation function for such short-term correlated data. However, it (m) occurs at times s that are different from the original s used for the generation of the data and that depend on the detrending order m. This systematic deviation is most significant in the DFAm with higher m. Extensive numerical simulations (see Fig. 3 in [35]) show that the ratios of (m) /s are 1.6, 2.6, 3.6, 4.5, and 5.4 for DFA1, DFA2, . . . , s DFA5, with an error bar of approximately 0.1. Note, however, that the precise value of this ratio will depend on the (m) method used for fitting the crossover times s (and the method used for generating the data if generated data is analyzed). If results for different orders of DFA shall be (m) compared, an observed crossover s can be systematically corrected dividing by the ratio for the corresponding DFAm. If several orders of DFA are used in the procedure, several estimates for the real s will be obtained, which can be checked for consistency or used for an error approximation. A real crossover can thus be well distinguished from the effects of non-stationarities in the data, which lead to a different dependence of an apparent crossover on m. The procedure is also required if the characteristic time scale of short-term correlations shall be studied with DFA. If consistent (corrected) s values are obtained based on DFAm with different m, the existence of a real characteristic correlation time scale is positively confirmed. Note that lower detrending orders are advantageous in this case, (m) might besince the observed crossover time scale s come quite large and nearly reach one forth of the total series length (N/4), where the results become statistically inaccurate. We would like to note that studies showing scaling long-term correlations should not be based on DFA or variants of this method alone in most applications. In particular, if it is not clear whether a given time series is indeed long-term correlated or just short-term correlated with a fairly large crossover time scale, results of DFA should be compared with other methods. For example, one can employ wavelet methods (see, e. g., Subsect. “Discrete Wavelet Transform (WT) Approach”). Another option is to remove short-term correlations by considering averaged series for comparison. For a time series with daily observations and possible short-term correlations up to two years, for example, one might consider the series of two-

year averages and apply DFA together with FA, binned power spectra analysis, and/or wavelet analysis. Only if these methods still indicate long-term correlations, one can be sure that the data are indeed long-term correlated. As discussed in Subsect. “Crossovers and Non-stationarities in Time Series”, records from real measurements are often affected by non-stationarities, and in particular by trends. They have to be well distinguished from the intrinsic fluctuations of the system. To investigate the effect of trends on the DFAm fluctuation functions, one can generate artificial series (x˜ i ) with smooth monotonous trends by adding polynomials of different power p to the original record (x i ), x˜ i D x i C Ax p

with

x D i/N :

(22)

For the DFAm, such trends in the data can lead to an artificial crossover in the scaling behavior of F2 (s), i. e., the slope ˛ is strongly increased for large time scales s. The position of this artificial crossover depends on the strength A and the power p of the trend. Evidently, no artificial crossover is observed, if the detrending order m is larger than p and p is integer. The order p of the trends in the data can be determined easily by applying the different DFAm. If p is larger than m or p is not an integer, an artificial crossover is observed, the slope ˛trend in the large s regime strongly depends on m, and the position of the artificial crossover also depends strongly on m. The artificial crossover can thus be clearly distinguished from real crossovers in the correlation behavior, which result in identical slopes ˛ and rather similar crossover positions for all detrending orders m. For more extensive studies of trends with non-integer powers we refer to [35,36]. The effects of periodic trends are also studied in [35]. If the functional form of the trend in given data is not known a priori, the fluctuation function F2 (s) should be calculated for several orders m of the fitting polynomial. If m is too low, F2 (s) will show a pronounced crossover to a regime with larger slope for large scales s [35,36]. The maximum slope of log F2 (s) versus log s is m C 1. The crossover will move to larger scales s or disappear when m is increased, unless it is a real crossover not due to trends. Hence, one can find m such that detrending is sufficient. However, m should not be larger than necessary, because shifts of the observed crossover time scales and deviations on short scales s increase with increasing m. Sign and Magnitude (Volatility) DFA To study the origin of long-term fractal correlations in a time series, the series can be split into two parts which are analyzed separately. It is particularly useful to split the

473

474

Fractal and Multifractal Time Series

series of increments, x i D x i  x i1 , i D 1; : : : ; N, into a series of signs x˜ i D s i D signx i and a series of magnitudes x˜ i D m i D jx i j [32,47,48]. There is an extensive interest in the magnitude time series in economics [49,50]. These data, usually called volatility, represent the absolute variations in stock (or commodity) prices and are used as a measure quantifying the risk of investments. While the actual prices are only short-term correlated, long-term correlations have been observed in volatility series [49,50]. Time series having identical distributions and longrange correlation properties can exhibit quite different temporal organizations of the magnitude and sign sub-series. The DFA method can be applied independently to both of these series. Since in particular the signs are often rather strongly anti-correlated and DFA will give incorrect results if ˛ is too close to zero, one often studies integrated sign and magnitude series. As mentioned above, integraP tion x˜ i ! ijD1 x˜ j increases ˛ by one. Most published results report short-term anti-correlations and no long-term correlations in the sign series, i. e., ˛sign < 1/2 for the non-integrated signs si (or ˛sign < 3/2 for the integrated signs) on low time scales and ˛sign ! 1/2 asymptotically for large s. The magnitude series, on the other hand, are usually either uncorrelated ˛magn D 1/2 (or 3/2) or positively long-term correlated ˛magn > 1/2 (or 3/2). It has been suggested that findings of ˛magn > 1/2 are related with nonlinear properties of the data and in particular multifractality [32,47,48], if ˛ < 1:5 in standard DFA. Specifically, the results suggest that the correlation exponent of the magnitude series is a monotonically increasing function of the multifractal spectrum (i. e., the singularity spectrum) width of the original series (see Subsect. “The Structure Function Approach and Singularity Spectra”). On the other hand, the sign series mainly relates to linear properties of the original series. At small time scales s < 16 the standard ˛ is approximately the average of ˛sign and ˛magn , if integrated sign and magnitude series are analyzed. For ˛ > 1:5 in the original series, the integrated magnitude and sign series have approximately the same two-point scaling exponents [47]. An analytical treatment is presented in [48]. Further Detrending Approaches A possible drawback of the DFA method is the occurrence of abrupt jumps in the detrended profile Y˜s ( j) (Eq. (20)) at the boundaries between the segments, since the fitting polynomials in neighboring segments are not related. A possible way to avoid these jumps would be the calculation of F2 (s) based on polynomial fits in overlapping windows. However, this is rather time consuming due to the polynomial fit in each segment and is consequently not

done in most applications. To overcome the problem of jumps several modifications and extensions of the FA and DFA methods have been suggested in recent years. These methods include  The detrended moving average technique [51,52,53], which we denote by the backward moving average (BMA) technique (following [54]),  The centered moving average (CMA) method [54], an essentially improved version of BMA,  The modified detrended fluctuation analysis (MDFA) [55], which is essentially a mixture of old FA and DFA,  The continuous DFA (CDFA) technique [56,57], which is particularly useful for the detection of crossovers,  The Fourier DFA [58],  A variant of DFA based on empirical mode decomposition (EMD) [59],  A variant of DFA based on singular value decomposition (SVD) [60,61], and  A variant of DFA based on high-pass filtering [62]. Detrended moving average techniques will be thoroughly described and discussed in the next section. A study comparing DFA with CMA and MDFA can be found in [43]. For studies comparing DFA and BMA, see [63,64]; note that [64] also discusses CMA. The method we denote as modified detrended fluctuation analysis (MDFA) [55], eliminates trends similar to the DFA method. A polynomial is fitted to the profile function Y( j) in each segment  and the deviation between the profile function and the polynomial fit is calculated, p Y˜s ( j) D Y( j)  y;s ( j) (Eq. (20)). To estimate correlations in the data, this method uses a derivative of Y˜s ( j), obtained for each segment , by Y˜s ( j) D Y˜s ( jCs/2)Y˜s ( j). Hence, the fluctuation function (compare with Eqs. (13) and (21)) is calculated as follows: 31/2 2 N X 2  1 (23) Y˜s ( j C s/2)  Y˜s ( j) 5 : F2 (s) D 4 N jD1

As in case of DFA, MDFA can easily be generalized to remove higher order trends in the data. Since the fitting polynomials in adjacent segments are not related, Y˜s ( j) shows abrupt jumps on their boundaries as well. This leads to fluctuations of F2 (s) for large segment sizes s and limits the maximum usable scale to s < N/4 as for DFA. The detection of crossovers in the data, however, is more exact with MDFA (compared with DFA), since no correction of the estimated crossover time scales seems to be needed [43]. The Fourier-detrended fluctuation analysis [58] aims to eliminate slow oscillatory trends which are found espe-

Fractal and Multifractal Time Series

cially in weather and climate series due to seasonal influences. The character of these trends can be rather periodic and regular or irregular, and their influence on the detection of long-range correlations by means of DFA was systematically studied previously [35]. Among other things it has been shown that slowly varying periodic trends disturb the scaling behavior of the results much stronger than quickly oscillating trends and thus have to be removed prior to the analysis. In the case of periodic and regular oscillations, e. g., in temperature fluctuations, one simply removes the low frequency seasonal trend by subtracting the daily mean temperatures from the data. Another way, which the Fourier-detrended fluctuation analysis suggests, is to filter out the relevant frequencies in the signals’ Fourier spectrum before applying DFA to the filtered signal. Nevertheless, this method faces several difficulties especially its limitation to periodic and regular trends and the need for a priori knowledge of the interfering frequency band. To study correlations in data with quasi-periodic or irregular oscillating trends, empirical mode decomposition (EMD) was suggested [59]. The EMD algorithm breaks down the signal into its intrinsic mode functions (IMFs) which can be used to distinguish between fluctuations and background. The background, estimated by a quasi-periodic fit containing the dominating frequencies of a sufficiently large number of IMFs, is subtracted from the data, yielding a slightly better scaling behavior in the DFA curves. However, we believe that the method might be too complicated for wide-spread applications. Another method which was shown to minimize the effect of periodic and quasi-periodic trends is based on singular value decomposition (SVD) [60,61]. In this approach, one first embeds the original signal in a matrix whose dimension has to be much larger than the number of frequency components of the periodic or quasi-periodic trends obtained in the power spectrum. Applying SVD yields a diagonal matrix which can be manipulated by setting the dominant eigenvalues (associated with the trends) to zero. The filtered matrix finally leads to the filtered data, and it has been shown that subsequent application of DFA determines the expected scaling behavior if the embedding dimension is sufficiently large. None the less, the performance of this rather complex method seems to decrease for larger values of the scaling exponent. Furthermore SVD-DFA assumes that trends are deterministic and narrow banded. The detrending procedure in DFA (Eq. (20)) can be regarded as a scale-dependent high-pass filter since (low-frequency) fluctuations exceeding a specific scale s are eliminated. Therefore, it has been suggested to obtain the de-

trended profile Y˜s ( j) for each scale s directly by applying digital high-pass filters [62]. In particular, Butterworth, Chebyshev-I, Chebyshev-II, and an elliptical filter were suggested. While the elliptical filter showed the best performance in detecting long-range correlations in artificial data, the Chebyshev-II filter was found to be problematic. Additionally, in order to avoid a time shift between filtered and original profile, the average of the directly filtered signal and the time reversed filtered signal is considered. The effects of these complicated filters on the scaling behavior are, however, not fully understood. Finally, a continuous DFA method has been suggested in the context of studying heartbeat data during sleep [56, 57]. The method compares unnormalized fluctuation functions F2 (s) for increasing length of the data. I. e., one starts with a very short recording and subsequently adds more points of data. The method is particularly suitable for the detection of change points in the data, e. g., physiological transitions between different activity or sleep stages. Since the main objective of the method is not the study of scaling behavior, we do not discuss it in detail here. Centered Moving Average (CMA) Analysis Particular attractive modifications of DFA are the detrended moving average (DMA) methods, where running averages replace the polynomial fits. The first suggested version, the backward moving average (BMA) method [51,52,53], however, suffers from severe problems, because an artificial time shift of s between the original signal and the moving average is introduced. This time shift leads to an additional contribution to the detrended profile Y˜s ( j), which causes a larger fluctuation function F2 (s) in particular for small scales in the case of long-term correlated data. Hence, the scaling exponent ˛ is systematically underestimated [63]. In addition, the BMA method preforms even worse for data with trends [64], and its slope is limited by ˛ < 1 just as for the non-detrending method FA. It was soon recognized that the intrinsic error of BMA can be overcome by eliminating the artificial time shift. This leads to the centered moving average (CMA) method [54], where Y˜s ( j) is calculated as 1 Y˜s ( j) D Y( j)  s

(s1)/2 X

Y( j C i) ;

(24)

iD(s1)/2

replacing Eq. (20) while Eq. (21) and the rest of the DFA procedure described in Subsect. “Detrended Fluctuation Analysis (DFA)” stay the same. Unlike DFA, the CMA method cannot easily be generalized to remove linear and higher order trends in the data.

475

476

Fractal and Multifractal Time Series

It was recently proposed [43] that the scaling behavior of the CMA method is more stable than for DFA1 and MDFA1, suggesting that CMA could be used for reliable computation of ˛ even for scales s < 10 (without correction of any systematic deviations needed in DFA for this regime) and up to smax D N/2. The standard errors in determining the scaling exponent ˛ by fitting straight lines to the double logarithmic plots of F2 (s) have been studied in [43]; they are comparable with DFA1 (see end of Subsect. “Detrended Fluctuation Analysis (DFA)”). Regarding the determination of crossovers, CMA is comparable to DFA1. Ultimately, the CMA seems to be a good alternative to DFA1 when analyzing the scaling properties in short data sets without trends. Nevertheless for data with possible unknown trends we recommend the application of standard DFA with several different detrending polynomial orders in order to distinguish real crossovers from artificial crossovers due to trends. In addition, an independent approach (e. g., wavelet analysis) should be used to confirm findings of long-term correlations (see also Subsect. “Detection of Trends and Crossovers with DFA”). Methods for Multifractal Time Series Analysis This section describes the multifractal characterization of time series, for an introduction, see Subsect. “Multifractal Time Series”. The simplest type of multifractal analysis is based upon the standard partition function multifractal formalism, which has been developed for the multifractal characterization of normalized, stationary measures [6,12,65,66]. Unfortunately, this standard formalism does not give correct results for non-stationary time series that are affected by trends or that cannot be normalized. Thus, in the early 1990s an improved multifractal formalism has been developed, the wavelet transform modulus maxima (WTMM) method [67,68,69,70,71], which is based on wavelet analysis and involves tracing the maxima lines in the continuous wavelet transform over all scales. An important alternative is the multifractal DFA (MFDFA) algorithm [33], which does not require the modulus maxima procedure, and hence involves little more effort in programming than the conventional DFA. For studies comparing methods for detrending multifractal analysis (MFDFA and WTMM, see [33,72,73]. The Structure Function Approach and Singularity Spectra In the general multifractal formalism, one considers a normalized measure (t), t 2 [0; 1], and defines the box R tCs/2 ˜ s (t) D ts/2 (t 0 ) dt 0 in neighborhoods of probabilities 

(scale) length s  1 around t. The multifractal approach is then introduced by the partition function Z q (s) D

1/s1 X

˜ s [( C 1/2)s] s (q)  q

for s  1 ; (25)

D0

where (q) is the Renyi scaling exponent and q is a real parameter that can take positive as well as negative values. Note that (q) is sometimes defined with opposite sign (see, e. g., [6]). A record is called monofractal (or selfaffine), when the Renyi scaling exponent (q) depends linearly on q; otherwise it is called multifractal. The generalized multifractal dimensions D(q) (see also Subsect. “Multifractal Time Series”) are related to (q) by D(q) D (q)/(q  1), such that the fractal dimension of the support is D(0) D (0) and the correlation dimension is D(2) D (2). In time series, a discrete version has to be used, and the considered data (x i ), i D 1; : : : ; N may usually include negative values. Hence, setting N s D int(N/s) and P X(; s) D siD1 xsCi for  D 0; : : : ; N s  1 we can define [6,12], NX s 1

Z q (s) D

jX(; s)j q s (q)

for s > 1 :

(26)

D0

Inserting the profile Y( j) and FFA (; s) from Eqs. (10) and (11), respectively, we obtain Z q (s) D

NX s 1

˚

[Y(( C 1)s)  Y(s)]2

q/2

D0

D

Ns X

jFFA (; s)j :

(27)

D1

Comparing Eq. (27) with (13), we see that this multifractal approach can be considered as a generalized version of the fluctuation analysis (FA) method, where the exponent 2 is replaced by q. In particular we find (disregarding the summation over the second partition of the time series)

F2 (s)

1/2 1 Z2 (s) s [1C(2)]/2 Ns ) 2˛ D 1 C (2) D 1 C D(2):

(28)

We thus see that all methods for (mono-)fractal time analysis (discussed in Sect. “Methods for Stationary Fractal Time Series Analysis”and Sect. “Methods for Non-stationary Fractal Time Series Analysis”) in fact study the correlation dimension D(2) D 2˛ 1 D ˇ D 1 (see Eq. (15)).

Fractal and Multifractal Time Series

It is straightforward to define a generalized (multifractal) Hurst exponent h(q) for the scaling behavior of the qth moments of the fluctuations [65,66],

1/q 1 Fq (s) D Z2 (s) s [1C(q)]/q D s h(q) Ns 1 C (q) ) h(q) D q

d (q) dq

and

f (˛) D q˛  (q) :

(29)

(30)

Here, ˛ is the singularity strength or Hölder exponent (see also  Fractals and Multifractals, Introduction to in the encyclopedia), while f (˛) denotes the dimension of the subset of the series that is characterized by ˛. Note that ˛ is not the fluctuation scaling exponent in this section, although the same letter is traditionally used for both. Using Eq. (29), we can directly relate ˛ and f (˛) to h(q), ˛ D h(q)C qh0 (q)

and

Z(q; s) D

j max X

jL ( j ; s)j q :

(32)

jD1

with h(2) D ˛  H. In the following, we will use only h(2) for the standard fluctuation exponent (denoted by ˛ in the previous chapters), and reserve the letter ˛ for the Hölder exponent. Another way to characterize a multifractal series is the singularity spectrum f (˛), that is related to (q) via a Legendre transform [6,12], ˛D

up the qth power of the maxima,

The reason for the maxima procedure is that the absolute wavelet coefficients jL (; s)j can become arbitrarily small. The analyzing wavelet (x) must always have positive values for some x and negative values for other x, since it has to be orthogonal to possible constant trends. Hence there are always positive and negative terms in the sum (16), and these terms might cancel. If that happens, jL (; s)j can become close to zero. Since such small terms would spoil the calculation of negative moments in Eq. (32), they have to be eliminated by the maxima procedure. In fluctuation analysis, on the other hand, the calculation of the variances F 2 (; s), e. g. in Eq. (11), involves only positive terms under the summation. The variances cannot become arbitrarily small, and hence no maximum procedure is required for series with compact support. In addition, the variances will increase if the segment length s is increased, because the fit will usually be worse for a longer segment. In the WTMM method, in contrast, the absolute

f (˛) D q[˛  h(q)]C1: (31)

Wavelet Transform Modulus Maxima (WTMM) Method The wavelet transform modulus maxima (WTMM) method [67,68,69,70,71] is a well-known method to investigate the multifractal scaling properties of fractal and selfaffine objects in the presence of non-stationarities. For applications, see e. g. [74,75]. It is based upon the wavelet transform with continuous basis functions as defined in Subsect. “Wavelet Analysis”, Eq. (16). Note that in this case the series x˜ i are analyzed directly instead of the profile Y( j) defined in Eq. (10). Using wavelets orthogonal to mth order polynomials, the corresponding trends are eliminated. Instead of averaging over all wavelet coefficients L (; s), one averages, within the modulo-maxima method, only the local maxima of jL (; s)j. First, one determines for a given scale s, the positions  j of the local maxima of jW(; s)j as a function of , so that jL ( j  1; s)j < jL ( j ; s)j  jL ( j C1; s)j for j D 1; : : : ; jmax . This maxima procedure is demonstrated in Fig. 7. Then one sums

Fractal and Multifractal Time Series, Figure 7 Example of the wavelet transform modulus maxima (WTMM) method, showing the original data (top), its continuous wavelet transform (gray scale coded amplitude of wavelet coefficients, middle), and the extracted maxima lines (bottom) (figure taken from [68])

477

478

Fractal and Multifractal Time Series

wavelet coefficients jL (; s)j need not increase with increasing scale s, even if only the local maxima are considered. The values jL (; s)j might become smaller for increasing s since just more (positive and negative) terms are included in the summation (16), and these might cancel even better. Thus, an additional supremum procedure has been introduced in the WTMM method in order to keep the dependence of Z(q; s) on s monotonous. If, for a given scale s, a maximum at a certain position  j happens to be smaller than a maximum at  j0   j for a lower scale s 0 < s, then L ( j ; s) is replaced by L ( j0 ; s 0 ) in Eq. (32). Often, scaling behavior is observed for Z(q; s), and scaling exponents ˆ (q) can be defined that describe how Z(q; s) scales with s, Z(q; s) sˆ (q) :

(33)

The exponents ˆ (q) characterize the multifractal properties of the series under investigation, and theoretically they are identical with the (q) defined in Eq. (26) [67,68,69,71] and related to h(q) by Eq. (29). Multifractal Detrended Fluctuation Analysis (MFDFA) The multifractal DFA (MFDFA) procedure consists of five steps [33]. The first three steps are essentially identical to the conventional DFA procedure (see Subsect. “Detrended Fluctuation Analysis (DFA)” and Fig. 4). Let us assume that (x˜ i ) is a series of length N, and that this series is of compact support. The support can be defined as the set of the indices j with nonzero values x˜ j , and it is compact if x˜ j D 0 for an insignificant fraction of the series only. The value of x˜ j D 0 is interpreted as having no value at this j. Note that we are not discussing the fractal or multifractal features of the plot of the time series in a two-dimensional graph (see also the discussion in Subsect. “Fractality, SelfAffinity, and Scaling”), but analyzing time series as onedimensional structures with values assigned to each point. Since real time series always have finite length N, we explicitly want to determine the multifractality of finite series, and we are not discussing the limit for N ! 1 here (see also Subsect. “The Structure Function Approach and Singularity Spectra”). Step 1 Calculate the profile Y( j), Eq. (10), by integrating the time series. Step 2 Divide the profile Y( j) into N s D int(N/s) nonoverlapping segments of equal length s. Since the length N of the series is often not a multiple of the considered time scale s, the same procedure can be repeated starting from the opposite end. Thereby, 2N s segments are obtained altogether.

