Smooth Invariant Manifolds and Normal Forms (World Scientific Series on Nonlinear Science. Series a, Monographs and Treatises, V. 7)

NONLINEAR SCIENC E ~~'""10~ WORLD SCIENTIFIC SERIES ON

Series A Vol. 7

Series Editor: Leon O. Chua

SMOOTH INVRRIRNT MRNIFO~OS RNO NORMR~ FORMS

I. U. Bronstein and A. Va. Kopanskii Institute of Mathematics Academy of Sciences of Moldova

h

'

World Scientific

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SMoom INVARIANT MANIFOLDS AND NORMAL FORMS Copyright C 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts tMreof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be inllented, without wriUen permission from tM PublisMr.

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ISBN: 981-02-1S72-X

v

CONTENTS

Introduction

vii

Chapter I. Topological properties of nows and cascades in the vicinity of a rest point and a periodic trajectory § 1. Basic definitions and facts

1

§ 2. Hyperbolic rest points § 3. Floquet-Lyapunov theory § 4. Hadamard-Bohl-Perron theory

8

Chapter

n.

13 21

Finitely smooth normal forms of vector fields and diffeomorphisms

c:c

§ 1. The problem on reducing a ~ vector field to normal form in the vicinity of a hyperbolic rest point § 2. Normalization of jets of vector fields and diffeomorphisms § 3. Polynomial normal forms

29 34 47

§ 4. Simplification of the resonant normal form via finitely

67

smooth transformations § S. A general condition for § 6.

c:c linearizability

c:c linearization theorems

§ 7. Some sufficient conditions for

76 89

c:c linearizability

c:c

§ 8. Theorems on normal forms § 9. Linearization of finitely smooth vector fields and diffeomorphisms § 10. Normal forms (a supplement to § 8) § 11. Summary of results on finitely smooth normal forms

108 136 143 178 182

vi

Chapter

m.

Linear extensions of dynamical systems

§ 1. Basic notions and facts § 2. Exponential separation and exponential splitting § 3. The structure of linear extensions § 4. Quadratic Lyapunov functions § S. Weak regularity and Green-Samoilenko functions § 6. Smooth linear extensions

193 200 206 224 230 251

Chapter IV. Invariant subbundles of weakly non-linear extensions § 1. Invariant subbundles and their intrinsic characterization § 2. The decomposition theorem § 3. The Grobman-Hartman theorem § 4. Smooth invariant subbundles

261 268 274 278

Chapter V. Invariant manifolds

§ 3. Necessary condition for persistence

290 293 300

§ 4. Asymptotic phase

304

§ 1. Persistence of invariant manifolds § 2. Normal hyperbolicity and persistence

Chapter VI. Normal forms in the vicinity of an invariant manifold § 1. Polynomial normal forms (the nodal case) § 2. Polynomial normal forms (the saddle case)

318 332

Appendix. Some facts from global analysis

343

Bibliography

364

Subject index

380

List of symbols

384

vii

INTRODUCTION

This book is related to the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Much attention is given by specialists in differential equations to the investigation of invariant manifolds. There are several reasons for this. Firstly, the collection of all compact invariant manifolds (in particular, eqUilibria, periodic orbits, invariant tori etc.) constitutes, so to speak, the skeleton of the dynamical system. Therefore, one would like to know whether these manifolds persist under perturbations of the vector field, and what happens in their vicinity (for example, do the nearby solutions tend to the manifold, or stay nearby, or leave the neighbourhood?) Secondly, the existence, for example, of an exponentially stable invariant sub manifold permits one to reduce the investigation of nearby motions to that of points in the manifold itself and thereby to lower the dimension of the phase space. Thirdly, the possibility of reducing a dynamical system to normal form is intimately related to the existence of invariant manifolds. The following simple observation serves as an illustration. Two differential equations = ft.x) and y = g(y) are conjugate via a smooth change of variables y = h(x) if and only if the system = ft.x), = g(y) admits a smooth invariant sub manifold of the form {(x, y): y = h(x)}. Various interrelations between conjugacies of dynamical systems, on the one hand, and invariant sections of certain extensions, on the other hand, are repeatedly used in this work. The main purpose of the book is to present, as fully as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent achievements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory. Besides, an attempt is made to summarize the not numerous results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold. The choice of material is stipulated by the wish to reflect, in the first place, the typical, generic properties of dynamical systems. That is why we consider normal forms relative only to the hyperbolic variables (i.e., in the direction transverse to the center

x

x

y

viii

manifold), whereas the subtle problem concerning further simplifications along the center manifold is beyond the scope of our considerations. The first two chapters deal with dynamical systems near an equilibrium and a periodic orbit. Several important results are stated here without proof because they easily follow from more general theorems concerning arbitrary compact invariant manifolds which are presented (with full proofs) in the last four chapters. This way of presentation has allowed us to essentially shorten the text, but, as can be expected, it will not be approved by readers interested only in classical topics. Let us note in excuse that, suprisingly enough, the proofs for a rest point are not much easier as compared with the general case (cf., for example, the papers by Takens [1] and Robinson [1]). There is a vast array of papers and books devoted to the questions touched upon in this book. When speaking about invariant manifolds, one should first of all mention the fundamental investigations of Lyapunov [1] and Poincare [2] mainly devoted to the analytic case. Further progress was achieved by Hadamard [1], Bohl [1], and Perron [13]. Hadamard [1] proposed a highly useful method for proving the existence of invariant manifolds now called tM graph transform method. Another approach close to the method of Green's functions was developed by Bohl [1] and Perron [1-3]. The Hadamard-Bohl-Perron tMory was further elaborated and extended by Anosov [1-3], Smale [1], Kelley [I], Kupka [1], Neimark [1-3], Pliss [1-2], Reizins [1], Samoilenko [1,2], Takens [1] and many others. A great number of theorems about integral manifolds was established by applying asymptotic methods due to Bogolyubov and Mitropolskii (see the book by Mitropolskii and Lykova [1]). Grobman [1] and Hartman [1] have shown that a vector field near a hyperbolic singular point is topologically linearizable. This result was extented by Pugh and Shub [1] to the case of an arbitrary normally hyperbolic compact invariant submanifold. On the basis of previous results obtained by McCarthy [1], Kyner [I], Hale [1], Moser [1] and others, Sacker [1,2] proposed a rather general condition sufficient for a compact invariant manifold to persist under perturbations. In the seventies, the Hadamard-Bohl-Perron theory was summed up and brought to its final form (see Hirsch, Pugh and Shub [1] and Fenichel [1-3]). Unfortunately, the style of presentation in these works can hardly be acknowledged as fully satisfactory because many proofs are only sketched and their accomplishment (left to the reader) needs in fact a deep insight into global analysis on manifolds. The method of normal forms founded by Poincare [1] was further developed by Dulac, Siegel, Sternberg, Kolmogorov, Arnold, Moser, Bruno and others (see the books by Arnold

ix [4], Hartman [3] and Bruno [2]). These investigations are chiefly devoted to formal, analytic, and infinitely differentiable normal forms. The problem on finitely smooth normal forms was studied by Belitskii [1], Samovol [1-10] and Sell [1-3]. Let us briefly review the contents of the book. In the first chapter, we present the well-known facts on the structure of flows and cascades near an equilibrium and a closed orbit. In § 1, we recall the relationships between differential equations, vector fields, and phase flows. The second section, § 2, is devoted to the Grobman-Hartman linearization theorem in the vicinity of a hyperbolic singular point. § 3 is concerned with the Floquet-Lyapunov normal form of a vector field near a periodic orbit. The next section contains the main results on the existence of local smooth manifolds in the vicinity of an equilibrium and a periodic trajectory (the so-called Hadamtlrd-Bohl-Pemm theory). These results are used to derive some theorems on preliminary normal forms which serve as the starting point of the next chapter. Chapter n, central to this book, deals with normal forms of vector fields and diffeomorphisms in the neighbourhood of a fixed point with respect to the group of finitely smooth changes of variables. In recent years, it was acknowledged that these nonnal forms are essential for the non-local bifUrcation theory (see Arnold, Afrajmovich, Il'yashenko and Shil'nikov [1], D'yashenko and Yakovenko [1]) because they are stable under perturbations, in contrast to the classical resonant normal forms. The first section serves as an introduction. We pose here the problem on reducing vector fields near a hyperbolic equilibrium to normal form and sketch the research objects pursued in this chapter. In § 2, we present the classical results due to Poincare and Dulac on normalization of jets of vector fields and diffeomorphisms at a rest point. The next section contains several important theorems on polynomial (weakly) resonant normal forms. We show, in particular, that if two vector fields have contact of a sufficiently high order at the equilibrium, then they are locally ~ conjugate with one another. In § 4, we discuss the possibility of further simplification of the resonant normal form and consider a number of examples which demonstrate that certain monomials entering the normal form can be killed by ~ changes of variables. In § S, we propose a new, very general condition, S(k) , imposed on a monomial xT that enables one

to delete xT out of the resonant normal form. This condition is used in § 6 to prove a deep theorem on d< linearization. Because the condition S(k) is rather involved, it is desirable to have some relatively simple conditions each implying S(k). Several spch

x conditions are established in § 7. The next section contains theorems on c! normal forms expressed in terms of the condition i!(k). These results are supplemented in § 9 and § 10 by some theorems based, besides i!(k) , on some other principles. The last section gives a survey of all the results obtained in Chapter II. The third chapter is concerned with linear extensions of dynamical systems. Such objects occur, for example, when linearizing a dynamical system near an invariant submanifold. In §§ 1-3, we give a brief review of the main results obtained in this area (for a detailed exposition, the reader is referred to the book by Bronstein [4]). Although these results may appear somewhat far from our subject, they are basic to many constructions and proofs in the sequel. In order to describe two important classes of linear extensions (namely, linear extensions satisfying the transversality condition and those with no non-trivial bounded motions), we use in § 4 quadratic Lyapunov junctions dermed on the underlying vector bundle. Various kinds of weak regularity of linear extensions are investigated in § S. Some relationships between weak regularity , are transversality, hyperbolicity, and the existence of a Green-Samoilenko function established. In particular, it is shown that a c! Green-Samoilenko function exists if and only if the k-jet transversality condition is fulfilled. In Chapter IV, we investigate invariant sub bundles of weakly non-linear extensions of dynamical systems. Some results on the existence of invariant subbundles of extensions close to exponentially splitted linear extensions are presented in § 1. In particular, a theorem which generalizes the classical result of Hadamard [1] is proved. In § 2, we show that any non-linear extension sufficiently close to an exponentially separated linear extension can be decomposed into a Whitney sum of two extensions. The GrobmanHartman linearization theorem is generalized in § 3 to weakly non-linear hyperbolic extensions. In § 4, we examine the question on smoothness of invariant subbundles. The proof of the main theorem is based on the now traditional graph transform method. It also makes use of the smooth invariant section theorem which is presented (with a detailed proof) in the Appendix. The application of global analysis methods and results enables us to avoid the use of local coordinates and to control all stages of the proof. Chapter V deals with smooth invariant sub manifolds satisfying the so-called normal k-hyperbolicity condition introduced in § 1. We prove in § 2 that such a submanifold is

c!

persistent under perturbations. We also establish the existence of its stable and unstable local manifolds. These results constitute the kernel of the general Hadamard-Bohl-Perron theory. Besides that, we present a theorem on topological

xi

linearization near the given submanifold which is a direct generalization of the Grobman-Hartman theorem. As it is shown in § 3, normal k-hyperbolicity is not only sufficient, but also necessary for a submanifold to be ~ persistent. The notion of

asymptotic phase for an exponentially stable invariant sub manifold is studied in § 4, and some theorems on smoothness of the asymptotic phase are proved. Besides, it is shown that the stable manifold W of a normally k-hyperbolic compact invariant submanifold A is invariantly fibered by ~ submanifolds~,

x e

A (of

course,

a

similar

result

is valid for the unstable manifold W'). These statements may be considered as a supplement to the Hadamard-Bohl-Perron theory. In the final part of this section, we present proofs of several theorems stated (but not proved) in Chapter I. Chapter VI is concerned with the question of whether two dynamical systems are smoothly conjugate to one another in the vicinity of their common smooth invariant submanifold. It is assumed that these systems have contact of high order at all points of the submanifold. In § I, we consider the case when this sub manifold is exponentially stable and prove a generalization of Sternberg's [1] theorem on linearization of contractions. The general case is handled in § 2, and a theorem due to Robinson [1] is presented which extends some results previously obtained by Sternberg [1,2], Chen [1] and Takens [1]. We deduce from these theorems some results (stated without proof in Chapter IT) concerning polynomial resonant normal forms of dynamical systems near an equilibrium and a periodic orbit with respect to finitely smooth changes of coordinates. The book is adressed to specialists in the qualitative theory of differential equations and, especially, in bifurcation theory. Although written for mathematicians, it may prove to be helpful to all those who use normal forms when investigating concrete differential equations. While research workers will find in the book an up-to-date account of recent developments in the theory of finitely smooth normal forms, the authors have tried to make the first part (Chapters I and II) accessible to non-specialists in this field. Such readers should use Chapter I as a summary (or, rather, a glossary) and take the classical results presented in this chapter on faith. The background material needed to understand Chapter II is differential calculus of several variables and ordinary differential equations. To be more precise, the main tools used here are the Taylor expansion formula and Banach's fixed point theorem for contractions (applied to operators in some special functional spaces). As to the second part, the reader is assumed to be familiar with the fundamentals of global analysis on manifolds (Bourbaki

xii [1], Leng [1], Hirsch [1]) and fiber bundle theory (Husemoller [1]). For the reader's convenience, at the end of the book an Appendix is given which contains some definitions and facts from differential calculus and the theory of smooth manifolds, as well as some more special results repeatedly used in the course of the book. We adopt standart notation: the group of real numbers is denoted by IR, the group of integers is denoted by Z, z+ is the set of non-negative integers. Given a mapping X -+ Y, graph(f) denotes the set {(x, j(x»: x E Xl. The symbol ~ marks the beginning and marks the end of a proof. The sections are divided into subsections, each numbered (within a chapter) by a pair of numbers, where the first one refers to the number of section and the second one

f.

refers to the number of the subsection in this section. If necessary, we add, in front of these two numbers, the number of the chapter. So, the triple m.2.4, for example, denotes subsection 4 of section 2 of chapter m. The Appendix is also divided in subsections numbered consecutively and marked with a capital A. The authors are thankful to G.R.Belitskii, A.D.Bruno, Yu.S.n'yashenko, and V.S.Samovol for many helpful conversations on subjects considered in the book. Special thanks are due to V.A.Glavan for a number of useful suggestions and comments on the text and to a. Yu.Demidova for the help rendered in preparing the camera-ready manuscript. We are extremely grateful to Leon a.Chua, the editor of the World Scientific Series in Nonlinear Sciences for his kind offer to include our work in this series. Finally, we should like to acknowledge our great debt to the late K.S.Sibirskii for the instruction, interest in our work, and encouragement he has offered over many years.

1

CHAPTER I TOPOLOGICAL PROPERTmS OF FLOWS AND CASCADES IN THE VICINITY OF A REST POINT AND A PERIODIC TRAJECTORY

§ 1. Basic Definitions and Facts

In this section, we recall the basic terminology and properties of differential equations, vector fields and flows. We establish relations between flows and cascades and discuss a general approach to the notion of normal form.

Differential equations, vector fields, flows and cascades 1.1. Differential equations. Let U be an open subset of IRn and f. U -+ IRn be a continuous map. A relation of the form

.

x. -

dx =j(x) dt

(x

E

(1.1)

U)

is called an ordinary autonomous differential equation. Let x E U and let I be an open interval of the real axis IR containing O. A differentiable function ,,: I -+ U is said to be a solution of the equation (1.1) with initial condition (x, 0) if the following equalities hold: d

dt ,,(t)

= j(,,(t))

(t

E

I),

,,(0)

= x.

According to the well-known Local Existence and Uniqueness Theorem, if the function

f. U -+

IRn

is continuously differentiable then for each point x

E

U there exists a

2 solution Ill: I -+ U with the initial condition (x, 0) and, moreover, if 1/1: J -+ U is also a solution satisfying the same initial condition then lIl(t) = I/I(t) for all t e l n J. Among the solutions of (1.1) with initial condition (x, 0) there is a solution IIlx: Ix -+ U defined on the maximal interval Ix c: lit The set {(x, t): x E U, t E Ix} is open and the map Ill: (x, t) 1-+ IIlx(t) is continuous. Moreover, the equality IIl(X, t + s) = lIl(ep(X, t), s) holds for all x, t, s such that both the right hand and the left hand sides are defined. If / e C-(U, IRn ), then the map ep is also of class C-. Strictly speaking, the right hand side of (1.1) is not a function. To see this, let us introduce a new variable y

= ~(x),

where ~: U -+ IR n is a diffeomorphism. Then (y

E ~(U)

whereas the function x t-+ j(x) after the coordinate change becomes y t-+ t(/(t- 1(y»). From the geometric viewpoint, the right hand side of (1.1) is an object different from a function. Namely, it is a vector field. 1.2. Vector fields. Let M be a smooth (boundaryless) manifold, (TM, tangent bundle and r be a positive integer. A vector field to be a

C smooth section

such that TM

0

I;

I; of the tangent bundle TM: TM -+ M, i.e., a

= idM • The set of all C

C topology is denoted by

0/ class C

TM'

M) be its

on M is defined

C map 1;: M -+ TM

vector fields 1;: M -+ TM provided with the

rl"(TM)'

1.3. Velocity vectors. Let I be an interval of the real axis and Ill: 1 -+ M be a differentiable map. The pair (Ill, 1) is called a (local) motion. Let Tep: TI I x IR -+ TM denote the tangent map. The tangent vector Tep(t, 1) e TM is called the velocity vector of the motion ep: I -+ M at the moment t E I. Denote

=

~(t)

= Tep(t,

1)

!!!

TIIl(t)·1

(t

E

I)

1.4. Motions of a vector field. Let ~ e rl"(TM)' A differentiable map ep: I -+ M is said to be a motion 0/ the vector field 1;, if the velocity vector of ep at t t=- I coincides with the value of the vector field ~: M -+ TM at the point ep(t), i.e., ~(t)

= ~(ep(t»

(I

E

1).

(1.2)

3

1.5. Global Existence and Uniqueness Theorem. Let M be a compact smooth manifold and ~: M ~ TM be a vector field of class

(1) Given a point x

E

C, r

1. The following assenions hold: M, there exists a motion IPx: IR ~ M of the vector field i!:

~

such

E C+ 1(1R,

M); that IPx(O) = x and IPx (2) If IP: I ~ M and 1/1: I ~ M are two motions of number to E I n I, then rp(t) = I/I(t) for all tel n 1.

1.6. Flows. Let M be a smooth manifold and {lP t :

C diffeomorphisms IPt: M ~ M, i.e., IP: M x IR ~ M by lP(x, t) = IPt(x) (x

1P 0 E

= id

M, t

E

t

E

and rpt

~

and lP(to) = I/I(tO> for some

IR} be an one-parameter group of 0

IPs

=

IPt+s (t, s

E

IR).

Define

IR). If the map IP: M x IR ~ M is continuous

then {lP t : t E IR} is said to be a continuous flow. The flow {lP t : t E IR} will also be denoted by (M, IR, IP). The function IPx: IR ~ M defmed by IPx(t) = cp(x, t) (t E IR) is called the motion of the point x E M. The set IPx(lR) is the trajectory (or the orbit) of x, and M is the phase space. The flow (M, IR, IP) is said to be smooth if for each x E M the motion CPx: IR ~ M is smooth. In this case one can define the velocity vector ~x(t) = Trp(t, 1) == Trp(t)·1 (t E IR). Thus, a smooth flow (M, IR, tp) gives rise to a vector field ~tp: M ~ TM, where ~cp(x) = ~x(O) (x

E

velocity vector field

M).

~tp

We say that a smooth flow (M, IR, IP) is of class C is

C smooth.

if its

It is easy to see that in this case the map

rp: M x IR ~ M belongs to the class C,r+l in the sense of Bourbaki [1] and, moreover, to

the class C. 1.7. Theorem. Let M be a compact smooth manifold and ~: M ~ TM be a vector field of class C, r i!: l.Define IP: M x IR ~ M by rp(x,t) = rpx(t) (x e M, t e IR), where rpx: IR ~ M is the motion of the vector field ~ with initial condition rpx(O) = x. Then (M, IR, IP) is a flow of class C.r+l.

Thus, if M is a compact (boundary less) smooth manifold, then there is a one-to-one correspondence between C vector fields

~:

M ~ TM and c!",r+1 flows (M, IR, IP).

1.8. Cascades. Along with flows, the theory of dynamical systems is concerned with

4 the study of cascades. Let M be a smooth manifold and {g": n e z} be a group of

C

diffeomorphisms g": M ... M homomorphic to the group of integers. In such a case we say that {g": n

E

z} is a C

determined by the

g-"

= g-I

0 ... 0

g-I

cascade. Clearly, the cascade {g": n

C diffeomorphism g _ gl:

E

z} is uniquely

g" = g

0 ... 0

g,

(in both cases the composition consists of n multipliers). Every

C

M ... M

j

namely,

smooth flow (M, IR, g) determines in a natural way a cascade on M, namely, {g": n E z}. The converse statement does not, in general, hold, i.e., not for each diffeomorphism g: M ... M there exists a flow

{l:

IE IR} on M such that

i = g.

For example, if the diffeomorphism g: M ... M is not homotopic to the identity mapping, then it cannot be embedded in a flow with the same phase space M. However, to any diffeomorphism g: M ... M we can put in correspondence a flow (M, IR, 11') defined on some enlarged manifold M. This manifold can be obtained from the Cartesian product M x R by identifying points according to the rule: (X, t + n) E. (g"(x) , t) (x E M, nEZ, t E IR). The shift flow (M x R, R, 11'0), where tpo(x, t, s) = (x, t + s) (x E M; I, s E IR), determines the required flow (M, IR, 11') on the quotient manifold M = (M x IR) I E. The submanifold

{O} of M is a global cross-section of the flow (M, IR, 11') so that 11'1: Mo ... Mo can be identified with g: M ... M. The flow (M, R, 11') is called the Smale suspension of the diffeomorphism g. Thus, flows and cascades are closely related with each other. It is reasonable to expect that results obtained for flows are usually valid for cascades and vice versa. In what follows, the presentation of material is carried out in parallel for flows and cascades, but, as a rule, proofs are given for only one kind of dynamical systems.

Mo

III

M

x

Conjugation of flows and cascades in the vicinity of an invariant manifold The main objects of investigation in this book are flows and cascades near an invariant manifold. 1.9. Dermition. Let M and N be smooth manifolds, A be a sub manifold of M. Let us define the following equivalence relation on the set of all maps F: M ... N: FI - Fl , if

6 there exists such an open neighbourhood U of A in M that F\(x) = F2 (x) for all x E U. The corresponding equivalence classes are called A-germs (or, simply, germs, when there is no chance of ambiguity). A A-germ is said to be a local C- diffeomorphism (local homeomorphism) if there are a representative F of this germ and a neighbourhood U of the submanifold A such that F: U ... F(U) c N is a C- diffeomorphism (homeomorphism, respectively). Sometimes, the mapping F: U... F(U) itself will be called a local diffeomorphism (or local homeomorphism). Let Diff,;(M) denote the set of all local C- diffeomorphisms F: M ... M satisfying FI A = id. Let HomeoA(M) be the class of all local homeomorphisms with the same property. 1.10. Def"mition. Let M be a smooth manifold and A be a sub manifold of M. Let

(M, R, fI) and (M, R, "') be flows such that fit (x)

= ,,l(x)

E

A (x

E

A, t

E

~), i.e. A is

their common invariant set. We say that the fI and '" are locally C- conjugate flows in the neighbourhood of the invariant sub manifold A if there exists an element

h

E

Diff,;(M) such

that the

A x {O}-germs

of

the maps (X, t)

(x,t) t-+ h(flt(x» coincide. If we replace the condition h we get the definition of local topological conjugacy.

E

t-+

Dif~(M) by h

E

",t(h(x» and HomeoA(M),

1.11. Notation. Let M be a smooth manifold, (M, R, ,,) be a flow and A be a submanifold of M invariant under fl. Let U be an open neighbourhood of A in M and x E M. Let 1x (" , U) denote the maximal connected interval of R containing the number 0 and such that

"E' E

1x ('" U) implies "T(x)

E

U.

1.12. Lemma. The flows (M, IR, ,,) and (M, IR, "') are locally topologically conjugate with one another near their common compact invariant submanjfold A iff there exist an open neighbourhood Uo of A and an element h E HomeoA(M) such that (1.3) An analogous statement holds in the case of local

• (X, t)

Suppose that the t-+

A x {O}-germs

of

C- conjugacy, as well. the

maps

(X, t)

t-+

",t(h(x»

and

h("t(x» coincide for some element h. HomeoA(M). Then there exist an open

6 neighbourhood U of A in M and a number E > 0 such that ,,/(h(x» = h(c/(x» (x E U. III < E) and (- E. E) c: Ix(cp. U) for all points x in a sufficiently small neighbourhood Uo c: U. Let x holds. Suppose III

ITI

2:: E.

E

Ua and

I

E

Ix(cp. Ua). If III

<

E

then the equality (1.3)

then I can be represented in the form I = n ~

Ej2. For definiteness. suppose n > O. Then rplel2+ T (x) E Uo

:S

+ 't,

where n

E

l.

(I = 0..... n) and

I/IT(h(x» = h(cpT(x». Hence

Employing this argument n times. we get that (1.3) is valid. Conversely. suppose (1.3) holds. Choose a neighbourhood U of the compact sub manifold A in

M and a number E

= h(cpt(x»

I/It(h(x» (x, I)

t-+

I/It(h(x»

> 0 in

(x E U.

and (x. I)

such a way that cpt(U) c: Ua for all I E (- E. E). Then

III t-+

<

E).

Therefore the A

x

{O}-germs of the maps

h(cpt(x» coincide.

1.13. Lemma. LeI M be a compact smooth manifold, r 2:: 1, 1;: M -+ TM and

C smoolh veclor fields. Let (M.

1):

M -+ TM

and (M. IR. 1/1) be the corresponding flows. Suppose that A is a smooth submanifold of M invariant under both cp and 1/1. The flows

be

IR. cp)

and 1/1 are locally C conjugate with one another iff the vector fields i; and 1) are C conjugate with one another near the submanifold A, i.e.. I; 0 h = 111 0 1) for some cp

h

Dif~(M).

E

~

This assertion follows from Theorem 1.5.

1.14. Dermition. Let M be a smooth manifold and Ft : M -+ M (i = 1. 2) be C diffeomorphisms. Suppose that A is a sub manifold of M invariant under both Fl and F2•

i.e .• Ft(x)

E

A

(x

E

A; i

=

1. 2). We say that Fl is locally C conjugate to F2 in

the neighbourhood of A if for some element h E Dif~(M) the germs of maps Fl 0 h and h 0 F2 coincide. The notion of local topological conjugacy of homeomorphisms (and of the corresponding cascades) is defined similarly.

7

Normal forms Normal forms are met in various branches of mathematics. For example, recall the Jordan normal form of matrices, normal forms of conics and surfaces of second order. In the most general setting, the notion of normal form can be introduced in the following way. Let m be a set, G be a group and (m, G, n) be a transformation group, i.e., n: m x G -+ m satisfies the conditions:

(2) n(x, e)

=x

(x

E

m; e is the identity element of the group G).

Then the set m can be represented as the union of non-empty mutually disjoint classes, the orbits of the transformation group (m, G, n). This leads to the following two problems. The first problem deals with the classification of objects belonging to M with respect to the action of G. In other words, the question is whether or not two given elements x, Y E M belong to the same G-orbit. The second problem is to select from each orbit a representative of the most simple kind. These representatives are just called

normal forms. In some cases one succeeds only to find conditions sufficient for a given element to belong to a certain orbit (and so these conditions divide the set M in parts maybe smaller than orbits). In such a situation some orbits contain more than one representative. Although the classification problem is not completely solved in this case, we shall yet consider these representatives as normal forms, thus enlarging the meaning of this notion. In this book, the set m is usually asummed to consist of dynamical systems (flows and cascades) considered in the vicinity of a smooth invariant submanifold I\. c M. The group

G is supposed to be equal to Dif~(M) (r = 1, 2, ... ) or to Homeol\(M). As a rule, we assume that the degree of smoothness of the dynamical systems is greater than r.

8 I 2. Hyperbolic rest points In this section, we investigate topological normal forms of flows and cascades in the neighbourhood of the most simple invariant sub manifold consisting of a single point. Consider a vector field (: M ~ TM. A point x E M is called an equilibrium of ( if (X) 0, where 0 denotes here the origin of TxM. The point x is also called the zero,

=

or critical point, or singular point of (. Let {I/} denote the local flow determined by (. Clearly, x is a singular point of (iff "t(x) = x for all I. For this reason, x is also called a .fixed point, or a slationary point, or a resl point. Taking into consideration the local nature of the problem, we may assume without loss of generality that the phase space is simply a finite dimensional Banach space and the rest point is situated at the origin. This allows to simplify the notation and terminology. At the same time, we shall try to present the material in invariant terms (i.e., not depending on the choice of coordinates). This may serve as a training for the reader and make easier the further examination of the case of an arbitrary smooth invariant manifold. 2.1. Notation. Let E be a finite dimensional Banach space. Let x E E, TxE be the tangent space at x and (TE, T E' E) be the tangent vector bundle. There exist canonical isomorphisms TxE ,. E, TE AI E • E. Given a diffeomorphism F: E ~ E, the tangent map TF: TE ~ TE can be written (via the isomorphism TE ,. E • E) in the form TF(x, y) = (F(x), DF(x)y). In other words, the tangent map TF: TxE ~ TF(x)E can be canonically identified with the derivative DF(x): E -+ E (x

e,

E

E). Let (

E

r(E) • r('tE)

e

be a vector field of class t.e., a smooth section of the tangent bundle (TE, T E , E). Taking into account the canonical isomorphism (TE, TE' E) ,. (E. E,pr., E),

we get ~(x)

= (x,

~(x», where ~

E

e(E, E). Thus, r(E) ,. e(E, E).

Diffeomorphisms 2.2. Definition. Let F E deE, E), F(O) = O. The fixed point 0 E E is said to be hyperbolic if L Ii Dj(O): E -+ E is a hyperbolic linear operator, i.e., an invertible

9 operator, which has no eigenvalue with modulus one. By the Inverse Mapping Theorem, F is a local C diffeomorphism. In what follows, when speaking about F we really have in mind its germ at the point O.

2.3. Remark. The fixed point 0 is hyperbolic iff there exist L-invariant linear subspaces

F, It"

and an appropriate norm on E such that ilL IF

II

< 1, ilL-II It" II < 1.

2.4. Def"mition. Let F E e(E, E), F(O) = o. We say that the mapping F is topologically linearizable in the vicinity 0/ the fixed point 0, if there exist neighbourhoods U = U(O) , V = YeO) and a homeomorphism H: U -+ V such that H(L(x» = F(H(x»

(x

E

U n CI(U)

(2.1)

This means that F is locally topologically conjugate with its linear part L = DF(O). 2.5. Grobman-Hartman Linearization Theorem for Fixed Points. Every local diffeomorphism near a hyperbolic fixed point is topologically linearizable. ~

This follows from Theorem 2.7 below.

2.6. Notation. Let ~(E, E) denote the Banach space of all continuous mappings 9': E -+ E satisfying the condition 119'110 E sup {1I9'(X)II: x E E} < 00. Let 1f(0) be the group of all homeomorphisms H: E -+ E such that H(O) = O. By 1f(O, id) we shall denote the subgroup of the group 1f(O) consisting of all such homeomorphisms H: E -+ E that H

= id +

h, where h

E

~(E, E), h(O)

= O.

2.7. Theorem. Let 0 be a hyperbolic fixed point 0/ FE C(E, E) and IIDF(x) - LII < I: (x E E), where I: is a small enough positive number depending only on L E DF(O). 71zen there is a unique element H E 1f(0, id) such that H 0 L = F 0 H . • This assertion is a sequel of Lemma IV.3.2. Now let us show how Theorem 2.5 follows from Theorem 2.7. Let FE C(E, E), F(O) = 0, and suppose that L;;; DF(O) is a hyperbolic linear operator. There exists a smooth function ex: IR -+ [0, 1] with the following properties: ex(t) = 1 whenever III !is 114;

10

oe(t) = 0 whenever It I IPe: E -+ E as follows:

2:

<

1; IIDoe(t)II

2 (t

E

IR). For a given e

>

0 define a mapping

for IIzll ::s e; for IIzll Clearly, 'Pc Fe

E

E

>

e.

C~(E, E) and IPe(z) = IP(z) whenever IIzll::S e/2. Set Fe = L + IPe' Then

C(E, E) and Fe(z)

= F(z)

sup {IIDFe(x) - LII:

for all z with IIzll::s e/2. It is easy to verify that

x

E

E} • sup {IiDIPe(x)II:

x

E

E} ::s Ce

where C is a constant. To end the proof, apply Theorem 2.7 to the function Fe, where e is sufficiently small. It 2.S. Remark. Theorem 2.7 gives a complete classification (up to the group

~(O,

id»

of mappings F E C(E, E) such that L II DF(O) is a hyperbolic operator and sup {IIDF(x) - LII: x E E} is sufficiently small. In fact, F is conjugate to L by a homeomorphism H E ~(O, id). On the other hand, if LI and ~ are hyperbolic linear operators and H

0

LI

=~

0

H for some homeomorphism H

E

~(O,

id) then LI =~. Indeed,

= x + h(x) where h E ~(E, E), hence Llx + h(Llx) = L,;c + ~h(x), (~ - LI)(x) = h(Llx) - L'J,h(x) (x E E). Clearly, h LI - ~ h E ~(E, (~ - L I ) E ~(E, E), but this is possible only when L'J, = L I .

H(x)

0

0

consequently, E). Therefore

Widening the class of admissible transformations allows further simplification of topological normal forms of local diffeomorphisms in the vicinity of a hyperbolic fixed point. In view of Theorem 2.5, the problem is really concerned with normal forms of linear operators. One can show that for every hyperbolic linear operator there exist uniquely determined L-invariant linear subspaces ~

~,~,

Et, Et

L: E -+ E

such that

• ~ = E', Et. Et = E'

and, moreover, the dimensions of these subspaces completely determine the topological structure of L. Namely, L is topologically conjugate with the

er,

operator (XI' x'J" YI' Y~ 1-+ (2x I , - 2x'J" y1/2, - Y2/2) (Xt E Yt E Eft, t = 1, 2). As a corollary, we obtain that the set of all topological normal forms of local diffeomorphisms near a hyperbolic fixed point is finite.

11 Vector fields 2.9. Dermition. Let ~: E -+ TE be a

C

smooth vector field and ~(O) = O. Denote

the linear operator D~(O): E -+ E by A (we assume that rl(E) is identified with C(E,E». The singular point 0 of the vector field ~ is said to be hyperbolic if the spectrum of A (i.e., the set of its eigenvalues) do not meet the imaginary axis. In this case one also says that the linear vector field y = Ay is hyperbolic. 2.10. Remark. As far as we consider the problem on topological classification of vector fields in the vicinity of a hyperbolic singular point, we may assume, without loss of generality, that the vector field ~ determines a flow (E, IR, ~~). In fact, it suffices to replace ~ by a vector field ~ such that ~(x) = ~(x) whenever IIxll:s 112 and ~(x) = Ax whenever IIxll ~ 1.

2.11. Grobman-Hartman Linearization Theorem for Singular Points. Let 0 be a hyperbolic singular point of the vector field ~ E rl(E). Then the flow generated lTy ~ is topologically linearizable in the neighbourhood of the point 0, i.e., there exists a local homeomorphism transfering motions of t; to motions of the linear vector field = Ay, A = Dt;(O).

y

• This assertion follows from the next theorem.

2.12. Theorem. Assume that 0 is a hyperbolic singular point of the vector field ~ E rl(E) ,. C(E, E)

and, besides, that sup {IIDt;(x) - All: x E E} is SUfficiently small. Then there exists an uniquely determined element H E H(O, id) that conjugates the flow (E, IR, ~~) with the flow of y = Ay (y E E) . • Theorem 2.12 is a particular case of Theorem N.3.S. 2.13. Remark. In addition to Theorem 2.11, let us state some results concerning the topological classification of linear vector fields. Assume that A is a linear operator with no pure imaginary eigenvalues. Then there exists an A-invariant splitting E = E+ e K

such that the spectrum of A I E+ (A I K) lies to the right (respectively, to

12 the left) of the imaginary axis. It turns out that the flow of the differential equation

x = Ax is topologically conjugate with the flow of the model vector field y = y, i = - z (y • E+, Z E). If the vector fields x = Ax and x = Bx are both hyperbolic then E

their flows arc topologically conjugate

with one another iff dim E+ (A) = dim E+ (B)

[1».

(or, equivalently, dim E(A) = dim E(B» (see Vaisbord Moreover, the conjugacy B. can be chosen in If(O, id) iff A As to non-hyperbolic linear vector fields, the question on topological conjugacy can

=

be answered as follows. By n+(A) (n-(A» denote the number of eigenvalues of the operator A with positive (recpectively, negative) real parts. The flows generated by

x = Ax

and

x = Bx

arc topologically equivalent iff n+(A)

= n+(B),

n-(A)

= n-(B)

and the restrictions of A and B to their invariant subspaces that correspond to the pure imaginary eigenvalues arc linearly equivalent (see Latiis

[1».

2.14. Remark. Our interest in hyperbolic singular points of vector fields is certainly motivated not only by the fact that there is a simple topological classification of these points. The main reason is that vector fields all of whose singular points are hyperbolic form a generic subset. To be more precise, the following assertions hold. ~ e rl(E). Then

(1) Let 0 E E be a hyperbolic singular point of the vector field there exist a neighbourhood every vector field hyperbolic.

'"

U(O) c E and a neighbourhood

E H(~)

(2) Given a vector field '"

vector

field

IID~(x)

- D1I(x)1I

~ e rl(E)

<

I:

(x

such that

has exactly one singular point in U(O) , and this point is

E

rl(E), ",(0)

= 0,

then for each

E

>

0 there exists a

II~ - ",III < E (i.e., 1I~(x) - ",(X)II E) and 0 is a hyperbolic singUlar point of ~. such

E

H(~) c rl(E)

that

(3) Let M be a compact smooth manifold. The set of vector fields ~

singular points are all hyperbolic is open and dense in the space assertion constitutes a part of the well-known Kupka-Smale theorem. (4) Assume that 0 E E is a hyperbolic singular point of ~ field ~ is locally structurally stable near 0, i.e., for each

E E

E

<

E,

rl(TM) whose rl(TM)'

This

rl(E). Then the vector

>

0

there exists a

number a > 0 such that every vector field '" E rl(E), II", - f;1I1 < a, possesses a unique singular point pEE, lip II < E, and the flows generated by ~ and 11 near 0 and p are

13 topologically conjugate with one another. Similar results are valid for cascades. Assertions (I), (2) and (4) follow easily from Theorem 2.11 and Remark 2.13. The Kupka-Smale theorem can be found in the following books: Nitecki [1], Abraham and Robbin [1], Palis and de Melo [1].

2.1S. Example. Consider the mapping F: R2 -+ R2,

FCz, y)

=(21', 4y

+ ~).

Clearly,

(0, 0) is a hyperbolic fixed point. According to Theorem 2.5, the local diffeomorphism F is topologically linearizable near (0, 0), i.e., there exists a local homeomorphism H: R2 -+ 1R2 such that F

0

H

=H

0

L,

where

L(u, v)

= (2u,

generality we may assume that H has the form H(u, v)=(u, v that !p(u)

'II

=

4v).

+

Without loss of

!p(u». It is easily seen

must satisfy the functional equation f1(2u) = 4qI(u) + u2, whence In1ul 221nlul 2 Cu + u 4 In 2· Taking C = 0, we get that H(u, v)=(u, v + u 4 In 2) is a C

smooth conjugacy.

§ 3. FJoquet-LyapuDov Theory

This section is devoted to the Floquet-Lyapunov normal form of a smooth vector field near a periodic trajectory and to the Grobman-Hartman linearization theorem for hyperbolic periodic orbits. 3.1. Notation. Let Mbe a compact smooth manifold, dim M =

C of

smooth vector field and 'II

'II

11

+

1; (: M -+ TM be a

be the corresponding flow. Let S be a periodic trajectory

and w be its smallest positive period.

Denote by S! the circle obtained from the segment

[0, w]

by glueing its ends

S! = R I wZ. The tangent space TS! is canonically isomorphic Therefore T(S!)( R") = S! )( R" )( RI )( IR". Here it is assumed that the

together. In other words, to S!)(

IR.

projection of the tangent bundle of the manifold S!)( IR" coincides with the projection onto the first two coordinates. Let U be some open neighbourhood of S. Suppose that there exists a suJjective diffeomorphism lit: U -+ S! )( R". It carries the vector field ( 1U onto S!)( R". The resulting vector field Tilt. ~ • lit-I can be written in the form

14 T«

0

I;

The mapping 1;",__ SlW T«

0

I;'

0

«-I

0

«-1(9, X)

= (9,

x IR " ~ IR I x IR"

X, 1;",(9, X»

(9 E S~,

X E IR")_

is called the principal part of the vector field

near S.

3.2. Dermition. The periodic trajectory S is said to be twisted (untwisted) if for some point pES the tangent map TprpW: T~ ~ T~ is orientation reversing (respectively, orientation preserving). This definition is independent of the choice of pES.

If the manifold M is orientable, then every periodic orbit ScM is untwisted.

3.3. Lemma. Suppose that S is untwisted. Then there exist an open neighbourhood U of

S and a C diffeomorphism ex: U ~ S! x (8

E

IR"

such that «(S)

= S! x {OJ and ~",(8, 0) = (1, 0)

I

Sw).

Denote TM[S] = {v E TM: "I:M(v) E S}. Let NS be the normal vector bundle TM[S] I TS and L(I;') be the vector subbundle of TM[S] whose fiber [L(I;)])C is equal to {1l1;'(X): Il E IR} (x E S). Clearly, L(I;') = TS. The normal bundle NS = TM[S] / TS'" TM[S] I L(I;) can be ~

naturally provided with the structure of, a vector bundle of class commutative diagram

we obtain the following commutative diagram:

rp

t

C.

From the

15 So we get the normal floW untwisted, the map

(NS, IR, Nrp)

C-I.

of class

Since the trajectory

S is

Tprp"': TpM ... T,)1 is orientation preserving. Hence it follows that

the linear operator Nprp"': NpS ... NpS is also orientation preserving for all pES. Fix pES. Since the set of orientation preserving linear automorphisms of a finite

c»

dimensional vector space is open and pathwise connected, one can construct a smooth mapping A: IR ... L(NpS, NpS), t 1-+ At, such that: (1) Ao(v) = v (v E NpS);

= (Nprp"')-I = Nprp""'. For t E [0, w] and q = rpt(p) , let us define a mapping Bt : NpS ... NqS by Bt = Nprpt At. Then Bo = B", = id. Bearing in mind that [0, w) is (2) A",

0

identified with S! , we can define a mapping g: S! x NpS ... NS as follows:

Clearly, g is of class

C- I for a '" O. By changing the map g slightly, we get a C

g

vector bundle isomorphism

from NS to the trivial vector bundle (S!

x

IR", prl' S!).

Now let U be a sufficiently small open neighbourhood of S in M. There exists a C normal tubular neighbourhood of S, i.e., a C diffeomorphism f. NS ... U such that j(NS) = U and I 0 Z = ids, where Z: S ... NS denotes the zero section. Set 0:

= g-I 0/: It

is

U ... S! x IR".

easily

seen

that

0:

is a

C diffeomorphism

and

the

t 1-+ 0: 0 rpt 0 o:-I(a, 0) coincides with the motion of the vector field To: issuing from the point (a,O). From the definition of it follows that ~ex(a, 0)

g

(a

E

mapping 0

(

0

= (1,

0:- 1

0)

S!>.

3.4. Remark. Lemma 3.3 shows that the vector field .; E r('t'M) in a small enough neighbourhood of an untwisted periodic orbit S can be reduced via a suitable C diffeomorphism

0::

U ... S! x IR" to the form (ex

= (.;!, .;,!), where the maps

~!: S! x IR" ... IR

and (!: S! x IR" ... IR" are given by (!(9, x)

=1+

Q(a, x),

2

(ex(a, x)

Here B: S! ... L(IR", IR"), Q(a, 0)

to consider

(!

and

(!

= B(a)x + R(a, x)

= R(a,

0)

= D 2R(a,

0)

as functions defined on IR x IR"

(a

=0 !!

E

(a

IR"+I

S!, x E

E

IR").

(3.1)

S!). It is convenient

and w-periodic in the

16 first argument. Similarly, the map cc'\: S! x R" -+ U induces a map cr: IR x R" -+ U with the same periodicity property.

3.S. Definition. A periodic C pseudochart for an w-periodic untwisted trajectory S is a pair (U, cr), where U is an open neighbourhood of S in M and cr: IR x IR" -+ U is a map such that: (1) cr(a

+ mw, x) = cr(a,

;.: S! x IR" -+ U induced by cr is a

C

x)

(a

E

R, x

E

IR", mEl);

diffeomorphism; (3) cr(1R x {O})

C

(2) the map

= S.

Thus, the formulas (3.1) represent the principal part of the vector field

~

in the

vicinity of an untwisted periodic trajectory by means of a periodic pseudochart. As it will be shown below, one can obtain a similar local representative also in the case of a twisted trajectory by going twice around the normal vector bundle. Let M denote an orientable two-fold covering of M. For ~ E r"(orM) , let ~: M -+ TM denote the uniquely determined vector field, the lift of ~. If F is a twisted w-periodic trajectory of ~, then there exists a 2w-periodic trajectory covering of ~. Observe that

l

is untwisted because

M is

l

of (, which is a double

orientable. Hence, for

l

there

exists a 2w-periodic C pseudochart. It possesses some additional properties indicated in the following definition.

3.6. Dermition. Let S be an a w-periodic trajectory of the vector field and U be an open neighbourhood of S in M. Let cr: R (1) cr(a

i.e.,

+

w, x)

= cr(a,

Jx)

(a

E

IR, X

E

IR" -+ U be a

R"), where J

.r = id; (2) cr is a local diffeomorphism; (3) cr(1R

is called an w-demiperiodic

x

x

E

C

~

E

r"(or M)

map such that:

L(IR", IR") is an involution,

{O})

= S.

Then the pair (U, cr)

C pseudochart.

It follows from this definition that the map cr is 2w-periodic in the first argument. = id, Definition 3.6 turns into Definition 3.5.

In the case where J

Let (U, cr) be a w-demiperiodic

C

pseudochart for S. Define cr.~: IR x IR" -+ T(R x R")

by cr.~(a, x)

=

[Tcr(a, x)],\ ~(cr(a, x».

17 Let (cr: IR x IR" .... IR x IR" be the principal part of the vector field 0".(.

3.7. Theorem. Let M be a (n + l)-dimensional smooth 11IIJ1Jifold, (E r'"('rM), S be a periodic trajectory (twisted or untwisted), w be its smallest positive period. Then

there is a w-tiemiperiodic C" pseudochart (U, 0") such that (cr = (~!, ~!), where ~!(a, x)

~!(a, x)

= 1 + Q(a, x);

= B(a)x + R(a, x) Q(a, 0)

= R(a,

0)

(3.2)

= D~(a,

= o.

0)

The maps B: R .... L(IR", IR"), Q and R are w-tiemiperiodic. More exactly, Q(a

+ w, x) = Q(a, Jx), R(a + w, x) = JR(a, Jx), B(a + w) = JB(a)J,

where J is the involution corresponding to the pseudochart (U, then J = id. ~

be a normal tubular neighbourhood of class C", Po

motion of the vector field v: NS .... S by", i.e. E

«u, v)

E

E).

vector bundle

(

= leu,

issuing from Po. v)

E

R x NS: fI(U)

0

I

= " ".10'.

Observe that v

".10':

is untwisted

S

0

E

S and ,,:

IR ....

M be the

Consider the pullback ".v:E .... IR

= v(v)},

,,-v(u, v)

= u.

Put I(U, v)

of

=v

Since the space R is contractible, the

E .... R is trivializable. Let g: E .... IR x R" be some C" smooth

global trivialization of 0

If

Let (NS, v, S) be the normal bundle, U be an open neighbourhood of S in M,

f. NS .... U

pr2

0").

(3.3)

".10'.

Let Ee denote the fiber of ".v over the point a. Clearly,

glEe: Ee .... IR" and pr2

0

gIEe+w: Ee+w .... IR" are linear isomorphisms. Denote

If the trajectory S is untwisted, then orientation reversing. Since the sets {A

"e is orientation preserving. Otherwise, "e is E

L(IR", IR"): det A

> O}

and

{A

E

L(IR",

IR"):

det A < O} are open and pathwise connected, the trivialization g: E .... IR x IR" can be modified so that we get "e = id (a E R) whenever S is untwisted, and IPe = J (a E R)

18 in the opposite

case.

involution. Let

g:

way.

cr =

Denote

J e L(IR" , IR")

Here

E -+ IR x IR"

f

~

0

some

fixed

orientation

reversing

C smooth trivialization obtained in this -+ U. Then (U, cr) is a w-demiperiodic C

denote the

gOI:

0

is

IR x IR"

pseudochart. As crllR x {OJ = cp, we get ~.,.(a, 0) = (1, 0). Define (3: IR x IR" -+ IR x IR" by (3(a, x) = (a seen that cr

0

(3

+ w, Jx). From the = cr. Consequently,

l)2(a

+

(3.11

[1l3rl

5!!

(cr

0

(3).~

l): IR x IR" -+ T(IR x IR"),

given vector field equality

consruction of the pseudochart (U, cr) it is easily

11

0

f3 - 11

0

= (3.cr.~ = cr.~. l}(a, x)

=

just means that

Now observe that for a

(a, x, 1I1(a, x), 1I2(a, x», the

1I1(a

+

w, x) = 1I1(a, Jx)

and

w, x) = Jl}2(a, Jx). Therefore ~.,. is really of the form (3.2), (3.3).

3.8. Lemma. Let B: IR -+ L(IR", IR") be a C smooth map, J e L(IR", IR") be an involution

and B(a + w) = JB(aJ). Then there exists a C map P: (1) for each a e IR the map P(a) is invertible; (2) P(a

+

L(IR", IR") such that:

(a e IR);

2w) = p(a)

(3) p(a)B(a)[p(a)]"1

IR -+

+ DP(a)[p(a)r l does not in fact depend on

a e IR •

• Let l) be the vector field on IR x IR" defined by lI(a, x) = (a, x, 1, B(a)x) (a e IR,

x e IR"). We look for a diffeomorphism g: 1R"+1 -+ IR"+I such that the local

representative

g.lI

of l) near S is as simple as possible. Clearly, the flow of l} can be

written as IJ/(a, x, t) = (a and

cpHs(a)

= cpt (a + s)

+ 0

t, cpt(a)x), where cpt(a):

cpS (a)

IR" -+ IR"

is a linear isomorphism

for all t, s, a e IR.

From B(a

(t, a e IR). Putting a

=0

+ w) = JB(a)J it s = t = w, we get

follows that cpt(a

+

cp2w(O) = Jcpw(O)

Jcpw(O). Thus the element cp2w(O) has a square root.

0

w)

= Jcpt(a)J

(see, for example, Pontryagin [1, p. 327])

A e L(IR" ,

IR")

satisfying

p(a) = exp (SA) [11'9(0)]"1

that

in

= rp2W(0).

(a e IR).

It is easy to verify that P is

Now define

+

an operator

P: IR -+ L(IR" ,

C

1': IR x IR" -+ IR x IR"

We have

=(a, x, 1, {DP(a)[P(a)rl

It is well-known

case there is

exp (2wA)

satisfies the conditions (1) and (2). Consider the map 1'(a, x) = (a, P(a)x).

this

and

p(a)B(a)[p(e)ri}x).

IR")

by

smooth and defined by

19

The diffeomorphism r sends the motion of the motion I

t-+

~

issuing at t

=0

from the point (0, x). to

+ p(e)B(e)[p(e)r 1

(t, exp (tA)x) , hence, DP(e) [p(e)r 1

Ii

A.

3.9. Floquet-Lyapunov Theorem. Let M be a smooth manifold, E' E r"(TM)' S be a closed

trajectory of E' and w be its smallest positive period.

Then there exists a

w-demiperiodic C pseudochart (U, cr) for S such that the principal part ~cr of the local representative cr.E' of the vector field ~ with respect to (U, cr) has the form

E'!(e, x)

=

= Ax +

1 + Q(e, x), E';,(e, x)

where A E L(lRn , IRn ), Q(e, 0) = R(e, 0) are 2w-periodic in the first argument.

(e

R(e, x)

= D2R(e, 0) = 0

(e

E

E

IR, X

IR),

E

IR n ),

(3.4)

and the maps Q and R

~

In order to prove this theorem one needs to apply Lemma 3.8 to the preliminary normal form (3.2), (3.3). 3.10. Definition. Let ~

E

r"(M), (M, IR, rp) be the flow generated by E' and S be a

closed orbit of E'. We say that S is hyperbolic if there exist vector subbundles ~ and

N' of 1M[S] invariant under 1M[S] = TS E9 ~ e N' and IITrpt(p)vlI

:is

Trpt for all t

c IIvll exp (- a.t)

E

T and numbers c

(p

S, t

E

> 0,

>

0, a.

>

0 such that

V E ~),

(3.5) II Trp-t(p)VII

:is

c IIvll exp (- a.t)

(p

E

S, t

> 0,

V E ~).

3.11. Remark. The trajectory S is hyperbolic iff the spectrum of the operator A entering the normal form (3.4) has no points on the imaginary axis (i.e., Re ;>'t ¢ 0 for all eigenvalues ;>'1I"',;>'n of A). 3.12. Remark. Even in the case when S is untwisted and the vector bundles TM[S] and

NS are trivial the subbundles ~ and N' may fail to be trivial. This can be confirmed by the following simple example. Let SI

= IR

I 21tZ,

M

= SI

x 1R2 and let P(e): 1R2 -+ 1R2

denote the rotation through an angle e. Consider the following ~ smooth flow 1R3

= 1R2 x

IR;

It

on

20

.'ex, y, 0) Remark that xt(x, y, e

sl,

+

(1'(012) •

2x)

1i:). The set S

II

[~'

:.,l·

1"(012)

~l'

0

+ I).

xt<x, y, e), so the flow (1R1 x IR, IR, x) induces a flow

= {(O,

0, e): e E Sl} is a 2x-periodic trajectory of the latter flow. It is easy to see that S is hyperbolic and (1R1 x

IR,

~ = {(p(e/2) ~}

e): e

E

Sl, Y E R},

~

e): e

E

Sl, x

= {(p(e/2)

[~}

E

IR}.

Clearly, ~ and ~ are Mobius bands (see Figure 3.1)

NS

Figure 3.1.

3.13. Grobman-Hartman Linearization Theorem for Periodic Orbits. Let M be a smooth

manifold, (E rl(TM) and S be a hyperbolic closed trajectory of (. Then the flow Ip( is topologically linearizable in a sUfficiently small neighbourhood U of S, i.e., there exists a homeomorphism h: U -+ NS transfering the motions of Ip( to motions of (NS, R, Nip).

21 • This follows from Theorem 1V.3.S. 3.14. Remark. Suppose that the vector field ~ near the hyperbolic periodic trajectory

S is reduced via some demiperiodic pseudochart to the Floquet-Lyapunov normal form ~l(e, x)

~2(e, x)

=1+

Q(e, x),

= Ax + R(e, x)

(e E IR, x E IR").

It follows from the preceding theorem that, in this pseudochart, the flow topologically equivalent to the flow ,,/(e, x)

= (e + t,

near S is

tpo(

exp (At)x).

§ 4. Hadamard-Bohl-Perron Theory

In this section, we shall be concerned with the existence of some smooth local

manifolds in the vicinity of a rest point and a periodic orbit, namely, the stable, unstable, center, center-stable, and center-unstable manifolds. These manifolds allow us to obtain more detailed local normal forms of a vector field. Similar results hold for cascades, as well. We leave to the reader to formulate the corresponding definitions and theorems. The case of cascades can be reduced to that of flows by means of the Smale suspension.

Rest points 4.1. Notation. Let E be a finite dimensional Banach space and ~: E -+ TE be a smooth vector field. Let 0 be a singular point of ~. Without loss of generality, assume that ~ generates a (global) flow (E, IR, tp). Recall that the seemingly more general setting of vector fields on manifolds in the vicinity of an equilibrium can be easily reduced to this case. Let us identify

~

E

rr (E)

with

the corresponding

function

~

E

C (E,

E)

(see

subsection 2.1). Denote A = D~(O). Decompose the space EasE = E'. Jt= .~, where

E', Jt= and

~ are A-invariant linear subspaces such that AS'

II

AI E' (Au

Ii

AI~) has

22 eigenvalues with negative (respectively, positive) real parts and imaginary eigenValues.

Ac!! A I Jt=

has pure

4.2. Definition. A smooth submanifold W c: E is said to be locally invariant with respect to the vector field ~ if for each point x E W the vector ~(x) belongs to the tangent space Tx W. In other words, for each point x that " (x, t)

E

W for all t

E

W there is a number

C vector field E. Then in some neighbourhood U of 0 there exist C manifolds W, W", we, locally invariant with respect to

~,

t > 0 and Ip(x, [0, t]) c:

~

defined on

we·, and we'"

each containing the point 0 and such that:

(2) there exists a ~-invariant C smooth foliation

= Wo

> 0 such

E (- £, E).

4.3. Theorem. Assume that 0 is a singular point of the

such a way that W

E:

we· = u [ W;:

x

E

we]

defined in

and there are numbers C > 0 and a: > 0 so that if

we

YEW;,

then d(c/(X) , "tey» :s C d(x, y) exp (- a:t);

C smooth foliation we'" = u [ ~: x E we] such and there exist numbers C > 0 and a: > 0 with the following property:

(3) similarly, there is a

~-invariant

that W" = ~ ify E W;, t > 0 and Ip(x, [- t, 0]) c: ~

we then

d(lp-t(x) , Ip-tey» :s C d(x, y) exp (- a:t).

The proof is given in Chapter V (see subsection 4.9).

4.4. Remark. The manifolds W (W", we, we·, we"') are called the stable (unstable, center, center-stable and center-unstable, respectively) local manifolds of the singular point O. The stable and unstable manifolds are completely determined by the property (1) of the Theorem 4.3 and, according to the properties (2) and (3), they coincide with the sets of points x E U that exponentially approach the point 0 as t -+ + 00 and t -+ - 00, respectively. As to the center, center-stable and center-unstable manifolds, they are not, in general, unique. This is manifested by the following example. 4.5. Example (see Anosov [2]). Define a vector field ~ on ~2 by ~(x, y) Integrating the equation dx / dy

= - x / l,

we obtain x

= c e"P

= (- x,

y2).

(1 / y) ey '" 0, c is

23 the constant of integration), y 4.1:

= O.

The phase portrait of

~

is presented in Figure

x

/' Figure 4.1. Let

uCy, c)

M(c)

Since u(O, c)

= au ay

= {(x,

(0, c)

the singular point (0, 0). serve as

a center

={

=0

c exp (1 I y)

for y < 0,

o

for y

y): x

= uCy,

for all c

c), Y

E

E

of class

(c

IR}

E

IR).

IR, the curve M(c) is tangent to x

Hence it follows that for each

manifold

> 0,

c

E

IR

= 0 at

the curve M(c) can

C for (0, 0). Similarly, for the vector field

1)(x, y, z) = (- x, l, z), the center-stable singular point (0, 0, 0) are also nonunique.

and

center-unstable

manifolds

4.6. Remark. The assertions (2) and (3) of Theorem 4.3 allow us to defme

of the

C

maps

WC.. -+ WC and tu: WCU -+ WC as follows: t ..Cy) = x Cy E W!, X E WC), tuCy) = x Cy E ~, X E WC). These maps are called the stable (respectively, unstable) asymptotic t .. :

phase.

24 Theorem 4.3 permits also to introduce some special coordinates in the vicinity of 0 that straighten out the manifolds WC· and This fact is expressed in the following

WCu as well as the corresponding foliations.

c: smooth vector field (

4.7. Theorem. Let 0 be a singular point of the

there exists a c: diffeomorphism field 0:.( has the fonn: ~(x,

y, z)

= (A,.x

0::

on E. Then

E ... E such that the principal part of the vector

+ P(x, y, z), AuY + Q(x, y, z), AcZ + p(z) + R(x, y, z» (x

E

~,

Y E It', z E Jt=),

where P, Q, R and pare C /Unctions and P(O, y, z) ~

= 0,

Q(x, 0, z)

= 0,

R(O, y, z)

= R(x,

0, z)

= 0,

p(O)

= 0,

Dp(O)

= o.

The proof is given in Chapter V (see subsection 4.10).

vector field ( on E. The flow IP~ of ( near the point 0 is topologically conjugate with the flow of the following vector field (0: 4.8. Theorem. Let 0 be a singular point of the

(o(x, y, z) ~

= (A,.x,

AuY, AcZ

C

+ p(z»

(x

E

~, Y

E

It',

z

E

Jt=).

For a proof, see subsection V.4.11.

Periodic trajectories 4.9. Notation. Let M be a (n + 1)-dimensional compact manifold, (E rr('tM)' S be a closed orbit of (and III be its smallest positive period. Let (M, IR, IP) denote the flow generated by (. As before, let TM[S] = {v E TM: 'tM(V) E S}, and let NS = TM[S] I TS be the normal bundle. For every number I, the transformation

linear morphism NIPt: NS ... NS.

Pix a point b

E

TIPt: TM ... TM induces a

S and consider the linear operator

NbIPw: NbS ... NbS. The space N~ can be represented as a direct sum of NbIPw·invariant

25 linear subspaces N:" III:, and ~ such that the spectrum of N,:/pw1N:, (NbrpwlIII:,) lies inside (respectively, outside) the unit circle, and the eigenvalues of the operator Nbrpw I~ belong to this circle. Denote

U N:"

~ =

U 111:"

~ =

beS

~ =

beS

U ~. beS

It is easy to see that ~,~ and ~ are vector subbundles of the normal bundle NS invariant under the flow (NS, IR, Nrp).

4.10. Theorem. Let S be a periodic orbit of the vector field ~ E r""(TM). Then in a small enough neighbourhood U of S there exist locally invariant manifolds W,

WC·, WC

of class

U

(1) W[S] WU[S]

=

~

we,

C each containing S and satisfying the following conditions:

= TS$~,

TS ,.

WU,

,.

:rW'[S]

= TS,.~,

W[S]

= TS.~,

W·[S]

= TS,.~,.~,

~j

C smooth foliation WC· = U [ K/!: x e WC] such that W'" = U [ b E S] and there exist numbers C > 0 and« > 0 with the follOWing property: if x e WC, y e ~, t > 0 and rp(x, [0, t]) c WC then «(rpt(y), rpt(x» s Cd(x, y) exp(-«/)j (3) there exist a i;.-invariant C smooth foliation WC U = U [ ~: x E WC] with similar (2) there is a i;.-invariant

w::

properties. • The proof is presented in subsection V.4.9.

we

One can define the asymptotic phases I.: WC· .. WC and Iu: WC U .. for the periodic orbit S in the same manner as for singular points (see Remark 4.6). The mappings I. and lu are

C smooth.

4.11. Theorem. Let S be a closed orbit of the vector field i;. E r""(TM). Then there exists an w-demiperiodic C pseudochart such that the principal part of i;. with respect to this pseudorhan is of the form: ~(x,

y, Z, 9)

= (A,.x +

p(x, y, Z, 9), AuY

+

Q(x, y, Z, 9),

26 Ac:Z

+

+ R(x, y,

IPo(z, e)

+

Z, 9), 1

l/Io(Z, 9)

+

V(x, y, z, 9»

where the eigenvalues of As: IRk -+ IRk (Au: IRI -+ 1R1, Ac: IR m -+ IRm) satisfy the condition

°(

Re ~ < Re ~ > 0, Re ~ = 0, respectively). The functions P, Q, R, V, IPo, 1/10 are C smooth and 2w-periodic in 9. The following equalities hold:

= 0,

P(O, y, Z, 9) V(O, y, Z, 9)

=

= 0,

Q(x, 0, z, 9)

V(x, 0, Z, 9)

= 0,

R(O, y, z, 9)

IPo(O, 9)

= 0,

= R(x,

1/10(0, 9)

0, z, 9)

= 0,

= 0,

D 1IPo(0, 9)

= 0.

• The proof will be given in Chapter V, subsection 4.12. 4.12. Theorem. Let S be a closed orbit of the vector field ~

r 1 (T M ). Then there

E

exists an w-demiperiodic C pseudochart for S such that the flow of the vector field ~ described in Theorem 4.11 is topologically equivalent near S to the flow generated by the following vector field ~o: ~o(x,

y, Z, 9)

= (A-s-X,

AuY, Ac:Z

+

IPo(Z, 9), 1

+

l/Io(Z, 9»

~o

refered to in Theorems

Im ) (XEIRk ,YEIR,ZEIR ,9EIR,

• See V.4.13. 4.13. Example. Let us show that the vector fields

~

and

4.11 and 4.12 are not, in general, C conjugate with each other. For this purpose let us examine the following vector field: ~(X, y, 9)

= (x,

- y, 1

+ xy)

(x E IR, Y e IR, 9 e IR).

Clearly, it can be regarded as the Floquet-Lyapunov normal form of a vector field defined on 1R2 x SI, SI

=

IR I Z, and having S

=

{(O, 0, 9): 9

E

SI}

orbit. Our goal is to show that ~ is not C conjugate with the vector field ~o(X,

y, 9)

= (x,

- y, 1)

(x E IR, Y E IR, 9 E IR).

~

as a periodic

27 Suppose the contrary holds. Then there exists a carries the motions

C

mapping

«x, y, a) to periodic motions I//(a) or I

Set ,.,.(x, y, e)

= I(x,

=e

+

1 (a

E

Sl, 1

(xet , ye- t , a + 1 + xyt)

E

IR), that is,

= I(X,

I: 1R1 x Sl .... Sl

E

1R1 x Sl, t

I

0

tpt

E

= !/It

which

IR) 0

I

(I

E

IR),

y, e) + I.

y, a) - a. We obtain the following functional equation for ,.,.:

lJ.(xet , ye- t ,

e + t + xyt) + xyt

= J.I(x, y,

e).

(4.1)

Denote I

vex, y)

=

I

J.I(X, y, e) de.

o

Integrating (4.1) with respect to e 9, we get

E

Sl

and taking into account that

J.I

is periodic in (4.2)

Note that VEe, by our assumption. The general solution of the equation (4.2) is of the form vex, y) = - xy In Ix I + ~(xy), where ~ is an arbitrary function. Observe that ~ is differentiable at every point (X, y) with x ¢ 0, Y "* O. Therefore Bv

-By = - x In

Ixl

+ x D~(xy),

Bv

-Bx = - y In

Ixl

+ y + y D~(xy).

Let x be a fixed non-zero number, then lim Bv(x, y)

= -x

In Ixl

By

y-+O

+x

lim

D~(z),

z-+ 0

consequently, D~(z) (z '" 0) is bounded as that z .... O. On the other hand, if y is fixed, Y '" 0, then lim

x-+~

Bv(x, y) By

= -y

lim In Ixl x-+o

+ y + Y lim z-+O

D~(z),

28 hence, DI;(z) -+

C

CD

We get a contradiction. Thus, the vector field ~ is not

as z -+ O.

conjugate to 1;0. According to Theorem 4.12, there are continuous functions .: 1R1 x Sl -+ Sl satisfying

the condition •

0

rpt = I/I t

.(x, y,

9)

o.

=9

(t -

E

IR). For example, we can take

xy In Ixl for x

~

0, .(0, y, 9)

= 9.

Bibliographical Notes and Remarks to Chapter I Theorems 2.5 and 2.11 are due to Grobman [1] and Hartman [1]. The presentation of the Floquet-Lyapunov theory (see Floquet [1], Lyapunov [1]) is adapted from the books by Abraham and Robbin [1], Reizin5 [1]. Theorems 4.3 and 4.10 have a long history (see the historical comments by Anosov [2], but note that the works of Bohl [1] are not reflected there). The Hadamard-Bohl-Perron theory is presented in the books by Abraham and Robbin [1], Anosov [2], Nitecki [1], Hirsch, Pugh and Shub [1], Palis and de Melo [1].

29

CHAPTER II FINITELY SMOOTH NORMAL FORMS OF VECTOR FIELDS AND DIFFEOMORPHISMS

The Problem on Reducing a ex> Vector Field to in the Vicinity of a Hyperbolic Rest Point

§ 1.

CC

Normal Form

We consider the problem on reducing an infinitely smooth vector field at a hyperbolic rest point to normal form by the aid of coordinate changes of class ~, where k is a fixed positive integer. This section contains a short review of the results presented in Chapter II, together with some explanations and examples.

1.1. Preliminaries. Let E be a real Banach space of finite dimension n. Let ~ E r""(E) be an infinitely smooth vector field on the space E, and ~(O) = O. We assume that the spectrum of the operator A == D~(O) lies out of the imaginary axis. In order to simplify the subsequent discussion we suppose that the operator A is diagonal, i.e,

= diag[9 1,

where 9, are real numbers and restriction is not essential and will be dropped in the sequel.

A

... , 9 n ],

We shall use the following notation:

· co11ection

0f

== vi

+ ... +

vn

0 (i

= 1,

... , n). This

v = (vI, ... , v n ) denotes a multiindex,

. .10tegers v '('I = 1" , " n) ; xa- = non-negative

The number Ivl

9,"

a- I

X I '. , , ' X.,.n n

i.e., a

, a monomI'al , IS

is said to be the degree of the monomial xv.

1.2. Topological and smooth linearization. As it was shown in Chapter I (see Theorem 1.2.11), the topological normal form of (a germ of) a smooth vector field at a hyperbolic rest point is linear. Unfortunately, the linearizing homeomorphism may fail to be smooth. This fact can be confirmed by the following example (see Hartman [2]) of a polynomial vector field with a hyperbolic rest point at the origin which does not admit any

C

linearization,

30 1.3. Example. We consider two differential systems

x = 2x, and

Y = y + Xl, Z = - z;

u = 2u, v =

v,

W=

(1.1)

w.

-

(1.2)

The formulas x(t)

= xe2t ,

y(t)

= [y + tu]et ,

z(t)

= ze-t

(1.3)

and

= ue2t ,

U(t)

v(t)

= vet,

= we-t

w(t)

(1.4)

give the solutions of these systems. According to the Grobman-Hartman Theorem, the systems (1.1) and (1.2) are topologically equivalent in the vicinity of the origin, i.e., there exists a local homeomorphism transfering solutions (1.3) to solutions (1.4). Suppose this homeomorphism can be chosen differentiable. Then, without fail, there exists a conjugating diffeomorphism of the following structure:

x

= U,

Y

= v + rp(u,

w), z

= w.

(1.5)

°

In fact, the system (1.2) leaves the plane v = invariant. In virtue of our assumption, the system (1.1) has a locally invariant smooth manifold y = rp(x, z) tangent at the origin to the plane y = 0. Then (l.5) is a smooth conjugacy. Therefore, the identity

is valid, i.e., 2t

rp(ue ,

Let u

¢

0,

W

¢

we-t ) =

+ tuw]et •

[rp(u, w)

0. Take a sequence {tn} -+

+

00

(l.6)

and set (l.7)

Let rpn denote the solution of the system (l.2) beginning at the point (un' 0, w) t 0, i.e.,

=

rpn(t)

= (une2t , 0,

The identity (1.6), for t = tn and

-t

we )

Ii

(ue

u = Un' gives

2(t-t )

-t

n, 0, we ).

for

31

(1.8)

It is clear that 1p(0, 0) = O. The map (1.5) transfers the stable manifold of the point u == 0, v = 0, W = 0 for the system (1.2) to the stable manifold of the point x = 0, y == 0, z = 0 for the system (1.1). Consequently, 1p(0, w) = O. According to our assumption, the function Ip is differentiable. Therefore

Note that unwetn

= uwe- tn

by (1.7). Thus the equality (1.8) becomes (1.9)

Tending

11

to infinity, we get Ip(u, 0)

= O.

Consequently, (1.10)

Divide both parts of the equality (1.9) by tne- tn . Taking into account the relations (1. 7) and (1.10) and passing to the limit, we obtain uw = 0, contradicting the choice of u and w. This contradiction shows that the system (1.1) does not admit smooth linearization in the vicinity of the origin. 1.4. Polynomial resonant nonnal fonn. Consider the family of all smooth vector fields ~: E -+ TE each having the origin as a hyperbolic singular point. The group Dif~(E) of

O-germs Ip E

of (!t diffeomorphisms, k ~ 1, acts in a natural way on vector fields, namely,

Dif~(E) transfers ~ to Ip.~, where Ip.~

= Tip

0

~

0

Ip -I.

As it was shown above, the

orbit {Ip.~: Ip E Dif~(E)} may fail to contain a linear vector field. So we face the somewhat vague problem of finding an element of this orbit which has the most simple form. Such and element is referred to as (!t normal form of the vector .field ~.

e

Sternberg [2] and Chen [1] have shown that for a given vector field ~: E -+ TE with the origin being a hyperbolic equilibrium there exists a number Q = Q(k, 8 1"", 8 n ) such that ~ is locally (!t equivalent to its Taylor

expansion of order Q. In other

words, the terms of degree greater than Q can be deleted from the (!t normal form. Thus, when dealing with finitely smooth normal forms of C" vector fields at a hyperbolic rest

32 point, we can confine ourselves by considering only polynomial vector fields. As Poincare [l,2] and Dulac [1] have shown, by the aid of polynomial coordinate changes, the vector field ~ can be reduced to the following form Q

= 9,rj

Xj

+

L p/,x<1' + rp~Vc)

(j

= 1, ... , n),

(1.11)

111'1=2

Irp~(X)1

where x = (XI'"'' xn );

= O(IIXIIQ)

as

X -+

0; II'

E

z~;

p; are

real numbers and,

p;

besides, ;t 0 implies 9 j = <11',9> 511'19\ +... + II'n9n• Combining the abovementioned statements, we get that an infinitely smooth vector field in the vicinity of a hyperbolic rest point can be reduced to the so-called polynomial resonant normal form, i.t., to the polynomial normal form Q

Xj

= 9,rJ

+

L p/,x<1'

(1.12)

(j = 1, ... , n),

111'1=2

that contains only resonant terms, i.e., monomials p/,x<1' such that the resonance condition 9 j = <11',9> is fulfilled. In order to clarify the sense of this condition, consider a resonant monomial p/,x<1'

and denote y = x<1'. Then •

y

I

= (II' 9 1

+ ... +

n

<1'1

II' 9 n )i1 ...

...n xn + ...

= 9 jY

+ ... ,

where dots mean a polynomial in XI"'" xn of order higher than 111'1. Thus, a resonant monomial x<1' entering the j-th equation of (1.12) has the same Lyapunov exponent as the variable xJ does. 1.5. Example. Let us try to relate the condition 9 j

to the notion of resonance adopted in physics. Assume that the eigenvalues 9 j in (1.12) are pure imaginary. Such systems occur in the theory of small nonlinear oscillations. For example, consider the following system

z=

iz + ;-. z, i == - iz + z· Z2

(z

== < II' ,9 >

E

c)

(1.13)

33 that corresponds to the real system

x = Y + x(i" + i), Clearly, the monomials

r!.1,

y=

-x

+ y(i" + i).

and z·1,'1. entering the system (1.13) satisfy the resonance

condition. The solutions of the linearized system are 2n-periodic. The nonlinear terms

t·1,

and z· 1,'J, can be regarded as small perturbations. When considered along the

solutions of the linearized system, they are also 2n-periodic. But the solutions of (1.13) are not periodic. 1.6. Example. Consider the following polynomial vector field on the plane (1.14)

Make the change of variables YI

= XI'

Y2

= ~ - ~,

then

(1.15)

Thus by using a polynomial coordinate change, we succeeded in deleting the non-resonant term ~ from the second equation. However, it is impossible to linearize the vector field (1.15) by the aid of analytical coordinate changes. In fact, making Zl

= YI + ... ,

l'J,

= Y2 +

ay~

+ ...

. . . l'J, = Y'J, + 2aylYI Assuming ~

= 2z'J"

substitutions

in (1.15), we get 'J,

+ ... = 2y'J, + YI +

'J, 2ayl

+ ...

we obtain the identity

2y'J,

+ y~ + 2ay~ + ... = 2(y'J, + ~ + ...),

whence we get the impossible relation 2a

+

1

= 2a.

Therefore the system (1.15) does not

admit any further simplification via analytical coordinate changes. As a matter of fact, the same reasoning shows that the system (1.15) does not admit any the vicinity of the origin. At the same time, it is easy to verify that the coordinate change

C

linearization in

34

is a local

C

diffeomorphism reducing the system (1.15) to the linear form

1.7. Further simplification of the polynomial resonant nonnal fonn. The problem on further simplification of the polynomial resonant normal form by using finitely smooth changes of variables was considered first by Samovol [1-3]. He has found some sufficient condition (the so-called condition S(k» allowing one to delete resonant monomials from the normal form by a transformation of prescribed smoothness k. In some cases, this condition cannot be improved (see Samovol [3, 9, 10]). Recently, Samovol [5-9] has shown that, in general, condition S(k) can be weakened. The authors have suggested and thoroughly investigated an essentially more general condition of this kind (denoted by 2l(k». These results are presented in §§ 4-8. 1.S. Nonnal forms of finitely smooth vector fields. Up to here, we dealt with C< normal forms of infinitely smooth vector fields. From now on, we assume the vector fields to be finitely smooth (say, of class ~, K < 00). We shall establish some conditions involving the numbers K and k, and the spectrum of the linear part so that their fulfilment guarantees that the vector field can be reduced to polynomial normal form by a C< local change of variables. Results obtained in this direction by Sternberg [1,2], Chen [1], Robinson [1], Takens [1], Belitskii [1], Sell [1-3] and the authors are presented in §§ 9-11.

§ 2. Nonnalization of Jets of Vector Fields and Diffeomorphisms

e

Following Poincare and Dulac, in order to simplify smooth vector fields and diffeomorphisms near a fixed point, we use polynomial changes of variables. In fact, we discuss how polynomial transformations act on jets (i.e., Taylor expansions of finite order) of our objects. In this section, we prove that jets of vector fields and diffeomorphisms can be reduced, in this way, to resonant normal form.

35 Vector fields 2.1. Preliminaries. Let E be a finite dimensional linear space and Q be a positive integer. Let r~(E) denote the space of all

cf1

smooth vector fields i; on E having the

origin as a rest point. In other words, r~(E) is the collection of cf1 smooth sections E': E -+ TE of the tangent bundle (TE, T E , E) such that E'(O) = o. Two vector fields E' and l/ are said to have contact of order Q at the origin of the space E if

= o(lIxn Q)

as IIxll -+ O. This relation is an equivalence relation. We define Q-jets of vector fieltb al the origin to be the corresponding equivalence classes (see IIE'(X) - l/(x)1I

subsection A.14). For E' E r~(E), let }.Q(E') denote the Q-jet of the vector field E' at the origin, i.e., the equivalence class containing the element E'. When the coordinate system is fixed, the vector field E': E -+ TE can be identified with its principal part (: E -+ E.

Therefore, two vector fields E' and l/ in r~(E) have contact of order Q at

the origin if and only if d(O)

= d'~(O)

for p

= 1, ... ,

Q. Thus, in a fixed coordinate

system, every jet }.Q(E') contains exactly one vector field l/ whose principal part polynomial of degree Q without free terms, namely,

li

is a

Q

~(x)

= r

_1 d'(O)? l.. p!

(x

E

E).

p=1

In

other

words,

the

choice

of a

coordinate

system

establishes

a

one-to-one

correspondence between Q-jets of vector fields E' E r~(E) at the origin and polynomial mappings of degree Q without free terms from E to E. In what follows, we shall consider only jets of vector fields at the origin. 2.2. Normalization of Q-jets of vector fields. Given a vector field E' having the origin as a singular point, we look for a

cf1

smooth coordinate system in the vicinity of

the origin such that the jet }'?,(E') being expressed as a polynomial in the new variables has the most simple form. Let us recall that a smooth change of variables y = t(x) transfers the vector field E': E -+ TE to the vector field t.E', where

36

(we identify the vector field ~: E -+ TE with its principal part ~: E ... E ). Henceforth. we shall deal with local diffeomorphisms y = 1(0) satisfying 1(0) = 0 and DI(O) = id. The second condition looks unnatural but. for our purposes. it suffices to consider only such transformations. Note that the Q-jet of the transformed vector field I.~ is determined completely by the Q-jet of I. so we conclude that in the search of a normalizing transformation we can confme ourselves by the class of mappings of the form t(x) = x + PQ(x). where PQ is a polynomial of degree Q with no free and linear terms.

2.3. Notation. Fix a coordinate system on E and consider a polynomial vector field

~

on E: Q

~(z)

!!

~(z)

L PwZw

= Az +

(Z

E

E

Zd. +.

E).

(2.1)

Iwl-:z

..' = (ZI ..... Zd) E IRd -E'. = (p! ..... P=> E Rd; ZW denotes.

Here

- ("' I ..... '"d)

'" -

·tt.

zi

l .... as usual. Let "I ..... "d be the eigenvalues of the linear operator A: E -+ E. Let 9 1..... 9 n denote all distinct values in the collection {Re "l: 1 1..... d}; 9 (9 1..... 9 n ). Let Xl ..... Xn be the A-invariant linear subspaces corresponding to the numbers 9 1..... 9 n .i.e.. the spectrum of the operator A IXl lies on the straight line Re " = a, (I = 1..... n). The coordinates on the subspace X, will be denoted by Z'.I ..... z'.m,.

Pw

=

where

ml

= dim X,. We shall also use the notation x, =

=

(Z'.I .....

z'.m,) (i = 1..... n).

i.e .•

To each

multiindex

multiindex v = cr(",) pair (i. cr) P;

E

E

E

'"

= ("'I .....

",d) E z:.

z~ defined by v' =

{l ..... n} x z~.

there

exists

",'.1

we put

+ ... + ",'.m,

an

in

corespondence

the

(i = 1..... n). For every

unambiguously defined polynomial

Pcr(XI ..... Xn; X,) (see subsection A.1) such that

37

For i E {I •...• n} and x, E X,. let x,e, be the element in E having (in the new agregated coordinates) the i-th component equal to x, and the others equal to O. Using the new notation. we can write the vector field ( in the form Q

n

~(x) = Ax

2.4. Dermition. Let

L

+

(x

p!;zw

E. tr

E

Z~).

(2.2)

z.:,) be a coordinate system on E that brings the matrix

(ZI.....

A to the Jordan normal form. Let j

the monomial equality holds:

E

E

{I •...• d} and w

E

z~. The pair

V.

w) as well as

are said to be resonant (in the sense of Poincarl) if the following

2.S. Dermition. Let s E {I ..... n} and tr (s. v) is (weakly) resonant if

== (tr\ .... trn ) E z~. We say that the pair

(2.3) The integer Iv I =

vI

+ ... +

vn

is called the order of resonance. The term p;Xtr is also

said to be (weakly) resonant. Let Re v J

=

9.

and v

= v(w).

It is easily seen that whenever the pair

V.

w)

is

resonant in the sense of Poincare then the pair (s. v) is (weakly) resonant. The converse statement is false. as the following example shows. 2.6. Example.

ZJ =

2z3

+

Consider

ZI~' ~ = 3Z4

+

the vector

field

il =

(1

+

2l)zl'

~

=

(1 - 2l)~.

i.~. The monomial ZI~ is resonant in the usual sense; the

monomial z~~ is weakly resonant. but not resonant in the sense of Poincare. 2.7. Remark. According to the well-known Poincare-Dulac Theorem (see Poincare [1]. Dulac [1]). any formal vector field having a singular point at the origin is formally

38 equivalent to a vector field whose linear part is brought to the Jordan normal form and which contains only resonant terms (such a vector field is said to be a resonant normal form of the given vector field). Here the word formal refers to formal power series. Let us emphasize that recognition of Poincar~ resonant terms presupposes that the linear part of the vector field is brought to the Jordan normal form, which, in tum, is tied with complexification of the vector field. Bruno [1-3] has introduced the notion of real normal form. By definition, the real normal form is obtained from the resonant normal form via a standard linear transformation. We don't know any intrinsic characterization of real normal forms. The most unsatisfactory feature of both the resonant normal form and the real normal form is that they may fail to depend continuously on the linear part of the vector field. The notion of weak resonance does not require the linear part to be pteviously reduced to Jordan normal form. We only need to assume that the space E is decomposed as E = XI 111 ... 111 Xn , where the subspaces XI'"'' Xn are described above. This decomposition is stable under C perturbations of the vector field. Therefore we may expect that the notion of weak resonance will be of use in bifurcation theory. It should also be noted that the classical notion of resonance reflects adequately the obstacles to linearization of a formal complex vector field near a singUlar point by the aid of polynomial transformations. But whenever we investigate the more general problem on linearization of a vector field in the vicinity of a smooth invariant manifold and seek, to this end, conjugating maps in more wide function classes, it becomes clear that the notion of weak resonance, based on comparison of Lyapunov exponents, turns out to be more suitable. That is why we use henceforth only Definition 2.5, omitting the adjective weak (cf. Takens [1]). 2.S. Lemma. Let Q and q be positive integers, 2 ~ q ~ Q, and ~ e r~(E) be a vector field given by q

~(x)

= Ax +

L PaXv

+ .f(x)

(x e E),

(2.4)

Iv 1-2

where Pv E Pv(XI , ... , Xn; X), Ilf{X)1I = o(lIxll q ) as IIxll -+ O. Let s E {I, ... , n}, 't E z~, l't I = q and the pair (s, or) be non-resonant, i.e., 905''' < 't, 9>. Then there exists a polynomial h~ E P-.:(XI , ... , Xn; Xs) such that the coordinate change x = y brings the vector field (2.4) to the form

+

~y-':es

39 q

n

~(y)

= Ay +

L

L p;y'''et

+ l(y)

(y

E

(2.5)

E),

1... 1-2

t-I (t ....)-=(S.T)

where 1 E CJ(E, E) and n](y)1I

= O(lIyllq)

• After the indicated change

as lIyll -+ o.

of variables

is

accomplished,

the

vector

field

X = ~(x) takes the form n

LDJ[~yT]j~S

j +

= Ay + ~yTes + p~(y +

~yTes)Tes

J-I n

+

q

r r

+ h~yTes)"'et + fly + ~yTes).

p;(y

1... 1-2

t-I (t ....);t(s. T)

Try to choose a polynomial ~ in such a way that n

Ay

+

t-I

L

L p;y"'et + 1(J) + LDJ[~yT] AJYJ + 1... 1-2

p;"y'"

+ lJ(y)]es

1... 1-2

J-I

(t ....);t(S.T)

then we have

q

n

q

r

y = ~(y) be satisfied,

U .... )-=(S.T)

(2.6)

+

n

q

t-I

1... 1-2

r r

p;(y

+ ~yTes)"'et + fly + h~yTes)·

Equate the coefficients of yT in both sides of the s-th line and write the auxiliary equation for ~: n

-r

DJ[~yT]A;YJ

+

A~yT

+ p~yT = o.

(2.7)

J-I

Let Ss,T: PT(X1, ... , Xn; Xs) -+ PT(X1, .•. , Xn; Xs) be the linear operator defined as fOllows:

40

[21.,T9']yT

= A.9'y

"

T

-

LDjP(y)A..iYJ

J-I

According to the differentiation formula for a polynomial (see subsection A.7),

Here ~ denotes the 't'-linear map associated with the 't'-homogeneous polynomial 9'. Hence it follows that the real parts of the eigenvalues of the operator 2l.,T are equal to e. - 't'le l - ... - 't'"e" .. O. Consequently, the operator 2l.,T is invertible. Put (2.8)

Substitute the solution h! of equation (2.7) into (2.6) and observe that the coefficients of ya' coincide in both parts of the equation (2.6) for every For i III $, set

tr

(2 s Itr I s q).

q

1,(Y)

L IP!(Y + h!yTe.)a' - p!ya'] + ft(y

=

+ h!yTe.).

(2.9)

1a'I-l Finally, put

1.(Y)

= lid + D.h~(y)rl

q

"

{- L D ;h~(y) J-I Jill.

q

- D~(y)

L P;'ya'

+ 'J(y)]

1a'1-2

q

L p!ya' + L IP!(Y 1a'I-l cr_T

[

Icrl-l

+

~yT e.)a' _ p!ya']

(2.10)

41 It is not difficult to check that substituting the expressions (2.8),

(2.10) into (2.6) gives a true equality. lIyll ...

Moreover,

1 E CJ

and

IIl(y) II

(2.9) and

= o(lIyllq)

as

O.

2.9. Theorem. Let .f denote the Q-jet of a smooth vector field on E with 0 equilibrium point. There exist a vector field 'II

E

/2

and a

C"

E

E as

smooth coordinate system

t: E ... IRd such that ~, the principal part of 1.'11, is of the fonn ~(x)

= Ax +

and, moreover, p; - 0 implies at

(2.11)

= < 0',

a>.

The vector field (2.11) is called the resonant nonnal fonn of the jet jQ. ~ Let ~ belong to the jetjQ. By Lemma 2.8, every non-resonant monomial of degree

q :s Q entering the Taylor expansion of the vector field ~ can be deleted with the help of polynomial changes of variables. It should be noted that these transfonnations do not influence other monomials of degree not greater than q. Applying successively Lemma 2.8 to all non-resonant monomials of degree q = 2, ... , Q, we conclude that ~ can be reduced to the fonn Q

'II(y)

= Ay +

r

PaXtr

+ fly)

(y

E

(2.12)

E),

113'1-1 where

p; = 0

whenever

at - <0', a>, and

1lf{y)1I

= o(lIyIlQ)

as lIyll ... O. This means

that the polynomial resonant vector field (2.11) belongs to the jet J.Q(~). Clearly, the resulting coordinate change is a local diffeomorphism of class

C".

2.10. Remark. In general, the resonant normal form of a jet of a vector field is not uniquely detennined. Let us present two examples in favour of this statement.

2.11. Example. Consider the vector field

42

x = x + y,

= y, z = 2z - 2yl.

Y

(2.13)

The non-linear term in the third equation is resonant. However, the polynomial change of variables

u

= x,

= y,

v

W

= % + 2xy - 1

linearizes the system (2.13). In fact,

w= z + 2Xy + 2xY - 2» = 2z - 2y +2(x + y)y + 2xy - 21 = 2(% + 2xy - I)

= 2w.

Note that the linear part of (2.13) cannot be reduced to diagonal form. It is easy to show that, provided the linear part is diagonal and the vector field contains only one resonant monomial of degree k, then there is no coordinate change linearizing the k-jet of this vector field. 2.12. Example. Let us show that even in the diagonal case there exist some possibilities for further simplification of a vector field at a hyperbolic rest point. Consider the vector field on-the plane (2.14) Here all non-linear terms are resonant. We shall show that the 7-jet of the vector field (2.14) can be reduced to the form .

2

Yt

3 1

= Yt + Y1Y2 + YtYl,

Make the coordinate change Yt

= xt

1 = Xt + XtXl

-

Yl

~~, Y2

-

4_.3

2x tXl

= - Y2'

(2.15)

= x2• Then

+ ... ,

where dots mean a sum of terms of degree greater than 7. On the other hand, Yt

+

1

YtYl

+

3 1

YtYl

_~

= Xt + XtXl

-

~_43

"""tXi

+ ... ,

43 i.e., YI

= YI + Y~Y2. + Y~Y~ + ... ,

where dots have the same meaning as before. Hence, the 7-jet of the vector field (2.14) in the new coordinate system contains the vector field (2.15). Thus, we see that some resonant monomials can be deleted via polynomial coordinate changes of degree less than the degree of the monomial to be deleted, in contrast with the proof of Lemma 2.8, were we have used polynomial transformations of the same degree. In connection with the above examples, we face the problem of describing all resonant normal forms of a given Q-jet of a vector field. Unfortunately, this problem is far from being solved (see Bruno [2, pp.154-155]).

1.13. Remarks. (1) Jets of generic smooth vector fields at a singular point are linearizable. In fact, if the real parts of the corresponding eigenvalues are rationally independent, then we have no resonant monomials. Since this can be achieved by arbitrarily small perturbations of the given vector field, we see that non-resonant vector fields are everywhere dense in the space of all smooth vector fields. On the other hand, for each fixed number q > 0, the set of smooth vector fields having no resonances of order less than q is open in the c! topology. (2) Consider a hyperbolic singular point, i.e., such that 9, '" 0 (i = 1, ... , n). Suppose all the numbers 9, have the same sign (in this case the equilibrium is said to be nodal). Then the maximal order of resonances does not exceed the value max 19,1 / min 19,1. In the case of a saddle type point (i.e., when among 91, ... , 9n

,

,

there are both positive and negative numbers), resonances of arbitrarily high orders may occur. (3) Theorem 2.9 can be extended to the case of jets of vector fields in the vicinity of a periodic trajectory. The formulation and proof of such a proposition are left to the reader. Diffeomorphisms

1.14. Preliminaries. Let E be a finite dimensional linear space and Q be a positive integer. In this Chapter,

Diffg(E)

denotes the space of all local

CJ

diffeomorphisms

from E into E with a fixed point at the origin. Let F, G E Diffg(E). The diffeomorphisms F and G are said to have contact of order Q at the origin of the space E if

44 IIF(x) - G(X)II

= O(IIXII O)

as IIXII -+ O. The corresponging equivalence classes are called

Q-jets of diffeomorphisms at the origin. For F E Dif{g(E), let l(F) denote the Q-jet F at the origin. With respect to a fixed coordinate system, every

of the diffeomorphism

jet l(F) contains only one polynomial of degree Q without free terms, namely, Q

L-J,r ifF(O)?

p(x) =

(x

E

E).

p-I

We are looking for such a

CJ

smooth coordinate system in the vicinity of the origin that

the polynomial representive of the Q-jet of the given diffeomorphism F in this coordinate system has the most simple form. Let us recall that a smooth coordinate change y = I(x) transfers the diffeomorphism F to the diffeomorphism I.F, where I.F = I • F • 1,1.

As in the case of vector fields, it suffices to consider only norma-

lizing transformations of the form I(X) = x

+ PQ(x),

where Po is a polynomial of degree

Q without free and linear terms. 1.IS. Notation. Fix a coordinate system on E and consider a polynomial

o F(z) = Lz

L P~w

+

(z

E

(2.16)

E),

Iwl-2

where L is an invertible linear operator. operator L.

Let

denote

9... .. , 9 n

Let all

VI' ... , Vd

be the eigenValUes of the

distinct values in the

collection

v,

{In I I:

I = 1, ... , d}. Let XI' ... , Xn be the L-invariant linear subspaces that correspond to the numbers 2.3, we get

9"

and L,

55

LIX, (i = 1, ... , n). Agregating the coordinates likewise in Q

n

F(x) = Lx

+ rl.. (-I

where L Let i

= diag E

rl..

'rre, PaX

(x

E

E,

0' E

Z~),

(2.17)

Irrl-2

(L .. ... , Ln).

{I, ... , n} and

0'

=

(0'1, ... , 0'") E

resonaTll if the equality (2.3) is valid.

z~.

The pair (i,

0')

is

said to be

45 2.16. Lemma. Let Q and q be positive illlegers, 2 ~ q ~ Q, and FE Diffo'(E) be a

diffeomorphism given by

q

= Lx +

F(x)

where P<3' s

E

E

P<3'(XI ,

{I, ... , n}, 't

9. . .

<'t,

... , E

Xn; X),

Z~

and

L

PvX<3' 1<3'1-1

+ j(x)

(x

E

IE

CJ(E, E), and 1lf{x)1I

l'tl

= q. q

= O(IIXll q)

coordi1lllle change y

FCy)

= x + Jz;xTe•

= Ly +

"

as

IIXIl ...

O.

Let

the pair (s, 't) is non-resolUUll (i.e.,

then there exists a polynomial h: E PT(XI ,

9»,

(2.18)

E),

... ,

Xn; X.) such that the

brings the dU/eomorphism (2.18) to the form

q

L L p;y<3'e, + lCy) '-I 1<3'1-1 (', <3')_(.,

cy

E

E),

(2.19)

T)

where

1E

CJ(E, E) and IIlCy)1I

= O(lIyllq)

• Note first that the mapping y

x

= !{ICy)

as

lIyll ... O.

= x + ~Te.

is a local diffeomorphism.

denote the inverse transformation. It is easy to verify that !{ICy)

+ O(lIy II q)e..

Perform the change of variables in formula (2.19): n

FCy)

= Lx + Lh!xTe. + "

q

L L

+

q

L L p!x<3'e, '-I 1<3'1-1 (', <3')"(., T)

1'P;(x + Jz;xTe.)<3' - p!x<3']e, + lex + Jz;xTe.).

'-I

1<3'1-1 (',<3')-(.,T) On the other hand,

+

~(F(x»Te.

= Lx +

Q

r r

n

F(x)

'-I

p!x<3'e,

1<3'1-1

=Y-

Let

JJ!yTe•

46 Q

+ ./{x) +

~(Lxfc es

+

[h~(Lx

L PaXa- + j{x»T: - h~(Lx)T:]es.

+

la-I -2 Write down the equation (2.20) Then q

n

Lh~T: es

+

L L

p!.(x

+ h~T: est

- p!xa-]e t

t-IIa-I=2 (t , a-);t (s ,

T:)

(2.21)

q

+

[h~(Lx

L PaXa- + j{x»T: - ~(Lx)T:]es'

+

1a-1-2 Consider the auxiliary equation (2.22) Let !fs,T:: PT:(X 1, follows:

... ,

Xn; Xs) -+ PT:(XI, ... , Xn; Xs) [ !fs,T:'P]xT: = L'I s 'P (Lx)T:

+

be the affine operator defined as

L'I S T: s P-r:X .

(2.23)

It is not difficult to verify that the spectrum of the linear part of !fs,T: lies on the

circle of radius r = exp (-

9s

+
;t

1 centered at the origin. Therefore the

operator !fs,"C has an attracting or repelling fixed point evidently a solution of equation (2.22). Put

~.

The polynomial

q

I(Y)

= ./{I/J(y»

+

[h~(LI/J(y)

+

L Pa-(I/J(y»a- + ./{I/J(y») la-I =2

T:

~

is

47 n

q

L L

- h~(L/J(y)>""]es -

1P;y'" - p;(I/J(y»""]e,.

(2.24)

'-I Icrl-2 ('. cr)"(s. T)

It is easy to show that inserting the polynomial

~

and the expression (2.24) into

equation (2.21), we obtain a true equality. Besides, as lIyll -+ O.

1E

CJ(E, E) and n](y)1I = o(lIyllq)

2.17. Theorem. Let l

be the Q-jet of a CJ smooth diffeomorphism from E into itself

with the origin 0 being a fixed point. Then there exist a diffeomorphism G E j'Q, a neighbourhood U of 0 and an irifinitely smooth change of variables, t: (U, 0) -+ (E, 0), such that Q

t· 1

•

G

0

t(x)

= Lx +

L PaX

cr

(x

E

U),

(2.25)

Icrl-2 and, moreover, p; .. 0 implies at

. The expression (2.25) is called the

resonant normal form of the j'et j'Q . • This follows from the arguments used in the proof of Theorem 2.9. The only difference is that now we must use Lemma 2.16 instead of Lemma 2.8.

§ 3. Polynomial Nonnal Forms

In this section, we establish some sufficient conditions for the finitely smooth conjugacies of vector fields in the vicinity of a rest periodic orbit. These results allow to reduce the problem on normalization vector field to the same problem for the associated jet, which was discussed also consider normal forms of diffeomorphisms near a fixed point.

existence of point and a of a smooth in § 2. We

Vector fields in the neighbourhood of an eqUilibrium 3.1. Definition. Let Q, K, and k be positive integers, k:s min {Q, K}. Let I; and

11

48 be mappings from E to E of class

order (Q, k)

at

CC.

= O(IIXIIQ-P)

lIif(x) - if'll(x)1I as IIxll

~

The mappings ( and 'II are said to have COlllact of

the origin if

= Q :5 K, ( and if DP(O) = vP'II(O)

O. For k

(p

= 0,

1, ... , k)

'II have contact of order (Q, k)

II

(Q, Q) at the

(p = 0, 1, ... , Q). In this case, the mappings origin if and only ( and 'II are also said to have contact of order Q (see Definition 2.1).

y = Ay

3.2. Notation. Let

be

hyperbolic vector field;

let v .. ... ,

Vd

be

the

eigenvalUes of the operator A. As in the previous section, let 9.. ..., 9 n be all distinct values contained in the collection {Re Vt: i = 1, ... , d}. By the hyperbolicity assumption, 9 t '" 0 (i = 1, ... , n). Introduce new notation for the numbers 9 ..... , en

as follows: -

~l <

Given a number a

E

{9 11 ... , en}

= {- ~lI

... < - ~I < 0 <

< ... <

/oIm'

m+ I

where

= n.

IR, let [a] denote, as usual, the integral part of a.

3.3. Theorem. Let k ~ 1; ~ and 'II be D~(O) = D1l(0) = A. Set Qo ;;; Qo(k)

/oil

... , - ~I' /oil' ... , /oIm }

~l + k ( X7 /oIm = [ XI +

If ( and 'II have COlllact of order diffeomorphism conjugating

~

and

CC smooth vector fields on E;

1 )] +

~(O)

~l + 1)] + 2. ji'j' + k ( /oil

[ /oIm

= '11(0) = 0; (3.1)

(Qo, k) at the origin, then there exists a local

CC

1).

• For a proof, see Chapter IV, Theorem 2.7.

3.4. Theorem. Let k ~ 1; ( and 1) be CC smooth vector fields on E having a nodal rest poilll (for definiteness, of stable type) at the origin. Put (3.2)

49

if ~ and

71 have contact of order (QI> min {k, QI}) at the origin, then ~ and conjugate in the vicinity of the origin .

71

are c;k

• Apply Theorem VI.1.3 to the case where A is a rest point.

3.S. Theorem. Let K and k be positive integers; ~ be a vector field on E of class c;k with the origin being a saddle rest point; A = D~(O). if K 2:: Qo(k) (see (3.1», then the vector field ~ can be reduced in the vicinity of the origin I7y the aid of a c;k transformation to the resonant polynomial normal form n

00

y = Ay +

(y

E

E) •

(3.3)

• Let ~ E r~(E) and K I: Qo(k). By Theorem 2.9, there exists an infinitely smooth coordinate change y = !pCx) reducing the Qo-jet of the vector field ~ to the resonant normal form (3.3). With respect to the new coordinates, the vector field ~ takes the form = ~(y) II !P.~(y), and since the vector field (3.3) belongs to the jet lo(~), the vector fields (3.3) and ( have contact of order Q at the origin. By Theorem 3.2, the

y

vector fields ( and (3.3) are locally c;k conjugate. Hence, ~ and (3.3) are also locally

r::c conjugate with

one another.

3.6. Theorem. Let k be a positive integer; ~ be a vector field of class c;k having the origin as a nodal rest point; A. D~(O). if k I: QI (see (3.2», then the vector

field ~ can be reduced in the vicinity of the origin I7y means of a c;k transformation to the resonant polynomial normal form n

01

Y = Ay + E E p;yf1'e,

'-I

(y

E

E).

(3.3)

1f1'1-2

• This statement follows from Theorems 2.9 and 3.3. 3.7. Remark. Theorems 3.5 and 3.6 give sufficient conditions for reducibility of a

50

vector field in the vicinity of a hyperbolic rest point to the resonant polynomial normal form via finitely smooth changes of variables. The condition K 2: Qo(k) of Theorem 3.5 involves the numbers K, the smoothness of the vector field, and k, the smoothness of the conjugation map, as well as the spectrum of the linear approximation operator. Let the number k be fixed. Because K 2: Qo(k) , formula (3.1) shows that the number K strongly depends on the value ~z Ilm ~z Ilm} max { ;;-. X-' X-';;- , "I

I

1"1

which might be called the spectral spread of the operator A. The situation in Theorem 3.6 is different: the smoothness of the normalizing mapping coincides with that of the vector field. The following reason can partly account for this difference: in the saddle case, resonances of arbitrarily high orders may occur, whereas in the nodal case resonances of order greater than QI are absent (see Remark 2.13(2». We note in passing that Qo(k) :s 2

[ (2k + 1) max {Az, {A It mln

Ilm} ] } III

+ 2 II

Q.(k).

(3.5)

3.S. Example. It should be pointed out that there exist smooth vector fields which cannot be reduced near a hyperbolic rest point to polynomial form by any transformation of class

d.

For instance, the following

C system

x = 2x, Y = y + xz(i + £-rl/3,

.z

= - .z

(3.6)

is not locally C conjugate with a polynomial vector field. Suppose the contrary holds. By some results presented below (see §§ 4-8), the polynomial normal form corresponding to the linear part of (3.6) with respect to the class of all C transformations is as follows: (3.7) = 2u, = \I + auw, W= - w, a = const.

u

v

By the assumption, there exists a local x

= j(u,

v, w),

y

conjugating (3.6) and (3.7). Clearly

C diffeomorphism

= g(u,

v, w),

111(0, 0, 0)

z

=

III

= h(u,

v, w)

(0, 0, 0). Since III transfers the

51

stable (unstable) manifold of the point (0, 0, 0) with respect to the system (3.7) to the corresponding manifold for the system (3.6), one has

=0,

j{O, 0, w)

h(u, v, 0) Because

~

=0;

g(O, 0, w)

(3.8)

O.

==

(3.9)

is a local homeomorphism, it follows from (3.8) that h(O, 0, w) '" 0

if w '"

o.

(3.10)

Write out the solutions of (3.6) and (3.7): t

[y

+ xz

I (ile

4t

+ le-2tr l13 dt] et ,

o z(t)

= ze-t ; u(t)

=

lie

U

v(t) = (v

,

+ auwt) e,t wet)

~

= we .

Let u and w be fixed sufficiently small non-zero numbers. The map solution

!p(t)

=

(ui t , auwttl, we-t)

of (3.7) with initial condition

~

!p(0)

transfers the

=

(u, 0, w)

to a solution of (3.6). Hence 2t

-t

t

j{1Ie , auwte , we )

= f(u,

g(ue2t , auwtet , we-t )

=

{

2t

0, w) e ;

g(u, 0, w) (3.11)

t

+ j{u, 0 ,w,'\h(u, 0 ,w,,\

I [g2(u,'0 w,e,\

4t

+ h2(U,

0, w'\e, 2t ]-1/3 dt} et .,

o (3.12) Let {tn } -+

+

III.

Denote

-2t

U"

= lie

(3.13)

n.

Then 2t

t

-t

(u"e , aU"wte , we )

= (ue 2(t-tn >,auwtet-2t ",

t

we-).

52 Replace, in the identity (3.12), t by tn' and U by Un' then

(3.14) tn

+ ft.u n'

0 , w'\h( , un' 0 ,w,~

J

[g2(Un"

0 w,e \ 4t

+

h2(Un , 0, w~e-2tl-\13 dt} etn • ,

o

Since g(O, 0, w) = 0 (see (3.8», one has (3.15) Further,

and by using the equalities (3.11) and (3.13) one obtains

By virtue of (3.15), the equality (3.14) becomes

+ ft.u,

-tn

auwtne

-tn

,we

tn -tnJ

)h(um 0, w)e

2 [g (U m

4t

0, w)e

o

The following estimates are valid: tn

In

==

J[g2(U o

m

0, w)e4t

+

h2(Uno 0, w)e-lt r\13 dt

(3.16)

53 tn

S

J[h(u

n , 0,

w)r2l3elt13 cit

const./tn13 ,

s

o since

Ii m h(uno 0, w)

= h(O,

0, w)

;I:

0

n-++oo

by (3.10) (recall that

W;l:

0). Hence it follows from (3.16) as n -+ g(u, 0, 0)

00

= 0,

that (3.17)

therefore

(3.18)

-t

Divide both parts of the equality (3.16) by tnt n, then by force of (3.18) one obtains

g~(u, 0, O)auw

+ g:"(u, 0,

O)wt~ 1

+

o(t~ I) (3.19)

The left-hand part of the last equality and the first term of the right-hand part are bounded as n -+ 00 (see (3.13». Let t E [/n/4, 3/n/4], then by (3.1S) and (3.13) one gets

Consequently, (n -+ .).

It therefore follows from (3.19) that J(u, 0, O)h(O, 0, w)

= o.

54 Since w '" 0, one has h(O, 0, w) '" 0 by (3.10), hence, j{u, 0, 0) = O. Using (3.9) and (3.17), one concludes that lII(u, 0, 0) = (0, 0, 0). Because U"# 0, q,(O, 0, 0)

=

(0, 0, 0)

and q,

is locally one-to-one, we get a contradiction.

This contradiction

shows that the vector field (3.6) cannot be reduced to polynomial form by a local C diffeomorphism. This example answers affirmatively the question raised by A.D.Myshkis. A similar example of a 4-dimensional vector field was proposed by Belitskii [1]. 3.9. Remark. Let us remind the reader that a vector field

~

is said to be

Q-delermined with respect to a group G of coordinate changes if every vector field 11, belonging to the Q-jet of the vector field ~, lies in the G-orbit of ~. From this point of view, Theorems 3.3 and 3.4 affirm that every C" smooth vector field with saddle (nodal) linear part at the rest point 0 is Qo-determined (Qcdetermined) with respect to the group Difta'(E).

Vector fields near an equilibrium of general type 3.10. Statement of the problem. If a rest point of a smooth vector field is not hyperbolic, then the phase flow in the vicinity of this point is not finitely determined even with respect to the group of homeomorphisms. For instance, an equilibrium of center type of a planar linear vector field can be transformed into a focus by polynomial perturbations of arbitrarily high order. Therefore, the problem on topological (and, moreover, smooth) normalization of a vector field near an equilibrium of general type cannot be reduced to the normalization of finite order jets. By Theorem 1.4.8, the dynamical behaviour in the vicinity of such a rest point is to a great extent determined by the properties of the dynamical system restricted to the center manifold. When investigating rest points with pure imaginary spectrum, it is reasonable to discuss either normalization of finite order jets or topological normalization of vector field families of finite co-dimension (see Arnold and n'yashenko [1]). These problems are outside the scope of this book. Our aim is to investigate (up to finitely smooth changes of variables) dynamical systems in the vicinity of the center manifold. Therefore, in the sequel, we shall assume that all changes of coordinates are of the form y = x + h(x) , where h vanishes along the center manifold. We shall obtain normal

55 forms expressed as polynomials in the hyperbolic variables with coefficients being functions defined on the center manifold. 3.11. Preliminaries. Let t; e r~(E), A

linear subspaces such that E

= F!' III E,

= Dt;(O).

Let

F!'

and E be the A-invariant

the eigenvalues of the operator Ah

= A IF!'

lie

out of the imaginary axis and the eigenvalues of the operator Ac = A IIf lie on the imaginary axis. Let 9\1 ••• , 9 n denote all distinct values of the real parts of the eigenvalues of the operator

Ah ;

let

XI' ... , Xn

be the corresponding Ah-invariant

linear subspaces. By Theorem 1.4.7, there exist coordinates of class principal part ~ of the vector field to the form ~(x, z)

=

(A,.x

+

F(x, z), AcZ

+

+

p(z)

R(x, z»

where the mappings F, R and p are of class

CJ,

(x e

F(O, z)

F!',

Z e

= R(O,

CJ

bringing the

E), z)

= 0,

(3.20) p(O)

= 0,

=

O. Without loss of generality assume that the supports of the functions F, R and II (X, z)1I :s c, where c is a small positive number to be specified later.

Dp(O)

p are contained in the ball

3.12. Definition. Let / subspace of E; /I W = g I W. order Q along W, if

and g be

CJ

smooth mappings from E to E; W be a linear

The mappings / and g are said to have venical contact l!f(y) - g(y)1I

=

0/

0([P(Y, W)]Q)

uniformly on Easy tends to W (here p is a metric on E). This is an equivalence relation. We define venical Q-jets of mappings from E into E with respect to W to be the corresponding equivalence classes. 3.13. Example. In order to illustrate the significance of the above notion, let us return to Example 1.3. The system (1.1) contains only one resonant monomial yz. Since the linear part of (1.1) is diagonal, it is clear that this system admits no

C

linearization. Moreover, we know that there is no C linearization. The fact that the 2jet at the origin (or, in other words, the quadratic term yz) turns out to be an obstacle to

C

linearization seems to be a surprise. We shall see in a moment that the

56 problem on linearization of (1.1) near the origin is equivalent to the same problem in the vicinity of the invariant manifold z = O. Therefore the monomial yz, the obstacle to

C

z

= O.

linearization,

is

best

to regard as vertical

I-jet with respect to the plane

Next we shall give another proof of the fact that

C

(1.1) is not

linearizable

(based on the notion of vertical jet).

Suppose, to obtain a contradiction, that (1.1) and (1.2) are locally Then, as it was shown in subsection 1.3, there exists a x

and .,(0, w)

= !p(u,

invariant manifold

=

u, y

=v+

= O. The vertical = 0 is of the fonn

0)

w

C conjugacy

!p(u, w), z

C

conjugate.

of the fonn

=w

I-jet of the mapping ., with respect to the 1/I(u)w, where 1/1 is a continuous function. In

other words, !p(u, w) = 1/I(u)w + A(u, w), where I A(u, w) I to determine the functions 1/1 and A, consider the equation

= o( I wi)

as w -+ O. In order

(I

In what follows, we shall deal with the domain u

E

IR).

> 0, w > O. The last equality implies (3.21)

A(U, wJ~

The equation (3.21) has

-t) -t = A(2t Ill! ,we e •

a particular solution 1/Io(u)

=i

(3.22) u In u. The general solution of

the homogeneous equation

on the half-line u > 0 is .,(u) = CU, where C is a constant. Hence it follows that the general solution of the equation (3.21) in the domain u > 0 can be written as

_ 1

1/I(u) - 2: u In u

+

cu.

By continuity, set 1/1(0) = O. In order to find A, let us first note that Ao(u, w) = UW is a particular solution of (3.22). The general solution of (3.22) will be sought in the fann A(U, w)

= UWJ.L(u,

w) (u

> 0,

W

> 0). Then J.I(u, w) = J.I(uWZ, 1). Thus, the general

57 solution of equation (3.22) in the domain u > 0, A(U, w)

> 0

W

can be written as

= uwt(uwl)

where I is an arbitrary function. Since A(u, w) = o(w) as w -+ 0, we get ~ -+ O. Therefore, we may put 1(0) = O. So we have

x

=

u, y

l

=w

=

+

v

r

In u

+

Cuw

+

UWl(uWZ)

= "'(u,

I(~) -+

0 as

v, w),

(3.23) (u

> 0, w > 0).

Let us show that the function y = ",(u, v, w) is not differentiable with respect to u at u = O. In fact, if u > 0 and u -+ 0, then I(uwl) -+ O. Hence, the map

u ...... UWI(uWZ) is differentiable at the point (0, w), the derivative being equal to O. On the other hand, the function ./(u) = u In I u I (u'" 0), differentiable at u = O. Consequently, the partial derivative exist. We have reached a contradiction.

./(0) = 0, is not a",/au I u-o does not

3.14. Vertical jets. Let us dwell on the notion of vertical Q-jet of the vector field (3.20) with respect to the center manifold W'. Let have one and the same center manifold and

11

MP, and

1;, 11

I; I W

r~(E). Suppose that I; and

E

= 11 I W'.

11

It is easy to verify that I;

have vertical contact of order Q along W if and only if there is a coordinate

system of class CJ such that the function ~ is given by formula (3.20) and the principal part ~ of the vector field 11 is given by the equality ~(x, z)

=

(A,.x

+

G(x, z), A.,.z

where lIG(x, z) - F(x, l)1I IIxli -+ O. For

I;

E

=

r~(E),

+

p(z)

O(IIXllo),

+

ex If',

Sex, z»

E

IIS(x, l) - R(x, Z)II

=

ff),

(3.24)

O(IIXllo), uniformly on

ff as

z

E

let )'$(1;) denote the vertical jet of the vector field I; along

the center manifold W. The vector fields I; and

11

have vertical contact of order Q with

respect to W if and only if D~G(O,

(p

=

I, ... , Q; z e ff,

z)

= D,(O,

z),

D~(O,

z)

= D,(O,

z)

IIzlI:s E). Therefore the vertical jet J~(I;) contains one and

58 only one vector field 11 with Q

~(x.

z)

= (A~ +

L J,r D~F(O. z)?

AcZ

+

p(z)

p-I

(3.25) Q

+

L ftr D~(O. z)?)

(x

E

E'. z E

F)

p-I

3.1S. Theorem. Let l,(f.) be the vertical Q-jet of a CJ+k. smooth vector field f. with respect to the center manifold at the origin. There exist a vector field

11 E

J'?,(f.)

and

~ smooth coordinates bringing the principal part ~ of 11 to the form Q

Q

(3.26)

where Ah

E

CJ+k.·I; t

E

CJ+q; Ptr' qtr E CJ+k.-ltrl; Ah(O) = Ah ; teO) = 0; Dt(O) = Ac;

P;' ;! 0 implies at = (i = 1•...• n); qtr II! 0 implies = O. The vector field (3.26) is said to be the generalized polynomial resonant normal form of the vertical jet J'?,. • The validity of this theorem follows from the lemma below.

3.16. Lenuna. Let K and q be positive integers. K

2:

q. ~

E

r~(E) be

a vector field

given by Q

X = Ah(z)x +

L

P...(z)x'" Itr,1-2

+

rp(x. z).

(3.27)

z=

Q

t(z)

+

L qtr(z)xtr + !/I(X. z)

(x

E

E'.

Z

E

F).

I tr 1-\

y

= Ah(O)y

be a hyperbolic linear vector field;

9\ ••••• 9 n

the real parts of the eigenvalues of the operator Ah(O). Let

E'

be all distinct values = X\

El ••• El

0/

Xn be the

59 corresponding direct sum decomposition into Ah(O)-invariant subspaces;

=

t(O)

0;

be a linear operator with pure imaginary eigenvalues; t E c"+l; Ah, PO', qO", rp, I/J E c" (IT E z~, 1:s IITI :s q); IIrp(x, z)1I = o(lIxll q); III/J(x, z)1I = o(lIxll q) Dt(O): g= -+ g=

uniformly on g= as IIXIl -+ (1)

~

If

s

o.

{1, ••• , n}, or

E

E

z~,

lorl

=q

and 9. "" <1',9>, then there exists a map

= y + ~(w)y'f:e••

c"(g=, P'f:(XIl ••• , Xn; X.» such that the change of variables x reduces the vector field (3.27) to the form

E

=w

z

n

y = Ah(w)y

q

L L p;(w)yO"et

+

t-I

+ ~(y,

w),

1.,.1-2

Ct. "'J""C •• 'f:)

(3.28)

q

w= t(w)

L qO"(w)yO" + ~(y, w)

+

(y

E

E',

w

E

g=),

10"1-1 where ~

~

E'),

c;K(E,

E

c"(E, g=); 1I~(y, W)II

E

= o(lIyllq);

1I~(y, W)II

= o(lIyllq)

uni-

formly on g= as lIyll -+ O. (2) g'f: y

E

If

l' E

z~,

11'1

=q

and

<1', 9> - 0,

then

there

c"(g=, P'f:(X1, ••• , Xn; g=» such that the change of variables reduces the vector field (3.27) to the form

exists

a

map

z = w + g'f:(w)y'f:,

=x

q

L pO"(w)yO" + ~(y, w),

y = Ah(w)y +

10"1-2

w= t(w)

(3.29)

q

+

L qO"(w)yO" + ~(y, w)

(y

E

E',

WE

g=),

10"1-1 O"""'f:

where ~

nyn

E

c;K,

~

E

c";

1I~(y, W)II

= o(lIyllq);

1I~(y, W)II

= o(lIyllq)

-+ O•

• (1) Applying the first of the above two transformations, we get

uniformly on g= as

60

n

= Ah(y) +

q

E E p;(w)y'"e, + ~(y, w) '-I 1.,.1-1

(' • .,.).( •• "&')

q

+ D~(w)[I(w) +

E q.,.(w)y'" + ~(y, w)]y"&'e. 1.,.1-1 n

+ Dy[~(W)y"&'][Ah(W)Y +

q

E E p;(W)y'"e, + ~(y, w)]e. '-I 1.,.1-1

(' • .,.);1:( •• "&')

q

= Ah(w)y + Ah(W)[~(w)y"&'es] +

L p.,.(W)[y + ~(w)y"&'est' 1.,.1-1

q

W=

I(w)

L q.,.(w)[y + ~(w)y"&'e.t' + 1/J(y + ~(w)y"&'esl w).

+

(3.30)

1.,.1-. We have to use (3.30) in order to determine the unknown functions ~, ~, ~. Consider the auxiliary equation

where 71:.: If' .. X. is the natural projection. We shall solve the equation (3.31) in the class of functions ~ Define a mapping

E

c"(~; P"&,(X, ,

according to the following rule:

"'1

Xn; XIII»'

61

Consider the characteristic system for the equation (3.31): (3.32) System (3.32) determines a non-homogeneous linear extension with respect to the vector bundle

As it follows from the proof of Lemma 2.8, the real parts of the eigenvalues of the operator l!I.,T • i.,T(O) are all equal to e. - 'rIel - ... - 'r"e" ;6 O. Taking into account that the operator Ac. Dt(O) has only pure imaginary eigenvalues and choosing the number £ > 0 to be sufficienUy small, we conclude that this extension is K-con-

c<

tracting (in the positive or the negative direction). By Theorem A.33, there exists a smooth invariant section of the extension (3.32). It is not difficult to verify that this section satisfies the equation (3.31). Taking the solution ~ of the equation (3.31) and repeating the computations carried out in the proof of Lemma 2.8, we get the required functions q; and ~. (2) In order to determine the functions gT' q; and ~, consider the following relations: q

y = Ah(w +

gT(W)yT)y

+

L p~(w + gT(W)yT)y~

1~1-2

q

+

,,(y, w

+

gT(W)yT)

= Ah(w)y

+

E p~(w)y~ + q;(y, w); 1~1-2

q II

t(w)

+

E q~(w)y~ + ~(y, w)

+ D~T(W)[t(w)

62

q

L qv(w)yv + ~(y, W)]yT

+

+ Dy[gT(W)yT][Ah(W)y

q

+

L Pv(w)yV + ~(y, w)]

= I(W + gT(W)yT)

Ivl-l q

+

L qv(w + gT(W)yT)yV + 1/I(y, w + gT(W)yT).

lvi-I To find the function gT' we construct the following auxiliary equation:

(3.33)

The solvability of this equation in the class of functions Xn;~»

E

c«~, PT(Xh ... ,

can be proved in the same way as the solvability of equation (3.31), by using

the relation G(w

gT

< 't, 9>

+ g(W)yT)

- G(w)

;I: II

O. Let us note that if G and g are G1(y, w) is also

c< smooth mappings, then

c< smooth and, besides, IIG1(y, w)1I = O(lIyIlITI-I)

as lIyll -+ O. Taking into account this remark, we easily find the functions ~ and ~ with the required properties. Applying successively this lemma for q

= 1,

... , Q, we get (3.26).

3.17. Definitions. Let Q, K and k be positive integers, k:s min {Q, K}. The vector fields (3.20) and (3.24) are said to have venical contact of order (Q, k) along their common center manifold at the origin if there exists a positive number c such that

IIDS[R(x, z) - Sex, z)] II :s C IIXIIQ-.S' (s

= 0,

1, ... , k;

II (x, z)1I :s E).

.{3.34)

63 If k = Q :s K. then the notion of vertical contact of order Q is equivalent to the notion of vertical contact of order (Q. k) Ii (Q. Q). In some cases it is useful to sharpen the definition of vertical contact. Namely. replace the inequalities (3.34) by the conditions IID~~[F(x. z) - G(x. z)]11 :s c IID~~[R(x. z) - Sex. z)]11 :s c

(p

If K

= Q + k.

E Z+.

q

E Z+.

P

IIXllmax{Q-q,O}.

(3.35)

IIXllmax{Q-q,O}

+ q = O.....

k;

II(X. Z)II

:s E:).

then (3.34) and (3.35) are equivalent.

3.1S. Theorem Let k be a positive integer. ~ and 11 be c!' smooth vector fields with principal parts represented in the form (3.20) and (3.24). Define the numbers >-lo >-1. "'I. and Qo(k) for the operator A Ii A" as it was done in 3.2 and (3.1). U ~ and 11 have vertical contact of order (Qo(k). k) along the center manifold at the origin. then

"'m

they are c!'conjugate in the vicinity of the origin .

• The proof will be given in § 2 of Chapter VI (see VI.2.8). 3.19. Theorem. Let k be a positive integer. ~ and 11 be c!' smooth vector fields with principal parts of the form (3.20) and (3.24). Suppose that all the eigenvalues of the operator Ah lie in one and the same side of the imaginary axis (for definiteness. in the left side). Define the number QI by formula (3.3). U ~ and 11 have vertical contact (in the strong sense) of order (QI' k) along the center manifold at the origin. then they are

c!'conjugate

with one another near the origin .

• The proof will be presented in § 1 of Chapter VI (see VI.1.6). 3.20. Theorem. Let k 2: 1. ~ E r~(E) and K 2: Qo(k) + k. There exists a c!' smooth coordinate system near the origin bringing the principal part ~ of the vector field ~ to the generalized resonant polynomial normal form

64

(3.36)

• Let Q = Qo(k). By Theorem 3.15, there exist a ~ smooth coordinate system and a vector field 11 E J~(I:.) such that the principal part ~ of the vector field 11 with respect to this coordinate system is of the form (3.26). In virtue of Theorem 3.18, the vector fields I:. and

are locally ~ conjugate.

11

3.21. Theorem. Let K and k be positive illlegers and I:.

the numbers

9 ..... , 9 n

have the same sign (for definiteness,

E

9,

r~(E). Suppose thal all

< 0 (i

=

1, ... , n».

Define the number Q1 by formula (3.3). Assume that K!! Q1 + k. Then there exists a ~ smooth coordinate system near the origin which brings t:. to the generalized resona1ll polynomial normal form

°1

L Pa-(z)ra-,

X = Ah(z)r +

.i:

= tCz)

Cx

E

E', z E F)

C3.37)

la-I =2 • The validity of this theorem follows from Theorems 3.15 and 3.19 since
E

z~,

Icr I

2:

9>

~

0

1.

Vector fields near a hyperbolic periodic orbit 3.22. Statement of the problem. Let M be a smooth manifold of dimension (d be a positive integer and

I:.

E

+

1), K

rKCTM)' Let S be a closed trajectory of the vector field

1:., with prime period w. By Theorem 1.3.9, there exists a w-demiperiodic

c<

pseudochart for S, such that the principal part of the local representative cr.1:. of the vector field I:. with respect to this pseudochart takes the form (U, cr)

1 I:.a-Ca, x) = 1

+ Q(a, x), (3.38)

E'!Ca,

x)

= Ax +

RCa, x)

(9

E

IR,

X E IRd ),

65 where A. e L{rRd, IRd ), Q(a, 0) = R(a, 0) = DIl(a, 0) = 0 (a e IR), and, besides, the functions Q and R are 2w-periodic in the first argument. In other words, the relations (3.38) define a vector field on the Cartesian product of the circle IR I 2wz and the linear space IRd. Our purpose is to obtain the resonant normal form of a vector field near a periodic orbit. Suppose the periodic orbit S to be hyperbolic (see Definition 1.3.10). In other words, the spectrum of the operator A. entering the normal form (3.28) is assumed to lie ai' ... , en, AI' Al , "'I and Ilm as in

out of the imaginary axis. Define the numbers subsection 3.2.

3.23. Theorem. ut K and k be positive integers, ( be a vector field of class having a hyperbolic w-periodic orbit S. Define thl! number Qo(k) by formula (3.1).

c:c 1/

K II:: Qo(k) + k then there exists a IIHkmiperiodic c" pseudochart (U, cr) for S such thai thl! principal part of the representative cr.E' of thl! vector field E' with respect to this pseudochart takes the form

ooele)

E';(9, x)

=1+

r

q".(9Pt',

1".1-1 °oele)

E';(9, x)

= Ax +

r

p".(e)x"'.

1".1-2. Here p". and q". are 2w-periodic fimctions of class (i = 1, ... , n); q". iii! 0 implies = O.

c";

p;. 0 implies at

= < cr, e>

3.24. Theorem. Suppose thai the conditions of thl! previous theorem are fulfilled. Suppose in addition the spectrum of the operator A. to lie to thl! left of thl! imaginary axis. Define the number Q I by formula (3.2). 1/ K II:: Q I + k then there exists a r.rdemiperiodic

c" pseudochart

(U, cr) such that thl! principal part of creE' gets IhI! form

66 QI

~~(9, x) = 1,

~;(9, x)

=

Ax

L Per(9)xer,

+

lerl-2 where Per are 2w-periodic functions of class (i = 1, ... , n).

C<,

and p~ ~ 0

implies



9t

The proofs of Theorems 3.23 and 3.24 are left to the reader. One can also formulate and prove a theorem on the generalized polynomial resonant normal form of a vector field near a periodic orbit of general type.

Diffeomorpbisms near a fIXed point 3.25. Statement of tbe problem. Let E be a finite dimensional linear space,

f E -+ E

be a diffeomorphism of class CC with AO) = O. Let X and W be DAO)-invariant linear subspaces such that E = X iii W, L DAO)IX is a hyperbolic operator (i.e., a invertible operator having the spectrum out of the unit circle), and the spectrum of the operator DAO) IW lies on the unit circle. Let dimX = d; VI' ••• ' V d be the eigenvalues of the operator L and 9 1, ••• , 9 n be all distinct numbers contained in the collection {In IVt I: i = 1, ... , d}. By our assumption, 9 t *' 0 (i = 1, ... , n). Define the numbers ;\1' ;\t, fJ\I fJ m , as in subsection 3.2. Let X\I ... , Xn be L-invariant linear subspaces corresponding to the numbers 9\1 ••• , 9 n (i.e., the spectrum of the operator LIXt lies on the circle of radius exp 9 t ). Let us consider the problem on reducing the diffeomorphism f near the origin to the generalized resonant polynomial normal form. We give only two theorems similar to Theorems 3.20 and 3.21.

=

3.26. Theorem. Let k

2:

/II: E

1 =/II

has the form

f

0

/11.1

f

-+ E of class

diffeomorphism 0

1,

E

Dif~(E) and

C<-

K

2:

such that /11(0)

Qo(k)

=

+

0 and the diffeomorphism

(x

ler 1=2

lerl-I

There exists a

k.

E

X,

WE

W),

67 where L h , rp, p~ and q~ are functions of class eft, p;. 0 implies 9, = q~ • 0 implies = 0, Lh(O) = L, 9'(0) = 0, D9'(O) = Dj{O) I W.

<0',

9:;:',

3.27. Theorem. Let k 2: 1 and f E Dif~(E). Suppose all the numbers 9 1, ••• , 8 n to have the same sign (for definiteness, 9, < 0 (i 1, ... , n». Define the number QI I1y formula (3.2). if K 2: QI + k then there exists a diffeomorphism ~: E -+ E of class

=

c!

such that ~(O)

=0

i!E

and the diffeomorphism

~

0

f

0

~-I takes the form

QI

j(x, w)

L p~(w)x~, 9'(w» ,

= (Lh(w)x +

1~1-2

where L h , p~, q~, 9' are of class

11',

c!,

9'(0)

= 0,

Drp(O)

= Dj{O) I W,

p;

iE

0

implies

= <11',9>.

3.28. Remark. In the hyperbolic case (i.e., W = {OJ), the requirements K 2: Qo(k) 2: QI + k in Theorems 3.26 and 3.27 may be replaced by K 2: Qo(k) and K 2: QI, respectively.

+ k and K

§ 4. Simplification of the Resonant Nonnal Fonn

via Finitely Smooth Transfonnations

In this section, by considering a number of examples, we discuss the problem on deleting some monomials out of the resonant polynomial normal form by means of finitely many times differentiable changes of variables. 4.1. Model example. Consider the vector field •

Z

",l

= 9Z + Xl

",1_,,91 ... XI YI ...

13m

Ym , (4.1)

Xt = - ;>.,X, where XII

... , Xl' YI' ... ,

Ym'

(i

= 1,

Z E IR, ;>'l

... , l), Yj

= ""jYj

> ... >

> 0,

;>'1

(j

=

1, ... , m),

""m > '" >

""1

> 0;

9

= - «l~l

68 I I m « • ./3 _ «l «1 • ./3 1 13 m - ... - a. ~I + 13 JlI + ... + 13 Jlm & <13, Jl> - , I.e., x / - Xl ... Xl/I'" Ym is a resonant monomial. It is easy to show that system (4.1) cannot be linearized by any

cI «I + 1131

smooth transformation. On the other hand, according to the Grobman-Hartman

Theorem, (4.1) is topologically linearizable. We would like to obtain an estimate from below for the highest smoothness class of linearization maps.

x«1 satisfies the condition S(k) (see Samovol [1-3]) if at

We say that the monomial

least one of the following inequalities r k a.I ~I + .. . + a. ~r > ;>'r

(1

:II

r

I),

:II

(4.2) 13 I JlI

+ ... +

13• Jl. >

kJl.

(1

holds. Let us show that the condition S(k) guarantees near the origin,

:II

s

c!

:II

m)

linearizability of system (4.1)

i.e., the existence of a c! conjugacy between (4.1) and the linear

system

w

= aw,

Ut

= - ~tUt

(i

= 1,

... , I),

vJ

= JlJvJ

= 1,

(j

... , m).

(4.3)

For definiteness, assume that the inequality

holds. Select a number M such that M

Z

= W + ~ u«"

>

k~r'

We shall check that the change of variables

r

In (

L1

M/~t

Ut 1

),

Ut

= Xt

(i

= 1,

... , I),

t-I

VJ

is of class

c!

Denote v

= YJ

(4.4) (j

= 1,

... , m)

and conjugates (4.3) and (4.1).

= a.I ~I +

•••

+

a.r ~r and consider the auxiliary differential system

(I

Since v

>

= 1,

... , r).

(4.5)

k;>'r and by Theorem A.33, the extension (4.5) has a uniquely determined local

69 invariant section function

Ip

=

Ip(UI'

Ip

•••

,ur ) of class c! . It is easy to check that the

= ~ "~I

... "~rln (

rl"tIM/~t r

)

t-I

satisfies the system (4.5). Hence it follows that (4.4) is of class c! (one can also refer to Proposition 5.13 below). A straightforward calculation shows that (4.4) conjugates (4.3) and (4.1). Let us note that the transformation (4.4) agrees with the identity map when

"r

M/~I

•

M/~r

restricted to the surface I"II + ... + I I = 1. This property uniquely determines the change (4.4) because almost all trajectories of system (4.3) (more exactly, all trajectories except those in the subspace this surface exactly once.

"I

= ... =

u,.,

=

0) intersect

Samovol [3] has shown that the condition S(k) is sufficient for c! linearizability in a much more general setting than (4.1). If I = 0 or m = 0 in (4.1), then the estimate (4.2) of the smoothness class of a linearization cannot be improved. 4.2. Example. Consider the system

(4.6) The monomial

x8y~y~

is resonant and satisfies the condition

S(7).

By the above

arguments, the system admits a C7 smooth linearizing transformation, namely

4.3. Example. Let 11 be a positive integer. Let us show that the system

x = x, y = ny + x" is C'.I linearizable but does not admit C' linearizations. A conjugation between the initial system and the linear system

u

=",

v

= nv

70 can be chosen, without loss of generality, in the form x

Clearly,

= u,

Y

=v+

rp(u).

satisfies the equation

rp

nv + ntJI + un

= nv +

T

dcp u

du

'

or

u dcp

du

Hence, rp(u) =

un

In lui

+

= nrp +

un.

cun , c = const, completing the demonstration.