Step 3 Calculate the local trend for each of the 2N s segments by a least-square fit of the profile. Then determine the variance by Eqs. (20) and (21) for each segment  D 1; : : : ; 2Ns . Again, linear, quadratic, cubic, or higher order polynomials can be used in the fitting procedure, and the corresponding methods are thus called MFDFA1, MFDFA2, MFDFA3, . . . ) [33]. In (MF-)DFAm [mth order (MF-)DFA] trends of order m in the profile (or, equivalently, of order m  1 in the original series) are eliminated. Thus a comparison of the results for different orders of DFA allows one to estimate the type of the polynomial trend in the time series [35,36]. Step 4 Average over all segments to obtain the qth order fluctuation function ( ) 1/q 2N q/2 1 Xs  2 F (; s) : (34) Fq (s) D 2N s D1 DFAm This is the generalization of Eq. (13) suggested by the relations derived in Subsect. “The Structure Function Approach and Singularity Spectra”. For q D 2, the standard DFA procedure is retrieved. One is interested in how the generalized q dependent fluctuation functions Fq (s) depend on the time scale s for different values of q. Hence, we must repeat steps 2 to 4 for several time scales s. It is apparent that Fq (s) will increase with increasing s. Of course, Fq (s) depends on the order m. By construction, Fq (s) is only defined for s  m C 2. Step 5 Determine the scaling behavior of the fluctuation functions by analyzing log-log plots Fq (s) versus s for each value of q. If the series x˜ i are long-range powerlaw correlated, Fq (s) increases, for large values of s, as a power-law, Fq (s) s h(q)

with

h(q) D

1 C (q) : q

(35)

For very large scales, s > N/4, Fq (s) becomes statistically unreliable because the number of segments N s for the averaging procedure in step 4 becomes very small. Thus, scales s > N/4 should be excluded from the fitting procedure determining h(q). Besides that, systematic deviations from the scaling behavior in Eq. (35), which can be corrected, occur for small scales s  10. The value of h(0), which corresponds to the limit h(q) for q ! 0, cannot be determined directly using the averaging procedure in Eq. (34) because of the diverging exponent. Instead, a logarithmic averaging procedure has to be employed, ( ) 2N 1 Xs  2 F0 (s) D exp ln F (; s) s h(0) : (36) 4N s D1

Fractal and Multifractal Time Series

Note that h(0) cannot be defined for time series with fractal support, where h(q) diverges for q ! 0. For monofractal time series with compact support, h(q) is independent of q, since the scaling behavior of the 2 variances FDFAm (; s) is identical for all segments , and the averaging procedure in Eq. (34) will give just this identical scaling behavior for all values of q. Only if small and large fluctuations scale differently, there will be a significant dependence of h(q) on q. If we consider positive values of q, the segments  with large variance F 2 (; s) (i. e. large deviations from the corresponding fit) will dominate the average Fq (s). Thus, for positive values of q, h(q) describes the scaling behavior of the segments with large fluctuations. On the contrary, for negative values of q, the 2 segments  with small variance FDFAm (; s) will dominate the average Fq (s). Hence, for negative values of q, h(q) describes the scaling behavior of the segments with small fluctuations. Figure 8 shows typical results obtained for Fq (s) in the MFDFA procedure. Usually the large fluctuations are characterized by a smaller scaling exponent h(q) for multifractal series than the small fluctuations. This can be understood from the following arguments. For the maximum scale s D N the fluctuation function Fq (s) is independent of q, since the sum in Eq. (34) runs over only two identical segments. For smaller scales s  N the averaging procedure runs over several segments, and the average value Fq (s) will be dominated by the F 2 (; s) from the segments with small (large) fluctuations if q < 0 (q > 0). Thus, for s  N, Fq (s) with

Fractal and Multifractal Time Series, Figure 8 Multifractal detrended fluctuation analysis (MFDFA) of data from the binomial multifractal model (see Subsect. “The Extended Binomial Multifractal Model”) with a D 0:75. Fq (s) is plotted versus s for the q values given in the legend; the slopes of the curves correspond to the values of h(q). The dashed lines have the slopes of the theoretical slopes h(˙1) from Eq. (42). 100 configurations have been averaged

q < 0 will be smaller than Fq (s) with q > 0, while both become equal for s D N. Hence, if we assume an homogeneous scaling behavior of Fq (s) following Eq. (35), the slope h(q) in a log-log plot of Fq (s) with q < 0 versus s must be larger than the corresponding slope for Fq (s) with q > 0. Thus, h(q) for q < 0 will usually be larger than h(q) for q > 0. However, the MFDFA method can only determine positive generalized Hurst exponents h(q), and it already becomes inaccurate for strongly anti-correlated signals when h(q) is close to zero. In such cases, a modified (MF-)DFA technique has to be used. The most simple way to analyze such data is to integrate the time series before the MFDFA procedure. Following the MFDFA procedure as described above, we obtain a generalized fluctuation ˜ functions described by a scaling law with h(q) D h(q) C 1. The scaling behavior can thus be accurately determined even for h(q) which are smaller than zero for some values of q. The accuracy of h(q) determined by MFDFA certainly depends on the length N of the data. For q D ˙10 and data with N D 10,000 and 100,000, systematic and statistical error bars (standard deviations) up to h(q)  ˙0:1 and  ˙0:05 should be expected, respectively [33]. A difference of h(10)  h(C10) D 0:2, corresponding to an even larger width ˛ of the singularity spectrum f (˛) defined in Eq. (30) is thus not significant unless the data was longer than N D 10,000 points. Hence, one has to be very careful when concluding multifractal properties from differences in h(q). As already mentioned in the introduction, two types of multifractality in time series can be distinguished. Both of them require a multitude of scaling exponents for small and large fluctuations: (i) Multifractality of a time series can be due to a broad probability density function for the values of the time series, and (ii) multifractality can also be due to different long-range correlations for small and large fluctuations. The most easy way to distinguish between these two types is by analyzing also the corresponding randomly shuffled series [33]. In the shuffling procedure the values are put into random order, and thus all correlations are destroyed. Hence the shuffled series from multifractals of type (ii) will exhibit simple random behavior, hshuf (q) D 0:5, i. e. non-multifractal scaling. For multifractals of type (i), on the contrary, the original h(q) dependence is not changed, h(q) D hshuf (q), since the multifractality is due to the probability density, which is not affected by the shuffling procedure. If both kinds of multifractality are present in a given series, the shuffled series will show weaker multifractality than the original one.

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Fractal and Multifractal Time Series, Figure 9 Illustration for the definition of return intervals rq between extreme events above two quantiles (thresholds) q1 and q2 (figure by Jan Eichner)

Comparison of WTMM and MFDFA The MFDFA results turn out to be slightly more reliable than the WTMM results [33,72,73]. In particular, the MFDFA has slight advantages for negative q values and short series. In the other cases the results of the two methods are rather equivalent. Besides that, the main advantage of the MFDFA method compared with the WTMM method lies in the simplicity of the MFDFA method. However, contrary to WTMM, MFDFA is restricted to studies of data with full one-dimensional support, while WTMM is not. Both WTMM and MFDFA have been generalized for higher dimensional data, see [34] for higher dimensional MFDFA and, e. g., [71] for higher dimensional WTMM. Studies of other generalizations of detrending methods like the discrete WT approach (see Subsect. “Discrete Wavelet Transform (WT) Approach”) and the CMA method (see Subsect. “Centered Moving Average (CMA) Analysis”) are currently under investigation [76]. Statistics of Extreme Events in Fractal Time Series The statistics of return intervals between well defined extremal events is a powerful tool to characterize the temporal scaling properties of observed time series and to derive quantities for the estimation of the risk for hazardous events like floods, very high temperatures, or earthquakes. It was shown recently that long-term correlations represent a natural mechanism for the clustering of the hazardous events [77]. In this section we will discuss the most important consequences of long-term correlations and fractal scaling of time series upon the statistics of extreme events [77,78,79,80,81]. Corresponding work regarding multifractal data [82] is not discussed here. Return Intervals Between Extreme Events To study the statistics of return intervals we consider again a time series (x i ), i D 1; : : : ; N with fractal scaling behavior, sampled homogeneously and normalized to zero mean

and unit variance. For describing the reoccurrence of rare events exceeding a certain threshold q, we investigate the return intervals rq between these events, see Fig. 9. The average return interval R q D hr q i increases as a function of the threshold q (see, e. g. [83]). It is known that for uncorrelated records (‘white noise’), the return intervals are also uncorrelated and distributed according to the Poisson distribution, Pq (r) D R1q exp(r/R q ). For fractal (long-term correlated) data with auto-correlations following Eq. (5), we obtain a stretched exponential [77,78,79,80,84], Pq (r) D

 a exp b (r/R q ) : Rq

(37)

This behavior is shown in Fig. 10. The exponent is the correlation exponent from C(s), and the parameters a and b are independent of q. They can be determined from R (r), i. e., P (r) dr D1 the normalization conditions for P q q R and rPq (r) dr D R q . The form of the distribution (37) indicates that return intervals both well below and well above their average value Rq (which is independent of ) are considerably more frequent for long-term correlated than for uncorrelated data. It has to be noted that there are deviations from the stretched exponential law (37) for very small r (discretization effects and an additional power-law regime) and for very large r (finite size effects), see Fig. 10. The extent of the deviations from Eq. (37) depends on the distribution of the values xi of the time series. For a discussion of these effects, see [80]. Equation (37) does not quantify, however, if the return intervals themselves are arranged in a correlated or in an uncorrelated fashion, and if clustering of rare events may be induced by long-term correlations. To study this question, one has to evaluate the auto-covariance function Cr (s) D hr q (l)r q (l C s)i  R2q of the return intervals. The results for model data suggests that also the return intervals are long-term power-law correlated, with the same exponent as the original record. Accordingly, large and small return intervals are not arranged in a random fash-

Fractal and Multifractal Time Series

Fractal and Multifractal Time Series, Figure 10 Normalized rescaled distribution density functions Rq Pq (r) of r values with Rq D 100 as a function of r/Rq for long-term correlated data with  D 0:4 (open symbols) and  D 0:2 (filled symbols; we multiplied the data for the filled symbols by a factor 100 to avoid overlapping curves). In a the original data were Gaussian distributed, in b exponentially distributed, in c power-law distributed with power 5.5, and in d log-normally distributed. All four figures follow quite well stretched exponential curves (solid lines) over several decades. For small r/Rq values a power-law regime seems to dominate, while on large scales deviations from the stretched exponential behavior are due to finite-size effects (figure by Jan Eichner)

ion but are expected to form clusters. As a consequence, the probability of finding a certain return interval r depends on the value of the preceding interval r0 , and this effect has to be taken into account in predictions and risk estimations [77,80]. The conditional distribution function Pq (rjr0 ) is a basic quantity, from which the relevant quantities in risk estimations can be derived [83]. For example, the first moment of Pq (rjr0 ) is the average value R q (r0 ) of those return intervals that directly follow r0 . By definition, R q (r0 ) is the expected waiting time to the next event, when the two events before were separated by r0 . The more general quantity is the expected waiting time q (xjr0 ) to the next event, when the time x has elapsed. For x D 0, q (0jr0 ) is identical to R q (r0 ). In general, q (xjr0 ) is related to Pq (rjr0 ) by Z 1 (r  x)Pq (rjr0 )dr : (38) q (xjr0 ) D x Z 1 Pq (rjr0 )dr

Distribution of Extreme Events In this section we describe how the presence of fractal long-term correlations affects the statistics of the extreme events, i. e., maxima within time segments of fixed duration R, see Fig. 11 for illustration. By definition, extreme events are rare occurrences of extraordinary nature, such as, e. g. floods, very high temperatures, or earthquakes. In hydrological engineering such conventional extreme value statistics are commonly applied to decide what building projects are required to protect river-side areas against typical floods that occur, for example, once in 100 years. Most of these results are based on statistically independent values xi and hold only in the limit R ! 1. However, both of these assumptions are not strictly fulfilled for correlated fractal scaling data. In classical extreme value statistics one assumes that records (x i ) consist of i.i.d. data, described by density distributions P(x), which can be, e. g., a Gaussian or an exponential distribution. One is interested in the distribution

x

For uncorrelated records, q (xjr0 )/R q D 1 (except for discreteness effects that lead to q (xjr0 )/R q > 1 for x > 0, see [85]). Due to the scaling of Pq (rjr0 ), also q (xjr0 )/R q scales with r0 /R q and x/R q . Small and large return intervals are more likely to be followed by small and large ones, respectively, and hence q (0jr0 )/R q D R q (r0 )/R q is well below (above) one for r0 /R q well below (above) one. With increasing x, the expected residual time to the next event increases. Note that only for an infinite long-term correlated record, the value of q (xjr0 ) will increase indefinitely with x and r0 . For real (finite) records, there exists a maximum return interval which limits the values of x, r0 and q (xjr0 ).

Fractal and Multifractal Time Series, Figure 11 Illustration for the definition of maxima mR within periods of R D 365 values (figure by Jan Eichner)

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density function PR (m) of the maxima (m j ) determined in segments of length R in the original series (x i ), see Fig. 11. Note that all maxima are also elements of the original data. The corresponding integrated maxima distribution G R (m) is defined as Z m G R (m) D 1  E R (m) D PR (m0 )dm0 : (39)

previous maximum [81]. The last item implies that conditional mean maxima and conditional maxima distributions should be considered for improved extreme event predictions. Simple Models for Fractal and Multifractal Time Series

1

Fourier Filtering Since G R (m) is the probability of finding a maximum smaller than m, E R (m) denotes the probability of finding a maximum that exceeds m. One of the main results of traditional extreme value statistics states that for independently and identically distributed (i.i.d.) data (x i ) with Gaussian or exponential distribution density function P(x) the integrated distribution G R (m) converges to a double exponential (Fisher–Tippet–Gumbel) distribution (often labeled as Type I) [86,87,88,89,90], i. e., G R (m) ! G

m  u ˛

h  m  u i D exp  exp  (40) ˛

for R ! 1, where ˛ is the scale parameter and u the location parameter. By the method of moments those pap rameters are given by ˛ D 6/ R and u D m R  n e ˛ with the Euler constant n e D 0:577216 [89,91,92,93]. Here mR and  R denote the (R-dependent) mean maximum and the standard deviation, respectively. Note that different asymptotics will be reached for broader distributions of data (x i ) that belong to other domains of attraction [89]. For example, for data following a power-law distribution (or Pareto distribution), P(x) D (x/x0 )k , G R (m) converges to a Fréchet distribution, often labeled as Type II. For data following a distribution with finite upper endpoint, for example the uniform distribution P(x) D 1 for 0  x  1, G R (m) converges to a Weibull distribution, often labeled as Type III. We do not consider the latter two types of asymptotics here. Numerical studies of fractal model data have recently shown that the distribution P(x) of the original data has a much stronger effect upon the convergence towards the Gumbel distribution than the long-term correlations in the data. Long-term correlations just slightly delay the convergence of G R (m) towards the Gumbel distribution (40). This can be observed very clearly in a plot of the integrated and scaled distribution G R (m) on logarithmic scale [81]. Furthermore, it was found numerically that (i) the maxima series (m j ) exhibit long-term correlations similar to those of the original data (x i ), and most notably (ii) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the

Fractal scaling with long-term correlations can be introduced most easily into time series by the Fourier-filtering technique, see, e. g., [94,95,96]. The Fourier filtering technique is not limited to the generation of long-term correlated data characterized by a power-law auto-correlation function C(s) x  with 0 < < 1. All values of the scaling exponents ˛ D h(2)  H or ˇ D 2˛  1 can be obtained, even those that cannot be found directly by the fractal analysis techniques described in Sect. “Methods for Stationary Fractal Time Series Analysis” and Sect. “Methods for Non-stationary Fractal Time Series Analysis” (e. g. ˛ < 0). Note, however, that Fourier filtering will always yield Gaussian distributed data values and that no nonlinear or multifractal properties can be achieved (see also Subsect. “Multifractal Time Series”, Subsect. “Sign and Magnitude (Volatility) DFA”, and Sect. “Methods for Multifractal Time Series Analysis”). In Subsect. “Detection of Trends and Crossovers with DFA”, we have briefly described a modification of Fourier filtering for obtaining reliable short-term correlated data. For the generation of data characterized by fractal scaling with ˇ D 2˛  1 [94,95] we start with uncorrelated Gaussian distributed random numbers xi from an i.i.d. generator. Transforming a series of such numbers into frequency space with discrete Fourier transform or FFT (fast Fourier transform, for suitable series lengths N) yields a flat power spectrum, since random numbers correspond to white noise. Multiplying the (complex) Fourier coefficients by f ˇ /2 , where f / 1/s is the frequency, will rescale the power spectrum S( f ) to follow Eq. (6), as expected for time series with fractal scaling. After transforming back to the time domain (using inverse Fourier transform or inverse FFT) we will thus obtain the desired longterm correlated data x˜i . The final step is the normalization of this data. The Fourier filtering method can be improved using modified Bessel functions instead of the simple factors f ˇ /2 in modifying the Fourier coefficients [96]. This way problems with the divergence of the autocorrelation function C(s) at s D 0 can be avoided. An alternative method to the Fourier filtering technique, the random midpoint displacement method, is

Fractal and Multifractal Time Series

based on the construction of self-affine surfaces by an iterative procedure, see, e. g. [6]. Starting with one interval with constant values, the intervals are iterative split in the middle and the midpoint is displaced by a random offset. The amplitude of this offset is scaled according to the length of the interval. Since the method generates a selfaffine surface xi characterized by a Hurst exponent H, the differentiated series x i can be used as long-term correlated or anti-correlated random numbers. Note, however, that the correlations do not persist for the whole length of the data generated this way. Another option is the use of wavelet synthesis, the reverse of wavelet analysis described in Subsect. “Wavelet Analysis”. In that method, the scaling law is introduced by setting the magnitudes of the wavelet coefficients according to the corresponding time scale s. The Schmitz–Schreiber Method When long-term correlations in random numbers are introduced by the Fourier-filtering technique (see previous section), the original distribution P(x) of the time series values xi is always modified such that it becomes closer to a Gaussian. Hence, no series (x i ) with broad distributions of the values and fractal scaling can be generated. In these cases an iterative algorithm introduced by Schreiber and Schmitz [98,99] must be applied. The algorithm consists of the following steps: First one creates a Gaussian distributed long-term correlated data set with the desired correlation exponent by standard Fourier-filtering [96]. The power spectrum SG ( f ) D FG ( f )FG ( f ) of this data set is considered as reference spectrum (where f denotes the frequency in Fourier space and the FG ( f ) are the complex Fourier coefficients). Next one creates an uncorrelated sequence of random numbers (x ref i ), following a desired distribution P(x). The (complex) Fourier transform F( f ) of the (x ref i ) is now divided by its absolute value and multiplied by the square root of the reference spectrum, p F( f ) SG ( f ) : (41) Fnew( f ) D jF( f )j After the Fourier back-transformation of Fnew ( f ), the new sequence (x new i ) has the desired correlations (i. e. the desired ), but the shape of the distribution has changed towards a (more or less) Gaussian distribution. In order to enforce the desired distribution, we exchange the (x new i ) by the (x ref ), such that the largest value of the new set is i replaced by the largest value of the reference set, the second largest of the new set by the second largest of the reference set and so on. After this the new sequence has the desired distribution and is clearly correlated. However, due

to the exchange algorithm the perfect long-term correlations of the new data sequence were slightly altered again. So the procedure is repeated: the new sequence is Fourier transformed followed by spectrum adjustment, and the exchange algorithm is applied to the Fourier back-transformed data set. These steps are repeated several times, until the desired quality (or the best possible quality) of the spectrum of the new data series is achieved. The Extended Binomial Multifractal Model The multifractal cascade model [6,33,65] is a standard model for multifractal data, which is often applied, e. g., in hydrology [97]. In the model, a record xi of length N D 2n max is constructed recursively as follows. In generation n D 0, the record elements are constant, i. e. x i D 1 for all i D 1; : : : ; N. In the first step of the cascade (generation n D 1), the first half of the series is multiplied by a factor a and the second half of the series is multiplied by a factor b. This yields x i D a for i D 1; : : : ; N/2 and x i D b for i D N/2 C 1; : : : ; N. The parameters a and b are between zero and one, 0 < a < b < 1. One need not restrict the model to b D 1  a as is often done in the literature [6]. In the second step (generation n D 2), we apply the process of step 1 to the two subseries, yielding x i D a2 for i D 1; : : : ; N/4, x i D ab for i D N/4 C 1; : : : ; N/2, x i D ba D ab for i D N/2 C 1; : : : ; 3N/4, and x i D b2 for i D 3N/4 C 1; : : : ; N. In general, in step n C 1, each subseries of step n is divided into two subseries of equal length, and the first half of the xi is multiplied by a while the second half is multiplied by b. For example, in generation n D 3 the values in the eight subseries are a3 ; a2 b; a2 b; ab2 ; a2 b; ab2 ; ab2 ; b3 . After nmax steps, the final generation has been reached, where all subseries have length 1 and no more splitting is possible. We note that the final record can be written as x i D a n max n(i1) b n(i1) , where n(i) is the number of digits 1 in the binary representation of the index i, e. g. n(13) D 3, since 13 corresponds to binary 1101. For this multiplicative cascade model, the formula for (q) has been derived earlier [6,33,65]. The result is (q) D [ ln(a q C b q ) C q ln(a C b)]/ ln 2 or h(q) D

1 ln(a q C b q ) ln(a C b)  C : q q ln 2 ln 2

(42)

It is easy to see that h(1) D 1 for all values of a and b. Thus, in this form the model is limited to cases where h(1), which is the exponent Hurst defined originally in the R/S method, is equal to one. In order to generalize this multifractal cascade process such that any value of h(1) is possible, one can sub-

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tract the offset h D ln(a C b)/ ln(2) from h(q) [100]. The constant offset h corresponds to additional longterm correlations incorporated in the multiplicative cascade model. For generating records without this offset, we rescale the power spectrum. First, we transform (FFT) the simple multiplicative cascade data into the frequency domain. Then, we multiply all Fourier coefficients by f h , where f is the frequency. This way, the slope ˇ of the power spectra S( f ) f ˇ is decreased from ˇ D 2h(2)  1 D [2 ln(a C b)  ln(a2 C b2 )]/ ln 2 into ˇ 0 D 2[h(2)  h]  1 D  ln(a 2 C b2 )/ ln 2. Finally, backward FFT is employed to transform the signal back into the time domain. The Bi-fractal Model In some cases a simple bi-fractal model is already sufficient for modeling apparently multifractal data [101]. For bifractal records the Renyi exponents (q) are characterized by two distinct slopes ˛ 1 and ˛ 2 , ( q  q q˛1  1 (43) (q) D q˛2 C q (˛1  ˛2 )  1 q > q or ( (q) D

q˛1 C q (˛2  ˛1 )  1

q  q

q˛2  1

q > q

:

(44)

If this behavior is translated into the h(q) picture using Eq. (29), we obtain that h(q) exhibits a plateau from q D 1 up to a certain q and decays hyperbolically for q > q , ( q  q ˛1 ; (45) h(q) D 1 q (˛1  ˛2 ) q C ˛2 q > q or vice versa, ( q (˛2  ˛1 ) 1q C ˛1 h(q) D ˛2

q  q q > q

:

(46)

Both versions of this bi-fractal model require three parameters. The multifractal spectrum is degenerated to two single points, thus its width can be defined as ˛ D ˛1  ˛2 . Future Directions The most straightforward future direction is to analyze more types of time series from other complex systems than those listed in Sect. “Introduction” to check for the presence of fractal scaling and in particular long-term correlations. Such applications may include (i) data that are not

recorded as a function of time but as a function of another parameter and (ii) higher dimensional data. In particular, the inter-relationship between fractal time series and spatially fractal structures can be studied. Studies of fields with fractal scaling in time and space have already been performed in Geophysics. In some cases studying new types of data will require dealing with more difficult types of non-stationarities and transient behavior, making further development of the methods necessary. In many studies, detrending methods have not been applied yet. However, discovering fractal scaling in more and more systems cannot be an aim on its own. Up to now, the reasons for observed fractal or multifractal scaling are not clear in most applications. It is thus highly desirable to study causes for fractal and multifractal correlations in time series, which is a difficult task, of course. One approach might be based on modeling and comparing the fractal aspects of real and modeled time series by applying the methods described in this article. The fractal or multifractal characterization can thus be helpful in improving the models. For many applications, practically usable models which display fractal or transient fractal scaling still have to be developed. One example for a model explaining fractal scaling might be a precipitation, storage and runoff model, in which the fractal scaling of runoff time series could be explained by fractional integration of rainfall in soil, groundwater reservoirs, or river networks characterized by a fractal structure. Also studies regarding the inter-relationship between fractal scaling and complex networks, representing the structure of a complex system, are desirable. This way one could gain an interpretation of the causes for fractal behavior. Another direction of future research is regarding the linear and especially non-linear inter-relationships between several time series. There is great need for improved methods characterizing cross-correlations and similar statistical inter-relationships between several non-stationary time series. Most methods available so far are reserved to stationary data, which is, however, hardly found in natural recordings. An even more ambitious aim is the (timedependent) characterization of a larger network of signals. In such a network, the signals themselves would represent the nodes, while the (possibly directed) inter-relationships between each pair represent the links (or bonds) between the nodes. The properties of both nodes and links can vary with time or change abruptly, when the represented complex system goes through a phase transition. Finally, more work will have to be invested in studying the practical consequences of fractal scaling in time series. Studies should particularly focus on predictions of future values and behavior of time series and whole complex

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systems. This is very relevant, not only in hydrology and climate research, where a clear distinguishing of trends and natural fluctuations is crucial, but also for predicting dangerous medical events on-line in patients based on the continuous recording of time series. Acknowledgment We thank Ronny Bartsch, Amir Bashan, Mikhail Bogachev, Armin Bunde, Jan Eichner, Shlomo Havlin, Diego Rybski, Aicko Schumann, and Stephan Zschiegner for helpful discussions and contribution. This work has been supported by the Deutsche Forschungsgemeinschaft (grant KA 1676/3) and the European Union (STREP project DAPHNet, grant 018474-2). Bibliography 1. Mandelbrot BB, van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10: 422 2. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Water Resour Res 5:321–340 3. Mandelbrot BB (1999) Multifractals and 1/f noise: wild selfaffinity in physics. Springer, Berlin 4. Hurst HE (1951) Long-term storage capacity of reservoirs. Tran Amer Soc Civ Eng 116:770 5. Hurst HE, Black RP, Simaika YM (1965) Long-term storage: an experimental study. Constable, London 6. Feder J (1988) Fractals. Plenum Press, New York 7. Barnsley MF (1993) Fractals everywhere. Academic Press, San Diego 8. Bunde A, Havlin S (1994) Fractals in science. Springer, Berlin 9. Jorgenssen PET (2000) Analysis and probability: Wavelets, signals, fractals. Springer, Berlin 10. Bunde A, Kropp J, Schellnhuber HJ (2002) The science of disasters – climate disruptions, heart attacks, and market crashes. Springer, Berlin 11. Kantz H, Schreiber T (2003) Nonlinear time series analysis. Cambridge University Press, Cambridge 12. Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals. Springer, Berlin 13. Sornette D (2004) Critical phenomena in natural sciences. Springer, Berlin 14. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anti-correlations and non-Gaussian behaviour of the heartbeat. Phys Rev Lett 70: 1343 15. Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter JH, Voigt K (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736 16. Vyushin D, Zhidkov I, Havlin S, Bunde A, Brenner S (2004) Volcanic forcing improves atmosphere-ocean coupled general circulation model scaling performance. Geophys Res Lett 31:L10206 17. Koscielny-Bunde E, Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber HJ (1998) Indication of a universal persistence law governing atmospheric variability. Phys Rev Lett 81:729

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Fractals in Biology SERGEY V. BULDYREV Department of Physics, Yeshiva University, New York, USA

Article Outline Glossary Definition of the Subject Introduction Self-similar Branching Structures Fractal Metabolic Rates Physical Models of Biological Fractals Diffusion Limited Aggregation and Bacterial Colonies Measuring Fractal Dimension of Real Biological Fractals Percolation and Forest Fires Critical Point and Long-Range Correlations Lévy Flight Foraging Dynamic Fractals Fractals and Time Series SOC and Biological Evolution Fractal Features of DNA Sequences Future Directions Bibliography Glossary Allometric laws An allometric law describes the relationship between two attributes of living organisms y and x, and is usually expressed as a power-law: y x ˛ , where ˛ is the scaling exponent of the law. For example, x can represent total body mass M and y can represent the mass of a brain mb . In this case mb M 3/4 . Another example of an allometric law: B M 3/4 where B is metabolic rate and M is body mass. Allometric laws can be also found in ecology: the number of different species N found in a habitat of area A scales as N A1/4 . Radial distribution function Radial distribution function g(r) describes how the average density of points of a set behaves as function of distance r from a point of this set. For an empirical set of N data points, the distances between all pair of points are computed and the number of pairs Np (r) such that their distance is less then r is found. Then M(r) D 2Np (r)/N gives the average number of the neighbors (mass) of the set within a distance r. For a certain distance bin r1 < r < r2 , we define g[(r2 C r1 )/2] D [M(r2 )  M(r1 )]/[Vd (r2 )  Vd (r1 )], where

Vd (r) D 2 d/2 r d /[d (d/2)] is the volume/area/length of a d-dimensional sphere/circle/interval of radius r. Fractal set We define a fractal set with the fractal dimension 0 < df < d as a set for which M(r) r d f for r ! 1. Accordingly, for such a set g(r) decreases as a power law of the distance g(r) r , where  D d  df . Correlation function For a superposition of a fractal set and a set with a finite density defined as D limr!1 M(r)/Vd (r), the correlation function is defined as h(r)  g(r)  . Long-range power law correlations The set of points has long-range power law correlations (LRPLC) if h(r) r for r ! 1 with 0 <  < d. LRPLC indicate the presence of a fractal set with fractal dimension df D d   superposed with a uniform set. Critical point Critical point is defined as a point in the system parameter space (e. g. temperature, T D Tc , and pressure P D Pc ), near which the system acquires LRPLC h(r)

1 r d2C

exp(r/) ;

(1)

where  is the correlation length which diverges near the critical point as jT  Tc j . Here  > 0 and  > 0 are critical exponents which depend on the few system characteristics such as dimensionality of space. Accordingly, the system is characterized by fractal density fluctuations with df D 2  . Self-organized criticality Self-organized criticality (SOC) is a term which describes a system for which the critical behavior characterized by a large correlation length is achieved for a wide range of parameters and thus does not require special tuning. This usually occurs when a critical point corresponds to an infinite value of a system parameter, such as a ratio of the characteristic time of the stress build up and a characteristic time of the stress release. Morphogenesis Morphogenesis is a branch of developmental biology concerned with the shapes of organs and the entire organisms. Several types of molecules are particularly important during morphogenesis. Morphogens are soluble molecules that can diffuse and carry signals that control cell differentiation decisions in a concentration-dependent fashion. Morphogens typically act through binding to specific protein receptors. An important class of molecules involved in morphogenesis are transcription factor proteins that determine the fate of cells by interacting with DNA. The morphogenesis of the branching fractal-like structures such as lungs involves a dozen of morpho-

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genes. The mechanism for keeping self-similarity of the branches at different levels of branching hierarchy is not yet fully understood. The experiments with transgenic mice with certain genes knocked-out produce mice without limbs and lungs or without terminal buds.

for each model and therefore may shed light on the origin of a particular biological phenomenon. In Sect. “Diffusion Limited Aggregation and Bacterial Colonies”, we discuss the techniques for measuring fractal dimension and their limitations. Introduction

Definition of the Subject Fractals occur in a wide range of biological applications: 1) In morphology when the shape of an organism (tree) or an organ (vertebrate lung) has a self-similar brunching structure which can be approximated by a fractal set (Sect. “Self-Similar Branching Structures”). 2) In allometry when the allometric power laws can be deduced from the fractal nature of the circulatory system (Sect. “Fractal Metabolic Rates”). 3) In ecology when a colony or a habitat acquire fractal shapes due to some SOC processes such as diffusion limited aggregation (DLA) or percolation which describes forest fires (Sects. “Physical Models of Biological Fractals”–“Percolation and Forest Fires”). 4) In epidemiology when some of the features of the epidemics is described by percolation which in turn leads to fractal behavior (Sect. “Percolation and Forest Fires”). 5) In behavioral sciences, when a trajectory of foraging animal acquires fractal features (Sect. “Lévy Flight Foraging”). 6) In population dynamics, when the population size fluctuates chaotically (Sect. “Dynamic Fractals”). 7) In physiology, when time series have LRPLC (Sect. “Fractals and Time Series”). 8) In evolution theory, which may have some features described by SOC (Sect. “SOC and Biological Evolution”). 9) In bioinformatics when a DNA sequence has a LRPLC or a network describing protein interactions has a selfsimilar fractal behavior (Sect. “Fractal Features of DNA Sequences”). Fractal geometry along with Euclidian geometry became a part of general culture which any scientist must be familiar with. Fractals often originate in the theory of complex systems describing the behavior of many interacting elements and therefore have a great number of biological applications. Complex systems have a general tendency for self-organization and complex pattern formation. Some of these patterns have certain nontrivial symmetries, for example fractals are characterized by scale invariance i. e. they look similarly on different magnification. Fractals are characterized by their fractal dimension, which is specific

The fact that simple objects of Euclidian geometry such as straight lines, circles, cubes, and spheres are not sufficient to describe complex biological shapes has been known for centuries. Physicists were always accused by biologists for introducing a “spherical cow”. Nevertheless people from antiquity to our days were fascinated by finding simple mathematical regularities which can describe the anatomy and physiology of leaving creatures. Five centuries ago, Leonardo da Vinci observed that “the branches of a tree at every stage of its height when put together are equal in thickness to the trunk” [107]. Another famous but still poorly understood phenomenon is the emergence of the Fibonacci numbers in certain types of pine cones and composite flowers [31,134]. In the middle of the seventies a new concept of fractal geometry was introduced by Mandelbrot [79]. This concept was readily accepted for analysis of the complex shapes in the biological world. However, after initial splash of enthusiasm, [14,40,41,74,75,76,88,122] the application of fractals in biology significantly dwindled and today the general consensus is that a “fractal cow” is often not much better than a “spherical cow”. Nature is always more complex than mathematical abstractions. Strictly speaking the fractal is an object whose mass, M, grows as a fractional power law of its linear dimension, L, M Ldf ;

(2)

where df is a non-integer quantity called fractal dimension. Simple examples of fractal objects of various fractal dimensions are given by iterative self-similar constructs such as a Cantor set, a Koch curve, a Sierpinski gasket, and a Menger sponge [94]. In all this constructs a next iteration of the object is created by arranging p exact copies of the previous iteration of the object (Fig. 1) in such a way that the linear size of the next iteration is q times larger than the linear size of the previous iteration. Thus the mass, M n , and the length, Ln , of the nth iteration scale as M n D p n M0 L n D q n L0 ;

(3)

where M 0 and L0 are mass and length of the zero order

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Fractals in Biology, Figure 1 a Cantor set (p D 2, q D 3, df D ln 2/ ln 3  0:63) is an example of fractal “dust” with fractal dimension less then 1; b Fractal tree (p D 3, q D 2, df D ln 3/ ln 2  1:58) with branches removed so that only the terminal points (leaves) can be seen. c The same tree with the trunks added. Both trees are produced by recursive combining of the three smaller trees: one serving as the top of the tree and the other two rotated by 90ı clockwise and counter-clockwise serving as two brunches joined with the top at the middle. In c a vertical segment representing a trunk of the length equal to the diagonal of the branches is added at each recursive step. Mathematically, the fractal dimensions of sets b and c are the same, because the mass of the tree with trunks (number of black pixels) for the system of linear size 2n grows as 3nC1  2nC1 , while for the tree without trunks mass scales simply as 3n . In the limit n ! 1 this leads to the same fractal dimension ln 3/ ln 2. However, visual inspection suggests that the tree with branches has a larger fractal dimension. This is in accord with the box counting method, which produces a higher value of the slope for the finite tree with the trunks. The slope slowly converges to the theoretical value as the number of recursive steps increases. d Sierpinski gasket has the same fractal dimension as the fractal tree (p D 3, q D 2) but has totally different topology and visual appearance than the tree

iteration. Excluding n from Eq. (3), we get  d f Ln ; M n D M0 L0

Self-similar Branching Structures (4)

where df D ln p/ ln q

(5)

can be identified as fractal dimension. The above-described objects have the property of self-similarity, the previous iteration magnified q times looks exactly like the next iteration once we neglect coarse-graining on the lowest iteration level, which can be assumed to be infinitely small. An interesting feature of such fractals is the power law distribution of their parts. For example, the cumulative distribution of distances L between the points of Cantor set and the branch lengths of the fractal tree (Fig. 1) follows a power law: P(L > x) x d f :

(6)

Thus the emergence of power law distributions and other power law dependencies is often associated with fractals and often these power law regularities not necessarily related to geometrical fractals are loosely referred as fractal properties.

Perfect deterministic fractals described above are never observed in nature, where all shapes are subjects to random variations. The natural examples closest to the deterministic fractals are structures of trees, certain plants such as a cauliflower, lungs and a cardiovascular system. The control of branching morphogenesis [85,114,138] involves determining when and where a branch will occur, how long the tube grows before branching again, and at what angle the branch will form. The development of different organs (such as salivary gland, mammary gland, kidney, and lung) creates branching patterns easily distinguished from each other. Moreover, during the development of a particular organ, the form of branching often changes, depending on the place or time when the branching occurs. Morphogenesis is controlled by complex molecular interactions of morphogenes. Let us illustrate the idea of calculating fractal dimension of a branching object such as a cauliflower [51,64]. If we assume (which is not quite correct, see Fig. 2) that each branch of a cauliflower give rise to exactly p branches of the next generation exactly q times smaller than the original branch applying Eq. (5), we get df D ln q/ ln p. Note

Fractals in Biology

Fractals in Biology, Figure 2 a Cauliflower anatomy. A complete head of a cauliflower (left) and after it is taken apart (right). We remove 17 branches until the diameter of the remaining part becomes half of the diameter of the original head. b Diameters of the branches presented in a. c Estimation of the fractal dimension for asymmetric lungs ( D 3) and trees ( D 2) with branching parameter r using Eq. (7). For r > 0:855 or r < 0:145 the fractal dimension is not defined due to the “infrared catastrophe”: the number of large branches diverge in the limit of an infinite tree

that this is not the fractal dimension of the cauliflower itself, but of its skeleton in which each brunch is represented by the elements of its daughter branches, with an addition of a straight line connecting the daughter branches as in the example of a fractal tree (Fig. 1b, c). Because this addition does not change the fractal dimension formula, the fractal dimension of the skeleton is equal to the

fractal dimension of the surface of a cauliflower, which can be represented as the set of the terminal branches. As a physical object, the cauliflower is not a fractal but a three-dimensional body so that the mass of a branch of length L scales as L3 . This is because the diameter of each branch is proportional to the length of the branch.

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The distribution of lengths of the collection of all the branches of a cauliflower is also a power law given by P k n a simple idea that there are N n D n1 kD0 p  p /(p  1) branches of length larger than L n D L0 qn . Excluding n d gives N n (L > L n ) L n f . In reality, however (Fig. 2a, b), the branching structure of a cauliflower is more complex. Simple measurements show that there are about p D 18 branches of sizes L k D L0  L0 [0:5  0:2(k  1)] for k D 1; 2; : : :; p, where L0 is the diameter of the complete head of the cauliflower. (We keep removing branches until the diameter of the central remaining part is equal to the half diameter of the complete head and we assume that it is similar to the rest of the branches). Subsequent generations (we count at least eight) obey similar rules, however p has a tendency to decrease with the generation number. To find fractal dimension of a cauliflower we will count the number of branches N(L) larger than certain size L. As in case of equal branches, this number scales as Ld f . Calculations, similar to those presented in [77] show that in this case the fractal dimension is equal to 0s 1  @ 2 2 1  ln p 2  1A ; (7) df D  2   where  and  are the mean and the standard deviation of ln q k  ln(L0 /L k ). For the particular cauliflower shown in Fig. 2a the measurements presented in Fig. 2b give df D 2:75, which is very close to the estimate df D 2:8, of [64]. The physiological reason for such a peculiar branching pattern of cauliflower is not known. It is probably designed to store energy for a quick production of a dense inflorescence. For a tree, [142] the sum of the cross-section areas of the two daughter branches according to Leonardo is equal to the cross-section area of the mother branch. This can be understood because the number of capillary bundles going from the mother to the daughters is conserved. Accordingly, we have a relation between the daughter branch diameters d1 and d2 and the mother branch diameter d0 , 





d1 C d2 D d0 ;

(8)

where  D 2. We can assume that the branch diameter of the largest daughter branch scales as d1 D rd0 , where r is asymmetry ratio maintaining for each generation of branches. In case of a symmetric tree d1 D d2 , r D 1/2. If we assume that the branch length L D sd, where s is a constant which is the same for any branch of the tree, then we can relate our tree to a fractal model with p D 2 and q D 21/ . Using Eq. (5) we get df D  D 2, i. e. the tree skeleton or the surface of the terminal branches is

an object of fractal dimension two embedded in the threedimensional space. This is quite natural because all the leaves whose number is proportional to the number of terminal branches must be exposed to the sunlight and the easiest way to achieve this is to place all the leaves on the surface of a sphere which is a two-dimensional object. For an asymmetric tree the fractal dimension can be computed using Eq. (7) with  D j ln[r(1  r)]/2j

(9)

and  D j ln[r/(1  r)]/2j :

(10)

This value is slightly larger than 2 for a wide range of r (Fig. 2c). This property may be related to a tendency of a tree to maximize the surface of it leaves, which must not be necessarily exposed to the direct sunlight but can suffice on the light reflected by the outer leaves. For a lung [56,66,77,102,117], the flow is not a capillary but can be assumed viscous. According to flow conservation, the sum of the air flows of the daughter branches must be equal to the flow of the mother branch: Q1 C Q2 D Q0 :

(11)

For the Poiseuille flow, Q Pd 4 /L, where P is the pressure drop, which is supposed to be the same for the airways of all sizes. Assuming that the lung maintains the ratio s D L/d in all generations of branches, we conclude that the diameters of the daughter and mother branches must satisfy Eq. (8) with  D 4  1 D 3. Accordingly, for a symmetrically branching lung df D  D 3, which means that the surface of the alveoli which is proportional to the number of terminal airways scales as L3 , i. e. it is a space filling object. Again this prediction is quite reasonable because nature tends to maximize the gas exchange area so that it completely fills the volume of the lung. In reality, the flow in the large airways is turbulent, and the parameter  of lungs of different species varies between 2 and 3 [77]. Also the lungs are known to be asymmetric and the ratio r D Q1 /Q0 ¤ 1/2 changes from one generation of the airways to the next [77]. However, Eq. (7) with  and  given by Eqs. (9) and (10) shows that the fractal dimension of an asymmetric tree remains very close to 3 for a wide range of 0:146 < r < 0:854 (Fig. 2c). The fact that the estimated fractal dimension is slightly larger than 3 does not contradict common sense because the branching stops when the airway diameter becomes smaller than some critical cutoff. Other implications of the lung asymmetry are discussed in [77]. An interesting idea

Fractals in Biology

was proposed in [117], according to which the dependence of the branching pattern on the generation of the airways can be derived from optimization principles and give rise to the complex value of the fractal dimension. An interesting property of the crackle sound produced by diseased lungs is that during inflation the resistance to airflow of the small airways decreases in discrete jumps. Airways do not open individually, but in a sequence of bursts or avalanches involving many airways; both the size of these jumps and the time intervals between jumps follow power-law distributions [128]. These avalanches are not related to the SOC as one might expect, but their power law distributions directly follow from the branching structure of lungs (see [3] and references therein).

Fractal Metabolic Rates An important question in biology is the scaling of the metabolic rate with respect to the body mass (Kleiber’s Law). It turns out that for almost all species of animals, which differ in terms of mass by 21 orders of magnitudes, the metabolic rate B scales as B M 3/4 [28,112,140]. The scatter plot ln B vs ln M is a narrow cloud concentrated around the straight line with a slope 3/4 (Fig. 3). This is one of the many examples of the allometric laws which describe the dependence of various biological parameters on the body mass or population size [1,9,87,105,106]. A simple argument based on the idea that the thermal energy loss and hence the metabolic rate should be propor-

Fractals in Biology, Figure 3 Dependence of metabolic rate on body mass for different types of organisms (Kleiber’s Law, after [112]). The least square fit lines have slopes 0:76 ˙ 0:01

tional to the surface area predicts however that B L2 M 2/3 . Postulating that the metabolic rate is proportional to some effective metabolic area leads to a strange prediction that this area should have a fractal dimension of 9/4. Many attempts has been made to explain this interesting fact [10,32,142,143]. One particular attempt [142] was based on the ideas of energy optimization and the fractal organization of the cardiovascular system. However, the arguments were crucially dependent on such details as turbulent vs laminar flow, the elasticity of the blood vessels and the pulsatory nature of the cardiovascular system. The fact that this derivation does not work for species with different types of circulatory systems suggests that there might be a completely different and quite general explanation of this phenomenon. Recently [10], as the theory of networks has became a hot subject, a general explanation of the metabolic rates was provided using a mathematical theorem that the total flow Q (e. g. of the blood) in the most efficient supply network must scale as Q BL/u, where L is the linear size of the network, B is the total consumption rate (e. g. metabolic rate), and u is the linear size of each consumption unit (e. g. cell). The authors argue that the total flow is proportional to the total amount of the liquid in the network, which must scale as the body mass M. On the other hand, L M 1/3 . If one assumes that u is independent of the body mass, then B M 2/3 , which is identical to the simple but incorrect prediction based on the surface area. In order to make ends meet, the authors postulate that some combination of parameters must be independent of the body size, from where it follows that u M 1/12 . If one identifies a consumption unit with the cell, it seems that the cell size must scale as L1/4 . Thus the cells of a whale (L D 30 m) must be about 12 times larger than those of a C. elegance (L D 1 mm), which more or less consistent with the empirical data [110] for slowly dividing cells such as neurons in mammals. However, the issue of the metabolic scaling is still not fully resolved [11,33,144]. It is likely that the universal explanation of Kleiber’s law is impossible. Moreover, recently it was observed that Kleiber’s law does not hold for plants [105]. An interesting example of an allometric law which may have some fractal implications is the scaling of the brain size with the body mass Mb M 3/4 [1]. Assuming that the mass of the brain is proportional to the mass of the neurons in the body, we can conclude that if the average mass of a neuron does not depend on the body mass, neurons must form a fractal set with the fractal dimension 9/4. However, if we assume that the mass of a neuron must scale with the body mass as M 1/12 , as it follows from [10,110], we can conclude that the number of neu-

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rons in the body scales simply as M 2/3 L2 , which means that the number of neurons is proportional to the surface area of the body. The latter conclusion is physiologically more plausible than the former, since it is obvious that the neurons are more likely to be located near the surface of an organism. From the point of view of comparative zoology, the universal scaling of the brain mass is not as important as the deviations from it. It is useful [1] to characterize an organism by the ratio of its actual brain mass to its expected brain mass which is defined as Eb D AM 3/4 , where A is a constant measured by the intercept of the log–log graph of brain mass versus body mass. Homo sapiens have the largest Mb /Eb D 8, several times larger than those of gorillas and chimpanzees. Physical Models of Biological Fractals The fractals discussed in Sect. “Self-Similar Branching Structures”, which are based on the explicit rules of construction and self-similarity, are useful concepts for analyzing branching structures in the living organisms whose development and growth are programmed according to these rules. However, in nature, there are many instances when fractal shapes emerge just from general physical principles without a special fractal “blueprint” [26,34, 58,124,125,131]. One of such examples particularly relevant to biology is the diffusion limited aggregation (DLA) [145]. Another phenomenon, described by the same equations as DLA and thus producing similar shapes is viscous fingering [17,26,131]. Other examples are a random walk (RW) [42,58,104,141], a self-avoiding walk (SAW) [34,58], and a percolation cluster [26,58,127]. DLA explains the growth of ramified inorganic aggregates which sometimes can be found in rock cracks. The aggregates grow because certain ions or molecules deposit on the surface of the aggregate. These ions come to the surface due to diffusion which is equivalent on the microscopic level to Brownian motion of individual ions. The Brownian trajectories can be modeled as random walks in which the direction of each next step is independent of the previous steps. The Brownian trajectory itself has fractal properties expressed by the Einstein formula: r2 D 2dtD ;

(12)

where r2 is the average square displacement of the Brownian particle during time t, D is the diffusion coefficient and d is the dimensionality of the embedding space. The number of steps, n, of the random walk is proportional to time of the Brownian motion t D n, where  is the

Fractals in Biology, Figure 4 A random walk of n D 216 steps is surrounded by its p D p 4 parts of m D n/p D 214 steps magnified by factor of q D 4 D 2. One can see that the shapes and sizes of each of the magnified parts are similar to the shape and size of the whole

average duration of the random walk step. The diffusion coefficient can be expressed as D D r2 (m)/(2dm), where r2 (m) is the average square displacement during m time steps. Accordingly, p r(n) D pr(m) ; (13) which shows that the random walk of n steps consists of p D n/m copies of the one-step walks arranged in space in such a way pthat the average linear size of this arrangement is q D n/m times larger than the size of the m-step walk (Fig. 4). Applying Eq. (5), we get df D 2 which does not depend on the embedding space. It is the same in one-dimensional, two-dimensional, and three-dimensional space. Note that Brownian trajectory is self-similar only in a statistical sense: each of the n concatenated copies of a m-step trajectory are different from each other, but their average square displacements are the same. Our eye can easily catch this statistical self-similarity. A random walk is a cloudy object of an elongated shape, its inertia ellipsoid is characterized by the average ratios of the squares of its axis: 12.1/2.71/1 [18]. The Brownian trajectory itself has a relevance in biology, since it may describe the changing of the electrical potential of a neuron [46], spreading of a colony [68], the foraging trajectory of a bacteria or an animal as well as the motion of proteins and other molecules in the cell. The self-avoiding walk (SAW) which has a smaller fractal dimension (df,SAW D 4/3 in d D 2 and df,SAW  1:7 in

Fractals in Biology

Fractals in Biology, Figure 5 A self-avoiding walk (a) of n D 104 in comparison with a random walk (b) of the same number of steps, both in d D 3. Both walks are produced by molecular dynamic simulations of the bead-on-a-string model of polymers. The hard core diameter of monomers in the SAW is equal to the bond length `, while for the RW it is zero. Comparing their fractal dimensions df,SAW  1:7, and df,RW D 2, one can predict that the average size (radius of inertia) of SAW must be n1/df,SAW 1/df,RW  2:3 times larger than that of RW. Indeed the average radius of inertia of SAW and RW are 98` and 40`, respectively. The fact that their complex shapes resemble leaving creatures was noticed by P. G. de Gennes in a cartoon published in his book “Scaling Concepts in Polymer Physics” [34]

d D 3) is a model of a polymer (Fig. 5) in a good solvent, such as for example a random coil conformation of a protein. Thus a SAW provides another example of a fractal object which has certain relevance in molecular biology. The fractal properties of SAWs are well established in the works of Nobel Laureates Flory and de Gennes [34,44,63]. Diffusion Limited Aggregation and Bacterial Colonies The properties of a Brownian trajectory explain the ramified structure of the DLA cluster. Since the Brownian trajectory is not straight, it is very difficult for it to penetrate deep into the fiords of a DLA cluster. With a much higher probability it hits the tips of the branches. Thus DLA cluster (Fig. 6a) has a tree-like structure with usually 5 main branches in 2 dimensions. The analytical determination of its fractal dimension is one of the most challenging questions in modern mathematics. In computer simulations it is defined by measuring the number of aggregated particles versus the radius of gyration of the aggregate (Fig. 6c). The fractal dimension thus found is approximately 1.71