4.4. Example. Let us return to Example 4.2. The monomial

x8y~y~

satisfies the

condition S(7) but does not satisfy the condition S(8). Nevertheless, we can C linearize the system (4.6), In order to do this, introduce an additional variable

Yo = y~. Then

i=

15z + Y~M, x

= - x,

Yo

= Yo,

YI

= Yh

Y2

= 3Y2'

The monomial y~~y~ satisfies the condition S(8) for r = 1, namely, 4· 1 In accordance with subsection 4.1, the system (4.7) admits a Z

where

Vo

=W + =

V2U 2 •

110

4J.i

I~ ,

= x,

C

(4.7)

+

5· 1

>

linearization

= YJ'

= 0,

10 (vo

+

VI

Hence, we get a

C

smooth linearizing transformation for (4.6):

Vo 1 2

In

U

vJ

8· 1.

(j•

1, 2),

Thus, by introducing one extra monomial variable, we produced a new linearization map (as compared with (4.4» and thereby improved the smoothness class. This motivates the following 4.5. Dermition. A monomial xOl.y'l is said to satisfy the condition MS(k) with respect to system (4.1) if there exist multiindices

C( E

z~,

Kt

E

Z':'

(i = 1, ... , p)

and

71

positive integers

(2) letting

<0

and

9,

(519 1

(i

(5'

= 1,

= -
... , p) such that

+

~>

IJ.>


+ ... + (5P9 p < Jeep

or

(i

= 1,

9 p 2: ... 2: 9 1

... , p), we get either

>

0 and

(519 1

9 p :S ... :S 9 1

+ .. , + (5P9 p > Jeep.

xOl.I satisfy the condition MS(k)

4.6. The linearizing transformation. Let a monomial and, for definiteness, let 9 p 2: ... 2: 9 1 > O. Introduce

£,

c,

1

variables ~,

I

= Uz I

p

the

additional

m

Ie,

Ie ,

UI VI'"

divisible by ~~ ... ~!

Z

I

vm

(i

= 1,

... ,

pl.

Then the monomial

1 M

01._

U

13

v In (

1 I I c, c, L. I U z ... u l VI'"

rP

Ie,

m

Ie l

vm I

M/5,

c!

is

),

'-I is a

01._13 U v

. In virtue of subsection 4.1, the coordinate change

=W+ x,

monomial

= u,

(i

=

(4.8)

1, ... , I), YJ

= vJ

(j

= 1,

... , m)

smooth linearizing transformation.

4.7. Remark. As a matter of fact, it is shown in Example 4.4 that the monomial

x·yM 15 1

satisfies the condition MS(8) for £1

= 5,

(5z

= 0,

£z

= 2,

"I

=

(1, 0), "z

= (0,

1),

= 4.

4.8. Example. Consider the system (4.9)

Here the condition S(9) is fulfilled, but the condition MS(IO) does not hold. Nevertheless, there exist linearizing transformations of class

C3 ,

for instance,

(4.10)

72 (this will be shown later in subsection 7.21). 4.9. Remark. The coordinate change (4.10) is of the form (4.8) for a definite choice of non-negative numbers c~ ..... c~. K~ . . . . . K~ (i = 1. 2). satisfying the conditions (1) and (2) of Definition 4.5. but they are no longer integers. as contrasted to the condition MS(k). Thus. it seems reasonable to further enlarge the class of possible linearization maps. Let p be a positive integer. c, vectors with non-negative components

9, • - .> + -

= (i

(c~. .. .• c!)

= 1.

and K, = (K!. . ... K~) be . ..• p) such that all the numbers

0 have one and the same sign. One can directly verify that

the map

(4.11)

x,

= u,

(I

= 1.....

I). YJ

= vJ

(j

= 1.....

m)

conjugates (4.3) and (4.1). We are thus led to the problem of how to determine the smoothness class of the map (4.11). Surprisingly enough. this turns out to be a difficult task. The next section establishes a new condition. l!I(k). which is weaker than MS(k) and nevertheless ensures

c!

smoothness of the map (4.11).

4.10. Example. Up to this point. we considered differential systems (4.1) having the following particular structure: only one component of the vector field. say Z. contains a non-linear term which does not depend on the variable z. If this condition is violated. the situation becomes substantially more involved. In fact. let us consider the system (4.12) = x. y = 2y + xyz. Z = - z

x

and the corresponding linear system

u= u. The system

cd

(4.1~)

V = 2v.

W

= - w.

~4.13)

satisfies the condition S(I). According to Samovol [3]. there exists a

smooth mapping which conjugates (4.12) and (4.13). Let us try to find such a

73 coordinate change. Make the change of variables suggested by (4.4):

~ This is a local

C

= x, 'II = Y -

(.i + l),

XY1. In

l: =

(4.14)

1..

diffeomorphism. The inverse map can be written in the form

x where I "'(~, 'II, l:) I

t = ~,

i

= 0('11)

=~,

y =

"'(~, 'II, l:),

-

i ~2l:2",(~,

= l:,

O. Then

as'll ....

~ = 2y + XY1. - ~ XY1. In (x" + l)

= 2'11

1.

- XY1. - i ryr In

(X"

+ l) (4.15)

'II, 1;)

In [~2

+ ",2(~,

'II, l:)],

~ = - 1;.

Thus, the change (4.14) reduces (4.12) to the form (4.15). The system (4.15) contains only one non-linear term of order 4. One can easily verify that the coordinate change

where A is a suitable real number, reduces (4.15) to a system, which is linear up to order 6. Using this process inductively, we can show that there exists a map x = u, y = v + cp(u, v, w), 1. = w, transforming (4.13) into (4.12). The function cp satisfies the equation 2cp

+

u(v

+

cp)w

= 8IP 8u

u

+

8IP 2v 8V

+

8IP (- w). 8w

The corresponding characteristic system has the form du _ dv _ dw

U -

2v -

_

dIP

:-w - 2IP + u(v + cp)w •

C

conjugation

74

Using the first integrals v = ct u2 and

Hence, rp CI

= vu-2

VW = C2'

we get the equation

= - ctu2 + C3Uc2+2 , where c3 is an arbitrary function in the variables and c2 = VW. Thus, Y = UUW+2~(VU-2, uw). We have to find a function ~ so

that the corresponding coordinate change is of class ~(~, 11)

= ~(1 + r.2)Tl/4.

ct.

this end,

put

Then uw

y

To

uw

= uuw +2vu-2(1 + ~u~'" = v(u4 + ~).,..

It is not difficult to check that the transformation uw X

= u,

= v(u4 +

y

~).,.,

z=w

is a local Ct conjugation between (4.13) and (4.12). Let us consider one more approach to constructing conjugation maps. Introduce the function F(u, v)

is

fulfilled

u2

+~ >

= (u4 +~)

along

the orbits of system (4.13).

0, and let to

surface F(u, v)

= 1,

= - ~ In

+ ~).

to

u4 + ~

(u4

= 1,

suggested by formula (4.4). Note that the equality

= to(u,

be a point with

v, w) denote the moment when (u, v, w) reaches the

vt~

F(uto ,

i.e.,

Let (u, v, w)

II

e4to (u4

+ ~) =

1. Hence it follows that

Assume that the conjugation, when restricted to the surface

agrees with the identity map. Note that U

(u

4

+

2 1/4'

v)

Yo

2to

II

ve

=

V

(u 4

+v

2)

1/2'

Zo

5

we

The solution of system (4.12) is x(t)

= xet,

y(t)

= yeCJ.+XZ)t,

z(t)

= ze-t .

-to

4 2 1/4

= w(u + v)

.

75 Substituting x = xo, y = Yo, z = Zo, 1 = - 10 into the last formula, we conclude that the conjugation map sends (u, v, w) to the point (x, y, z), where

X

If

u

= 0,

= u, v

y

= 0,

1 4 2 (2+uw)!ln(u +v) _

V

= -(u-:4-+-';"-:-)"-I/:-::"Z e then

x

= 0,

y

-

4

v(u

uw

+

_.2 """'1"

v)

,

z = w.

= O.

4.11. Example. Consider the following systems of differential equations:

x = x + rye, u = u,

y = 2y, i = - 2%;

(4.16)

V = 2v, W = - 2w.

(4.17)

Write down the solutions of these systems: (4.18)

U(/)

= uet ,

v(1)

= ve2t ,

= we"2t.

wet)

(4.19) ()

It is easy to check that the system (4.16) satisfies the condition S(2) and does not satisfy the condition MS(3). Nevertheless, the system (4.16) admits a c! linearization. In order to prove this, we use the second method described in Example 4.10 above. We set F(u, v, w) = u20 + U2ZWI2 • Let the point (u, v, w) satisfy F(u, v, w) '" O. Note that F(uet , vi t , we"2t)

= eZOtF(u,

v, w). Therefore 10

= - ~O

In (u zo

+

';ZWI2).

By formula

(4.18) and (4.19), we get x

= u(1 - !.. ;wl' In 10

(u20

+ y2ZwI2»"II2,

y = v, z = w.

It is not difficult to verify that (4.20) is a local diffeomorphism of class can be deduced from Proposition 5.13 below). Using the formula (1 -

0:)"112

= 1 + ~ _!.:1 0:2 + 2

2.4

1· 3 ·5

2.4.6

0:3 _ •••

we conclude that the transformation (4.20) can be written as

(I I < 1)

0:,

(4.20)

c!

(this

76 x

= u + ~o u3 w/- In + _S_ uVw6 16000

(uzo + ~2W11

[In (u20

+ ~2W12)]3

-

8~ uVw· - ... ,

Y

= lI,

Observe that the first two summands in the last expression for (4.11).

c!

§ S. A General Condition for

This section is technical in nature.

[In (u zo

z;

+ ~WI2)]2

= w.

x correspond to formula

Linearizability For a multiindex

"t' E

z~,

we introduce three

pairwise equivalent conditions I!l(k) , I!lo(k) and !!I1(k) such that if "t' satisfies these conditions and L: E ... E is a hyperbolic linear operator, then the polynomial map

x

1-+

Lx

+ p~TeJ

x = Ax + p~TeJ'

is

locally

c!

linearizable (the same is true for the vector field

if the linear vector field

x = Ax

is hyperbolic). We also show that

in the particular case when "t' J = 0 there is a large class of linearizing transformations whose smoothness class can be completely characterized in terms of I!l(k). The proof of the linearization theorems is postponed until § 6.

Condition I!l(k) and a

c!

linearization theorem

We introduce here a condition, I!l(k), imposed on a resonant multiindex

"t'

under which a

polynomial diffeomorphism (vector field) which contains the monomial xT non-linear term admits a local

c!

linearization.

5.1. Preliminaries. For x, Y ERn, we shall write x

for every j

~ = 0 Let it was = Ax

x

E

as the only

l!:

y (and y:s x) if ~

{1, ..• , n}. Given a positive integer k, denote 1R~(k)

=

{x

E

l!:

r

R~: either

or ~ l!: k (j = 1, ... , n)}. L: E ... E be a hyperbolic linear operator. Define the numbers 9 1, ••• , 9~. as done in subsection 2.1S. Similarly, given a hyperbolic linear vector field (x E 8), let 9 1, ••• , 9 n denote the numbers defined in subsection 2.3.

77

. A collection

t1'

=

0f

(t1'\, ... , t1'p)

n-vectors crl

) =( crll\..." , crl

" 9-collection if all the quantities

Lcr{eJ

==

= 1,

(i

E

IR+"'call 1S ed a

... , p) are non-zero and

J-\

of one and the same sign. A 9-collection cr = (cr\, ... , crp ) if I I = 1 for every i E {1, ... , pl. Given a e-collection cr = (cr\, ... , cr p) denote

is said to be normalized

p

rlc(cr)

= {("',

p):

'"

Ii

z~, P

E

z:, 0 ~ Ipl ~ 1",1 ~

k,

T

+

Lplcrl - '"

2:

O}. (5.1)

1-\

5.2. Def"mition. Let v = (v\, ... , v p ), VI E IR~ (i say that T satisfies the condition A(k, cr) (and write T

= I, E

... ,p) and

A(k,

T E

IR~.

v» if the inequality

We

(5.2) holds for every vertex by the relations UJ 2:

0

u = (u\, ... , un>

V = 1,

... , n),

of the polyhedral convex domain D determined


U> it

1 (i

= 1,

... , p).

(5.3)

5.3. Definition. We say that the multiindex T E z~ satisfies the condition S(k) (and write T E S(k» if there exists a normalized 9-collection t1' = (v\, ... , v p ), crt

E

R~

5.4.

(i

= I,

... ,p), such that

T E

A(k, cr).

c:c Linearization Tbeorem for a Map. Let k be a positive integer,

L: E -+ E be a

hyperbolic linear operator, j Ii {1, ... , n}, and T E z~ be a multiindex such that the pair V, T) is resonant, i.e., 9J = . (f the multiindex T satisfies the condition S(k) , then the map x 1-+ Lx + p-f.x~eJ is conjugate to the linear map y 1-+ Ly near the origin.

c:c

5.5. Remarks. (1) The proof of Theorem 5.4 will be given in § 6. Indeed, we shall prove a more general theorem (see Theorem 6.2). To this end, we need to introduce two other conditions, SoCk) and SICk), which are, in fact, equivalent to S(k). Although

78

these new conditions look more complicated as compared with i!I(k), they naturally arise in the proof of Theorem 6.2. (2) Given a multiindex T E z~, a collection cr

=

(crh ... , crp ),

crt E IR~ (i = 1, T satisfies the

... ,p), and a number k, it is not a trivial task to check whether or not

condition A(k, cr). We have no inclination to dwell upon this problem and confme ourselves by noting that in convex analysis there exist algorithms for enumerating all vertices of a polyhedral convex domain (see, for example, the papers by Balinski [1], Altherr [1], Dyer and Proll [1, 2], Dyer [1]).

Condition i!lo(k) and its equivalence to i!I(k) 5.6. Defmition. Let cr = (crl' ... , crp ) be a e-collection. The multiindex T E z~ is said to satisfy the condition Ao(k, cr) if for every pair (or, p) E rk(cr) (see (5.1» the conditions UJ

i!:

0

U=

1, ... , n),



i!:

I I

(i

= 1,

... , p)

(5.4)

imply p

l(u) •
+

p

Lp'cr, - or, U> - Lp'l I >

5.7. Defmition. The multiindex

T

z~

E

crt E 1R~(k)

exists a e-collection cr = (crl' ... , crp ), T E Ao(k, cr).

5.S. Proposition. The multiindex

T

satisfies

(5.5)

O.

the condition

i!lo(k)

(i = 1, ... ,p),

if there

such

that

satisfies the condition i!I(k) iff there exists a

e-collection cr = (crl' ... , cr p ), crt E 1R~(k) (i = 1, ... ,p), such that the inequality (5.2) holds for every vertex U = (UI' ... , Un) of the polyhedral convex domain D' detennined by the inequalities (5.4). ~ Let

T E i!I(k) ,

i.e.,

there exists a normalized

e-collection

cr

= (erh

... , erp ),

er, E IR~ (i = 1, .. , ,p), such that T E A(k, er). Define the required collection cr = (crl' ... , cr p ) by the rule: crt = (i = 1, ... ,p), where

Mer,

79

=k/

M

min {v{ > 0: i

=

1, ... ,p; j

= 1,

... , n}.

k if cr{ > 0, hence, crt E 1R~(k) (i = 1, ... ,p). Note that I I = MI I = M, and cr is a a-collection. Let U = (u t , ... , un) be a vertex of the domain D'. Then v = ~ U is a vertex of the domain D defined by the relations (5.3). Since or E S(k, v), we have Then cr{ = Mv{ = I <Mv" a> I

i!:

Conversely, let cr = (cr\> ... , crp ), crt E 1R~(k), be a a-collection inequality (5.2) holds for every vertex u of the domain D'. Put

=

It,

1 crt (i I I

= 1,

such that the

... , p).

Then It = (Itt, ... , It p ) is a normalized a-collection. Let us show that or E A(k, It). In fact, the relation i!: 1 can be rewritten as i!: I I and, consequently, the domains D and D' coincide. Therefore inequality (5.2) holds for every vertex of the domain

u

Uj i!:

Thus, or

E

0 (j

=

1, ... , n),


U> i!:

1 (i

=

1, ... , p).

A(k, It).

5.9. Lenuna. The conditions S(k) and So(k) are equivalent.

• Let or

a',

1R~(k)

E

S(k).

=

By Proposition 5.8, there exists a a-collection cr

=

(crt, ... , crp ),

such that inequality (5.2) holds for every vertex U = (Ut, ... , un) of the polyhedral convex domain D' described by relations (5.4). Let (r, p) E rJc(cr) and U be a vertex of the domain D'. From (5.4) we get E

(i

1, ... , p),

p


+

p

Lp'cr, - r, u> - Lptl I

80 p

= <'t -

r ptCI

U> +

7,

U> I - 1<0'" 9> I)

<0'"

(-1

Therefore

't E

Ao(k, 0'). hence 't

E

2Io(k).

Conversely. let 't" E 2Io(k) , and 0' = (0'1' ...• O'p). 0', E 1R~(k) (I = 1, ... , pl. be a e-collection such that 't E Ao(k, 0'). Consider an arbitrary vertex U of the domain DO. There exist two sets I c {I •... , p} and J c: {I, ...• n} such that card I + card J = n and the vector U is the unique solution of the system of linear equations =

Let

J E l~. Ijl

otherwise. For

Is

= III

$

S

implies

I I (t

k. For every j

E

r

= o.

In fact. if O'~

solution of (5.6), then without fail Us

(j

E

7-1 = yl

{I .... , n}. put

E {I ..... ra} denote Is S

=0

I). UJ

E

J).

if

= {tEl: O'~ > OJ. = 0 for every tEl

= O.

(5.6)

uJ

u is

and

r = O.

hence

the unique

S

r

S

p(

=1

(r. p)

E

p

lact. if 'I-s I:.

S

~ 't" •

then

't"

s

+ Lrp' 0'( - 'I-s

s

~ 't"

-

-s

7

2:

0• Let -s s 7 > 't" • then

(=1

p

_s

•

Since 't

E

+ r ' s L p 0', - 'I

Ao(k. 0') and

(r,

~ T

s

+

ts

s

-s

p O't s - 7

2:

'ts + k -

'I

s

2:

p) e rl«O'), we have p

p

o <
r. u> - Ep'I1 p

=

+

Ep'«O't. u> - 1<0'" 9> I) (-1

=0

It is easy to show that

> 'ts ~ O. then Is '" O. Denote ts = min {t: t E Is}. Put for some s with :;s > 't"s. and p' = 0 otherwise. Let us show that Let

> 0, and 7-1

O.

if i

= ts

rk(O'). In

81

=

<'t -

r, U>

+ ~>'«"'h U> -1<"",e>l)

(5.7)

'EI

=

<'t - r, U>

=

L('t J - rJ)uJ

=

= for every 7

E

<'t -

7,

U>

Z~, 171 s k, and every vertex

that Us = max {UI' ... , un}. By putting 7' deduce from (5.7) that

<'t, U> - k Hence, by Proposition 5.8,

't

L(orJ - 7J)UJ J~J

Jf.J

E

U of the domain IY. Let s be a number such

=k

max {u l ,

for 1 = s and 7'

... ,

=0

otherwise, we

un} > O.

21(k).

we have to show that the nonhomogeneous linear functionals (5.5) are positive at every point of the convex polyhedral domain (5.4). Since the coefficients of these functionals are non-negative, it suffices to check the condition (5.5) at the vertices of the domain (5.4) only. Lemma 5.9 shows that doing this we may restrict ourselves by considering only one special

5.10 Remark. To verify the condition Ao(k, "')'

choice of values of multiindices 7 and p. for j = s (here (5.5) turns into (5.2). 7'

=k

Us

= max

Indeed, putting p'

{u l , ... , un}), and 7'

=0

=0

p

<'t +

=

("'I' ... , "'p), "" E 1R~(k)

't

(I

satisfies the

= 1,

... ,p),

P

LP''''' - 7, U> - Lp'

holds for every pair (7, p) E r Jc (",) (5.3).

... , p),

otherwise, we see that

5.11. Remark. It is easy to deduce from the proof of Lemma 5.9 that condition A(k, "') for a given collection '" iff the inequality

= 1,

(I

> 0

and every point U of the domain D determined by

82 Checking smoothness of linearization maps (a model example) In order to clarify the essence of the condition 2l(k) , let us consider the following model mapping (compare with (4.1»:

1, ... , n), n

where 11,9\0 ••• , 9 p are non-zero real numbers and 11

=

L J 9J .

II

T

Due to

J-I

simplicity of this map, we can present a sufficiently large family of linearizing coordinate changes and, by using the condition 2l0(k) , determine the class of smoothness of transformations belonging to this family.

S.12. Notation. For

multiindex

T

e IR~

define a function t:

t(x)

=

"t' {

x

o

n

n

and a collection IRn

In (Ixl

=

Ixll

= (a'1o ••• , f1'p),

a'(

x e IR, f1' e IR+, denote Ixl a'

tr'

tr'1

... IXn I

e 1R~(k)

~

.

Given a

(i = 1, ... , p),

-+ IR as follows: tr'1

+ ... +

Ixl

tr'p

),

•

If Ixl

tr'1

+ ... +

Ixl

tr'p

> 0;

(5.8)

otherwise.

The reader can directly check that the transformation z

=w+

C t(x)

= _1_

(5.9)

for every i e {I, ... , pl. Ce Tl We note in passing that most of the linearizing coordinate changes used in § 4 are of the form (5.9) for an appropriate choice of the collection a' = (f1'I' .•. , f1'p). The following proposition establishes the exact value of the smoothness class ,of the

linearizes the model map iff

83 function (5.8). 5.13. Proposition. The function t is c" smooth • Let

"t' E

A(k, 0') .

A(k, 0'). Clearly, t is infinitely many times differentiable at every point

"t' E

of the domain M = {x Let

iff

r E z~, I r I

!!

> 0 (j = 1, ... , n)}. We shall prove that t E c"(lRn , rl + ... + rn :s k. For x E M, the following formula is valid: E

IRn : xJ

+

~

+ ... +

D7[XT In (lx(1

Lc(P) p

Ixi p)]

= D7(XT)

I~I +

...

+

In (Ixl

pP~p

~I

xT: +

P

(X~I

~ I P + ... + x p)p + .. . +p

0

+ ... +

Ixl

IR).

~p

)

7

(5.10)

,

where c(P) are certain non-zero real numbers and the sum is taken over all multiindices p E z~ such that (r, p) E fJt(O'). Consider the convex polyhedral domain D determined by (5.3). Without loss of generality, we may suppose that for every j E {I, ... , n} there

exists a number i

E

{I, ... , p} with O'{ > O. Then there exists a number L > 0 such

UO II

that the point (L, ... , L) belongs to the domain D. Denote Do (j = 1, ... , n)}. Clearly, Do c: D.

= {u

E

IRn :

UJ it

L

Fix a number c, 0 < c < 1, and put c5 = c L • For every point x E IRn , which fulfills the inequalities 0 < xJ < c5 (j = 1, ... , n), we can choose a vector ;:; E Do in such a way that ;:;

it

Uo and x J

= c V. J

(j

= 1,

... , n). Furthermore, there exist numbers s

and 1 E {I, ... , p} such that the vector u

= sol;:;

it

1

belongs to the domain D and,

besides, Denote t

= CS

<0'/,

u> =

1.

(5.11)

tJ

(j = 1, ... , n).

(5.12)

:s c. Then XJ

=

To summarize, for every point x E IRn , 0 < xJ < c5 (j = 1, ... , n), we get a point U E D and numbers 1 and t, 0 < t:s c < 1, such that the equalities (5.11) and (5.12) hold. Since

"t' E

A(k, 0') for (r, p)

E

rk(O') and

U

E D, we have

84

S ==

<'t'

P

P

t-I

t-I

Lptrrt - 'I, u> - Lp'

+

> O.

(5.13)

From (5.13) and (5.1) we deduce the existence of a number II' > 0 such that the minimum of S over all ('1, p) E r ,,(rr) and U E D (denoted by SmiJ is greater than 211'. In order to prove CC smoothness of ., it suffices to show that sup ~VI

+ ... + xVPfv

D'I'[x"C'

In ~VI

+ '" +

<

XVp)]

III.

xEM

In virtue of (5.9), we must prove that ... + pPvp • 'I'

x"C' + P IVI +

II

==

sup xEM

for (-I', p)

E

+ ...

(XVI

==

sup

III

For x

E

+ ... + XVp) I < +... + ."vP)v .

x"C"'I'lln(x vl

xEM

't' 2: 'I.

<

r,,(rr) , and

11 if

+XVp)pl+ ... +pP+v

( vI

X

III,

M choose a point U E D and numbers I and I satisfying (5.11),

(5.12). Taking into account the relation inequality (5.13), and the condition Smin

Similar arguments show that 11 Conversely, let us prove that

<

XVI

>

211'

+ >

+

xVp

2: X Vl ,

equality

(5.11),

0, we get

III.

't' II!

A(k, rr) implies •

II!

CC(Rn , R). If

't' II!

A(k, rr),

then there exist a vertex U of the domain D and multiindices 'I E z~, P E z~ such that ('I, p) E r ,,(rr) and the value S of the left-hand side of the formula (5.13) is nonpositive. Choose u, 'I and p in such a way that S = Smin' Consider first the case SmiD < O. Let xJ(t) = t J (j = I, ... , n) and G be the curve I 1-+ ~I(t), ... , xn(t» (0 < t < 1). Let us examine the behaviour of the derivative

D'I'.

along the curve

G.

There exists a number I such that (5.1l) is valid.

85

x

Therefore. the inequality C

= p-(pI+ ... +pPj.

+

(XVI

CI

+... +

Vp X It

Vl

px

is fulfilled

G.

along

Denote

In virtue of (5.10). we get

xT

It

VI

P IVI

+

. .. + pPvp -

+ ... +

'I"

XVp)pl+ ... +pP


+ ... + pPj

'1". u> _ (pI

s . = CI nun -+

CD

since SmiD < O. Other summands entering the right-hand side of the formula s· (5.10) either are bounded or tend to infinity slower than t DIIn. This means that the

as I -+ O.

=

derivative D'I"I(a). where a

lim x(t). does not exist. t-+ 0

Now let Smin

=

along the curves t

O. If P 1--+

I!:

2 then by considering the behaviour of the derivative D'I" I

(xl(t), .... xn(t» (0

<

t

< 1), where

for uJ > 0, cJ for uJ we get finite partial limits as

=

= const;

0,

t -+ O. which depend on the choice of the numbers

Hence, D'I"tea) does not exist. If p

=

CJ'

1. the problem reduces to the investigation of

the function 1m: R -+ R, where tmCx):::::; J!" In Ixl smoothness class of this function is (m - 1).

if x '" O.

=

and tm(O)

O. The

5.14. Example. Consider the linearizing transformation constructed in Example 4.4. Here 11 = 3. 11 = 15. 9 1 = 1, 9 2 = 3. satisfies the condition 1lI(8). In fact, put

9] ITI

= -1.

= (5,

6, 8). Let us show that T = (10, O. 0), IT2 = (0, 10. 20). It is T

D: uJ

easy to check (see Figure 5.1) that the convex polyhedral domain .... 3).

~

=

lOUI I!:

1.

1 1 (10' 10' 0).

1O~ +

20&1]

I!:

1 has exactly two verties:

It

u = (~O' l

0 O.

Ci = 1, ~o> and

86

1/20

Figure 5.1. Hence, or E A(8; crl' crJ. According to Proposition 5.13, the linearizing transformation

is of class

C. Condition i!ll(k)

Now let us return to Theorem 5.4, which is concerned with smooth linearization of the map x 1-+ Lx + p!.:xT eJ' The model example discussed above corresponds to the case where orJ = 0 and admits, as we have seen, very simple linearization maps of the form (5.8), (5.9). If orJ '" 0, the situation in Theorem 5.4 is much more complicated, and we were not able to give a more or less explicit description of the required linearization maps. Nevertheless, we shall see in § 6 that even in this case the proof makes use of successive transformations of the form

87 _

"£'

+x

y - x

qJ(lxl

crt

+ ... +

Ixl

crp

),

(5.14)

(as suggested by the instructive Examples 4.10 and 4.11). It will be proved that the condition T E A(k, cr) ensures c!- smoothness of (5.14). To do this, we must introduce one more equivalent form of the condition S(k).

cr "" (crt, ... , crp ), we say that the multiindex

5.15. Defmition. Given a a-collection

IR~ satisfies the condition At(k, cr) (and write T E At(k, cr» if for every pair (r, p) E rJc(cr) there exist real numbers At, ... , Ap satisfying the following inequalities: T E

At

~

0

(i "" 1, ... ,

p

p);

p

LAtcrt ::s

T

+

Lptcrt - r;

(5.15)

p

L(At - pt)1 I

> O.

t-t

5.16. Definition. A multiindex

there exists a a-collection T E At(k, cr).

T E

IR~

is said to satisfy the condition

cr "" (crt, ... , crp ),

crt

E

1R~(k)

(i "" 1, ... ,p),

St (k)

if

such that

5.17. Lemma. The conditions SoCk) and SI(k) are equivalent.

• According to Chernikov [1, pp. 123, 189], the following duality principle is valid: (1) The system of linear inequalities

(5.16) is compatible iff the system of linear equations

has no non-negative solutions;

88 (2) Let the system (5.16) be compatible. Then the system

is compatible iff the system of equations

has no non-negative solutions such that

UI

+ ... +

Um'

> O.

Therefore the system (5.15) is compatible iff the following system of relations Vt

l!!


(i = 1, ... , p),

0

U> - vt -

uJ

l!!

wi I

p

(j = 1, ... , n),

0

=0

(i

W

> O.

= 1•...• pl.

p

- wL/I1:s 0 has no solutions or. equivalently. the conditions UJ

l!!

0

(j

= 1•...•

n).

W

> 0, (5.17)


U> - wi I

l!!

0

(i = 1, ... , p)

imply the inequality p

p

- wLptl1> O. Note that if for a collection (u l , and

t > O.

setting t =

.!,

••• ,

then for the collection

Uno

(5.18)

w) the relations (5.17) and (5.18) are valid

(tUI.. .• ' tum tw)

these relations also hold. By

we get the relations (5.4) and (5.5).

W

5.18. Remark. Thus, we have sllown that the conditions !!l(k). !!loCk) and !!ll(k) are pairwise equivalent. We have also established that these conditions give an exact

89 estimate of smoothness when applied to an important class of functions occuring in the linearization theory. Now we would like to emphasize that the property 1: E I!(k)

C

persists under small fact, the numbers system (S.IS).

perturbations of the given diffeomorphism (vector field). In are met only in the third (strict) inequality of the

9 1" " , 9 n

p

5.19. Lemma.

Ec,

(i = 1, ... ,p),

Let

=

1,

P

O'p+1

=

E '-I

c,O',.

Then

1: E

A(k;

iff

0'1' ... , O'p)

1: E

A(k;

0'1' ... , O'p, O'P+I)'

• In order to prove this assertion, it suffices to observe that the domains

D: uJ

I:

(j

0

= 1,

... , n),

<0'"

U>

1 (i

I:

=

1, ... , p)

and D·: uJ

I:

0

(j

=

1, ... , n),

<0'"

U>

I:

1 (i

=

1, ... , P

+

1)

coincide.

§ 6. c;k Linearization Theorems

In this section, we use the condition 2I(k) to formulate and prove theorems on c;k linearizability of resonant polynomial maps and vector fields. 6.1. Preliminaries. Let L: E -+ E be a hyperbolic linear operator, and be the corresponding real numbers (see subsection 2.1S). For a given e-collection

0'

= (0'1'

... , O'p),

I

0', E

R~(k)

(i

= 1,

... , p), defme

n

IX!"" = IIXIII"" ... uXnu"" Ixi '" -- (ixi "'I , ... , Ixl "'p),

(i

=

1, ... , p);

rLX]

=

Ixl "'I

+ ... +

9 1, ... , 9 n

Ixl"'P.

90 Let Q be a positive integer. Q ~ 2. A polynomial

p(x)

Q

Q

Iwl-2

Iwl-2

L p~w. .... L p~W)

=(

(6.1)

is said to be resonant if the condition p~ =- 0 implies the equality a J Let

T E

z~.

the inequality

= < w.

a>.

IT I ~ 2. The polynomial (6.1) is called T-divisible if p~ - 0 implies

T:S

w.

Denote J(T)

= {w

E

z~:

3j

E

{1 ..... n}: a J

= <w,

a>;

1.01

~ T.

11.011 :S

Qo(k)

+

I"r!

+

1}

(for the definition of Qo(k) see formula (3.1».

6.2.

eft

multiindex.

T E z~ be a 2, and p: E .... E be a T-divisible resonant polynomial of degree

Linearization Theorem for a Map. Let k be a positive integer. IT I

~

Qo = Qo(k). If T satisfies the condition 5!l(k). then the map x conjugate to the linear map y H Ly near the origin.

H

Lx

~ The proof will be divided into several parts. First. we shall find a

+ p(x) is eft

c!

map H such

that the local diffeomorphisms HI • (L + p) • H and L have contact of order (Qo, k) at the origin. This will be done by solving some triangular system of affine functional equations in certain special functional spaces. Then. in order to prove that HI • (L

+ p)

• H and L are

c!

conjugate, we shall apply Theorem 3.3.

Analysis of the linearization problem Let T E 5!l(k) = SI(k) and rr corresponding a-collection. i.e.. follows. we shall assume that

= (rrl .... , T E

rr p ). rrt E 1R~(k) (i = 1 • ....p). be the Without loss of generality, in what

A(k. rr).

91

"

erie J = - 1

sign ~

(i

= 1,

pl.

... ,

J-l

We try to find a II(L

CC

+ p)

diffeomorphism H: E .... E meeting the requirement 0

H(x) - H(Lx)1I

= o(IIXIl Qo+ h:1 +1 )

as

IIXIl ....

0

and having the following form:

H(x)

= x + hex)

:; x

~ hw(lxla)xw,

+

(6.2)

weJ(T)

where the mappings hw: IRP .... Pw(E, 8), hw = (h!, ... , h:), are such that implies 9 J = <W, 9>. We have (L

+ p)

0

H(x)

= (L + p)

0

(id

+

h)(x)

= Lx + Lh(x) + p(x +

h(x» ,

where

" ",eJ(T)

t-l

wel~

weJ(T)

(j = 1, ... , n).

6.3. Lemma. if q~ • 0, then 9 J = <W, 9> and W2: T-divisible resonant polynomial with variable coefficients.

T,

i.e.,

p(x

+

hex»~ is a

• In virtue of the binomial theorem, we get

(6.3)

Where s =

"

~

t-l

St,

CIt

( {3 11 ,

... , {3

l,sl

" 1

, ... , {3 , , ... , (3

",Sn

)

92

s"

SI

E

Z:;

A= (

L 13 t-I

o ~ A ~ IX;

L 13"t);

1t , ••• ,

wtt

E

J(T)

1, ... , n; t

(i

= 1, ... , St);

t_1

(j, IX) and (i,

Wtt)

are resonant pairs, i.e., 9J

=

, 9 t

=

<Wtt, 9>

(i = 1, ... , n; t = 1, ... , St); c~ E Pa.-Il.(3(E, The sum in (6.3) is taken over all IX and f3 such that Sr

wr

=

fir -

"S(

L f3M

+

t-I

L L W~t/3(t

(r = 1, ... , n),

(-I t-I

Sr

L 13

M

~ IXr

(r = 1, •. , n).

t-I

Let us show that the pair (i, w) is resonant. In fact,

<W,

9>

=



"St

=



L L /3 tt

(-I t-I

" (9( -

L W~t9r)

=

< IX, 9 >

r-I

since both (i, IX) and (i, Wtt) are resonant. It is easily seen that W ~ T. p(x + h(x» is a T-divisible resonant polynomial with variable coefficients.

Thus,

6.4. Notation. Let q denote the sum of all terms entering the expression p(x + h(x» and having degree less than Qo + I T I + 1, i. e. , q is of the form q(x) = (ql (x) , ... , q"(x» , where - p(x)

93 rr(x)

r q;!(lxl")xW

=

(j

=

1..... n).

wEJ(or)

Put r(x)

= p(x + h(x»

- p(x) - q(x).

6.5. An auxiliary system of functional equations. Set down the functional equation h(Lx)

= Lh(x) + p(x) + q(x)

(x

E

E).

(6.4)

or. in the coordinate form.

L h~(lLxI")(Lx)w = L LJh~(lxl")XW wEJ(or)

wEJ(or)

L p-!;cW + L q;!(lxl")xw

+

wEJ(T)

=

(j

1..... n).

wEJ(T)

Consider the related system of functional equations

(6.5) (j

= 1.....

n; WE J(T);

9J

= <W. 9».

Clearly. substituting the solutions h~ of system (6.5) into formula (6.2). we get the needed solution of equation (6.4). Consider first the case when the operator L is diagonal (this somewhat simplifies the further arguments; the general case will be considered separately in the final part of this section). Let few: Pw(E. XJ ) -+ Pw(E. XJ ) be the linear operator defined by

Let t.: IR P -+ IR P

be the linear operator having the following matrix form n

n

94

Rewrite system (6.5) as follows: (6.6) Along with (6.6), consider the following system of functional equations:

(u e IR P ;

x e E; j

=

1, ... , n; we J('r); BJ

=

<w, B»,

or, equivalently,

(6.7) (j

=

1, ... , n; w e J(T); BJ

=

<w, B».

Clearly, every solution of (6.7) satisfies (6.5). Note that the functions q~ depend only on h~ with 1cr.1 < 1w 1 (see formula (6.3». Therefore the system of equations (6.7) has block triangular structure with respect to the blocks {h~: 1w 1 = r} (r = 1T I, ... , Qo + 1T 1 + 1) and, besides, the equations within every fixed block are independent. Consequently, system (6.7) can be solved inductively on the number

=

1w 1

Consider first the case r in (6.7) takes the form

=

r.

1T I. Since q~;;; 0, w

(B J

=

= T,

the corresponding equation

(6.8)


Solution of equation (6.8) 6.6. Notation. Let xp denote the set of maps 91: IRP \ {OJ -+ PT(E, X J ). Let be the linear operator defined by

"'0:

xp -+ Xp

95 6.7. Discussion. Our aim is to pick out a special norm on Xp in such a way that: (1) If a function !P is bounded with respect to this norm, then the map

x ~ !p(lxlcr)xT admits a ~ smooth extension to the unit ball B(O, 1) (2) The operator .40 is contracting with respect to this norm;

= {x E E: x

(3) Near the origin, there exists a local representative of the map the form

p!:x = p(lx1cr)xT, T

and, moreover, the map

p

= .40 + l~1 P

x

~ p~(lxlcr)xT is a ~ smooth local solution of the equation (6.8).

Xp -+ Xp has an attracting fixed point ~, therefore the map

6.S. Notation. Let (r, p) Let

I:

Tt

+

pl.,.~

E

rk(v). Choose real numbers St

+ ... +

pPv; - rt

(i

= 1,

= st(r,

p) in such a way

... , n). Denote S

be a positive number. By n~,£ denote the Banach space of maps

PT(E, XJ )

of class

of

then the affine operator

t4

that 0 ~ St ~

p;,.xT

is bounded with respect to this

norm. If there exists a norm satisfying the above requirements, tI- l :

H

IIxll ~ 1};

= (SI'

... , sn)'

tp: IR P \

{OJ -+

~ having their support in the unit ball and satisfying the

condition

then the map x ~ !p(lx1cr)xT from {x E E, [x]. O} to XJ can be extended to a ~ smooth map from the unit ball B(O, 1) to XJ • 6.9. Lemma. If

!P E n~,£

~ By the differentiation formula for a polynomial with variable coefficients (see subsection A.8), we have

where c5 E z~, r E z~, 1:s I r I :s k, da are non-negative numbers. Recall that to derive this formula, we introduced two functions, ~: E -+ PT(E, XJ ) and A: E -+ X J ' as follows:

96 ;\(x)

= tp(lxl a),

A(X)

= ;\(X)x"C'.

Then

Let us explain the meaning of the last formula. Da;\(x)

E

L:(E, P"C'(E, XJ»'

consequently,

D"""~(x) ;;; D"""[Da;\(x)va](x) 7-a~T weget

hence, D"""~(x)v"-a

E

E

L:",,(E, Xj)'

~

;\: E -+ P"C'(E, Xj), we have Da;\(x)va E P"C'(E, Xj)' Therefore,

Since ii

Clearly,

~

if

E

P"C'(E, Xj),

P"C'-" +a (E, Xj)' If there exists such a number k

E

then

for

{1, ... , n} that

71e - ale > Tie, then D"""~ ;;; O. Finally,note that D""": P"C'(E, Xj) -+ L:_a(E, P"C'_,,+a(E, XJ» is a bounded operator. For I a I ~ 1, by the composite mapping formula (see subsection A. 6), the following equality holds:

Da[cp(lxl a )]

=

r

P

e(p, ~)[ifcp](lxla\ ... , Ixi"'P) IT

(i = 1, ... , p; j = 1, ... , pt), e(p,~) = const P

sum is taken over all p and

~ = (~tj)

Therefore

p

Hence it follows that

such that

1 ~ Ipl ~ 1151,

and the

p'

rr

~tj = a.

97 (6.10) P

= const and the summation extends on all

where C(p)

+

I

T+pcrl

p

E

z~ with 0 sip I s lor I and

p

.. ·+pcrpit:r.

Let IIxll s 1, [x] ~ O. In accordance with formulas (6.9) and (6.10), if rp 7 E z~,

then for every

E

fi~,.,

0 s Irl s k, we have (6.11)

Thus, the map

X t-+

CC

rp(lxl")x'" can be extended

1P(lxlcr>x... ;;;; 0 (IIXII s 1,

smoothly to B{O, 1) by putting

[x] = 0).

6.10. Lemma. Let T E AI{k, IT). Then/or every (r, p) E rJ«cr), a vector s = S(7, p)

p( Ix I cr)

with

IR P -+ P-.:(E, X J ),

i.e.,

can be chosen in such a way that there exists a local representative p; = -

P

E

-J<

fI ... ,.,.

• Take a smooth truncation p(u) = p~

lIuli s 1/2

for

and

p

of the constant map

p(u)

=

0

for

p;:

lIuli it: 1. Note that the property

sup {lIifp(lxlcrll x E E} < co can be ensured only by assuming that the mapping p has compact support. In order to use the latter property, recall that according to the condition AI(k, IT) for every (7, p) E rJ«cr) there exist non-negative real numbers Ixl s :

AI' ... , Ap

such that

Set (i = 1, .... p).

Then

Ixl S =

AI

ul

Ap

... Up •

Therefore.

sup IIDPp(lxlcr)1I Ixl s = sup lIifp(u h xEE

uEIR~

....

Up)IIU~1

'"

u;P

<

co,

98

since

p is

a smooth function with compact support. Thus

pe

;i~,.,.

6.11. Notation. By !11~,., we shall denote the Banach space of ~ smooth mappings

to}

q>: RP \

~ P-.;(E, X J )

having their support in the unit ball and satisfying the

property IIflll., =

sup

{lliffl(lxl")1I IXIAlcrl+ .. ·+Apcrp

[xr":

x e E,

[x] '" 0,

('1, p) e rJt(IT)}

< ...

6.12. Lemma. The norm of the operator Ao acting in the space !11~,., does not exceed the

value n

p

110 ;: exp [sup {

L (At - pt) L IT{9 J

t-I

• By the condition

n

9J

-

t= I ,p J-I

ilL J!f.~ III = 1. Therefore

< T, e>, we get

=

L IT{e J:

min

E

J=I

Ix I Alcrl + ... +Apcr p

IL - I x I A I crl + ... +Apcrp p

n

n

p

L / L IT{e J + L 9J LAtIT{ - E m~ IT{e J

::s IIflll., exp [sup { -

(=1

}]

t= I ,p

J=I

p

= IIflll., exp [sup {

L (At - pt) -

6.l 3• Lemma. There exists a number

m~ }].

E

t

t=1

EO

=I

,p

> 0 such that 0 <

E

::s

EO

implies

IlAoII.,

< 1.

99 ~

Set (r, p) e rJc(cr)}

If 0

<

C :S Co

IlAolic < 1.

then

6.14. Lemma. There exists a local solution h~ e !Il~.c corresponding map x

1-+

~(lxlC7)xT from B(O, 1) to XJ

of the equation (6.8).

The

is ~ smooth.

~ Since the operator Ao from n~.c into itself is contracting, the equation (6.12)

.

has a unique solution ~ e !Il~ c' The map

p

is a truncation of the constant map

p~: IRP -+ PT(E, X), hence the function h~ is a local solution of the equation (6.8). By Lemma 6.9, the map x

1-+

~(lxlC7)xT has.a ~ smooth extension to B(O, 1).

Solution or system (6.7) Consider the system of functional equations

h'!, = LJ fl.;} h'!,

to· 1

+ fl.;} p'!,

to· 1

+ fl.;} q'!,

to· l•

LJ fl.;} h'!,

to-I

+ fl.;',! q'!,

to-I

(6.13) (j

Where

p;

= 1,

... , n;

we J(T);

are truncations of the constant maps

9J

p;:

= <w, 9»,

Proceeding inductively, suppose that the equations of system (6.13) with solutions

h; in the spaces !Il~.c for some sufficiently small

defined in the same manner as !Il~.c). (Iwl :s r; j = 1, ... , n).

r it: I T I. Iwl:s r have

IR P -+ PweE, XJ ). Let

C

>

°

(the spaces !Il~.c are

Let H denote the set of all such solutions

h;

100

6.1S. Definition. Let N(r) denote the set of all truncations of polynomials 'Pt with

ht

Iwl s r and of all functions that can be represented as resonant polynomials in E H. Let nCr) denote the linear hull of the set N(r). For elements of the set n(r) , define inductively the notion of weight as follows: (1) Truncations 'Pt of polynomials have weight 0; (2) If the functions I/JI' ... , I/Jt have the weights K 1,

••• ,

Kt , respectively,

CS E z~,

• is a polylinear map and 'II = .'I/J~I ... I/J:t, then {I is of weight K = Kl + ... + KtCS t ; (3) The weight of a sum of functions in nCr) is equal to the maximal weight of its summands; (4) If a function the solution

I/It

(I wI s r) belongs to nCr) and has weight K then the weight of

h; of the equation

is equal to (K

+

1).

6.16. Lemma. If the function q/", I wI s r, defined by formula (6.3) has weight K, then it can be represented as a sum of solutions q of equations of the form

q = A.q + 1/1, where A. are contracting operators on K - 1 belonging to nCr) .

• Formula (6.3) shows that

qt

n=.c and I/J are functions of weight

is a polynomial in x and h~ (I v I

linear and free terms. When I ~ 1 = 1, we get the term assertion in virtue of the property (4).

< Iwi) without

c-'xw-",h~( 1x I"')x'"

satisfying the

Let I /31 II: 2. Then, by the property (2), the weight of each such function h~ does not exceed K - 1 and (6.14) where q~ K - 2.

E

nCr). According to (4), the weight of any function q~ is not greater than

Without loss of generality and for simplicity suppose that q~ is a multi-

homogeneous polynomial in h~. By substituting (6.14) in the equality q~ = P({h~}), we get

101 ..J _

{

,

q", - P( A, h".

-, + l".01 q".

ol} t II ) . P({A, h".})

+

R,

where R is a resonant polynomial in h~ and q~, and consequently R operator J.. by J..

q!,

Since q~

= P({h~})

Thus, q~

= L,

= P({A, h!}). Then

J..

q!,

+ R.

!TI(r). Define an

= P({L, l~1 h! llol}) = P({L,l~1 h~})llol.

and the pair (i, w) is resonant, i.e., sJ

l~ I q~ II 01

E

= <w,

s>, we have

Taking into account the above discussion on weights of h~

and q~ and the property (2), we conclude that the weight of the function R does not exceed K - 1. This completes the proof. 6.17.

Lemma.

The junctions ~ (Iwl ~ r) belong to the spaces n~.c .

• The proof is by induction on the number K and relies on Lemmas 6.14 and 6.16. 6.18. Discussion. Consider the equation (6.13) for Iwl function q~

E

nCr)

belongs to

n=.c.

=r +

1. By Lemma 6.17, the

According to Lemma 6.14, there exists a solution

h~ of the equation (6.13) which belongs to the same space. Assume that the weight of the function q~ E nCr) is equal to K. Assign the weight K + 1 to the solution of this equation. It is clear how to extend the notion of weight to all functions in the set n(r + 1) c: nCr) so that to keep the requirements of Definition 6.15. Thus, the solvability of system (6.13) in the spaces

n=.c

(c ~ co> is proved.

The

solutions h~ of system (6.13) are, clearly, local solutions of system (6.7).

Completion of the proof of Theorem 6.2 (including the non-diagonal case) Thus, system (6.7) can be solved in the functional spaces !naps x

1-+

h~(lxlc'')xT from {x

E

!ll=.c'

By Lemma 6.9, the

E: [xl - O} into Pw-T(E, XJ ) can be extended to

CC

102

smooth maps defined on B(O, 1). Since '" defined by the formula

=x +

H(x)

't' for every '"

it:

h(x) • x

J('t'), the function H

E

= id + h

L h~(IXI")xweJ

+

wEJ("C') J-I,n

is a local

(L

c!

+ p)

smooth solution of equation (6.4). We have 0

H

=L + L

0

h

+P +

q

+ r, H L = L + L 0

0

h

+P +

q,

consequently, (L

+ p)

H

0

=H

0

L

+ r,

+

r

i.e., L

+P =H L 0

0

HI

0

HI,

where ,J(x)

=

(j

=

1, ... , n).

According to the definition of the function r(x) and by Lemmas 6.16 and 6.17, we get !Jl~,.,. In virtue of Lemma 6.9, the map rp~(x) = ri!(lxl")x"C' is c! smooth and, besides, ~(Ix I"').xw = rp~(x)xw-"C', I", - 't'l it: Qo + 1. Denote R = r 0 HI. Then ri!

E

(p

= 0,

1, ... , k)

in a small enough neighbourhood of the origin. By Theorem 3.3, the map L

+

p

is

locally c! conjugate to L. Thus, if L a diagonal operator, Theorem 6.2 is proved. Now let us tum to the non-diagonal case. At first, we would like to find out the complexities that can arise in the new situation. To this end, let us look through the previous parts of the proof. Evidently, the operator I:J. introduced in subsection 6.4 fails now to be diagonal. Similarly, the assertion of Lemma 6.12 is no longer true. Nevertheless, Lemma 6.13 holds in the non-diagonal case, as well. In fact, although Ins not diagonal, the spectrum of I:J. lies, as before, on the circles of radii

103 n

n

exp

LIr{Sj,

... ,

exp

j-I

LIr~Sj'

j-I

Further, let us remind the reader that given a linear operator L: E ... E and a number 6 > 0, there exists a norm on E such that IILII:S r(L) + c5, where r(L) denotes the spectral radius of L (this follows from the Spectral Radius Theorem). Therefore IILJ f~11I :5 1 + 6, and the norm of the operator Ao (see subsections 6.6 and 6.12) can be estimated from above by "0 + c5. Consequently, Lemma 6.13 is still valid. These considerations show that dropping the diagonality assumption does not cause essential changes in the proof. We hope that these comments will enable the reader to fill in the necessary details.

Linearization of a polynomial vector field 6.18. Theorem. Let k be a positive integer,

y = Ay

be a hyperbolic linear vector

field; T E z~ be a multiindex, I T I 2: 2; p: E ... E be a T-divisible resonant polynomial of degree Qo = Qo(k). If T satisfies the condition ~(k) then the vector field

x = Ax + p(x)

(x

E

(6.15)

E)

is locally ~ conjugate to the linear vector field

y = Ay

(y

E

(6.16)

E)

~ Suppose for simplicity that the operator A is diagonal. conjugation mapping. Then

Let x

=y +

H(y)

be a

x = y + DH(y)y = Ay + DH(y)Ay = Ay + AH(y) + p(y + H(y», or DH(y)Ay

= AH(y) + p(y + H(y».

We shall show that solving (6.17) is equivalent to finding a ~ smooth manifold H invariant under the flow defined by

(6.17)

= t(y)

104

H=

AH

+ pry + 11), y =

Ay

(y

E

E,

H

E

E).

Since (6.18) represents a (non-linear) extension, we may also say that section of this extension. Clearly, H

11 i.e., (9 J

= tM

(6.18)

H

=

try) is a

is an invariant section if and only if

= Dt(y)Ay = At(y) + pry + t(y»,

(6.17). Let Hw: E -+ E (w E J(T), (Ht - 0) .. < w, 9>)) be some mappings. Let Qw: E -+ E be functions expressed in terms of

t is a solution of equation

=

functions

p!

and

{

II!

~w y

likewise (6.3). Consider the extension

= AHw + Pw(y) + =

(w

Qw(Y,1I)

E

J(T»,

(6.19)

Ay

(y

E

E,

Hw

E

E).

Since the functions Qw depend only on those HI(, for which 11(.1 < 1wi, the extension (6.19) has block triangular structure and, moreover, the equations within every fixed block are independent. The search of an invariant section Hw = tw(y) (w E J(T» is thus reduced to solving succesively the functional equations t

tw(eAty)

= eAt[twM + I e-As(pw(eAsy) +

Qw(eAsy»ds).

(6.20)

o Let 21 be the linear operator on the space of maps from E to E defined as follows: I

21 Fey)

=

eA

I

e-AsF(eAsy)ds

(y

E

E).

o Since we are looking for a local invariant section, the polynomial p can be replaced by its truncation. In this case, the invariant section is uniquely determined. suffices to find a solution of (6.20) for t .w(Ly)

where L

= eA ,

Pw = 21

= Ltw(y) + Pw , Qw

=$

Hence it

= 1, t.e.,

P",,(y)

+

Qw'

We try to find tw expressed in the form

Qw(y)

(y

E

E),

(6.21)

105

~",(y)

= ""'( Iy I")y"',

where ,,'"

E

~~ •.,. Note that P/.,(y)

= pty'"

and the pair (j, 101) is

resonant. Replacing the polynomial p~ by a truncation p~(z), we get P",(y) = Pw(lyl")yW, !Jl~ •.,. Similarly, Q",(Y) = q",(lyl")yW, Qw E !Jl~ •.,. Thus, the original problem is reduced to solving the system of functional equations of the form (6.7) in the spaces

Pw

E

!J1~ •.,. The latter problem is settled in the first part of this section. Let Hw = ~w(y) (w E J(T» be the invariant section found above.

Consider the

section

L t",(y).

Ho = ~o(y) =

wEJ(T)

It is invariant with respect to the extension

{

~o Y

where Q(Y, Ho)

=

= ABo

=

+ P(y) +

Q(Y, HO>,

(6.22)

Ay,

L Qw(Y, HO>.

It remains to note that the vector fields (6.22) and

WEJ(T)

(6.18) have contact of order (Qo, k)

at the origin and,

consequently, they are

conjugate with one another. Therefore, the extension (6.18) has a section.

c!<

c!<

smooth invariant

6.19. Remark. Theorems 6.2 and 6.18 can be extented to the case of maps as well as vector fields in the vicinity of an arbitrary (i.e., non-hyperbolic) rest point. Consider, for instance, the case of vector fields. We shall keep the notation adopted in subsections 3.11 and 3.15. Let ~ be a vector field with principal part of the form

where L h , ", P and q are c!< smooth functions, p' iii 0 implies at = < T, a>, q iii 0 implies <~, a> = 0, Ah(O) = A h , 11'(0) = 0 and Drp(O) = Ac. Suppose the multiindices T and ~ satisfy the condition i!l(k) with respect to the operator A h • Then the vector field E is locally

c!<

conjugate to the vector field

106

The proof of this assertion is similar to that of Theorem 6.2. In this case, the unknown functions

h! depend

not only on x

E'

E

Jt=. By !I1~.£(Jt=) we now {O}) x Jt= -+ Pw(E', E) of class c!-,

but also on

Z E

denote the space of all mappings 11>: supp II> C {(x, z): IIxll:S I}, endowed with the norm (IRP \

r

E

z~,

P E Z!, 11 E Z+o X E

E',

[x] .. 0, Z

E

Jt=, Irl +

11 :S k, (r, p)

E

rJc(cr)}.

Since the eigenvalues of the operator Drp(O) lie on the imaginary axis, all the methods and estimates used in the proof of Theorem 6.2 continue to be efficient in this situation. 6.20. Remark. As it was already pointed out (see Introduction), the problem on conjugation of diffeomorphisms (vector fields) is closely tied with the problem on the existence of invariant manifolds. For example, in order to conjugate the mappings x t-+ Lx + ~(x) and y t-+ Ly + ~Cy) with one another, it suffices to find an invariant section of the following extension:

[x] Y

In fact, let x

= hCy)

~ rLx + ~(x + lLY + IIICy)

y) - IIICy)]

be an invariant section then

= h(Ly + IIICy»,

LhCy)

+

ICy

+

hCy» - IIICy)

LhCy)

+

ICy

+

hCy»

= Ly +

+

= (id + h)

hence

Ly

+

IIICy)

+

h(Ly

(L

+

+

i.e., (L

+

I)

Thus the coordinate change x

0

(id

=y +

h)

0

hCy) is a conjugation.

III).

'IIcy»,

107 The converse statement does not, in general hold, i.e., not every problem concerning the existence of an invariant section of a non-linear extension can be reduced to a conjugacy problem. Note that Theorem 6.2 can be rephrased as follows: if a multiindex 1: satisfies the condition l!I(k) and p: E -+ E is a 1:-divisible resonant polynomial, then the extension

has a

C<

smooth invariant section. Let us show that a more general assertion is true.

6.21. Theorem. Let E, F be finite dimensional linear spaces, L: E -+ E and C: F -+ F be hyperbolic linear operators. Assume, besides, that the set of moduli of eigenvalues of the operator C is contained in the corresponding set for the operator L. Let k be a positive integer, 1: be a multiindex, p: E -+ E is a 1:-divisible resonant polynomial and t: E -+ E be a resonant polynomial with teO, u) = 0 (u E F). U 1: E l!I(k) then the extension

.- - - - . lfIxCu +

rXyJ '--------"

~

possesses a local ~

t(x, u)

+ P(U)]

(x

E

E.

uE

F)

C< smooth invariant section.

Observe that x

= h(u) Lh(u)

is an invariant section iff

+ t(h(u), u) + p(u) = h(Cu).

It is, undoubtedly, clear that the required section can be represented in the form

h(u)

=

L hw( I u Itr)uw wEJ("C')

(we maintain the notation introduced in subsection 6.1). In the case under consideration, the map u 1-+ t(h(u), u) has much the same structure as (6.3). Actually, the problem reduces to solving the equation (6.4). The final stage of the proof makes use of Theorem 3.3. We leave the details to the interested reader.

108 § 7. Some Sumcient Conditions for

c!

LinearizabiUty

We would like to draw the reader's attention to the fact that the general condition !(k) introduced in § 5 depends on infinitely many parameters, namely, on the integer p

E

z~ and on the a-collection

IT

= (crl ,

... , ITp)

= (cr~,

of n-vectors crt

... , cr7)

E

IR~.

Given a multiindex "t' e z~ and a hyperbolic operator L: E ... E, our purpose should be to find a a-collection IT = cro that realizes the largest value k with T E A(k, cr). So far we were able to solve this problem only when n = 2 and n = 3 (these results are established at the end of this section). In general, this optimization problem seems to be enormously involved and awaits future investigation. That is why we are looking for relatively simple conditions ensuring c! linearizability (although stronger than !(k». In this section, a number of such conditions (including the already mentioned conditions S(k) and MS(k») 'is presented and examined.

Conditions S(k) and MS(k) 7.1. Notation. In what follows, we shall use new notation for the numbers ai' ... , an (introduced in subsection 2.15), namely, {9 t , ... , 9 n } = {- ;\" ... , - ;\1' "'11 ... , "'m}, where (7.1) - ;\, < ... < -;\1 < 0 < "'I < ... < "'m' m + I = n. , Similarly, gIven "t' according to (7,1).

E

n

Z+,

we shall write

"t'

with respect to the linear operator L: E ... E C

=

(cI' ... ,

15" ... , c5P such that:

c p ),

Ct

e z~

/3)

= (a.,,

I

I

.. " a. , /3, ... , /3

m)

Z~

satisfies the condition MS(k) (and write "t' E MS(k» if there exist a

7.2. Definition. We say that a multiindex a-collectiom

= (a.,

"t'

(i

=

E

1, ... , p),

and positive integers

109 (1)

+ ... +

aici

aPc p :5

T;

P

Lat 1< Ct, S> 1 > k

(2)

max 1< C t ,

S> I.

7.3. Theorem. The condition MS(k) implies S(k). • Let

e MS (k) , and Ct, a

T

t

(i

(i = 1, ... ,p).

o < SI• :5 tI'

= (ti'l ,

In fact,

... :5

... , tl'p),

s·

=


tI',

= -.s,

s>

(i

aPs; > k s;.

= 1,

tl't 2:

e 1R~(k)

0 (i

=

Let us show that

Te AI(k, tI'), where

... , p) and R is a large enough positive number.

(i = 1, ... , p) and

sufficiently large, then there exist numbers t;t

1, ... , p) be the corresponding elements. Denote

Without loss of generality, we may suppose that

sp• and a I SI• + ... + RCt

=

=

(i

1,

s· > 0, ... , pl.

hence, Let

tI'

is a e-collection. If R is

(" p) e rlc(tI').

Show that

1, ... , p) such that

(7.2)

t'

..

:5 '" .t

+ p'....!i... (.r = 1, ... , p.)

Consider the convex domain Q c

s;

RP

defined by the inequalities (7.3)

The set of vertices (angular points) of this domain is contained in the set of nonnegative solutions of all subsystems

t;'

=0 (7.4)

P

t' ttc J, = ,J l....

(j e J '"

UI'

'}) (s ... J",

1, ... , min {P,

nIl,

110

each of which has a unique solution. Let (0

= «(~, ... , (~)

p

be an angular point. Then

p

LrJ = L L~~ e{ = L(~ Le{. JEJ

Besides, if

Le{ = 0

JEJ t-1

(j = 0

then

(since the point (0 is a vertex). and

JEJ

Le{ - 0 JEJ

Le{ ~ 1. Therefore,

implies

JEJ p

p

L(~ Le{ ~ L(~. t-1

JEJ

t-1

p

Hence. 1(0 1

=

L(~~ LrJ ~ 1r I. t-1

Thus. the origin and all the angular points of the

JEJ

= 1r I.

The condition

+ pP ~)

belongs to the

domain Q lie in one and the same side of the hyperplane 1(I (r. p) E rk(er) implies that the point A

= (.s1 + pi

~ ..... .sP al

ap

domain Q. Consequently. the point B. the intersection of the straight line OA with the boundary of the domain Q. satisfies (7.2). Set At = gla; (.st - (t) p

+

pt ~ 0 (i = 1.... ,

pl.

Then

p

LAtert = L.stet + t-I

'-I

Further. p

L(A, - p')<ert. a>

=:

•

p

La;(.s' - (t) ~:

•

p

(

L.sta; - k a;)

> O.

7.4. Definition. We say that a multiindex 't E z~. 't = (a.. 13). satisfies the condition S(k) (and write 't E S(k» if at least one of the following n inequalities holds:

111

f3 I

+ ... + f3II"'II >

"'I

k

"'II (1

S

ssm).

7.5 Theorem. The condition S(k) implies MS(k) .

< 't

°

• Rearrange the numbers 9 1, ••• , 9" in such a way that 9 1 < ... < 9 t < < 9 t + 1 ... < 9" and change accordingly the numeration of components of multiindices. Let

E

Suppose, for definiteness, that the inequality Ch'l

S(k).

fulfilled. Set £~

£~

=

°

= 1,

(s '" 1 - r

9 *t

=

1* a 91

Hence, 't

E

+

£~

1), 15 1

<£h 9>

+ ... +

=

°

(s '" I), £~.I

= a. 1,

= - 7I. t <

f"* a 9f"

... , af"

°

= - ex. I 71.1

(i

= 1, £~ = = a.f". Then

= 1,

+ ... +

a.f"7I.f"

>

k 7I.f" is

°

(s '" 1 - 1), ... , £~""+I == 1,

7I.f"

= k* 9f'"

... , r);

f" - ... - a. 7I.f"

< -k

MS(k).

7.6. Remark. Thus, the conditions S(k), MS(k) , and 2I(k) introduced above satisfy the

relations S(k) .. MS(k) .. 2I(k).

Therefore, each of these conditions ensures the possibility of CC smooth linearization of the diffeomorphism X 1-+ Lx + p(X) , where p(x) is a 't-divisible resonant polynomial. It is easy to show that the converse implications do not take place. In fact, the multiindex (5, 6, 0, 8) satisfies the condition MS(8) (see examples 4.2, 4.4 and remark 4.6) but does not satisfy the condition S(8). Hence MS(k) does not imply S(k). Let us show that 2I(k) is, indeed, weaker than MS(k). 7.7. Example. Consider the multiindex

't

= (0,

resonant monomial ;y:Oy~ (see Example 4.7). Here 9.

= 200.

Show that 't

E

21(13).

Set

IT1

= (0,

5, 10, 4) 91

that corresponds to the

== - 690,

0, 1, 0),

IT2

92

= - 300,

= (0, 1~'

0,

~).

93

= 1,

Then

112

= = 1. Check that determined by the inequalities

T

e A(13; cr), cr:J. Consider the convex domain Q c ~

(j ~ 2::

= 2,

3, 4),

1,

~Il_ + ~u 100-":

4

The domain Q has exactly two vertices: A

4

z:

1.

= (1~,

1, 0) and B

=

(0, 1,

~).

We get

4


A>

=

LTJAJ

=5

100 83

J-2



= 10 . 1

+ +

10 . 1 = 1330 > 1300 = 13 . 100 = k max A J 83 83 83 J-l,3,4 ' 4 . -4

5

= -66 > -65 5

=

13

5

=

k max BJe J-2,3,4

Thus, T e A(13; cr), cr:J and, consequently, T e 21(13). Nevertheless, the maximal value of k such that T e MS(k) is equal to 9. Therefore l!I(k) does not imply MS(k). 7.S. Theorem. If the operator L is contracting or expanding, then the conditions l!I(k) and S(k) are equivalent. ~

By Remark 7.6, it suffices to prove that in this case l!I(k) .. S(k). For definiteness, let L be an expanding operator. If T e l!I(k) , then T e A(k, cr) for some normalized 9-collection cr = (cr), ... , crp )' Since L is expanding, we have m = n. Rearrange the invariant subspaces XIo ... , Xn in such a way that 0 < 9) < ... < 9 n • By s denote the maximal number j e {I, .... n} such that e {I, ... , pl. Let us verify the inequality

Since

=

1 (i

=

cr{ > 0

1, .... p), the point

U=

(9) •.•.• 9 s ,

0, ... , 0)

is a vertex of the domain D defined by (5.3). Hence it follows that

for some i = io

113

= Tiel + ... + TSes > k max e, = k es ' ,- 1 ,s

Therefore,

T E

S(k).

7.9. Derlnition. The multiindex T = (ex, 13) satisfies the condition i!'1(k) if at least one of the following four properties is valid: (1) there are a number r, 1 s r s I - 1, and positive real numbers I:r+1 S ... S I:l such that (r

+

1s i

:5

I)},

(7.S)

(p

(2) there are a number s, such that

1:5

= r + 2,

ssm -

1,

.. " I);

and POsitive real numbers I:s +I:5 ...

S

I:m

(7.6)

(p

=s +

2, .. " Ill);

114

m

(4)

L

(fJJJ.lJ

>

k J.lm •

J-=I

7.10. Definition. The multiindex T = (a., (fJ) satisfies the condition Irl(k) if at least one of the following properties is fulfilled: (1) there exist a number r, 1 ~ r ~ I - 1, and positive real numbers C 1 ~ ... ~ Cm such that J

r

m

t-I

J-I

La.t~t + L ~ cllJ

> k max {~r;

Il J

(1

~j ~

m)},

cJ

J

(7.7)

= 2,

(q

(2) there exist a number s, such that s

L

(fJJJ.lJ

1

+

~

s

m - 1, and positive real numbers

t

>

k max

~t {J.l s ; -

(1

~ j ~

CI ~ .. , ~ C1

l)},

Ct

Ct

m

L~J"'J + L 1~J+ "'J

J-I

~

L a. ~t t-I

J-I s

1

... , m);

J-s+I

CI

>

k max

{"'s;

1

"'m

+

}, CI

(7.8)

115 s

m

(q =

2, ... , I);

m

(4) [(3JftJ

> k

ftm .

J~I

7.11. Tbeorem. The condition !!'I(k) implies fI(k). Fix a number r, 1:s r:s 1- 1. Let EtJ denote some positive numbers (i = r + 1, and p = m(1 - r) + r. By 11' = (cr l , •.. , I1'p) denote the normalized a-collection such that ~

..• , I; j = 1, ... , m)

... ,

The condition A(k, 0") can be rephrased in the following way: if •.. , vm ) is a vertex of the polyhedral convex domain D determined by U, 2:

0 (i

=

r

+

vJ ~

1, ... , I),

0 (j

(Ur+I' ••. ,

uz ,

VI'

1, ... , m), (7.9)

u(

1

+

c'J

~t

+

EtJ

vJ -

2:

1

(i

=

r

+

1, ... , /; j= 1, ... , m),

IlJ

then r

Lcx(~( (=1

m

+

(

L a. U, t-r+1

+

L{3JvJ

>

k max {~r; ur + l , •.. , uz ,

J=I

Let the property (1) of Definition 7.9 be fulfilled. Set

VI' ••• , v m }.

(7.10)

116

=

Ctj

c,

(i = r

+

1, ... , 1; j

= 1,

... , m).

Then system (7.9) takes the form

r + 1, .... I),

U,

a: 0 (i =

VJ

a: 0 (j = 1.... , m),

1

+

U,

c,

(7.11)

E, a: 1

+

vJ -

(i

I'J

A,

=r +

1, ... , 1; j

= 1,

... , m).

Let us show that the convex domain D c: IRn -r defined by (7.11) has no more than I - r

+

1

vertices of the following structure:

Ao = (

Ar+l =

A r+l

l+c r +l

AI

..... 1

)

' 0, ... , 0 ,

+£1

(0.... , 0, ~, .... Cr + 1

= (Ar+ 1 (

1

Er+ 1) •

CP -

+

I'm ), Er+ 1

£r+1

(p = r

+ 2.... ,

I).

A = (U~+h ... , U~, v~,

... , v!) be a vertex. Suppose there exists a number t E {I, .... m} such that v~ = O. Then, by (7.ll). u~ = A,/(1 + E,) (i = r + 1, .... I). hence v~ = ... = v! = 0 (thus, we get the vertex Ao). Let v~ > 0 (j = 1,

In fact. let

.... m). Assume u! (i = r B

+

such that

P

E

{r + 1..... I}. Show that p < I implies u~ = 0 in virtue of (7 .ll). hence,

R.ea1lY,

1.... , I).

= (u~+lt

u!+1 = .. ,

= 0 for some

... , U~h 0, ,,_, 0, v~, ... , v!)

= u~ = O.

U!.I

$

If U~+1

0, P VJ

it

r .f-

a: ~ f:p

also belongs to D.

= 0, we get the vertex

the point

Since A. is a vertex

A r + 1• Letp be the maximal number

2. Then system (7.11) is equivalent to the following one: (j

=

1.... , m),

up

= ... = Ul

= O.

117

Because these relations are independent. we get the vertex Ap. Let us show that there are no other vertices. In fact. let A be a vertex and u~

=r

= 1•...•

1•...• I). v~ > 0 (j of some subsystem of the system

(i

+

(i

> 0

m). Then this vertex is the unique solution

=r +

1•...• I;

J = 1•... ,

(7.12)

m).

But (7.12) has the solution A o, a contradiction. Substituting the coordinates of the vertices Ao. A,.+1o •..• Ap into (7.10) and applying inequality (7.S), we establish that 't' E A(k, 0'). It remains to note that the property (2) of Definition 7.9 is equivalent to property (1) for the operator L- I , and, besides, (3) ~ S(k), (4) ~ S(k). 7.12. Theorem. The condition I!:l(k) implies S(k). ~

E'J

Let

= EJ

't' E

(i

=

Suppose the property (1) of Definition 7.10 is fulfilled. Set r + 1, ... , I; J = 1, ... , m). Then system (7.9) takes the form

I!:l(k).

U, it

0 (i

=r +

VJ it

0 (j

= 1,

u, 1 +

EJ

~t

+ vJ

1, ... , I),

... , m). EJ -

it

1

"'J

(7.13) (i

= r + 1,

... , I;

J = 1,

... , m).

Using the same arguments as in the proof of Theorem 7.11, we can check that the convex polyhedral domain Dc: the following structure:

Bo

=

(0, ... , 0,

IRn-r

determined by (7.13) has no more than m

"'I. ... , "'m ), EI

Em

+

1 vertices of

118

(q

= 2,

... , m).

Substituting the coordinates of the vertices Bo, BI , inequality (7.7), we establish that 't E A(k, 0').

... ,

Bm

into (7.10) and applying

7.13. Dermitions. We say that the multiindex 't = (a., /3) satisfies the condition It'(k) if T E It'1(k) or 't E It'2(k). The multiindex 't is said to satisfy the condition C(k) if at least one of the following properties holds: (1) there are numbers I: > 0 and r, 1 :s r :s I - 1, such that ,.

l

( ( + -1- L 0::>', L0::>', 1+1:

(-I

:>'z > k max {:>',., - },

1+1:

(-,.+1

(7.14) ,.

m

Lo:'i\, +! L/3JjJ.J

> k max {:>',.,

jJ.m };

I:

(2) there are numbers

I:

J-I

(-I

> 0

I:

and s, 1 :s

S

:s m - 1, such that

(7.15) IS

L/3 JjJ.J

I:

L0:':>., (-I

> k max

> k:>.z or

L/3JjJ.J

> k

{jJ.s, i\z }; I:

(-I

m

Z

(3)

La.'i\,

+!

J-I

jJ.m'

J-I

7.14. Theorem. The condition C(k) implies Il:'(k).

119

• It suffices to note that putting £( = £ (i (7.14) (and similarly, (7.6) goes over into (7.1S».

=r+

1, ... , I),

(7.S) turns

into

7.1S. Theorem. The condition S(k) implies C(k).

• For definiteness, suppose that the following inequality holds: a: 1~I

Put

£

= max { ~l

~r

-

+ ". +

a:r ~r

> k

~r'

1, /-1m }. Then the condition C(k) is fulfllied. ~r

7.16. Notation. Fix an integer r, 1::s r ::s I. Set

For

£

Let

£1

> 0 put

be the solution of the equation

that belongs to the segment [(;\l - ~r) I ~r' /-1m I ~r] and

7.17. Theorem. Let T

=

leo

£'1.

be the solution of

be defined as indicated in Table 7.1. Then the multiindex (a:, (3) satisfies the condition C(k) for every integer k < leo.

120

Conditions

1\ I" - + -

?r?,

leo ?, -

:5

?r

1\ I" -+-

13

1\ - - + ;>'r f.Lm

f.L m

I'].

1 :5 1 ?r

+

?r

I" +

£

?r

;>.,

13 1\ 13 >- -+-

f.Lm

f.L m

?r

f.L m

?, -

?r

-;>'rf.Lm

-

;>'r

?, - ?r 1\ 13 I" -+- >1\--+?r?'

?r f.Lm

?r

1\

f.Lm

+

?r

f.Lm

1> 1 J\

+

?r

f.Lm

?,

K\(£\)

£

\

f.Lm

?r

f.Lm

1\ 13 -+?r

f.Lm

13 I" -<--

1\ I" -+-

J3 I" -it--

K\(£")

?,

?r

f.Lm

13 I" 1\ + -it-+?,

?,

13 :5-

?, -

?r

?r

?,

f.Lm

-?r

;>., - ?r

-;>'r

1\ 13 I" +-<-+?,

?r

f.Lm

?,

Table 7.1

?, -

?r

£"

121

• Note that JI').;I

K I (£)

+

+ £»)"1,

J 2[').,. (1

if

£

2

if

£

<

').1 -

').,.

').1

').,.

={ JI').il(l

+ £) +

J 2 ').i l ,

-

').,.

Denote Ko(£) = min {KI (£), K2(£)}' By Theorem 7.14, to prove the assertion, it suffices to put leo = max {KQ(£): £ E (0, + co}}. Since the function KI (K-J is monotonically increasing on the interval

( 0,

tonically decreasing on the interval (').1 - ').,.,

~I ~

,.

Besides, the function ;

/

').1

- ').,.,

",.

(2) Let

KI

and is mono-

').,.

+ co) «( j.Lm, + co),

').,.

the function Ko(£) on the interval between

«( 0, ~ )

').,. )

we have to maximize

').,. ').1

- ').,. ~,.

and

j.Lm ').,.

is decreasing and the function K2 is increasing on the interval

j.Lm ]. ~,.

KI (')., ",.

~,.) >

K2 (

').1 ').,.

~,. ).

If in this case

KI (

j.Lm) '). ,.

>

K2 (

~m ), i.e., n. ,.

122

and let

(b) Let 1

£1

be the solution of the equation

> 1. Then

Observe that the function Kl is increasing and the function K2 is decreasing on the segment [IJ.m, ~t ~r

Let

12 _ ~

~t

-

~r

].

~r

13

~t - ~r '

and let

£

2

denote the solution of the equation

123

7.1S. Dermition (see Samovol [5]). Let 1:s r :s I, duce the following notation: r

1:s s :s m,

= (<<,

T

(3). Intro-

S

LatAt - k A,.,

p~".(-t)

t-I

We say that T satisfies the condition SoCk) if Pk.(T) i!:: 0 and, besides, implies that at least one of the following two conditions holds: (1)lal >0,1131 >

oand eitherpic(T) =0,

mina.,.< lal,orp~(T)=O, minf3s < 1131; "C'EM_("C')

(2)

pic(T) = p~(T) = 0,

Pk.(T) = 0

SEM+("C')

max {I a.1, If3l} > k.

7.19. Theorem. The condition SoCk) implies C(k) . • If Pk.(T) > 0 then the condition S(k) is fulfilled, and it remains to apply Theorem 7.15. Assume Pk.(T) = O. Let the condition (1) take place. Suppose, for definiteness, that pic(T)"= p~.,.(T) == 0, r < I, 1a I > Ia,.l. Then J I = k A,., J2 > 0, J3 > O. By Theorem 7.17, the relation ko > k holds, i.e., T E C(k). Suppose now that the property (2) is valid. Without loss of generality, we may assume that lal > k,

alAI

+ ... + alAl = k Al,

f31J.1.1

besides,

ri

< k, J I =

L«tAt,

+ ... + f3mJ.l.m

= k J.l.m,

al

'"

0

and,

m

l-I

J 2 = alAl, J 3 =

t-I

we have J I = (k - «l)A 1• J3 = k"m. Therefore,

L{3JJ.I.J '

J-I

Since

p~(T) = pic(T)

= 0,

124

-

JI

>"-1

-

J" >.,

+ - -k =

JI

>'1-1

J ("

>'1-1

+

(k - « )>'1

Jl.m

>"-1

+

cr. - k

= (k - «, )(>',

- >"-1)

> 0,

,

J3

+ - -k =

I ", - I

Jl.m

Al_ I A,

By Theorem 7.17,

(k - cr.')>."

ko >

) +....! J _k = A,

>

0,

(k _ «')

Jl.m

> O.

A'_I

k and, consequently,

T E

C(k).

7.20. Remark. Thus, we have obtained several relatively simple conditions each implying the condition 21(k) and, consequently, sufficient for c! linearizability of the corresponding diffeomorphism (vector field). The main logical hierarchy between these notions is indicated in Table 7.2 (where arrows denote, as usual, implications).

SoCk) -

C(k) -

~ S(k) _

It' (k)

~ MS(k) _

21(k)

Table 7.2 7.21. Example. Let us apply Theorem 7.17 to the vector fields (4.6) and (4.9).

(1) Reversing the time direction in (4.6), we get «I = 5, «" = 6, 13 1 = 8, AI = 1, >.,. = 3, Jl.1 =1. Put r = 1, I = 2. Then J I = 5, J" = 18, J3 = 8, 1 = 2 > 1. Hence

+ Jl.1) J" 10 18 S JI J" J" J3 -......;;....-......;...-+-=-+-<-+8=-+-=6>4=-~A I >." A" 3 3 3 >." Jl.1' >." A" - AI

JI(>'I

According to Table 7.1,

125

Thus, the condition C(9) is fulfilled. In fact, for

• «:>..

£

= 1

1 2 +- «~= 5 + 9 > -272 = 9 max {I, -32 } = k 1+£"2 1 «:>'1

+ -£1 (3 1J.il = 5 +

= 13 >

8

9

=9

max {I, I}

:>. 2 max {:>.. - - } '1+£'

=k

max

J.il {;\I' }. £

The coordinate change

is

f!

smooth and conjugates (4.6) with the linear system

(2) Reversing the time direction in (4.9), we get ~

= 200,

J.il

1 = 199/300

= 300,

r

= 1,

I

= 2.

Therefore,

= 10, «2 = 4, (31 = 5, ;\1 = I, II = 10, 12 = 800, 13 = 1500,

«I

< 1. Hence it follows

Thus, leo = K 1(£\ where £1 is the solution of the equation £2 - 149£ - 24150 = 0 that lies between 199 and 300. Take the approximate solution £ = 246. Then I 1 2 «:>.. + ~ «~

=

800 10 + 247 > 13

i.e., the condition C(13) holds.

=

200 13 max {I, 247 }

=k

:>. 2 max {:>." ~ },

126

7.22. Example. Let us show that t!(k) does not imply C(k). Consider the vector field

= 3, cr.3 = 3, ~I = 6, ~I = 1, J 2 = 75, J 3 = 36, 1 = 1~ > 1.

= 3, = 3,

cr.2

=2

then J 1

Here cr. 1 then J 1

~

= 5,

~3

= 20,

fl.1

= 1.

If r

=1

Besides,

hence,

If r

= 18,

J

J

~2

~3

= (£2'

£3)

J2

leo = ....!. + ..! =

Consequently,

= 60,

J3

= 36,

1

= 1~ >

1, and

6.6. Hence, the condition C(6) is fulfilled (but C(7)

does not hold). Let r

1

cr. ~I

+ -~

= 1,

£

= (4,

19). Show that

1

fl.1 £2

= 3 + 9 = 12 >

10.5

=7

3

• -2

't E

= k max

t!(7). In fact,

{~I'

fl.1

-

},

£2

Thus, the condition t!(k) does not imply, in general, the condition C(k). The change of variables

127

conjugates the vector field under consideration to its linear part and is of class C7 •

The condition S(k) for n

= 2, 3

Samovol [3] has shown that in the nodal (diagonal) case the condition S(k) gives the best estimate of smoothness of a linearizing map. For such equilibria, S(k) is equivalent to S(k) (see Theorem 7.8). Our next goal is to examine the condition S(k) for n = 2, 3, assuming that the equilibrium is of saddle type.

7.23. Lemma.

1fT E

= 1,

for every i

A(k, cr) and there is a number j

... , P then k <

<

(crl' ... , cr p )' Assume, for definiteness, that 0

=0

(i

It}),

uJ

a>

1
=

{I, ... , n} such that cr~

It

0

TJ •

• By Lemma 5.9, we may suppose that

Set U,

E

1

crJ

Ao(k, cr) for some a-collection cr

T E

crJ

cr J 1
I ,

a> 1

S

1
'

a> 1

(i

=

= 2, ... , p).

. Then

I

hence the point A. = (0, ... , 0, uJ' 0, ... , 0) lies in the domain D defined by (5.4). Since T E Ao(k, cr), the following inequality is fulfilled for ('1, p) E rk(cr): p

(T J

+

plcr{

+ ... + pPcr~ - rJ)uJ

> I>'Il.

'-I Put '1'

=0 T

SI

(i

+

It}), lSI

p crl

r!

+ ... +

In fact, for s. j, We get because crT

it

= k,

TSI

p pSI

p cr p

+

k. Hence, ('1, p)

= (1,

E

SI

=

T

'1S1

=

TS

- '1

cr~ -

0, ... ,0). Then SI

+

cr l - '1

+

cr~

SI

it

rk(cr) and therefore

SI

it

0

(s = 1, ... , n).

O. If s = j then

TJ

+ crT - k it 0

128

Consequently,

'tJ

> k.

7.24. Theorem. 't2

= 2,

Let n

<

9,

<

0

92

and

't

e l!I(k). Then either

't'

>

k or

> k. ~

Let

't

e l!I(k) , then

is a 9-collection and 9,

e A(k, cr) for some 9-collection cr = (cr" ... , crp). Since cr

't

<

0

<

we have either cr! '" 0 (i = 1, ... , p) or cr~ '" 0

9 2,

(i = 1, ... , p). Then, by Lemma 7.23, either

't'

>k

T? >

or

k.

Now let us consider the saddle case for n = 3. Suppose, for definiteness, that 92

<

9,

<

0

<

9 3•

Denote A, = - 9"

7.2S. Theorem. If n = 3 and

't

~

= - 9 2, 1.1 = 9 3 .

e l!I(k) , then at least one of the following

statements holds:

(1) (2)

't

e A(k, Ie\),

't

e A(k, Ie" Ie~, where K, is defined above and K2 = (0, 1IA2' 0);

(3)

't

e A(k,

(4) some ~

05

't E

A(k,

>

1IA2.

1e3), Ie"

where Ie, = (1IA" 0, 0); where 1e4),

= (0, 0, 111.1);

1e3

where

Ie,

is defined above,

1e4

= (0,

05, (ClA2 -

1)/1.1) for

The proof of Theorem 7.25 is based on two lemmas stated below.

7.26. Lemma. that crs

= (<<,

If

n = 3,

1 - «A,

't

e A(k, cr) and there is a number s e {I, ... , p} such

,0), O:s«

A2

~ It is easily seen that

T E

<

1IA"

A(k; Ie" Ie~

convex polyhedral domain defined by

then

iff

't E

't'i\,

A(k;

+

Ie"

't1Az

Ie~.

>

kAl.

Let D

be the

129 3

UJ ~

0 (j

= I.

2. 3).

Lu.p{ ~ 1

(i

=

I •...• p).

J-\

Consider the point B

= ().\.

).2'

0). Observe that

=1

and

Therefore. BED. and since the segment U ~ O. = 1 is compact. we see that the point B belongs to the convex hull of vertices of the domain D. Hence.
B> - k max B, > O. ,- \ ,2,3

i.e .• \

T ).\

Thus.

T E

A(k;

+ T2~ >

k).2'

IC\I IC~.

7.27. Lemma. If n = 3. T E A(k. cr). crt = (1/).\. o. 0). cr2 = (0. and E > 1/).2' then there exists a number IS such that T E A(k; IC\. 1C4)'

E.

(E~ - 1)/Jt)

• Denote

If there exists a number i E {I. .. .• p} such that ~,.- 0 and «,).\ + ~').2 = I. then the assertion holds by virtue of Lemma 7.26 for IS = 1/).2' Suppose now that ~,.- 0 implies «,).\ + ~').2 > 1. Consider separately two cases: (a) «,E).\ + ~, ~ E (I = 3..... p). Then the domain D determined by

(7.16) U\«,

+ ~, + ~(<<,).\ + ~').2

has exactly two vertices: A\ if u\ ~).\I ~ ~ lIE - ~(E~

= ().\I -

1/E. 0)

-

I)/Jt

and A2

1)IJtE and ~ ~ O. then

~

I.

= ().\.

O. JtI(E~ -

1».

In fact.

130

Hence, (7.16) is equivalent to (7.17) Consequently, the domain D has the vertices AI and A2• Put a = E. Since 't' E A(k, cr), the relations (5.2) for AI and A2 take the form

i.e.,

't' E A(k; "I' "4)' (b) There is a number S

~s

>

{3, ... , p} such that OI:sEi\1 + ~s < E. In this case ¢ 0, because ~s = 0 would imply asi\1 I: 1 giving asEi\1 + ~s I: E. If (1 - 01:Ji\I)/fj J (1 - OI:ti\I)/fjt for i, j E {3, ... , p}, ~t ¢ 0, fj J ¢ 0 then E

In fact, the last inequality may be rewritten as

i.e., Without loss of generality, we shall suppose that (1 - "-3i\1)/~3 3 :s i :s p, ~t ¢ OJ. Then in virtue of the above arguments

Consider the points

=

max {(I -

OI:ti\I)/fjt:

131

Let us show that B J and B2 are vertices of the domain D. In fact, for B J we get


B.>

<17'3'

BJ>

{3, ..

0, i

E

pl.

{4, ... ,

«,i\.

If

{3,

+

= I:

BJ>

1-

a.Ji\.

it I:

1-

fl3

(1 - a.3i\.)

= "Ji\. + {3J

BJ> =

<17'"

if

= 1,
a.si\J fls

> 1;

= 1;

{33

1-

{3t

"Ji\.

+

it a.ti\J

{33

=0

then

i\.a.,

l!:

1-

"ti\J

f3, =

1

{3,

1.

For B2 :

(1 - "si\ J)(£i\2 - 1) l!:

+

Q:~.

{3~2 -

> f3~2 + a:~. «,5'i\. + f3 s i\2

{3,

=- 0, i

Put a =

= (1

B2 >

= a:ti\J +

a:ti\.

f33 -~-. 1 - «3i\J

or e A(k, 17') yields

pl.

If

(3,

s i\J)£i\2

' a.Ji\J

{4, ... ,

Q:

+f3 s i\2 - 1

= 1.

1 1

= "Ji\J +

E

-

a.si\J

<17'3' ~>

<17'(0

if

-

1

+

(3Ji\2 - 1

I'

+

(3,i\2 -

f.I

1

f.I(1

-

a:Ji\J)

a: 3 i\1

+

f3 3 i\2 -

1'(1

-

~i\.)

a:Ji\.

+

{3Ji\2 -

1

= 1·,

1

= 0 then

Let us check that T E A(k; K., KJ. Really, for B. the condition

132

i.e .•

For B'}.. we get

,ho. +

't 3

>

JL(l - ot]A.) ot]AI

+

~]A'}. -

k max {A.. J.L(1 - ot3AI)

I

+

ot]A.

}.

1i3A'}. - I

i.e .•

This just means that

't E

A(k. " ••

",.>.

Now let us tum to the proof of Theorem 7.25. Let number i E {I •...• p} such that cr~ 1 - otA. - - - • 0). where 0 s ot < I/A 1• and

= O.

't E

A(k. cr).

Suppose there is a

In this case. if cr~;I: 0 then crt

= (ot.

•

't

A'}.

e A(k; " •• ":z} accordmg to Lemma 7.26. If

cr~ = O. then crt = (lIA •• O. 0). Consider the following two possibilities: (1) cr!;I: 0 for every

S E

{I •. , .•

pl.

Then k

<

by Lemma 7.23;

'tl

(2) There is a number S;I: i such that cr! = O. Then crs = (0. E. (£~ - I)/JL). If. in addition. £~ = I. then 't E A(k; " .. K:z} by Lemma 7.26. If £~ > I then 't E A(k; " •• '(4). in accordance with Lemma 7.27. It remains to notice that when

i.e..

't E

cr!;I: 0

for every

S

E {I •...• p}. then

A(k. "3) by Lemma 7.23.

solution of the equation

lying between (A'}. - AI)/AI and J.LIA I ; '}.

't+

J.(I

£'}.

+ A'}.

be the solution of the equation £) __

~.

J

£

I .+.

JL

k

< i.

133

belonging to the segment [piAl'

KI(c)

=

{

~I

+

(~ - ;\1)/;\1);

J

2 [;\I(1

+

c)]

JI( 1

+ c)/A'}. +

~1

'1.

~I(A'}. - AI)

3 +T,

Consider the domains:

I

~+T~

fJ.

I

'1.

T+T~

TI(~ - AI) fJ.

3 +T,

134

Example Domain

leo

£0

;\1 3

1:

7)1

~ I~

1

2

1: 1

2

I

1:2

I

1:3

0

1

2

3

0

1

:>'2 - ;\1 7)2

1: 1 +

1:2

1

2

1: 1 +

1:

3

1

3

2

1

2

2

1

4

4

0

3

2

1

2

2

3

1

2

1

2

2

0

4

3

1

4

1

0

1

3

1

5

3

3

2

4

1

4

2

2

1

1

1

3

1

0

2

3

:>'1 ~

7)3 :>'1

I ;\1

7)5

2

1:-+1:

7)4

;\2

£

I

KI (£1)

I ;\1 1:

7)6

-

2 T

+

;\2 1:3

7)1 ;\2 - ;\.1

TI +

7)8

T

2

;\.1

7)9

7)10

£

2

2 K I (£ )

T

3

Table 7.3 7.29. Theorem. For given values ;\.1' ;\.2' ~. according to Table 7.3. If k < ka. then T e 21(k).

I T.

2 T •

3 T •

define the number

leo

135

~ Note first that in order to exhaust the whole space of parameters

~I' ~2'

"', TI,

we have to consider, besides the domains D, - D,o, the following domain D,,:

T?, T 3 ,

T'(~,

> "',

~ -~,

~2

the value of ~,T' three cases, fact, ~2 - ~,

+ "')

",T3 it:

> '"

+

2

+T >

~2T2

- ",T3

T

,

3

+ T.

However, by virtue of the resonance condition,

belongs to

the set {~" ~2'

- ",}.

In each of these

~,T' + ~2T2 - ~ it: ~2(T2 - 1). Show that the domain D" is empty. In implies ~/", > 1 and (~, + "')/~2 < 1. Therefore, the relation

T'(~,

+ "')/~2 + T2 > T' + T3 gives T2 > T3. Besides, from T2 > 1 we get T3 it: ;>'2(T2 - 1)/", > T2 - 1. Thus, if T2 > 1, then T2 > T3 > T2 - 1, contradicting the fact that T2 and T3 are integers. Consider now the case T 2 !iS 1. Since T2 > T 3, we have T2

= 1,

= O. The resonance condition implies

T3

T'

= O. Therefore,

= 1, contradicting the assumption I T I it: 2. Consequently, D" = Table 7.3 establishes examples showing that all the sets D, - DIO finish the proof, we refer to Theorems 7.17 and 7.25.

IT I

= T' + T2 + T3

We point out that are non-empty. To

Ill.

7.30. Example. Let us consider in detail. one of the examples given in Table 7.3, namely, the following vector field: _.3 2' ' = 2y. x,• = - x, + x,xV' , X2 = - 22'x Y

Here

= "';

;>.,

= 1,

T'

+

T2

i\2

= 2,

=4 >

(7.18)

= 2, T' = 3, T2 = 1, T3 = 2. Therefore ;>'2 - ~, = 1 < 2 = T'(i\2 - ;>.,)/", + T3; ~T2/(;>., + "') = 2/3 < 2 = T3. Consider

'"

7/2

the equation

3

+ _2_ = 3£ + 2 1

Let

= [1;

Thus,

£'

be

the solution

2]. Clearly, 7/6

(;>."

<

+

3£2

+

£- 6

= O.

of this equation lying in the segment

£'

;>'2' "', TI, T2, T 3)

<: 4, according

2

£

ii

<

[(;>'2 - ;>.,)/i\"

"";>'1]

4/3, hence,

(I, 2, 2, 3, I, 2) e Ds. Therefore, 3 <

to Table 7.3. Take the approximate solution

£'

0$

£ ii

ko =

KI (£')

6/5. Then we have

136 "I = (1/:>'10 0, 0) = (1, 0, 0), "4 = (0, £, (£:>'2 - l)/J.I) = (0, 6/5, 7/10). The corresponding domain D (see (5.3» is defined by UJ 2:

0 (j

= 1,

2, 3),

and has the following two vertices: A. ditions (5.2) take the form

3 Thus,

+

-r II (-rl, -r2, -r3) E A(3, "I'

= (1,

>k

20 7

ul

"J.

2:

~ ~ + .1 ~

1,

5

0, 1017) and B

10 3 7 '

+

5 -6

Consequently,

2:

10

1

= (1,

5/6, 0). The con-

> k. system

(7.18) is locally

C

linearizable. 7.31. Remarks. (1) We conjecture that Theorem 7.29 gives the best possible value of the smoothness class of linearizing coordinate changes (in the diagonal case).

(2) Recently we have obtained some results analogous to Theorem 7.25 for n = 4, but we have not included them because they are only fragmentary. Investigation of this case is in progress.

f 8. Theorems on

ck

Normal Fonns

In this section, the results obtained thus far are applied to prove several theorems concerning ck normal forms of finitely smooth vector fields (diffeomorphisms) near an equilibrium (fixed point, respectively).

8.1. Lemma. Let L: E ... E be a hyperbolic linear operator; q: E ... E be a resonant polynomial of order "(q) not less than 2; -r be a multiindex satisfying the condition S(k); p: E ... E be a -r-divisible resonant polynomial. Then the mappings F(x) = Lx + q(x) and G(y)

= Ly + q(y) + p(y)

are

ck

conjugate near the origin.

~ Define the number Qo = Qo(k) by formula (3.1). When proving the lemma, it will be useful to return to the scalar variables %1' ••• , Zct (see subsection 2.3). Denote

137

E(w)

Let

VI'

=

{r

E

Z!:

==

{r

E

z~: er(r) = er(w)}

••• , I'd

rl.1

+ ... +

rt.ml

= Wt.1 + ... + Wt.mt

be the eigenvalues of the operator L.

i - j if I V t I = IvJ I. For every multiindex for at least one number i E {I, ... , n},

For i, j

(i

E

=

1, ... , n)}

{I, ... , d}, write

which satisfies the condition q~;e 0 introduce additional monomial variables

W

= i~

°

(r E E(w» and form the vector ul == fil(Z) = {u!: r E E(w)}. Let 1 be the dimension of this vector and BI denote the d x 1 matrix composed of the elements

u!

btJ =

°

q! (.,.

E

E(w),

i

= 1,

... , d).

Describe the transformation of the vector ul induced by the map F. Since the subspaces E I , .... En are invariant under the operator L = {IU}l.J-I .....d we have

Zt 1-+

r

Qo

It;ZJ

+

r

q!zw

(i = 1, ... , d),

I W I =1C(q)

t- J

and, consequently, d

u!(Lz + q(z» I W 1-IC(q)

Hence it follows that the induced transformation is of the form

where CI is a linear operator such that the moduli of its eigenvalues take values of the fiorm

..,1

d

11"11 ... IVdl"', and ql(z) is a resonant polynomial. Moreover. since q is resonant and 1.,.1 2: 2. we conclude that these moduli belong to the set {lvII, ... , 1",,1}, and K(ql) 2: rc:(q) + 1. Thus. we get the mapping

Continue the process of introducing additional monomial variables untill the least degree of monomial terms that are not T-divisible exceeds the number Qo(k). As a result.

138

we get a mapping of the form

where u' are gcvectors (i

= 1,

... , r); the moduli of the eigenvalues of the operator

(where the blank entries are zeros) belong to the set {exp 9 1, ... , exp 9 n }; qr is a resonant polynomial having contact of order (Qo(k) , k) with the zero map. Besides, it is easily seen that the

c! smooth manifold

is invariant under t r • It follows from Theorem 3.3 (reformulated for diffeomorphisms) that conjugate with the map

~r(~' v)

= (L~ + Bv,

Cv)

(~E E, v

= (VI,

tr

c!

is locally

... , v».

Let (z, u) = he!;, v) II (hl(~' v), ~(~, v» be the conjugating map. Then the map has an invariant manifold v = ~(~), where graph tp = h(graph ~), t.e.,

~r

(8.1)

Put H(F.) = hl(F., ~(F.», R = HI 0 F 0 H. Since h has contact of order (Qo, k) with the identity map, the same is true for the mapping H. Therefore, R(F.) = ~ + q(F.) + 1(F.) , where I is a c! smooth function having contact of order (Qo, k) with the zero map. Since h = (hI' ~ conjugates ~r and t r , we have

(8.2)

139

+

C~(~, v)

Letting v

= ~(I;),

q,.(hl(~' v»

=

h,,(1.(

+

Bv, Cv).

(8.3)

we deduce from (8.2) that

Taking into account (8.1), we get

or, equivalently, (8.4)

That F and R are conjugate means

Comparing (8.4) and 8.5) yields h,(Lt;

+

B~(I;), cq;(~»

The equality (8.3) for v

whence it follows U

= "(l»

=

= ~(I;)

hl(Lt;

+

+

q(l;)

1(1;), ~(LI;

+

q(l;)

+

1(1;))).

(8.6)

gives

(because of (8.1)

and

the invariancy condition for the manifold

that

In virtue of (8.5), h,,(Lt;

+

B~(I;), C~(I;»

= ,,(hl(LI; + q(l;) + 1(1;),

~(Lt;

+

q(l;)

+

1(1;»))).

Applying once more equality (8.1), we get h1(Lt;

+

B~(I;), C~(t;»

= h,,(Lt; +

q(~)

+

I(t;) , ~(I.(

+

q(l;)

+

Since h is a local diffeomorphism, it follows from (8.7) and (8.6) that

1(1;»).

(8.7)

140 B~(~) = q(t;)

+

1(t;) ,

(8.8) C~(~) = fi(Lt;

+

+

q(t;)

Let us show that the mappings G and R are locally

1(t;».

eft conjugate with

one another. By

Remark 6.20, it suffices to find an invariant section of the extension

[

Y ]1---+

[LY

t;

+ q(y + t;) + p(y + + q(t;) + l(~)

L(

Since I has contact of order

t;) - q(t;) - 1(t;) ].

with the zero map,

(Qo, k)

(8.9)

the extension (8.9) is

eft

conjugate to the extension

1---+ [Lx + q(x + t;) + p(x + t;)

x]

[

t;

L(

+

Consider the additional variable v, (S.lO) to the extension

q(t;)

+

- q(t;) ].

(8.10)

l(t;)

v = ~(t;).

Taking into account (8.S), pass from

Lx + [q(x + t;) - q(t;) + p(x + ~) - p(t;)] + p(t;)

1---+ [ Lt;+Bv

(S.l1)

Cv The expression in square brackets is a resonant polynomial vanishing for x = 0, and p is a or-divisible resonant polynomial. In virtue of Theorem 6.21, (S.ll) has a invariant section x = g(t;, v). invariant section x = g(t;, f(t;».

Therefore, Thus,

the extension

(8.10)

has a

the extension (S.9) also has a

invariant section and, consequently, the mappings G and R are locally Because F = H origin.

0

R

0

8.2. Lemma. Let ~

HI, we conclude that F and G are also

= At;

eft

eft smooth eft smooth eft smooth conjugate.

eft conjugate near the

be a hyperbolic linear vector field, q: E -+ E be a resonant polynomial with K(q) ~ 2, or e l!I(k) , and p: E -+ E be a or-divisible resonant

141

polynomial. Then the vector fields conjugate near the origin .

x = Ax + q(x)

and

Y = Ay +

q(y) +p(y) are ~

• Use the method of introducing additional monomial variables, likewise in Lemma 8.1 above.

8.3. Theorem. Let K and k be positive integers, ~ be a c< vector field on E, the origin being a saddle rest point, A = D~(O). If K it Qo(k) then the vector field ~ can be reduced, by means of a ~ coordinate change near the origin, to the resonant polynomial normal form n

Qo

y = Ay + E E p;l'et t-I

where p; - 0 implies at

=

(y

E

1.,.1-2

and cr _ S(k) .

• By Theorem 3.S, the vector field ~ is locally (3.3) which, in tum, is 8.2.

(8.12)

E),

~ equivalent to the vector field

~ conjugate to the vector field (8.12), by virtue of Lemma

8.4. Theorem. Let k be a positive integer, ~ be a ~ vector field on E having the origin as a nodal rest point, and A = D~(O). If k it Q1 then the vector field ~ can be reduced, by the aid of a ~ coordinate change near the origin, to the resonant polynomial normal form n

Y = Ay +

E E p;y'"et t-I

where p! - 0 implies at

QI

(y

E

E),

(8.13)

1.,.1-2

=

and cr _ S(k) .

• The validity of Theorem 8.4 follows from Theorems 3.6, 7.8, and Lemma 8.2. The next two theorems are completely analogous to Theorems 8.3 and 8.4. Therefore

142 their proofs will be omitted.

8.5. Theorem. Let k I!: 1, IE Difto(E), K I!: Qo(k) and L. Df(O) be a hyperbolic linear operator of saddle type. Then near the origin the diffeomorphism f is c! conjugate with the map n

fly)

00

= Ly + rL rL PerY'ere,

(y

E

E),

(8.14)

,=\ 1.,.1-1 where P;;I: 0 implies a,

=
9> and

8.6. Theorem. Let IE Difto(E), k

I!:

IT

tl

21(k).

QIo and L

rator 01 nodal type. Then the diffeomorphism I is resonant polynomial normal form

Ii

c! reducible near the origin to the

(y

where P;;I: 0 implies

9,

=


a> and

IT

tl

E

(8.1S)

E),

S(k).

8.7. Theorem. Let k be a positive integer, ~ be a

origin, and K I!: Qo(k)

Df(O) be a hyperbolic linear ope-

c<

vector field vanishing at the

+ k. Then ~ can be locally reduced to the c! generalized resonant

polynomial normal lorm (3.36) where p; .. 0 implies a, q.,.(Z) • 0 implies 0 = < IT, a > , IT tl 21(k).

= ,

IT

tl

21(k), and

~

The proof of Theorem 8.7 is similar to that of Theorem 8.3 (see Remark 6.19 and the method of introducing additional monomial variables exposed in Lemma 8.1).

8.8. Theorem. Let k be a positive integer, ~ be a c< vector field on E, and 0 be an equilibrium all 01 whose eigenvalues have non-positive real parts. if K I!: Q\ + k, then the c! generalized resonant polynomial normal form (3.37) of ~ contains only such resonant terms Per(z)xer that IT II! S(k).

143

We leave to the reader as an exercise to formulate and prove similar theorems for local diffeomorphisms near a fixed point and for vector fields in the vicinity of a periodic orbit (without hyperbolicity assumptions).

§ 9. Linearization of Finitely Smooth Vector Fields

and Diffeomorphisms In § 9, 10, we shall prove several theorems on linearization as well as a few more general theorems concerning normal forms of fmitely smooth vector fields and diffeomorphisms in the neighbourhood of a fixed point. These theorems supplement the results presented in the previous sections, but essentially differ from them in the techiniques of proof. The last section, § 11, is devoted to a comparison of all the results presented in this chapter. In what follows, we shall freely use notation and definitions introduced in § 1 - 7.

9.1. Theorem. Let L: E -+ E be a hyperbolic linear operator and M, N, k be positive illlegers satisfYing M;>'I > k ;>." N J.l1 > k J.lm • Denote K = M + N + k, Q = M + N

+ max

{M, N}

>

K.

If f.

E -+ E is a

c" mapping which has contact of order (Q, K) with

the zero mapping at 0 e E, then the local diffeomorphism F(x)

= Lx + ft.x)

is

F c c"(E, E)

defined by

c! linearizable near O.

• Let U and V denote the contracting and expanding invariant linear subspaces of the hyperbolic operator L: E -+ E. Set LI = L I U, ~ = L I V. Then E = E $ V and L = LI $ ~. Denote Ji = pru • f, h = pry • f, where pru: U $ V -+ U and pry: U $ V -+ V are the canonical projections. By Theorem 1.4.7, we may assume with no loss of generality that Ji (0, v) • 0, h(U, 0) • O. For every numbers ro > 0 and £ > 0, one can find a > 0 and a mapping fa: E -+ E of class

c"

so that:

(1) fa(x) = j(x) if

IIxll

< a; (2) fa(x) = 0 if

IIXII it

ro;

(3) sup {IIDrfa(X)II: x E E} < £ (r = 0, I, ... , K). Henceforth, we shall assume that I is replaced by the truncation fa, where a corresponds to the numbers TO and £ to be specified later. According to Taylor's formula, write

144 j(U, v) = ./(0 ,VI~

+

D V\ no , V) U

+

•••

+

M M I (M _11) 1 D I ·1'{O , V)U •

(9.1)

J I

+

o Note that D~O, v) .•. , K - i;

i

= 0,

M.I

(1 - t) DM'j(tu v)uMdt (M - 1) I I' .

c""

E

and IID~D~O,

= O(IIVIIQ-q·,)

V)II

as

IIvll

-+ 0 (q = 0, 1,

1, ... , M - 1);

J(1(M- _ 1) I I

IIDq [ 1

M.I

t)

DM'j(tu V\UMdt]1I I

'I

= O(IIUII M )

o as

IIUII

-+ 0 (q

= 0,

1, ... , K - M);

[J (~M-_tL I I

IIDr

M.I

D~./(tu,

= O(IIXII Q·r )

v)uMdt]1I

(x

= (u,

v»

o as

IIXII

-+ 0 (r

= 0,

1, .... K - M).

Using once again Taylor's theorem and taking into account that Dkj(O, 0) (k = 0, 1, ... , K), we get

D'no v) V\'

=

J(1 I

=0

N I

- s)1)1• D'DNnO sv)~ds (N12.1\'

(.r

= 0, 1, ... , M - 1).

o Finally, apply Taylor's expansion in (9.1). As a result, we obtain M·I

v up

to order

N - 1 to the remainder term in

N·I

where

= ...! - s) • i! J(1(N-l)1 I

rp ( v)

,

o

N I

D'DNnO sv\ds 12.1\'

I

0 1, ... , M - 1) ; r=,

(.

145

( ) _

1

I/JJ U - j!

J(1(M- _ I)! I

t)

M I

-

o

= J(1 I

:t:(u v) ,

M

o

J

D ,Dlf(IU. O)~

J(1

(j = 0, 1, ... , N - 1);

I

-

S)H-I

(N - I)!

- t)M-I

(M - I ) ! ~"tDr:.tc.tu, sV)dtds • UM~;

0

(q = 0, 1, ... , M - i

+

k;

(q = 0, 1, ... , N - j

+

k; j

i:,.

=:

°,

1, ... , M _ 1) ,

°,

1, ... , N - 1) ,

(9.2)

(9.3)

(9.4) (IiUIi -+

0,

IIvll -+

0; p

+

q = 0, 1, ... , k).

Consider a mapping T: E -+ E of the form M-I

T(x)

=X + t(x) =x

+

+

h(x)

g(x) = x

+

H-I

Lht(v)ut~ + LgiU)uMyl, J-O

where ~(v)

=°and

g~(u)

=0

(the superscripts 1 and 2 refer to the projections onto

U and V, respectively). Then M-I

F

0

T(x)

= LT(x) +

LIPt(v + r(x»(tI + I ,(x»t(v +

:J. H 1 (x»

H-I

+

LI/Jiu + tl(x»(u + tl(x)t(V + r(x»J +

;t •

T(x).

J-O

Let us examine an arbitrary term of the form Applying Taylor's theorem to

IPs(v

+

r(x»

IPs(V

=IPs("

+ r(x»(u + t'(x»S"(v + r(x»H. + ~(v)~ + (r(x) - h~(v)~» at

146

the point v

+ h~(v)~,

we get

M-s-\

L c(q)Dqrps(v + h~(v)~)(u + i(x)t(v + r(x»N[r(x) - h~(v)~]q

=

qaO

M-s-\

M-\

L Dqrps(v + h~(V)VN) Lc(t;) n[h~(V)U(VNt(

=

q=O N-\

M-\

N-\

}=\

t=o

}=\

n[g~(U)UMv't} n[h~(V)U(VNt( n[g~(U)UMv't}

ueyr

(9.5) where the inner summation is taken over all multiindices t; = (€la, •.. , aM_to b\, ... , bN _\, c\' ... , CM_to do, ... , dN _\, e, f, q)

such that M-\

M-\

N-\

N-\

Lat + Lb} ~ q, Lat + Lb} + f (=\

t=o

)=\

}=\

M-\

= N

N-\

Lc( + Ld} + e =

+ q,

t=\

s;

}=o

c(t;) are certain non-negative constants.

Pick out the terms in (9.5) with b} = d} = 0 and associate them in groups according to the power of the variable u ranging from 0 to M - I inclusively. All other terms include in the remainder. Thus, we obtain

M-\ M-s-\

M-\

L L Dqrps(v + h~(V)VN) LC(II) n[h~(v)tt l-O

q=O

t-O

147 M-I

E N(4 t +ct)+4o!i+f

M-I

n[h~(v){Vv

t -I

+ iAu, v),

t-I

where the summation is taken over all

1)

= (lla, ,,-, aM_I, CI' ••• , CM_I, J, q)

such

that M-I

M-I

M-I

M-I

s

(note that the last equality implies form

~ l). The function

is(u, v) = 4I;(u, V)UM~, where 41; e C«E;

as lIuli -+ 0,

IIvll -+ 0

(0

N-I

=

~

+

p

q

~

M-I

is can be expressed in the

PM,N(U

Ell

V; E),

therefore

k). Similarly,

N-r-I

N-I

m=O p=O

J=O

L L vPI/I,.(u + g~(U)UM) Lc(~) n[gj(u){J N-I

N-I

n[gJ2(U)]d

E M(bJ+d j>+boM+f! Ju J-I

_m V

+ ;;;"',.(u, v" ,\

J=I

where the summation is over all multiindices such that N-I

J

;!:

p,

Ld J=I

J

LbJ + e =

~ r,

(b l ,

bN _1> dl> ... , dN _1> e, p)

.'"

N-I

M

IP; e C<-(E; PM N(U

N-I

Ljb

+ p,

J

J-O

~,.(u, v) = IP;(X)UM~, where

Thus, we get

=

N-!

N-I

Lb

~

+

J=I Ell

V; E).

LU - l)dJ

J='l

+r

= m,

148 M.I

F • T(x)

N·I

= LT(x) + E~1(v)i'" + E([/m uMyn + F(x) ,

where I M.s·I

~1(v) =

I

E L Ec(1J)D .-0 q=O

Q

rps(v

+ ~(v)"')

(=0

'"

I 1+1

n[h~(v)t'

[

I

a, +

n[h~{v){' v'-o

1+1

I

0, -

I)N+f

(-I

'-I 1+1

Lc, s S, '=1 1+1

Lia, + L{i - 1)c, + s = '-I '-I

I

(for I

I:

1).

We have used the fact that the equality M·I

M·I

Lia,

+

'=1

L(i - 1)c, + s =

I

'-I

= aM.I = c 1+2 = ... = CM.I = O.

takes place only when a1+1

Clearly,

1+1

[ La, + Lc, - 1]N + f

I:

O.

The functions ([/m{u) (m = 0, 1, ... , N - 1) can be expressed similarly. Let us introduce the follOwing auxiliary equation for the mapping T: M·I

T • Lx

= LT(x) +

N·I

L~1(V)Ul'" + L([/m(u)uMyn,

(9.6)

149

or. in a more detailed form. M-I

Lx

r

+

N-I hz(L"v)(LIU)Z(L"V)N

r

+

gm(Llu)(Llu)M(L"V)m

m-O

M-I

= Lx

+

N-I

L[Lhz(v)uz~ + tz(v)i~] + L[Lgm(u)uMv'" + IJtm(U)UMv'").

In order to solve (9.6). it suffices to find functions hz and gm satisfying the

equations hz(L"V)(LIui(L"V)N = Lhz(v)uz~

+

tz(v)i~

(I = O..... M - 1);

(9.7) gm(LIU)(LJu)M(L,.v)m

= Lgm(u)uMv'" + IJtm(u)uMv'"

(m

= O.....

But previously we need to express explicitly how the functions t z and and gJ' respectively. First let 1 = O. Consider the equation

tm

N - 1).

depend on h,

(9.8) Because ~(v)

Ii

O. the equation (9.8) becomes

According to the definition.

rr

M-I

t~(v)

=

Where cI:S O. tlo

c('IJ)Dq9'~(v + h~(v)~)[~(v)to[h:(v){JV(·O"'CI-J)N+f.

q-O

11

+I

= N

.~(v)

=

+ q.

r

q = O. whence

cJ =

O. tlo

+I

N

c(ao)Dq9'~(v + ~(v)~)[h~(v)tO/O(N-I)

·0-0

= N. Thus

150

and equation (9.8) can be rewritten as N

h~(~V)(~V)N For I

= 1,

= ~h~(v)~ +

L c(ao)tp~(v + ~(v)~)[~(v)(O/OCN-I)+N.

(9.9)

we get the equation (9.10)

where I M-.-I

fl(V)

=

I

2

L L LC('Il)Dqfl.(V + h~(v)~) n[h~(v)(t n[h~(v){t VC"'0+"'I+ CI+ C2-I)N+f t=O

and the inner summation is taken over all al it

q, c i

+ c2 :s S,

can be written as

fl

S E

{O, I},

al

= fl,o + fl,l'

= (ao, ai' CI , C2' f, q) such that + a l + f = N + q. Therefore fl

ao

where

M-I

fl,o(V)

11

+ c2 + S = 1, I

2

= L LC(1I)Dqflo(V + h~(v)~) n[h~(v)(t n[h~(v){t

vcaO+"'I+CI+Crl)N+f,

q-O 11

a l it

q

E

q, CI + C2 :S 0, {O, I}, ao + f

a l + C2 = 1, ao + a l + f = N + q, = N + q - 1. Thus,

whence, c i

= c2 = 0, a l = 1,

N

+ rL.

I

c2(ao)D tpo(v

+

2

N

2

ho(v)v· ) [ho(v)]

"'0 2

CaO+I)CN-I)+l

hi (v) v

.

Further, M-2

fl,I(V)

=

where a l it q, c i + c2 :s 1, a l = 0, c1 E {O, I}, ao + f

q

I

2

L LC(1I)Dqfll(V + h~(v)~) n[h~(v)(t n[h~(v){t vcao+al+CI+C2-I)N+f, 1, ao Therefore

+ c2 + 1 =

= N.

+ at + f= N + q, whence

al

= c2

= 0,

151

Thus, the equation (9.10) becomes

40 =0

40 -0

(9.11)

Now let I 1

2:

2. Then

1+1

tat

2:

q,

t Ct t=1

Z+I :5

S,

Liat

+

L(i - I)Ct

+

S = I, O:s S :s I,

Lat

+f

= N

+ q.

(=0

t=2

Suppose Cz+1 ¢ 0, then Cz + I = 1, consequently, c2 = ... = Cz = 0, s = 0, contradicting the condition Cl+1:S s. Thus, CZ+I = 0. If az ¢ then al = ... = aZ_1 = c1 == ... = c1 = 0, az = 0, s = 0, q E {O, I}, ao + f = N + q - 1. If c1 ¢ then Cz 1, s = I and al = ... = al = CI = ... = CZ_I = 0, q = 0, ao + f = N. Thus,

°

=

°

152 we get the following equation for hi:

N

+

[L

C(ao)IPI(V

+ ~(v)~)[~(V)tOvaO("'-I)+N]h!(v)Ul~

aozO I

+[

N+q-I

L L C(flo)DqIPo(V + ~(V)~)[~(\I)]a%(N-I)+N+q-l]h~(v)ul~

q=O ao=O

+ where

.;(v)

DqIPs(v

+

• I vN , .1(V)U

(9.12)

is a polynomial in

with

ho(V) , ... , hl_l(lI)

coefficients

depending

on

~(v)~) (q = 0, 1, ... , M - 1), namely.

I-I

n

I-I

[

r

I-I at

[h~(v){i v t-O

+

r

I)N+f

c( -

(-I

(9.13)

i-I

where I-I

11 = (flo, ... , ai' cI ,

... ,

Cl , q,

n. Lat

I-I Ii:

q,

t-I

I-I

Lat + f t-O

I-I

I-I

=N +

q.

Lia

Lct :s S, t-I

t

+

L(i - l)ct +

S

= 1, O:s q :s M -

S -

1.

t=2

Thus, we get an independent non-linear equation (9.9) for h~ and a triangular affine system (9.12) - (9.13) for hi (l = 1, ... , M - 1) (note that equation (9.11) has the same structure as (9.12) - (9.13». In other words, we must first solve the non-linear equation (9.9), then substitute its solution ~(v) into the right-hand sides of (9.11) -

153

(9.13) and then succesively find the functions

by solving. at each

hl(v) • •..• hM.I(v)

stage. a non-homogeneous linear functional equation.

V»

So we begin with equation (9.9). For h e CC(V. PN(V;

=

IIhll

The set

ma = {h e

sup sup {IIDrh(v)IIIIVll r -N : "ev 1I"II'jto

CC(V. PN(V;

V»:

r

=

define

O•...• k}.

(9.14)

< III} endowed with the norm (9.14) becomes a

IIhll

Banach space. Let AN: PN(V; V) -+ PN(V; V) denote the linear operator defined by

(ANP)·~ = 'P·(LlIV)N. (LlI)

is equal

to

exp f.lm

spectral radius of AN .4: !lRo -+!/JI0

by

than l. Indeed.

It is clear that the spectral radius

"h

is

(exp (- f.l1)'

= L"ANh.L2I.

for h e mo.

lim! Inll"shll

=

Hili S

respectively).

not greater than

Hence

exp (- Nf.lI).

it

Define

follows a

linear

that

the

operator

Let us prove that the spectral radius of " is smaller

IIhll 'jt O. we have

lim! In {sup IIU("shHv) ": IIVII N - r

Hili S

:5

of the operator L,," LIV

IIvll 'jt O.

r

= O.

I ..... k}

lim! In {IIL;II IIAZ,II IIhll IILlS'II N} HillS

:5

lim ! {In IIL;II

+

In IIAZ,II

+ In

IIhll

+

N In IILlS'II}

HillS

Now we shall show that if

ho

e!IRo then the function

N

V 1---+

t' l..

c(ao)9Io(V

+

2

N

2

a O aoCN-I)

ho(v)v )[ho(v)] v

(9.15)

154

also belongs to mo. According to (9.2), CPo e mo. Moreover, the support of CPo is contained in the disc IIvll::s '0' where '0 is a sufficiently small number. We begin with the first summand in (9.15) that corresponds to llo = O. If , = 0, then IIcpo(V

sup

II VII N

IIvll*O lI!fio(

sup

IIv

II v 11"0

If ,

~

+ h~(v)~)11 v + h~(V)VN)1I + h ~(V)VNIIN

IIv

+ h~ (v)vNII IIVII N

I, then sup

IIVII N - r

IIvll*O

+

2

N N-q

IIv ho(v)v II r-q -----'-----lIvll

IIVII N - q r

::s CIIICPoll

q

Lsup { q_1

IT 11;.1

2

aO

IIDi;.S(v

+ h~(V)VN)II:

IIv

+ h~(v)~11

::s 'o}

s-I =r

2

_JV

then [ho(v)] belongs to mo' Because !fio(v + ho(v)v) also belongs to mo (see above), the proof of our assertion is complete. We note in passing that the exact Lipschitz constant of the operator .4 1: mo -+ m o, where

If llo ~ I

155 N

(Alh)(v)

=

L c(ao)IPo(V + h(v)~)[h(v)tovao(N.I),

can be made arbitrarily small (to this effect, replace f by a suitable truncation fa). Thus, equation (9.9) has exactly one solution h~ in the space

~: V ... PN(V; V) be a

ck

mo

V».

c: ck(V, PN(V;

function with support contained in the disc IIvll:S

agrees with ~ in a sufficiently small neighbourhood of the point

v

=

O.

To

Let

which

In what

follows, we shall drop the tilde placed over h~. Next we shall show the existence of ck smooth solutions of equations (9.11) - (9.13) for I = 2, ... , M - 1. These equations have the following structure:

where s(v) is a linear operator and ib depends only on ho(v) , ... , hb.l(v) (b ... , M - 1). We shall solve the system (9.16) by induction. Suppose that for b ... , d - 1, II:

= (11: 11

... ,

the equation (9.16) have solutions in «,), 1«1 = d, fil = ( fill' ... , film), Ifill

rcx,~: Pcx,~(U Ell V; E)'" Pcx,~(U Ell V; E) by (rcx,~",)·ucxJ3 operator A, let r(A) denote its spectral radius. Clearly, r(rcx,~)

= exp [-

The equation (9.16) with b hd.(v)ud.~

(II:, ~)

+ (fil,

= d can

"')],

r(r~:~)

ck(V, Pb,N(U

=

V;

= 2, = 2,

E».

Let N. Define a linear operator Ell

= fI'(Llu)cx(L,.v)~.

= exp

Given a linear

[(<<, ~) - (fil, ",)].

be rewritten in either of the following two forms:

= Lhd.(L;,IV)(Ljlu)d.(L;,IV)N +

S(z.;lv)hd.(z.;IV)(Ljlu)d(z.;IV)N

+

id(z.;IV)(Ljlu)d(LiIV)N

or

Given a function hd: V -+ Pd,N(U Ell V; E), let hcx,~: V -+ Pcx,~(U Ell V; E) denote the projection of hd onto the linear subspace PO(.,~(U Ell V; E) with respect to the direct sum decomposition

156

Pd,N(U

Vi E)

Ci)

=

Ci) Ci) Pot.,(3(U lot.l-d 1(3I-N

el

Vi E).

Let hot.,(3,j: V -+ Pot.,/3(U Ci) Vi E j ) denote the function defined by hot.,(3,j = prj 0 hot.,(3' where pri E a Et Ci) . . . Ci) En -+ E j is the canonical projector onto the space E j (that corresponds to the eigenvalues of L with modulus exp 9 j

U = 1, ... ,

n».

Denote 1

= {(<<,

f), J):

1«1

= d,

1f)1 = N, j = 1, ... , n},

It = {(<<, f), J) e 1:

9j

+

(<<, ~) - (f), IL)

< NlLt},

12 = {(<<, f), J)

9j

+

(<<, ~) - (f), IL)

;t:

E

1:

Nllt}.

The equation (9.16) for b = d is equivalent to the following system:

(9.17) hot.,(3,j(v)uot.J3 =

(L- 1

IEj)hot.,(3,J(L'1y)(Llut(~v)(3

- (D 1 IEj)8;
««, f), J) e I~. Given 1/101.,(3

E

("«V, Pot.,(3(U

Ci)

V; E), put n

11"' 01.,.. 11 'I' ..

= rL J-t;

where

11,1. II 'l'ot.,(3,j,

157

«0:, ~,J) e

I.J.

The set m ....#! = {I/I .... /3 e CC(V, p",.~(U IB l'; E): 111/1",,1311 linear operator

<

DO} is a Banach space. Define a

by

Further, define a linear operator

according to the rule:

i3a..(3.J { i3 -,

il "'.(3.) -

if

(o:,~,

if (IX,

a..(3.)

(3,

J) e I,; J) E 12 ,

Finally, set n

il"'.(3

=

IB

ila..13.j'

j~,

Let us show that the spectral radius of the operator il",,(3: than 1. In fact, if (0:, lim

(3,

!

J) e I,

and 1II/I"'./3,J II "'" 0, then

In lIil:,I3,j 1/1",,/3,)11

s-+DOs

= lim

.! In

s-+ ... s

{

sup IIvll"'"o

r_O. I •... ,Jc

m",.13

-+

m",,/3

is smaller

158

oS

If (ex., (3, J)

I",

E

+

9j

(<<, ~) -

«(3, j.&) -

Nj.&1

< O.

then we get similarly

· -1 1n lIil«/3j II 11m 1/I«/3jll •

rico S

oS

t

•

lim -sl In { IIL-sIEjll

I

IIr:,/311 II I/I«,,,,j II

IIL~IIJc

+

-

}

HOI

:s - 9 j

-

(<<, ~)

+

(f3, j.&)

kj.&m

< - 9j

(ex., ~)

+ «(3,

j.&)

+

Nj.&1

:s O.

Thus, the operator il«,,,: ID«,/3 -+ ID",,13 is contracting (with respect to an appropriate equivalent norm on ID«,,,)' For every £ > 0 one can find c5 > 0 so that if I is replaced by the truncation la, then

(9.18)

Recall that the operator !l«,/3 is contracting. Taking into consideration the special form

of the linear operator 8(v) (see (9.11) - (9.13», the equality (9.18) and the

C<

smoothness of h~(v), we conclude that the linear part of the affme operator defined by the right-hand side of the system (9.17) is also contracting, if £ is sufficiently small. According to the hypothesis, the functions ho(v) , ... , hd_1(V) are d'- smooth. It A easily follows from (9.2) that ~«,,,(v) e ID«,,, (1«1 = d, 1(31 = N). Therefore tho system (9.7) has a

d'-

smooth solution. One can similarly prove that the second part of

159 ~ functions.

(9.7) is also solvable in the class of ho(v), ... , hM.I(v), go(u) , ... , gN.I(U)

Thus, there exist

~ mappings

such that the function M·I

N·I

satisfies the equation (9.6). Now we shall prove that

as IIxll -+ O. In fact, F

=

0

T(x) - T

0

Lx

M·I M·s·1

M·I

s-o q=O

t=o

L{ L Dqrps(v + ~(v)~) Lc(~) n[h~(v)ut~tt

N·I

M·I

N·I

n[g](u)uMv't n[h~(v)i~{t n[g)(U)UMv']d J

J=I

+

J ut!!yf'

J=o

N·I Nor·1

N·I

,.-0 p=o

J-o

L{ LDPI/J,.(u + g~(U)UM) Lc(~) n[g)(u)uMv'tJ

M·I

M·I

n[h~(v)ut~l'~t n[h~(v)i~{t

N·I

n[g](u)uMv']" JutV

V

+

i,.(u, u)}

+

:t(T(x» ,

J-I

Where N·I

L(bJ

J-I

M·I

+

dJ ) ~ 1,

L(7t t-I

+

I3t) ~ 1.

+

ts(u, v)}

160 It follows from (9.2) and (9.3) that

+ ~(V)~)II = O(IIVII N-b)

IIDb+QlPs(V

= O..... N;

(b

IIvlI'" (».

This together with (9.4) gives the needed result. Next let us show that

(a

O. b

I:

I:

O. a

+ b :s k.

IIxll ... 0).

IID~D~

In fact. the following more general asssertion holds: If

as IIUIi ... O. IIvlI ... 0 IID~D~ l(e(x»11

a I: O. b

for

= O(IIUIiM-aIiVIIN~.

= O(IIUIIMIIVII N).

a+b

If

I:

O. a

I:

Indeed.

+ b :s k. if a

and

+ b = O.

e(x)1I "" O(IIUIiM-aIiVIIN-b)

IEc!'. then

1(0)

then

III(S(x»II:S clle(x)1I

1 and IIXII:S 1. then

a+b

IID~D~ l(e(x»II:S

= O.

L L c(r.~. 11) IIDrl(e(x»I,"DIID2IS(X)1 t;

71

..... II

Dt;r

7Ir

I D2 S(X)II.

I~I= .. 17II-b

Since

~t:S

a and

:S b. we get

lI t

Denote 1P(x) = ta- I reasoning shows that IIDfD~ IP(X)II

0

F

0

ta(x) - Lx.

I(X)

= O(IIUIiM-PllvnN-Cl)

as IIxlI ... O. Besides that. II'

E

(p

= ta- I I:

0

F

O. q

I:

0

ta(x)

O. P

II

Lx

+q

+ lP(x). :s k)

The above

(9.19)

c!'(E. E) and the support of II' is compact. It remains to

prove that the local diffeomorphisms I

and L are c!' conjugate with one another near

x = O. To this end. we need the following Lemma. Suppose that the above Irypotheses are fulfilled. Then

161

(9.20)

as IIXII

-+ 0 (p ~ 0,

• Denote 1/1 (L· I

+

q

~

+ q :s

0, p

= t· 1 - L·I •

k) .

From id = t

0

t·1

II

-I

+

(L

+ rp)

0

(D I

+ 1/1) = id + Up + I{J

0

1/1) we obtain 1/1

Since tl(O, v) == 0, sequently,

= - L- I 0)

t 2(u,

==

0,

0

I{J

0

we

IItil(u, V)II :s ellull,

in the vicinity of (0, 0) First let p

+

U

E

III

(L

get

1/1).

(9.21)

til(O, v) == 0,

t21(U, 0) == 0,

IItil(U, V)II :s ellvll

con(9.22)

V.

q = O. Then

therefore (9.19) and (9.22) imply that III/I(U, v)1I Now consider the case where p

+q

= O(IIUIIMIIVII N )

as

lIuli

+

IIvll -+ O.

= 1. Differentiating the equality (9.21), we

get [)yJ(u, v) = - L- I

0

Drp(t-I(u, v» . (L- I

+ [)yJ(u,

v»,

or whence

Given a linear operator Y, II Yn < 1, write the identity (id - f)-loY = Y from which it follows that

0

(id - f)-I,

162

Therefore

and the estimates (9.19) and (9.22) show that (9.20) holds when p + q = 1. Suppose that (9.20) is fulfilled for all p', if such that p':S p, if:S q, p' + q < P + q :s k. We must prove (9.20). According to the composite mapping formula (see subsection A.6), the equality (9.21) gives

....

where T = (TI' T:z) ranges over all such pairs of non-negative integers that

p~

+

q~ ~ 1 (i = 1, ... , TI), p~

+

q~ ~ 1 (i = 1, ... , T:z).

Rewrite (9.23) in the form

[id

+

L- I

0

Dcp(~-I(U, v))] DfDi I/I(u, v) = -

rl

L

c(T)D ....cp(~-I(U, v»

.... 1+ ....2>1

Denote the right-hand side of (9.24) by S(u, v).

= O(IIUIIM-PIIVIIN-q)

Let us show that

IIS(u, v)1I

as lIuli -+ 0, IIvll -+ O. We shall estimate each summand separately. If

163

TI :s p and T2

:S

q,

nDT,(~-I(u, v»n

then

= o(lIuIiM-Pnvn N-q )

< q. Suppose that q~ = 0 (i = 1, ... ,

(by (9_19) and (9_22».

Let

T\

> p,

then

L q~ = q

and

T2

< q

imply that 3 j e {I, ... ,

Thus, either 3 i E {I, ... , In the first case, we have

T 2}

T\).

Then

as """ -+ 0, IIvil -+ 0

T2

T 2}

for which q~

l!

2.

q~

l!

2.

[-I

because p!

+

q!

<

p

+

such that q~

E

2 or

3

j e {I, ,,_,

T 2}

with

q, p!:s p, q!:s q. Thus,

In the second case

consequently,

The case where Thus,

as IIuli

T2

IIS(u, V)II

+

> q,

=

T\

< p is handled similarly.

0(11 "" M-p II VII N -q ). Consequently,

IIvil -+ O.

Now we are going to prove that ~ is c!< conjugate with L. The conjugacy H will be represented as H = id + h. Then the equation H. ~ = L H gives Lx + ,(x) + h(Lx + = Lx + Lh(x), consequently, 0

,(x»

or, in other form,

164

= {I,

Denote J

"" n}, J I = {j

J"

= {j

J: 9 J

+

J:

+

E

E

9J

MAl - NJlI M).I - NJlI

:s OJ,

> OJ.

Define A = ~ AJ , where the mappings AJ : ecCE, EJ ) ~ ecCE, EJ ) are given by J-I

n

Set IIhll

= LIIhJII.

where hJ

= prj

0

h, and

J-I

,,- = sup

{ IlDiDt hJ(u. V) II

___1

{

II ,-J II

11,,-11 =

sup

IID~Dt hJ (u. V)II IIUII M

}

:

Ilvll",.

0; O:s p

+

q :s k

:

Ilull""

0; O:s p

+

q :s k}

IIVII N - q

-p

(j

E

J,).

Let M denote the set of all functions h E ecCE. E) such that IIhn < lID, Then At is an A-invariant Banach space. Our goal is to prove that the spectral radius T(A) of the linear operator A: M ~ M is smaller than 1. Actually. we shall prove that the operator A is contracting with respect to suitable (equivalent) norms in E I •..•• En. Recall the following simple fact. If A: F ~ F is a linear operator. whose spectrum lies within r > 0 of the origin o e c. then for each number I: > 0 one can find an equivalent norm II' lie in F so that IIAxll e

<

(r

+

e)IIX11e

ex e F).

Thus. let us show that A: At ~ At is a contracting operator whenever E I , .,.. En are provided with appropriate norms. Let j E J2 and I: > 0 be a small parameter which

165

will be specified later. According to the above remark, we may suppose that II (L -\ IEJ) II < exp (- 9 J + c). Then

Notice that if '1:\

=P

then

q! it 1, or 3 j E {t, ... , '1::z} truncation of rp, we get

'1::z

< q. Therefore either there exists i

such that

q}

it

2.