Fractals in Biology, Figure 6 a A DLA cluster of n D 214 particles produced by the aggregation of the random walks on the square lattice (top). A comparison of a small BD cluster (b) and a DLA cluster (c). The color in b and c indicates the deposition time of a particle. The slopes of the loglog graphs of the mass of the growing aggregates versus their gyration radius give the values of their fractal dimensions in the limit of large mass

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Fractals in Biology, Figure 7 A typical DLA-like colony grown in a Petri dish with a highly viscous substrate (top). A change in morphology from DLA-like colony growth to a swimming chiral pattern presumably due to a cell morphotype transition (bottom) (from [15])

Fractals in Biology

in two dimensions and 2.50 in three dimensions [34]. It seems that as the aggregate grows larger the fractal dimension slightly increases. Note that if the deposition is made by particles moving along straight lines (ballistic deposition) the aggregate changes its morphology and becomes almost compact and circular with small fluctuation on the boundaries and small holes in the interior (Fig. 6b). The fractal dimension of the ballistic aggregate coincides with the dimension of embedding space. Ballistic deposition (BD) belongs to the same universality class as the Eden model, the first model to describe the growth of cancer. In this model, each cell on the surface of the cluster can produce an offspring with equal probability. This cell does not diffuse but occupies one of the empty spaces neighboring to the parent cell [13,83]. DLA is supposed to be a good model for growth of bacterial colonies in the regime when nutrients are coming via diffusion in a viscous media from the exterior. Under different conditions bacteria colonies observe various transition in their morphology. If the nutrient supply is greater, the colonies increase their fractal dimension and start to resemble BD. An interesting phenomenon is observed upon changing the viscosity of the substrate [15]. If the viscosity is small it is profitable for bacteria to swim in order to get to the regions with high nutrient concentrations. If the viscosity is large it is more profitable to grow in a static colony which is supplied by the diffusion of the nutrient from the outside. It seems that the way how the colony grows is inherited in the bacteria gene expression. When a bacteria grown at high viscosity is planted into a low viscosity substrate, its descendants continue to grow in a DLA pattern until a transition in a gene-expression happens in a sufficiently large number of neighboring bacteria which change their morphotype to a swimming one (Fig. 7). All the descendants of these bacteria start to swim and thus quickly take over the slower growing morphotype. Conversely, when a swimming bacteria is planted into a viscous media its descendants continue to swim until a reverse transition of a morphotype happens and the descendants of the bacteria with this new morphotype start a DLA-like growing colony. The morphotype transition can be also induced by fungi. Thus, it is likely that although bacteria are unicellular organisms, they exchange chemical signals similarly to the cells in the multicellular organisms, which undergo a complex process of cell differentiation during organism development (morphogenesis). There is evidence that the tree roots also follow the DLA pattern, growing in the direction of diffusing nutrients [43]. Coral reefs [80] whose life depends on the diffu-

sion of oxygen and nutrients also may to a certain degree follow DLA or BD patterns. Another interesting conjecture is that neuronal dendrites grow in vivo obeying the same mechanism [29]. In this case, they follow signaling chemicals released by other cells. It was also conjectured that even fingers of vertebrate organisms may branch in a DLA controlled fashion, resembling the pattern of viscous fingering which are observed when a liquid of low viscosity is pushed into a liquid of higher viscosity. Measuring Fractal Dimension of Real Biological Fractals There were attempts to measure fractal dimension of growing neurons [29] and other biological objects from photographs using a box-counting method and a circle method developed for empirical determination of fractal dimension. The circle method is the simplified version of the radial distribution function analysis described in the definition section. It consists of placing a center of a circle or a sphere of a variable radius R at each point of an object and counting the average number of other points of this object M(R) found inside these circles or spheres. The slope of ln M(R) vs. ln R gives an estimate of the fractal dimension. The box counting method consists of placing a square grid of a given spacing ` on a photograph and counting number of boxes n(`) needed to cover all the points belonging to a fractal set under investigation. Each box containing at least one point of a fractal set is likely to contain on average N(`) D `d f other points of this set, and n(`)N(`) D N, where N is the total number of the points. Thus for a fractal image n(`) N/`d f `d f . Accordingly the fractal dimension of the image can be estimated by the slope of a graph of ln n(`) vs. ln `. The problem is that real biological objects do not have many orders of self-similarity, and also, as we see above, only a skeleton or a perimeter of a real object has fractal properties. The box counting method usually produces curvy lines on a log–log paper with at best one decade of approximately constant slope (Fig. 8). What is most disappointing is that almost any image analyzed in this fashion produces a graph of similar quality. For example, an Einstein cartoon presented in [93] has the same fractal dimension as some of the growing neurons. Therefore, experimental determination of the fractal dimension from the photographs is no longer in use in scientific publications. However, in the 1980s and early 1990s when the computer scanning of images became popular and the fractals were at the highest point of their career, these exercises were frequent.

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Fractals in Biology, Figure 8 An image of a neuron and its box-counting analysis (a) together with the analogous analysis of an Einstein cartoon (b). The log–log plot of the box-counting data has a shape of a curve with the changing slope. Bars represent the local slope of this graph, which may be interpreted as some effective fractal dimension. However, in both graphs the slopes change dramatically from 1 to almost 1.7 and it is obvious that any comparison to the DLA growth mechanism based on these graphs is invalid (from [93])

For example, R. Voss [137] analyzed fractal dimensions of Chinese graphics of various historical periods. He found that the fractal dimension of the drawings fluctuated from century to century and was about 1.3 at the time of the highest achievements of this technique. It was argued that the perimeters of many natural objects such as perimeters of the percolation clusters and random walks have similar fractal dimension; hence the works of the artists who were able to reproduce this feature were the most pleasant for our eyes. Recently, the box counting method was used to analyze complex networks such as protein interaction networks (PIN). In this case the size of the box was defined as the maximal number of edges needed to connect any two nodes belonging to the box. The box-counting method appears to be a powerful tool for analyzing self-similarity of the network structure [120,121].

Percolation and Forest Fires It was also conjectured that biological habitats of certain species may have fractal properties. However it is not clear whether we have a true self-similarity or just an apparent mosaic pattern which is due to complex topography and geology of the area. If there is no reasonable theoretical model of an ecological process, which displays true fractal properties the assertions of the fractality of a habitat are pointless. One such model, which produces fractal clusters is percolation [26,127]. Suppose a lightning hits a random tree in the forest. If the forest is sufficiently dry, and the inflammable trees are close enough the fire will spread from a tree to a tree and can burn the entire forest. If the inflammable trees form a connected cluster, defined as such that its any two trees are connected with a path along which the fire can spread, all the trees in this cluster will

Fractals in Biology

be burnt down. Percolation theory guarantees that there is a critical density pc of inflammable trees below which all the connected clusters are finite and the fire will naturally extinguish burning only an infinitesimally small portion of the forest. In contrast, above this threshold there exists a giant cluster which constitutes a finite portion of the forest so that the fire started by a random lightning will on average destroy a finite portion of the forest. Exactly at the critical threshold, the giant cluster is a fractal with a fractal dimension df D 91/48  1:89. The probability to burn a finite cluster of mass S follows a power law P(S) S d/d f . Above and below the percolation threshold, this distribution is truncated by an exponential factor which specifies that clusters of linear dimensions exceeding a characteristic size are exponentially rare. The structure of the clusters which do not exceed this characteristic scale is also fractal, so if their mass S is plotted versus their linear size R (e. g. radius of inertia) it follows a power law S Rdf In the natural environment the thunderstorms happen regularly and they produce fractal patterns of burned down patches. However, if the density of trees is small, the patches are finite and the fires are confined. As the density the trees reaches the critical threshold, the next thunderstorm is likely to destroy the giant cluster of the forest, which will produce a bare patch of a fractal shape spreading over the entire forest. No major forest fire will happen until the new forest covers this patch creating a new giant cluster, because the remaining disconnected groves are of finite size. There is an evidence that the forest is the system which drives itself to a critical point [30,78,111] (Fig. 9). Suppose that each year there is certain number of lightning strokes per unit area nl . The average number of trees in a cluster is hSi D ajp  pc j , where D 43/18 is one of the critical exponents describing percolation, a is a constant, and p is the number of inflammable trees per unit area. Thus the number of trees destroyed annually per unit area is nl ajpc  p] . On the other hand, since the trees are growing, the number of trees per unit area increases annually by nt , which is the parameter of the ecosystem. At equilibrium, nl ajpc  p] D nt . Accordingly, the equilibrium density of trees must reach pe D pc  (anl /nt )1/ . For very low nl , pe will be almost equal to pc , and the chance that in the next forest fire a giant cluster which spans the entire ecosystem will be burned is very high. It can be shown that for a given chance c of hitting such a cluster in a random lightning stroke, the density of trees must reach p D pc  f (c)L1/ , where L is the linear size of the forest,  D 4/3 is the correlation critical exponent and f (c) > 0 is a logarithmically growing function of c.

Accordingly, if nt /nl > bL / , where b is some constant, the chance of getting a devastating forest fire is close to 100%. We have here a paradoxical situation: the more frequent are the forest fires, the least dangerous they are. This implies that the fire fighters should not extinguish small forest fires which will be contained by themselves. Rather they should annually cut a certain fraction of trees to decrease nt . As we see, the forest fires can be regarded as a self-organized critical (SOC) system which drive themselves towards criticality. As in many SOC systems, here there are two processes one is by several orders of magnitudes faster than the other. In this case they are the tree growing process and the lightning striking process. The model reaches the critical point if a tuning parameter nt /nl ! 1 and in addition nt ! 0 and L ! 1, which are quite reasonable assumptions. In a regular critical system a tuning parameter (e. g. temperature) must be in the vicinity of a specific finite value. In a SOC system the tuning parameter must be just large enough. There is an evidence, that the forest fires follow a power law distribution [109]. One can also speculate that the areas burned down by the previous fires shape fractal habitats for light loving and fire-resistant trees such as pines. The attempts to measure a fractal dimension of such habitats from aerial photographs are dubious due to the limitations discussed above and also because by adjusting a color threshold one can produce fractal-like clusters in almost any image. This artifact is itself a trivial consequence of the percolation theory. The frequently reported Zipf’s law for the sizes of colonies of various species including the areas and populations of the cities [146] usually arises not from fractality of the habitats but from preferential attachment growth in which the old colonies grow in proportion to their present population, while the new colonies may form with a small probability. The preferential attachment model is a very simple mechanism which can create a power-law distribution of colony sizes P(S) S 2 , where  is a small correction which is proportional to the probability of formation of new colonies [25]. Other examples of power laws in biology [67], such as distributions of clusters in the metabolic networks, the distribution of families of proteins, etc. are also most likely come from the preferential attachment or also, in some cases, can arise as artifacts of specifically selected similarity threshold, which brings a network under consideration to a critical point of the percolation theory. The epidemic spreading can also be described by a percolation model. In this case the contagious disease spreads from a person to a person as the forest fire spreads from

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Fractals in Biology, Figure 9 A sequence of frames of a forest fire model. Each tree occupies a site on 500  500 square lattice. At each time step a tree (a colored site) is planted at a randomly chosen empty site (black). Each 10,000 time steps a lightning strikes a randomly chosen tree and the forest fire eliminates a connected cluster of trees. The frames are separated by 60,000 time steps. The color code indicates the age of the trees from blue (young) to red (old). The initial state of the system is an empty lattice. As the concentration of trees reaches percolation threshold (frame 3) a small finite cluster is burned. However it does not sufficiently decrease the concentration of trees and it continues to build up until a devastating forest fire occurs between frames 3 and 4, with only few green groves left. Between frames 4 and 5 several lightnings hit these groves and they are burned down, while surrounding patch of the old fire continues to be populated by the new trees. Between frame 5 and 6 a new devastating forest fire occurs. At the end of the movie huge intermittent forest fires produce gigantic patches of dense and rare groves of various ages

a tree to a tree. Analogously to the forest fire model, a person who caught the disease dies or recovers and becomes immune to the disease, so he or she cannot catch it again. This model is called susceptible-infective-removed (SIR) [62]. The main difference is that the epidemics spread not on a two-dimensional plane but on the network describing contacts among people [61,86]. This network is usually supposed to be scale free, i. e. the number of contacts different people have (degree) is distributed according to a an inverse power law [2,71]. As the epidemic spreads the number of connections in the susceptible population depletes [115] so the susceptible population comes to a percolation threshold after which the epidemic stops.

This model explains for example why the Black Death epidemic stopped in the late 14th century after killing about one third of the total European population. Critical Point and Long-Range Correlations Percolation critical threshold [26,127] is an example of critical phenomena the most well known of which is the liquid-gas critical point [124]. As one heats a sample of any liquid occupying a certain fraction of a closed rigid container, part of the liquid evaporates and the pressure in the container increases so that the sample of liquid remains at equilibrium with its vapor. However, at certain tempera-

Fractals in Biology

ture the visible boundary between the liquid at the bottom and the gas on the top becomes fuzzy and eventually the system becomes completely nontransparent as milk. This phenomena is called critical opalescence and the temperature, pressure, and density at which it happens is called a critical point of this liquid. For water, the critical point is Tc D 374 Cı , Pc D 220 atm and c D 330 kg/m3 . As the temperature goes above the critical point the system becomes transparent again, but the phase boundary disappears: liquid and gas cannot coexist above the critical temperature. They form a single phase, supercritical fluid, which has certain properties of both gas (high compressibility) and liquid (high density and slow diffusivity). At the critical point the system consists of regions of high density (liquid like) and low density (gas like) of all sizes from the size of a single molecule to several microns across which are larger than the wave length of visible light  0:5 m. These giant density fluctuations scatter visible light causing the critical opalescence. The characteristic linear size of the density fluctuations is called the correlation length  which diverges at critical temperature as  jT  Tc j . The shapes of these density fluctuations are self-similar, fractal-like. The system can be represented as a superposition of a uniform set with the density corresponding to the average density of the system and a fractal set with the fractal dimension df . The density correlation function h(r) decreases for r ! 1 as r exp(r/), where   d  2 C  D d  df and d D 1; 2; 3; : : : is the dimension of space in which the critical behavior is observed. The exponential cutoff sets the upper limit of the fractal density fluctuations to be equal to the correlation length. The lower limit of the fractal behavior of the density fluctuations is one molecule. As the system approaches the critical point, the range of fractal behavior increases and can reach at least three orders of magnitude in the narrow vicinity of the critical point. The intensity of light scattered by density fluctuations at certain angle can be expressed via the Fourier transform of the density R correlation function S(f) h(r) exp(ir  f)dr, where the integral is taken over d-dimensional space. This experimentally observed quantity S(f ) is called the structure factor. It is a powerful tool to study the fractal properties of matter since it can uncover the presence of a fractal set even if it is superposed with a set of uniform density. If the density correlation function has a power law behavior h(r) r , the structure factor also has a power law behavior S( f ) f d . For the true fractals with df < d, the average density over the entire embedding space is zero, so the correlation function coincides with the average density of points of the set at certain distance from a given point, h(r) rdC1 dM(r)/dr r d f d , where M(r) r d f is

the mass of the fractal within the radius r from a given point. Thus, for a fractal set of points with fractal dimension df , the structure factor has a simple power law form S( f ) f d f . Therefore whenever S( f ) f ˇ , ˇ < d is identified with the fractal dimension and the system is said to have fractal density fluctuations even if the average density of the system is not zero. There are several methods of detecting spatial correlations in ecology [45,108] including box-counting [101]. In addition one can study the density correlation function h(r) of certain species on the surface of the Earth defined as h(r) D N(r)/2 rr  N/A, where N(r) is the average number of the representatives of a species within a circular rim of radius r and width r from a given representative, N is the total population, and A is the total area. For certain species, the correlation function may follow about one order of magnitude of a fractal (power-law) behavior [9,54,101]. One can speculate that there is some effective attraction between the individuals such as cooperation (mimicking the van der Walls forces between molecules) and a tendency to spread over the larger territory (mimicking the thermal motion of the particles). The interplay of these two tendencies may produce a fractal behavior like in the vicinity of the critical point. However, there is no reason of why the ecological system although complex and chaotic [81] should drive itself to a critical point. Whether or not there is a fractal behavior, the density correlation function is a useful way to describe the spatial distribution of the population. It can be applied not only in ecology but also in physiology and anatomy to describe the distribution of cells, for example, neurons in the cortex [22]. Lévy Flight Foraging A different set of mathematical models which produce fractal patterns and may be relevant in ecology is the Lévy flight and Lévy walk models [118,147]. The Lévy flight model is a generalization of a random walk, in which the distribution of the steps (flights) of length ` follows a power law P(`) ` with  < 3. Such distributions do not have a finite variance. The probability density of the landing points of the Lévy flight converges not to a Gaussian as for a normal random walk with a finite step variance but to a Lévy stable distribution with the parameter ˛    1. The landing points of a Lévy flight form fractal dust similar to a Cantor set with the fractal dimension df D ˛. It was conjectured that certain animals may follow Lévy flights during foraging [103,116,132,133]. It is a mathematical theorem [23,24] that in case when targets

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are scarce but there is a high probability that a new target could be discovered in the vicinity of the previously found target, the optimal strategy for a forager is to perform a Lévy flight with  D 2 C , where  is a small correction which depends on the density of the target sites, the radius of sight of the forager and the probability to find a new target in the vicinity of the old one. In this type of “inverse square” foraging strategy a balance is reached between finding new rich random target areas and returning back to the previously visited area which although depleted, may still provide some food necessary for survival. The original report [132] of the distribution of the flight times of wandering albatross was in agreement with the theory. However, recent work [39] using much longer flight time records and more reliable analysis showed that the distribution of flight times is better described by a Poisson distribution corresponding to regular random walk rather then by a power law. Several other reports of the Lévy flight foraging are also found dubious. The theory [23] however predicts that the inverse square law of foraging is optimal only in case of scarce sources distribution. Thus if the harvest is good and the food is plenty there is no reason to discover the new target locations and the regular random walk strategy becomes the most efficient. Subsequent observations show that power laws exist in some other marine predator search behavior [119]. In order to find a definitive answer, the study must be repeated for the course of many successive years characterized by different harvests. Once the miniature electronic devices for tracing animals are becoming cheaper and more efficient, a new branch of electronic ecology is emerging with the goal of quantifying migratory patterns and foraging strategies of various species in various environments. Whether or not the foraging patterns are fractal, this novel approach will help to establish better conservational policy with scientifically sound borders of wild-life reservations. Recent observations indicate that human mobility patterns might also possess Lévy flight properties [49].

pose that there is a population N t of a certain species enclosed in a finite habitat (e. g. island) at time t. Here t is an integer index which may denote year. Suppose that at time t C 1 the population becomes

Dynamic Fractals

Fractals and Time Series

It is not necessary that a fractal is a real geometrical object embedded in the regular three-dimensional space. Fractals can be found in the properties of the time series describing the behavior of the biological objects. One classical example of a biological phenomenon which under certain condition may display fractal properties is the famous logistic map [38,82,95,123] based on the ideas of a great British economist and demographer Robert Malthus. Sup-

As we can see in the previous section, even simple systems characterized by nonlinear feedbacks may display complex temporal behavior, which often becomes chaotic and sometimes to fractal. Obviously, such features must be present in the behavior of the nervous system, in particular in the human brain which is probably the most complex system known to contemporary science. Nevertheless, the source of fractal behavior can be sometimes trivial with-

N tC1 D bN t  dN t2 ; where b is the natural birth rate and d is the death rate caused by the competition for limited resources. In the most primitive model, the animals are treated as particles randomly distributed over area A annihilating at each time step if the distance between them is less than r. In this case d D  r2 /A. The normalized population x t D dN t /b obeys a recursive relation with a single parameter b: x tC1 D bx t (1  x t ) : The behavior of the population is quite different for different b. For b  1 the population dies out as t ! 1. For 1 < b  b0 D 3 the population converges to a stable size. If b n1 < b  b n , the population repetitively visits 2n values 0 < x1 ; : : : ; x2n < 1 called attractors as t ! 1. The bifurcation points 3 D b0 < b1 < b2 <    < b n < b1  3:569945672 converge to a critical value b1 , at which the set of population sizes becomes a fractal with fractal dimension df  0:52 [50]. This fractal set resembles a Cantor set confined between 0 and 1. For b1 < b < 4, the behavior is extremely complex. At certain values b chaos emerges and the behavior of the population become unpredictable, i. e. exponentially sensitive to the initial conditions. At some intervals of parameter b, the predictable behavior with a finite attractor set is restored. For b > 4, the population inevitably dies out. Although the set of attractors becomes truly fractal only at certain values of the birth rate, and the particular value of the fractal dimension does not have any biological meaning, the logistic map has a paradigmatic value in the studies of population dynamics with an essentially Malthusian take home massage: excessive birth rate leads to disastrous consequences such as famines coming at unpredictable times, and in the case of Homo sapiens to devastating wars and revolutions.