E

{I, ... , '1:\},

In both cases, for a suitable

166

(9.25) Now let

"t l

< p. Then either

for which p~

{P:, ... , P!I}

l!::

3 j E {I, ... , "tl}

with p~

1. This implies (9.25). Finally, let

contains

TI -

TI

l!::

2 or 3 j

;t

{I, ... , "t2}

> p. Then the collection

P numbers equal to 0 (for definiteness,

= P!I_P = 0). Then q:;t 0, ... , q!l_p

E

let P:

= ...

O. Consequently,

IIUIi

if

is replaced by a suitable truncation. Summing up, we obtain

If)

(9.26) Since j e J 2, we have e J + MAl - NIlI > o. Because k 11m < NIlI, we get - e J - MAl 11m < O. Therefore (9.26) shows that if the number c > 0 is sufficiently small (and the norms in E l , ••• , En are appropriately chosen), then AJ is contracting. The case

+

k

167

where j

E

J I can be examined analogously. The proof is complete.

9.2. Theorem. Let the integers M, N, k, K, and Q satisfy the hypotheses of Theorem 9.1.

If the

~

vector field

°

E

rK(E)

has contact of order (Q, K)

E E, then the vector field field at the origin near the equilibrium x = O.

x=

Ax

+

with the zero vector

fix)

is

eft

linearizable

• The proof is similar to that of Theorem 9.1. We first make a preliminary change of variables M·I

T(x)

=X +

L h,(v)u'.ji

N-I

+

LgJ(u)uMyI, x J

'=0

=

(u, v).

~O

Instead of (9.6), we get the following first order quasilinear partial differential equation for h, and gJ: M-I

DT(x) . Ax

= AT(x) +

L ~,(v)u'.ji

N-I

+

L q,J(u)uMv-'.

i-O

J=O

It suffices to solve the following system of partial differential equations:

where the operators At

E

L(Pt,N(U, V; E), Pi,N(U, V; E)

= Arp·ui.ji -D[rp·ui.ji]Ax (i = 0, ... , M - 1) and 'B J

are given by ['BJrp).uMyI = Arp·uMyI - D[rp·uMyI]Ax corresponding characteristic system:

E

are defined by

[At'P) , i.ji

L(PM,;(U, V; E), PM,iU, V; E) (j

= 0,

... , N - 1).

Write the

It follows from the arguments used in the proof of the preceding theorem that this

system breaks into two independent subsystems each of which has triangular form. It is

168

not hard to prove that the affine extensions determined by these subsystems have c! smooth invariant sections which belong to the corresponding functional spaces introduced in the course of the proof of Theorem 9.1. The second stage is similar to the second part of the preceding theorem. The only difference is that we have now partial differential equations instead of functional equations. To solve the former, choose smooth invariant sections for the corresponding characteristic equations in the same functional spaces as in Theorem 9.1. The details are left to the reader (cf. the proof of Theorem 6.18).

9.3. Notation. Let Z be a finite dimensional linear space; L: Z .. Z be a hyperbolic linear operator; 9 1, ... , 9" be the set of all distinct values of 1n I~tl, where ~t ranges over all eigenvalues of L. Let ZI' ... , Z" denote the corresponding spectral subspaces of L (i.e., the spectrum of Lt • LIZt lies on the circle I~I = exp 9 t (i = 1, ... ,

n».

Let

p --

(pI , ... , P") E

z"+,

I p I --

pI

+ ... +" ' p, r- mm

{t. - , 1 p. i -

... ,

n} ,

t be a positive integer. For a collection c = {iI' ... , ill C {1, ... , n} • I, c '" 121, denote Ze = Ell Zt. Let Zc be the projection of z onto Ze' Given a multiindex r = (.,.1, tEe

... , .,."), denote

tEe

A(p,

C)

= {or.

tEe E

z~: or. t s pt - 1 (i

tEe E

1), or.t = 0 (i

E

c)}.

9.4. Lemma. Let • E cI P I +t(Z, Z) satisfy DP teO) = 0 for p = 0, 1, ... , I p I - n. Then for each non-empty collection c c I and for every multiindex or. E A(p, C) there

exist mappings .~.": Z .. Z, .~."(z)

= 9'~."(Zc)·z",

such that:

169 (3) t

E

=

eCI /¥EJ.{p,e)

~ Define differential operators

(s

= 1,

(s

a:

=

1, ... , n; p

= 0,

... , ps - 1) and

... , n) by

a:t(z)

= p. --.!, n: t(z) Iz -0 s

•

z:;

s

p -I

11~·

= id -

Ea:.

p-o Since lY't(O)

=0

= 0,

(p

1, ... , Ipl - n), it follows from Clt

E

l~,

ICltI:s Ipl - n,

that

(9.27)

tit

P -I

Further, since id

=

r a: +

li~

s

for each

S E

1, we get

p-o n

id

= IT [ Ea: + li~\

Taking into account that the operators

id

s

p-I

a: and

li~'

= E eCI O:SP':SP'-I 'Ee e"'fII (tEe) sEne

Let c

C

1 and Clt

E

"(p, c). Define

commute if s '" i, we obtain

(9.28)

170

sEc tEI\c

It follows from (9.27) and (9.28) that

-=

L ar.EA(p,c)

cCI

c'll:li!I

Clearly. -~,ar. can be represented as -~,ar.(z)

=

!P~,ar.(Zc)·zar.. Now we shall show that the

functions !P~,ar. satisfy the requirements (1) and (2). Indeed. by Taylor's theorem.

Denote tel

= (WI'

•.••

wn ). where (s

E

e).

(s

E

J\e).

We get

n .6~s_(z) sEc

(ac

=

= ac

I

I

o

o

I·,· I

u.;-(lel)

n[(1 - lsi,s-Idls]'~ .. ~~(z)·~ sEc

const). Thus.

n a~t[~~(z)'~] tEI\c

tEI\C -p

-t

P

where !Pc,ar. E L • Note that rpc,ar.(Zc)

=

-p

.J>

!Pc ar.(Zc)·'c.

hence.

171

It remains to show that ¥>~,or.

t~,or.

E

C+ n +t - I • Observe that

n

Ii

sEc tEIlc

sEc tEIlc

From this and (9.27), we derive that ¥>~,or. can be expressed in terms of partial derivatives of order not greater than

+

Ipl - r - n

1 (recall that r = min {pi, ... , pn}).

Therefore the class of smoothness of the functions ¥>~,or. is not less than (Ipl

+ t)

- (Ipl -

9.S. Lemma. Let k be a positive integer,

DPt(O) t~,or.(z)

=r+n+t

r - n + 1) IC.

= (k,

... , k)

E

- 1.

l~j t

E

CCn+I(Z, Z) and

= 0 for p = 0, 1, ... , (k - l)n + 1. Then there exist functions = ,,~,oe(Zc) ·l' (c c I, c - 121, «E 14(IC.,. c», such that:

(2) t

L L

=

t~,or.: Z -+ Z,

t~,or.'

cCI oeE.4{rc,c) c_12I

({3 E l~,

(3)

• Let

{3

E

l~,

I{31

!is

k. If s

E

I and i

I (31

it

{3

s

!is

k).

~f3

t

_

t _/3S ~f3

,then v ~st - ~s

v t. Indeed,

172

< fill,

If ;

then JjJa!

= O.

6:

Next let us prove the equality rfl •

=

6:-lJ s

rfl. In fact,

•

Ie-I

rfl •

6: t(z) = rfl[t(z) - r a~t(z)] Ie-I

= JjJt(z)

t rfla~t(z)]

-

Ie-I

= rflt(z)

-

t

a~-fJsJjJt(z)

le-lJs-1

=

lid -

r

a:lrJIt(z)

= 6:-fJ'"

• rflt(z).

q=D

Assume that the hypotheses of Lemma 9.5 are fulfilled. According to Lemma 9.4 (with P

= Ie)

and assuming for definiteness that f3'

=0

for i

E

1\ c, we have

sEe [Ene

sEe [Ene

therefore

i.e., the condition (3) is fulfilled. Applying Lemma 9.4 for p that (1) and (2) also hold.

=

It

and

I

=

I, we see

173

9.6. Theorem. Let I E c!(Z, Z) and ifl(O) = 0 for p = 0, I, .,,' K - 1. Denote k = [(K - 1) I n]. Then the local diffeomorphism F: Z -+ Z defined by F(z) = Lz + I(z)

is c;k linearizable near the .fixed point z ~

= O.

Without loss of generality, we may assume that n

> 0 and a function

exist a number 15

i(z) = 0 if IIzll ~ 1 and

~

2. For every c

> 0 there

= t(z)

if IIzll:s 15,

i E c!(Z, Z) such that i(z)

Z and p = 0, I, .'" K - 1. Since the problem under consideration is local in nature, we may replace I by its truncation

i. Further, let >.

o :s >.(z) :s (i

1 (z

E

= 1 if IIzlI:s 15, >.(z) Z; P = 0, I, .,,' K).

Z) and

lIif>'(Z)II:S 2 (z

E

Let 0 denote the set of all multiindices w

= (wi,

=

{I, .,,' n}

I, .,,' n) and there exists a number j

= {s E 1: w EO).

= 0 if IIzll ~ I,

c!'(Z, R) be such that >.(z)

E

E

lIifi(z)lI:S c for all z

w· = k} (w Denote :E =

$

EO).

E

.,,' w") 1&

E

E

I,

W EO],

such that w':s k

1 with wJ

S =

= k.

Put c(w)

s!". Pw(Z; ZJ)

Assume we are given polynomials

[Pc.,cZ; ZJ): j

z~

V. I,

Ls-!J.W, JEI wEn

g(z, 3')

= z + >.(z)S

(1) .J

all

C

0

E I,

(2) sup{

SEC,

g(z, S) C;II

II,

Z, S;;; {~}

E

:E).

(c c: I,

C;II

II,

j E I) of class c;k+1 such that:

= L~,oc,.(Zc; SEC,

«E .4(It,

S)·zoc

VE

I), where the summation is taken over

c);

II~~ (Z· S)- II c c,OC,. c' : IIz.1I ;II 0, I ZcIK·~IIZ.1I

(3) ~,oc,. does not depend on ~

E

~,oc,.: Zc x :E -+ Poc(Z; ZJ)

9.7. Lemma. There exist mappings « E A(IC, c),

(z

S:

if

IIz,lI;II 0 (i

c(w) \

C

;II II

E

c), fJ

E z~,

IfJl:s k}

<

00;

V, pEl).

Let • satisfy the hypotheses of the theorem. The mapping G(z, S) ;;; •

0

g(z,

3') is

174

c"- smooth

=0

IY'~(O) cc E

=0

in z and IY'G(O, 8) (p

= 0,

(p

= 0,

1, ... , k n)

because Kit k n

1, ... , k n). It follows from Lemma 9.5 that for c

"(Ie, c) there exist

c"-+2

smooth mappings Gj"OI.: Zc x

o'(z, 8) =

I

C

1, c

+

1 and

'It Ill,

and

-+ POI.(Z; ZJ) such that

(j

I),

E

eCI OI.E.I(IC,e)

sup {IID~Gj"OI.(z; 8)1I'lzl 13-IC: IIZcIl

s

1, IIz(1I > 0 (i

Let j, p E l , w E 0, C C 1 and cc E "(Ie, c). If r the composite mapping formula (see subsection A.6),

E

c), f:!

E

c(w)\c

E

z~,

'It Ill,

If:! I

s

then wr

k}

<

= k.

00.

By

1 =-

\

v\

. Dr (z\

v

vpP

T

v\

S cc

P

w

~(z)Sw'Z

+ ... )

" " v\ v T"

OI.r

+ ~(Z)S~·Zw + ... ) ... Dr (z" + ... ) ... Dr (z" + ... )lzr-o'Zr ,

= ('rt,

Since \.I~

P

\

+ ... ) ... D/: (z\ + ... ) ... Dr (Zp +

• DrT (zp

where

\

... ,

T") E

z~, \.I~

r < k and wr

= k,

E

z+ (I

E

1; m

= 1,

... ,

Tl)

and

" L

we infer that ~~r o'(z; 8) does not depend on S~.

Consequently, the 'same is true for Gj"OI.' Let c = {ill ... , ill C 1. According to the proof of Lemma 9.5, we have Gj"OI.(Zc)

Set

= O/:,OI.(Zc)·~,

where

0/:,01. E C.

175 - j j G (z) ~ Gc,,,,(O, c,cx,!2, c

'"

ZL 2 , ... , ZL Z) -

- j GC,,,,(O, 0,

ZL , .... Zt Z)' 3

............ , ................................................

j Gc,,,,,t ('7) ~ 0/:,,,,(0 • ... , 0, z --c

Because nKo!"",(O)

= O.

we get

Zt ). Z

0/:,,,,(0) = 0, consequently. z

G~,,,,

LG{""L

p '

p_1

Denote

Clearly, the functions

o!:,,,,,s possess the required properties.

Let IAl e n and ex("') e l~ denote the multiindex defined by if i

II!

c(w).

if i e c(w). Given a collection

c c: I,

c

wen such that c

= c(w)

and ex

'$ fZJ,

and a multiindex ex e A(K, c), there is a multiindex

= ex(w).

mapping defined by (S~'''''Z}'zCt = S~·zw.

I =

9)

{Pw-«(Z; P",(Z; Zp»: j e I. eel.

S~.Ct e Pw-«(Z; P",(Z; Zp»

Let Set c

'$

Ill, ex e A(K, c).

Clearly, there exists a one-to-one correspondence between I family {S~,,,,} e Zc x

i: -+

t.

P",(Z; ZJ)

and

C

= c(w), I.

It follows from Lemma 9.7 that one can find such of class

CC+ I

denote the

ex

= a;(IAl)}.

S denote the mappings Gf:,,,,,s: Let

that

"'j '" - GJ Gc,,,,,s(Zc; S) C,"',s ('7' --c. S)'"

and.

moreover,

tI!:,,,,,s

satisfies the conditions (1) and (2) of Lemma 9.7, and if

176

el

\

e

¢

121

then

~I

lTc,,,,,s

does not depend on

Ap

SCI,(3

(p e 1, fl e "(!C, el ».

We look for a linearizing transformation expressed in the form

L cc.x

Sf:,,,,(Zc)·Z'"

=zJ

+ S'(z).

cce.4(rc,c)

c¢/21

Recall that

t(z + ;\(z)S(z) < z »

n

L

=

cc.x

(9.29) cce.4(rc,c) J-I

We have the following functional equation for

S:

Lz + S(Lz) = L(z + S(z)
(9.30)

Consider also the equation

Lz + S(Lz)

= L(z + S(z)
t(z

+ A(Z)S(Z)
(9.31)

The equations (9.30) and (9.31) agree in the Cl-neighbourhood of the origin, therefore the solutions of (9.31) may serve as local linearization transformations. According to (9.29), in order to find a solution of (9.31), it suffices to solve the following system of functional equations:

(9.32)

(j e 1, c c. 1, e

¢

121,

a. e "(!C, e),

see).

Then the functions

(e c. 1, a. e "(!C, e» sec

will determine the required change of variables. There exists a number T, 0 < T < 1, such that 9J -

(w,

9) - T as

¢

0

(j e I,

w e C,

S

e

e(w».

177

Let j

E

I, eel, e ""

cr.

RI,

E

"(/C, c) and

SEC.

Let

and cr.(w) = cr.. Let !IJl~,cr.,... denote the space of all satisfying

Define a linear operator ~: m~,CII,5' -+ m~,CII,5' by (I. E Z, C E Zc). Let ,,(21) denote In I'(:B), where It is easily seen that

,,(~)

=

lim! In

m-+III

m

(IISmll~cu) ' .

= 9J

be such that e(w) = c

W E g

d'

mappings 9': Zc -+ Pcr.(Z; ZJ)

(~9')(C)·l' = L jP(DIC)·(L-II.)cr. is the spectral radius of ~.

I'(~)

- (w, 8) -

T

9....

Evidently, the operator 21 is invertible. The further consideration make essential use of the fact that ,,(:B- 1) = - ,,(:B). Suppose, for definiteness, that 9 J - (w, 9) - T 8 s

For all j

E

I,

eel, e ""

cr.

RI,

E

"(/C, c).

< O.

(9.33) let the mapping I.e

SEC.

1---+

~,cr., ...(I.e)

be an element of m~,.,.. Applying Lemma 9.7 and the composite mapping formula (see

subsection A.6), it is not hard to establish that the mapping

I.e

1---+

bf:.•,s(I.e,

~(I..»

also belongs to the space m{cr..s. Taking into consideration the specific way in which the functions 1.1' ....

card

1.n

and S~,tJ,r,

tJ!:.•.s

depend on

we see that the system (9.32) can be solved by induction.

e being the induction parameter. For a fixed collection e, the mapping {~.cr.,s: cr.

E

"(/C. c). j (p

E

E

I,

I, del.

SEC}

~ E

II

~c

1---+

ti:,tJ,r(" ~c(·»

.4(/C, 4), rED)

satisfies the Lipschitz condition with respect to the norm defined in m~.tJ,r' Moreover, the Lipschitz constant of this mapping can be made arbitrarily small if £ is small

178

enough (see the first paragraph of the proof). Thus, the non-linear operator

where b(sc)(i':)'z" = {~./3.,.(L"I~, Sc(L-l~»'(L-1Z)": pel, del,

«E

,4(1(, c),

rED},

has a fixed point in the space

!Dld

=•

{!Dl~./3.": pEl, ~ E ,4(1(, d), rED}.

Therefore (9.30) has a solution of class

c!'.

9.8. Remark. Arbitrarily small C perturbations of the mapping F can lead to the increase of the number n. card {I i'll : i'l E cr(L)} , where cr(L) denotes the spectrum of the operator L. DF(O), so that n can even attain the value dim Z. Therefore the estimate of smoothness of the linearizing transformation given in Theorem 9.6 at first sight seems to be unstable under perturbations. But it is easily seen from the proof of this theorem that this estimate is valid for all diffeomorphisms sufficiently

d

close

to F. Indeed, /J(~) and /J(~-I) depend continuo sly on L.

9.9. Theorem. Let ~ be a c! smooth vector field on Z and DP~(O) = 0 for p = 0, 2, ... , K - 1. Assume that 0 is a hyperbolic singular point. Let n denote the number of distinct values in the collection {Re i'll, where i't ranges over all eigenvalues of the operator D~(O). Denote k near the point O. ~

=

[(K - 1) I n). Then the vector field ~ is

c!'

linearizable

The proof is similar to that of Theorem 9_6 and therefore omitted.

§ 10. Normal Fonns (a Supplement to § 8)

In this section, we combine the methods and results exposed in § 3 and § 9 and obtain, in this way, some new results on finitely smooth normal forms.

179

10.1. Theorem. Let F: E -+ E be a local diffeomorphism of class hyperbolic operator; k

~

ct<;

1; M and N be positive integers satisfying

L!Ii DF(O) be a M ~I > k ~l'

N III > k Ilm • If K ~ Q == M + N + max {M, N}, then there exists a local morphism which brings the diffeomorphism F to the polynomial normal form n

Fo(x)

t-I

where p; - 0 implies that at

=

M+N-I

L L p!xa'e"

= Lx +

C< diffeo-

(10.1)

Ia'i =2

<(7', a> and (7' does not satisfy the condition !(k) .

• Let F meet the hypotheses of the theorem. Then F can be brought (by a variables) to the form ~(x) = Lx + q(x) + p(x),

e

change of (10.2)

Q

where q(x) =

Lql!l'~

is a resonant polynomial and p is a

of order Q with the zero mapping. We shall show that diffeomorphism G(y)

= Ly +

q(y)

(y

E

ct< mapping which has contact

F is ck

conjugate with the local (10.3)

E).

As it is seen from the proof of Lemma 8.1, one needs to show the existence of a smooth invariant section of the extension defined by

[ :] ........ [ : +

qf.<

+ () - q(E) + pf.< + E) ].

C<

(10.4)

where C is a linear operator such that the set of moduli of its eigenvalues is contained in the corresponding set for L. If x = H(t;.) is an invariant section of (10.4), then LH(F.)

+

q(F.

+

H(F.» - q(t;.)

+ p(t;. +

H(t;.»

= H(CF.).

(10.5)

Recall that the linearizing transformation we looked for in Theorem 9.1 was expressed as x = F. + H(t;.) , where H is a solution of the equation LH(F.)

+ p(F. +

H(t;.»

= H(Lx).

(10.6)

180

The functional equation (10.5) can be solved just like equation (10.6). Therefore we confine ourselves by a brief outline of the proof. We make first a preliminary transformation 1(1;)

=

M·I

N·I

M·I

N·I

Eht(v)ut,ji + LgiU)uMyl.

Then

+

p(1;

1(1;»

= Etl(v)Ul,ji +

LifIm(U)UM.;n + p(i!)

(cf. (9.6». Further. M·I

q(i!

+

I(i!» - q(l;)

N·I

= Eal(v)hl(v)Ul,ji + Ebm(u)gm(u)uM.;n M·I

+

N·I

LAl(v)ul,ji + LBm(u)uM.;n

+ P(i!).

where al and bm are polynomials in v and u. respectively. of order not lower than 1; Al and Bm are polynomials in v. ho, .... hl . 1 and u. go • .... gm.l; P is a polynomial in u. v. hJ. gJ such that its order in u is not less than M and the order in v is not less than N. Write down the following auxiliary equation: M·I

L

N·I

+

M·I

M·I

N-I

LifIm(u)uM.;n + Lal(v)hl(v)UI,ji + Lbm(u)gm(u)uM.;n

M·I

+

N·I

Lhl(v)UI,ji + L Lgm(u)uM.;n + Ltl(V)UlyY

N·I

LAI(v)Ul,ji + LBm(u)uM.;n

181

r

r

M-I

=

N-I

+

hl(CV)(Cu)'(CV)N

(10.7)

gm(Cu)(CU)M(CV)m.

m-O

To find a solution of (10.7), it suffices to solve the system

= hl(CV)(Cu)'(CV)N

(I

= 0,

1, ... , M - 1); (10.8)

= gm(Cu)(Cu)M(CV)m

(m

= 0,

1, ... , N - 1).

System (10.8) has the same structure as (9.7) and can be solved in the functional spaces

moe•tJ used before. The solution t(~) is c! smooth. The mapping H will be represented in the form H(~) = t(~) + r(~). From (10.S) and (10.7), we get the following functional equation for r: Lr(~)

+ P(~) + p(~) +

[q(~

+ t(~) +

r(~» - q(~

+ t(~))] (10.9)

+

[P(~

+ t(~) +

r(~» - p(~

+ t(~))] = r(~).

The existence of a c! solution of (10.9) can be proved analogously to the last stage of the proof of Theorem 9.1. Evidently, Theorem 10.1 can be translated into the context of vector fields. We invite the reader to formulate and to prove the corresponding theorem.

10.2. Theorem. Let ~ E r"(E) and 0 be a hyperbolic singular poilll.of~. Let n denote the number of distinct values of Re A" where At ranges over all eigenvalues of

the operator A = D~(O). Let k = [(K - l)/n]. Then the vector field ~ near 0 is conjugate with its resonant normal polynomial form

y = Ay +

n

L

t-I

r p!ll'e "

J,

Icrl-:z

c!

182

where p; - 0 implies thai tS' does not satisfy the condition i!l(k) • • Apply Theorem 9.9 and the method of introducing additional monomial variables (see the proofs of Lemma 8.1 and Theorem 10.1).

§ 11. Summary of Results on Finitely Smooth Normal Forms

The time has arrived to gather the main results on linearization and normal forms obtained in this Chapter and to make a comparison between them. 11.1. Notation. Let E be a finite dimensional vector space, L: E -+ E be a hyperbolic linear operator, and tS'(L) be its spectrum. Let {ell ... , en} be the collection of all distinct values contained in the set {In I ~ I : ~ E tS'(L)}. As before, we shall use the notation {e l , ... , en} = {- ~L! ... , - ~II /JI' ... , /Jm }, where - ~l

< ... < - ~I <

0

<

/JI

< ... <

/Jm'

= n.

1+ m

Given a positive integer k, put Qo(k)

M = [k

i\l

I

= [ ~l + k ~I

i\1]

+

( /Jm ~I

+

1 )]

1, N = [k /Jm I /JI]

+ [ /Jm + +

11.2. Lemma. Let T e z~ be a multiindex. T

k

/J I

(~l +

T

= (ex,

13) and IT I

1, Qs(k) = M

If ITI

Ii:

+

2;

N

+

max {M, N};

min {Qo(k) , Qs(k) , QB(k)}, then

Ii:

Qs(k). Then either ex

Ii:

+ 1 or 1131 Ii: [k /Jm //Jll + ... + exl~l Ii: Iex I~I > k i\l'

[k ~l/~I]

For definiteness, let Iex I Ii: [k ~l/~tl + 1. Then exl~1 e S(k). Next, let I T I Ii: QB(k). Then either there exists a number r

+ 1. i.e.,

exr

+

satisfies the condition S(k) • • Let

exr

1 )]

/JI

T

> k, or there exists a number s E {I, ... , m} such that > k, then exli\1 + ... + exri\r Ii: exr~r > k ~r' i.e., T E Finally, let

IT I

Ii:

I3 s

E

{I, ... , I}

such that

> k. If, for instance,

S(k). Qo(k). Suppose the condition S(k) is not fulfilled, i.e.,

183

(1

~

r

~

I),

(1

~

s

~

m).

= «I + ... +

la:l

Consider the problem on maximization of the linear functional

in

«I

the polyhedral convex domain of IRI defined by the inequalities « I ~k ,

11 «~I « ~k ~ ~,

+

... ,« I ~I

+ ... +

ex I ~l

~

k

~l'

This domain has a unique vertex, namely,

Therefore,

Let us examine how the number «(~I'"'' ~l) depends on the variables ~h"" ~l under the assumption that the number ~l I ~I ;;; P is fixed. Using the notation p,

= ~, I

~t+1

(i

=

1, ... , I - 1), we get

«(i\I' ... , i\l) ~ k I - min {k(PI, ... , PI-I):

0 < p, < 1,

PI'"

PI_I

= 1 I pl. I

It can be easily checked that the minimum is attained at the point

(i

= 1,

P,

=P

... , 1- 1), i.e., I

«(i\I' ... ,

~I) ~

«(I) ;;; k I - (I - 1) k P-

The function «(I) is monotonically increasing and lim «(I) 1-++ DO

i\ I «I ~ k (In...!.

+

1).

"'m

+

1),

~I

Similarly, we get 1131

:5

k (In

"'I whence it follows that

r:t.

= (1 + 1n p)k.

Thus,

- r:t

184

I"t'l

).,

$

k (1n -

+

).1

But the condition

Il

In ...!!! Il I

+

2)

). I

11.3. Corollary. Let L: E

c! ~

+

E

S(k).

~ E be a hyperbolic linear operator, M

= [k Ilm/lll] + 1, k a= 1. If f. E ~ E

has contact of order M

is

III

I "t' I a= Qo(k) implies

contradicting the previous relation. Thus, "t'

N

l)., = k (1n I ...!!! + 10 - + 2).

= [k ).,/).1] + 1,

cr<

is a smooth map, K a= M + N + max {M, N} andf N with the zero map, then the diffeomorphism F(x) = Lx + .f(x)

linearizable.

The assertion is valid in virtue of Theorem 10.1 and Lemma 11.2.

11.4. Comparison between c! linearization theorems. Three main theorems concerning c! linearization of finitely smooth vector fields have been proved, namely, Theorems 3.3, 9.1 and 9.9. They have a great deal in common and can be unified as follows: if ( is a

cr<

smooth vector field which has contact of order (Q, R),

at 0, the origin being a hyperbolic equilibrium, then ~ is Specifically, in Theorem 3.3: K a= k, Q a= Qo(k) , R a= k;

R

c!

$

K, with its linear part

linearizable near O.

in Theorem 9.1: K a= M + N + k, Q a= M + N + max {M, N}, R = K; in Theorem 9.6: K a= kn + 1, Q a= kn, R = Q. By Lemma 11.2, if ~ is an infinitely smooth vector field, then these three theorems follow from Theorem 6.2. Let us show that Theorems 3.3, 9.1 and 9.6 are mutually independent. More exactly, we shall give examples of vector fields such that their

c!

linearizability (for a certain

k) can be established by virtue of one of these theorems whereas two other theorems give

no defmite answer.

11.!. Example. Consider the vector field

185

Xt = - (16.4 .... YJ

= (-

3.6i}tt

=

(i

11.8 .... 21.8J)YJ

=

(j

Here n Then

I, ... , 10),

11

11

YlI = 228YII

I, ... , 11),

.... ( L~

11

Ly~ )

+

= [ 56 +

Qs (14) = [14. 56 ] 20

Jtl

= 10,

+ [ 228 +

1)]

+

+

Jtn

= 228,

K

= Q = R = 296.

+

2

= 292

+

14(56 10

10

2 [14. 228 ] 10

+

+

Qs (14) = 14·22

+

14(228 20

20

LIXl I

141 (

= 22, m = 11, I == 11, AI = 20, An = 56, Q (14) o

(11.1)

1)]

'

3 = 680,

1 = 309.

Therefore, the vector field (11.1) is cf4 linearizable by Theorem 3.3, but Theorems 9.1

C 4 linearizability.

and 9.6 do not guarantee

11.6. Example. Consider the vector field

Xt

= -(0.75

+

YJ

=

25J)YJ

(75

+

(i = I, ... , 5),

0.25i)%t (j

=

5

. y, Here n Then

= 10,

=

200y,

M

,

~+ 1

= 2·10

= 5,

2(200 1

of- 1

N === 5,

+

(t"_2 L Kt

1=5, ", == 5, AI

Qo(2) = [ Qs(2)

+

(11.2)

I, ... , 4),

= I,

+

1)]

,

+

t" yJl )1 (t"I'W't L L ... l

A,

= 2,

Jtl

+ [ 200 + 100

= 100, 2 2( 00 1

Jt,

+

= 200,

1)]

+

=M +

N

+

max {M, N}

=

= Q = R = 6.

2 = 410,

= 21,

Qs(2)

K

15.

186

Thus, the vector field (11.2) can be

C

linearized according to Theorem 9.1, but Theo-

rems 3.3 and 9.6 do not ensure the possibility of

C

linearization.

11.7. Example. Consider the vector field

(11.3)

Here n Then

= 4,

1= 2, m

Qo(2)

= 2,

~I

= 1,

200 = [ I4 + 2(-1 +

1)]

~l

= 4,

J'I

= 100,

4 + [ 200 100 + 2(100 +

Qs(2)

= 2[2'1 ] + [2';~ ] + 3 = 23,

Qs(2)

= 2·4 + 1 = 9.

Therefore, the vector field (11.3) is and 9.1 do not provide

C

C

J'l

= 200, 1)]

K

= Q = R = 12.

+ 2 = 412,

linearizable by Theorem 9.6, while Theorems 3.3

linearization.

11.S. Remark. As to the nodal case, it is easily seen that the methods of Theorem 9.1 do not lead to any improvement of Theorem 3.4. As a matter of fact, for nodal equilibria, we have two main results, namely, Theorem 3.4: K ~ k, Q ~ Q1 = [~l I ~I] + 1, R = min {Q., k}, Theorem 9.9: Kit: k I + 1, Q it: k I, R = Q (for definiteness, it is assumed that the equilibrium is attracting). One can easily show.by examples that these two theorems are independent one from another. 11.9. Polynomial Normal Form Theorem (the saddle case). Let K and k be positive integers, ~ be a CC smooth vector field defined on a finite dimensional linear space E and having the origin as a saddle type equilibrium. Let A. = D~(O). 1/ K it: min {Qo(k) , Qs(k) , Qs(k)} , then the vector field ~ can be reduced near the origin by means of a

C-

change of variables to the resonant polynomial normal form

187 n

N(k)

L L p;y'''et,

Y = Ay +

t-I

where N(k) is the integral part of the number k ( In implies that 9,

=
9>

and cr

II!

(11.4)

1.,.1-2

~tJ.lm + 2 ). Moreover, p;" 0

~1J.l1

2I(k) .

• The assertion follows from Theorems 8.3, 10.1, 10.2 and Lemma 11.2.

11.10. Polynomial Normal Form Theorem (the nodal case). Let K and k be posi-

tive integers, ~ E r'(E) and the origin be an attracting equilibrium of ~. If K I: min {[~t / ~I] + 1, k I + I}, K I: k, then the vector field ~ can be locally reduced via a

c!

coordinate change to the resonant polynomial normal form n

Y = Ay +

N1(k)

L L p;y'"et, t-I

1.,.1-2 ~

where NI (k) is the integral part of the number k ( 1n ...!. ~I

that

~t

=
~>

and cr

II!

+

1 ), and p; '" 0 implies

S(k) .

• This follows from Theorems 8.4, 10.2, 7.8 and Lemma 11.2.

Bibliographical Notes and Remarks to Chapter

n

The method of normal forms in the theory of vector fields and mappings defined by power series (convergent or formal) is developed by Poincare [1] and Dulac [1]. The problem on reducing analytical vector fields near an equilibrium to the resonant normal form (as well as many other classification problems in the theory of analytical differential equations) naturally splits into two parts. At the first stage, one looks for a normalizing change of variables represented by a formal power series, the second part deals with convergence. Various problems concerning analytical normalization are investigated by Birkhoff [1], Siegel [1], Kolmogorov [1, 2], Arnold [1 - 3], Moser [1],

188 Bogolyubov, Mitripolskii and Samoilenko [1], Bruno [2], n'yashenko [1] and many others. This vast research field is outside the scope of our book. Some theorems on smooth conjugacy of ~ The question equilibrium was They introduced

vector fields are obtained by Sternberg [1, 2] and Chen [1]. on finitely smooth linearization of polynomial vector fields near an first considered by Samovol [l - 3] and Kondratyev and Samovol [1]. the condition S(k) and proved several interesting results. Samovol [4]

has also investigated the more general problem on CC smooth normal forms, but, as we have observed, his proof is incomplete (it fits only under the additional assumption

that all the coordinate hyperplanes are invariant). As it is shown by Samovol [3, 9, 10], the condition S(k) cannot be weakened at least in the following three cases: (1) the equilibrium is a node; (2) the equilibrium is a two-dimensional saddle; (3) k = 1. It was discovered later by Samovol [5 - 9]

condition S(k) is not necessary for

CC

that for saddle points in

linearizability.

R", n

it:

3,

the

Samovol [5 - 9] proposed new

conditions sufficient for CC linearizability (we have denoted them as C(k) and So(k); see Definitions 7.9, 7.10, 7.13 and 7.18). Samovol [4] obtained also a theorem like Theorem 3.3, but with the estimate Q.(k) instead of Qo(k) (see formula (3.5». The method of introducing additional monomial variables was suggested by the authors. It was applied by Samovol [9] to fl11 the gap occured in the proof of the main theorem on normal forms given in Samovol [4]. The condition l!I(k) is introduced by the authors (see Bronstein and Kopanskii [5 - 7]). The question of whether this condition can be weakened is still open. It seems unexpected and mysterious that the study of finitely smooth normal forms led us to the notion l!I(k) which involves convex polyhedral domains. This is a surprise even against the fact that in the local theory of differential equations as well as in algebraic geometry one often meets Newton polyhedra (see, for example, Bruno and Soleev [1]). In fact, these two situations differ in that the Newton polyhedron is uniquely determined once the problem is stated, whereas in our situation the convex polyhedral domains are not given but must be suitably chosen. However it may be, Proposition 5.13 confirms that the condition l!I(k) is to the point. Recently, Bruno published two papers devoted to finitely smooth linearization of differential equations (see Bruno [4, 5]). Unfortunately, the first paper is erroneous. For example, it follows from Lemma 3 of this paper that the function f,

189

j{x)

is of class

= {;'X~~4 In (~o + ~x!0 + ~o~<), o d.

But really,

fEe

if

~o + x~x!° + ~~o _ 0;

otherwise,

according to our Proposition 5.13. One can directly

check that a'll a~ does not exist at the origin. In Bruno [5], a new, corrected version of his condition, T(k) , is proposed. But it is immediately seen that T(k) is equivalent to A(k, ero), the collection ero being chosen as indicated by Bruno [5]. One might conjecture that T(k) is equivalent to 2l(k) , but we do not think so. In any case, the problem on optimal choice of er (i.e., giving the maximal smoothness class) is still unsettled when n > 3 (for n = 2, 3, see Theorems 7.24 and 7.25). Belitskii [1] and Sell [1 - 3] tackled the problem on linearization of finitely smooth vector fields and diffeomoIphisms. To (f linearize the local diffeomorphism F(x) = Lx + j{x) (x E Rd , j(0) = 0), one must find a solution h E (f(R- d , Rd ), h(O) = 0, Dh(O) = 0, of the functional equation h 0 L - L 0 h = f 0 (id + h). At first, one has to show that the linear operator "(h) = h 0 L - L 0 h is invertible in a certain subspace n of (f(R- d , Rd ). To do this, one usually decomposes the space n into a direct sum of some II-invariant subspaces E1, ••• , Ep and provides each of these subspaces with an appropriate norm so that the restriction of " (or ,(1) becomes contracting. For example, when proving Theorem 9.1, we first looked for a function h(x) Ii h(u, v) expressed likewise in Sell [3]: M-I

N-I

but later we were forced to use a more refined expansion "

h(u, v) =

M-I

N-I

E (E E E h!.f3(v)u«~ + EEL g! ..,(U)u"'va ). '-I --01«1-- 1f3I-N I,.I-M lal-r r-O

After proving that the operator " is invertible in the space n = EI • •.• • E p , one turns to solving the homological equation .4h = tp for tp E !J1. Write the equation h 0 L - L 0 h = f 0 (id + h) in the form "(h) + ~(h) = f, where ~(h) = f 0 (id + h) - f.

190

One naturally attempts to prove that I. is a Lipschitzian operator with Lipschitz constant arbitrarily small when restricted to a sufficiently small neighbourhood of the origin. This necessitates a thorough examination of the projections of the function (id + hi + ... + hv) - / onto the subspaces E I , ... , Ev' Such scheme of proof is proposed by Sell [3]. Let us present the main results announced in this paper.

/0

Theorem 1. Let M, N, and k be positive integers satisfying the conditions M?t.1

>

k~l'

= M + N + max {M, N} + k. If /e c!'(Rd , Rd ), D"j(O) = 0

(r

= 0,

N 1-11

>

k I-Im' Denote P

1, ... , M

+N-

1) and there are no resonant terms o/degree M

/or all j e {I, ... , n} F(x)

= Lx + j(x),

«E z~,

and all

admits a

1«1

+N

= M + N),

(i.e., 9J

;I:

<<<,9>

then the mapping F,

c! linearization near the fixed point O.

Theorem 2. Suppose that the spectrum 0/ the operator L lies in one side 0/ the imaginary axis (for definiteness, in the left side). Let the numbers M and k be such

that M ~I > k ~,. If /

E

c!"+k(Rd, Rd ) and D"j(O)

the mapping x ~ Lx + j(x) is

=0

(r

= 0,

1, ... , M - 1), then

c! linearizable near x = O.

Unfortunately, the proofs of these theorems given by the author are unsatisfactory. Firstly, it is not proved that the quantity IID~D~XIIOll entering the inequality on p.l082 of the paper by Sell [3] is bounded. Therefore it is questionable whether the operator J defined by the right-hand side of formula (6.42) of this paper transforms the functional spaces defined in Lemma 13 into itself. Secondly, it is not proved that the operator J is contracting in these spaces (it is only shown that J is contracting with respect to the cfJ norm). But the main shortcoming is that one needs a more refined direct sum decomposition than that used by Sell [3]. We point out that our Theorems 9.1 and 3.4 strengthen the assertions formulated by Sell [3]. Let us present the results on linearization of finitely smooth dynamical systems near a rest point obtained by Belitskii [1]. For the sake of brevity, we confine ourselves by local diffeomorphisms. Let L: Rn -+ Rn be an invertible linear operator and Pq , PI < ... < Pq' be the moduli of its eigenValues. Put J(L ,k) =

Ie-] U[P tPJ'+ PJPt ie

t<J

(k

=

1, 2, ... ),

PI' ... ,

191 where p+

= max {p,

denote the set of all

I}, P-

= min {p,

ce smooth

I}, and [a, b] is empty whenever b

f.

functions

IRn

-+ IRn

such that Dk.f

<

a. Let ce,l

satisfies the

Lipschitz condition with Lip D~ s 1. Theorem 3. Suppose that the set I(L, k) does not contain the numbers Pt

... , q). Then the germ of each is

ce

ce,l

diffeomorphism F, F(x)

(i

= 1,

= Lx + .f(x), .f(x) = O(IIXIlk.),

linearizable at the origin.

= Lx +

Theorem 4. Let F(x)

diffeomorphism. Then F is

O(IIXII Jc)

ce,l

conjugate to L.

(!k.lq]

Bruno [1] has shown that if PI

denote the germ of a hyperbolic

< 1 and Pq > 1, then Theorem 3 works only for k

Belitskii [1] presents also the following result on

C

linearization.

Theorem S. Let L: IRn -+ IRn be an invertible linear operator. In order that every

diffeomorphism F, F(x) = Lx sl4fficient and necessary that

+ .f(x), ft.0) = 0, (Pt

Pt" PJ·Pk.

< 1,

Dft.O)

PJc

= 1.

= 0,

be

C

C

linearizable, it is

> 1).

(1)

This theorem implies the well-known result obtained by Hartman [2] that every twodimensional hyperbolic C diffeomorphism is C linearizable. A number of interesting results concerning smooth linearization of two-dimensional flows and diffeomorphisms is obtained by Stowe [1]. In the case where F let x = (u, v), where I:

E

U E

(!», Theorem S follows from results by Samovol [3]. In fact, E:(L), v

3, as well as uOl. and "

for

E

E"(L). The monomials

1«I

I:

2,

XW

Ii

uOI.~ with 1«I

+ 1~ 1

1~ 1 I: 2, satisfy the condition S(I). The

condition (1) just means that the monomials uOl.~ with 1«I = 1, 1~ 1 = 1 are nonresonant. Theorem 9.4 coincides, in fact, with Theorem S, but the proof is new. In our opinion, the proofs of Theorems 3 - S given by Belitskii [1] should be recognized as insufficient and incomprehensible.

192 Recently, Rychlik [1] proposed a new estimate of the class of smoothness needed to linearize a vector field near a saddle point. But this estimate seems to be incorrect.

c!-

193

CHAPTER III LINEAR EXTENSIONS OF DYNAMICAL SYSTEMS

I 1. Basic Notions and Fads For the reader's convenience, let us briefly present some definitions and results from the theory of dynamical systems which will be used later. A more detailed exposition of this material can be found, for example, in the book by Bronstein [4].

Attractors. RepeUers. Chain recurrence 1.1. Definitions. Let M be a compact metric space with metric p. Let T denote the group IR of reals or the group Z of integers. Let (M, T, /J be a dynamical system (or, in other words, a topological transformation group). This means that f. M x T -+ M is a continuous mapping and the following identities hold: (1) j(x, 0) = x (x E M); (2) ./(x, t\ + toJ = j(f(x, '\), toJ (x E M; til t z E 7). Along with (M, T,

/J,

l(x, I) = j(x, -t) (x E M, t Let Y c M. The set

we shall consider the system (M, T, /) defined by E

7).

,.,(Y, /J

=

n

j(Y, [t, t>o

+ ID»

I)

(1.1)

is called the w-limit set of Y. The set ,.,(Y, is said to be the a.-limit set of Y and iii denoted as a.(Y, /J. These sets are closed and invariant under the transfonnation group (M, T, /J. The subset A. c M is called an attractor if there exists a neighbourhood S of A such that A. = ,.,(S, /J. An attractor for 1 is said to be a repeller (for /J. Let A. c M be a closed invariant set. Denote

194 W(A. /) • W(A) WU(A. /) - WU(A)

=

{x

M: w(x. /) c: A}

E

= {X E

= {x

E

= {x

M: «(x. /) c: A}

E

M: p(f{x. t). A) ... 0 M: p(f{x. t). A) ... 0

If A c: M is an attractor for the dynamical system A·

IE

M \ W(A. /)

is non-empty and is a repeller for

f.

f

as t ... + Go}; as t ... - Go}.

and A;t M. then the set

In this case.

A·

is called

the repeller dual to the attractor A. Every repeller is dual to some attractor. Let A be an 'II

II

attractor of the

(M. T. /).

system

'IIA •.f': M ... [0. 1] with the following properties:

<

'II(x)

(x

M \ (A u A*>. t

There exists a

(1) '11- 1(0)

= A.

function

'11- 1(1)

= A·;

>

0). The function 'II will be referred to as the Lyapunov junction constructed for the attractor A. (2) 'II(f{x. t))

E

By 1U,f) we shall denote the intersection of all sets of the form A u A·. where A is any attractor of the system f. The point x E M is said to be chain recurrent if for every numbers c there exists a collection points x t E M and numbers (i

= O.

1..... k - 1).

1.2.

Theorem.

'R.(f)

>

0 and t

{xo = X. XI • .... Xle_I' xle = x; to. t l • .... tle _l } tt E T such that tt ill: t and p(f{xt • tt). Xt+I)

>

0 of

<

I:

coincides with the set of all chain recurrent points of the

system f. Filtrations and Morse collections 1.3. Dermitions. A Morse set is the intersection of an attractor and a repeller. Let Y c: M and let n(Y. /) denote the intersection of all attractors containig w(Y. /) (see the formula (1.1». A fmite ordered collection {AI' .... Ale} of closed dynamical system (M. T. /) is called a Morse collection if: (1)

«(x. /) u w(x. /) c: AI

U ... U

Ale

(x

E

M);

invariant

subsets

of the

195 A flltration for / is a finite sequence 1/1 == Ao C AI c: ••• c: A k _1 c: A.Ie == M of attractors At (i = 1•...• k). There exists a one-to-one correspondence between filtrations and Morse collections. namely.

At

= n(AI

U ••• U

Ato /)

(i

=

1, ...• k). Ao

= 1/1.

Consequently. each set entering a Morse collection is a Morse set and. moreover. AI is an attractor and A.Ie is a repeller. If {AI' •••• Ale} is a Morse collection. then

The set ~(f) of chain recurrent points of (M. T. /) coincides with the intersection of the family of sets AI U ••• U A.Ie ranging over all Morse collections {AI' •.• , Ale}. The theory of the chain recurrence is due to Conley [1]. Let N be a Morse set of the system (M. IR. /). i.e., N

=A

n B·. where A and B are

attractors of / and B· is the repeller dual to B. Choose a neighbourhood U of the set N so that U c: WS(A) n ~(B·). Let V

== lIIA ,f:

M -+ [0. 1] be a Lyapunov function constructed

for the attactor A. If the number d is small enough then D set D is said to be fundamental. I

1.4. Theorem. Let Q be an arbitrary neighbourhood ~(N)

U

J(Q, IR) is a neighbourhood

0/

== V

0/

-I(d) n WS(N) c: U. The

D in

V -I(d).

Then

~(N) in M.

1.5. Lemma. Let {All .•• , Ale} be a Morse collection for the system (M, T, /). Then

J:St

Jl:.l

196

1.6. Lemma. If

is a Morse collection of the system (M. T• ./). then there exists a continuous junction I: M ... [1, k] such that:

(1) 1-1([1,

{AI' ••.• Ale}

m = UWU(AJ);

1-1(1)::> At;

J:5t

(2) the junction t is strictly decreasing along the trajectories of the system f out of the set A i! AI V .,. v Ale'

Bundles. Vector bundles

1.7. Def"mitions. Let us later on. A bundle is a triple ~ = p: X ... B is a continuous the total space of the

recall some notions from the theory of

bundles

needed

where X and B are topological spaces. and surjective (i.e .• p(X) = B) map. The space X is called bundle. B is called the base, and the preimage (X. P. B).

aE {x E X: p(x) = b} == Xb is called the fiber of the bundle lying over the point bE B. Let (X, P. B) and (X. P'. B) be bundles. A pair (t. 9') consisting of continuous mappings I: X ... X and 9': B ... B such that p' 0 I = 9' 0 P is said to be a morphism from the bundle (X. P. B) into the bundle (X. P'. B). Because tp is uniquely determined by t. we shall sometimes denote the bundle morphism (t. 9') simply as t. Let :BUll. denote the category of bundles. as defined above. Let ~ = (X. P. B) be a bundle and f. BI ... B be a continuous map. The bundle (XI> PI. B I ) where

p-I(b)

XI = {(b. x): b

E

B I, X

E

X, p(x) = ftb)},

PI(b, x) = b,

It:..

is called the pullback of the bundle (X, p, B) by f. BI ... B and denoted Let (X, p, B) and (X, p', B) be bundles with the same base B. A continuous map I: X ... X is called a B-morphism of the bundle (X, p, B) into (X. p', B) if p' 0 I = p. The B-morphism I: X ... X is called a B-isomorphism if the map I is invertible and I-I: X ... X

is continuous.

197

Let Suns denote the category of bundles with a fixed base B and B-morphisms as morphisms. A bundle (X, p', B) is said to be a subbundle of the bundle (X, p, B) if X is a subspace of X and p' = piX. A bundle (X, p, B) is called an n-dimensional vector bundle if each fiber Xb (b E B) is provided with the structure of a real n-dimensional vector space in such a way that the following property of local triviality holds: for each point p E B there exist a neighbourhood

V

= V(b)

and a

h: V x R" ... pol(Y)

V-isomorphism

such that the

restriction of h to x x IR", X E V, is a vector space isomorphism. Let (X, p, B) and (X, p', B) be vector bundles. A bundle morphism (L, 91) is said to be a vector bundle morphism if for each point b E B the restriction

L: pol (b) ... (P')ol(9I(b» of L: X ... X is a linear map. Let VS denote the just defined category of vector bundles. Given an arbitrary paracompact space B, let vS s denote the category of all vector bundles over B and let VS~ be the category of all vector bundles with the base B and fiber R".

=

Let (X, p, B) be a vector bundle and Xo be a subspace of X such that p(~ B and (Xo, plXo, B) is vector bundle, too. We say that (Xo, plXo, B) is a (vector) subbundle of the vector bundle (X, p, B) if the inclusion Xo c X is a vector bundle B-morphism. Given vector bundles (X, p, B) and (X,]i, B), one can define a new vector bundle L(X, X) Ii Hom(X, X). The fiber of Hom(X, X) over the point b Ii B is equal to the vector space L(Xb' Xi,) of all linear mappings 91: Xb ... Xi,. There is a one-to-one correspondence between sections of the bundle Hom(X, X) and vector bundle B-morphisms from X into X. The Whitney sum of bundles (X, p, B) and (X, p', B) is defined to be the bundle (X. X, p • p', B), where

X. X

= {(x,

(P • p') (x, x')

x'): x

E

X, X

E

= p(x) = p'(x')

X, p(x)

= p'(x)},

«x, x) EX. X).

Note that (p. p')"l(b) = Xb x Xi, (b Ii B). If (X, p, B) and (X, p', B) are vector bundles, the (X. X, p • p', B) is naturally endowed with the structure of a vector bundle. A Riemannian metric on a vector bundle (X, p, B) is a continuous map g: X • X ... IR

198

such that for each point b

E

B

the restriction g IXb

X

Xb

is an inner product on the

fiber Xb. The number IIxll = V'g(x, x) is called the norm of x E X. Usually, we shall write <x, y> instead of g(x, y). Whenever the base is paracompact, Reimannian metrics exist. Let (X, p, B) be a vector bundle and IRs E (B x IR, prl' B) be the trivial onedimensional vector bundle over B. The vector bundle L(X, IRs) is said to be dual to (X, p, B) and will be denoted (X·, p" B). The fiber X~ is equal to the space of all linear functionals I: Xb -+ IR. Fix some Riemannian metric <.,. > on the vector bundle (X, p, B). To every element

y (x

EX E

put in correspondence the element

X,

p(x) = p(y».

(X, p, B)

ot, p.,

The map

y

1-+

l

y.

E/y

eX· defined by /y(x) = <x, y>

is a vector bundle

B-isomorphism from

into (X·, p., B). Thus, every Riemannian metric allows us to identify

with (X, p, B). We also note that the vector bundle dual to (X·, p., B) coincides with (X, p, B). Let Y be a vector subbundle of (X, p, B). By yl- we shall denote the set B)

yl- = {~ It is easy to verify

that

E

X·: ~(y) = 0 (y

E

Y)}.

yl- is a vector subbundle of the dual vector bundle

(X·, p., B).

Given a vector bundle (X, p, B) and a set Me B, XIM (or X[M]) will denote the

restriction of (X, p, B) to M (i.e, the vector bundle (p-I(M), plp-I(M), M).

Extensions of transformation groups 1.S. Definitions. Let (X, T, n) and (B, T, p) be transformation groups. Let p: X -+ B be a continuous map such that p(n(x, t» = p(P(x) , t) for all x E X and t E T. In this case, we shall say that p: X -+ B is a homomorphism and write p: (X, T, n) -+ (B, T, p). If, moreover, p(X) = B, then (X, T, n) is said to be an extension of (B, T, p) and the map p: (X, T, n) -+ (B, T, p) itself is also called an

extension. Let (X, p, B) be a vector bundle and let (X, T, n) and (B, T, p) be topological

199 transformation groups so that p: X -+ B is a homomorphism. The extension p: (X, T, x) -+ (B, T, p) is said to be linear if for every b E B and t E T the map XtlXb: Xb -+ Xp t (b) is linear.

Linear extensions of dynamical systems occur, for example, when linearizing smooth dynamical systems. Let B be a smooth manifold and (B, IR, p) be a smooth flow. Let (TB, 'tB' B) denote the tangent vector bundle. The flow (TB, IR, x) defined by X

t

(v)

=

t

(Tp )b(V)

(b

B, v

E

E T~)

is said to be tangent to (B, T, p). Clearly, 'tB: (TB' IR, x) -+ (B, IR, p) extension. It will be referred to as the tangent linear extension. If, in particular, B

=

IRn and the flow

(B, IR, p)

is a linear

is generated by the differential

x

equation = j(x) (x E IRn ), then the tangent linear extension (TB, IR, 'It) corresponds to the so-called equation 01 first variation:

x = j(x), t

= Dj(x)~

(x

E

IR n , ~

E

IRn ).

We would like to warn the reader that whenever B is an arbitrary smooth manifold and the fiow (B, IR, p) is determined by a vector field ~ E rr('tB) then the flow (TB, IR, x) corresponds not to the vector field n: TB -+ T(TB), as it might be conjectured, but to Jon;, where J: T(TB) -+ T(TB) is some involution defined in a standard way. A morphism of the linear extension p: (X, T, x) -+ (B, T, p) into the linear extension PI: (XI' T, XI) -+ (BI' T, PI) is defined to be a vector bundle morphism of (X, p, B) into (X.. PI' B) such that L(xt(x»

= x~(L(x»

(x

E

X, t

E

1),

Clearly, the class of all linear extensions forms a category. Therefore, it makes sense to speak about the Whitney sum of linear extensions, the pullback and so on. For (W, T, p) be a example, let p: (X, T, x) -+ (B, T, p) be a linear extension, transformation group and h: (W, T, :>.) -+ (B, T, p) be a homomorphism. Denote N = {(x, w): x E X, WE W, p(x) = hew)} and define a transformation group (N, T, p.) by p.t(x, w)

=

(xt(x) , ;>.t(w».

Then

q: (N, T, p.) -+ (W, T, :>.),

q(x, w)

=

w, is a linear

200 extension called the pullback of h: (W, T, ~) .. (B, T, p).

ot, p.,

Let p: (X, T, tr) .. (B, T, p) be a linear extension and (X, p, B).

bundle dual to

tr.(~, t)

It

can

E

by the homomorphism

p: (X, T, n) .. (B, T, p)

For ~

E

(X)b ,. (Xb)· and t

E

B)

be the vector

T. defme an element

X;t(b) by

be

easily

verified

that

(X,

T, tr.)

is

a

transformation

group

and

l: (X,

T, tr.) .. (B, T, p) is a linear extension. The latter is said to be dual to the linear extension p: (X, T, tr) .. (B, T, p).

Identify

the

dual

vector

bundle

fixed Riemannian metric <',' > on p: (X, T, tr.) .. (B, T, p) is given by

(X,

p., B)

(b

Consequently, for

with

(X, p, B). Then

E

B, t

(X, p, B)

the

E

by

using

a

dual linear extension

7).

x, Y E Xb , we have

The last equality can be expresed in an invariant (i.e., not depending on the choice of Riemannian metric) form: t

tr.(~)(tr

t

(x» =

-t

~(tr

t

(tr (x») = ~(x)

(b

E

B, x

E

Xb , ~

.,.

E Ab'

t

E

7) •

• 2. Exponential Separation and Exponential Splitting 2.1. Notation. Let B be a compact space, T = IR or T = z, (X, p, B) be a vector bundle and p: (X, T, n) .. (B, T, j..) be a linear extension. Set

201 (I e T, b e B).

In the theory of linear extensions of great use are the following Lyapunov exponents (see Lyapunov [I], Bylov, Vinograd, Grobman and Nemytskii [1]): g(n, b)

= lim

sup! In IIn!1I

t-++11111

101(n,

b)

iii

· sup -1 1n = - 1t-++11111 1m ==

inf {II: IIn!1I exp (- Ill) -+ 0 lin . tt

pCb)

II

1·1m sup -1 In

iii -

t-++11111

sup {II: IIn·~ II exp (Ill) -+ 0 P (b)

It is easy to verify that

+

(1-+

(I -+

+

III)};

11(11't,,>·1 II

III)}.

and w('If, .) are constant along the trajectories of (B, T, p) and do not depend on the w(n, b) :s g(n, b)

(b

E

B). The functions

g(n, .)

(X, p, B). Identify the dual vector bundle

choice of Riemannian metric on

(X·,

l,

B)

with (X, p, B) by using some Riemannian metric. Let p.: (X, T, n.) -+ (B, T, p) be the dual linear extension, then

=

IIn·tlX t

p (b)

(b e B, IE 7).

II

Therefore g(n., b)

= - w(n,

b),

w(n., b)

= - g(n,

b)

(b e B).

Although the exponential rates g(n, b) and w(n, b) are asymptotic in nature, they are intimately tied to uniform behaviour. This can be seen from the next two lemmas obtained by Fenichel [1-3].

2.2. Lemma. LeI lhe number II be such lhal lim IIn!1I exp (- Ill) = 0

(b e B)

t-++III

lhen lhere exisl numbers c > 0 and ; , ; < II, satisfying Ihe inequality (b

E

B, I

~

0) .

• For every point b e B, choose a number r(b) , l(b) > 0, so that

202 1In:~(b)1I

<

exp [cxt(b)].

By continuity, there is a neighbourhood U(b) , for which (y E

Choose a finite covering {U(b l ), such that

U(b».

...

U(b l )} of the compact space B and a number a.

... ,

= 1,

(yE U(bJ),j

= max

Set L

{t(b J ): j

= 1,

C

= sup

<

a.

... ,1).

... , I} and

{1In:~1I exp [- ~]: b

E

B, t

E

[0, Ln.

b E B, construct a sequence j(l), j(2),... in the following way. Let j(1) denote a number satisfying b E U(bJ(I»' Now define 1(1) = t(b J(1» , T(O) = 0, T(l) = t(l). Proceeding inductively, we define j(n) so that p(b, T(n - 1» E U(b J(rI» and put I(n) = t(b J(rI» , T(n) = T(n - 1) + t(n). Every number t > 0 can be presented in the form t = T(n) + r, where 0 s r < L. Hence, we get Given a point

s c exp { ~ [r

+

+

l(n)

t(n - 1)

+ ... +

t(l)]}

=c

exp (~).

It follows from Lemma 2.2 and from the definitions of O(n:, b) and w(n:, b) that for each

£

>

0 there exists a number

c; I exp

[w(n:, b) - £]1

s

Cc

1In:~1I

>

s Cc

Moreover, for large enough numbers I

>

0 such that exp [o(n:, b)

• Denote «0

= sup { O(n:,

H

b): b

O(n:, b) E

£]t

(b E B, I

+

£]1

(b

>

0).

0,

exp [w(n:, b) - £]t s 1In:~1I s exp [O(n:, b)

2.3. Lemma. The function b

+

(b

E

E

B).

B) attains its supremum .

B}. We shall show that there exists a point bo E B

203

such that C(n, bo) Then

= CXo.

Suppose, to the contrary, that C(n, b)

Ii m IIn:1I exp (- exot) H+oa

=0

IIn:1I :s

C

exp (~)

Consequently, we can choose a number

«'1'

A

<

«'1

<

A

and «,«

(b e B, t

~

CXo

<

CXo, such that

>

0).

CXo so that

According to the definition of C(n, b), this means that C(n, b) :s b E B. We get a contradiction: CXo < «0. Similar arguments prove that the function b .....

for all b e B.

(b e B).

> 0

According to Lemma 2.2, there exist numbers c

<

wen,

0:1

<

CXo

for all

b) attains its infimum.

2.4. Defwition. Let XI and X2 be vector subbundles invariant under (X, T, n), and X = XI Ell X2• Suppose that, for some Riemannian metric on (X, p, B), there exist numbers d > 0 and « > 0 satisfying lint (x 2 ) II IX2 "

In this case, we shall say that the linear extension p: (X, T, n) -+ (B, T, p) is exponentially separated relative to XI and X2• Because B is compact, this property does not depend on the choice of Riemannian metric on (X, p, B). Denote

2.5. Proposition. The following assertions are pairwise equivalent: (1) the linear extension p: (X, T, n) -+ (B, T, p) is exponentially separated relative to the invariant vector subbundles XI and X2 ;

204

(3) tMre is a number

T

> 0 satisfying the inequality

(t> 0, Z, EX, (i = 1,2), P(ZI) = p(z,J, (5) tMre exist numbers

(6) n(nl' b)

<

> 0

T

w(n2' b)

and q

> 0

Ilzl"I~1

'#-

0);

such that

(b E 8).

In (3) and (5), it is additionally assumed that tM subbundles Xl and X2 are mutually onhogo1/ll1 with respect to 1M Riemannian metric on (X, p, 8) . • For proof, see Bronstein [4, Proposition 6.11].

1.6. Def"mitiOD. As before, let B be a compact space and p: (X, T, n) .... (B, T, p) be a linear extension. Suppose that there exist vector subbundles Xl' ... , Xk. invariant under (X, T, n) and such that X = XI • ••• • Xk.. We shall say that the linear extension p is exponentially splitted (or that X = XI ••.•• Xk. is an exponential splitting) if and II > 0 satisfying there exist numbers d >

°

t lin (XI)II

lint (x 2 ) II

--IIXI"

IX2 "

(xl

E

Xu

~

E

Suppose that X

X'+I' IIX1"11~1I

=

XI • ~

'#-

for

0, P(xl)

~

d exp (Ill)

= p~,

I

> 0, i = 1, ... , k - 1).

some invariant vector subbundles. If there exist

205

>

numbers d

0,

01:

>

0 such that

then the linear extension p is said to be hyperbolic. We shall also say in this situation that X Xl • Xl determines an exponential dichotomy. Because B is compact, this property does not depend on the norm on X. Without loss of generality, we may assume that d = 1 (this can be achieved by an appropriate choice of the Riemannian metric). Such a metric is said to be a Lyapunov metric (or adapted to the hyperbolic linear extension). A continuous function F: B x T -+ IR is called an (additive) cocycle of the dynamical system (B, T, p) if the following identity holds:

=

F(b, I When T

= IR,

+

'1:)

= F(b,

t)

+

F(Pt(b) , '1:)

(b

E

B; I, '1:

E

7).

a wide class of cocycles can be expressed in the form t

F(b, t)

= JG(pT(b»d'1:

(b

E

B, t

(2.1)

IR),

E

o where G: B -+ R is an arbitrary continuous function. In fact, if the cocycle F(b, I) is differentiable with respect to I at t = 0 and, moreover, the function G: B -+ R defined by the formula

G(b)

= !. (F(b, dt

t)) 1t

-

0

(b

E

B)

is continuous, then (2.1) holds. Indeed,

!. (F(b, dt

= lim

t»1 t-O

riO

F(b, '1:

+ s)

- F(b, '1:) S

= lim riO

F(pT(b) , s)

= G(pT(b».

S

2.7. Theorem. Let p: (X, T, n) .. (B, T, p) be a linear extension exponentially splitted into invariant vector subbundles XII"" Xk • Then there exist cocycles r,: B x T -+ IR, IR,: B x T .. IR (i = 1, ... , k) and a number fJ > 0 such thai a point x

206

belongs to the subbundle X,

W (b

for all large enough numbers t

= p(x»

> O. Moreover.

rt+l(b. t) - R,(b. t)

it

(b

fit

E

B. t

E

T. i

=

1..... k - 1) .

• See Bronstein [4. Theorem 6.33].

2.B. Theorem. Suppose that X

= XI

is an expo~ntial splitting of the li~ar extension p: (X. T. '11') -+ (B. T. pl. Given a number h. 0 < h < 1. o~ can choose numbers TO > 0 and EO > 0 so that every linear extension p: (X. T. A) -+ (B. T. p) satisfying the inequality III ••• ..

XI<.

can be represented as the Whitney sum of uniquely subbundles ~ ..... ~. Moreover. X T. '11') -+ (B. T. p). and

p:

ex.

~ c C(X,. h)

={x

+ y:

= ~ ....... ~

x EX,. Y .L

X,. p(x)

determi~d

A-invariant vector

is an expo~ntial splitting for

= p(y).

lIyll :s hIlXII} •

• See Bronstein [4. Theorem 9.26].

§ 3. The Structure of Linear Extensions In this section. we shall present the main results concerning the structure of two

important classes of linear extensions. namely. extensions without non-trivial bounded motions and extensions satisfying the transversality condition. 3.1. Def"mition. Let B be a compact metric space. (X. P. B) be a vector bundle and

p: (X. T. '11') -+ (B. T. p) be a linear extension. Let II· II denote a fixed norm on

eX. P.

B).

The motion of a point x E X is called bounded if sup {1I,l(x)II: t e T} < '"'. and non-trivial. if x does not belong to the zero section Z = Z(B) of (X. P. B).

207 Denote

r

Ii

r(n) = {x

~ ;;; ~(n) S • S(n)

E

X: lim IInt(x)1I = O}, t-++OII

= {x E X: = {x

lim IInt(x)1I t-+ - 011

= O},

e X: sup IInt(x)1I < OIl}. teT

Clearly, these sets are n-invariant and vectorial (i.e., their intersection with any fiber Xb (b e B) is a linear subspace). The set r is called the stable set and ~ is called the unstable set of the linear extension p: (X, T, n) -+ (B, T, p). Denote

3.2. Theorem. Assume that p: (X, T, n) -+ (B, T, p) satisfies the condition i.e., X has no non-trivial bounded motions. Suppose that dim Xb does not depend on b E B (this is true, for example, when the space B is connected). Denote S

= Z(B) ,

n = dim Xb , A](.

Ii

A](.(n)

= {b

E

B: dim ~

= k, (k

dim

(b

E

B),

x: = n - k}

= {b

E

A: dim ~

= k}

= O,l, ... ,n).

Then {Ao, Alo ... , An} is a Morse collection and the restriction of the linear extension p: (X, T, n) -+ (B, T, p) to the closed invariant set A is hyperbolic. Since ~(p) c Ao U AI U ... U An II A, the extension p is hyperbolic over ~(P) .

• For a proof, see Sacker and Sell [1] or Bronstein [4, Theorem 8.11]. 3.3. Def"mition. One says that the linear extension satisfies the transversality condition if X

= r + ~,

p: (X, T, n) i.e., Xb =

-+ (B, T, p)

x: + ~

for all

be B.

3.4. Theorem. If p: (X, T, n) -+ (B, T, p) satisfies the transversality condition then {Ao, AI' ... , An} is a Morse collection and the extension p is hyperbolic over A

::I ~(P).

208 ~

See Bronstein [4, Theorem 8.47].

3.S. Theorem. A. linear extension satisfies the transversality condition if! the dual extension has no non-trivial bounded motions. ~

See Bronstein [4, Theorem 8.46].

3.6. Theorem. A. linear extension is hyperbolic if! it satisfies the transversality condition and, at the same time, has no non-trivial bounded motions. ~

This follows from Theorems 3.2 and 3.4.