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out evidence of any cognitive ability. An a example of such a trivial fractal behavior is the random firing of a neuron [46] which integrates through its dendritic synapses inhibitory and excitatory signals from its neighbors. The action potential of such a neuron can be viewed as performing a one-dimensional random walk going down if an inhibitory signal comes from a synapse or going up if an excitatory signal comes form a different synapse. As soon as the action potential reaches a firing threshold the neuron fires, its action potential drops to an original value and the random walk starts again. Thus the time intervals between the firing spikes of such a neuron are distributed as the returning times of a one-dimensional random walk to an origin. It is well known [42,58,104,141], that the probability density P(t) of the random walk returns scales as (t) with  D 3/2. Accordingly, the spikes on the time axis form a fractal dust with the fractal dimension df D   1 D 1/2. A useful way to study the correlations in the time series is to compute its autocorrelation function and its Fourier transform which is called power spectrum, analogous to the structure factor for the spatial correlation analysis. Due to the property of the Fourier transform to convert a convolution into a product, the power spectrum is also equal to the square of the Fourier transform of the original time series. Accordingly it has a simple physical meaning telling how much energy is carried in a certain frequency range. In case of a completely uncorrelated signal, the power spectrum is completely flat, which means that all frequencies carry the same energy as in white light which is the mixture of all the rainbow colors of different frequencies. Accordingly a signal which has a flat power spectrum is called white noise. If the autocorrelation function C(t) decays for t ! 1 as C(t) t  , where 0 <  < 1, the power spectrum S(f ) of this time series diverges as f ˇ , with ˇ D 1  . Thus the LRPLC in the time series can be detected by studying the power spectrum. There are alternative ways of detecting LRPLC in time series, such as Hurst analysis [41], detrended fluctuation analysis (DFA) [19,98] and wavelet analysis [4]. These methods are useful for studying short time series for which the power spectrum is too noisy. They measure the Hurst exponent ˛ D (1 C ˇ)/2 D 1  /2. It can be shown [123] that for a time series which is equal to zero everywhere except at points t1 ; t2 ; : : : ; t n ; : : : , at which it is equal to unity and these points form a fractal set with fractal dimension df , the time autocorrelation function C(t) decreases for t ! 1 as C(t) t  , where  D 1  df . Therefore the power spectrum S(f ) of this time series diverges for f ! 0 as f ˇ , with ˇ D 1   D df . Accordingly, for the random

walk model of neuron firing, the power spectrum is characterized by ˇ D 1/2. In the more general case, the distribution of the intervals t n D t n  t n1 can decay as a power law P(t) (t) , with 1 <  < 3. For  < 2, the set t1 ; t2 ; : : : ; t n is a fractal, and the power spectrum decreases as power law S( f ) f d f D f C1 . For 2 <  < 3, the set t1 ; t2 ; : : : ; t n has a finite density, with df D D D 1. However, the power spectrum and the correlation function maintain their power law behavior for  < 3. This behavior indicates that although the time series itself is uniform, the temporal fluctuations remain fractal. In this case, the exponent ˇ characterizing the low frequency limit is given by ˇ D 3  . The maximal value of ˇ D 1 is achieved when  D 2. This type of signal is called 1/ f noise or “red” noise. If   3, ˇ D 0 in the limit of low frequencies and we again recover white noise. The physical meaning of 1/ f noise is that the temporal correlations are of infinite range. For the majority of the processes in nature, temporal correlations decay exponentially C(t) exp(t/), where the characteristic memory span  is called relaxation or correlation time. For a time series with a finite correlation time , the power spectrum has a Lorentzian shape, namely it stays constant for f < 1/ and decreases as 1/ f 2 for f > 1/. The signal in which S( f ) 1/ f 2 is called brown noise, because it describes the time behavior of the one-dimensional Brownian motion. Thus for the majority of the natural processes, the time spectrum has a crossover from a white noise at low frequencies to a brown noise at high frequencies. The relatively unusual case when S( f ) f ˇ , where 0 < ˇ < 1 is called fractal noise because as we see above it describes the behavior of the fractal time series. 1/ f -noise is the special type of the fractal noise corresponding to the maximal value of ˇ, which can be achieved in the limit of low frequencies. R. Voss and J. Clarke [135,136] had analyzed the music written by different composers and found that it follows 1/ f noise over at least three orders of magnitude. It means that music does not have a characteristic time-scale. There is an evidence that physiological process such as heart-beat, gate, breath and sleeping patterns as well as certain types of human activity such as sending e-mails has certain fractal features [6,16,20,27,47,55,57, 72,73,97,99,100,113,126,129]. A. Goldberger [99,126] suggested that music is pleasant for us because it mimics the fractal features of our physiology. It has to be pointed out that in all these physiological time series there is no clear power law behavior expanding over many orders of magnitude. There is also no simple clear explanation of the origins of fractality. One possible mechanism could be due

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the distribution of the return times of a random walk, which has been used to explain the sleeping patterns and response to the e-mails. SOC and Biological Evolution Self-organized criticality (SOC) [8,92] describes the behavior of the systems far form equilibrium, the general feature of which is a slow increase of strain which is interrupted by an avalanche-like stress release. These avalanches are distributed in a power law fashion. The power spectrum of an activity at a given spatial site is described by a fractal noise. One of the most successful application of SOC is the explanation of the power-law distribution of the magnitudes of the earthquakes (Gutenberg–Richter’s law) [89,130]. The simple physical models of self-organized criticality are invasion percolation [26,127], sand-pile model [8], and Bak–Sneppen model of biological evolution [7]. In the one-dimensional Bak–Sneppen model, an ecosystem is represented by a linear chain of the pray-predator relationships in with each species is represented by a site on a straight line surrounded by its predator (a site to the right) and a pray (a site to the left). Each site is characterized by its fitness f which at the beginning is uniformly distributed between 0 and 1. At each time step, a site with the lowest fitness becomes extinct and is replaced by a mutated species with a new fitness randomly taken from a uniform distribution between 0 and 1. The fitnesses of its two neighbors (predator and prey) are also changed at random. After certain equilibration time, the fitness of almost all the species except a few becomes larger than a certain critical value fc . These few active species with low fitness which can spontaneously mutate form a fractal set on the pray-predator line. The activity of each site can be represented by a time series of mutations shown as spikes corresponding to the times of individual mutations at this site. The power spectrum of this time series indicates the presence of the fractal noise. At a steady state the minimal fitness value which spontaneously mutates fluctuates below fc with a small probability P( ) comes into the interval between fc  and fc . The distribution of the first return times t to a given -vicinity of fc follows a power law with an -dependent exponential cut-off. Since each time step corresponds to a mutation, the time interval for which f stays below fc  corresponds to an avalanche of mutations caused by a mutation of a very stable species with f > fc  . Accordingly one can speculate, that evolution goes as a punctuated equilibrium so that an extinction of a stable species causes a gigantic extinction of many other species which hitherto have been well adopted. The problem with this SOC model is the def-

inition of a time step. In order to develop a realistic model of evolution one needs to assume that the real time needed for a spontaneous mutation of species with different fitness dramatically increases with f , going for example as exp( f A), where A is a large value. Unfortunately, it is impossible to verify the predictions of this model because paleontological records do not provide us with a sufficient statistics. Fractal Features of DNA Sequences DNA molecules are probably the largest molecules in nature [139]. Each strand of DNA in large human chromosomes consist of about 108 monomers or base-pairs (bp) which are adenine (A), cytosine (C), guanine (G), and thymine (T). The length of this molecule if stretched would reach several centimeters. The geometrical packing of DNA in a cell resembles a self-similar structure with at least 6 levels of packing: a turn of the double helix (10 bp), a nucleosome (200 bp), a unit of 30 nm fiber (6 nucleosomes), a loop domain ( 100 units of 30 nm fiber), a turn of a metaphase chromosome ( 100 loop domains), a metaphase chromosome ( 100 turns). The packing principle is quite similar to the organization of the information in the library: letters form lines, lines form pages, pages form books, books are placed on the shelves, shelves form bookcases, bookcases form rows, and rows are placed in different rooms. This structure however is not a rigorous fractal, because packing of the units on different levels follows different organization principles. The DNA sequence treated as a sequence of letters also has certain fractal properties [4,5,19,69,70,96]. This sequence can be transformed into a numerical sequence by several mapping rules: for example A rule, in which A is replaced by 1 and C, T, G are replaced by 0, or SW rule in which strongly bonded bp (C and G) are replaced by 1 and weakly bonded bp (A and T) are replaced by  1. Purine-Pyrimidine (RY) mapping rule (A,G: +1; C,T:1) and KM mapping rule (A,C:+1; G,T -1) are also possible. Power spectra of such sequences display large regions of approximate power law behavior in the range from f D 102 to f D 108 . For SW mapping rule we has almost perfect 1/ f noise in the region of low frequencies (Fig. 10). This is not surprising because chromosomes are organized in a large CG rich patches followed by AT rich patches called isochors which extend over millions of bp. The changing slope of the power spectra for different frequency ranges clearly indicates that DNA sequences are also not rigorous fractals but rather a mosaic structures with different organization principles on different length-scales [60,98]. A possible relation between the frac-

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Fractals in Biology, Figure 10 a Power spectra for seven different mapping rules computed for the Homo sapiens chromosome XIV, genomic contig NT_026437. The result is obtained by averaging 1330 power spectra computed by fast Fourier transform for non-overlapping segments of length N D 216 D 65536. b Power spectra for SW, RY, and KM mapping rules for the same contig extended to the low frequency region characterizing extremely long range correlations. The extension is obtained by extracting low frequencies from the power spectra computed by FFT with N D 224  16  106 bp. Three distinct correlation regimes can be identified. High frequency regime (f < 0:003) is characterized by small sharp peaks. Medium frequency regime (0:5  105 < f < 0:003) is characterized by approximate power-law behavior for RY and SW mapping rules with exponent ˇM D 0:57. Low frequency regime (f < 0:5  105 ) is characterized by ˇ D 1:00 for SW rule. The high frequency regime for RY rule can be approximated by ˇH D 0:16 in agreement with the data of Fig. 11. c RY Power spectra for the entire genome of E. coli (bacteria), S. cerevisae (yeast) chromosome IV, H. sapiens (human) chromosome XIV and the largest contig (NT_032977.6) on the chromosome I; and C. elegans (worm) chromosome X. It can be clearly seen that the high frequency peaks for the two different human chromosomes are exactly the same, while they are totally different from the high frequency peaks for other organisms. These high frequency peaks are associated with the interspersed repeats. One can also notice the presence of enormous peaks for f D 1/3 in E. coli and yeast, indicating that their genomes do not have introns, so that the lengths of coding segments are very large. The C. elegans data can be very well approximated by power law correlations S(f) f 0:28 for 104 < f < 102 . d Log–log plot of the RY power spectrum for E. coli with subtracted white noise level versus jf 1/3j. It shows a typical behavior for a signal with finite correlation length, indicating that the distribution of the coding segments in E. coli has finite average square length of approximately 3  103 bp

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Fractals in Biology, Figure 11 RY Power spectra averaged over all eukaryotic sequences longer than 512 bp, obtained by FFT with window size 512. Upper curve is average over 29,453 coding sequences; lower curve is average over 33,301 noncoding sequences. For clarity, the power spectra are shifted vertically by arbitrary quantities. The straight lines are least squares fits for second decade (Region M). The values of ˇM for coding and noncoding DNA obtained from the slopes of the fits are 0.03 and 0.21, respectively (from [21])

tal correlations of DNA sequences and packing of DNA molecules was suggested in [52,53]. An intriguing property of the DNA of multicellular organisms is that an overwhelming portion of it (97% in case of humans) is not used for coding proteins. It is interesting that the percent of non-coding DNA increases with the complexity of an organism. Bacteria practically do not have non-coding DNA, and Yeast has only 30% of it. The coding sequences form genes ( 104 bp) which carry information for one protein. The genes of multicellular organisms are broken by many noncoding intervening sequences (introns). Only exons which are short ( 102 bp) coding sequences located between introns ( 103 bp) are eventually translated into a protein. The genes themselves are separated by very long intergenic sequences  105 bp. Thus the coding structure of DNA resembles a Cantor set. The purpose and properties of coding DNA are well understood. Each three consequent bp form a codon, which is translated into one amino acid of the protein sequence. Accordingly, the power spectrum computed for the coding DNA has a characteristic peak at f D 1/3 corresponding to the inverse codon length (Figs. 11 and 12). Coding DNA is highly conserved and the power spectra of coding DNA of different organisms are very similar and in case of mammals are indistinguishable (Fig. 12). The properties of noncoding DNA are very different. Non coding DNA contains a lot of useful information in-

Fractals in Biology, Figure 12 Comparison of the correlation properties of coding and non-coding DNA of different mammals. Shown RY power spectra averaged over all complimentary DNA sequences of Homo sapiens (HS-C) and Mus musculus (mouse) (MM-C). The complimentary DNA sequences are obtained from messenger RNA by reverse transcriptase and thus lack non-coding elements. They are characterized by huge peaks at f D 1/3, corresponding to the inverse codon length 3 bp. The power spectra for human and mouse are almost indistinguishable. Also shown RY power spectra of large continuously sequenced segments of chromosomes (contigs) of about 107 bp long for mouse (MM), human (HS) and chimpanzee (PT, Pan troglodytes). Their power spectra have different high frequency peaks absent in the coding DNA power spectra: a peak at f D 1/2, corresponding to simple repeats and several large peaks in the range from f D 1/3 to f D 1/100 corresponding to interspersed repeats. Notice that the magnitudes of these peaks are similar for humans and chimpanzees (although for humans they are slightly larger, especially the peak at 80 bp corresponding to the long interspersed repeats) and much larger than those of mouse. This means that mouse has much smaller number of the interspersed repeats than primates. On the other hand, mouse has much larger fraction of dimeric simple repeats indicated by the peak at f D 1/2

cluding protein binding sites controlling gene transcription and expression and other regulatory sequences. However its overwhelming fraction lacks any known purpose. This “junk” DNA is full of simple repeats such as CACACACA. . . as well as the interspersed repeats or retroposons which are virus-like sequences inserting themselves in great number of copies into the intergenic DNA. It is a great challenge of molecular biology and genetics to understand the meaning of non-coding DNA and even to learn how to manipulate it. It would be very interesting to create transgenic animals without non-coding DNA and test if their phenotype will differ from that of the wild type species. The non-coding DNA even for very closely

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related species can significantly differ (Fig. 12). The length of simple repeats varies even for close relatives. That is why simple repeats are used in forensic studies. The power spectra of non-coding DNA significantly differ from those of coding DNA. Non-coding DNA does not have a peak at f D 1/3. The presence of simple repeats make non-coding DNA more correlated than coding DNA on a scale from 10 to 100 bp [21] (Fig. 11). This difference observed in 1992 [70,96] lead to the hypothesis that noncoding DNA is governed by some mutation-duplication stochastic process which creates long-range (fractal) correlations, while the coding DNA lacks long-range correlations because it is highly conserved. Almost any mutation which happens in coding DNA alter the sequence of the corresponding protein and thus may negatively affect its function and lead to a non-viable or less fit organism. Researchers have proposed using the difference in the long-range correlations to find the coding sequences in the sea of non-coding DNA [91]. However this method appears to be unreliable and today the non-coding sequences are found with much greater accuracy by the bioinformatics methods based on sequence similarity to the known proteins. Bioinformatics has developed powerful methods for comparing DNA of different species. Even a few hundred bp stretch of the mitochondrial DNA of a Neanderthal man can tell that Neanderthals diverged from humans about 500 000 years ago [65]. The power spectra and other correlation methods such as DFA or wavelet analysis does not have such an accuracy. Never the less, power spectra of the large stretches of DNA carry important information on the evolutionary processes in the DNA. They are similar for different chromosomes of the same organism, but differ even for closely related species (Fig. 12). In this sense they can play a role similar to the role played by X-ray or Raman spectra for chemical substances. A quick look at them can tell a human and a mouse and even a human and a monkey apart. Especially interesting is the difference in peak heights produced by different interspersed repeats. The height of these peaks is proportional to the number of the copies of the interspersed repeats. According to this simple criterion, the main difference between humans and chimps is the insertion of hundreds of thousands of extra copies of interspersed repeats into human DNA [59]. Future Directions All the above examples show that there are no rigorous fractals in biology. First of all, there are always a lower and an upper cutoff of the fractal behavior. For example, a polymer in a good solvent which is probably the most

rigorous of all real fractals in biology has the lower cutoff corresponding to the persistence length comprised of a few monomers and the upper cutoff corresponding to the length of the entire polymer or to the size of the compartment in which it is confined. For the hierarchical structures such as trees and lungs, in addition to the obvious lower and upper cutoffs, the branching pattern changes from one generation to the next. In the DNA, the packing principles employ different mechanisms on the different levels of packing. In bacterial colonies, the lower cutoff is due to some tendency of bacteria to clump together analogous to surface tension, and the upper cutoff is due to the finite concentration of the nutrients, which originally are uniformly distributed on the Petri dish. In the ideal DLA case, the concentration of the diffusing particles is infinitesimally small, so at any given moment of time there is only one particle in the vicinity of the aggregate. The temporal physiological series are also not exactly self-similar, but are strongly affected by the daily schedule with a characteristic frequency of 24 h and shorter overtones. Some of these signals can be described with help of multifractals. Fractals in ecology are limited by the topographical features of the land which may be also fractal due to complex geological processes, so it is very difficult to distinguish whether certain features are caused by biology or geology. The measurements of the fractal dimension are hampered by the lack of statistics, the noise in the image or signal, and by the crossovers due to intrinsic features of the system which is differently organized on different length scales. So very often the mosaic organization of the system with patches of several fixed length scales can be mistakenly identified with the fractal behavior. Thus the use of fractal dimension or Hurst exponent for diagnostics or distinguishing some parts of the system from one another has a limited value. After all, the fractal dimension is only one number which is usually obtained from a slope of a least square linear fit of a log–log graph over a subjectively identified range of values. Other features of the power spectrum, such as peaks at certain characteristic frequencies may have more biological information than fractal dimension. Moreover, the presence of certain fractal behavior may originate from simple physical principles, while the deviation from it may indicate the presence of a nontrivial biological phenomenon. On the other hand, fractal geometry is an important concept which can be used for qualitative understanding of the mechanisms behind certain biological processes. The use of similar organization principles on different length scales is a remarkable feature, which is certainly em-

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ployed by nature to design the shape and behavior of living organisms. Though fractals themselves may have a limited value in biology, the theory of complex systems in which they often emerge continues to be a leading approach in understanding life. One of the most impelling challenges of modern interdisciplinary science which involves biology, chemistry, physics, mathematics, computer science, and bioinformatics is to build a comprehensive theory of morphogenesis. Such a great mind as de Gennes turned to this subject in his late years [35,36,37]. In these studies the theory of complex networks [2,12,48,67,90] which describe interactions of biomolecules with many complex positive and negative feedbacks will take a leading part. Challenges of the same magnitude face researches in neuroscience, behavioral science, and ecology. Only complex interdisciplinary approach involving the specialists in the theory of complex systems may lead to new breakthroughs in this field.

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Fractals and Economics MISAKO TAKAYASU1 , HIDEKI TAKAYASU2 1 Tokyo Institute of Technology, Tokyo, Japan 2 Sony Computer Science Laboratories Inc, Tokyo, Japan Article Outline Glossary Definition of the Subject Introduction Examples in Economics Basic Models of Power Laws Market Models Income Distribution Models Future Directions Bibliography Glossary Fractal An adjective or a noun representing complex configurations having scale-free characteristics or selfsimilar properties. Mathematically, any fractal can be characterized by a power law distribution. Power law distribution For this distribution the probability density is given by a power law, p(r) D c  r˛1 , where c and ˛ are positive constants. Foreign exchange market A free market of currencies, exchanging money in one currency for other, such as purchasing a United States dollar (USD) with Japanese yen (JPY). The major banks of the world are trading 24 hours and it is the largest market in the world. Definition of the Subject Market price fluctuation was the very first example of fractals, and since then many examples of fractals have been found in the field of Economics. Fractals are everywhere in economics. In this article the main attention is focused on real world examples of fractals in the field of economics, especially market properties, income distributions, money flow, sales data and network structures. Basic mathematics and physics models of power law distributions are reviewed so that readers can start reading without any special knowledge. Introduction Fractal is the scientific word coined by B.B. Mandelbrot in 1975 from the Latin word fractus, meaning “fractured” [25]. However, fractal does not directly mean fracture itself. As an image of a fractal Fig. 1 shows a photo of

Fractals and Economics, Figure 1 Fractured pieces of plaster fallen on a hard floor (provided by H. Inaoka)

fractured pieces of plaster fallen on a hard floor. There are several large pieces, many middle size pieces and countless fine pieces. If you have a microscope and observe a part of floor carefully then you will find in your vision several large pieces, many small pieces and countless fine pieces, again in the microscopic world. Such scale-invariant nature is the heart of the fractal. There is no explicit definition on the word fractal, it generally means a complicated scale-invariant configuration. Scale-invariance can be defined mathematically [42]. Let P( r) denote the probability that the diameter of a randomly chosen fractured piece is larger than r, then this distribution is called scale-invariant if this function satisfies the following proportional relation for any positive scale factor  in a considering scale range: P( r) / P( r) :

(1)

The proportional factor should be a function of , so we can re-write Eq. (1) as P( r) D C()P( r) :

(2)

Assuming that P( r) is a differentiable function, and differentiate Eq. (2) by , and then let  D 1. rP 0 ( r) D C 0 (1)P( r)

(3)

As C 0 (1) is a constant this differential equation is readily integrated as P( r) D c0 r C

0 (1)

:

(4)

P( r) is a cumulative distribution and it is a non-increasing function in general, the exponent C 0 (1) can be replaced by ˛ where ˛ is a positive constant. Namely, from

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the scale-invariance with the assumption of differentiability we have the following power law: P( r) D c0 r˛ :

(5)

The reversed logic also holds, namely for any power law distribution there is a fractal configuration or a scale-invariant state. In the case of real impact fracture, the size distribution of pieces is experimentally obtained by repeating sieves of various sizes, and it is empirically well-known that a fractured piece’s diameter follows a power law with the exponent about ˛ D 2 independent of the details about the material or the way of impact [14]. This law is one of the most stubborn physical laws in nature as it is known to hold from 106 m to 105 m, from glass pieces around us to asteroids. From theoretical viewpoint this phenomenon is known to be described by a scale-free dynamics of crack propagation and the universal properties of the exponent value are well understood [19]. Usually fractal is considered geometric concept introducing the quantity fractal dimension or the concept of self-similarity. However, in economics there are very few geometric objects, so, the concept of fractals in economics are mostly used in the sense of power law distributions. It should be noted that any geometrical fractal object accompanies a power law distribution even a deterministic fractal such as Sierpinski gasket. Figure 2 shows Sierpinski gasket which is usually characterized by the fractal dimension D given by DD

log 3 : log 2

(6)

Paying attention to the distribution of length r of white triangles in this figure, it is easy to show that the probability that a randomly chosen white triangle’s side is larger than r, P( r), follows the power law, P( r) / r ˛ ;

˛DDD

Fractals and Economics, Figure 2 Sierpinski gasket

log 3 : log 2

(7)

Here, the power law exponent of distribution equals to the fractal dimension; however, such coincidence occurs only when the considering distribution is for a length distribution. For example, in Sierpinski gasket the area s of white triangles follow the power law, P( s) / s ˛ ;

˛D

log 3 : log 4

(8)

The fractal dimension is applicable only for geometric fractals, however, power law distributions are applicable for any fractal phenomena including shapeless quantities. In such cases the power law exponent is the most important quantity for quantitative characterization of fractals. According to Mandelbrot’s own review on his life the concept of fractal was inspired when he was studying economics data [26]. At that time he found two basic properties in the time series data of daily prices of New York cotton market [24]: (A) Geometrical similarity between large scale chart and an expanded chart. (B) Power law distribution of price changes in a unit time interval, which is independent of the time scale of the unit. He thought such scale invariance in both shape and distribution is a quite general property, not only in price charts but also in nature at large. His inspiration was correct and the concept of fractals spread over physics first and then over almost all fields of science. In the history of science it is a rare event that a concept originally born in economics has been spread widely to all area of sciences. Basic mathematical properties of cumulative distribution can be summarized as follows (here we consider distribution of non-negative quantity for simplicity): 1. P( 0) D 1 , P( 1) D 0. 2. P( r) is a non-increasing function of r. 3. The probability density is given as p(r)   drd P( r). As for power law distributions there are three peculiar characteristics: 4. Difficulty in normalization. Assuming that P( r) D c0 r˛ for all in the range 0  r < 1, then the normalization factor c0 must be 0 considering the limit of r ! 0. To avoid this difficulty it is generally assumed that the power law does not hold in the vicinity of r D 0. In the case of observing distribution from real data there are naturally lower and upper bounds, so this difficulty should be necessary only for theoretical treatment. 5. Divergence R 1of moments. As for moments defined by hr n i  0 r n p(r)dr, hr n i D 1 for n  ˛. In the

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special case of 2  ˛ > 0 the basic statistical quantity, the variance, diverges,  2  hr2 i  hri2 D 1. In the case of 1  ˛ > 0 even the average can not be defined as hri D 1. 6. Stationary or non-stationary? In view of the data analysis, the above characteristics of diverging moments is likely to cause a wrong conclusion that the phenomenon is non-stationary by observing its averaged value. For example, assume that we observe k samples fr1 ; r2 ; : : : ; r k g independently from the power law distribution with the exponent, 1  ˛ > 0. Then, the sample average, hri k  1k fr1 C r2 C    C r k g, is shown to diverge as, hri k / k 1/˛ . Such tendency of monotonic increase of averaged quantity might be regarded as a result of non-stationarity, however, this is simply a general property of a power law distribution. The best way to avoid such confusion is to observe the distribution directly from the data. Other than the power law distribution there is another important statistical quantity in the study of fractals, that is, the autocorrelation. For given time series, fx(t)g, the autocorrelation is defined as, C(T) 

hx(t C T)x(t)i  hx(t)i2 ; hx(t)2 i  hx(t)i2

(9)

where h   i denotes an average over realizations. The autocorrelation can be defined only for stationary time series with finite variance, in which any statistical quantities do not depend on the location of the origin of time axis. For any case, the autocorrelation satisfies the following basic properties, 1. 2. 3.

C(0) D 1 and C(1) D 0 jC(T)j  1 for any T  0. The R 1 Wiener–Khinchin theorem holds, C(T) D 0 S( f ) cos 2 f d f , where S(f ) is the power spectrum defined by S( f R)  hb x( f )b x( f )i, with the Fourier transform, b x( f )  x(t)e2 i f t dt.

In the case that the autocorrelation function is characterized by a power law, C(T) / T ˇ , ˇ > 0, then the time series fx(t)g is said to have a fractal property, in the sense that the autocorrelation function is scale-independent for any scale-factor,  > 0, C(T) / C(T). In the case 1 > ˇ > 0 the corresponding power spectrum is given as S( f ) / f 1Cˇ . The power spectrum can be applied to any time series including non-stationary situations. A simple way of telling non-stationary situation is to check the power law exponent of S( f ) / f 1Cˇ in the vicinity of f D 0, for 0 > ˇ the time series is non-stationary.

Three basic examples of fractal time series are the followings: 1. White noise. In the case that fx(t)g is a stationary independent noise, the autocorrelation is given by the Kronecker’s function, C(T) D ıT , where ( 1; T D 0 ıT D 0; T ¤ 0: The corresponding power spectrum is S( f ) / f 0 . This case is called white noise from an analogy that superposition of all frequency lights with the same amplitude make a colorless white light. White noise is a plausible model of random phenomena in general including economic activities. 2. Random walk. This is defined by summation of a white Pt x(s), and the power specnoise, X(t) D X(0) C sD0 trum is given by S( f ) / f 2 . In this case the autocorrelation function can not be defined because the data is non-stationary. Random walks are quite generic models widely used from Brownian motions of colloid to market prices. The graph of a random walk has a fractal property such that an expansion of any part of the graph looks similar to the whole graph. 3. The 1/f noise. The boundary of stationary and nonstationary states is given by the so-called 1/f noise, S( f ) / f 1 . This type of power spectrum is also widely observed in various fields of sciences from electrical circuit noise [16] to information traffics in the Internet [53]. The graph of this 1/f noise also has the fractal property. Examples in Economics In this chapter fractals observed in real economic activities are reviewed. Mathematical models derived from these empirical findings will be summarized in the next chapter. As mentioned in the previous chapter the very first example of a fractal was the price fluctuation of the New York cotton market analyzed by Mandelbrot with the daily data for a period of more than a hundred years [24]. This research attracted much attention at that time, however, there was no other good market data available for scientific analysis, and no intensive follow-up research was done until the 1990s. Instead of earnest scientific data analysis artificial mathematical models of market prices based on random walk theory became popular by the name of Financial Technology during the years 1960–1980. Fractal properties of market prices are confirmed with huge amount of high resolution market data since the 1990s [26,43,44]. This is due to informationization of fi-

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nancial markets in which transaction orders are processed by computers and detail information is recorded automatically, while until the 1980s many people gathered at a market and prices are determined by shouting and screaming which could not be recorded. Now there are more than 100 financial market providers in the world and the number of transacted items exceeds one million. Namely, millions of prices in financial markets are changing with time scale in seconds, and you can access any market price at real time if you have a financial provider’s terminal on your desk via the Internet. Among these millions of items one of the most representative financial markets is the US Dollar-Japanese Yen (USD-JPY) market. In this market Dollar and Yen are exchanged among dealers of major international banks. Unlike the case of stock markets there is no physical trading place, but major international banks are linked by computer networks and orders are emitted from each dealer’s terminal and transactions are done at an electronic broking system. Such a broking system and the computer networks are provided by financial provider companies like Reuters. The foreign exchange markets are open 24 hours and deals are done whenever buy- and sell-orders meet. The minimum unit of a deal is one million USD (called a bar), and about three million bars are traded everyday in the whole foreign exchange markets in which more than 100 kinds of currencies are exchanged continuously. The total amount of money flow is about 100 times bigger than the total amount of daily world trade, so it is believed that most of deals are done not for the real world’s needs, but they are based on speculative strategy or risk hedge, that is, to get profit by buying at a low price and selling at a high price, or to avoid financial loss by selling decreasing currency. In Fig. 3 the price of one US Dollar paid by Japanese Yen in the foreign exchange markets is shown for 13 years [30]. The total number of data points is about 20 million, that is, about 10 thousand per day or the averaged transaction interval is seven seconds. A magnified part of the top figure for one year is shown in the second figure. The third figure is the enlargement of one month in the second figure. The bottom figure is again a part of the third figure, here the width is one day. It seems that at least the top three figures look quite similar. This is one of the fractal properties of market price (A) introduced in the previous chapter. This geometrical fractal property can be found in any market, so that this is a very universal market property. However, it should be noted that this geometrical fractal property breaks down for very short time scale as typi-

Fractals and Economics, Figure 3 Dollar-Yen rate for 13 years (Top). Dark areas are enlarged in the following figure [30]

Fractals and Economics, Figure 4 Market price changes in 10 minutes

cally shown in Fig. 4. In this figure the abscissa is 10 minutes range and we can observe each transaction separately. Obviously the price up down is more zigzag and more discrete than the large scale continuous market fluctuations

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Fractals and Economics, Figure 5 Log–log plot of cumulative distribution of rate change [30]

shown in Fig. 3. In the case of USD-JPY market the time scale that this breakdown of scale invariance occurs typically at time scale of several hours. The distribution of rate change in a unit time (one minute) is shown in Fig. 5. Here, there are two plots of cumulative distributions, P(> x) for positive rate changes and P(> jxj) for negative rate changes, which are almost identical meaning that the up-down symmetry of rate changes is nearly perfect. In this log–log plot the estimated power law distribution’s exponent is 2.5. In the original finding of Mandelbrot, (B) in the previous chapter, the reported exponent value is about 1.7 for cotton prices. In the case of stock markets power laws are confirmed universally for all items, however, the power exponents are not universal, taking value from near one to near five, typically around three [15]. Also the exponent values change in time year by year. In order to demonstrate the importance of large fluctuations, Fig. 6 shows a comparison of three market prices. The top figure is the original rate changes for a week. The middle figure is produced from the same data, but it is consisted of rate changes of which absolute values are larger than 2, that is, about 5 % of all the data. In the bottom curve such large rate changes are omitted and the residue of 95 % of small changes makes the fluctuations. As known from these figures the middle figure is much closer to the original market price changes. Namely, the contribution from the power law tails of price change distribution is very large for macro-scale market prices. Power law distribution of market price changes is also a quite general property which can be confirmed for any market. Up-down symmetry also holds universally in short time scale in general, however, for larger unit time the distribution of price changes gradually deforms and for very large unit time the distribution becomes closer to

Fractals and Economics, Figure 6 USD-JPY exchange rate for a week (top) Rate changes smaller than 2 are neglected (middle) Rate changes larger than 2 are neglected (bottom)

a Gaussian distribution. It should be noted that in special cases of market crashes or bubbles or hyper-inflations the up-down symmetry breaks down and the power law distribution is also likely to be deformed. The autocorrelation of the time sequence of price changes generally decays quickly to zero, sometimes accompanied by a negative correlation in a very short time. This result implies that the market price changes are apparently approximated by white noise, and market prices are known to follow nearly a random walk as a result. However, market price is not a simple random walk. In Fig. 7 the autocorrelation of volatility, which is defined by the square of price change, is shown in log–log scale. In the case of a simple random walk this autocorrelation should also decay quickly. The actual volatility autocorrelation nearly satisfies a power law implying that the volatility time series has a fractal clustering property. (See also Fig. 31 representing an example of price change clustering.) Another fractal nature of markets can be found in the intervals of transactions. As shown in Fig. 8 the transaction intervals fluctuate a lot in very short time scale. It is known that the intervals make clusters, namely, shorter intervals tend to gather. To characterize such clustering effect we can make a time sequence consisted of 0 and 1, where 0 denotes no deal was done at that time, and 1 denotes a deal was done. The corresponding power spectrum follows a 1/f power spectrum as shown in Fig. 9 [50].

Fractals and Economics

Fractals and Economics, Figure 7 Autocorrelation of volatility [30]

Fractals and Economics, Figure 10 Income distribution of companies in Japan

Fractals and Economics, Figure 8 Clustering of transaction intervals

Fractal properties are found not only in financial markets. Company’s income distribution is known to follow also a power law [35]. A company’s income is roughly given by subtraction of incoming money flow minus outgoing money flow, which can take both positive and negative values. There are about six million companies in Japan and Fig. 10 shows the cumulative distribution of annual income of these companies. Clearly we have a power law distribution of income I with the exponent very close to 1 in the middle size range, so-called the Zipf’s law, P(> I) / I ˇ ;

Fractals and Economics, Figure 9 Power spectrum of transaction intervals [50]

ˇ D 1:

(10)

Although in each year every company’s income fluctuates, and some percentage of companies disappear or are newly born, this power law is known to hold for more than 30 years. Similar power laws are confirmed in various countries, the case of France is plotted in Fig. 11 [13]. Observing more details by categorizing the companies, it is found that the income distribution in each job category follows nearly a power law with the exponent depending on the job category as shown in Fig. 12 [29]. The implication of this phenomenon will be discussed in Sect. “Income Distribution Models”. A company’s size can also be viewed by the amount of whole sale or the number of employee. In Figs. 13 and 14 distributions of these quantities are plotted [34]. In both cases clear power laws are confirmed. The size distribution of debts of bankrupted companies is also known to follow a power law as shown Fig. 15 [12].

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Fractals and Economics, Figure 11 Income distribution of companies in France [13]

Fractals and Economics, Figure 14 The distribution of employee numbers [34]

Fractals and Economics, Figure 12 Income distribution of companies in each category [29]

Fractals and Economics, Figure 15 The size distribution of debts of bankrupted companies [12]

Fractals and Economics, Figure 13 The distribution of whole sales [34]

Fractals and Economics

Fractals and Economics, Figure 16 Personal income distribution in Japan [1]

Fractals and Economics, Figure 17 The distribution of the amount of transferred money [21]

A power law distribution can also be found in personal income. Figure 16 shows the personal income distribution in Japan in a log–log plot [1]. The distribution is clearly separated into two parts. The majority of people’s incomes are well approximated by a log-normal distribution (the left top part of the graph), and the top few percent of people’s income distribution is nicely characterized by a power law (the linear line in the left part of the graph). The majority of people are getting salaries from companies. This type of composite of two distributions is well-known from the pioneering study by Pareto about 100 years ago and it holds in various countries [8,22]. A typical value of the power exponent is about two, significantly larger than the income distribution of companies. However, the exponent of the power law seems to be not universal and the value changes county by country or year by year. There is a tendency that the exponent is smaller, meaning more rich people, when the economy is improving [40]. Another fractal in economics can be found in a network of economic agents such as banks’ money transfer network. As a daily activity banks transfer money to other banks for various reasons. In Japan all of these interbank money transfers are done via a special computer network provided by the Bank of Japan. Detailed data of actual money transfer among banks are recorded and analyzed for the basic study. The total amount of money flow among banks in a day is about 30  1012 yen with the number of transactions about 10 000. Figure 17 shows the distribution of the amount of money at a transaction. The range is not wide enough but we can find a power law with an exponent about 1.3 [20].

The number of banks is about 600, so the daily transaction number is only a few percent of the theoretically possible combinations. It is confirmed that there are many pairs of banks which never transact directly. We can define active links between banks for pairs with the averaged number of transaction larger than one per day. By this criterion the number of links becomes about 2000, that is, about 0.5 percent of all possible link numbers. Compared with the complete network, the actual network topologies are much more sparse. In Fig. 18 the number distribution of active links per site are plotted in log–log plot [21]. As is known from this graph, there is an intermediate range in which the link number distribution follows a power law. In the terminology of recent complex network study, this property is called the scale-free network [5]. The scale-free network structure among these intermediate banks is shown in Fig. 19. There are about 10 banks with large link numbers which deviate from the power law, also small link number banks with link number less than four are out of the power law. Such small banks are known to make a satellite structure that many banks linked to one large link number banks. It is yet to clarify why intermediate banks make fractal network, and also to clarify the role of large banks and small banks which are out of the fractal configuration. In relation with the banks, there are fractal properties other than cash flow and the transaction network. The distribution of the whole amount of deposit of Japanese bank is approximated by a power law as shown in Fig. 20 [57]. In recent years large banks merged making a few mega banks and the distribution is a little deformed. Historically there were more than 6000 banks in Japan, however,

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Fractals and Economics, Figure 20 Distribution of total deposit for Japanese banks [57] Power law breaks down from 1999 Fractals and Economics, Figure 18 The number distribution of active links per site [20]

Fractals and Economics, Figure 19 Scale-free network of intermediate banks [20]

now we have about 600 as mentioned. It is very rare that a bank disappears, instead banks are merged or absorbed. The number distribution of banks which are historically behind a present bank is plotted in Fig. 21, again a power law can be confirmed. Other than the example of the bank network, network structures are very important generally in economics. In production process from materials, through various parts to final products the network structure is recently studied in view of complex network analysis [18]. Trade networks among companies can also be described by network terminology. Recently, network characterization quantities such as link numbers (Fig. 22), degrees of authority, and Pageranks are found to follow power laws from real trade data for nearly a million of companies in Japan [34]. Still more power laws in economics can be found in sales data. A recent study on the distribution of expen-

Fractals and Economics, Figure 21 Distribution of bank numbers historically behind a present bank [57]

diture at convenience stores in one shopping trip shows a clear power law distribution with the exponent close to two as shown in Fig. 23 [33]. Also, book sales, movie hits, news paper sales are known to be approximated by power laws [39]. Viewing all these data in economics, we may say that fractals are everywhere in economics. In order to understand why fractals appear so frequently, we firstly need to make simple toy models of fractals which can be analyzed completely, and then, based on such basic models we can make more realistic models which can be directly comparable with real data. At that level of study we will be able to predict or control the complex real world economy.

Fractals and Economics

a power law, P(> y) D y ˛ for y  1. This is a useful transformation in case of numerical simulation using random variable following power laws. 2. Let x be a stochastic variable following an exponential distribution, P(> x) D ex , for positive x, then, y  e x/˛ satisfies a power law, P(> y) / y ˛ . As exponential distributions occur frequently in random process such as the Poisson process, or energy distribution in thermal equilibrium, this simple exponential variable transformation can make it a power law. Superposition of Basic Distributions

Fractals and Economics, Figure 22 Distribution of in-degrees and out-degrees in Japanese company network [34]

A power law distribution can also be easily produced by superposition of basic distributions. Let x be a Gaussian distribution with the probability density given by p R 2 R p R (x) D p e 2 x ; (11) 2 and R be a 2 distribution with degrees of freedom ˛,  1 ˛/2 w(R) D

2



˛ 2

˛

R

 R 2 1 e 2 :

(12)

Then, the superposition of Gaussian distribution, Eq. (11), with the weight given by Eq. (12) becomes the T-distribution having power law tails: Z1 p(x) D

W(R)p R (x)dR 0

   ˛C1 1 2  / jxj˛1 ; Dp ˛  2 (1 C x 2 ) ˛C1 2 Fractals and Economics, Figure 23 Distribution of expenditure in one shopping trip [33]

Basic Models of Power Laws In this chapter we introduce general mathematical and physical models which produce power law distributions. By solving these simple and basic cases we can deepen our understanding of the underlying mechanism of fractals or power law distributions in economics. Transformation of Basic Distributions A power law distribution can be easily produced by variable transformation from basic distributions. 1. Let x be a stochastic variable following a uniform distribution in the range (0, 1], then, y  x 1/˛ satisfies

(13)

which is P(> jxj) / jxj˛ in cumulative distribution. In the special case that R, the inverse of variance of the normal distribution, distributes exponentially, the value of ˛ is 2. Similar super-position can be considered for any basic distributions and power law distributions can be produced by such superposition. Stable Distributions Assume that stochastic variables, x1 ; x2 ; : : : ; x n , are independent and follow the same distribution, p(x), then consider the following normalized summation; Xn 

x1 C x2 C    C x n  n : n1/˛

(14)

If there exists ˛ > 0 and n , such that the distribution of X n is identical to p(x), then, the distribution belongs to

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one of the Levy stable distributions [10]. The parameter ˛ is called the characteristic exponent which takes a value in the range (0, 2]. The stable distribution is characterized by four continuous parameters, the characteristic exponent, an asymmetry parameter which takes a value in [1, 1], the scale factor which takes a positive value and the location parameter which takes any real number. Here, we introduce just a simple case of symmetric distribution around the origin with the unit scale factor. The probability density is then given as p(x; ˛) D

1 2

Z1

˛

ei x ej j d :

(15)

the power law is obtained,  P( x) D

1  Mlog x

0

:

(18)

In other words, a power law distribution maximizes the entropy in the situation where products are conserved. To be more precise, consider two time dependent random variables interacting each other satisfying the relation, x1 (t)  x2 (t) D x1 (t 0 )  x2 (t 0 ), then the equilibrium distribution follows a power law. Another entropy approach to the power laws is to generalize the entropy by the following form [56],

1

For large jxj the cumulative distribution follows the power law, P(> x; ˛) / jxj˛ except the case of ˛ D 2. The stable distribution with ˛ D 2 is the Gaussian distribution. The most important property of the stable distribution is the generalized central limit theorem: If the distribution of sum of any independent identically distributed random variables like X n in Eq. (14) converges in the limit of n ! 1 for some value of ˛, then the limit distribution is a stable distribution with the characteristic exponent ˛. For any distribution with finite variance, the ordinary central limit theory holds, that is, the special case of ˛ D 2. For any infinite variance distribution the limit distribution is ˛ ¤ 2 with a power law tail. Namely, a power law realizes simply by summing up infinitely many stochastic variables with diverging variance.

x x0

1 Sq 

1 R

p(x)q dx

x0

q1

;

(19)

where q is a real number. This function is called the q-entropy and the ordinary entropy, Eq. (15), recovers in the limit of q ! 1. Maximizing the q-entropy keeping the variance constant, so-called a q-Gaussian distribution is obtained, which has the same functional form with the T-distribution, Eq. (12), with the exponent ˛ given by ˛D

q3 : 1q

(20)

This generalized entropy formulation is often applied to nonlinear systems having long correlations, in which power law distributions play the central role.

Entropy Approaches Let x0 be a positive constant and consider a probability density p(x) defined in the interval [x0 ; 1), the entropy of this distribution is given by Z1

Here, we find a distribution that maximizes the entropy with a constraint such that the expectation of logarithm of x is a constant, hlog xi D M. Then, applying the variational principle to the following function,

L x0

01 1 Z p(x) log p(x)dx  1 @ p(x)dx  1A 0 C 2 @

x0

Z1

x0

(21)

(16)

x0

Z1

Stochastic time evolution described by the following formulation is called the multiplicative process, x(t C 1) D b(t)x(t) C f (t) ;

p(x) log p(x)dx :

S

Random Multiplicative Process

P(> x) / jxj˛ ;

1

p(x) log xdx  M A

where b(t) and f (t) are both independent random variables [17]. In the case that b(t) is a constant, the distribution of x(t) depends on the distribution of f (t), for example, if f (t) follows a Gaussian distribution, then the distribution of x(t) is also a Gaussian. However, in the case that b(t) fluctuates randomly, the resulting distribution of x(t) is known to follows a power law independent of f (t),

(17)

(22)

where the exponent ˛ is determined by solving the following equation [48], hjb(t)j˛ i D 1 :

(23)

Fractals and Economics

This steady distribution exists when hlog jb(t)ji < 0 and f (t) is not identically 0. As a special case that b(t) D 0 with a finite probability, then a steady state exists. It is proved rigorously that there exists only one steady state, and starting from any initial distribution the system converges to the power law steady state. In the case hlog jb(t)ji  0 there is no statistically steady state, intuitively the value of jb(t)j is so large that x(t) is likely to diverge. Also in the case f (t) is identically 0 there is no steady state as known from Eq. (21) that log jx(t)j follows a simple random walk with random noise term, log jb(t)j. The reason why this random multiplicative process produces a power law can be understood easily by considering a special case that b(t) D b > 1 with probability 0.5 and b(t) D 0 otherwise, with a constant value of f (t) D 1. In such a situation the value of x(t) is 1 C b C b2 C   C b K with probability (0:5)K . From this we can directly evaluate the distribution of x(t),   b KC1  1 D 2KC1 i: e: P  b1 (24) log 2 P( x) D 4(1 C (b  1)x)˛ ; ˛D : log b As is known from this discussion, the mechanism of this power law is deeply related to the above mentioned transformation of exponential distribution in Sect. “Transformation of Basic Distributions”. The power law distribution of a random multiplicative process can also be confirmed experimentally by an electrical circuit in which resistivity fluctuates randomly [38]. In an ordinary electrical circuit the voltage fluctuations in thermal equilibrium is nearly Gaussian, however, for a circuit with random resistivity a power law distribution holds. Aggregation with Injection Assume the situation that many particles are moving randomly and when two particles collide they coalesce making a particle with mass conserved. Without any injection of particles the system converges to the trivial state that only one particle remains. In the presence of continuous injection of small mass particles there exists a non-trivial statistically steady state in which mass distribution follows a power law [41]. Actually, the mass distribution of aerosol in the atmosphere is known to follow a power law in general [11]. The above system of aggregation with injection can be described by the following model. Let j be the discrete space, and x j (t) be the mass on site j at time t, then choose

one site and let the particle move to another site and particles on the visited site merge, then add small mass particles to all sites, this process can be mathematically given as, 8 ˆ <x j (t) C x k (t) C f j (t) ; x j (tC1) D x j (t) C f j (t) ; ˆ : f j (t) ;

prob D 1/N prob D (N  2)/N prob D 1/N (25)

where N is the total number of sites and f j (t) is the injected mass to the site j. The characteristic function, Z( ; t)  he x j (t) i, which is the Laplace transform of the probability density, satisfies the following equation by assuming uniformity, Z( ; t C 1)

 N 2 1 1 D Z( ; t)2 C Z( ; t) C he f j (t) i : N N N (26) The steady state solution in the vicinity of D 0 is obtained as p (27) Z( ) D 1  h f i 1/2 C    : From this behavior the following power law steady distribution is obtained. P( x) / x ˛ ;

˛D

1 : 2

(28)

By introducing a collision coefficient depending on the size of particles power laws with various values of exponents realized in the steady state of such aggregation with injection system [46]. Critical Point of a Branching Process Consider the situation that a branch grows and splits with probability q or stops growing with probability 1  q as shown in Fig. 24. What is the size distribution of the branch? This problem can be solved in the following way. Let p(r) be the probability of finding a branch of size r, then the next relation holds. p(r C 1) D q

r1 X

p(s)p(r  s) :

(29)

sD1

Multiplying y rC1 and summing up by r from 0 to 1, a closed equation of the generating function, M(y), is obtained, M(y)1Cq D q y  M(y)2 ;

M(y) 

1 X rD0

y r p(r) : (30)

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tributed in N-sites. At each time step choose one site randomly, and transport a finite portion, x j (t), to another randomly chosen site, where is a parameter in the range [0, 1]. x j (t C 1) D (1  )x j (t) ; x k (t C 1) D x k (t) C x j (t) :

Fractals and Economics, Figure 24 Branching process (from left to right)

Solving this quadratic equation and expanding in terms of y, we have the probability density, p(r) / r3/2 eQ(q)r ;

Q(q)  log 4q(1  q) :

(31)

For q < 0:5 the probability decays exponentially for large r, in this case all branches has a finite size. At q D 0:5 the branch size follows the power law, P( r) / r1/2 , and the average size of branch becomes infinity. For q > 0:5 there is a finite probability that a branch grows infinitely. The probability of having an infinite branch, p(1) D 1M(1), is given as, p 2q  1 C 1  4q(1  q) p(1) D ; (32) 2q which grows monotonically from zero to one in the range q D [0:5; 1]. It should be noted that the power law distribution realizes at the critical point between the finite-size phase and the infinite-size phase [42]. Compared with the preceding model of aggregation with injection, Eq. (28), the mass distribution is the same as the branch size distribution at the critical point in Eq. (31). This coincidence is not an accident, but it is known that aggregation with injection automatically chooses the critical point parameter. Aggregation and branching are reversed process and the steady occurrence of aggregation implies that branching numbers keep a constant value on average and this requires the critical point condition. This type of critical behaviors is called the self-organized criticality and examples are found in various fields [4].

It is known that for small positive the statistically steady distribution x is well approximated by a Gaussian like the case of thermal fluctuations. For close to 1 the fluctuation of x is very large and its distribution is close to a power law. In the limit goes to 1 and the distribution converges to Eq. (28), the aggregation with injection case. For intermediate values of the distribution accompanies a fat tail between Gaussian and a power law [49]. Fractal Tiling A fractal tiling is introduced as the final basic model. Figure 25 shows an example of fractal tiling of a plane by squares. Like this case Euclidean space is covered by various sizes of simple shapes like squares, triangles, circles etc. The area size distribution of squares in Fig. 25 follows the power law, P( x) / x ˛ ;

˛D 1/2 :

(34)

Generalizing this model in d-dimensional space, the distribution of d-dimensional volume x is characterized by a power law distribution with an exponent, ˛ D (d  1)/d, therefore, the Zipf’s law which is the case of ˛ D 1 realizes in the limit of d D 1. The fracture size distribution measured in mass introduced in the beginning of this article corresponds to the case of d D 3.

Finite Portion Transport Here, a kind of mixture of aggregation and branching is considered. Assume that conserved quantities are dis-

(33)

Fractals and Economics, Figure 25 An example of fractal tiling

Fractals and Economics

ence as, x(t) D  (t) f (t) ;

(35)

where f (t) is a random variable following a Gaussian distribution with 0 mean and variance unity, the local variance  (t) is given as  (t)2 D c0 C

k X

c k (x(t  k))2 ;

(36)

jD1

Fractals and Economics, Figure 26 Fractal tiling by river patterns [45]

A classical example of fractal tiling is the Apollonian gasket, that is, a plane is covered totally by infinite number of circles which are tangent each other. For a given river pattern like Fig. 26 the basin area distribution follows a power law with exponent about ˛ D 0:4 [45]. Although these are very simple geometric models, simple models may sometimes help our intuitive understanding of fractal phenomena in economics.