Let p: (X, T, 1t) ... (B, T, p) be a linear extension and Y be an invariant vector subbundle of p. Let X I Y denote the corresponding quotient vector bundle and q: (X I Y, T, 1t) ... (B, T, p) be the naturally defined quotient linear extension.

3.7. Theorem. Suppose that the linear extensions pi Y: (Y, T, 1t) ... (B, T, p) and q: (X I Y, T, 1t) ... (B, T, p) have no non-trivial bounded motions, then the same is true for p: (X, T, 1t) ... (B, T, p). Similarly, if pi Y and q satisfy the transversality condition, then p: (X, T, 1t) ... (B, T, p) satisfies the same condition. ~

See Bronstein [4, Theorem 8.52].

The transversality condition is preserved when passing to quotient linear extension, but restrictions to invariant vector subbundles do not, in general, inherit this

property. 3.B. Corollary. If pi Y: (Y, T, 1t) ... (B, T, p) and q: (X I Y, T, 1t) ... (B, T, p) are hyperbolic, then p: (X, T, 1t) ... (B, T, p) is also hyperbolic. ~

This follows immediately from Theorems 3.6 and 3.7.

3.9. Lemma. A. linear extension is hyperbolic if! its dual linear extension is hyper-

bolic. 3.10. Remark.

The linear extension

p: (X, R, 1t) ... (B, IR, p)

satisfies

the

209 transversality condition iff p: (X, z, n) ... (B, z, p) has the same property.

The vectorial sets X- and ~ in the vicinity or a Morse set 3.11. Lemma. Let M be a Morse set of the system (B, T, p). If the linear extension ~IW'(M) and X-IWS'(M)

p: (X, T, It) ... (B, T, p) is hyperbolic over M, then (continuous) subbundles .

are

• Since the extension pIX(M): (X(M) , T, It) ... (M, T, p) is hyperbolic, we see that X-(M)

and ~(M) are subbundles, X(M)

= X-(M)

• ~(M) and

for a suitable Riemannian metric on (X, p, B) and some

number~,

<

0

<

~

1.

Extend the subbundles X-(M) and ~(M) continuously over some neighbourhood U of the set M and denote the so obtained subbundles as r(V) and XU(V), respectively. If the neighbourhood U is small enough, then r(V). ~(V) numbers satisfying the conditions ~ small, then Illtt(X)1I

< <

jj

<

jjt IIxll

1,

T

>

ex

0, E

jj-r:

= XCV).

Let

< 112.

If U is sufficiently

jj

and

T

be some

r(V),

(3.1)

for all 1 E

[T, T

of M so that III

+ C

1]. By using a Lyapunov function, we can select a neighbourhood UI U,

III n W'(M) is closed and negatively invariant, i.e.,

(I

~

0).

Let !J1 1 denote the space of all vector bundle morphisms XI =

r(lIl n W'(M»

and

X2

(3.2)

L:

X2 ... XI'

where

= ~(lIl n W'(M», satisfying the following conditions:

L covers the identity map of III n W'(M),

IILII:s 1, and

LIXe:. LIXe!

=0

(b

E

M). For

210

t

E [T, T

+

~

1], we shall now defime a translormation

Nt

•

1[: !Ill -+ !Ill. Namely, given

Nt

t

L E !Ill' let L IE 1[ (L) denote the morphism in !Ill determined by the relation graph(Lt ) = 1[t(graph(L». We must show that 1[t(graph(L» is, in fact, the graph of some morphism belonging to !Ill. Observe first that the map

is a vector bundle morphism covering pt (see (3.2». With respect to the decomposition

X(UI n ~(M»

= XI til X2 ,

1[t can be represented in the form

F~sl

t Fuu

'

x:; -+ X:~(b) (11', = S, U; b E p.t(UI ) n ~(M». X" are invariant and x:; = x:; (b M; = s, u), we get where

F!"'.:

Because the sets

11"

E

r

and

11'

Taking into account the inequalities (3.1) and assuming UI to be chosen small enough, we

F~u: ~ -+ 'it't b p ()

conclude that the map

t

E [T, T

+

(b

VI) is invertible and for b

E

and

UI

E

1] the following inequalities hold: II

II

t'UU

FuslAb II

+

II

t!US

t!US

FsulAb II :s

t

J.I ,

t

(3.3)

FsslAb II :s J.I ,

!US ).1 J.I:S t 1• ( I - J.ItII t FsulAb II

To simplify the presentation, let us denote E = graph(L) and

Eb

= {(Lx,

E~

x): x

= {1[t(Lx,

Observe that

E

~}

x): x

E

~

It

= 1[t(graph(L». Then

VI n ~(M», therefore

(b

E

}

= {«F~s + F,!.s

0

L)(x) , (F~u

+

F,!.u

0

L)(x»: x

E

~

}.

211 t ( Fuu

t

+

Fsu

~

0

L)I.Ab

t = Fuu

0

[/

+

t

(Fuu)

t

-I 0

Fsu

~

0

L] lAb'

where / denotes the identity map. Hence it follows from (3.3) that the operator (3.4) is invertible for all L E !Ill' b E UI and t E [T, T

Indeed, since

IILII ~

+

1]. Moreover,

I, we have

+

hence, the operator / known estimate, 11(/

+

t

(F~url • F!u • L: ~ -+ ~

(Fuu)

-I

t

0

-I

~

Fsu • L) lAb II ~ (

1

-

is invertible and, by the well-

jJ.

t

t 'bB -I II FsulAb II)

for all b E UI , t E [T, T + 1]. Thus, the mapping (3.4) is invertible, therefore

is a vector bundle morphism and, besides,

for all t

E [T, T

then p-t(b)

E

UI n

we get Lt(~)

+

1] (here the supremum is taken over all points b e UI n W'(M),

w"'(M) by (3.2». Because the subbundle X'(M)

=0

(b e M,

t e [T, T

+

Ii

r(M) is invariant,

1]).

The above considerations show that jit(!IlI) E!IlI for all t e [T, T prove that i and (3.5), we obtain

t :!J1 1 -+!J1 1

is contracting. In fact, let L I ,

L"

E

+

1]. Now let us

!J1 I , then by (3.1), (3.3)

212

In the last lines, we have used the inequality 11(1 + elrl - (1 ("el" :is 112, "el" :s 112) ,which follows from the identity

where

el

and

el

+

C2)-11l :s

411el - elll

are linear operators.

Clearly, !fll is a complete metric space. Since the operators contracting for all I E ['t, 't

+

it:!fll .... !fl l

are

1], each map it has a unique fixed point. Because these

maps form a commutative family, we conclude that they have a common fixed point, say L· (see Lemma A.30). Denote

If

= graph(L\

Hence it follows that

If

Clearly,

If

is invariant under all

is invariant under

Tet

for all

I

vector subbundle defined over the negatively invariant set

I:

Tet

(t

E

['t, 't

O. Observe that

fll n

+ 1]). If is a

W'(M). It can be

continued by invariancy over W'(M), and thus we obtain an invariant vector subbundle

= 0 for all L E!fll and bE M, E(M) coincides with X"(M). To fmish the proof, it suffices to show that

E(W'(M». Since L(~)

X"(W'(M»

= E(W'(M».

(3.6)

Recall that E(M) = X"(M) and the linear extension p: (X, T, Te) -+ (B, T, p) is hyperbolic over M. Hence, there exists a small enough neighbourhood Ul of M such that

213

(x

Let b

E

E(U-J).

WU(M). Then a.(b, p) c M, therefore pt(b) -+ Mast -+ -

an invariant subbundle, it follows that n

E

= 1, 2, ...), hence

for all d

E

and

= X(M),

E(M)

E(w"(M» c

E(W"(M» c

Further,

a.(b, p).

we

.xu(WU(M»

110.

Since E(WU(M» is

(x E E(U." n W"(M»,

IIn""T(x)lI:S ,,"T IIXIl

.xu(w"(M». It is not hard to show that dim since E(w"(M»

conclude

implies

that

(3.6).

subbundle. The statement concerning

r

dim

Thus,

re: = dim x:;:

is a (continuous) vector subbundle

re: = dim Eb

(b

w"(M»,

E

therefore

.xu I w"(M) is a (continuous) vector

I W'(M) follows from the above arguments by

reversing the time direction.

Continuation of subbundles Let B be a compact metric space and p: (X, T, n) -+ (B, T, p) be a linear extension. Suppose that M is a Morse subset of the system (B, IR, p), i.e., M

= AI

and A." are attractors of the system p. Let V: W'(A I ) -+ IR and

V:

n A;, where AI

W"(A;' -+ IR

be

Lyapunov functions constructed for the attractor AI and the repeller A;, respectively. Further, let d be a sufficiently small positive number and D

= {b E

Fix a positive number

f

W'(A I ) n w"(A;': V(b)

= d,

V(b)

= OJ.

and denote

Q

= {b E B:

V(b)

= d,

V(b)

< f}.

According to Theorem 1.4, w"(M) v p(Q, IR) is a neighbourhood of the set W"(M) in B. Assume that the extension p is hyperbolic over M. Let (continuous) subbundle of X(Q) satisfying the condition (b ED)

G!! (f' • (f'(Q)

be a

(3.7)

(observe that Dc W'(M), hence riD is a subbundle by Lemma 3.11). Continue the subbundle G over the set w"(M) u p(Q, R) by letting

214

This definition is unambiguous, since each trajectory

p(q, IR), q

E

Q, intersects the

set Q in one and only one point, because B is a Lyapunov function. Let GI WU(M) v p(Q, IR) denote the so defined vectorial set. 3.12. Theorem.

If the

extension XIM is hyperbolic, then GI WU(M) v p(Q, IR) is a

(continuous) invariant vector subbundle of XI WU(M) v p(Q, IR). ~

The proof is much the same as that of Lemma 3.11 and, for this reason, some details will be omitted. The Riemannian metric on (X, p, B) can be chosen so that IInt(x)1I

oS

~hlXII

(x

E

XS(M), t

> 0), (3.8)

for some ~, 0 < ~ < 1. Besides, we may assume that ~ and X"(M) are vector subbundles and XS (M) &I X"(M)

J.

~ (b

E

M). Recall that XS(M)

= X(M). By Lemma 3.11,

XS'I WS (M)

and X" I WU(M) are (continuous) subbundles. Extend the subbundles XS I WS(M) and X" I WU(M) continuously over some neighbourhood U of WS(M) contained in WU(M) v p(Q, IR+) (see Theorem 1.4) and denote the so

obtained subbundles by XS'(U) and

X'(U), respectively.

Shrinking, if necessary, the

neighbourhood U, we may suppose that XS'(U) &I X'(U) = X(U). Without loss of generality, we may (and do) assume that Q c U. Indeed, there exist numbers '1\, 0 < '1\ < '1, and TO > 0 such that p(Q\, TO> c U, where Q\

= {b E B:

V(b)

< d, V(b) < r\}. Since

Q\ c Q, we can define subbundles G(Q\) and

n(G(Q\), TO)' Replace Q by p(Q\, TO)' D by p(D, TO> and G by n(G(Q\), TO)' Since XS is an invariant vectorial set, the equality (3.7) will remain true.

Thus, assume Q c U. Taking the number '1 small enough, we get that Xb = ~ 81 Gb for all b E Q. Hence, the subbundle can be represented in the form G(Q) = graph(gO>, where

go: X'(Q) ... XS'(Q) is a vector bundle morphism covering the identity map of Q.

215

>

Fix some number t

O. The mapping

t,...

'II: : .II.

•

~

.II.

IU n

t p- (ll) -+

'bS

.II.

•

~

.II.

I U n p t (ll)

is a vector bundle morphism covering pt. With respect to the direct sum decomposition XI U

= (rl U)

• ('it'l ll), it can be written as

F~sl t

Fuu

where F;....: ~ -+ ~;(b) subbundles and ~

= x:

«(1', (b

(1"

E

" = 'II:tlAb'

t'lP'

F.,.".IAb

Select a number Jl so that i\

'

= s, u; b E Un p-t(ll). M, (1' = s, u) we have t

~

FuslAb =

<

Jl

<

t

Since

'bS

r

and 'Jt' are invariant

(b EM).

0, FsulAb = 0

1. Taking into account the inequalities (3.8) and

shrinking, if necessary, the neighbourhood U, we get that the map F~u I~ is invertible and (b

Denote K 10

>

= max {ligoll,

E

ll).

(3.9)

I}. Assume that the neighbourhood U is so small and the number

0 is so large that

(3.10) (b

E

U, I

2:

to>.

Defme a vector bundle morphism

covering the identity mapping by the formula graph(gt) Gpt(b)

= 'll:t(Gb )

(b

E

Q) and GI Q

= graph (go) ,

hence,

= GI U "

pt(Q). Recall that

216

Gpt(b)

= {«F~.. + F!. • 80>(V) ,

(F~u +

F!u • 80>(V»:

vE ~

}

(b

E

Q).

From (3.9) and (3.10), it easily follows that the operator

has an inverse for all b

E

B and, moreover, (b

U).

E

Therefore, 8t

consequently,

18t"

Recall that Gb

S

t t = (Fu. + F...

",t K

= Xe:!

(t

ill:

\ (Ft • 80)·

t uu + F.u

to>. Thus, 18t"" 0 as

for all b

E

W'(M),

vectorial set GI W'(M) u p(Q, R). Hence, GI U morphism is defined by 81 Un pt(Q)

Because 18t"" 0 as I ..

= 8t +

00,

(I

ill:

,0>,

-1

• 80> ,

I ..

+

00.

according to the definition of the

= graph (g) ,

where the vector bundle

81 Un W'(M)

= O.

we conclude that 8 is continuous at all points of

Un W'(M). Whenever bl , b" E U \ W'(M), then there exist uniquely defined numbers 'I' I" ill: 0 and points '1"" E Q such that b, = p(y" t,) (i = I, 2). Hence, if b" .. bl , then ,,," 'I' ',," 'I' Since G(Q) is a subbundle, we conclude that the map 81i"(P(Q, R+» determines a (continuous) subbundle.

Lower seml-contlnulty or the vectorial sets ~ and

X"

Let (X, p, B) be a vector bundle. A vectorial subset E c: X, p(E) = B, is said to be lower semi-continuous at the point b E B if there exist a neighbourhood U of b and a subbundle G of the vector bundle XI U such that Gb = Eb and Gz c: Ez for all Z E U. The vectorial subset E is called low" semi-continuous if it is lower semi-continuous at each point b E B.

217

If the linear extension p:

3.13. Theorem.

transversality condition then the vectorial sets

r

(X. T. n) -+ (B. T. p)

and

salisjies the

Jt' are lower semi-conrinuous .

• For a proof. see Bronstein [4. Theorem 8.48].

Coherent families of stable and unstable subbundles Let B be a compact metric space and p: (X. R. n) -+ (B. R. p) be a linear extension satisfying the transversality condition X A

=

{b

E

B:

= r + Jt'.

x: • r,,: = X

b}

According to Theorem 3.4. the set

== {b E B:

x:

n

r,,: =

{O}}

is closed and invariant. Moreover. the linear extension p is hyperbolic over A. Denote At

=

{b e A: dim

x: = i}

(i

=

O. 1•...• n).

(3.11)

Then A = "0 U AI U ••• U An. and {Ano An_I' ••.• "o} is a Morse collection. Observe that the numeration (3.11) is opposite to that used in Theorem 3.4. hence (3.12) To each point b

E

B. one can assign uniquely determined numbers i. j e {O. 1•...• n}

such that b e W(At) n ~(AJ). Since p is hyperbolic over A. there exists a Riemannian metric on (X. P. B) adapted to A. i.e.• such that for some number Ao. 0 < Ao < 1. the following inequalities hold:

(3.13)

ex E Jt'IA.

t ~ 0).

The proof of the following result is based on the technique developed by Robbin [1] and Robinson [2].

3.14. Theorem. Let p: (X. IR. n) -+ (B. IR. p) be a linear extension satisfying the At and transversality condition. ~n there exist neighbourhoods Ut of the sets

218

(continuous) vector subbundles ~ and Et i, j = 0, I, .... n the following assertions hold:

(d) ~

CD

Et

(e) IInt(x)1I

=

:s

(x

"At IIxll

~b c~,

such

that

for

Xlp(Ut , IR);

IIn-t(x)1I s "At IIxll

(0

of XI p(U" IR)

~b

C

(x ~

~I U" 0 :s t

E

E

Etl Ut , o:s t

(b

E

p(Uk

,

1),

s

s

1) for some number

"A,

0

<

"A

< 1;

IR), k = 0, I, ... , n).

Proceeding inductively, we shall, firstly, construct vector bundles ~ and Et satisfying the conditions (a), (b) and (c). After that, we shall show that the conditions (d), (e) and (0 hold, as well. We shall restrict ourselves by considering ~

the case rr = u (the subbundles ~ can be built in a similar way). Let IX be the smallest number such that Aa. ~ fZl. Then Aa. is a repeller of the flow (B, IR,

pl. Let Ua. be a neighbourhood of Aa. contained

Without loss of generality. we may assume that vector subbundle.

IX

= O.

in WU(Aa.>. Define

By Lemma 3.11, X" In(Uo, IR) is a

Suppose that the neighbourhoods U t = Ut(A t ) and the subbundles Et are already defined for all O:s i < 1 so that the conditions (a), (b) and (c) are satisfied whenever 0 S i. j < I. Besides, suppose that for O:s j < 1 the equality (g)

E;b

+

~ == Xb

(b E p(UJ , IR»

holds. Let us show that there exist a neighbourhood Ul and a subbundle

Itt

satisfying the

219

conditions (a). Let t be a (B. IR. p) and The function

(b). (c) and (g) for O:s i. j :s 1. Lyapunov function constructed in accordance with Lemma 1.6 for the flow the Morse collection {An. An_I' .... "o}. Set L(b) = n - feb) (b e B). L: B -+ IR is increasing along the trajectories of (B, IR. p) out of the set

A and

L"1(z)

At

C

(i = 0, .... n).

neighbourhoods Ut so that At

C

L,

By continuity of

Ut c L-I(i - 114, i

+ 114)

we can

shrink the

(i = 0, ... , I - 1). We

choose an open neighbourhood Ul of 1.1 with Ul c L-I(l - 114. 1 + 114). Since 1.1 is a Morse set and the linear extension p is hyperbolic over 1.1• it follows from Lemma 3.11 that X"'I W"(A1) (cr = s. u) are vector subbundles. Hence there exist subbundles

E'

of XI Ul

es and

such that (3.14)

(3.15) provided Ul is sufficiently small. Select a number d, 0

<

By monotonicity of L, every trajectory in one point. It is not hard to show that

intersects the set D in exactly

WeAl) \ 1.1

d

Therefore jj c L-I(l - d) n ( U [ W(A J): j :s l]). Since 1 - d

< 114, satisfying

>

1 - 1, it follows from

the properties of L that jj c W(A l ) , whence jj = D. Thus, D is a compact set. From )3.16) it is easily seen that

(3.17)

In

fact, if b e WeAl) n WU(A,).

then

i:s I according to (3.12).

Taking into

220

consideration that D c: W'(A,)\A, and {An ..... Ao} is a Morse collection. (3.17). Let •• denote the Lyapunov function constructed according to Lemma 1.6 flow (B. IR. p.).

where

{Ao.

1.\ .....

L(b)

= i ••·(b) = i}

p·(b. t)

= pCb.

-t)

(b

B. t

E

E

IR).

we get for

the

and the collection

An}. The function •• is decreasing along the trajectories of p. and. consequently. increasing with respect to the given flow p. Note that A, = {b e B: (i

= {b

W,

where the number

£

E

= O.

1..... n). Put

B:

i - £ < L(b).

(3.18)

< £ < d and W, c U, (0 sis I). U [ p(W" IR+): 0 s i < I). Given U c: B and q ill: O. let Since D is a compact set. there exists an integer q > 0

is assumed to be so small that 0

It follows from (3.17) that D c: us denote r}q) satisfying

< i + £}.

.·(b)

= p(U.

[0. q]).

(3.19) Next we shall prove that D n p(W" IR+) = III. Suppose. to the contrary. that there are a point b E W, and a number t ill: O. for which pCb. t) E D. Then I - d = L(p(b. t» ill: L(b) ill: I - £. according to the properties of L and (3.18). but this contradicts the condition

W1q) =

£

< d.

Thus.

D n p(W" IR+)

= III.

consequently.

Given a point bED. let j

= j(b)

denote the smallest integet j.

I. such that b E W)q) and b til ~q) (j the point bED c U, so that

< is

I). Choose a neighbourhood Vb of

D n

III.

os j <

(3.20)

(recall that

WJ

c:

UJ and. consequently. W)q)

v.b

- + n p(W,. IR )

= III

(j

c:

U)q». Now let us show that

< i

(3.21)

s I).

Suppose. contrariwise. that there exist a number i. j < i s I. and a point Y satisfying the condition Y number s > q. hence

E

p(W,. IR+) \ ~q) (see (3.20». Then y

p-ll(y) E

W,. Since L(p°s(y»

E

E

E

Vb c U>q)

p(Wto s) for SOme

L(W,) c: L(U,) c [i - 114. i

+ 1/4]

221

and the function L is nondecreasing along the trajectories, we get J:

i - 1/4 > j

+ 114

L(Pt(y» ~ L(p's(y»

(-s s t sO), consequently, pt(y). UJ for t

E [-

q, 0], i.e.,

y • ujq)j a contradiction. Thus, (3.21) holds. Our next step is to construct the subbundle

Et.

Let b

E

B and j

= j(b)

be defmed

as above. Since j < I, the induction hypothesis (d) holds. From D C WS"(I\,) and (3.14), it follows that there exists a vector subbundle G of XI Vb satisfying the conditions (3.22) Let p: XI Vb -+ E'I Vb denote the orthogonal projection (relative to the Riemannian metric adapted to 1\). Denote

db)

(b)

Gy

= peG).

.,.u C l:.Jy

Then

(y

E

V(b».

(3.23)

Taking into consideration (3.22) and assuming the neighbourhood Vb to be small enough, we get (3.24) Let UD be a closed neighbourhood of D in L·1(1 - d) satisfying the condition UD C U, n (U [ Vb: bED]). Find a continuous partition of unity subordinate to the cover {Vb: bED}. Since UD is a compact set, we may suppose that all but a finite number of the functions fJb are identically O. Because UD C Ult (3.15) and (3.24) imply that (3.25) As it is seen from (3.25),

db)

can be represented as the graph of a vector bundle

morphism gb covering the identity map:

Define gD: E"I UD ... ~I UD ,

gD

=

L fJb8 bED

b,

222 and set

if =

graph (gD), then

if is a vector subbundle of

XI UD. It follows from (3.22)

and (3.14) that (3.26) Now let us prove the inclusion (3.27) Since the fiber ~ of the vector bundle (b Y

E E

D),


is a convex combination of the fibers G~ b)

G~b) c

it suffices to show that

Vb n p(W{o IR+), then j

if

Et J .

Let

bED and

j

= j(b).

If

1 by (3.21). Taking into account (3.23) and the

induction hypothesis, we get G;'b) c

EJy c Et y.

Now set (3.28)

and extend

Itt

over p(UD , IR) by virtue of (X, IR, 'Ir). Finally, put (3.29)

According to Theorem 1.4, 'Ir(UD , IR) Wz is sufficiently small so that

U

W'(A z) is a neighbourhood of Az in B. Assume that

(3.30)

E't I Wz is a vector subbundle. show that E't I Wz is the required subbundle.

By Theorem 3.12, Now let us prove that for

To be more precise, we shall O:s i, j :s 1 the assertions (a), (b), (c) and (g) hold with Ut replaced

by Wt . Since E't = ~ over W'(A1), and the subbundle (X, IR, 'Ir), we have

Next let us show that (c) holds for

O:s i, j :s I. If

Itt I p(UD, IR)

is invariant under

O:s i, j < 1, this condition

223

the induction hypothesis. So it remains to verify (c) for j = 1 and

takes place by

°

< I. Because the subbundles Et and

:s i

Et

are invariant, and taking into account

(3.27) and (3.28), we get (3.31) Let Y

p(W" IR+) n p(W,. IR). Observe first that

E

(3.32) For if x L(y)

>

E

then L(x)

W"

1- 114

>i+

< i + 114 according to (3.18), and if

Y

E

114. Thus, (3.32) holds. Since peW"~ IR) c: p(UD , IR)

U

W'(A,) , then W'(A,) and

pew"~ IR+) n W'(A,)

peW"~ IR+) n p(UD , IR), hence it follows

from (3.31) that

(c) is fulfilled.

Next we shall

= 121 by (3.32), we get Y E /tty c: Et y. Thus, the condition prove that /tt satisfies (g), i.e.,

(3.33) Let bE peW"~ IR) c: p(UD, IR)

U

/ttb = ~

W'(A,). If be W'(A,), then

(3.33) follows from the transversality condition

Xb = ~

+

by (3.29), hence,

~. According to Theorem

3.13, the vectorial set E' is lower semi-continuous. By making, if necessary, the neighbourhood UD of D smaller, we therefore may assume, in view of (3.26) and (3.28), that

Because ~ and Et are n:-invariant subbundles, we get Hence, the condition (3.33) is really fulfilled.

Xb = Etb + ~

(b e p(UD , IR».

/ Thus, we have find neighbourhoods U, of At and subbundles ~ of the bundles Xlp(U" IR) such that conditions (a), (b) and (c) hold for i, j = 0, 1, ... , n and 0'

= s,

u.

By

shrinking,

XI U,

= (erl U,) •

if

necessary,

(Etl U,)

(i

the

= 0,

neighbourhoods

1, ... , n).

Indeed,

UtI

(a)

we

can show that

implies

ErIA,

= xr

(0' = S, u; i = 0, 1, ... , n). Recall that (X-IA) • (.xuIA) = XIA. These considerations together with property (b) show that (d) is true. Similarly, employing (3.13), we get,

224

by continuity. that (e) holds for some ;\. E Finally. verify the property (t).

i. J E {OJ I ..... n} such that b Assume. b

for

E

definiteness.

I). Let b

(;\'0'

E

p(UJc • IR).

There exist numbers

W(AJ ) n WU(A,). hence. b E p(UJ • IR) n p(U" R+). that b E p(UJc • s). where s ~ O. Then

p(UJc • IR+) n p(UJ • IR). and from (c) and (a) we deduce that ~b c ~b

E

> s and

number to hence. y ~y c

E

= p(b. - to) E p(UJc • s - to>.

denote y

p(U,. IR+) n p(UJc • R).

Et y = X:.

Since

Then y

E

= X:. Fix some

p(UJc • R) n WU(A,).

By the properties (c) and (a). we therefore get

Jt' is invariant. we deduce from

(b) that ~b ere:.

I 4. Quadratic LyapuDov Functions To describe the structure of some classes of linear extensions. quadratic Lyapunov functions are often of use. Let B be a compact metric space. (X. P. B) be a finite dimensional real vector bundle. p: (X. T. n) -+ (B. T. p) be a linear extension. As usual. T denotes the group IR or the group Z. Let 11'11 be some Riemannian norm defined on (X. p. B). A function t: X -+ IR is said to be quadratic if it is continuous and the restriction tlXb is a quadratic form for all b E B. A quadratic function t is said to be nondegenerate (positive definite and so on) whenever the quadratic form tlXb have the respective property for all b E B.

4.1. Lemma.

1/ (X. T. n)

has no non-trivial bounded motions. then there is an integer

N such that (4.1) ~

x

E

It suffices to establish X. IIxll '" O. and for all or

the existence of a number N > 0 such that for all ~ N at least one of the following inequalities holds:

Suppose. to the contrary. that for each number n. one can find a point xn IIxn II = 1. and a number mn > n so that

E

X.

225 (4.2)

= max {I"r(x", S)II: - mn :s S :s mn} 2:

Denote An

1. Let us p~ve sup {An}

="'.

Indeed,

there exists a subsequence of {xn} converging to some point x· , IIX·II == 1. Whenever sup {An} < "', the motion of the point x· is bounded, contradicting our assumption. Hence sup {An} = "'. Select a number In E [- mn, mn] SO that An = IIIf(xn, kn)lI. Without loss of generality, we may suppose that the sequence {Yn}, Yn some point {-mn - kn }

i,

IIi II

= 1.

kn), converges to

It is easily seen from (4.2) that {mn - kn } ....

therefore the motion of

.... - "',

= A~IIf(xn'

i

+ "',

is bounded; a contradiction. --

4.2. Corollary. 71ze quadraric form Q defined by

is positive definite.

4.3. Proposition. q (X, IR, If) has no non-trivial bouncled motions, then there auts a quadraric function I: X .... IR such thlll the derivative dl(If(x, s» I dsls=o aists and represents a positive definite quadraric function .

• Set N

l(x)

=I

{1I1f(X,

1

+

N)u l

-

UIf(x, l)U I } dI,

o where N is taken from Lemma 4.1. Then

1(1f(x,

s»

s

= l(x) +

J{UIf(x, 2N + T)U

I -

2UIf(x, N

+

T)U I

+

UIf(X, T)U I } dT,

o

hence, dl(If(x,

s» I dsls=o =

UIf(x, 2N)u l

-

2UIf(X, N)u l

+

uxu l

iii

Q(x).

4.4. Proposition. If (X, z, If) has no non-trivial bounded motions, then there aisls Q

quadratic function IP: X .... R such thlll IP(IfI(x» - IP(x)

= Q(x)

(x

E

X).

226

• Put N-\

L {lIn(X, k + N)1I2 - IIn(X, k)1I2},

'II(X) =

Jc=o

where N is the same as in Lemma 4.1. 4.5. Lemma. Assume that there exist a number T > 0 and a quadratic function ~: X -+ IR such that ~(n(x, ~(x) > 0 (x e X, IIxll '" 0). Then the following equalities are true:

T» -

r = {x e Jt'

x:

~(n(x,

t»

::s 0

(t e 7)},

= {x E x:

~(n(x,

t»

~

(t e 7)} .

0

• Let us check, for example, the first relation. Given x e (t e 7). Suppose, to the contrary, that 3;\ e T: ~(n(x,

nT +

;\»

>

~(n(x,

;\» >

0

r,

prove ~(n(x, t)) ::s 0 IIxll '" 0 and

> O. Then

~(n(x,;\»

(n = 1, 2, ... ).

(4.3)

Because x e r, we have IIn(X, nT + ;\)11 -+ 0 as n -+ CII. Since ~ is continuous and B is compact, we get ~(n(x, m: + ;\» -+ 0, contradicting (4.3). Conversely, let the element x e X be chosen so that ~(n(x, t)) ::s 0 for all t e T.

Prove x e

r.

Suppose the

contrary holds, i.e., 3 tJc e T, tJc > k, 3 p > 0: IIn(x, tJc)1I ~ p (k = 1, 2, ... ). It is no loss of generality in assuming that each number tJc is a multiple of T. According to our hypothesis, the quadratic form P, defined by P(x) = ~(n(x, ~(x) (x e X),

T» -

is positive definite. Since B is compact, we get P(x) ~ and all x e X. Then

C IIXll 2

n-I

n-\

k=O

Jc=o

for some number c > 0

L P(n(x, h» ~ c L IIn(x, h)1I2

for all positive integers n. Because IIn(x, tk)1I ~ p for all k = 1, 2, ... , we conclude that the right-hand side of the above inequality tends to infinity as n -+ DO. Hence ~(n(x, m» -+ + CII, contradicting the condition ~(n(x, t» ::s 0 (t e 1).

227

4.6. Corollary. Denote Z valid:

r

IZ

Ii

Z(B)

=

= {x

= O}.

e X: IIxll

{x e X: I(n(x,

t»

< 0

The following relations are

(t e 7)},

(4.4)

It' I Z =

{x e X: l(n(X, t»

> 0 (t

e 7)}.

4.7. Lemma. Suppose the conditions of Lemma 4.5 are fulfilled. IIn(x, t)1I -+ 0 or IIn(X, t)1I -+ III as It I -+ ± III.

Then either

• Suppose IIn(x, t)1I does not tend to 0 as t -+ + III. Repeating the arguments from the second part of the proof of Lemma 4.5, we get I(n(x, t» -+ III as t -+ + III. Because I is cOntinuous and B is compact, this is possible only when IIn(X, t)1I -+ III (I -+ +111).

4.8. Proposition. Suppose that there exist a number T > 0 and a quadratic function I: X -+ IR such Ihat I(n:(x, I(X) > 0 (x e X, IIxll ,;. 0). Then (X, T, n) has no non-trivial bounded motions.

T» -

• Suppose,

to

the contrary,

that there is a point x e X

so that

IIXIl,;. 0, but

sup {lIn(x, 1)11: t e 7} < III. By Lemma 4.7, x must belong to r n It'. According to we have Lemma 4.5, I(n(x, t» == 0, in particular, l(x) = 0. Since IIxll,;. I(n(x, I(X) > 0, hence, I(n(x, > 0, contradicting I(n:(x, t» == O.

T» -

°

T»

4.9. Theorem. The system (X, T, n) has no non-trivial bounded mOlions there exists a quadratic JUnction I: X -+ IR such that I(n(x, t» - I(X) > ex e X, IIxll ,;. 0, 1 > 0).

iff

°

• This follows from Propositions 4.3, 4.4 and 4.8. Let p.: (X, T, n.) -+ (B, T, p) be the dual linear extension. 3.5, Theorem 4.9 implies the following statement.

4.10. Theorem. transversality condition

The linear

extension

p: (X, T, n)

iff there exists a quadratic function

I(n.(x, t)) - I(X)

> 0

I:

-+

X

According to Theorem

(B, T, p)

satisfies the

-+ IR such that

(x eX·, IIxll ,;. 0, t > 0).

228

ot,

Recall that p., B) can be identified with (X, p, B), once a Riemannian metric on (X, p, B) is given. It follows from Theorems 3.5 and 3.6 that a linear extension

p: (X, T, 1t) .. (B, T, p) is hyperbolic if and only if both (X, T, 1t) and

ot,

T, 1t.)

have no non-trivial bounded motions. Therefore we get

4.11. Theorem. A linear extension p: (X, T, 1t) .. (B, T, p) is hyperbolic Iff there exist quadratic jimctions 1\ and 12 such that

4.12. Theorem. A linear extension p: (X, T, 1t) .. (B, T, p) is hyperbolic Iff there exists a non-degenerate quadratic jUnction I: X .. IR such that 1(1t(X, t)) - I(X)

>

0

(x

• We shall consider only the case where T are vector subbundles and X

X-

= X- • X"'.

E

X, IIxll - 0, t

= IR.

>

0) .

Hyperbolicity means that

X-

and ~

Choose a Riemannian metric on (X, p, B) such

X"'. Define the function I in the same manner as it was done in Proposition 4.3. By Corollary 4.6, tlX- is negative definite and IIX"' is positive definite, hence that

.L

is a non-degenerate quadratic function. Conversely, let I be a non-degen~rate quadratic function satisfying the inequality 1(1t(X, t» - I(X) > 0 (x E X, IIxll - 0, t > 0). According to Proposition 4.8, the system (X, T, 1t) has no non-trivial bounded motions. Without loss of generality, assume that the space B is connected. It follows from Theorem 3.2 that

I

x: •

E B: Xe:! = Xb } is a closed invariant set of the system (B, T, p), and the extension p: (X, T, 1t) .. (B, T, p) is hyperbolic over A. Since I is a non-degenerate

A - {b

quadratic function, we conclude from Corollary 4.6 that the functions b b

1-+

dim

Xe:!

1-+

dim

x:

and

are constant on A, therefore Theorem 3.2 implies that the extension

p: (X, T, 1t) .. (B, T, p) is hyperbolic. Now consider the particular case when B is a smooth compact manifold, (B, IR, p) is a flow determined by a smooth vector field

Col,

X

= B x R",

and (X, IR, 1t) is generated by

229 the following vector field:

- = w(b) , -d~ = A(b)~ dt dt db

~ e IR"),

(b e B,

(4.5)

where A = A(b) is a linear operator depending continuously on b e B. The dual linear extension is determined by the equations db dt

- = w(b)

'

-d~ = - A°(b)~ dt

(b e B, ~ e IR"),

where the asterisk denotes the transpose of a matrix. From Theorems 4.11 and 4.12, we get the following results.

4.13. Theorem. The linear extension p: (X, IR, n:) -+ (B, IR, p) determined by the equations (4.5) is hyperbolic iff there exist quadratic functions t.(b,~);;; (S.(b)~, ~) and t2(b,~) = (S2(b)~,~) (b e B, ~ e IR"), where S.(b) and S2(b) are symmetric matrices depending smoothly on b e B such that for each point b e B the matrices

are positive definite. ~

In fact,

dt 2 (n:!(b, dt

~» I t-O

= (D (b)~, ~). 2

4.14. Theorem. The linear extension determined by (4.5) is hyperbolic iff there exists a symmetric matrix S(b) depending smoothly on b e B, non-degenerate for all b e B and such that DS(b)w(b) is positive definite (b

E

B).

+

S(b)A(b)

+ AO(b)S(b)

230 § 5. Weak Regularity and Green-Samoilenko Functions

In this section, we investigate several types of weak regularity of a linear extension. We establish relationships between weak regularity, transversality, and the existence of Green-Samoilenko functions.

5.1. Definitions. Let B be a compact metric space, T =

IR

or T = I, (X, p, B) be

a real n-dimensional vector bundle, and p: (X, T, 'If) -+ (B, T, p) be a linear extension. Further, let f B x T -+ X be a continuous map satisfying the following conditions: (b

E

B, t

1),

E

(5.1)

Define

/-I:

X

= j(/(b),

fib, t

+

x T -+

X by

/-I

t

-r)

(x)

=

t

'If (x)

-r)

+

+ ftp(x),

'1fT

(f{b , t»

(x

t)

E

(b

X,

E

B;

t, -r

E

1).

t e 1).

It is easy to verify that (X, T, /-I) is a dynamical system and

is a homomorphism. The extension p: (X, T, J.I) -+ (B, T, p)

(5.2)

p: (X, T,

/-I) -+ (B, T,

p)

is said to be affine (or

non-homogeneous linear). Consider first the case T = IR. Given a continuous section (X, p, B), the function

f

11:

B -+ X of the bundle

B x IR -+ X defined by t

j(b, t) = 'lft(

J

('If"

t;

0

11

0

pt;)(b) d~),

(5.3)

o satisfies the

conditions

(5.1). The corresponding affine extension will be denoted as p: (X, IR, 'lf1'l) -+ (B, IR, p) and called the extension associated with the linear extension p: (X, IR, 'If) -+ (B, IR, p) by means of the section 11: B -+ X. S.2. Example. Let B be a compact space, (B, IR, p) be a flow, X = B x IRn , and P: B -+ End IRn

be a continuous function. Let ~(b) denote the Cauchy operator of the

231

equation

y=

Define Tl(b, y) = (/(b) , ~(b)y),

(y E IRn , t E IR, b E B).

P(/(b»y

p(b, y) = b. Then p: (X, IR, ll) -+ (B, IR, p) is a linear extension. Let r: B -+ IR n

y=

be a

+

r(pt(b»

Set ,/(b, y) = (/(b) , rfo(b, y», p: (X, IR, 11) -+ (B, IR, p) is an affine extension of the above type (5.2), (5.3).

then

continuous function and ifo(b, y) denote the solution of with

initial

condition

P(pt(b»y

ifo(b, y) = y.

Now let T = I. Given a continuous section 11: B -+ X, define f.,,: B x I -+ X as follows: n-I

L

II

Z

o

11

0

pn-z(b)

for n

~

for n

= 0,

for n

s

1,

z=o f.,,(b, n)

=

0 n

L

-

II

Z

o

11

0

pn-z(b)

- 1.

Z=-1

It is easy to check that

f

=

=

II

satisfies the conditions (5.1). The equality (5.2), in

f."

this case, becomes 1

11 (x)

1

(x)

+

1

lI(p (P(x»)

=ll.,,(x 1

)

(x

E

X).

5.3. Definitions. The linear extension p: (X, IR, ll) -+ (B, IR, p) is said to be weakly regular at the point bo E B if for every uniformly continuous bounded map s: IR -+ X Xo E

satisfying the condition p(s(t» X such that p(xQ) = bo and

=

/(bQ)

(t

E

IR)

there

exists

a point

t

sup 111lt(XQ) tEIR

+

llt(

J

ll- 1;0

s(~) d~)11 <

IXI.

(5.4)

o

A weakly regular linear extension is called regular if the point Xo E X is uniquely determined. The notions of weak regularity and regUlarity of a linear extension

232

p: (X, I, 'It)

are defined similarly.

-+ (B, I, p)

=

p(s(k»

bounded map satisfying replaced by

plt(bo) (k

sup 117l(xo>

It is assumed that

s: z

-+ X is

a

Z). The inequality (5.4) should be

E

+ fs(k) II <

III,

ItEZ

where It-I

E n (s(k - l)) l

for k

~

1,

l=O

=

fs(k)

for k = 0,

0 It

- E nl(s(k - l))

for k

~

- 1.

l--I

A linear extension is said to be weakly regular (respectively, regular) at every point b E B.

(regular)

if it is weakly regular

5.4. Lemma. The notions of weak regularity and regularity do not alter their meaning when we consider only junctions s: T -+ X satisfying the additional property lim IIs(t)1I = O. I t I -+III

• The proof will be carried out only for T = IR (the case T = I can be handled in a similar way). Let EI denote the Banach space of all continuous maps r: IR -+ X such that p(r(t» IIrll

=

= pt(bo)

sup {1Ir(t)lI: t

(t E

E

IR}.

IR)

and lIr(t)1I -+ 0 as

It I -+ III,

endowed with the norm

Let E'}. denote the set of all pairs (x, r)

E

p-l(bO> x E\

such that the function t

!/Ix .r : IR -+ X,

!/Ix.r(t)

= nt(x) +

nt(

Jn- (. r(~) d~),

(5.5)

o

is bounded lI(x, r)1I

(i.e.,

= IIXIl +

sup { lI!/Ix,r(t)lI: t IIrll + sup {II!/Ix.r(t)lI: t

there exist elements Xo

E

p-I(bo), ro

E

E

E

Provide Ez with the norm IR}. Given a Cauchy sequence {(x", rn)} c E, IR }

<

III).

E\ such that {xn} -+ xo, {rn} -+ ro and {!/Ixn.rn}

233 is a Cauchy sequence in the space

{!/Ix-n.rn}

c:fl(IR, X), therefore

converges (uniformly

on IR) to a certain continuous bounded on IR function "': IR -+ X. But it is easily seen uniformly on each segment. Hence, from (5.5) that {!/Ix r} converges to !/Ix r

o·

n' n

!/Ixo.ro = IJ, i.e., the function

0

!/Ixo.ro is bounded on IR, whence (xo, ro)

E

E,.. Thus, E,.

is a Banach space. Derme a continuous linear operator A: E,. -+ EI by A(x, r) = r. The modified notion of weak regularity of the extension p at the point bo means that A is surjective. By the Banach Open Map Theorem, we can find a number k > 0 such that, for every element rEEl'

there

exists a point x

II (x, r)1I s k IIrli.

p-I(bo)

E

IIxll s k IIrll

Whence,

satisfying the conditions

(x, r)

E

E,. and

and

t

sup IInt(x)

+ nt(

tEIR

s:

Let p(s(t»

so

IR -+ X

= pt(bo)

that

rn(t)

be (t

E

J n- ~o

some

uniformly

continuous

IR). For every positive integer

= set)

(It I s n),

IIrn(t)II

p(rn(t» = pt(bo) (t E IR). Then rn E EI according to the modified notion of "Xn" s k"rn" s kllsll

r(~) d~)11

s k IIrli.

0

(n

n construct a function

=0 = 1, 2,

weak

bounded function such that (It I

2:

n

+

1),

rn: IR -+ X "rn" s lis II ,

... ). For rn select a point xn regularity of p at boo Then

and t

sup IInt(Xn)

+

nt(

Jn- ~o rn(~) d~)11 :s k"rn" :s kllsll.

tEIR

0

Consequently, t

IInt(Xn)

+

nt(

J

n-

(0

s(~) d~)11 s kllsll

(It I :s n).

(5.6)

o Suppose, Xo

E

with no loss of generality, that the sequence {xn}

p-I(bo)' Taking the limit as 11 -+

00,

we infer from (5.6) that

converges to some point

234 t

+ n:t(

lIn:t(XO)

J

n:- ~. S(~) d~)11

::s

kllsll

(t e IR).

o Thus, the linear extension p is weakly regular at bo in the initial sense.

5.5. Lemma.

If the linear extension p: (X,

T, n:) -+ (B, T, p)

the point bo e B, then the dual linear extension p.: (X·, IR, n:.) non-trivial bounded orbits intersecting the fiber over the point boo

is weakly regular

at

has no

-+ (B, IR, p)

~ Let T = IR (the case T = Z is handled in a similar way). Identify (X·, p., B)

<., . >.

with (X, p, B) via some Riemannian metric

Suppose that the linear extension

p: (X, IR, n:) -+ (B, IR, p) is weakly regular at some point bo e B, but there exists a point Yo e p-l(bo) such that

IIYoll '" 0 and sup {1In:!(Yo)lI: t e IR} <

v(t)

= n:!(yo)

IXI.

Define

(t e IR).

(5.7)

Clearly, v: IR -+ X is bounded and uniformly continuous. Observe that p(v(t)) (t e IR).

According to the

above

hypothesis,

= /(bO>

there exists a point Xo e p-l(bo)

such

that t

sup lIn:t(Xo)

+

n:t(

telR

J

n:" ~. n:~(yo) d~)11 <

IXI.

o

Now set t

s(t) = n:\~o) + n:t(

J

n:- ~. n:;(yo) d~)

(5.8)

(t e IR).

o

Clearly, s: IR -+ X is a bounded uniformly continuous mapping and From (5.8) we get

p(s(t» = /(bO>

(t e IR).

t

<s(t), v(t»

=


+

J
n:!(yo»

dt.

(5.9)

235

Recall the equality

< n:t(x) , n:!(y) >

<x, Y>

=

(x, Y

E

X; p(X) = p(y), t

E

IR).

(5.10)

It follows from (5.9) and (5.10) that t

< s(t) ,

vet»~

=

<xo, Yo> +

J1Iv(t;)1I

2

dt;.

(5.11)

o Indeed, denote v(t;) == n:~(yo)

Now let !pet)

= < s(t) ,

= Yi;

and write

vet) >. As it is seen from (5.ll),

+

!p(t

'1:) - !pet)

;!:

0

(t

E

IR, or z: 0).

In other words, the function !p: IR -+ IR is non-decreasing. Hence, there exist a = lim !p(t) , b = lim !pet). t-++GO

t-+-GO

= b = O. Suppose, for definiteness, that {tn } -+ +GO and I/In(t) = !pet + tn). The sequence {I/I n } converges to the constant

Next we shall prove that a lim n:.(yo, tn)

= Zo.

Set

It follows from (5.11) that ,,' (t) = IIv(t)112

function I/I(t) i i a uniformly on segments. (t E ~). Therefore

uniformly

on segments.

Because

{I/In } -+ 1/1 II a,

we have

lim I/I~(t) E O.

Hence,

t

1In:.(.zo)1I == 0, whence IIZoIl = O. Since s is a bounded function, a == 1im !p(tn ) n~..

Similar arguments show that b

= O.

=

1im < s(tn ) , v(tn ) >

n~

..

= O.

Taking into account that "

is non-decreasing, we

236

get

rp(t)

condition

!II

1Iv(~)1I

O. Hence, (5.11) implies

"Yo"

= 0,

i.e.,

IIn!(yo)II

ii

0, contradicting the

O.

¢

p: (X, T, n) .. (B, T, p)

5.6. Defmitions. We might say that the linear extension is uniformly weakly regular if for each continuous section continuous section s: B .. X of (X, p, B) such that t ·t f[vosop

=s

(t

E

B .. X

IT:

there exists a

7),

(5.12)

i.e., the image s(B) is invariant under (X, T, nv)'

= IR,

Whenever T

the relation (5.12) can be written as t

f[t(s(B»

+

f[t(

J

(n- (

0

IT

0

p()(b) d~)

= s(/(b»

(b

E

B, I

E

IR).

o For T

= z,

we get (b

E

B).

Surprisingly enough, uniform weak regularity, as defined above, does not imply weak regularity. In fact, consider the simplest linear extension with the one-point base (in other words, let X = IRn and f[t(x) = exp (At)·x). This trivial extension is uniformly weakly regular iff det A ¢ O. It is weakly regular iff A has no pure imaginary eigenvalues. It can be shown that if the system (8, IR, p) does not contain rest points and periodic orbits, then these two definitions are, in fact, equivalent. In the general case, it is reasonable to strengthen the notion of uniform weak regularity in the following way. We shall say that p: (X, T, f[) .. (8, T, p) is uniformly weakly regular, if the following condition is fulfilled. Suppose that W is an arbitrary compact space, (W, T,~) is any dynamical system and h: (W, T, ~) .. (B, T, p) is a homomorphism. We demand the linear extension q: (N, T, v) .. (W, T, ;\), the pullback of p by the homomorphism h, to be uniformly weakly regular in the previous sense. Recall that

X, W E W, p(x) = h(w)} , vt(x, w) = (nt(x) , ~t(w», q(x, w) = w. Let us show that uniform weak regularity (in the adopted strong sense) implies weak regularity. Let s: T .. X be a bounded uniformly continuous map such that

N = {(x, w): x

p(s(t»

= pt(bo)

E

(t

E

7), where bo is some point of B.

Let W denote the closure of the

237

family {s-r!': T E 1} of shifts S-r!'(I) = S(I + T) (I E 1) with respect to the compactopen topology (i.e., the topology of uniform convergence on compact sets). Clearly, W is a compact space. Define the shift dynamical system (W, T,:\) by :\t(!p)(E;) = !pet + E;) (!p E W, t E T, E; E 1). Evidently, the map h: W ~ B, helP) = p(!p(O» (!p E W), is a homomorphism. Let q: (N, T, v) ~ (W, T,:\) denote the pullback of p: (X, T, x) ~ (B, T, p) by h: (W, T, :\) ~ (B, T, p). The map 5: W ~ X, 5(!p) = !p(O) , is continuous. Therefore the map 0': W ~ N, O'(rp) = (5(rp) , !p), is a continuous section of the vector bundle (N, q, W), and !T(ST) = (p\b o), ST) (T E 1). By hypothesis, there exists a continuous section

/: W ~ N which is invariant under (N, T, vcr). Since the

image of / under the projection pr\: N ~ X is bounded, we get (in the case that (5.4) holds with Xo = pr\ 0 O'(s). The notion of uniform regularity (in the strong sense) can be introduced in way. We leave this to the reader. Now let us present the definition of a Green-Samoilenko function of extension p: (X, T, x) ~ (B, T, p). Let c: X ~ X denote a continuous

T = IR)

a similar a linear function

satisfying the conditions p 0 C = p, xt. C = C • xt (t E 1) and CIXb E L(Xb' X b ). Note that we do not suppose, in general, the function b..." dim Ker (CIXb ) (b E B) to be locally constant. In other words, C: X ~ X is not necessarily a vector bundle morphism. Put C-r!' = C for

T

~

0 and CT = C - I for

map). Further, set CT = xT 0 CT c > 0 and v > 0 such that

== CT

0

1[T

(T

E

(b

T

< 0 (here I is the identity

1). Assume that there exist numbers

E

B,

T E

(5.13)

1).

Then the mapping (b

E

B,

T E

1)

is called the Green-Samoilenko function. It is easy to show that if a given linear extension has a Green-Samoilenko function, then the same is true for any pullback. In fact, given C: X ~ X, we define C: N ~ N by C(x, w) = (C(x) , w). Observe that the existence of a Green-Samoilenko function implies uniform weak regUlarity. Indeed, if T = IR and l}: B ~ X is a continuous section then the section 0: B ~ X, where

238

+ c5(b)

=

00

JG (11 T

(p-T (b») d-r

(b e B),

- 00

is also continuous and invariant under (X, IR, nl)' In the case section can be written as

c5(b)

=

L

if.

lJ •

p-n(b)

T

=

I, the required

(b e B).

Let p: (X, T, n) ~ (B, T, p) be a linear extension and rO(X) be the Banach space of all continuous sections ~: B ~ X of (X, p, B) endowed with the norm

II~II = sup {1I~(b)lI: b e B}. Let (rO(X) , T, nil) denote the transformation group defined by n~(~)

= nt

• ~ • p-t

~ e rO(X». As usual, denote

(t e T,

5.7. Theorem. Let B be a compact metric space, (X, p, B) be a finite dimensional vector bundle, and p: (X, T, n) ~ (B, T, p) be a linear extension. The following statements are pairwise equivalent: (1) p is weakly regular; (2) p is uniformly weakly regular; (3) p satisfies the transversality condition; (4) rO(X) = rOs(X) + rOu(X); (S) the dual linear extension

p": (X·, T,

n.) ~ (B, T, p)

has no non-trivial bounded motions; (6) p admits a Green-Samoilenko jimction; (7) there exists a quadratic junction ~: )( ~ IR such that ~(n!(x» - ~(x)

> 0

0, t > 0) . • We confine ourselves by considering the case T = IR. Theorem 3.S shows that the statements (3) and (S) are mutually equivalent. The implication (1) .. (S) holds by Lemma S.S. It was shown above that (2) implies (1). Clearly, (4) implies (3). According to Theorem 4.10, (3) _ (7). As it was already mentioned, (6) implies (2). So it remains to show that (3) .. (4) and (3) ~ (6). Thus, assume that the linear extension p: (X, IR, n) ~ (B, IR, p) satisfies Ute transversality condition. We shall use Theorem 3.14. Let the neighbourhoods Ut of the

(x

e

X·,

IIxll;l:

239

Et

of Xlp(U" IR) (i = 0, 1, ... , n) satisfy the The sets {p(U" IR)} cover the base B. Let {a,}

sets 1\, and the subbundles ~ and conditions (a) - (f) of this theorem.

be a partition of unity subordinate to this cover. Given 11 E rO(X) , denote By using the property (d), represent the sections ll, in the form ll, = ll,. + ll'cr(b)

E

~b

(b

E

p(U" IR); i = 0, 1, ... , n;

liS

"

L ll,. ,

=

llu

tJ'

= s, u).

=

ll, = a,ll. ll,u,

where

Set

" L ll,u .

It follows immediately from the definition that liS and llu are continuous. Let us prove

that llcr E rOcr(X) (tJ' = s, u). The proof will be carried out only for tJ' = u. It is based on the methods due to Robbin [1] and Robinson [2]. Define neighbourhoods W, of 1\, by formula (3.18). If e > is sufficiently small, then W, c U, (i = 0, 1, ... , n). Because W, is defmed by the aid of two Lyapunov functions, the trajectory of each point b E B can enter W, (and leave Wi) only once. Recall that the ex- and ",-limit sets of all points b E B under (B, IR, p) are contained in the set 1\ = Ao V 1\1 v ... V 1\". The same is true for the cascade generated by the

°

powers of the diffeomorphism

f =

Therefore {p(W, , Z): i

pl.

= 0,

1, ... , n}

is an

open cover of the compact metric space B. Let {aa denote a partition of unity subordinate to this cover. Pick numbers ~ and e, < e < 1, A < ~ < 1, satisfying

°

(5.14)

Choose a finite atlas .4

= {(U,

each chart (U, ex, IR") and each number j ( U" WJ

E

E

{O, 1, ... , n} the following relations hold:

;t f2I ) .. (

(1 - e) IIvll :s IIxll :s (1

Where the element v

for the vector bundle (X, p, B) such that for

ex, IR")}

+

e)

(5.15)

V c: UJ ),

(5.16)

IIVIl

IR" is determined from the equality ex(x)

According to the definition of {a~}, we have

= (P(x) ,

v)

(x

E

p.I(U).

240 supp (e)

!!!

{b

e~(b)

B:

E

¢

O} c p(Wj

•

U = O.

Z)

1..... n).

r such that

Because the set supp (9) is compact. there is a number r

UI(W

supp (9) C

U = O.

j )

(5.17)

1..... n).

For b E B and 'II e rO(X) define 11'11. bll = max 1I'II«(b)lI. where 0:('11. b) and the maximum is taken over all charts (U. 0:) e" with b e U. Put

= sup

11'1111 0

It is easy to verify that

'II«(b»

{ 11'11. bll: b e B}.

rO(X)

is a norm on

II • 11°

= (b.

equivalent to the initial norm

11.11 0 ,

Fix some number i E {O. 1..... n}. For 'II e rO(X) define 'Ilk

Then for all k

i!:

=

'It

ok

0

'IItu

0

pk

E

ok('IItu)

(k

'ltl/

E

z).

O. we have (5.18)

Indeed. if b e supp ('Ilk). then pk(b) e supp ('IItu). hence. pk(b) e U [pl(Wt ): -r sis r] according to (5.17). whence b for all k

i!:

E

U [plok(Wt ): - r sis r] c U [ps(Wt ): -

00

< s :s

r]

Q

E

O.

Let Vo = sup 1I'lt0IIXbli.

v = max { vo(l

+

e)(1 - ert, 1 }.

bE'S

Define r: B ~ IR by the following conditions: reb) b e p(Wt • Z) n (Wo

U

WI

U ... U

W,,);

5.8. Lemma. Let the atlas " 'II e rO(X).

r(b)

=v

=0

for b e Q; reb)

=,.,.

for

otherwise.

and the number

15

satisfy the above conditions

0

If

then (b

E

B. k

= O.

1•. 0.).

(5.19)

241

~ Let 'II e rO(X) and b e B. Find two charts (U, cr.) and (V, (3) from b e U, j(b) e V. Then

.4

so that

Recalling (S .16) and the definition of v, we get

Thus, the inequality (S.19) holds at least when r(f{b»

= v.

= Il, i.e., j(b) e p(W" Z) n WJ for some number j e to, ... , n}. '11M 1 = n;l~. Since j(b) e p(Wt' Z) n Wj C P(Ut, IR) n p(UJ , IR+),

Assume now r(f{b»

Let ~ = 'IIx' then then the property (c) yields

Et.f(b) C EJ.f(b)'

But

~(f{b» = 'IIx(f{b» = (n- x • 'IItu because 'IItu(z) e (e), we get

Etz

0

/)(f{b» e EJ.f(b)

for all z e P(Wh IR). Therefore, by (5.14), (S.16) and condition

(recall that j(b) e j(U) n V n WJ , hence V c UJ , according to (S.lS». Thus, (S.19) is true in the second case, too. Suppose now that r(f{b» = 0, i.e., j(b) fl Q. Then 'IIx(f{b» = 0 by (S.18), therefore, 'IIx+l(f)

= n-1'llk(f{b» = O.

5.9. Lemma. Denote q rcf(b»

~ Let

=v

Thus, (S.19) holds in this case, as well.

= 2n + 3r.

Given an arbitrary point b e B, the equality

is valid for no more than q values of k

b e B.

Since

{a;}

2:

O.

is a partition of unity, it follows from (5.17) that the

242 family of sets {f(WJ ): for each Wo u ...

number U

- r

t e z,

k

:5

r; j

:5

the

= 0,

collection

1, ... , n} covers the space B. Therefore

{/'''(b) , ... ,I+"(b)}

meets

the

set

WJ no 2(n + l)r

Wn . The orbit of b e B under the flow (B, IR, p) enters (and leaves)

more than once. Consequently, f(b) e Wo u ... values of k 2: O. Four cases can occur. Case 1: b

I!

Q. Then f(b)

I!

U

Wn for all but no more than

Q, i.e., rl/(b»

Case 2: f(b) e p(W" Z) for all k

2:

=0 <

v for all k

2:

O.

O. Then

= J.I

< v ) for all but no more than 2(n + l)r values of k 2: O. Case 3: b e p(W" Z), but there exists a number I > 0 such that /(b) I! p(Wt' Z).

(i.e.,

rl/(b»

Let I be the smallest number with this property. Then

(i.e.,

rl/(b»

= J.I <

for

v )

k e {O, 1, ... , I - I}, and f(b) the equality rl/(b»

=v

Case 4: b e Q, but b k 2: r.

all I!

= 0)

Q (i.e., rl/(b»

holds for no more than 2(n I!

p(W" z). Then f(b)

I!

+

2(n

but no more than

for all k

+ l)r + r

E

values

l)r 2:

I + r. Hence,

q values k

Q (i.e., rl/(b»

of

= 0)

2:

O.

for all

Let us continue the proof of the theorem. Using (5.19) and proceeding inductively, we obtain (b e B, k

2:

0).

Lemma 5.9 enables us to conclude that (b e B, k

2:

0),

(5.20)

243

Given a number

1

e

IR +,

denote k

-t()

1[, TlLu

=

= 1[-t

0

[t], TlLu

IX

=

1 - k.

t

0

P

= 1[

We have

-
....

• Tlk • P ,

therefore (5.20) guarantees the existence of a constant c 1 > 0 such that (I

~

0).

(5.21)

n

Thus,

TlLu

e rou(Xlp(UL,

IR»,

therefore

Tl u -

[TliU

e

rOu(X).

Similarly,

TIs

e

rOs(X).

i=O

Since TI = TIs It remains

p: (X, IR,

1[) ~

+

Tl u ,

we see that (4) holds. to prove that (3) implies (6). Let the linear extension (B, IR, p) satisfy the transversality condition. We shall use Theorem

3.14 once again. By the property (d), Xlp(UiI IR) = Et Q) Itt. Let Pi: Xlp(U" IR) ~ Et denote the projection corresponding to this direct sum decomposition. Let {Bi} be a partition of unity subordinate to the cover {p(UiI IR)} of the base B. Let BLPt : X ~ X denote the map defined by BiP,IXb = Bt(b)(Pi IXb) for b e p(UiI IR), and by n

aLPt IXb

=0

otherwise. Define a continuous map C: X ~ X, C

= [

aLP•. Show that C

t-O

satisfies all the needed conditions. Clearly, p 0 C = p, 1[t 0 C = C 0 1[t (I e IR) and CIXb is a linear map for all b e B. Check that (5.13) holds. In fact, by using Lemma 5.9, it is not hard to show (in the same way as (5.21) was established) that there is a number CI > 0 satisfying

for all result.

iff

T

~

0,

b e p(Ut , IR),

i

=

0, 1, ... , n.

5.10. CoroUary. The linear extension p: (X, IR, p: (X, I, 1[) ~ (B, I, p) has the same property.

This immediately gives the required

1[) ~

(B, IR, p)

is weakly regular

244 ~

This follows from Theorem 5.7 and Remark 3.10.

5.11. Lemma. Let p: (X, T, n) -+ (B, T, p) be a hyperbolic linear extension, and Y c X be an invariant vector subbundle. If the quotient linear extension q: (X / Y, T, n) -+ (B, T, p) is hyperbolic, then pi Y is also hyperbolic. For definiteness, suppose T = IR. Since Y contains no non-trivial bounded motions, it suffices, by Theorems 3.6 and 5.7, to show that pi Y is weakly regular. Choose some ~

vector subbundle Z c: X so that X = Y
= [nt(y)

+

= y + z = (y, z),

nt(z) _ Ilt(Z»)

Let

bo e H,

the

condition

+

Ilt(Z) , therefore

s: IR -+ Y

and

where y e Y, z e Z. Then nt(x)

=

nt(y, z)

+

(nt(y)

= nt(y) + nt(z)

nt(z) - Ilt(Z) , Ilt(Z».

be a bounded uniformly continuous map

p(s(t» = /(bO>

(t e IR).

Define

a

s: IR -+ X

map

==

satisfying Ye Z

by

s(t) = (s(t) , 0),

where 0 denotes the origin of the fiber of Z over the point pt(bO>. According to Theorem 3.6, the linear extension p: (X, IR, n) -+ (B, IR, p) satisfies the transversality condition, hence p is weakly regular, by Theorem 5.7. Thus, there exists an element Xo e X such that p(xo) = bo and t

~~~

IInt(xo)

+

nt(

I

n- .;

0

s(~) d~)11 <

(5.22)

00.

o Write the element Xo e X p: (Z, IR, Il) -+ (H, IR, p) is n- .(

0

s(~)

= (n-

.(

0

as Xo = (Yo, Zo> e Y (f) Z. Since the extension hyperbolic, it follows that hence, Zo = 0,

s(~), 0), therefore (5.22) implies t

~~~ IInt(yo) + nt(

I

n- .(

0

s(~) d~)11 <

00

o Thus, the extension plY is weakly regular and, consequently, hyperbolic. 5.12. Lemma. If Y is an invariant vector subbundle of the linear extension p: (X, T, n) -+ (H, T, p), then the linear extension q: (X I Y, T, n) -+ (B, T, p) and

p.1

r

are mutually dual.

The same

is

true

for

the

extensions

plY

and

245

q.: pt I ~

r,

T, '11:.) -+ (B, T,

pl.

For a proof, see Bronstein [4, Lemma 8.41].

5.13. Lemma. Let p: (X, T, '11:) -+ (B, T, p) be a hyperbolic linear extension and Y c X be an invariant vector subbundle. If the restriction plY is hyperbolic, then q: (X I Y, T, '11:) -+ (B, T, p) is also hyperbolic. ~ According to Lemma 3.9, the dual extension

hyperbolic. By Lemma 5.12, the extension q.:

(X

I

p.:

T, '11:.) -+ (B, T, p)

is

r, T, '11:.) -+ (B,

too. Applying Lemma 5.11 to the linear extension p.

r,

pt,

T, p) is hyperbolic, and to its 'II:.-invariant vector

r

we get that p.1 is hyperbolic. Taking into account Lemma 3.9, we subbundle deduce from Lemma 5.12 that q is also hyperbolic. 5.14. Theorem. Let p: (X, T, '11:) -+ (B, T, p) be a linear extension, Y c X be an invariant vector subbundle and q: (X I Y, T, '11:) -+ (B, T, p) be the quotient linear extension. If any two of these three extensions, p, pi Y and q, are hyperbolic, then the third one is also hyperbolic. ~

This follows from Corollary 3.8 and Lemmas 5.11 and 5.13.

The next proposition is a slight generalization of a result due to Mather [1]. 5.15. Lemma. A linear extension 'II:~: rO(X) -+ rO(X)

p: (X, T, '11:) -+ (B, T, p)

is hyperbolic

iff

is a hyperbolic operator, i.e., the spectrum cr('II:~) does not meet

the circle Iz I = 1. • Let P be hyperbolic. Choose a Lyapunov norm on X and provide rO(X) with the ~rresponding norm. Because p is hyperbolic, we have X = XI $ Xi, where XI and Xi are U;variant vector subbundles. Consequently,

rO(X)

direct sum decomposition. Since 1I'II:llXlbll o < A < I, and for all b E B, we see that

$

~

rO(XI) $ rO(X2l is a 'II:~-invariant and 11'11:.1 IXib II $ ~ for some A,

=

246 This, clearly, ensures that 1l~ is a hyperbolic operator. Conversely, suppose 1l~ is hyperbolic. Prove that the extension p is hyperbolic. Since

1l~: rO(X) ... rO(X)

there exist linear subspaces p;S and It" of rO(X)

is hyperbolic,

invariant under 1l~, and a norm on rO(X) such that

Firstly, let us show that p;S and E' are sub modules of the module rO(X) over the ring

cfl(B, IR), i.e., if v

E

p;S (v

E

E') and rp

E

cfl(B, IR), then rpv

E

p;S (respectively,

v E It"). It suffices to prove the first assertion. By the Spectral Radius Theorem, v E p;S if and only if

Because 111l~(rpV)II:S IIrpll 111l~(V)II, we have

hence, rpv E p;S. For y E B and X"(y)

(I'

= {x

= s, u E

X: p(x)

Next we shall show that Xy on y

E

= y,

= XS'(y)

3 ~

E

ff,

~(y)

= x}.

(5.23)

• X"(y) (y E B), and X"(y) depends continuously

B. In other words, we shall show that XS'(y) and X"(y) determine two vector

subbundles Indeed, where y

r

and X" of (X, p, B), so that X

for every

= ~I(y)

+

XS'(y)

+ Jt'(y) = Xy

Prove r(Y) n X"(y)

b\l ... , b l

+ ~2

~ E rO(X)

such that ~(y)

with respect to the decomposition rO(X)

~2(y)' hence, ~I(y)

x

= r • X".

x E X there exists a section

= p(x). Write ~ = ~I

Then

and

denote

E

XS'(y) and ~2(y)

E

X"(y).