Market Models In this chapter market price models are reviewed in view of fractals. There are two approaches for construction of market models. One is modeling the time sequences directly by some stochastic model, and the other is modeling markets by agent models which are artificial markets in computer consisted of programmed dealers. The first market price model was proposed by Bachelier in 1900 written as his Ph.D thesis [3], that is, five years before the model of Einstein’s random walk model of colloid particles. His idea was forgotten for nearly 50 years. In 1950’s Markowitz developed the portfolio theory based on a random walk model of market prices [28]. The theory of option prices by Black and Scholes was introduced in the 1970s, which is also based on random walk model of market prices, or to be more precise a logarithm of market prices in continuum description [7]. In 1982 Engle introduced a modification of the simple random walk model, the ARCH model, which is the abbreviation of auto-regressive conditional heteroscedasticity [9]. This model is formulated for market price differ-

with adjustable positive parameters, fc0 ; c1 ; : : : ; c k g. By the effect of this modulation on variance, the distribution of price difference becomes superposition of Gaussian distribution with various values of variance, and the distribution becomes closer to a power law. Also, volatility clustering occurs automatically so that the volatility autocorrelation becomes longer. There are many variants of ARCH models, such as GARCH and IGARCH, but all of them are based on purely probabilistic modeling, and the probability of prices going up and that of going down are identical. Another type of market price model has been proposed from physics view point [53]. The model is called the PUCK model, an abbreviation of potentials of unbalanced complex kinetics, which assumes the existence of market’s time-dependent potential force, UM (x; t), and the time evolution of market price is given by the following set of equations; ˇ ˇ d C f (t) ; (37) x(tC1)x(t) D  UM (x; t)ˇˇ dx xDx(t)x M (t) b(t) x 2 ; (38) UM (x; t)  M1 2 where M is the number of moving average needed to define the center of potential force, xM (t) 

M1 1 X x(t  k) : M

(39)

kD0

In this model f (t) is the external noise and b(t) is the curvature of quadratic potential which changes with time. When b(t) D 0 the model is identical to the simple random walk model. When b(t) > 0 the market prices are attracted to the moving averaged price, xM (t), the market is stable, and when b(t) < 0 prices are repelled from xM (t) so that the price fluctuation is large and the market is unstable. For b(t) < 2 the price motion becomes an exponential function of time, which can describe singular behavior such as bubbles and crashes very nicely.

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In the simplest case of M D 2 the time evolution equation becomes, x(t C 1) D 

b(t) x(t) C f (t) : 2

(40)

As is known from this functional form in the case b(t) fluctuates randomly, the distribution of price difference follows a power law as mentioned in the previous Sect. “Random Multiplicative Process”, Random multiplicative process. Especially the PUCK model derives the ARCH model by introducing a random nonlinear potential function [54]. The value of b(t) can be estimated from the data and most of known empirical statistical laws including fractal properties are fulfilled as a result [55]. The peculiar difference of this model compared with financial technology models is that directional prediction is possible in some sense. Actually, from the data it is known that b(t) changes slowly in time, and for non-zero b(t) the autocorrelation is not zero implying that the up-down statistics in the near future is not symmetric. Moreover in the case of b(t) < 2 the price motion show an exponential dynamical growth hence predictable. As introduced in Sect. “Examples in Economics” the tick interval fluctuations can be characterized by the 1/f power spectrum. This power law can be explained by a model called the self-modulation model [52]. Let t j be the jth tick interval, and we assume that the tick interval can be approximated by the following random process, t jC1 D  j

K1 1 X t jk C g j ; K

Fractals and Economics, Figure 27 Tick intervals of Poisson process (top) and the self-modulation process (bottom) [52]

(41)

kD0

where j is a positive random number following an exponential distribution with the mean value 1, and K is an integer which means the number of moving average, g j is a positive random variable. Due to the moving average term in Eq. (41) the tick interval automatically make clusters as shown in Fig. 27, and the corresponding power spectrum is proved to be proportional to 1/ f as typically represented in Fig. 28. The market data of tick intervals are tested whether Eq. (41) really works or not. In Fig. 29 the cumulative probability of estimated value of j from market data is plotted where the moving average size is determined by the physical time of 150 seconds and 400 seconds. As known from this figure, the distribution fits very nicely with the exponential distribution when the moving average size is 150 seconds. This result implies that dealers in the market are mostly paying attention to the latest transaction for about a few minutes only. And the dealers’ clocks in their minds move quicker if the market becomes busier. By this

Fractals and Economics, Figure 28 The power spectrum of the self-modulation process [52]

Fractals and Economics, Figure 29 The distribution of normalized time interval [50]

Fractals and Economics

self-modulation effect transactions in markets automatically make a fractal configuration. Next, we introduce a dealer model approach to the market [47]. In any financial market dealers’ final goal is to gain profit from the market. To this end dealers try to buy at the lowest price and to sell at the highest price. Assume that there are N dealers at a market, and let the jth dealer’s buying and selling prices in their mind B j (t) and S j (t). For each dealer the inequality, B j (t) < S j (t), always holds. We pay attention to the maximum price of fB j (t)g called the best bid, and to the minimum price of fS j (t)g called the best ask. Transactions occur in the market if there exists a pair of dealers, j and k, who give the best bid and best ask respectively, and they fulfill the following condition, B j (t)  S k (t) :

(42)

Fractals and Economics, Figure 30 Price evolution of a market with deterministic three dealers

In the model the market price is given by the mean value of these two prices. As a simple situation we consider a deterministic time evolution rule for these dealers. For all dealers the spread, S j (t) B j (t), is set to be a constant L. Each dealer has a position, either a seller or a buyer. When the jth dealer’s position is a seller the selling price in mind, S j (t), decreases every time step until he can actually sell. Similar dynamics is applied to a buyer with the opposite direction of motion. In addition we assume that all dealers shift their prices in mind proportional to a market price change. When this proportional coefficient is positive, the dealer is categorized as a trend-follower. If this coefficient is negative, the dealer is called a contrarian. These rules are summarized by the following time evolution equations. B j (t C 1) D B j (t) C a j S j C b j x(t) ;

(43)

where Sj takes either +1 or 1 meaning the buyer position or seller position, respectively, x(t) gives the latest market price change, fa j g are positive numbers given initially, fb j g are also parameters given initially. Figure 30 shows an example of market price evolution in the case of three dealers. It should be noted that although the system is deterministic, namely, the future price is determined uniquely by the set of initial values, resulting market price fluctuates almost randomly even in the minimum case of three dealers. The case of N D 2 gives only periodic time evolution as expected, while for N  3 the system can produce market price fluctuations similar to the real market price fluctuations, for example, the fractal properties of price chart and the power law distribution of price difference are realized. In the case that the value of fb j g are identical for all dealers, b, then the distribution of market price difference follows a power law where the exponent is controllable by

Fractals and Economics, Figure 31 Cumulative distribution of a dealer model for different values of b. For weaker trend-follow the slope is steeper [38]

this trend-follow parameter, b as shown in Fig. 31 [37]. The volatility clustering is also observed automatically for large dealer number case as shown in Fig. 32 (bottom) which looks quite similar to a real price difference time series Fig. 32 (top). By adding a few features to this basic dealer model it is now possible to reproduce almost all statistical characteristics of market, such as tick-interval fluctuations, abnormal diffusions etc. [58]. In this sense the study of market behaviors are now available by computer simulations based on the dealer model. Experiments on the market is either impossible or very difficult for a real market, however, in an artificial market we can repeat occurrence of bubbles and crashes any times, so that we might be able to find a way to avoid catastrophic market behaviors by numerical simulation.

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This mathematical answer implies that even X is one million dollars this lottery is generous enough and you should buy because expectation is infinity. But, would you dare to buy this lottery, in which you will win only one dollar with probability 0.5, and two dollars with probability 0.25, . . . ? Bernoulli’s answer to this paradox is to introduce the human feeling of value, or utility, which is proportional to the logarithm of price, for example. Based on this expected utility hypothesis the fair value of X is given as follows, XD

1 1 X X U(2n ) log(2n ) D D 1 C log 2 nC1 2 2nC1 nD0 nD0

 1:69 ;

Fractals and Economics, Figure 32 Price difference time series for a real market (top) and a dealer model (bottom)

Income Distribution Models Let us start with a famous historical problem, the St. Petersburg Paradox, as a model of income. This paradox was named after Daniel Bernoulli’s paper written when he was staying in the Russian city, Saint Petersburg, in 1738 [6]. This paradox treats a simple lottery as described in the following, which is deeply related to the infinite expected value problem in probability theory and also it has been attracting a lot of economists’ interest in relation with the essential concept in economics, the utility [2]. Assume that you enjoy a game of chance, you pay a fixed fee, X dollars, to enter, and then you toss a fair coin repeatedly until a tail firstly appears. You win 2n dollars where n is the number of heads. What is the fair price of the entrance fee, X? Mathematically a fair price should be equal to the expectation value, therefore, it should be given as, 1 X 1 XD 2n  nC1 D 1 : (44) 2 nD0

(45)

where the utility function, U(x) D 1Clog x, is normalized to satisfy U(1) D 1. This result implies that the appropriate entry fee X should be about two dollars. The idea of utility was highly developed in economics for description of human behavior, in the way that human preference is determined by maximal point of utility function, the physics concept of the variational principle applied to human action. Recently, in the field of behavioral finance which emerged from psychology the actual observation of human behaviors about money is the main task and the St. Petersburg paradox is attracting attention [36]. Although Bernoulli’s solution may explain the human behavior, the fee X D 2 is obviously so small that the bookmaker of this lottery will bankrupt immediately if the entrance fee is actually fixed as two dollars and if a lot of people actually buy it. The paradox is still a paradox. To clarify what is the problem we calculate the distribution of income of a gambler. As an income is 2 n with probability 2n1 , the cumulative distribution of income is readily obtained as, P( x) /

1 : x

(46)

This is the power law which we observed for income distribution of companies in Sect. “Examples in Economics”. The key of this lottery is the mechanism that the prize money doubles at each time a head appears and the coin toss stops when a tail appears. By denoting the number of coin toss by t, we can introduce a stochastic process or a new lottery which is very much related to the St. Petersburg lottery. x(t C 1) D b(t)x(t) C 1 ;

(47)

where b(t) is 2 with probability 0.5 and is 0 otherwise. As introduced in Sect. “Random Multiplicative Process”, this problem is solved easily and it is confirmed that the steady

Fractals and Economics

state cumulative distribution of x(t) also follows Eq. (46). The difference between the St. Petersburg lottery and the new lottery Eq. (47) is the way of payment of entrance fee. In the case of St. Petersburg lottery the entrance fee X is paid in advance, while in the case of new lottery you have to add one dollar each time you toss a coin. This new lottery is fair from both the gambler side and the bookmaker side because the expectation of income is given by hx(t)i D t and the amount of paid fee is also t. Now we introduce a company’s income model by generalizing this new fair lottery in the following way, I(t C 1) D b(t)I(t) C f (t) ;

(48)

where I(t) denotes the annual income of a company, b(t) represents the growth rate which is given randomly from a growth rate distribution g(b), and f (t) is a random noise. Readily from the results of Sect. “Random Multiplicative Process”, we have a condition to satisfy the empirical relation, Eq. (10), Z hb(t)i D bg(b) D 1 : (49)

An implication of this result is that if a job category is expanding, namely, hb(t)i > 1, then the power law exponent determined by Eq. (50) is smaller than 1. On the other hand if a job category is shrinking, we have an exponent that is larger than 1. This type of company’s income model can be generalized to take into account the effect of company’s size dependence on the distribution of growth rate. Also, the magnitude of the random force term can be estimated from the probability of occurrence of negative income. Then, assuming that the present growth rate distribution continues we can perform a numerical simulation of company’s income distribution starting from a uniform distribution as shown in Fig. 34 for Japan and in Fig. 35 for USA. It is shown that in the case of Japan, the company size distribution converges to the power law with exponent 1 in 20 years, while in the case of USA the steady power law’s slope is about 0.7 and it takes about 100 years to converge [31]. According to this result extremely large com-

This relation is confirmed to hold approximately in actual company data [32]. In order to explain the job category dependence of the company’s income distribution already shown in Fig. 12, we plot the comparison of exponents in Fig. 33. Empirically estimated exponents are plotted in the ordinate and the solutions of the following equation calculated in each job category are plotted in the abscissa, hb(t)ˇ i D 1 :

(50)

The data points are roughly on a straight line demonstrating that the simple growth model of Eq. (48) seems to be meaningful.

Fractals and Economics, Figure 33 Theoretical predicted exponent value vs. observed value [29]

Fractals and Economics, Figure 34 Numerical simulation of income distribution evolution of Japanese companies [32]

Fractals and Economics, Figure 35 Numerical simulation of income distribution evolution of USA companies [32]

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panies with size about 10 times bigger than the present biggest company will appear in USA in this century. Of course the growth rate distribution will change faster than this prediction, however, this model can tell the qualitative direction and the speed of change in very macroscopic economical conditions. Other than this simple random multiplicative model approach there are various approaches to explain empirical facts about company’s statistics assuming a hierarchical structure of organization, for example [23]. Future Directions Fractal properties generally appear in almost any huge data in economics. As for financial market models, empirical fractal laws are reproduced and the frontier of study is now at the level of practical applications. However, there are more than a million markets in the world and little is known about their interaction. More research on market interaction will be promising. Company data so far analyzed show various fractal properties as introduced in Sect. “Examples in Economics”, however, they are just a few cross-sections of global economics. Especially, companies’ interaction data are inevitable to analyze the underlying network structures. Not only money flow data it will be very important to observe material flow data in manufacturing and consumption processes. From the viewpoint of environmental study, such material flow network will be of special importance in the near future. Detail sales data analysis is a new topic and progress is expected. Bibliography Primary Literature 1. Aoyama H, Nagahara Y, Okazaki MP, Souma W, Takayasu H, Takayasu M (2000) Pareto’s law for income of individuals and debt of bankrupt companies. Fractals 8:293–300 2. Aumann RJ (1977) The St. Petersburg paradox: A discussion of some recent comments. J Econ Theory 14:443–445 http://en. wikipedia.org/wiki/Robert_Aumann 3. Bachelier L (1900) Theory of Speculation. In: Cootner PH (ed) The Random Character of Stock Market Prices. MIT Press, Cambridge (translated in English) 4. Bak P (1996) How Nature Works. In: The Science of Self-Organized Criticality. Springer, New York 5. Barabási AL, Réka A (1999) Emergence of scaling in random networks. Science 286:509–512; http://arxiv.org/abs/ cond-mat/9910332 6. Bernoulli D (1738) Exposition of a New Theory on the Measurement of Risk; Translation in: (1954) Econometrica 22:22–36 7. Black F, Scholes M (1973) The Pricing of Options and Corporate Liabilities. J Political Econ 81:637–654

8. Brenner YS, Kaelble H, Thomas M (1991) Income Distribution in Historical Perspective. Cambridge University Press, Cambridge 9. Engle RF (1982) Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of UK Inflation. Econometrica 50:987–1008 10. Feller W (1971) An Introduction to Probability Theory and Its Applications, 2nd edn, vol 2. Wiley, New York 11. Friedlander SK (1977) Smoke, dust and haze: Fundamentals of aerosol behavior. Wiley-Interscience, New York 12. Fujiwara Y (2004) Zipf Law in Firms Bankruptcy. Phys A 337:219–230 13. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Phys A 335:197–216 14. Gilvarry JJ, Bergstrom BH (1961) Fracture of Brittle Solids. II Distribution Function for Fragment Size in Single Fracture (Experimental). J Appl Phys 32:400–410 15. Gopikrishnan P, Meyer M, Amaral LAN, Stanley HE (1998) Inverse Cubic Law for the Distribution of Stock Price Variations. Eur Phys J B 3:139–143 16. Handel PH (1975) 1/f Noise-An “Infrared” Phenomenon. Phys Rev Lett 34:1492–1495 17. Havlin S, Selinger RB, Schwartz M, Stanley HE, Bunde A (1988) Random Multiplicative Processes and Transport in Structures with Correlated Spatial Disorder. Phys Rev Lett 61: 1438–1441 18. Hidalgo CA, Klinger RB, Barabasi AL, Hausmann R (2007) The Product Space Conditions the Development of Nations. Science 317:482–487 19. Inaoka H, Toyosawa E, Takayasu H (1997) Aspect Ratio Dependence of Impact Fragmentation. Phys Rev Lett 78:3455–3458 20. Inaoka H, Ninomiya T, Taniguchi K, Shimizu T, Takayasu H (2004) Fractal Network derived from banking transaction – An analysis of network structures formed by financial institutions. Bank of Japan Working Paper. http://www.boj.or.jp/en/ ronbun/04/data/wp04e04.pdf 21. Inaoka H, Takayasu H, Shimizu T, Ninomiya T, Taniguchi K (2004) Self-similarity of bank banking network. Phys A 339: 621–634 22. Klass OS, Biham O, Levy M, Malcai O, Solomon S (2006) The Forbes 400 and the Pareto wealth distribution. Econ Lett 90:290–295 23. Lee Y, Amaral LAN, Canning D, Meyer M, Stanley HE (1998) Universal Features in the Growth Dynamics of Complex Organizations. Phys Rev Lett 81:3275–3278 24. Mandelbrot BB (1963) The variation of certain speculative prices. J Bus 36:394–419 25. Mandelbrot BB (1982) The Fractal Geometry of Nature. W.H. Freeman, New York 26. Mandelbrot BB (2004) The (mis)behavior of markets. Basic Books, New York 27. Mantegna RN, Stanley HE (2000) An Introduction to Economics: Correlations and Complexity in Finance. Cambridge Univ Press, Cambridge 28. Markowitz HM (1952) Portfolio Selection. J Finance 7:77–91 29. Mizuno T, Katori M, Takayasu H, Takayasu M (2001) Statistical laws in the income of Japanese companies. In: Empirical Science of Financial Fluctuations. Springer, Tokyo, pp 321–330 30. Mizuno T, Kurihara S, Takayasu M, Takayasu H (2003) Analysis of high-resolution foreign exchange data of USD-JPY for 13 years. Phys A 324:296–302

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31. Mizuno T, Kurihara S, Takayasu M, Takayasu H (2003) Investment strategy based on a company growth model. In: Takayasu H (ed) Application of Econophysics. Springer, Tokyo, pp 256–261 32. Mizuno T, Takayasu M, Takayasu H (2004) The mean-field approximation model of company’s income growth. Phys A 332:403–411 33. Mizuno T, Toriyama M, Terano T, Takayasu M (2008) Pareto law of the expenditure of a person in convenience stores. Phys A 387:3931–3935 34. Ohnishi T, Takayasu H, Takayasu M (in preparation) 35. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Phys A 269:125– 131. http://www.ingentaconnect.com/content/els/03784371; jsessionid=5e5wq937wfsqu.victoria 36. Rieger MO, Wang M (2006) Cumulative prospect theory and the St. Petersburg paradox. Econ Theory 28:665–679 37. Sato AH, Takayasu H (1998) Dynamic numerical models of stock market price: from microscopic determinism to macroscopic randomness. Phys A 250:231–252 38. Sato AH, Takayasu H, Sawada Y (2000) Power law fluctuation generator based on analog electrical circuit. Fractals 8: 219–225 39. Sinha S, Pan RK (2008) How a “Hit” is Born: The Emergence of Popularity from the Dynamics of Collective Choice. http:// arxiv.org/PS_cache/arxiv/pdf/0704/0704.2955v1.pdf 40. Souma W (2001) Universal structure of the personal income distribution. Fractals 9:463–470; http://www.nslij-genetics. org/j/fractals.html 41. Takayasu H (1989) Steady-state distribution of generalized aggregation system with injection. Phys Rev Lett 63: 2563–2566 42. Takayasu H (1990) Fractals in the physical sciences. Manchester University Press, Manchester 43. Takayasu H (ed) (2002) Empirical Science of Financial Fluctuations–The Advent of Econophysics. Springer, Tokyo 44. Takayasu H (ed) (2003) Application of Econophysics. Springer, Tokyo 45. Takayasu H, Inaoka H (1992) New type of self-organized criticality in a model of erosion. Phys Rev Lett 68:966–969

46. Takayasu H, Takayasu M, Provata A, Huber G (1991) Statistical properties of aggregation with injection. J Stat Phys 65: 725–745 47. Takayasu H, Miura H, Hirabayashi T, Hamada K (1992) Statistical properties of deterministic threshold elements–The case of market price. Phys A 184:127–134 48. Takayasu H, Sato AH, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys Rev Lett 79:966–969 49. Takayasu M, Taguchi Y, Takayasu H (1994) Non-Gaussian distribution in random transport dynamics. Inter J Mod Phys B 8:3887–3961 50. Takayasu M (2003) Self-modulation processes in financial market. In: Takayasu H (ed) Application of Econophysics. Springer, Tokyo, pp 155–160 51. Takayasu M (2005) Dynamics Complexity in Internet Traffic. In: Kocarev K, Vatty G (eds) Complex Dynamics in Communication Networks. Springer, New York, pp 329–359 52. Takayasu M, Takayasu H (2003) Self-modulation processes and resulting generic 1/f fluctuations. Phys A 324:101–107 53. Takayasu M, Mizuno T, Takayasu H (2006) Potentials force observed in market dynamics. Phys A 370:91–97 54. Takayasu M, Mizuno T, Takayasu H (2007) Theoretical analysis of potential forces in markets. Phys A 383:115–119 55. Takayasu M, Mizuno T, Watanabe K, Takayasu H (preprint) 56. Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52:479–487; http://en.wikipedia.org/wiki/ Boltzmann_entropy 57. Ueno H, Mizuno T, Takayasu M (2007) Analysis of Japanese bank’s historical network. Phys A 383:164–168 58. Yamada K, Takayasu H, Takayasu M (2007) Characterization of foreign exchange market using the threshold-dealer-model. Phys A 382:340–346

Books and Reviews Takayasu H (2006) Practical Fruits of Econophysics. Springer, Tokyo Chatterjee A, Chakrabarti BK (2007) Econophysics of Markets and Business Networks (New Economic Windows). Springer, New York

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Fractals in Geology and Geophysics DONALD L. TURCOTTE Department of Geology, University of California, Davis, USA Article Outline Glossary Definition of the Subject Introduction Drainage Networks Fragmentation Earthquakes Volcanic Eruptions Landslides Floods Self-Affine Fractals Topography Earth’s Magnetic Field Future Directions Bibliography Glossary Fractal A collection of objects that have a power-law dependence of number on size. Fractal dimension The power-law exponent in a fractal distribution. Definition of the Subject The scale invariance of geological phenomena is one of the first concepts taught to a student of geology. When a photograph of a geological feature is taken, it is essential to include an object that defines the scale, for example, a coin or a person. It was in this context that Mandelbrot [7] introduced the concept of fractals. The length of a rocky coastline is obtained using a measuring rod with a specified length. Because of scale invariance, the length of the coastline increases as the length of the measuring rod decreases according to a power law. It is not possible to obtain a specific value for the length of a coastline due to small indentations down to a scale of millimeters or less. A fractal distribution requires that the number of objects N with a linear size greater than r has an inverse power-law dependence on r so that C (1) rD where C is a constant and the power D is the fractal dimension. This power-law scaling is the only distribution that is ND

scale invariant. However, the power-law dependence cannot be used to define a statistical distribution because the integral of the distribution diverges to infinity either for large values or small values of r. Thus fractal distributions never appear in compilations of statistical distributions. A variety of statistical distributions have power-law behavior either at large scales or small scales, but not both. An example is the Pareto distribution. Many geological phenomena are scale invariant. Examples include the frequency-size distributions of fragments, faults, earthquakes, volcanic eruptions, and landslides. Stream networks and landforms exhibit scale invariance. In terms of these applications there must always be upper and lower cutoffs to the applicability of a fractal distribution. As a specific application consider earthquakes on the Earth. The number of earthquakes has a power-law dependence on the size of the rupture over a wide range of sizes. But the largest earthquake cannot exceed the size of the Earth, say 104 km. Also, the smallest earthquake cannot be smaller than the grain size of rocks, say 1 mm. But this range of scales is 1010 . Actual earthquakes appear to satisfy fractal scaling over the range 1 m to 103 km. An example of fractal scaling is the number-area distribution of lakes [10], this example is illustrated in Fig. 1. Excellent agreement with the fractal relation given in Eq. (1) is obtained taking D D 1:90. The linear dimension r is taken to be the square root of the area A and the power-law (fractal) scaling extends from r D 100 m to r D 300 km. Introduction Fractal scaling evolved primarily as an empirical means of correlating data. A number of examples are given below. More recently a theoretical basis has evolved for the applicability of fractal distributions. The foundation of this basis is the concept of self-organized criticality. A number of simple computational models have been shown to yield fractal distributions. Examples include the sand-pile model, the forest-fire model, and the slider-block model. Drainage Networks Drainage networks are a universal feature of landscapes on the Earth. Small streams merge to form larger streams, large streams merge to form rivers, and so forth. Strahler [16] quantified stream networks by introducing an ordering system. When two like-order streams of order i merge they form a stream of order i C 1. Thus two i D 1 streams merge to form a i D 2 stream, two i D 2 streams merge to from a i D 3 stream and so forth. A bifurcation