= x,

= p;S + E'.

This means that

(y E B). Clearly, XS'(y) and X"(y) are vector subspaces of Xy •

=

{O}.

In fact,

be a basis of X"(y).

let aI' ... , ale be a basis of the space XS'(y), By definition,

there exist continuous sections

247 ~h

... ,

bJ

= "'J(Y)

and

~le

E

XU(z)

E' (j

and

=

"'I' ... , "'l E

1, ... , l).

denote

the

It' such that

Let k"(z) linear

at

= ~t(y)

denote the linear

contrariwise, that the intersection of ~(y)

hull

"'I (z) , ... , "'l(Z)

hull of and

.xu(y)

=

(i

and

~I(Z), ... , ~le(z)

of (z

1, ... , k)

E

B).

Suppose,

is non-zero. Then there exist

continuous real-valued functions IPI'"'' IP le' 1/11' ... , 1/1, such that le

L IIPt(z) I

L II/IJ(z) I

> 0,

t-I

> 0,

J-I

(5.24) le

L IPt(z) ~t(z) = L I/Ilz) "'J(z) J-I

t-I

for all z in some neighbourhood V(y). Let v: B ... IR denote a continuous non-negative function such that v(y) = 1 and v(z) = 0 (z E B \ V(y». Define continuous functions ~t: B ... IR (i = 1, ... , k) and ~i B ... IR (j = 1, ... , l) as follows: ~t(z) = v(z) IPt(z) for z E V(y), ~t(z) = 0 for z. V(y), ~J(z) = v(z) 1/1iz) for z E V(y), and ~iZ) = 0 for z It V(y). It follows from (5.24) that le V!!

L 9i

t

~t =

t-I

L ~J '"J J-I

contradicting

E' and It' are submodules, we get VEE' nit", the hypothesis E' n It' = to}. Thus, ~(y) n .xu(y) = to}, hence

~(y) • .xu(y)

= Xy

is a non-zero section. Since (y

E

B).

It

follows

immediately from our definitions that

XS(y) c ~(y), r'(y) c .xu(y). Taking into account that dim k"(z)

for z in some neighbourhood of y, we conclude that XS(z) all z sufficiently near to y. Now define

gr = { ~

E rO(X):

~(z) E X"(z)

(z E B)}

+ dim XU(z) = dim Xz

= ~(z),

(IT

= S,

XU(z)

u).

= .xu(z)

for

248

Show that Y

= E".

such that r(z)

Evidently, E" c US'. Let y E B and V(y) be a neighbourhood of y

= X"(z)

for all z E V(y). Take a section ~

=

for all wEB \ V(y). Then ~ ~

E

L III, A" '-I

where

E

=0

are continuous functions, hence

III,

~. Let {VI' ... , Vm } be a finite subcover of the cover {V(y):

{I(.I , ... , I(.m}

~(w)

Ii: so that

y

E

be a partition of unity subordinate to {VI' ... , Vm }. For any

B}, and 11 E

Ii: ,

m

we have

Ii:

=

11

= ~.

L

I(.p

11,

and

I(.p

11 E

~, as it was already shown. Hence, 11 E~. Thus,

The same arguments prove that E:'"

= E'.

Now let us return to the proof of the claim that p is hyperbolic whenever Jr~ is a hyperbolic linear operator. Define XI

= r,

Xl

= X".

The above considerations show

that XI and Xl are vector subbundles of (X, p, B) and XI

Xz = X.

49

Since ~ and E' are

invariant under Jr~, the subbunles XI and Xl are invariant under (X, z, n). Next we shall show that XI and Xl have the same property with respect to (X, R, Jr) (in the case T

= IR). Let x

Denote

E

A

XI

=

II Jrn(.%) II

Ii

r,

y

= p(x).

max { IIJr~I~II,

= IIJr~

Find a section ~

IIJrillE'1i }.

~(pn(y»11

:S

E

~

such that ~(y)

By hypothesis,

sup II(Jr~ ~)(z)1I

A

= IIJr~(~)1I

= x,

II~II

= IIxli.

< 1. Then :S

An II~II

= An

IIXIi.

zEB

Similarly, IIJr"'(X)1I s An IIXIl (x E Xz, n = 1, 2, ... ). Thus, the linear extension p: (X, z, Jr) ~ (B, z, p) is hyperbolic. Bearing in mind that the stable subbundle XI and the unstable subbundle Xz are uniquely determined, we conclude that XI and Xl are invariant under all

Jrt

(t

E

7).

5.16. Theorem. Let B be a compact metric space, (X, p, B) be a .finite dimensio1UJl real vector bundle and p: (X, T, Jr) ~ (B, T, p) be a linear extension. The following assertions are pairwise equivalent: (1) p is hyperbolic; (2) the dual linear extension p. is hyperbolic; (3) X = r • YU; (4) rO(X) = ro.(X) • rOu(X); (5) p is regular; (6) p is uniformly regular (in the strong sense); (7) there exists a uniquely determined Green-Samoilenko function for p; (8) there exists a 1IOn-degenerate quadratic jUnction

249 I: X -+ IR such that I(n:t(x» - I(X)

> 0 (x

E X, IIxll ... 0, t

> 0); (9)

n:;: rO(X) -+ rO(X)

is a hyperbolic opera/or for all t ... O. ~

The proof that the properties (1) - (8) are pairwise equivalent is similar to that of Theorem 5.7. Lemma 5.15 implies that (1) • (9).

Let p: (X, T, n:) -+ (B, T, p) be a linear extension with no nontrivial bounded motions. Let p": pt, T, n:.) -+ (B, T, p) denote the dual linear 5.17. Theorem.

extension. There exists a hyperbolic linear extension p • p.: (X. 'It, T, n) -+ (B, T, p) such that: (1) the vector subbundle X. {OJ is n-invariant; (2) the system (X • {OJ, T, n) is isomorphic to (X, T, n:); (3) the quotient of

(X.

'It,

T, n)

defined on {O}. 'It is isomorphic to ('It, T, n:.). ~ Fix some Riemannian metric <',' > on (X, p, B) and identify ('It, p., B) with (X, p, B) via this metric. For definiteness, suppose T = IR (the case T = Z can be examined similarly). Set t

nt(x, y)

= ( n:t[x + I n:--

• n:!(y) ds], 1l'!(y) )

° for (x, y) EX. X· X • X and t E IR. It is easy to verify that (X. X, R, n) is a flow and p. p: (X. X, IR, n) -+ (B, IR, p) is a linear extension. Observe that the $Ij

subbundle X (I {OJ is invariant under (X (I X, R, n), and nt(x, 0) = (1l't(x) , 0) for x E X, t E IR. It is also clear that the quotient flow defmed on (X. X) I (X. {OJ) Ii {OJ (I X is isomorphic to n:.. By virtue of Theorem 4.12, it suffices to find a non-degenerate quadratic function V: X • X -+ IR such that

V(nt(x, y» - Vex, y) > 0 (1I(x, y)1I ... 0, t > 0). Since p: (X, IR, 1l') -+ (B,

p) has no non-trivial bounded motions, Theorem 4.9 ensures the existence of a quadratic function I: X -+ R satisfying the condition 1(1l't(x» - I(x) Let V(x, y)

«x, y)

E

X

= <x, y> + I(X) (I

X, t

E

«x, y)

IR). Therefore, for

E

> 0

(x

E

X, IIxll ... 0, t

X (I X). Recall that <1l't(x), 1l'!(y» and t > 0, we get

IIxll'" 0

IR,

> 0).

= <x, y>

250 t

=

v(iit(x, y»

< 'ltt[x

+

J

'It-s

0

+

'It:(y) ds], 1[!(y) >

I('ltt(x»

o t

=

<x

+

J1[-s

t

0

'It:(y) ds, y>

+

I('ltt(x»

>

<x

+

J

o

1[-$

+

I(x)

o t

=

'It!(y) ds, y>

0

vex, y)

+

t

J

1[!CY), y>

<1[-$ •

=

ds

+

V(x, y)

o

J

1I'It:(y)1I 2 ds.

o

Consequently, t

v(iit(x, y» - V(x, y)

>

J

111[:CY)1I 2 ds

2;

0

o

> O. Consider now the case IIXII,... O. Since

for all

Ilxll,... 0 and 1

II(X, y)11 ,... 0, we have lIyll"" 0, and by

repeating the above arguments once again, we get t

v(iit(x, y» - Vex, y)

=

J

111[:CY)1I 2 ds

> 0

(I

> 0).

D

Thus,

V(iit(x, y» - vex, y)

> 0

for all 1I(x, y)1I

#.

0, 1

> O.

The symmetric matrix, which corresponds to the quadratic form Vb be written as

= VIXb •

Xb , can

S(b)

o where S(b) corresponds to the quadratic form

»,

Ib = IIXb

(i.e.,

Ib(x) = <S(b)x,

x>

(x E Xb and In is the n x n identity matrix. Consequently, Vb is non-degenerate for all be B.

251

§ 6. Smooth Linear Extensions In this section, we consider the relationships between the notions of weak regularity, transversality and Green-Sarnoilenko function in the category of smooth linear extensions.

6.1. Def"mitions. Let B be a compact C manifold without boundary and (X, p, B) be a

C

vector bundle. As usual, T =

is said to be

C

IR

or T =

I.

smooth if 'Itt: X -+ X and pt: B -+ B are of class

Let O:s k :s r. By (P'«X, p, B), Pk.' B) sections of (X, p, B) (see A.14). a linear

A linear extension p: (X, T, 'It) -+ (B, T, p)

== P'«X)

t e T.

we shall denote the k-jet bundle of

The linear morphism

r('ltt): P'«X) -+ r(X)

morphism

C for all

thus

'Itt: X -+ X (t e 7) induces defining

a linear extension

PJc: (r(J{) , T, r('It» -+ (B, T, p) of class C-k., called the k-jet extension of p: (X, T, 'It) -+ (B, T, p). Fix some Riemannian metrics on (X, p, B) and (TB, T8, B). We say that the smooth linear extension p: (X, T, 'It) -+ (B, T, p) satisfies the k-jet transversality condition if the linear

extension Pk.: (r(X) , T, r('It» -+ satisfies the transversality condition (see Definition 3.3). The notion hyperbolicity is defined similarly. A linear extension p: (X, T, 'It) -+ (B, T, p) is called k-hyperbolic if 'It-invariant vector subbundles XI and ~ and positive integers c and ;\ X = XI til ~ and the following inequalities hold:

(B, T, p)

of

k-jet

there exist such that

(6.1)

(I ~ 0, b e B;

I

= 0,

1, ... , k).

Conditions (6.1) can be expressed in terms of Lyapunov numbers (see 2.1) as follows: O('lt l • b)

<

I w(Tp, b),

w(7[2' b)

>

I O(Tp, b)

(b e B),

262

=

=

=

nt lXI' n; 'Itt I~ (t ~ 0; I 0, 1, ... , k). where n~ Clearly, the notion of hyperbolicity coincides with that of O-hyperbolicity. In this

case, XI

= ](I,

~ =

r'. Since

B is compact, k-hyperbolicity does not depend on the

choice of Riemannian metrics on (X, p, B) and (TB,

'tB'

B).

6.2. Lemma. 1/ p: (X, T, n) -+ (B, T, p) is a k-hyperbolic linear extension of class ~, then

r

and K' are ~ vector subbundles.

~ Let us prove, for example, that

r' is a vector subbundle of class ~. Without loss the maps n: X -+ X and

of generality assume that T = Z. Replacing, if necessary, p: B -+ B by their iterates, we may assume that

< 1,

sup lI'ltlbll IITp·I(p(b»IIS' bEB

< 1

sup IIni!(b)1I IITp(b)lIS'

(s

= 0,

1, ... , k),

beB

where 'ltlb so that

-uS·1 = nI lAb' 'ltlp(b) = n.1 1Ap(b)' vU

p

= PI ,

b

EB.

S ber e ect i anum

C E

(0 , 1)

(6.2)

(6.3)

k, and kl be ~ smooth vector subbundles which approximate the subbundles XI. r and Xl. r', respectively, in the c! topology so that X = XI • Xl. Let P,: X -+ X,

Let (i

=

1, 2) be the corresponding projectors. Assume

(b) 11(1\ 111'1

0

0

'It I Xlbr '

II

S

'ltIXlbll :s c,

IIni!(b)1I

IIPl

0

+ c;

111'1

'ltIXlbll

<

0

I:

X,

,rlklb"

(b

E

is so near to X, (i

S

lI'ltlbll

B).

+ c;

= 1,

2) that

253 Since X = XI

Gl

X2 ,

the subbundle

Jt'

=X

2

can be represented as the graph of some

vector bundle morphism from Xl into XI' In other words, Xl can be written as

Xi

=

+ i)(X,),

(IT

where

1 = id: X2 -+ X2 • Let

vector bundle L(X2 , XI) and space rO(L(.\2, XI»

Banach satisfying

II IT II

rO(L(.\2, XI» (i.e.,

IT E

IT

'Eo

is a continuous section of the denote the closed subspace of the

consisting of all continuous sections

B -+ L(X2' XI)

IT:

1. Since the subbundle Xl is n-invariant, it is natural to use Theorem

:$

A.33. More exactly, apply this theorem to the case where E = L(Xl' XI), X E[I] = {IT E E: II IT II :$ I} and f. E[I] -+ E[I] is defined by

= B,

h =

pI,

(6.4)

Let us verify that

f.

E[I] -+ E[I]

is well-defined. Since

Pl

0

n IXlb

is an invertible

linear operator and

by (6.2), we conclude that the operator BTl = invertible and

P2

0

nb

0

('II

+ Ib ):

X2b -+

X2p(b)

is

(6.5)

according

to

the

well-known

J(TI) E L(X2P (b)' XIP(b», hence

:$

(IInlbll

+

E)IIB~111

f

+

estimate.

It is

easily

seen

from

(6.4)

that

covers p. Now let us show that 1!f{'II)II:Ii 1. In fact,

(IInlb ll + 2E)(lIn2~( b) II + E) EIIB~III :Ii - - - - - - - - - - - 1 - E (IIn2~(b) II + E)

according to (6.5) and the choice of E.

<

I,

254

Next we shall show that IIi",

0

Trb(XI

for all xt

e

+ x:z}

Xl'

p(X t )

-

ft:TI)

=b

0

1'2 • Trb(XI (i

=

1, 2),

+ X:z}1I

:S

(IITrlbll

+

Xlb),

111)11

1) e L(X2b ,

2£) 111)(x:z} - XI" :5

(6.6)

1. Indeed,

Further, prove the inequality sup Lip (flEb[I])

<

1.

(6.7)

beB

(IITrlbll + £) (lIlt2~( b)1I + £) :s - - - - - - - - - - - - 1 1 1 ) 2 -1)111 1 - £ (IITri~(b) II + £)

(6.8)

255 by (6.5). Clearly, (6.8) and (6.3) (s Now set a.b

=

= 0)

imply (6.7).

IITp·I(p(b»lI, ~b

= Lip (flEb[l]) < 1 and ).s = sup

(b

E

B). We see from (6.2), (6.3) and

{f3 b(a.b)s: b E B} < 1 (s = 1, ... , k). (6.8) that ).o!! sup {~b: b E B} Thus, all the hypotheses of Theorem A.33 are fulfilled (recall that the space B is compact). Hence, there exists an uniquelly defined invariant section Thus, the subbundle ~

ck.

therefore ~ is a

ck

Ii

~ coincides with the graph (t1'f)

= {(x,

to

E

of class

t1'f(X»: x

E

Xl}'

vector sub bundle.

6.3. Notation. Let ~(X) be the Banach space of all (X, p, B),

t1'f

ck

sections of the vector bundle

and rO(r(X» be the Banach space of continuous sections of the vector

bundle reX). There is a natural isometric embedding of the space rk(X) into ro(r(X». Let p: (X, T, n) -+ (B, T, p) be a linear extension of class

ck.

For each element t

E

T,

define a map n!: rk(X) -+ rk(X) by the formula n,t(11')

-t

n,(~)

t = nolI'

= Y...k (n)t

0

p ·t

0

~

0

p.

(11'

E r k (X»,

t

6.4. Lemma. The following four assertions are pairwise equivalent:

extension Pk: (r(X) , T, r(n» -+ (B, T, p)

is hyperbolic;

(1) the linear

(2) the invertible linear

operator n~: ~(X) -+ rk(X) is hyperbolic; (3) the operator ii~: rO(r(X» -+ rO(r(X» is hyperbolic; (4) the linear extension p is k-hyperbolic. ~

The assertions (1) and (3) are equivalent by Lemma 5.1S. Suppose (2) holds.

A slight modification of the proof of Lemma 5.15 shows that rk(X)

=F

and ~ are submodules of the ck(B, IR)-module rk(X). Define X"(y) (y by (5.23). Then

r

~

~

= rk(r) and

and ~ are

= rk(~).

ck

Gil E

~, where B;

vector subbundles and, moreover, X

ck

=r

= s, u) Gil~,

Applying Lemma A.29, we get that the extension p is

k-hyperbolic. Thus, (2) • (4). Conversely, let (4) be fulfllied. By Lemma 6.2, are

11'

F

vector subbundles. Therefore rk(X)

= rk(.r)

Gil

r

and ~

rk(~) coincides with the direct

256

sum decomposition of rk(X) into the stable and unstable subspaces, hence (2) holds. For each element ~ E r(X),

Thus, (2) .. (4).

there exists a section tr E rk(X) such

that ~ = j~(tr), where b = Pk(~)' Consequently, (2) .. (1). Let us show that (1) .. (4). The proof will be carried out by induction. If k = 0, our statement is trivial because ~(X) = X. Show that (1) .. (4) for k = 1 + 1.

Suppose this statement is true for all k:s 1. Thus, we have that the linear extensions

are hyperbolic. According to the induction hypothesis, the extension P is

PI' ... , Pl+ l

I-hyperbolic. Since the extensions PI and Pl+l are hyperbolic, Lemma 5.11 says that the

(see A.16) is also hyperbolic, therefore E;;; HI+I,I(X)

extension Pl+IIHI+1,l(X)

H1 + 1,1(X) =

be represented in the form

H1+ 1,1(X) s= P1+ I (TB, X) == P1+I(TB,

FeE'.

Besides, there exists an isomorphism

r) e PI+I(TB, Jt').

F = P 1+ I (TB, r), It' = P1+I(TB, r'). Because

It

remains

cF

w(a, pi)

point

Choose an element i; e P1 + I (TB, r) \

be the

",-limit set of the point a

b E w(a, pi),

i; e P1+I(TB, r ) \ IITp" I TJlII

E

w(a,

p\

F,

c&

F

and P1+ I (TB, r')

and denote a

=

eX".

is

not

Pl+l(i;). Let

with respect to the cascade pl. For every

PI +1(TJj, ~) c ~

or

PI+I(TJj, ~) c~.

Since

this means that Pl+I(TJj,~) c~. Hence it follows that

tends exponentially to zero as

;;; P1+I(TB, r )

b

either

show that

P1 + I (TB, ,r)

Suppose the contrary holds. For definiteness, suppose that

F.

to

P1 + I (TB, X) is a fmite dimensional

vector bundle, we need only to establish that P1 + I(TB, XS)

contained in

can

P1+I(TB,

n ~ +....

r'), we must conclude that

But the above arguments imply that

Recalling that

P 1 + I (TB, X)

PI+I(TJj,~) n ~

IITp-rlIT" BII ~... p

(b)

as

* III n

for all ~

+ ...

contradicting the preceding statement. This contradiction proves that (1) .. (4).

6.S. Lemma. Let (X, p, B) be a linear extension of class

C<

C<

vector bundle, p: (X, T, n:) ~ (B, T, p) be a

and M c B be a Morse set of (B, T, p).

over M, then rnY'(M) and r'1W'(M) are

C<

If P is k-hyperbolic

vector subbundles.

• By Lemma 3.11, the statement is true in the case k = O. For k > 0, the proof is similar to that of Lemma 3.11 and relies on Theorem A.36. The details are left to the reader.

257 6.6. Notation. Assume the hypotheses of the preceding lemma hold. Write M = AI n A;, where AI and A2 are attractors of (B, T, pl. For definiteness, assume T = IR. Let

V:

V: W'"(AI) ... IR and

WU(A~ ... IR be some Lyapunov functions constructed for the

attractor AI and the repeller A;. Let d be a small enough number, D

=

{b

E

W'"(AI) n WU(A~: V(b)

., be a positive number, and Q = {b

E

= d,

= O}

V(b)

< f}.

B: V(b) = d, V(b)

c W'"(M),

According to Theorem

1.4, WU(M) u p(Q, IR) is a neighbourhood of WU(M) in B.

Further, let G be some ~ vector subbundle of XI Q satisfying the condition Xb

= x: + Gb

(b

E

B). Since D c W'"(M), Lemma 6.S implies that riD is a

subbundle. Extend Gover WU(M) u p(Q, /R) by letting G

t

P (b)

= nt(Gb )

(b

E

c" vector Q, t

E

IR),

Gb = Xb (b e WU(M». By Theorem 3.12, the so defined vectorial set GI WU(M) u p(Q, IR) is, in fact, an invariant vector subbundle of XI WUCM) u p(Q, IR).

6.7. Theorem. Let p: (X, R, 'If) ... (B, IR, p) be a ~ linear extension and M be a Morse set of (B, IR, pl. If p is k-hyperbolic over M, then the above invariant vector subbundle GI WU(M) u p(Q, IR) is of class

c" .

• This is true by Theorem 3.12 and Theorem A.33.

6.8. Theorem. Let p: (X, IR, 'If) ... (B, IR, p) be a

c"

linear extension satisfYing the

k-jet trtlnsversalily condition. 'J"Mn there exist neighbourhoods subbundles

E:

and

Et

U, of A, and ~ vector

of XI p(U" IR) such that the assertions (a) - (f) of Theorem 3.14

hold. • The proof of this theorem follows the arguments used in the proof of Theorem 3.14

very closely and rests on Theorem 6.7 and Lemma 6.4. We leave the details to the reader.

6.9. Theorem. Let B be a compact vector bundle, and p: (X,

R, 'If) ...

c" manifold,

(B, IR, p) be a

(X, p, B) be a finite dimensional ~

c!'

linear extension.

'J"M following

258 assertions are pairwise equivalent: (1) the k-jet extension Pit: (p/«X) , T, pIt(n» -+ (B, T, p) is weakly regular; (2) Pit is uniformly weakly regular (in the strong sense); (3) Pit satisfies the transversality condition; (4) rlt(X) t IIn,(~)1I -+

(5)

= ~ + ~,where ~ = {~E rlt(X):

IIn:(~)1I -+ 0 as t -+

+ ID}, ~ = {~E rlt(X):

0 as t -+ - ID};

the dual linear extension p~ has no non-trivial bounded motions;

(6) the extension P admits a Green-Samoilenko function G: T x B -+ X of class

c!-

such that IID~G(T, b)1I s c exp (-

II'

IT I)

(b

E

B,

T

E

T;

I

=

0, I, ... , k)

for some constants c > 0 and" > 0; (7) there exists a quadratic function

for all ~

E

[r(X)]·, II~II '" 0, t

(8) P is uniformly weakly

c!-

t: [r(X)f -+ IR such that t(n!.(~» - t(~)

>

0

> 0; regular (in the strong sense) .

• The assertions (I), (2), (3), (5) and (7) are pairwise equivalent by virtue of Theorem 5.7. Let us show that (3) .. (4) and (3) .. (6). The proof of these implications consists, essentially, in repeating the reasonings used in the proof of Theorem 5.7. So we restrict ourselves by indicating the new points. Theorem 3.4 being applied to the linear extension Pit states that the restriction Pit I A is hyperbolic. It then follows from Lemma 6.4 that P I A is k-hyperbolic. Suppose, for definiteness, that T = IR. Let the neighbourhoods Ut of

At

and the subbundles ~ and

Itt

of Xlp(U h IR) (i

be taken from Theorem 6.8. Further, let {B t } be a

c!-

the cover {p(U h IR)} of the manifold B. Given '"

E

I, ... , n)

partition of unity subordinate to rJc(X), define

done in the proof of Theorem 5.7. Because the subbundles ~ and

=

= 0,

",s

and

Itt

are

",u

as it was

CC

smooth,

Since ",a' E rita' (X) , we see that (4) holds. The proof of the implication (3) .. (6) is carried out exactly as in Theorem 5.7, but since the projectors

",a'

E

rlt(X) (II"

s, u).

n

t=o

smooth. Clearly, this gives (6). It is not hard to see that (6) implies (1) and (8) and (4) implies (3) (because for

259 each element ~ E [r(X)]b there is a section rr E rl«X) such that j!;(rr) = ~). Now we shall prove that (8) implies (1). For convenience, let us suppose T = I. The condition (8) means that for every section S . f ymg . I -I = rr, I.e., . satis '/(& 0 rr 0 p - '/( I 0 rr

=

A: rl«x> -+ rl«x> by A(rr)

rr • pi -

'/(1 •

rl«X) there exists a section rr E rl«x> + rr p I = s. Defime a 1·mear operator

E

rr.

0

Condition (8) guarantees that A is a

surjective operator. Moreover, this is true not only for the given linear extension p but also for any pullback of p. Assume (8) holds. According to the well-known Banach theorem, there exists a constant L

D such that for each section

rl«x> satisfying the conditions

fmd a section rr

E

IIrrlll< :s LIISIIl<.

Let

h(rp(n»

>

= p"(bO>

(n

bo E B

11': I -+ r(X)

and

be

r(x>,

necessary, to a pullback) that the map ~: p(bo, I) -+ is well-defined and continuous.

~(b)

=

Db for

=

E

Extend

to a

Vi

function

Clearly,

there exists a section S

1, ... , m).

Let A

denote

E

pi -

=S

and

map such

that

'/(1

bounded

rr

0

~(p"(bO»

= rp(n)

~: B -+ r(X)

(n

E

I),

by letting

B \ p(bo, I), where Db denotes the origin of the fiber [r(X)]b. Let

B.

bu •.. , bm (i

b

a

0

rl«X) one can

With no loss of generality, we may assume (by passing, if

I).

E

A(cr);: rr

S E

the

collection of all finite subsets of B

ordered by set inclusion. There exists a net satisfying

lim 1!(S",)

=

rl«X) such that 1!(S)

E

=

{s",:

;PCb)

(b

IX

E

E

A}

of elements

;P(b()

partially E

rl«x>

rl«X) ,

i.e.,

s'"

B).

",EA

Let

{rr",:

IX E

A}

be

the

corresponding

net

of

elements

rr",

E

= s'" and IIII"",II:S L1is",lI. Since ;p: B -+ r(X) is a bounded section and we can apply Tikhonov's theorem on compactness of a Cartesian product of compact spaces. It follows that there exists a subnet {rr/3: ~ E AI} of {IT ... : IX E A} which 11"",

0

pi - '/(1 •

11"",

IIcr",1I :S LlIs",1I

converges pointwise to some bounded section I; of the bundle reX). Then Ib

I

~ • P ( ) - '/(l<

. J<. 0 ~(b) = hm [j • 1T/3(b) -

/3

I

'/(l<

01J<. • 1T/3(b)]

(b

E

B).

260 Denote

xo=

~(bO>.

It is clear that Xo satisfies the conditions of weak regularity of the

extension Pit at the point boo Thus, (8) ~ (1).

Bibllographical Notes and Remarks to Chapter

m

Readers interested in the topological theory of dynamical systems are referred to the following books: Nemytskii and Stepanov [1], Sibirskii [1], Bronstein [1]. The material sketched in the first three sections of this chapter is presented in detail in the book by Bronstein [4, §§ 1 - 8]. This book contains a vast bibliography on chain recurrence and linear extensions of dynamical systems. The method of quadratic Lyapunov functions in the theory of linear extensions is developed by Mitropolskii and Kulik [2], Mitropolskii, Samoilenko and Kulik [1], Kulik [1, 2], Lewowicz [1], Samoilenko [5]. The material presented in § 4 generalises some results of these papers and is due to Bronstein [3]. The method of Green's functions in the theory of linear extensions is developed by Samoilenko [1-5] and Kulik [2, 3]. § 5 is a slightly improved obtained by Bronstein [2]. Theorem S.17 was first proved by Man6 Samoilenko and Kulik [1]. For related results see: Mitropolskii and 5], Samoilenko and Kulik [2-4] and Trofimchuk [1-3]. Results presented in § 6 are obtained by Bronstein [7]. Theorem results due to Samoilenko [5].

exposition of results [1] and Mitropolskii, Kulik [1], Kulik [4, 6.9 generalises some

261

CHAPTER IV INVA~TSUBBUNDLES

OF WEAKLY NON-LINEAR EXTENSIONS

§ 1. Invariant Subbundles and Their Intrinsic Characterization

In this section we investigate invariant subbundles of extensions which are close to exponentially splitted linear extensions.

1.1. Dermitions. Standing assumptions. Let B be a compact metric space, (X, p, B) be a vector bundle, and rp be a morphism of this bundle into itself in the category :Bun, i.e., rp: X -+ X be a continuous mapping and p(rp(x» = p(rp(y» whenever p(x) = p(y). Let II· II be some (Finsler) norm on (X, p, B). One says that the morphism rp satisfies the Lipschitz condition if there exists a number ;\ ~ 0 such that IIrp(x) - rp(y)1I :s ;\ IIx - yll (x, y e X, p(x) = p(y». The smallest number satisfying this inequality is called the exact Lipschitz constant of the morphism rp and is denoted as Lip(rp). Clearly, Lip(rp) = sup {Lip (rp IXb ): b e B}. Let p: (X, T, A) -+ (B, T, p)

be a (not necessarily linear) extension.

We shall say

that it is a Lipschitz extension if for each number t e T the morphism from the bundle

(X, p, B)

sup {Lip (At): It I :s to} Let

<

OIl

section Z = Z(B)

of

X -+ X

into itself satisfies the Lipschitz condition and, moreover,

> O. p: (X, T, jJ) -+ (B, T, p)

for each number to

p: (X, T, ;\) -+ (B, T, p)

extensions. Suppose that both

At:

and

(X, T, A)

(X, p, B).

extension p: (X, T, A) -+ (B, T, p) p: (X, T, "') -+ (B, T, p) if

Let is

and

(X, T, jJ)

leave

be two Lipschitz invariant

the

zero

Co > 0, to > O. We shall say that the (to, £O>-close (in the Lipschitz sense) to

sup {Lip (At - ",t): It I :s to} :s

EO'

262

Thus, we have defined a topology in the space of all Lipschitz extensions which leave the zero section invariant. This topology corresponds to the metric Dd~, J,I)

=

sup min { sup Lip (~t - J,lt), 1 I T>O

't }

Itl:ST

and is stronger than the compact-open topology. In what follows, all extensions are assumed to satisfy the Lipschitz condition and to leave the zero section invariant. Fix a Riemannian metric on the vector bundle (X, p, B). Given a number h > 0 and a vector subbundle W of (X, p, B), let us write C(W,

h)

= {a +

b: a

E

W,

b

J.

W,

pea)

= pCb),

IIbll:S

h lIall}.

Assume that p: (X, T, 1t) ... (B, T, p) is a linear extension, and XI' ... , X k are 1t-invariant vector subbundles such that X = XI 81 ... 81 Xk is an exponential splitting (see Defmition III.2.6). Denote XlJ = Xl .... 81 XJ' Xu = Xl (1:s i < j :s k). Clearly X = XI ..... Xl_I 81 XlJ 81 X J +I 81 ... 81 XIt (1 < i < j < k) is also an exponential splitting. 1.2. Theorem. Assume thai the linear extension p: (X, T, 1t) ... (B, T, p) is exponentially splitted into invariant vector subbundles XI' ... , Xk • Then for each number h, 0 < h < 1, one can choose to > 0 and £0 > 0 so that: (1) if p: (X, T, ~) ... (B, T, p) is an arbitrary extension leaving the zero section invariant and (to, £O>-close (in the Lipschitz sense) to p: (X, T, 1t) ... (B, T, p),

then there exist subbundles ~'~J (l:s i < j :s k) of (X, p, B) in the category !3W1.B

invariant under (X, T,~) and such that X

=~

81 ... 81

~;

(2) ~ c C(Xh h), ~J c C(X'J' h)

(1:s i :s j :s k). If the number. h > 0 is sufficiently small, then ~J is the maximal ~-invariant set contained in C(XtJ , h), i.e.,

~J

=

n ~t[XlJ' h)); tET

(3) there exists a !3W1.B-isomorphism F of (X, p, B) into itself, FIZ(B)

= id,

which

263

carries X'J onto rtJ (1:1 I :I j :I k). The morphisms F - I and F· I - I, where I denotes the identity mapping, satisfy the Lipschitz condition and, moreover, Lip (F - I) and Lip (F ·1 - I) are O(h) as h -+ O. Besides thai, F .llrt coincides with the natural projection P,: X = XI Ell ••• 48 XJt -+ X, (i = 1, ... , k). ~

For a proof, see Bronstein [4, Theorem 9.28].

1.3. Theorem. Assume the hypotheses of the preceding theorem to be fulfilled. Let f3 be a positive number, and r, and R, (I = 1, ... , k) be the cocycles constructed in Theorem III.2.7. Then for every c, 0 < c < f3 12, there exist numbers to > 0 and 150 > 0 such that for each extension p: (X, T, ~) -+ (B, T, p) leaving the zero section invariant which is (to, c5O>-close to p: (X, T, 7f) -+ (B, T, p), the following statements hold: (1) the point IIXIl

r;

x belongs 10

exp [r,(b, I) - c/]

iff

lI~t(X)1I

:I

:I

IIxll

exp [R,(b, I)

+ ct]

(b

= p(x»

for all sl4iJiciently large numbers t > 0; (2) whenever 1:1 m

<

I

:I

k and

then IIXIl

exp [r,(b, t) - ct]

:I

lI~t(x)1I

:I

IIxll

exp [R,(b, t)

+ ct]

for all large enough t > 0, and IIXII

exp [Rm(b, t)

+ ct]

:I

lI~t(x)1I

:I

IIxll

exp [rm(b, t) - ct]

(b

= p(x»

for all negative numbers t with a large enough modulus. ~

See Bronstein [4, Theorem 9.29].

1.4. Notation. Let B a compact space, (E, p, B) be a vector bundle, and if, p) be a vector bundle automorphism of (E, p, B). Further, let X and Y be f-invariant vector

264 subbundles such that X

Y = E. We shall assume that (g, p) is an automorphism of

$

(E, p, B) in the category ~un. close, in a certain sense, to if, pl. The zero section Z(B) is not assumed to be g-invariant. We seek conditions ensuring the existence of

g-invariant subbundles Xg and Yg close to X and Y, respectively. The proof of the next theorem is based on the graph transform method. This means that the subset Xg , for example, will be represented as the graph of some idB-morphism

tl'g: X ... Y, i.e., Xg = {(x, crg(x»: x E Xl. Let PI: E ... X and P2: E ... Y denote the projectors corresponding to the direct sum decomposition E = X $ Y. Let cr: X ... Y be some idB-morphism in the category f3un.. The mapping g: E ... E carries graph(cr) onto the set g(graph(cr» which is not, in general, the graph of any morphism from X into Y. Since g(graph(cr» = {(PI

0

g(x, cr(x», P2 • g(x, cr(x))): x e

Xl,

g(graph(er» will be the graph of some morphism iff the mapping her == PI X ... X is invertible. In such a case, g(graph(er» = graph(g,(cr», where

0

g o(id

+ er):

The set graph(er) is invariant under g iff g,l(er) = er. Thus, in order to construct the required invariant subbundle Xg , we must find an idB-morphism er: X ... Y invariant under g,. Fix some Riemannian metric on (E, P, B) so that X.L Y, then IIPIII = IIpzll = 1. Denote (b e B).

(Ll)

1.5. Theorem. Assume

(1.2)

There exists a number

fJ.

> 0 such sup Lip

that if

«g - f) IEb )

<

fJ.

(1.3)

beB

and

sup hg(v) - j{V)1I veE

<

OIl,

(1.4)

265

then there is a uniquely determined continuous bounded idB-morphism a-g: X -+ Y in the category !JUII. satisfying the following conditions: (1) the set Xg

:;

graph (a- g) is invariant under g;

(2) sup Lip (a-gIXb) :s I;

(1.5)

bEB

(3) the mapping a-g depends continuously on g in the uniform (f -topology; (4) the set Xg is uniformly asymptotically stahle under the cascade (E, g) . • Given a bounded idB-morphism er: X -+ Y in the category !JUII., put lIerli = sup {1Ia-(X): x EX}. Let 1: denote the complete metric space of all continuous bounded morphisms provided with the metric p(er l , a-:z.} = sup {lIer,(x) - er2(X)II: X E X}. Further, let 1:, denote the closed subspace of 1: consisting of all morphisms er E 1: satisfying the condition Lip (IT) !!! sup Lip (IT IXb) :s 1. (1.6) bEB

Choose

~

> 0 so small that (1.7) (1.8)

Now let us show that the morphism ha-!!! p, 0 g 0 (id + IT) is invertible for all To this end, we shall use Lemma A.27. Define a mapping 9': X -+ X by , = p, 0 (g -.f) 0 (id + er). Observe that F:; PI 0 f 0 (id + IT) = f. It hence follows from (1.1) that ITE1:,.

According to Lemma A.27, the inequalities (1.7) imply that ha- is invertible and (1.9) Therefore g,(IT) is a continuous mapping. Further,

266

sup IIg,(v)(X)1I = sup "P2

xeX :s sup "P2

f

0

xeX

+ sup "P2

+

0

(id

+ 0')

(id

0

110'11 sup {13b: b

E

+ 0')

0

h.;.I(x)1I

h.;.I(x)1I

0

(g - j) • (id

0

xeX

:s (1

g

0

xeX

+ 0') • h.;.I(X)1I B}

+

sup {lIg(v) - .f{v) II: v

E

ED <

00

by virtue of (1.2) and (1.4). Hence g,(v) E l:. Thus, if 0' E l:1' then g(graph(v» = graph(g,(v», where g,(v) E l:. Show that g,(l:\) c II' Indeed, if 0' E II' b E B, and XI' X2 E Xp(b)' then by (1.3) and (1.9), we have

Taking into account (1.8), we get sup Lip (g,(v) IXb )

:5

1.

bEB

Now let us show that g,: l:1 -+ II is a contraction. Employing (1.1) and the estimate

we obtain, for 0' "P2

+

"P2

Ih b

E

E

g(~) - g,(v)

0

0

g

0

(id

+ IIg,(v) • PI

B and 0

+ 0')

PI 0

• g • (id

0

~ E

g(I;)1I

Eb , the following inequalities: :5

"P2

0

g(~) - P2

PI

PI(I;) - g,(v)

0

+ 0') • PI(I;)

- g,(v)

0

0

g

0

g(I;)1I

0

PI • g(t;)11

(id

+

0')

0

PI(~)II

267

+

+

:s ({3b

+

lip I

0

+

IIPI

0

:s (fJ b

+

E' = (id

+

+

/l) liE' - (id (g -f)

f

+

0

0

+

(id

0

+

0')

/l) liE' - (id

g

0')

0

+

+

('d

0

+

0')

0

E

B,

g(~)11

0

(g -f)(E')1I

t 0')

b

0

PI(E')II

0

+

0

I

j{e)1I

(g - /) IXb ) lie - (id

P2

P (E') - PI

0

PI(E') - PI

0

PI(E') - PI

0

11')

1

0

PI (~) II

11')

+ 0')

2/l) IIP2(e) -

t l,

0'2 E

vI)

(id

0

Lip (PI

:s (fJ b

Let vI'

0

Lip (g,(O') IXp(/)) IIPI

(id

x

and

0

PI(~)II

PI (~) II.

E XP(b)'

(1.10)

Applying (1.10) with

0'

= 0'2

and

h;" I(X), we get I

IIg,(O'I)(X) - g,(O'J(x) II

= IIP2 = lip']. :s (fJb

+

0

g

0

geE') - g,l(O'J

+

0

2/l)

(id

VI)

0

IIP2(~) - P2

h;"/(X) - P'].

0

PI 0

0

0

g

0

(id

+

O'J

0

h;"2 1(X)1I

g(~)11

(id

+

(1'J

0

PI(E')II

Hence it follows by (1.7) that g,: tl -+ tl is a contraction. Therefore the required morphism 0'11 coincides with the fixed point of 8,1' Consequently, 0'11 is a continuous bounded morphism satisfying (1.5). Let ro = II (1'g II and Y[r] = {y E Y: lIyll :s r}. It is not hard to show that 00

criX) =

n gn(X

$

Y[r])

for every r > 70' This just means that 11'g(X) is an uniformly asymptotically stable set. A similar result holds for flows, as well. The reader is invited to formulate and to prove the corresponding theorem.

268 § 2. The Decomposition Theorem

In this section, we prove a very useful theorem saying that every extension close enough (in the Lipschitz sense) to an exponentially separated linear extension can be decomposed into the Whitney sum of two extensions. 2.1. Standing assumptions. Let B be a compact metric space, (X, p, B) be a vector bundle, and p: (X, T, 'It) -+ (B, T, p) be a linear extension. Fix on (X, p, B) a Riemannian metric. Assume that the extension p satisfies the condition of exponential separation, i.e., X is decomposed as a direct sum of two 'It-invariant vector subbundles XI and Xl so that there exist numbers d > 0 and "> 0 satisfying sup lI'1tt IXlbll 1I'1t"t IX t 1I:Ii d exp (- ot.I)

bEB

(2.1)

l.p (b)

for all t > O. By virtue of Theorem m.2.S, the condition (2.1) can be rewritten in the form D('ltl' b)

<

w('ltl' b)

(b E B)

where 'It~ = 'Itt IXII 'It~ = 'Itt IXl (t E 7) (see subsection m.2.1). By Theorem 1.2, for each number h, 0 < h < 1, one can find numbers to > 0 and EO > 0 so that if p: (X, T, ~) -+ (B, T, p) is an arbitrary extension leaving the zero section invariant and (to, Eo)-close, in the Lipschitz sense, to p: (X, T, 'It) -+ (B, T, p), then there

r. and such that the bundle = r. ., ~ (in the category ~UI\.B) and ~

exist ~-invariant subbundles

as the Whitney sum X

~

(X, p, B)

can be represented

c: C(X" h) (i

= 1, 2).

If the number h > 0 is small enough, then ~ is equal to the maximal ~-invariant set contained in C(Xt' h) (I = 1, 2). By Theorem m.2.7, there exist cocycles 't, Rt (I = 1, 2) of the system (B, T, p) and a number ~ > 0 such that

(2.2)

269 for all sufficiently large I > 0 and all b E B. Given a number c, 0 < c < fJ I 3, one can choose 10 > 0 and ISo so that if p: (X, T, ~) -+ (B, T, p) is an extension (10' lSo>-close to IIXII

p: (X, T, n) -+ (B, T,

p), then

x

E

~ iff

exp [r,(b, I) - ct] s lI~t(X)1I s IIxli exp [R,(b, t)

for all large enough t > 0 (i R(b, t)

= 1,

= [RI(b,

t)

+

ct]

(b

= p(x»

2). Defme a cocycle R of (B, T, p) by

+ rl(b,

t)] I 2

It follows from the preceding statements that x

E

(b

E

B, t

E

r. iff lI~t(x)1I s

7).

IIxllexp [R(b, t) - ct]

for all sufficiently large t > O. Similarly, x E ~ iff lI~t(X)1I it IIxll exp [R(b, t) + ct] for all sufficiently large I > O. Without loss of generality, we shall assume that XI .L Xl' 2.2. Theorem. if p: (X, T, n) -+ (B, T, p) satisjies the above assumptions, then there exist numbers to > 0 'and ISo > 0 such that if p: (X, T, ~) -+ (B, T, p) is (to, lSo>-close 10 p: (X, T, n) -+ (B, T, p), and ~t(Z(B» = Z(B) (t E 7), is isomorphic to the Whitney extension p: (X, T, ~) -+ (B, T, p) p:

(r., T, ~) -+ (B, T, p)

lhen the sum of

and p: (~, T, ~) -+ (B, T, pl.

~

We must construct two systems of ~-invariant curvilinear coordinate surfaces in X. This construction will be carried out in such a way that the coordinate surfaces in Xb passing through the origin

Ob

will coincide with

r.

b

and ~b

(b

E

B).

Let O(b) be the orbit of the point b E B, i.e., O(b) = {pt(b): lET}. Let X, IO(b) denote the restriction of the subbundle X, to O(b) (i = 1, 2). Let m. !JJl(b) denote the set of all morphisms tp: XII O(b) -+ ~ IO(b) covering the identity mapping of O(b) and satisfying the conditions sup Lip(flIXld) s h < 1, fI(Od)

= Od

(d

E

(2.3)

O(b».

dEO(b)

We shall show that for every point b E B and every element y uniquely determined morphism fly E m(b) such that

E

~b' there exists a

270 (2.4)

where XI (/(b) , ~t(y»

= ~t(y) + (1 + rpy)(XI,pt(b»'

and 1 is the identity map. Observe

that ~t(y) E XI (i(b) , ~t(y», according to (2.3). We must show that for all t sufficiently large or > 0 the following equality holds:

E

T and

Provide the set m with the metric

Clearly, (m, p) is a complete metric space and the topology in (m, p) is stronger than that of uniform convergence on compact sets. Let P,: X .. X, (i = 1, 2) be the projectors corresponding to the decomposition X = XI • Xl' Define an operator AT(~, y): m .. m by AT(;>., y)rp = rpo, where

It will be shown below that AT (;>., y) is well-defined. Note that the equalities Pdn-T[nt(y) P2,{n-T[nt(y)

+ (1 + rp)(x)]

+ (1 + rp)(x)]

- nt-T(y)}

- nt-T(y)}

= PI

= Pl

0

0

n-T

n-T

0

(1

0

(1

+ rp)(x) = n-T(x) ,

+ rp)(x) = n-T

0

rp(x)

imply

hence it follows from (2.1) that AT (n, y): m .. m is a contraction for all sufficiently large or > 0 (say, for or such

iii:

'I)' There exist numbers t~, t~

that if (X, T,~) is

then the operators

AT (;>., y)

(t~, c5~)-close

iii:

tl

+ 1, and c5~, 0 < c5~ <

150,

to (X, T, n) in the Lipschitz sense,

are well-defined for all or

E

[to, to + 1]. Because

271

[to, to + I]} is a commutative family of operators, there exists, by Lemma A.30, a common fixed point 'Py E m. Moreover, for all t, or E T, we have {A"t'(;>., y):

or

E

Letting t = 0, we get

Let

Z E

X I •b and k be a positive integer. Since

there exists such a point t E XI,p1<:- I(b) that

Clearly,

(2.5)

Therefore, 1<: PI • ;>. £y

Assume

+ (/ +

= PI

0

;>.1{A1<:-I£y

= PI

0

;>.1{A1<:-I(y)

+ (/ +

15 0

> 0 to be so small that

150

11

+

hl '

+

1<: ) !py)(z)] - PI • A (y

+ (/ +

!py)(Z)]} - PI !Py)(t)} - PI

exp RI(b, 1) :s exp [RI(b, 1)

+

0

AI(;>.1<:-I(y»

0

c]

Al (A1<:-1 (y».

(2.6)

(b

(2.7)

E

B).

Then

for all i E XI' )i E ~ satisfying the conditions p(j) virtue of (2.2) and (2.7), we have

= p(i) = r,

E

O(b). In fact, by

272

Put deduce

z = t, then

Y = ;\It-I(y),

b = pit-I (b) ,

and from (2_5), (2.6) and (2.8), we

By induction, it

II PI • ~ [y

+

(1

+

9'y)(z)] - PI

It

~ (y)1I

0

According to the defmition of cocycle,

therefore,

for all b e B, y e ~b' It

~ [y

+

(1

e Xlb and k = 1, 2, .... Since

Z

+

It

+

IPy)(z)] - ~ (y) e (1

IPy)(XI It b) C C(XI , h), .P ( )

we have

II~Jc[y +

(1

+

lIly)(Z)]1I :s ./ 1

e X1(b. y). then x = y + (I all large enough· k > 0 we get Let

X

+

+

h2

lIly)(Z),

'

IIzll exp [RI(b, k)

+

b:].

where Z = PI(x - y). Consequently, for

(2.9) (b e B,

Since XI(b. y) ... Y

+

(I

+

y e ~b'

lIly)(Xlb ).

X

e X1(b, y».

we see that X1(b, y) is the graph of a Lipschitz

273 mapping from X lb into ~b whose inverse is equal to the projector PI' Define a mapping 1/1: X lb • ~b ... Xb by 1/1(1..)') = x. where x E XI(b.),) and PI(x) = 1.. The above reasonings show that 1/1 is one-to-one and satisfies the Lipschitz condition. hence. I/J is

a homeomorphism from X lb • ~b onto Xb • In other words. the fiber Xb is equal to the union of the sets XI (b. )'). )' E ~b' Let )'I.)'l E ~b and )'1 '" )'l' Prove )'1) n XI(b. )'~ = II. Suppose. to the contrary. that there exists a point x E XI(b. )'1) n XI(b. '1~. Then (2.9) implies

XI (b.

(2.10)

for all sufficiently large k > O. Here cl is a constant depending only on x. '11 and '1l (and not on k). On the other hand. by using (2.2) and taking into account that (X. T. ~) is (t~. c5~-close to (X. T. 'It) in the Lipschitz sense. it is not hard to check that (2.11)

for all large enough integers k > 0 (here cl is also a constant depending only on '11 and )'l. but not on k). Recall that

according to the choice of

This means that (2.10) contradicts (2.11) . Hence.

E.

Proceeding in a similar way. pairwise disjoint sets

we can represent each fiber Xb as the union of some (x E ~b)'

Xl(b. x)

Xl(b. x) being defined just like XI(b. ),).

Thus. to each point W E Xb • we can assign uniquely determined points x so that Define I: X ...

r. • ~

= XI(b. '1) n X2(b, x). = (X, )'). The mapping I

E

r.

b. '1

E

~b

{W} by I(W)

is one-to-one and, moreover,

the restriction Ib: Xb ... ~ b • ~b as well as the inverse mapping It. 1 satisfy the Lipschitz condition for all b E B. Because the sets XI(b.),) and X2(b. x) are constructed by applying the contraction principle, we conclude that they depend continuously on b E B, therefore I is a homeomorphism (or, more exactly, an isomorphism

274 is an in the category :Buns)' The property (2.4) implies that t: X -+ ~ • ~ then isomorphism in the category of dynamical systems, i.e., if t(w) = (x, y),

t(;\t(w»

= (;\t(x) ,

;\t(y».

1.3. Theorem. Let p: (X, T, 7t) -+ (B, T, p) be a linear extension and XII ... , Xk

be 7t-invariant vector subbundles of (X, p, B) such that X = XI • ••. • Xk determines an exponential splitting. Then there exist numbers to > 0 and 150 > 0 such that if the extension p: (X, T, ;\) -+ (B, T, p) is (to, c5J-c1ose to p: (X, T, 7t) -+ (B, T, p) and leaves the zero section Z(B) invariant, then there exist uniquely determined ;\-invariant subbundles ~, ... , ~ such that p: (X, T, ;\) -+ (B, T, p) to the Whitney sum of the extensions PI~ (i

= 1,

is isomorphic

... , k).

~ This follows immediately from Theorem 2.2. The subbundles

~, ... , ~

are

constructed in Theorem 1.2.

§ 3. The Grobman-Hartman Theorem

In this section, we generalize the Grobman-Hartman linearization theorem to the case

of weakly non-linear extensions. 3.1. Notation. Let B be a compact space and (X, p, B) be a vector bundle. Fix some Riemannian metric on (X, p, B). Let g be a homeomorphism of B onto itself. Let

f.

c!(X, X; g) be the set of all continuous mappings

X -+ X such that p

Let ~(X, X; g) denote the Banach space of all bounded mappings

f

0

f = gop.

E

c!(X, X; g)

endowed with the norm Ilfll = sup {1!f(X)II: x E X}. Finally, let "(X, X; g) be the Banach space of all vector bundle g-isomorphisms A: X -+ X provided with the norm IIAII

= sup

{IIA(x)lI:

x

E

X, IIxll

::5

I}.

3.2. Lemma. Let A E A(X, X; g) and IIAII

<

1. There exists a number Co

>0

such

that if the mappings rp and 1/1 belong to ~(X, X; g) and satisfy the Lipschitz condition with Lip(rp) < Co' Lip(l/I) < Co, then the jUnctional equation (A

+

1/1)

0

(1

+

h)

= (1 + h)

0

(A

+

rp),

(3.1)

275 where 1

= id:

X .. X, has a single solution h

~ According to Lemma A.27, if Lip(rp)

the category :Bun. and, consequently, A the form A • h - h • (A

<

E

cg(X, X; idB)'

+ rp, g) is an isomorphism in is invertible. Reduce the equation (3.1) to

IIAotll ot , then (A

+ rp

+ rp) = rp

- '" • (1

+ h).

(3.2)

Define linear operators

by

Further, given a mapping k cg(X, X; g) by a.Je(lI) kot

E

= 11

E

cg(X, X; g), define a linear operator a.Je: cg(X, X; idB) ..

• k.

Note that a.Je is invertible iff k is invertible and

cg(X, X; got). In this case, (<<0 ot

= «Jeot

and II«JeIi

= 1.

It is easy to see that (3.3) Here 1 denotes the identity transformation of the space cg(X, X; g). Prove that

L; is an invertible contracting linear operator. In fact,

Hence, it follows that the operator also invertible and

consequently,

(L; - 1) is invertible.

By (3.3), the operator Lrp is

276 The functional equation (3.2) can be rewritten as

(3.4) Define a mapping J.I: ~(X, Xi ids) ... ~(X, Xi ids) by

< 1-

and show that J.I is a contraction whenever Lip(l/I)

Thus, if Lip(!p) <

EO

and Lip(l/I) < EO

= mm.

A. In fact,

where

EO,

{A-I - I I- } II II, A I

(3.5)

then J.I has a unique fixed point h = h",." E ~(X, Xi ids). Note that h satisfies the equation (3.2) which is equivalent to the initial equation (3.1). Thus, we come to the conclusion that (3.1) has a unique solution h

3.3. Lemma. Let A

E

",(X, Xi

g) and

~(X, Xi ids).

E

IIAII

<

that for every mapping fI E ~(X, Xi g) Lip(!p) < EO, the .functional equation (J

+

has a uniquely determined solution

homeomorphism and [(I + hrl - I] • Define mapping hi

EO E

Further, letting

h E

E

EO

> 0 such

satisfying the Lipschil1. condition with

= (A +

h) • A

1. There exists a number

!p) •

(J

+

h)

~(X, X; ids). Moreover, (J

+ h) is a

~(X, Xi ids) .

by fonnula (3.5). Letting

1/1

== 0 and applying Lemma 3.2, we find a

~(X, Xi ids) such that

!p

= 0 and replacing 1/1 by

!p,

we get by the same lemma that there is a

277 mapping

hz

c!(X, X; idB) satisfying the equation

E

(/ +

= (A + ~) • (/ + hz).

hz) • A

From the last two equalities, we obtain

(/ + (/ + Since

both

hz) • (/

+

(/

+ hz)

hi) • (/

+

+

hi) • (A

hi) • (/

+

• A ~)

hz) - /

=A

• (/

+

hi) • (/

= (A + ~) • (/ + and

(/

+

+

hz),

hz) • (/

hz) • (/

+

+

hi).

hi) - /

belong

to

c!(X, X; idB) , we derive from Lemma 3.2 that

Hence, (/

+ hi)

+ hz)

as well as (/

remains to note that h

= hz

are homeomorphisms and (/

+ hz)-I = / + hi.

It

is the required solution.

3.4. Lemma. Let p: (X, T, 1() -+ (B, T, p) be an exponentially contracting linear extension. Let p: (X, T, ~) -+ (B, T, p) be an arbitrary extension such that for every

t E T the transfo11TlOJion ~t:X -+ X, where ~t(x) = ~t(x) - 1(t ex) ex E Xl, is bounded and satisfies the Lipschitz condition. There exist numbers EO > 0 and to > 0 such that if h

E

Lip(~t') <

EO,

then

one

can

jind

a uniquely determined mapping

c!(X, X; idB) satisfYing the equation

(/ + h) Moreover, (/

• 1(t

= ~t

• (/

+ h) is a homeomorphism

• Select to

>

0 so that l11( t oll -

+ h)

and (/

~ <

(t

E

7).

+ hrl

- /

(3.7) E

I, and denote A

c!(X, X; idB) .

=

1(

t o. Define the number

EO

by formula (3.5). According to Lemma 3.3, there exists a uniquely determined element

h E c!(X, X; idB) satisfying (3.6) with ~ = ~to, and (/ + h) is a homeomorphism. The equality (3.6) means that (/ + h) is a fixed point of the transformation

n(v)

= (A. + ~) • v • A-I

family {at: t

E

11

which can also be written as n\v)

= ~to

•

V •

1(-to. The

forms a commutative group. Therefore, Lemma A.30 guarantees that

278 (1

+

h) is a fixed point of all transformations

rl.

Hence, (3.7) holds.

3.S. Theorem. Let p: (X, T, It) ... (B, T, p) be a hyperbolic linear extension. Consider another extension p: (X, T, ~) ... (B, T, p) and assume that for every t E T, the mapping I/:X", X, where I/(x) = ~t(x) - nt(x) (x E X), is bounded and satisfies the Lipschitz condition. There exist numbers EO > 0 and to > 0 such that if t

< EO ( I t I :S to>, then the above extensions are topologically conjugate, i. e. , there exists an idB-isomorphism H: X ... X from (X, p, B) onto itself in the category ilwl. satisfying the equality H 0 nt = ~ t 0 H for all t E T.

Lip(!p)

~

This follows immediately from Lemma 3.4 and Theorem 2.2.

§ 4. Smooth Invariant Subbundles

In this section, we indicate conditions ensuring the existence of smooth invariant subbundles of weakly non-linear smooth extensions. 4.1. Standing assumptions. Let B be a compact smooth manifold, (E, q, B) be a smooth vector bundle, k

ill:

1, and if, p) be a ~ vector bundle automorphism of (E, q, B).

Further, let (g, p) be a ~ automorphism of (E, q, B) (in the category :Bun) sufficiently close to if, p) in the uniform

C

topology.

We shall assume that E is decomposed as the Whitney sum of two f-invariant vector subbundles

Jf and

yO. Choose a Riemannian metric on (E, q, B) such that

Jf .L

yO. Denote

In what follows, we assume that

sup ~b'l':

<

1

(s

= 0,

1, ... , k),

Clb:S 'l'b

(b

According to Theorem 1.5, the conditions (4.1) for s

= 0,

E

B).

bEB

1 guarantee that if

(4.1)

279

<

sup Lip (Cg - j) IE b )

sup IIg(v) - j(V)1I < "',

jJ.,

bes

where

jJ.

>

bes

0 is sufficiently small,

then there exists a uniquely determined continuous

bounded ids-morphism cr;: If ... fl (in the category ~UI1.) such that Xg ;;; graph (cr;) is a g-invariant subbundle. Henceforth, we shall assume that the support of the function g - / is compact, i.e., there is a number L > 0 such that g(v) = ft.y) for all IIvll l! L.

4.2. Theorem. Assume that the above conditions are fulfilled. Then there exists a > 0 such that if

number jJ.

IITg(v) - 1J{V)1I :s jJ.llvll then the subset Xg is

c:c smooth.

(ve TE),

(4.2)

Whenever g tends to go in the space

from E into E endowed with the uniform

c:c topology, then

c:c mappings Xgo in the c:c

0/ all

Xg tends to

topology .

• According to the proof of Theorem 1.5, o 1:\0 ... 1:\, 0 contraction g,: where g~(cr) p~: E ... I:~

If

= I:\(lf,

and

= p~

p~: E ...

D

Y'

g

0

(id

+

cr)

correspond

D

cr; e I:~

[p~

D

g

D

is

(id

+

the fixed point of

the

cr)r\,

to the decomposition

E =

If

Y) denotes the space of all continuous bounded morphisms

fl, and cr: If ... fl

&I

which satisfy the condition sup {Lip (crl~): b e B} :s 1. At first glance, it seems natural to reduce the question on smoothness of Xg to that

If ... fl. Unfortunately, a priori it is not known whether or not subbundles If and fl are c:c smooth. Therefore, we shall proceed as follows.

of cr::

c:c vector subbundles of

denote certain respectively. Let

1:\

=

g,(cr)

Define g,:

1:\ (X, 1').

= P2

D

g

(E, q, B)

D

(id

+

cr)

1:\ ...

D

[PI

sufficiently 1:\ by D

g

0

(id

+

C'

the vector Let X and Y

close to

If

and

fl,

cr)r l ,

where p\: E ... X and P2: E ... Y are projectors which correspond to the decomposition

280 E

= X.

Y. Assuming X and Y to be chosen close enough to}(l and

JP, respectively, we can

affmn that g, is well-defined and contracts the space II' Let tI'g E II denote the fixed point of the operator g,: II ... II, then Xg = graph [tI'g] Ii {(x, tl'ix»: x E X} • (id + tl'g)(X). Now let us prove, firstly, that tI'.: X ... Y is a that if g ... go in the space of all

c!

c!

smooth morphism and, secondly,

smooth p-morphisms endowed with the uniform

topology then tI'g ... tI'go with respect to the

c!

c!

topology.

Given a smooth mapping tI': X ... Y, let Ttl': TX ... TY denote the tangent mapping. Consider the vector bundle L(TX, TY; tI') with L(T,){, Tcr(x)y) as fiber over the point x

E

X. The mapping Ttl': TX ... TY allows us to define a section Ttl': X ... L(TX, TY; tI')

by the formula TtI'(x)

= TtI'(x).

Usually,

Ttl': TX ... TY

will be identified with

Clearly, tI' is of class c! iff Ttl' is a c!-I section. The proof will be carried out at first for k = 1. Denote by 8 the set of all pairs (tI', ~), where tI' E II and (~, tI') is a vector bundle morphism from TX into TY (i.e., ~: T,){ ... Tcr(.lC')Y is linear for all x E X) satisfying II~II Ii sup { II~I T,){II: x E X} s 1. Further, define t g: 8 ... 8 by Ttl': X ... L(TX, TY; tI').

(4.3) where

Fcr(~)

= Tp'}.

• Tg • (id

+ ~) •

[Tpl • Tg • (id

+ ~)rl.

(4.4)

Let us explain the meaning of id +~. Since E = X. Y, we have (TE, Tq, TB) = (TX, Tq, TB) • (TY, Tq, TB), hence id: TX ... TX and ~: TX ... TY should be considered as vector bundle idTs-morphisms. In particular, assuming tI': X ... Y to be a ids-morphism and letting I; = Ttl', we get Fcr(t;)

= Tp'}.

• Tg • (id

+

Ttl') • [Tpl • Tg • (id

+

TtI')r l

C

smooth

= 1Ig,(tI')].

This motivates our choice of formulas (4.3) and (4.4). Now, let us check that the mapping tg is well-defined whenever the condition (4.2), with a small enough J! > 0, is satisfied. Firstly, prove that the operator Tpl • Tg • (id + 1;) is invertible for II~II s 1. Clearly, (Tpl • Tg • (id

+

~), PI • g •

(id

+

tI'»

281

is a vector bundle morphism from

(TX,

T x,

into itself.

X)

Theorem I.S, we have established that p~. g

0

In the course of proof of

+ 0'<): X' ... f is invertible for to X' and f, respectively, we conclude (id

u.,.oU s 1. Assuming that X and Yare close enough that the mapping PI • g • (id + 0'): X -+ X is invertible for aU 0' suffices to show that the linear operator

+

7jJ1 • Tg • (id

E :1: 1,

Therefore, it

xX: TxX ... TozJ(,

(4.S)

~) I T

where z = PI 0 g • (id + 0') (x) , is invertible for all x vector subbundle complementary to vx {v E TX: Tq(v)

=

E

X. Let HX be a certain c!-I

= O}

in TX. As it is stated in

subsection A.IS, there exists a c!-I vector bundle isomorphism

and the following equalities hold:

iHX

(recall that

•

1P1 •

(7JlX) • ij,k(u, v, w)

= (l(u),

Tp(v), PI • CIO(u)v

+ PI

• DJ(u)w)

= (,f{u),

Tp(v), PI • CIO(u)v

+ PI

• Jt:w»

(4.6)

if, p) is a vector bundle morphism). If the vector subbundles X and Yare

sufficiently close to

X' and t,

is invertible for every is also invertible.

respectively, then the mapping

b E B, u

Because

E

Xb , therefore the map

1PI· 7J{u): TuX'" Tpl.,(ur

the set of all invertible linear operators is open, and

is a compact space, there is a number £ > 0 such that whenever uP11 fu < £, UP21 X'u 6 E B, u E Xb and L: T~ " Xb -+ Tp(b.,B "Xp(b) is a linear operator satisfying UL(u,

w) - (Tp(v), PI • CIO(u)v + PI • Jt:w»U <

<

B £,

£

(4.7)

282

then L is invertible. Let J.L < c/2. By (4.2), 117jJ. • Tg(v) - 7jJ. • 7J{v) II :s J.L

For

II~II:S

1 and v

E

TX,

117jJ. • Tg • (id

IIvll:s

+ ~)(v)

(v

E

TE, II vII :s 1).

I, we therefore have

- 7jJ. •

Tf. (id +

~)(v) II :s

2J.L < c.

(4.S)

From (4.7) and (4.S) we deduce that the operator (4.5) is invertible, as it was asserted. Thus, the mapping ~. is well-defined. Taking into account (4.1), select c > 0 so that sup

(fib

+ 4c)

('1b

+ 2c) <

(4.9)

1.

bES

Using (4.1), (4.6) and (4.S), we conclude that if J.L > 0 is small enough then the spectral radius of the operator (4.5) is smaller than '1b + c for all b E B, x e Xb and (cr,~) e 8. Choosing an appropriate Riemannian metric on (TX, 't x , Xl, we can therefore assume that (4.10) Again, assuming X and Y to be close enough to sufficiently small, we deduce from (4.2) and 117p•• Tg(W)1I :s cllWil

Jtl and Y', respectively, and J.L > 0 to be

Jt.Y1 = jP (W E

the inequality (4.11)

TY).

Now, let Z = Z(Y) be the zero section of (TY, 'ty, y), and HE = {(u, v): U E HX, v e Z, Tq(u) = Tq(v)}. Note that HE can be identified with a vector subbundle of (TE, Tq, TB) iii (TX, Tq, TB) e (TY, Tq, TB). Since 'tE. Tq = q • 'tE' we get that (HE,

'tE'

(TE,

'tE'

tHE:

(TE,

is. ~-. vector subbundle of (TE, E) = (HE, 'tE' E) e (VE, 'tE, E) (see E)

'tE'

E) -+ (E e TB e E,

pr.,

'tE'

E).

It is easy to verify that

subsection

A.lS).

Therefore

E) is a ~-. vector bundle isomorphism. It can

also be regarded as a ~-. ids-isomorphism tHE: (TE, q • 'tE' B) -+ E e TB e E = X .. Y. TB eX .. Y in the category :Bun.. Let P denote the mapping from TE into ,Y defined by

283

As it was noted above, (id + ~): TX ~ TE is a vector bundle Consequently, it can also be considered as an idB-morphism (id + X III Y til TB til X Ell Y in the category !lun.. Moreover,

Thus,

t;: TX ~

of

instead

II~II == sup {II~I T,xXII: X e X} :s

i~B-morphism. ~):

X

til

TB

1\1

X~

TY we get the mapping ~ == P ~: TX ~ Y with 1. So, we have defined ~: X Ell TB 1\1 X ~ Y. Accordingly, 0

(4.4) becomes -

-

F... (t;) = P

where 'P...,(: X

Ell

TB

1\1

X

~

X

1\1

0

Tg

0

Y Ell TB

'P... ,(

0

X

Gl

Ell

[7PI

0

Tg

0

'P...