Fractals in Geology and Geophysics

self-similar, side-branching topology was developed. Applications to drainage networks have been summarized by Peckham [11] and Pelletier [13]. Fragmentation

Fractals in Geology and Geophysics, Figure 1 Dependence of the cumulative number of lakes N with areas greater than A as a function of A. Also shown is the linear dimension r which is taken to be the square root of A. The straight-line correlation is with Eq. (1) taking the fractal dimension D D 1:90

ratio Rb is defined by Rb D

Ni N iC1

(2)

where N i is the number of streams of order i. A length order ratio Rr is defined by Rr D

r iC1 ri

(3)

where ri is the mean length of streams of order i. Empirically both Rb and Rr are found to be nearly constant for a range of stream orders in a drainage basin. From Eq. (1) the fractal dimension of a drainage basin DD

ln(N i /N iC1 ) ln Rb D ln(r iC1 /r i ) ln Rr

(4)

Typically Rb D 4:6, Rr D 2:2, and the corresponding fractal dimension is D D 1:9. This scale invariant scaling of drainage networks was recognized some 20 years before the concept of fractals was introduced in 1967. A major advance in the quantification of stream networks was made by Tokunaga [17]. This author was the first to recognize the importance of side branching, that is some i D 1 streams intersect i D 2, i D 3, and all higher-order streams. Similarly, i D 2 streams intersect i D 3 and higher- order streams and so forth. A fully

An important application of power-law (fractal) scaling is to fragmentation. In many examples the frequency-mass distributions of fragments are fractal. Explosive fragmentation of rocks (for example in mining) give fractal distributions. At the largest scale the frequency size distribution of the tectonic plates of plate tectonics are reasonably well approximated by a power-law distribution. Fault gouge is generated by the grinding process due to earthquakes on a fault. The frequency-mass distribution of the gouge fragments is fractal. Grinding (comminution) processes are common in tectonics. Thus it is not surprising that fractal distributions are ubiquitous in geology. As a specific example consider the frequency-mass distribution of asteroids. Direct measurements give a fractal distribution. Since asteroids are responsible for the impact craters on the moon, it is not surprising that the frequencyarea distribution of lunar craters is also fractal. Using evidence from the moon and a fractal extrapolation it is estimated that on average, a 1m diameter meteorite impacts the earth every year, that a 100 m diameter meteorite impacts every 10,000 years, and that a 10 km diameter meteorite impacts the earth every 100,000,000 years. The classic impact crater is Meteor Crater in Arizona, it is over 1 km wide and 200 m deep. Meteor Crater formed about 50,000 years ago and it is estimated that the impacting meteorite had a diameter of 30 m. The largest impact to occur in the 20th century was the June 30, 1908 Tunguska event in central Siberia. The impact was observed globally and destroyed over 1000 km2 of forest. It is believed that this event was the result of a 30 m diameter meteorite that exploded in the atmosphere. One of the major global extinctions occurred at the Cretaceous/Tertiary boundary 65 million years ago. Some 65% of the existing species were destroyed including dinosaurs. This extinction is attributed to a massive impact at the Chicxulub site on the Yucatan Peninsula, Mexico. It is estimated that the impacting meteorite had a 10 km diameter. In addition to the damage done by impacts there is evidence that impacts on the oceans have created massive tsunamis. The fractal power-law scaling can be used to quantify the risk of future impacts. Earthquakes Earthquakes universally satisfy several scaling laws. The most famous of these is Gutenberg–Richter frequency-

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magnitude scaling. The magnitude M of an earthquake is an empirical measure of the size of an earthquake. If the magnitude is increased by one unit it is observed that the cumulative number of earthquakes greater than the specified magnitude is reduced by a factor of 10. For the entire earth, on average, there is 1 magnitude 8 earthquake per year, 10 magnitude 7 earthquakes per year, and 100 magnitude 6 earthquakes per year. When magnitude is converted to rupture area a fractal relation is obtained. The numbers of earthquakes that occur in a specified region and time interval have a power-law dependence on the rupture area. The validity of this fractal scaling has important implications for probabilistic seismic risk assessment. The number of small earthquakes that occur in a region can be extrapolated to estimate the risk of larger earthquakes [1]. As an example consider southern California. On average there are 30 magnitude 4 or larger earthquakes per year. Using the fractal scaling it is estimated that the expected intervals between magnitude 6 earthquakes will be 3 years, between magnitude 7 earthquakes will be 30 years, and between magnitude 8 earthquakes will be 300 years. The fractal scaling of earthquakes illustrate a useful aspect of fractal distributions. The fractal distribution requires two parameters. The first parameter, the fractal dimension D (known as the b-value in seismology), gives the dependence of number on size (magnitude). For earthquakes the fractal dimension is almost constant independent of the tectonic setting. The second parameter gives the level of activity. For example, this can be the number of earthquakes greater than a specified magnitude in a region. This level of activity varies widely and is an accepted measure of seismic risk. The level is essentially zero in states like Minnesota and is a maximum in California. Volcanic Eruptions There is good evidence that the frequency-volume statistics of volcanic eruptions are also fractal [9]. Although it is difficult to quantify the volumes of magma and ash associated with older eruptions, the observations suggest that an eruption with a volume of 1 km3 would be expected each 10 years, 10 km3 each 100 years, and 100 km3 each 1000 years. For example, the 1991 Mount Pinatubo, Philippines eruption had an estimated volume of about 5 km3 . The most violent eruption in the last 200 years was the 1815 Tambora, Indonesia eruption with an estimated volume of 150 km3 . This eruption influenced the global climate in 1816 which was known as the year without a summer. It is estimated that the Long Valley, California eruption with an age of about 760,000 years had a volume of about

600 km3 and the Yellowstone eruptions of about 600,000 years ago had a volume of about 2000 km3 . Although the validity of the power-law (fractal) extrapolation of volcanic eruption volumes to long periods in the past can be questioned, the extrapolation does give some indication of the risk of future eruptions to global climate. There is no doubt that the large eruptions that are known to have occurred on time scales of 105 to 106 years would have a catastrophic impact on global agricultural production. Landslides Landslides are a complex natural phenomenon that constitutes a serious natural hazard in many countries. Landslides also play a major role in the evolution of landforms. Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt, or a large storm. The landslide event can include a single landslide or many thousands. The frequency-area distribution of a landslide event quantifies the number of landslides that occur at different sizes. It is generally accepted that the number of large landslides with area A has a power-law dependence on A with an exponent in the range 1.3 to 1.5 [5]. Unlike earthquakes, a complete statistical distribution can be defined for landslides. A universal fit to an inversegamma distribution has been found for a number of event inventories. This distribution has a power-law (fractal) behavior for large landslides and an exponential cut-off for small landslides. The most probable landslides have areas of about 40 m2 . Very few small landslides are generated. As a specific example we consider the 11,111 landslides generated by the magnitude 6.7 Northridge (California) earthquake on January 17, 1994. The total area of the landslides was 23.8 km2 and the area of the largest landslide was 0.26 km2 . The inventory of landslide areas had a good power-law dependence on area for areas greater than 103 m2 (103 km2 ). The number of landslides generated by earthquakes have a strong dependence on earthquake magnitude. Typically earthquakes with magnitudes M less than 4 do not generate any landslides [6]. Floods Floods are a major hazard to many cities and estimates of flood hazards have serious economic implications. The standard measure of the flood hazard is the 100-year flood. This is quantified as the river discharge Q100 expected during a 100 year period. Since there is seldom a long enough history to establish Q100 directly, it is necessary to extrapolate smaller floods that occur more often.

Fractals in Geology and Geophysics

One extrapolation approach is to assume that flood discharges are fractal (power-law) [3,19]. This scale invariant distribution can be expressed in terms of the ratio F of the peak discharge over a 10 year interval to the peak discharge over a 1 year interval, F D Q10 /Q1 . With self-similarity the parameter F is also the ratio of the 100 year peak discharge to the 10 year peak discharge, F D Q100 /Q10 . Values of F have a strong dependence on climate. In temperate climates such as the northeastern and northwestern US values are typically in the range F D 2–3. In arid and tropical climates such as the southwestern and southeastern US values are typically in the range F D 4–6. The applicability of fractal concepts to flood forecasting is certainly controversial. In 1982, the US government adopted the log-Pearson type 3 (LP3) distribution for the legal definition of the flood hazard [20]. The LP3 is a thintailed distribution relative to the thicker tailed power-law (fractal) distribution. Thus the forecast 100 year flood using LP3 is considerably smaller than the forecast using the fractal approach. This difference is illustrated by considering the great 1993 Mississippi River flood. Considering data at the Keukuk, Iowa gauging station [4] this flood was found to be a typical 100 year flood using the power-law (fractal) analysis and a 1000 to 10,000 year flood using the federal LP3 formulation. Concepts of self-similarity argue for the applicability of fractal concepts for flood-frequency forecasting. This applicability also has important implications for erosion. Erosion will be dominated by the very largest floods.

Adjacent values are not correlated with each other. In this case the spectrum is flat and the power spectral density coefficients are not a function of frequency, ˇ D 0. The classic example of a self-affine fractal is a Brownian walk. A Brownian walk is obtained by taking the running sum of a Gaussian white noise. In this case we have ˇ D 2. Another important self-affine time series is a red (or pink) noise with power spectral density coefficients proportional to 1/ f , that is ˇ D 1. We will see that the variability in the Earth’s magnetic field is well approximated by a 1/ f noise. Self-affine fractal time series in the range ˇ D 0 to 1 are known as fractional Gaussian noises. These noises are stationary and the standard deviation is a constant independent of the length of the time series. Self-affine time series with ˇ larger than 1 are known as fractional Brownian walk. These motions are not stationary and the standard deviation increases as a power of the length of the time series, there is a drift. For a Brownian walk the standard deviation increases with the square root of the length of the time series.

Self-Affine Fractals

Spectral expansions of global topography have been carried out, an example [15] is given in Fig. 2. Excellent agreement with the fractal relation given in Eq. (6) is obtained taking ˇ D 2, topography is well approximated by a Brownian walk. It has also shown that this fractal behavior of topography is found for the moon, Venus, and Mars [18].

Mandelbrot and Van Ness [8] extended the concept of fractals to time series. Examples of time series in geology and geophysics include global temperature, the strength of the Earth’s magnetic field, and the discharge rate in a river. After periodicities and trends have been removed, the remaining values are the stochastic (noise) component of the time series. The standard approach to quantifying the noise component is to carry out a Fourier transform on the time series [2]. The power-spectral density coefficients Si are proportional to the squares of the Fourier coefficients. The time series is a self-affine fractal if the power-spectral density coefficients have an inverse power-law dependence on frequency f i , that is Si D

C ˇ

(5)

fi

where C is a constant and ˇ is the power-law exponent. For a Gaussian white noise the values in the time series are selected randomly from a Gaussian distribution.

Topography The height of topography along linear tracks can be considered to be a continuous time series. In this case we consider the wave number ki (1/wave length) instead of frequency. Topography is a self-affine fractal if Si D

C k ˇi

(6)

Earth’s Magnetic Field Paleomagnetic studies have given the strength and polarity of the Earth’s magnetic field as a function of time over millions of years. These studies have also shown that the field has experienced a sequence of reversals. Spectral studies of the absolute amplitude of the field have been shown that it is a self-affine fractal [12,14]. The power-spectral density is proportional to one over the frequency, it is a 1/ f noise. When the fluctuations of the 1/ f noise take the magnitude to zero the polarity of the field reverses. The predicted distribution of polarity intervals is fractal and is in good agreement with the observed polarity intervals.

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Fractals in Geology and Geophysics, Figure 2 Power spectral density S as a function of wave number k for a spherical harmonic expansion of the Earth’s topography (degree l). The straight-line correlation is with Eq. (6) taking ˇ D 2, a Brownian walk

Future Directions There is no question that fractals are a useful empirical tool. They provide a rational means for the extrapolation and interpolation of observations. The wide applicability of power-law (fractal) distributions is generally accepted, but does this applicability have a more fundamental basis? Fractality appears to be fundamentally related to chaotic behavior and to numerical simulations exhibiting self-organized criticality. The entire area of fractals, chaos, selforganized criticality, and complexity remains extremely active, and it is impossible to predict with certainty what the future holds. Bibliography

4. Malamud BD, Turcotte DL, Barton CC (1996) The 1993 Mississippi river flood: A one hundred or a one thousand year event? Env Eng Geosci 2:479 5. Malamud BD, Turcotte DL, Guzzetti F, Reichenbach P (2004) Landslide inventories and their statistical properties. Earth Surf Process Landf 29:687 6. Malamud BD, Turcotte DL, Guzzetti F, Reichenbach P (2004) Landslides, earthquakes, and erosion. Earth Planet Sci Lett 229:45 7. Mandelbrot BB (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156:636 8. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422 9. McClelland L et al (1989) Global Volcanism 1975-1985. Prentice-Hall, Englewood Cliffs 10. Meybeck M (1995) Global distribution of lakes. In: Lerman A, Imboden DM, Gat JR (eds) Physics and Chemistry of Lakes, 2nd edn. Springer, Berlin, pp 1-35 11. Peckham SD (1989) New results for self-similar trees with applications to river networks. Water Resour Res 31:1023 12. Pelletier JD (1999) Paleointensity variations of Earth’s magnetic field and their relationship with polarity reversals. Phys Earth Planet Int 110:115 13. Pelletier JD (1999) Self-organization and scaling relationships of evolving river networks. J Geophys Res 104:7259 14. Pelletier JD, Turcotte DL (1999) Self-affine time series: II. Applications and models. Adv Geophys 40:91 15. Rapp RH (1989) The decay of the spectrum of the gravitational potential and the topography of the Earth. Geophys J Int 99:449 16. Strahler AN (1957) Quantitative analysis of watershed geomorphology. Trans Am Geophys Un 38:913 17. Tokunaga E (1978) Consideration on the composition of drainage networks and their evolution. Geogr Rep Tokyo Metro Univ 13:1 18. Turcotte DL (1987) A fractal interpretation of topography and geoid spectra on the earth, moon, Venus, and Mars. J Geophys Res 92:E597 19. Turcotte DL (1994) Fractal theory and the estimation of extreme floods. J Res Natl Inst Stand Technol 99:377 20. US Water Resources Council (1982) Guidelines for Determining Flood Flow Frequency. Bulletin 17B. US Geological Survey, Reston

Primary Literature 1. Kossobokov VG, Keilis-Borok VI, Turcotte DL, Malamud BD (2000) Implications of a statistical physics approach for earthquake hazard assessment and forecasting. Pure Appl Geophys 157:2323 2. Malamud BD, Turcotte DL (1999) Self-affine time series: I. Generation and analyses. Adv Geophys 40:1 3. Malamud BD, Turcotte DL (2006) The applicability of powerlaw frequency statistics to floods. J Hydrol 332:168

Books and Reviews Feder J (1988) Fractals. Plenum Press, New York Korvin G (1992) Fractal Models in the Earth Sciences. Elsevier, Amsterdam Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco Turcotte DL (1997) Fractals and Chaos in Geology and Geophysics, 2nd edn. Cambridge University Press, Cambridge

Fractals Meet Chaos

Fractals Meet Chaos TONY CRILLY Middlesex University, London, UK Article Outline Glossary Definition of the Subject Introduction Dynamical Systems Curves and Dimension Chaos Comes of Age The Advent of Fractals The Merger Future Directions Bibliography Glossary Dimension The traditional meaning of dimension in modern mathematics is “topological dimension” and is an extension of the classical Greek meaning. In modern concepts of this dimension can be defined in terms of a separable metric space. For the practicalities of Fractals and Chaos the notion of dimension can be limited to subsets of Euclidean n-space where n is an integer. The newly arrived “fractal dimension” is metrically based and can take on fractional values. Just as for topological dimension. As for topological dimension itself there is a profusion of different (but related) concepts of metrical dimension. These are widely used in the study of fractals, the ones of principal interest being:  Hausdorff dimension (more fully, Hausdorff–Besicovitch dimension),  Box dimension (often referred to as Minkowski– Bouligand dimension),  Correlation dimension (due to A. Rényi, P. Grassberger and I. Procaccia). Other types of metric dimension are also possible. There is “divider dimension” (based on ideas of an English mathematician/meteorologist L. F. Richardson in the 1920s); the “Kaplan–Yorke dimension” (1979) derived from Lyapunov exponents, known also as the “Lyapunov dimension”; “packing dimension” introduced by Tricot (1982). In addition there is an overall general dimension due to A. Rényi (1970) which admits box dimension, correlation dimension and information dimension as special cases. With many

of the concepts of dimension there are upper and lower refinements, for example, the separate upper and lower box dimensions. Key references to the vast (and highly technical) subject of mathematical dimension include [31,32,33,60,73,92,93]. Hausdorff dimension (Hausdorff–Besicovitch dimension). In the study of fractals, the most sophisticated concept of dimension is Hausdorff dimension, developed in the 1920s. The following definition of Hausdorff dimension is given for a subset A of the real number line. This is readily generalized to subsets of the plane, Euclidean 3-space and Euclidean n-space, and more abstractly to separable metric spaces by taking neighborhoods as disks instead of intervals. Let fU i g be an r-covering of A, (a covering of A where the width of all intervals U i , satisfies w(U i )  r). The measure mr is defined by mr (A) D inf

1 X

! w(U i )

;

iD1

where the infimum (or greatest of the minimum values) is taken over all r-coverings of A. The Hausdorff dimension DH of A is: DH D lim mr (A) ; r!0

provided the limit exists. The subset E D f1/n : n D 1; 2; ; 3; : : :g of the unit interval has DH D 0 (the Hausdorff dimension of a countable set is always zero). The Hausdorff dimension is the basis of “fractal dimension” but because it takes into account intervals of unequal widths it may be difficult to calculate in practice. Box dimension (or Minkowski–Bouligand dimension, known also as capacity dimension, cover dimension, grid dimension). The box counting dimension is a more direct and practical method for computing dimension in the case of fractals. To define it, we again confine our attention to the real number line in the knowledge that box dimension is readily extended to subsets of more general spaces. As before, let fU i g be an r-covering of A, and let N r (A) be the least number of sets in such a covering. The box dimension DB of A is defined by: log N r (A) : r!0 log 1/r

DB D lim

The box dimension of the subset E D f1/n : n D 1; 2; 3; : : :g of the unit interval can be calculated to give DB D 0:5.

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In general Hausdorff and Box dimensions are related to each other by the inequality DB  DH , as happens in the above example. The relationship between DH and DB is investigated in [49]. For compact, selfsimilar fractal sets DB D DH but there are fractal sets for which DB > DH [76]. Though Hausdorff dimension and Box dimension have similar properties, Box dimension is only finitely additive, while Hausdorff dimension is countably additive. Correlation dimension For a set of n points A on the real number line, let P(r) be the probability that two different points of A chosen at random are closer than r apart. For a large number of points n, the graph of v D log P(r) against u D log r is approximated to its slope for small values of r, and theoretically, a straight line. The correlation dimension DC is defined as its slope for small values of r, that is, dv : r!0 du

DC D lim

The correlation dimension involves the separation of points into “boxes”, whereas the box dimension merely counts the boxes that cover A. If Pi is the probability of a point of A being in box i (approximately n i /n where ni is the number of points in box i and n is the totality of points in A) then an alternative definition of correlation dimension is P log Pi2 i : DC D lim r!0 log r Attractor A point set in phase space which “attracts” trajectories in its vicinity. More formally, a bounded set A in phase space is called an attractor for the solution x(t) of a differential equation if  x(0) 2 A ) x(t) 2 A f or al l t. Thus, an attractor is invariant under the dynamics (trajectories which start in A remain in A).  There is a neighborhood U  A such that any trajectory starting in U is attracted to A (the trajectory gets closer and closer to A).  If B A and if B satisfies the above two properties then B D A. An attractor is therefore the minimal set of points A which attracts all orbits starting at some point in a neighborhood of A. Orbit is a sequence of points fx i g D x0 ; x1 ; x2 ; : : : defined by an iteration x n D f n (x0 ). If n is a positive it is called a forwards orbit, and if n is negative a backwards orbit. If x0 D x n for some finite value n, the orbit is periodic. In this case, the smallest value of n for

which this is true is called the period of the orbit. For an invertible function f , a point x is homoclinic to a if lim f n (x) D lim f n (x) D a

as n ! 1 :

and in this case the orbit f f n (x0 )g is called a homoclinic orbit – the orbit which converges to the same saddle point a forwards or backwards. This term was introduced by Poincaré. The terminology “orbit” may be regarded as applied to the solution of a difference equation, in a similar way the solution of a differential equation x(t) is termed a trajectory. Orbit is the term used for discrete dynamical system and trajectory for the continuous time case. Basin of attraction If a is an attractive fixed point of a function f , its basin of attraction B(a) is the subset of points defined by B(a) D fx : f k (x) ! a ; as k ! 1g : It is the subset containing all the initial points of orbits attracted to a. The basins of attraction may have a complicated structure. An important example applies to the case where a is a point in the complex plane C. Julia set A set J f is the boundary between the basins of attraction of a function f . For example, in the case where z D ˙1 are attracting points (solutions of z2  1 D 0), the Julia set of the “Newton–Fourier” function f (z) D z((z2 1)/2z) is the set of complex numbers which lie along the imaginary axis x D 0 (as proved by Schröder and Cayley in the 1870s). The case of the Julia set involved with the solutions of z3  1 D 0 was beyond these pioneers and is fractal in nature. An alternative definition for a Julia set is the closure of the subset of the complex plane whose orbits of f tend to infinity. Definition of the Subject Though “Chaos” and “Fractals” are yoked together to form “Fractals and Chaos” they have had separate lines of development. And though the names are modern, the mathematical ideas which lie behind them have taken more than a century to gain the prominence they enjoy today. Chaos carries an applied connotation and is linked to differential equations which model physical phenomena. Fractals is directly linked to subsets of Euclidean space which have a fractional dimension, which may be obtained by the iteration of functions. This brief survey seeks to highlight some of the significant points in the history of both of these subjects. There are brief academic histories of the field. A history of

Fractals Meet Chaos

Chaos has been shown attention [4,46], while an account of the early history of the iteration of complex functions (up to Julia and Fatou) is given in [3]. A broad survey of the whole field of fractals is given in [18,50]. Accounts of Chaos and Fractals tend to concentrate on one side at the expense of the other. Indeed, it is only quite recently that the two subjects have been seen to have a substantial bearing on each other [72]. This account treats a “prehistory” and a “modern period”. In the prehistory period, before about 1960, topics which contributed to the modern theory are not so prominent. There is a lack of impetus to create a new field. In the modern period there is a greater sense of continuity, driven on by the popular interest in the subject. The modern theory coincided with a rapid increase in the power of computers unavailable to the early workers in the prehistory period. Scientists, mathematicians, and a wider public could now “see” the beautiful geometrical shapes displayed before them [25]. The whole theory of “fractals and chaos” necessarily involves nonlinearity. It is a mathematical theory based on the properties of processes which are assumed to be modeled by nonlinear differential equations and nonlinear functions. Chaos shows itself when solutions to these differential equations become unstable. The study of stability is rooted in the mathematics of the nineteenth century. Fractals are derived from the geometric study of curves and sets of points generally, and from abstract iterative schemes. The modern theory of fractals is the outcome of explorations by mathematicians and scientists in the 1960s and 1970s, though, as we shall see, it too has an extensive prehistory. Recently, there has been an explosion of published work in Fractals and Chaos. Just in Chaos theory alone a bibliography of six hundred articles and books compiled by 1982, grew to over seven thousand by the end of 1990 [97]. This is most likely an excessive underestimate. Fractals and Chaos has since grown into a wide-ranging and variegated theory which is rapidly developing. It has widespread applications in such areas as Astronomy the motions of planets and galaxies Biology population dynamics chemistry chemical reactions Economics time series, analysis of financial markets Engineering capsize of ships, analysis of road traffic flow Geography measurement of coastlines, growth of cities, weather forecasting Graphic art analysis of early Chinese Landscape Paintings, fractal geometry of music Medicine dynamics of the brain, psychology, heart rhythms.

There are many works which address the application of Chaos and Fractals to a broad sweep of subjects. In particular, see [13,19,27,59,63,64,87,96]. Introduction “Fractals and Chaos” is the popular name for the subject which burst onto the scientific scene in the 1970s and 1980s and drew together practical scientists and mathematicians. The subject reached the popular ear and perhaps most importantly, its eye. Computers were becoming very powerful and capable of producing remarkable visual images. Quite elementary mathematics was capable of creating beautiful pictur