,d-I '

Y is given by

Next we shall verify the estimate

(4.12)

Identifying TE with X

&I

liP

Y &I TB 0

&I

X

Ell

Y via iHE , we get

7]{XI' YI' W,

X2'

whenever X and Yare sufficiently close to

yJ - j(yJII <

Jf and ]P,

C

IY2"

respectively. Hence, we may assume

that

Letting IJ. > 0 to be small enough and recalling (4.2), we infer (4.12). Now we shall prove that Ig(O',~) e 8 for all (O',~) e 8. It suffices to verify the ,inequality IIF...(~)II:S 1 (other properties are immediate). By (4.10) and (4.12), we get IIF ...(~)II = sup {liP

0

Tg

0

'P...,(

0

[1PI

0

Tg

0

'P...

,d-1(V)II:

ve TX, IIvll :s I}

284

:S

sup {~b

+ :S

+

3£) II~II 1I[1P, • Tg • 91cr,l;r'(V)1I

£ 1I[1P, • Tg • 91cr,t;1(V)II: IIvlI

sup {(~b

+

+

3£) (rb

2£)

+

E

TX, IIvll ~ 1, b

£(rb

+

2£): b

E

=q

0

'tx(v)}

B}.

Thus, lIJ'cr(~)1I < I, by virtue of (4.9). Our next step is to prove the inequality liP :S

~b

«cr,

Tg(w) - J'cr(~)

0

1)

+

4£)IIP[91cr,,,

e,

E

0

0

1P, •

Tg(w)1I

1P,(w) - w]lI

+ cr)(x) ,

w E :lEI (id

(4.13)

b

=q

0

'tE(w».

Observe first that the right-hand side of (4.13) is meaningful because

where w = (X" cr(x,) , w, ~, yi). Let us adduce several auxiliary statements which will be used in the proof of (4.13).

(b) (Fcr(~)' q 0 Pl 0 g,(O"» is a vector bundle morphism from (TX, 'tx, X) into (Y, q, 8). In fact, (Fcr(1) , g,(cr» is a vector bundle morphism from (TX, 'tx, X) into (:lE, 'tE, E) and (P, q 0 pi) is a vector bundle morphism from (TE, 'tE, E) into (Y, q, B). (c) 'tx

0

1P,

0

Tg

0

91cr,,,

0

1)J,(w)

= 'tx 1P, 0

0

Tg(w)

(III E TEl (id

+

cr)(X».

285

Recall fIrst that TX 0 '[pI == PI 0 T E , TE 0 Tg == g • (id + cr) • PI I (id + .,-)(X) = id. Therefore,

==

TX

TPI

0

0

(W

Tg(w)

E

TE' TE

0

'P"'.TI =

(id

+ cr)

0

TX

m'1 (id + cr)(X».

Now let us pass to the proof of (4.13). Using (b), (e) and the equality P

0

Tg

Fer(~)

'P"',TI =

0

Tpl • Tg

0

'P"'.TI'

0

we get

:S

+ :S

+

liP

0

Tg(w) - Fcr(~)

liP

0

Tg(w) - P • Tg • 'P.,..TI

liP liP

Tg

0

0

0

'Per •TI

0

Tpl

Tg(w)1I

0

TpI(w)1I

0

Tpl(w) - Fer(~)

0

0

Tpl

Tg(w)1I

0

Tg[w - 'P"'.TI • TpI(w)]11

IIFer(~)

Tpl • Tg

0

0

'P.,..TI

Tpl(w) - Fer(~)

0

TpI • Tg(w)1I

0

:S

liP

0

Tg[w - 'Per,TI

0

TpI(w)]11

+

IIF.,.(ii)1I 117p1

:S

liP

0

Tg[w - 'Per •TI

0

7p1(w)]11

+

IITpI

0

Tg['P"',TI

0

Tg['Per,TI 0

0

TPI(w) - w]1I

TPI(w) - w]lI.

Further, by (a), (4.11) and (4.12), we have liP :S

0

(~b

Tg(w) - Fer(~)

+

0

Tpl

3c) IIp[w - 'Per•TI

0

Tg(w)1I

TpI(w)]11

0

+

= (~b

+

3c) IIp[w - !Per,TI

0

7p1(w)]11

= (~b

+

4c) IIp[w -

0

TpI(W)] II.

!P.,.,TI

Thus, the proof of (4.13) is fmished.

+

C c

IIw - 'Per,TI

0

IIp[w - !Per,TI

TpI(w)1I 0

TpI(w)]11

and

286 Now let us show that sup {Lip(F.,.): cr

E

II} '" sup

To this end, let (cr, ~,)

Then Tpl

0

(i

B

E

{(tJb + 41:)

=

1, 2), v

+

(orb

E

TX,

v

E

21:): b b

=q

E

0

B}

< 1.

(4.14)

'rE(v) , and

Tg(w) = v and

= sup {liP

0

Observe that w

Tg(w) - F.,.(~z> E

TEl (id

= (id

+

Tpl

0

0

Tg(w):

TX,

IIvll:S

I}.

cr)(X). In fact,

+

cr)

0

'tx

0

[Tpl

0

Tg

0

rp.,..~/(V) e (id

+

Therefore, we can apply (4.13) with 11 = ~2' Then using the equality Tpl

cr)(X). 0

rp""~1 = idTx ,

we get

To finish the proof of (4.14), we only need to employ (4.9) and (4.10). Now apply Theorem A.25 with X =:EI> E = B, and j(cr, ~) = (g,,(cr), F.,.(~». Because (4.14) holds and crg is the globally attracting fixed point of gil: :EI -+ :E I , we conclude

287 that there exists an element ~ II ~g such that (tJ'g, ~g) is the globally attracting fixed point of the transformation f. Consequently, the element (tJ'g' ~g) is fixed and attracting under t g: e -+ e. Of course, here ~g: TX. -+ TY is defined by (4.15) Let tJ'0: X -+ E denote the zero section of (E, PI' X),

then (tJ'0, TtJ'O>

sequence {t~(tJ'o, TtJ'o)} converges to the element (tJ'g' ~g)

E

E

e, and the

e. On the other hand,

Since {g~(tJ'O>} converges to tJ'g uniformly on compact sets and {Tg~(tJ'o)} -+ ~g in the same sense, we see that TtJ'g exists and, moreover, TtJ'g = ~g. Consequently, (id + tJ'g) is a C section of (E, PI' X), therefore Xg = (id + tJ'g) (X) is a Cl submanifold of E. The second statement of our theorem for k = 1 follows immediately from the fact that the fixed point of a contracting operator depends continuously on the operator (see Lemma A.22). The proof for k it 2 will be by induction. Suppose that the assertion is valid for k = m. Let us prove that it is true for k = m + 1. Select £ > 0 so that sup {(~b

+

4£) (rb

+

2£t': b

E

< 1

B}

(s

= 0,

1, ... , m

+ 1).

(4.16)

en

By the induction hypothesis, tJ' g: X -+ Y is a smooth ids-morphism. The tangent mapping TtJ'g: TX. -+ TY coincides with ~g. Let ~B' = P 0 Tg. From (4.15) we see that

w, xz}

TtJ'B'(x I ,

= (tJ'iXI) , W,

~ixI'

W,

xz})'

By If we shall denote the naturally defined vector bundle with base X and L(TxX, Yq(X~ as fiber over x E X. There exists a canonical one-to-one correspondence between sections of

and vector bundle morphisms (~, q): (TX.,

If

given (~, q), define a section ~I: X -+ let us introduce a mapping gO: EO -+ set gO(71)

= Il

E

L(Tz){, Il

Ypoq(X» ,

=P

0

Tg

E'

E'

by ~I(X)(V)

TX '

= ~(v)

as follows: for x

E

X) -+ (Y, q, B).

(x E X, V E TxXJ. Now

X and

where 0

'P."

0

[1PI

0

Tg

0

Namely,

'P."rll Tz){,

71 E

L(TxX,

Yq(X»

288 Z = PI

g

0

+

(id

0

II"g) (x) ,

and "''\'I: TxX -+

"''\'I(X, w, u) = (x, II"g(X) , w, u, :;j(x, w, u»

Thus, g*: E* -+ E* covers the mapping section Sg

0

PI

S g: 0

mappings

g p*

X -+ E* defined by 0

(id

+

ITg)(x)

S g(x)

= g*[Sg(x»

and g* are also of class

p*

= P (x

E

is of the form

T(id+cr g)(x.fi

Ii

0

PI

«x, w, u) 0

g

0

(id

Til"g ITxX (x

+ E

E

X

II"g):

TB

Ell

(f)

X).

X -+ X. Note that the

is g*-invariant, that is,

X)

X). Since ITg is a morphism of class

en, the

en. It follows from (4.10) that

Looking through the proof of (4.14), we see that

Consequently, by (4.16) we get ~s ==

sup { (<<x*)s /3 x* : x

E

X} < 1

(S = 0, 1, ... , m).

en

According to Theorem A.33, the section Sg: X -+ E· is smooth. However, this theorem cannot be directly applied because the space X is not compact. To get over this difficulty, we compactify X by attaching, to each fiber Xb , the projective space PXb , and then extend the mapping p.: X -+ X in a standard way bearing in mind that the nonlinear mapping g: E -+ E agrees with the linear morphism f. E -+ E on the set {v E E: IIvll 2: L}. Since TITg(x, w, u) = (IT.ex), w, Sg(x)(x, w, u»

we conclude that TlT g : TX -+ TY is

«x, w, u)

E

X

en smooth. Hence, ITg: X -+ Y

(B

TB

(B

X),

is of class

en+ l •

Bibliographical Notes and Remarks to Chapter IV

Theorems 1.5 and 4.2 are taken from the preprint by Bronstein and Burdaev [l}; Related results (but with less detailed proofs) are presented in Hirsch, Pugh and Shub [1] and Fenichel [1-3]. Theorem 2.2 is obtained by Bronstein [5] (see also Pliss [1],

289 Rejnfeld [2,3], Shoshitaishvili [1]). Theorem 3.5 is due to Bronstein and Glavan [1]. Other modifications of the Grobman-Hartman theorem can be found in the papers by Kurata [1], Palmer [1], Rejnfeld [1, 4-7].

290

CHAPTER V INVARIANT MANIFOLDS

§ 1. Persistence of Invariant Manifolds

It is known that an exponentially stable smooth submanifold, subject to perturbations

of the vector field, can undergo various changes; for example, it can lose smoothness or tum into a topological manifold or even into a strange attractor. Let us present (or mention) a few examples which give a general idea of the wide spectrum of possibilities. 1.1. Example. (Hale [l]). Consider a system of differential equations in the plane which has an exponentially stable invariant set S diffeomorphic to the circle. Assume

that there exist exactly two rest points on S, namely, a stable node P and a saddle point Q. In other words, the manifold S consists of these two points and of the unstable separatrices of Q which tend to P. Perturb slightly the vector field near S. Assume, for simplicity, that this perturbation leaves fixed the points P and Q. It is well known that the separatrices of a saddle point represent a smooth curve, therefore the image S of the circle S is smooth at all points except, possibly, the point P. Let ~T denote the eigenvalue of the linear part of the given vector field at the singular point P which corresponds to the eigenspace tangent to S. Let ~N be the second eigenvalue. It is easy to see that whenever I~N I > I~T I, then S is smooth (Figure 1.1). But if I~N I < I~T I, then P can become a comer point of the perturbed invariant set S (Figure 1.2). 1.2. Example. (Sacker [2]). Let us show that even in the case where I~N I = I~T I, the invariant manifold, after perturbation, can fail to be smooth. To this end consider the following system of differential equations

x=

sinx - "y,

y=

- " sinx - y,

(1.1)

where " is a small parameter and x is a cyclic coordinate (i.e., x E IR / 2nZ). For " = 0, there is an invariant circle y = O. It contains two rest points, namely, P(O, 0)

291

p

Q

Q

Figure 1.1

Figure 1.2

and Q(n:, 0). Clearly, P is a saddle point and Q is a node with eigenvalues ~I = ~2 = 1. If '" > 0, then the point Q is a focus (at least for the linearized system). It is really a focus because the system (1.1) is C linearizable near Q (this easily follows from the results presented in Chapter 11). Thus, for '" > 0, the unstable separatrices of P are spirals tending to Q (Figure 1.3). We see that the invariant set arisen from the exponentially stable circle y = 0 for small '" > 0 fails to be smooth at the point Q, but is still homeomorphic to the circle.

p

Figure 1.3.

292 The extent of pathology is, in fact, much greater than the above examples reveal. As it was shown by Jamik and Kurzweil [1], an exponentially stable two-dimensional invariant manifold, when perturbed, can tum into a set which is not even a topological manifold. Moreover, Kaplan, Mallet-Parret and Yorke [1] have constructed an asymptotically stable torus which bifurcates into a set of non-integer Hausdorff dimension, a strange attractor. Thus, we face the problem of finding conditions for a smooth invariant manifold to persist under perturbations of the vector field. To be more exact, we seek conditions ensuring that the given isolated CC smooth invariant manifold, being perturbed, gives rise to a unique invariant manifold of the same smoothness class. This property will be refered to as

CC

persistence (precise definitions will be given later).

1.3. Notation and def'mitions. Let M be a smooth manifold, T

=R

or T

= Z,

r

~

1,

and (M, T, f) be a dynamical system of class C. Let 1 sis r and A be a compact c! smooth sub manifold of M invariant under (M, T, f). Let (TM, "CM' M) denote the tangent vector bundle and TII.M TM[A] = {v E TM: "CM(V) E A}. Because the tangent bundle (Til., "CII.' A) is embedded in TM[A], one can form the quotient bundle TM[A] / Til. which is

=

called the normal bundle of the submanifold A c M. Note that Til. is a c! smooth vector subbundle of the tangent bundle TM since the transition functions for the vector bundle Th can be obtained from that of TM by restriction to the c! submanifold heM. Hence, the normal bundle Nil. iii! TM[A]/TA can also be provided with the naturally defined

structure of a c! smooth vector bundle. The C dynamical system (M, T, f) induces a C· I smooth linear extension, the tangent linear extension "CM: (TM, T, T/) -+ (M, T, f), where T/(v) = T/(x)(v) (x E M, v E TxM, t E 7). Since the submanifold A is /-invariant, the vector subbundles Til. and TM[A] are invariant under (TM, T, T/). Therefore, (TM, T, T/) induces a dynamical system on Nh, denoted by (Nh, T, Nf), which is a linear extension of (A, T, /lh). The smoothness class of this extension is equal to min {I, r - I}. For each point x E M, let expx: TxM -+ M denote, as usual, the exponential mapping that corresponds to a certain Let I

=1

Riemannian metric on (TM,

"CM'

M).

C vector subbundle of TM[A] complementary to Th. Let rl(N) space of all C sections of the vector bundle (N, "C M' A) provided with

and N be a

denote the Banach

e

293 the

c: norm

Note that each

II' III'

the form A = {exPb(~(b»: b

E

c: smooth submanifold Anear A can be represented in

A} for a certain ~

E

rl(H).

c: submanifold invariant under the cascade generated by some diffeomorphism Difr(M). We shall say that A is c: persistent provided there exist a neighbourhood

Let A be /

E

U of A in M and a neighbourhood 'l.L(f) in Difr(M) such that for every mapping g there is a unique section 11

Ag

= "6 E rl(H) "

E

'l.L(f)

satisfying

n g"(ll) = {exPb(1I(b»: b

E

A},

"EZ

and, moreover, the mapping from 'l1(f) into rl(H) defined by g t-+ 1Ig is continuous. Note that Ag is the maximal g-invariant subset contained in U. It is assumed that A.f' = A, hence, A is supposed to be isolated. Now, let 12:2 and N be a

c: smooth vector subbundle of TM[A] complementary to TA. c!

Let rl(H) denote the Banach space of all endowed with the The

c!

norm

c! sub manifold

integer k, k

=

A

sections of the vector bundle (N,

TM'

A)

II' Ill'

is said to be

c! persistent if it is C persistent and for each

I, .,,' I, the mapping g

t-+ 1Ig

carries the set 'l.L(f) n Difrc(M) into

rc(H) and is continuous in the ~ topology.

The notion of formulations.

c!

persistence for flows can be defined similarly, but we omit precise

f 2. Normal Hyperbolicity and Persistence In this section, sufficient conditions for persistence of a submanifold are given.

smooth invariant

2.1. DermitioDS and notation. A smooth invariant submanifold A is said to be normally k-hyperbolic if the normal linear extension is k-hyperbolic (see Definition

m.6.1), i.e., there exist 1Vf-invariant vector subbundles JII and Jt4 of NA and positive numbers c and ~ such that JII. ~ = NA and

294

(2.1)

(t ~ 0, b e A; m = 0, 1, ... , k).

Let p: TM[A] .... NA be the canonical projection. Denote E: = p'I(~), Clearly, TA = f:S n

r r

El

TA

El

= p'l(i't').

By Lemma A.28 and (2.1), there exist Tf-invariant vector

r. TA,

and Jtl of 1M[A] such that ~ =

subbundles 1M[A] =

E'.

E'

E'

=

Jtl •

TA, consequently,

Jtl. Moreover, (2.1) can be rewritten as

(2.2)

~

(t

0, b e A; m

=

0, 1, ... , k).

Evidently, the just formulated condition is equivalent to k-hyperbolicity of A. Because A is compact, the conditions (2.1) and (2.2) do not depend on the choice of Riemannian metrics. Observe also that the submanifold A is normally k-hyperbolic under the flow (M, IR, 1) iff it has this property with regard to (M, I, 1). Therefore we shall confine ourselves by considering only the case of cascades. Denote (see 1lI.2.1)

rl(b)

= rl(7Jl TA,

b),

w(b)

= w(Tjl TA,

b).

The inequalities (2.2) become rl5(b)

Let

E

<

m w(b),

wl.l.(b)

be a positive number satisfying

>

m rl(b)

(b e A;

m = 0, I, ... , k).

295

< m w(b) - 4e,

cS(b)

>

wU(b)

m c(b)

+

4e

(b

E

= 0,

A; m

1, ... , k)

(such numbers exist by Lemma III.2.3). Further, there exists a number satisfying

c; I exp[w(b) - e]t

:S

II

if ITAli

Fix some Riemannian metric on (TM,

:S

c£ exp[c(b)

T M'

mutually orthogonal and approximate it by a

+

e]t

(b

t

E A,

M) such that the subbundles

e

I:

XS,

c£

>

0

0) TA and

Jtl are

Let d denote the

Riemannian metric.

metric on M that corresponds to the latter Riemannian metric. Let U be a fixed small enough neighbourhood of A in M. Denote

W'"(f) WU(f) Given x

E

U, let

tends to 0 as n -+

=

{x

E

= {x

U: .f(x) -+ A

E U: .f(x) -+

A

(n -+

+ GO)},

(n -+ - GO)}.

~(f) denote the set of all such points Y

+

E

GO faster than d(/,(x), .f(z», whichever Z

U E

that d(/,(x), .f(y» A, Z

'It

x, be chosen.

As it follows from the proof of the theorem below, ~(f)

= (y

E

U: d(/,(x), .f(y» exp[-cs(x) - e]n -+ 0

(n -+

+ GO)},

Similarly, we define the set w,:,(f):

w,:,(f)

= (y E

U: dif -"(x),

f -n(y»

exp[wu(x) - e]n -+ 0

(n -+

+ GO)}.

Note that by virtue of uniform integral continuity of the dynamical system on the compact subset U c M, we have

for each positive integer

v. Replacing, if necessary, the mapping

f by some iterate,r,

296 we may assume that

SUD n7J1X:n n7]'·tIT.f(b),\lIm:s 1/3

(m

bEh

= 0,

(2.3)

1, ... , k),

(2.4) Because we are investigating the behaviour of the dynamical system only near A, we shall assume, without loss of generality, that the manifold M is compact.

2.2. Theorem. Let M be a smooth manifold, T

= IR

or T

= z,

k

~

1, and (M, T,fJ be

a dynamical system of class ~. Let A be a C smooth submanifold of M invariant under (M, T, fJ and satiqying the condition of normal k-hyperbolicity. Then: (a) The sets A, W(f) and W'(f) are (b)

W(f) =

U W:(f), bEA

(c) The manifold A is

d'-

submanifolds;

W'(f) =

U ~(f), bEA

~ persistent, and if (M, T, g) tends to (M, T, go> in the ~

topology, then W(g) -+ W(go,) and W'(g) -+ W'(go,) in the same sense; (d) Near A, (M, T, fJ is topologically conjugate to (~ •

It', T, N/).

• At first sight, the most natural way of proving this theorem should be as follows. First, by using a tubular neighbourhood, the given dynamical system should be carried from a neighbourhood of A into a neighbourhood of the zero section of the normal bundle N- NA. Then one should apply Theorems IV.4.2, IV.2.2 and IV.3.S, and the proof would be complete. Unfortunately, the dynamical system induced on N is not, in general, an extension of the system defined on A, so these theorems are not directly applicable. Therefore, we are forced to proceed in a roundabout way, namely, instead of dealing with the neighbourhood of A, we must consider the neighbourhood of the diagonal tJ.(A) • {(x, x): x E A} in A x M. The proof will be carried out for cascades. Given r > 0, define U(r) = {(x, y): X" A, Y EM, d(x, y) < r}, TAM(r) = {v E TAM: nvn < r}. Clearly, U(r) is a neighbourhood of tJ.(A) in A x M. There exists a number Co > 0 such that the mapping

297

Exp: TAM(cO> -+ U(cO> defined by Exp(v) = (X, expxv) (x E A, v E TxM) is a C diffeomorphism. In general, the mapping Exp is as smooth as the submanifold A. Denote fa = f x /I A x M. Choose a neighbourhood V of lI(A) in A x M so that if x .f)(V) c U (recall the equality ft.A) = A). There exists a number C1' 0 < C1 < Co, such that the formula 70 70: TAM(c 1) -+ TAM(cO>.

= EXp-l

• if x.f)

0

Exp is meaningful and defines a mapping

Let g: M -+ M be a diffeomorphism close enough to f (in the

C topology) so that the

mapping go: TAM(cl) -+ TAM(co> , go = EXp-l • if x g) • Exp, is well-defined. Note that = w, where W is determined from the relation exp.f(lC)w = g(expxv) (here x = '1:M(V) and, consequently, J(x) = '1:M(V». Thus, '1:M 0 g = f· '1: M • go(v)

Let c be a small enough positive number. Choose a C smooth vector bundle isomorphism t: TAM -+ TAM so that l17J{x)(v) - t(V)1I < (c/lO)IIVIl (x E A, v E TxM). Assume IIgo(Ox)1I < c and IIDvgo(Ox)(v) - 7J{V)1I :5 (c/lO)IIVIl (x E A, v E TxM) (this can be achived by choosing g to be sufficiently C near to /). Define a mapping ~: TAM(cl) -+ TAM by ~(v) = go(v) - t(v). We have IIDv~(Ox)1I < cIS (x E A), hence there exists a number cr,

o < cr <

C\l such that IIDv~(W)1I < c/4 for IIwll with o:(t) = 1 for It I :5 112, 0:(1) = 0 for It I Let _

~(v)

=

{ a.(lIvll/cr)~(v)

o

< cr. Choose a ~

e

function 0:: R -+ IR 1 and IDo:(t) I < 3 for all t E R.

for IIv" < cr, for IIvll

~

cr.

Then f(V) = ~(v) whenever IIVII < cr12, and II~(V)II < c for all v E TAM. Set G(v) = t(v) + ~(v) (v E TAM). If II vII < cr12, then G(v) = go(v)' If IIvll > cr, then

= t(v).

Thus, G: TAM -+ TAM is a C diffeomorphism which agrees with go in TAM(cr/2) , and IIDvG(v) - til < c for all v E TAM. Let us emphasize that '1:M· G = f· ('1:M ITAM). / If the number c > 0 is sufficiently small, then there exists at-invariant G(v)

splitting TAM

= E" • ff .~,

where E", ff and ~ are vector bundles close to~, TA

and Jt', respectively, and the following inequalities hold:

(2.5)

29B

(2.6) For k

>

1, the above construction of the mapping G needs the following modification.

Let A be some

c!

smooth submanifold close enough to A in the

A is the image of a certain Using a

c!

c!

mapping j: A -+ M

d

a

c!

close to the embedding i: A c M).

smooth tubular neighbourhood of A, construct a

close to the identity mapping and satisfying J(f(A» diffeomorphism

c!

close to f, and ](A)

= A.

= A.

diffeomorphism. If .:

>

0

(f

go,

t

Exp. Instead of Choose a vector bundle morphism 0

X

g)

0

which approximates tj', then derive that G is a

c!

0 is small enough and the sub manifold A is sufficiently

C

t: TM[A) -+ TM[A) of class

c!

c! diffeomorphism J: M -+ M Set 1 = J f. Clearly, 1 is

Now repeat the construction of

go = EXp-l

and G with the following alterations. Set Tf. TAM -+ TAM consider tj': TM[A) -+ TM[A).

C topology (more exactly,

close to A, then TM[A) = ~ e It' e E' satisfies the previous conditions (of course, A should be replaced by A in the formulas (2.5) and (2.6». Now apply Theorem IV.4.2, letting E

= TM[A),

X

= It' GIl E',

Y = ~,f= t, g

= G. As

c! smooth G-invariant sub manifold Vt'(G). Similarly, X = ~ GIl Jt=, Y = It", f = t- l , g = (II and using once again Theorem IV.4.2, we a result, we obtain a

G-invariant sub manifold W(G) of class

c!.

Because W(G) is tangent to ~ e

taking find a

It', and

W'(G) is tangent to It" GIl Jt= at all points bEll., we conclude that W(G) and Vt'(G) cross transversely. Taking into account that G(v) = go(v) for II vII < (1'/2, we deduce that W(G) n Vt'(G) is a locally go-invariant sub manifold of class Exp [W(G) n W"'(G») is locally invariant under n W"'(G»)

= f, = A.

g

IE

Ag

is a g-invariant

we get that A Ii A" is a

c!

c!

1 x g,

hence the set pr2

c!. 0

The set

Exp [WS'(G)

smooth sub manifold of M. In particular, whenever

submanifold. Therefore, in what follows, we assume A

Now apply the Decomposition Theorem IV.2.2, letting It = t and ~ = F. According to this theorem, there exists a topological isomorphism of cascades h: (TAM, F) -+ (Xl' F l ) GIl (~, F~ GIl (X3 , F3), where Xl' Xl and X3 are some subbundles of (TAM, TM' A) in the category ~un.. Moreover, Xl' ~ and X3 are F-invariant and close (in the Lipschitz sense) to ~, Jt= and

It", respectively. There exists a number de > 0 such that

299

= {v E

X3

X2

TAM: IIF -n(V)1I :s de IIvll exp[- wl.l.(b)

= {v E

+

2c]n, b

= 'tMcr(V» ,

TAM: d;1 IIvll exp[w(b) - 2c]n :s IIF'(V)II :s de IIvll exp[n(b)

n

+

it

O}

2c]n,

According to the choice of c > 0, we have nS(b)

+

2c

<

+

web) - 2c, - wl.l.(b)

2c

< - neb)

- 2c

(b

E

(2.7)

A).

Hence it follows

Denote N = h'I(XI • X3 ), Nx = N n TxM (x E A). Let us show that U [exp Nx(cO): x E A] contains a neighbourhood of A in M. When proving this purely topological in nature assertion, we may assume that. = Nf, E' = N', If = Til., E' = It". The subbundles XI and ~ are close to

N' and It", respectively, in the Lipschitz sense. The needed result

follows from this by the Invariance of Domain Theorem. Thus, every point y of a small enough neighbourhood of A in M can be represented as y = expxv, where v is some element in Nx . Taking into account that h is an isomorphism of cascades and that Exp maps the cascade (TAM, F) onto (A x M, f x!J isomorphically, we conclude that every point of S(A, co) \ A leaves the neighbourhood S(A, co) as n .... + 00 (n .... - 00) at an exponential rate'" exp[wl.l.(b) - 2c]n (respectively, ,., exp[- nS(b) - 2c]n). At the same time, for v E X2 , we have IIF'(V)II :s de IIvll exp[n(b) + 2c]n, IIF -n(V)1I :s de IIvll exp[- web) + 2c]n /

,where n = 1, 2, ... , b prove the equality

= 'tM(V).

Therefore (2.7) implies Exp X2(co) c A x A. Now let us

(2.8)

Let

x

E

W(f) n S(A, cO).

Show

x

E

pr2 • Exp • h'I([XI .. ~(cO».

Suppose,

to

300 the

contrary,

v] '" O.

Then

that

0

= prl

the point x,

neighbourhood SeA, (a, b) e Exp

x

£0>,

Exp

0

0

h-I(VI, Vl' V3),

Vt

Xt

E

(i

=

1, 2, 3) and

moving in the positive direction of time,

contradicting the hypothesis

h-I(XI • X:z)

then

x

W(f).

E

there exist elements VI

E

leaves the

Conversely,

Xl and vl

E

X:z

if

such

that "E'M(VI) = "E'M(V:z} and (a, b) E Exp 0 h-l(vil v:z}. Since Exp and h are isomorphisms of the corresponding dynamical systems, the investigation of the behaviour of cr(a), I'(b» can be reduced to that of F'(vl , v:z}. Further, since 1"'(0, v:z}

and 111"'(0, v:z} SeA,

£0>

F'(vil V:z}1I

~ 0

as

PI ~

+

CD,

we see that the point

for all n it 0 and, moreover, dCf(b) , A) ~ 0 as n ..

true. Hence, for each point satisfying

x

E

WS'(f) n SeA,

£0>

there

reb)

+ 011.

exists

a

{OJ

E

X

Xl

remains in

Thus, (2.8) is point

yEA

Thus, we have proved the statements (a) and (b). Besides, we have established that the cascade (M, /) near A is isomorphic to the cascade (XII Fl ) • (Xl' F:z), therefore, locally (M, /) is an extension of the cascade (A, /1 A). Hence, the statement (c) follows from Theorem IV.4.2, and (d) follows from Theorem IV.3.S.

§ 3. Necessary Condition for Persistence

In this section, we show that normal k-hyperbolicity is not only sufficient but also necessary for a smooth invariant manifold to be

c!'

persistent.

3.1. Theorem. Let M be a smooth manifold, kit 1, fE Diff'<+I(M), and A be a compact

f-invariant submanifold of class

c!'.

The manifold A is

C"

persistent iff it is normally

k-hyperbolic . • Assume that A is normally k-hyperbolic. According to Theorem 2.2 (c), A is persistent. Conversely, let A be a

c!'

submanifold satisfying the condition of

~(N) we shall denote the Banach space of

C"

c!'

C"

persistence. By

smooth sections 11": A ~ N of the normal

301

bundle N .. 1M[A]ITA. Fix some (/it smooth tubular neighbourhood h: U -+ N, where U is a neighbourhood of the image Z(A) of the zero section Z: A -+ N. For each g let us define a local diffeomorphism

g by g = h·1

of (/it persistence, the mapping g .....

T/.

g

0

E

Diff't(M),

h. According to the definition

0

from 'U(f) n Dif("(M) into r'"(N) is continuous

= 0,

1, ... , k). Since J E Diff't+I(M), we can construct a (/it diffeomorphism U -+ N which agrees with NJ in some neighbourhood Uo c U of the zero section Z(A)

(m

g:

and such that g Let 1 s

5

hog

m !S k and

h-I

0

E

U(f).

v be an arbitrary element in rm(N). If the number c

sufficiently small, then the mapping

g

II

h


0

c .

tI'

U(f). Then there exists a uniquely determined section

0

se

P

0

E

rm(N)

g)

0

h-I

>

0 is

belongs to

satisfying the

condition


. v

0

p

g)

0

se

0

= se

0

P

0


c . v

0

p

0

g)

0

(3.1)

se'

which means that the image Se(A) of the section Se: A -+ N is invariant under g + c . v 0 p • g and consequently, Ai = {expx(se(x»: x E A}. It follows from (3.1) that g • Se + c . tI' 0 P 0 -g 0 Se = Se 0 P 0 g- • se' hence se - g- 0 se • (p 0 g- • ser I = C • tI'. If the number c > 0 is small enough, we get g 0 se = Nf 0 se' consequently,

p •

g

0

se

= P • NJ.

only one section

T/

se

= c-I

=/lA. . se 11 -

E

Thus, for each section v

E

r'"(N) there exists one and

r'"(N) satisfying the equality

NJ

0

11 • (j1 Arl

= v.

(3.2)

In other words, the operator id - /,: rm(N) -+ r'"(N) is invertible. Given

tI' E

rm(N), one can define the corresponding section j'n(v) of the vector

bundle p"(N) (see Definition A.14) by the formula x ..... J':(v) (x

E

A). Let rb(P"(N»

denote the Banach space of all bounded sections of p"(N) equipped with the norm 1I~lIb

= Sup{II~(X)II:

x

E

A}.

For each element

(E

rb(p"(N»

there

exists a net

{vex: « E A} of elements Vex E rm(N) such that J';(vCX> -+ (x) for all x E A.

Let

{1Jex: « E A} be the net of elements lIex E rm(N) satisfying the relation (id - /,)(1ICX> = Vex' By Tikhonov's theorem, there exists a subnet {1J/3: {3 E B} of the net {lIex: lie E A}

such that

{In.

11/3: {3 E

B} converges pointwise to a certain element (E rb(P"(N»

302

and, moreover, lim J";«(1"~)

= ~(x)

(x

= (id - fi)(-rr~).

(1""

A), where

E

Define

Then

= lim

[;11'1

1)~(x) -

0

~

Thus,

id - 1,11'1)

maps rb(P"(N)

r

onto itself.

0

Nf

0

1)~

0

(flArl(X)]

In other words, the linear extension

(P"(N) , p"(Nf» is weakly regular. By Theorem m.S.7, the dual linear extension has no non-trivial bounded motions. To finish the proof, it suffices (by Theorem m.3.6) to show that P"(Nf) does not have non-trivial bounded motions, too. With this end in view, let us prove the following proposition.

3.2. Lemma. Assume A to be I:

>

c! persistent.

Thenfor each a > 0 one canfind a number

0 and a neighbourhood ti of the diffeomorphism fin Difft(M) such that whenever

1 :s m :s k, ~

E

Nm,m-I(TM[A] I TA), II~II <

inequality 1Ip"(Ni)(~)1I <

I:

1:,

holds, then

= A, and, for some number I, the 1IP"(Ng')(I;)II:S a for all i = 1, ... , I. g

E

ti,

g(A)

~ Suppose, to the contrary, that there exists a number a o > 0 such that every c! neighbourhood of f contains an element g with the following property: for each I: > 0 one can find integers m, s, 1, satisfying

111;11

<

1:,

1:s m

1IP"(Ni)(l;) II

<

:S I:

k, 0

<s <

1, and an element t;

and II P" (NgS') (I;) II

E

Nm,m-I

> a o•

Let ti = ti(ao> be the neighbourhood mentioned in the definition of c! persistence. Pick out a diffeomorphism g E ti having the above property. Let XI E A be the source of the jet ~,and

X2

= i(xl)'

It suffices to consider the case when

g'(xl) '" XI

for

i '" O. Choose a small enough neighbourhood WI of XI in M and denote W2 = g'(WI ). There exist surjective c'< diffeomorphisms ",: W, -+ {y E IRP x IRq: lIyll < 1} satisfying the conditions ",(x,) = 0 and flt(Wt n A) = {y E IRP x {O}: lIyll < 1}. Here p is the dimension of A and p + q = dim M. Moreover, we may assume that T"t ITx,M are isometric

303

= 1,

operators (i Let i\

E

2).

(0, 1), ~

= {y E IR P

parameter i\ is sufficiently small, then

i\},

a ~ function 1/1: IR -+ IR such that 1/1(0) = 1, I/I(t) = 0 for for all t

E

IR. By

It I

it

II/I(t) I =s 1

1, and

t.;: IRP -+ IRq denote the homogeneous polynomial of degree m that

~ E H~;m-I

corresponds to

~ = i(Wi'). If the i = 1, ... , I. Choose

Wi' = "il(B~), Wi' n l(Wi') = ~ for all <

x IRq: lIyll

(or,

more exactly, to its representative in the chart "I)'

Define a mapping F: ~ -+ IRq by (3.3) Straightforward calculations show that the function F together with all its partial derivatives up to the m-th order inclusively tend to zero (uniformly relative to the parameter i\) as Set "

= "z

£ 0

i

-+ O. 0

"jl and define a mapping

H: "z(~ n A) -+ IRq so that 1)q1(graph(F»

= graph(H).

=

Further, define H: "z(~) -+ IRq by H(z\I ~ I/I(lIzll I i\)H(zl)' It is not hard to verify that the function H and all its partial derivatives up to the m-th order inclusively tend to zero as £ -+ 0 uniformly relative to i\. Now define a diffeomorphism ~

~(x)

= "jl

0

(id

+ F)

0

"I(X)

E

= x for x E M \ (Wi' u ~); = "z (id - 11) "z(x) for

Dif("(M) as follows: ~(x)

whenever

x E

Wi';

~(x)

0

0

en

E ~. It follows from the above considerations that ~ -+ 0 in the topology uniformly relative to i\ as I: -+ O. Consequently, if I: is small enough then the diffeomorphism g = E 0 g belongs to the neighbourhood U = U(~J off. According to

x

our construction, l

Ail

= (A

\

UgJ(~ n A» u (U J-O

where D

= f'jl(graph(F».

(3.4)

gJ(D» ,

J-O

It is easily seen from (3.3) that the m-jet of F at (0, 0)

coincides with (0, ... , 0, ~). Since the condition g E U(~J.

"p'"(Ng·)(~)"

~o, the equality (3.4) contradicts

HI,o, H Z•I ,

It follows from this lemma that the sets trivial bounded trajectories of the cascade

>

... ,

r,le-I

contain no non-

crCN) , rCNf). Recalling the first part

of the proof, we conclude that the morphism

rCNf)

is hyperbolic over these sets.

304

Using Theorem 111.5.14 and proceeding by induction, we deduce that

p"(NJ>

is

hyperbolic on p"(N). By Lemma III.6.4, A is normally k-hyperbolic.

§ 4. Asymptotic: Phase

In this section, the notion of asymptotic phase for an exponentially stable invariant submanifold is introduced and conditions ensuring smoothness of the asymptotic phase are given. Besides that, we present proofs of some theorems stated in Chapter I. 4.1. Standing assumptions. Let M be a smooth manifold; T

=R

or T

= Z;

r

I:

I, and

(M, T,/) be a dynamical system of class C. Further, let A be a C smooth compact f-invariant submanifold of M. Recall that 7M[A], TA, and N == 7M[A] I TA can be

provided in a natural way with

1/:

TM[A] -+ TM[A] induces a

M with a

C vector bundle structure. C- I morphism N/: N -+ N

The vector bundle morphism (I

E

1). Equip the manifold

c!'

Riemannian structure. Let d denote the corresponding metric on M. We assume that, for some numbers c > 0, cr. > 0 and (:J > 0, the following inequalities hold: IIN/INxll :s

C

exp(- cr.t)

(x

E

A, t

I:

(4.1)

0),

(4.2) Condition (4.1) means that A is exponentially stable in the positive direction with respect to (M, T, /). Condition (4.2) signifies that the exponential rate of contraction in the normal direction is greater than that along A. By Lemma A.28, it follows from

]f c 7M[A]

(4.2) that there exists an uniquely determined vector subbundle

if-invariant

for all

1

E

which is

T and complementary to TA. According to Theorem 2.2, there

exists a neighbourhood V of A in M which is invariantly fibered by w:. == W(x) (x E 1\) so that YEw:. iff difCy),/(x» :s c£ exp[(- cr.

+

e)l]

(I

I:

0)

305

for each

£

> O. It will be no loss of generality in assuming that I(V)

4.2. Dermition.

Y

Define a mapping

W'"'ex), then set . / = cf IA) • I

E

l(y)

= x.

as follows:

I: V -t A

Since 1(W'"'ex»

C

wcfex» (x

E

A, 1

C

V (I

2:

if

x

A

2:

E

0). and

0), we have

I (I it 0). The mapping I: V -t A is called the asymptotic phase. Note that II A = id. Reversing the direction of time, we get the notion of asymptotic phase for a submanifold exponentially stable in the negative direction.

4.3. Smoothness of the asymptotic phase. Let U be a neighbourhood of the zero section ZeAl of the normal bundle (N, p, A) and h: U -t M be a (partial) tubular neighbourhood

C. Let

of class

h(ll)

I: V -t A be the asymptotic phase. With no loss of generality, assume

Let k::5

= V.

T.

We shall say that

tubular neighbourhood h: U such that

I.

h

= pi U.

-t

V of class

I

C".r

is of class

C".r

whenever there exists a

(see Bourbaki [1, subsection 15.2.4])

Recall that h is of class

C".r

iff for each point lEU there

exist a neighbourhood WI of x == pel) in A, a neighbourhood W2 of w == hell in M, a vector bundle chart (WI,!p, Rq ) for N, a chart (WI' !/I, RP) on A, and a chart (W2' A, IRp + q ) on M such that the mapping H: !/I(WI )

=

)( Rq -t IRp

+q

defmed by H(a, b)

possesses partial derivatives D~D~H for all pairs (u, v) of non-negative integers such that U::5 k, u + V ::5 T. A • h • !p-I (!/I-I (a) , b)

Denote

= cf IA)

l = h-I

• I •

./ •

h: U -t U (I

it

0). Then p •

l =p

h = cf IA) • P (I > 0) and Wex) = I-lex)

Thus, if the asymptotic phase

1

is of class

C".r

2:

./ •

h

and h denotes the corresponding tubular

neighbourhood, then for the semidynamical system {l: I {/: I

= 1 ./ • h = hlp-lex) n ll) = h(Nx n ll). • h-I

2:

O}

O} via h, the sets Wex) coincide with Nx n U ex

E

which is conjugate with A). In orther words, the

mapping h straightens the subsets W'"'ex). Let

II

c TM[A] be some vector subbundle of class

sufficiently close to subbundle

II.

If.

The normal bundle N

E

C

complementary to TA and

TM[A] I TA is

Therefore we shall not distinguish between N and

Assume that the asymptotic phase idA-morphism !/I: U

-t

TA of class

I

is of class

c!'.r

such that

c!'.r.

C isomorphic to the

N.

Let us show that there exists an

306

= {expx(v,

W(x)

lII(v»:

v

E

Nx "V}

(x

E

(4.3)

A),

where expx: TxM .... M denotes the exponential mapping corresponding to the Riemannian structure on M. By Exp we shall denote the mapping defined by Exp(v) = ('t'M(V), exp v) for all v in a small enough neighbourhood Vo of the zero section of the tangent bundle

c!' diffeomorphism from Vo onto a neighbourhood of the diathat there exists a C vector bundle isomorphism TM[A] N \I TA.

TM. It is known that Exp is a gonal t. c: M x M. Recall

= pr2,

Define lII(v)

R$

EXp-1('t'M(V), h(v» (v

0

C!"'.

that III: U .... TA is of class

E

U). It follows from the above considerations

Since (v, III(V»

= EXp-I('t'M(V), h(v» , we have W(x) = h(Nx " U) = {expx(v,

(TM(V), h(v» = ('t'M(V, III(V» , exp(v, III(V»), therefore III(V»: v E Nx "V}. Conversely, if there exists an idx-morphism III: V .... TA of class

c!.r

such that (4.3) holds, then the mapping h: V .... V defined by h(v)

(v

E

Nx "U) is a morphism of class

I

=P

0

h-I •

c!.r

and

W(x)

= expx(v,

= h(Nx "U),

This means that the asymptotic phase I is of class

lII(v»

consequently,

c!.r.

Now let us look at the meaning of smoothness of the asymptotic phase from one more viewpoint which turns out to be most helpfull. But at first, we need to show that if (~, P2,' X) are

(EI' PI' X) and

can define a equal to

c!'

Let"t be a vector bundle atlas of class

(VI' !PI' IR")

~1(P1(y»(y»

(y

~I(X): E lx .... IR"

vector bundles then for every positive integer r one

vector bundle (C(EI' E2 ), tt, X) whose fiber [C(E I , ~]x at x

C(Elx , Elx ).

(i = 1, 2),

c!'

E

E"10

pjl(VI»,

and

(V2" !P2' IRm) E~, X

!P2(Z)

I: tt-I(VI " VJ .... (UI " VJ

by

(z

Ix(~)

0

0

:;I:

121.

Let us show that! is a

C(IR", IRm» and (VI" V2 , III, C(IR", IRm» and define ~: W .... L(IR" , IR") J1(x)

= ~2(X)

0

formula A(x)1I

and

Finally, define

= J1(x)

11

0

[~(x)rl

c!'

where

p;,I(UJ) ,

Further,

E

"1' (V2, !P2' IRm)

vector bundle atlas. Let both (UI L(lRm ,

IRm)

= VI "V2 ~(x) = ~I (x)

by

A: W .... L(C(IR", IR m), C(IR" , IRm»

(x E W). Then

define

1'It«()(~». By ! denote the fa-

"

E

~,

V2 , t,

"VI" Vz

belong to !. Set W

J1: W ....

[~2(X)]-I. 0

E

[~I(X)]-I.

mily of all triples (VI" V 2' I, C(IR" , IRm» where (VI' !PI' IR") and VI" V2

for (Et, Pt, X)

are linear isomorphisms. Define a mapping

= ~2(X) ~ x C(IR" , IRm) by I(~) = (tt(~),

Ix: C(Elx , ~) .... C(IR" , IRm)

X is

U I "V2• Then !PI(Y) = (PI (y) ,

(P2,(z) , ~2,(P2,(z»(z»

=

~2(X): ~ .... IRm

E

c!'

E

0

[~I (x)]"!, by the

307

Using some results from global analysis on manifolds (in particular, the ",-theorem; see, for example, Leng [I, p. 171-178]) it is not hard to prove that the transition function A is of class

e.

Recall that N ~

Thus, (C(EI' E,),

N

'11',

X)

is a

c: vector bundle.

and TA are vector bundles of class

c:

uniquely determined structure of a

C.

Provide Nand TA with the

vector bundle compatible with the structure of

C. Denote the so obtained vector bundles by Nand TA, respectively. Define the vector bundle (C(N, tA), '11', A), as described above. class

Now we shall show that the asymptotic phase

t: V -. A is of class

c!.r

e

iff there

exists a section 0': A -. C (N, tA) such that (4.4) and, moreover, for each integer p, 0 ~ p ~ k, the mapping cr: A -. COP(N, TA) is cP smooth. x

E

Indeed, let 111: N -. TA be an idA-morphism of class c!.r satisfying (4.3). Given A, define a mapping o'(x): Nx n U -. TxA by o'(x)(v) = 1II(v). According to Bourbaki

[1, subsection 15.3.7], the mapping

0': A -. COP(N, TA)

is a cP smooth section

(0 ~ p ~ k). Conversely, assume that the section 0': A -. C(N, TA) has the indicated properties. Then the mapping 111: N -. TA defined by 1II(V) = O'('rM(v»(v) is a morphism of class c!.r (see Bourbaki [I, subsection 15.3.7]). In the remainder of this section, we shall consider only cascades. Without loss of generality, we shall assume that (4.1) and (4.2) hold with c = 1 (to this end replace, if necessary, the diffeomorphism integer).

f.

M -. M

by

.r,

n being an appropriate positive

4.4. Theorem. Let M be a smooth manifold, r ~ I, and (M, T, /) be a dynamical system

308

of class C. Let A be an f-invariant C submanifold of M. Assume that there exist numbers « > 0 and ~ > 0 such that

for all x

E

and t

A

it

O. Then the asymptotic phase of (M, T, /) defined in the

vicinity of A is of class (fl'''. In other words, ~(f) are C manifolds depending continuously on x • Let I:

E

A.

Besides that, Tx~(f)

> 0 be sufficiently small and k:s

asymptotic phase is of class (fl.Jc. For x IIvll :S

El,

= N! • E!

E

A,

r.

(x

E

A) •

We shall prove inductively that the

let Nx(E) denote the set {v

E

Nx :

and ~(Nx(E), TxA) be the Banach space of all

satisfying !p(Ox)

= Ox

supplied with the norm

c!- mappings 11': Nx(l:) -+ TxA of uniform c!- convergence. Define a

Banach vector bundle ~(lV(I:), til.) with fiber [~(lV(E), tA)]x = ~(lVx(I:), TxA) (x E A) in a manner described above (henceforth, we shall omit the tildes over N(E) and TA). Given ~

E

~(Nx(E), TxA),

II~II:S 1, define an element

by where

prl and pr" correspond to the decomposition TM[A] = N. Til.. The element ~. is welldefined whenever I: > 0 is small enough. Since Nf is a contraction, in virtue of (4.1), by making the relation

E

> 0 smaller we get that

Consider first the case k

= O.

~.l is also contracting.

Note that I satisfies

Instead of the vector bundle ~(N(E), Til.) , we shall

309 use the bundle Coip(N(e), TA) whose fiber over the point X E A consists of all mappings '1': NAe) ~ TxA, !p(Ox) = Ox, satisfying the Lipschitz condition with Lip(!p) :s 1. Defme the norm of 'I' by the formula 11'1'11 = sup {II!p(V)1I I II vII : v E Nx(e),

c!

IIvll - O}. The corresponding topology is stronger than the topology of uniform convergence. It follows from (4.2) that if mapping

I:

Coip(N(e), TA) ~ CoiP(N(e), TA)

Lip(/) :s exp(- (3/2)

> 0

e

satisfies

is sufficiently small, then the the

Lipschitz

< 1. Hence there exists a continuous section

II IT II :s I, which is / -invariant, i.e., / (IT(x)) = lTif -I(x» satisfies the condition (4.4). For x

Next let us consider the case k = 1. I-jets, j~(i;),

denote the set of all condition t;;(v) = Ox x L(Nx, TxA),

E

TxA. Identify N with

N and

j~(t;;) = (v, t;;(v), Di;(v»

then

and

A

v

E

E

with

A ~ ~P(N(e), TA),

A). Consequently, Nx(e) ,

mappings (: Nx(e) ~ TxA

C

of

E

(x

IT:

condition

let

IT

H!,o(e)

satisfying the

i(Nx(e), Txh) with Nx(e) x TxA

t;;

for every

diffeomorphism f carries the mapping t;; to the mapping I(i;)

E

E

C~(Nx(e), TxA).

The

Co(Nrl (x)(e), Trl(x/)

and induces a mapping ie/) on the set of I-jets, which can be written as ie/lev, i;(v) , Di;(v»

= (~(v),

",(v), (v»,

where ~ and '" are defined above and the element i;(v)

E

L(Nrl(x)' Trl(x)A)

is given

by the formula i;(v) = pr2

0

Dexp~\x)(z)

0

Df -I(y)

0

Dexpx(v, t;;(v»

[pr l

0

Dexp - II

0

Df -I(y)

0

DexpAv, i;(v»r l ,

o

r

(x)

where y = expx(v, t;;(v» , z = f -I(y),

TxM

=

N!

III

TxA and Dexpx(Ox)

(z)

x E A.

v E Nx(e) and

= id (x

E

A). Whenever

N

Recall the equalities is invariant, we have

therefore

t;;(Ox) = Dif -I I A)(X) (here just like before, we identify N'"

0

Dt;;(Ox)

0

Nfl Nr

I

(x)

(x

E

A)

N). Thus, whenever N is invariant, we get

310

i(/)(o, 0, >.)

= Df'\

>.

0

0

Nf

(>.

E

x

L(Nx, TxA),

E

(4.5)

A).

It follows from (4.2) and (4.5) that

Let Po: TM -+ If and PI: TM -+ that IIPo - PIli < c, then

If

denote the idA-projectors with kernel TA. Assume

where A: N~ -+ TxA is a linear operator with sup {lIi(/)IN~,o(O)II: x close enough to

If.

E

<

A}

>

= O(c)

E

A}

< 1

0 is sufficiently small. By the fiber contraction theorem (see Theorem E

A) of the section

IT:

in fact, to the space C~(Nx(c), TxA) and, moreover, the section continuous.

> 1, the proof proceeds by induction. Let

k-jets j~(~) of

Therefore

By continuity,

A.25), we get that the values IT(x) (x

For k

as c -+ O.

1 even if the subbundle NI is notf-invariant but only

sup {lIi(/)IN~,o(c)lI: x whenever c

IIAII

c!

A -+ ~(N(c), TA) belong, IT:

N~,Jt'I(c)

A -+ ~(N(c), TA)

is

denote the set of all

mappings ~: Nx(c) -+ TxA having contact of order (k - 1) with the

zero mapping. Let N~,Jt'I(O) be' the subset of N~,Jt'I(c) consisting of all jets satisfying the condition v

= Ox'

The linear space N~,Jt'I(O) is isomorphic to the space PJt(Nx

of all k-homogeneous polynomials. Whenever

therefore

If

IIJJt(/) IN!,Jt·I(O) II

to be close enough to

If

S

exp(- (3)

and c

(x

If

E

, TxA)

is 1j-invariant, we clearly have

A) by virtue of (4.1) and (4.2). Assuming

> 0 to be sufficiently small, we get by continuity (4.6)

311

Suppose that o'(x)

E

~.l(Nx(e), TxA)

(x

E

A). The inequality (4.6) enables us to apply

once again Theorem A.25 and to prove,. in this way, that o'(x) (x

E

E

~(Nx(e), TxA)

A).

4.5. Notation. According to m.21, let

=

n(NJ, x)

lim sup

n-++CD

~ In

IINjINxll,

w(1J1TA, x) = - lim sup -nlln IIV·nIT nAil, n-++CD

n(1J1TA, x)

=

f

lim sup -nlln IITjlTxAll

n-++CD

(x)

(x

E

E

A)

A).

By Lemma II1.2.2 and Lemma III.2.3, the inequalities o'(Nf, x)

< 0,

o.(Nf, x) - w(7J1 Til., x)

<

°

(x

(4.7)

imply that (4.1) and (4.2) are valid.

4.6. Theorem. Let M, r, f and A be the same as in Theorem 4.4. Assume in addition that (4.7) holds. Let I denote the greatest integer satisfying l:s r and o.(NJ, x) - w(1J1 Til., x)

Then the asymptotic phase for

+

l·max {O, o.(1J1 Til., x)}

A

is of class

< 0

(x

E

(4.8)

A).

C,r .

• Let O:s k :s 1. By Theorem 4.4, there exists a continuous invariant section the bundle Co·k(N(e) , Til.)

C<

section.

4.7. Theorem. Assume the hypotheses of Theorem 2.2 to be fulfilled.

c!

of

satisfying (4.4). According to Theorem A.33, the inequality

(4.8) implies that 0': A -+ Co·k(N(e) , Til.) is a

~(f) and ~(f) are

IT

submanifolds depending continuously on x

• Apply Theorem 4.4 to W;:(f) and ~(f).

E

Then the sets A.

Besides,

312

4.8. Theorem. Assume that the hypotheses of Theorem 4.6 are fulfilled. There exists a

number c > 0 such thal if 0(1]1 TA, x) < c for all x E A, then the asymptotic phase is c!' smooth. The same conclusion holds, in particular, when O(NJ, x) < 0, 0(1]1 TA, x) < c and w(1]1 TA) > - c, where c is a small enough positive number. ~

Apply Theorem 4.6.

Proofs of some theorems stated in Chapter 1 At the end of this section, let us present proofs of Theorems 4.3, 4.7, 4.8, 4.11 and 4.12 stated (without proof) in Chapter I. 4.9. Proof of Theorem 1.4.3. Let t: E .... E denote the to-shift along the trajectories of the vector field ( for a sufficiently large number to > O. Set L = Dt(O), f = t - L, then .1(0) = 0, Dj(O) = O. Given a positive number IJ, there exist a number c5 = c5(IJ) and a function f/J. E c!'(E, E) such that f/J.(x) = j(x) for IIxll :IIi c5,f/J.(x) = 0 for IIXII ~ IJ and sup {max [llf/J.(x)II, IIDf/J.(x)II]: x E E} < c. We shall regard E as a trivial vector bundle over a singleton, then (L, id) is a vector bundle automorphism and (L + f,..., id) is a weakly non-linear automorphism of E. Apply Theorem IV.4.2. We have /30. IILI~II Take ~

$

~

< I,

«0

= IIL'II~

$

~II

:IIi

I, if to is sufficiently large.

and ~ in the capacity of L-invariant subbundles Xc, and Yo, respectively.

c!' subbundle WCu invariant under L + f/J. and tangent to ~ $ ~ at the origin O. Consequently, there exists a C submanifold W'u locally invariant under ( and such that To W'u = ~ $ ~. A similar reasoning proves the existence of the sub manifold W'S' with the required properties. Because the manifolds W'u and W'S' cross transversely at the point 0, we conclude that W'. W'u n W'S' is also a locally ~-invariant C submanifold, and ToW' = ~. By Theorem IV.4.2, there exists a unique

In order to prove the rest of Theorem 1.4.3, we must apply Theorems V.2.2 and V.4.7

taking the center manifold W' in the capacity of A. Unfortunately, we cannot apply this theorem directly because the submanifold W' is not compact. Therefore, we are forced first of all to compactify W'. This will be done as follows. According to the proof of

313

WC u

Theorem IV.4.2, the invariant manifold

C

it' • ff

smooth function 1/1: (x,

y, z)

carries the vector field

-+ P;-. The change of variables

(x - I/I(y, z),

H

~

can be represented as the graph of a certain

C

to a

y, z) (x

E

P;-, y ~

vector field

= ff.

WC

canonical form. Let B

IR P )

z

E

it'

WC·,

WC·

We shall also assume that the matrix Ac E L(RP ,

it',

having

unstable manifold. A similar reasoning can be applied to out loss of generality, we shall assume henceforth that

E

ff), (9

ff as its center-

as well. Therefore, with-

= P;- •

= A Iff

ff, WC U

=

it'

(9

ff,

is reduced to the real

be any block of the matrix Ac. Since the eigenvalues

of Ac are pure imaginary then either p

=2

and B

= [0 - Col

Col

1

0'

Col'"

0 or B is an

,

elementary nilpotent block. In the first case, the corresponding linear vector field on Rl is a center.

(z E IRl)

Compactify IRl by adding

to 1RIP3

=: IRl U 1R1P2

RlPl

in the standard way.

z = Bz

and extend the vector field

We get a vector field on 1RP3 all of

whose trajectories (except for the rest point 0 E 1R1 are periodic. The Lyapunov exponents of the linearized vector field are equal to zero for all points (in both directions). Consider now the case where B(u\I ... , up) = (~, ... , up, 0). The vector field is given by space

u = ~, ... , Up_I = up, l

RIP P

to

IRP ,

words, we embed

IRP

and in

up

= O. We compactify IR P

by attaching the projective

then extend projectively the indicated vector field. RP + I

In other

and extend the given vector field by adding the equation

v = 0; after that we pass to homogeneous coordinates. So we get a vector field on =: RP

In

IRtP p +1

u IRIPP , which can be written in each local chart as follows:

this

chart,

the vector

field

11

has

infinitely

many

equilibria

that

fill in

314

the axis Os. At each of these points, the eigenvalues are equal to O. (2) Let 2:s q :s p - 1,

... ,

=

wp

=v/

s

up / u q ,

u q '"

0,

=

WI

uq .

ul

wq _1

/ uq ,

= u q _1 /

Let us show that the vector field

=

=

Wp _1

=

uq + 1 / uq ,

2 Wq+2 -Wq+I' ••• ,

has no singular points in this chart. In fact, if

lJ

w

= 1. Otherwise, the existence of a singular point implies s = wp " = ... = Wq+2 = O, hence W· q + 1 = - Wq2 + 1 '" 0 ,a contrad"lction. 0 then

(3) Let ... ,

Wq+1

Then =

Wq+1

uq ,

q _1

0,

up '"

W p _2

=

Wp-I'

=

WI

Wp _I

UI / up, ••• ,

=

I,

s = O.

Wp-I

=

s = v / up. Then WI = W2, has no rest points in this part of

Up_I / up,

Clearly,

lJ

IRIPP+i.

Thus, the projective vector field

lJ

on IRIPP + 1 that corresponds to the Jordan block B

has only singular points with zero eigenvalues. All other trajectories tend toward rest points as I -+ + .... Let

A

be equal to the Cartesian product of all projective spaces that correspond (in

the above sense) to the Jordan blocks of Ac. Consider the naturally dermed vector field on A (i.e., the product of the corresponding projective vector fields).

Clearly, the

I -+

+ ...) of the tangent linear extension are equal to 0

IIxll ~ 11-,

we see from the above construction that the

Lyapunov exponential rates (as for all points of A.

=0

Since fll.(x)

for

diffeomorphism I/J == (L can choose II-

>

+ f~lJ?

117A(z)1I :s 1

Set M = ~ x

induces a C diffeomorphism A: 11.-+11.. Given e

If'"

x A.

+

e,

117A-I (z)lI:S 1

+

e

(z

(4.9)

A).

E

Define a vector bundle morphism r of (M, pr3' A) by

r(x, y, z) = (L.,x, LuY, A(z»

Along with

> 0, we

0 so small that

(x E~, Y

r, consider the following C diffeomorphism

'II

E

If'",

z e A).

from M to M:

315 _ { (L

+ f,.J(x,

y, z)

if (x, y, z)

E;

E

'II(x, y, z) -

rex, y,

if x

z)

The vector bundle TM[A] decomposes as

TM[A]

E

E", y

E

= 1:: elf' •

~, Z E A\E.

TA

where 1::

= A x E",

If' = A x ~ and, moreover, the subbundles 1::, If', and TA are invariant under IT. If c > 0 is sufficiently small, then (4.9) shows that TM[A] = 1:: $ If' e TA is an exponential splitting (see Definition m.2.6). Because'll tends to r in the C topology as Il -+ 0, we deduce from Theorem m.2.8 that for sufficiently small '" > 0, the vector

r,

bundle TM[A] can be represented as the Whitney sum of 7'{t-invariant vector subbundles

It', and

r

TA, where

is close to 1:: and

It' is close to If'. If

c

>

0 is small enough,

then by (4.9) we get

for all z

E

A and

m

= 0,

1, ... , r. In other words, the manifold A is normally

r-hyperbolic under the cascade (M, 'II). According to Theorems 2.2 and 4.7, there exist smooth manifolds W(A) and WU(A). Moreover, W(A) =

UW:,

WU(A) =

zEA

are

=~

Wo = ro- = E"

!Ii

and

=~. It is not hard to show that the manifold W(A) coincides with

WU(A) coincides with

and WU

U~; w: and ~ zEA

C smooth submanifolds depending continuously on z E A;

~

C

Wi:

d(¥'(z) , 'IIn(y»

are s

WC u in the vicinity of the point

c£ d(z, y) exp(- «zn) (n

one can show that inf

the

{«z: Z E

E

E c: A.

~-invariant. Besides, if

C smooth and

positive numbers. Repeating

0

=

arguments

A}

!!!

«

>

ZE A

and

YEW:,

and Cz are some in the proof of Lemma III.2.2,

1, 2, ... ), where used

«z

0 and that there exists a number c E~,

then

> 0

>

0

WC then d(rpt(z) , !pt(y» s c d(z, y) exp(- a:t). A similar statement holds for ~ (z E WC). If c > 0 is sufficiently small, then Theorem 4.8 implies that and ~ are C smooth in z E WC. satisfying

the condition: if

z

E

WC, Y

WC·, and The manifolds W;;; ~

t

and !p(z, [0, t]) c:

w:

316

4.10. Proof of Theorem 1.4.7. Let I; be a A

= Dt;(O):

C

smooth vector field on E, 1;(0) = 0,

E -+ E, and rp be the phase flow of 1;. Let ~. E', and If denote the

A-invariant linear subspaces of E

that correspond to the eigenValUes of A

Re ~ < 0, Re;\ > 0, and Re ~ = 0, respectively. Denote As = A I~,Au The principal part of the vector field t; can be expressed as I;(x,

C

E", y

E E', Z E If)

smooth functions of order O(IIXIl 2

+

lIyll

It follows from the above proof of Theorem 1.4.3 that after a suitable

C

where p, q and p are IIzll -+ 0.

= A IE', Ac = A IIf.

y, z) = (A,sX + p(x, y, z), AuY + q(x, y, z), AcZ + p(x, y, z» (x E

+

satisfying

change of variables, we get W's p(O,

y, z)

= q(x,

= E" 19 0, z)

If,

=0

W'u = E' (x E

+ 19

E", y

lIyll2

If,

+

IIZII2) as IIxll

W' = E", consequently,

E E', z Elf).

C smooth, If -+ If that

According to the same theorem, the asymptotic phases for W's and W'u are

i.e., there exist such C mappings ~I: E" ~\(O, z)

=

~2(y' u) =

~2(0, z)

=z

(z Elf),

19

If -+ If

and

~2: E'

1&

w: = {(x, 0, u): ~\(x, u) = z},

~

=

{(O, y, u):

z}. Make one more change of variables, namely, h: (x, y, z) ~ (x, y, ~2(y' ~\(x, z»)

(x E

E",

Y E E', z Elf).

Then h(W:)

=

h({(x, 0, u): ~\(x, u)

= z}) =

{h(x, 0, u): ~I(X, u)

=

= {(x, 0, ~2(0, ~\(x, u))): ~\(x, u) = z} = {(x, 0, z): x

Recall that rp(W:) =

W:(Z) ,

rp(~) = ~(Z) (z

E

z} E

r}.

If). Hence

tot t t) rpt{,x, 0 ,z) = ('P\(x, z), , IP3(Z» , 'Pt (0, y, z) = (0, 'P2(Y' z), 'P3(Z) .

Thus, passing to new coordinates, we obtain p(O, y, z) = p(x, 0, z) == p(z)

y E E', z

E

If). Set R(x, y, z)

=

p(x,

y, z) - p(z), p = p, Q = q.

(x

E

E",

317

4.11. Proof of Theorem 1.4.8. Apply Theorem 2.2 to the manifold A constructed in subsection 4.9. 4.12. Proof of Theorem 1.4.11. According to Theorem 1.3.9, the principal part of the local representative of the vector field ~ can be written as (x, y,

z, e) = (Ast + PVc, y, z, e), Au>' + AcZ Vc

E

+

E:, y

z, e),

y, z, e), 1 + cr(x, y, z, e»

p(x, E

q(x, y,

It',

Z

E

~,

e

E

IR),

where p, q, p and cr are c! smooth functions, 2w-periodic in e and such that p(O, 0, 0, e) ii 0, q(O, 0, 0, e) II 0, p(O, 0, 0, e) II 0, cr(O, 0, 0, e) ii 0, D,p(O, 0, 0, e) • 0, D 2q(0, 0, 0, e) E 0, D 3P(0, 0, 0, e) II O. As before (see the

weS' = ~

proof of Theorem 1.4.7), we may assume that therefore, p(O, y, z, e)

weS'

phases for = p(x, 0, z, e) proof, put P

ii

0, q(x, 0, z, e)

we

u and ii 91o(Z, e),

= p,

Q

= q,

ii

III

~

III

IR,

we = It' u

III

~

III

IR,

O. The existence of c! smooth asymptotic

allows to obtain the following equalities: p(O, y, z, e) cr(O, y, z, e) = cr(x, 0, z, e) II !/Jo(z, e). To finish the

R

= r - 910'

S

= cr - !/Jo.

4.13. Proof of Theorem 1.4.12. Compactify and then apply Theorems 1.4.11 and V.2.2.

we after the manner used in subsection 4.9

Bibliographical Notes and Remarks to Chapter V Theorem 2.2 can be found in the book by Hirsch, Pugh and Shub [1] and in the papers by Fenichel [1-3], but our proof is more detailed and more explicitly based on global analysis methods. Osipenko [1-3] has studied the notion of weak persistence (when the given invariant manifold is not isolated). Theorem 3.1 is proved by Mane [2] for k = 1. The case k > 1 is examined by Bronstein [6]. The notion of asymptotic phase for invariant manifolds is investigated by Hirsch, Pugh and Shub [1], Fenichel [1-3], Robinson [1], Kadyrov [1,2], Aulbach, Flockerzi and Knobloch [1]. Our exposition follows the paper of Bronstein and Kopanskii [4].

318

CHAPTER VI NORMAL FORMS IN THE VICINITY OF AN INVARIANT MANIFOLD

§ 1. Polynomial Normal Forms (the Nodal Case)

In this section, conditions for smooth conjugacy of two dynamical systems in the neighbourhood of their common smooth compact asymptotically stable invariant manifold are given. An illustrative example is presented. 1.1. Standing assumptions. Let M be a smooth manifold, k I!: 1, T

= IR or T = I, and

(M, T, /) be a dynamical system of class ~. Let A c: M be a compact sub manifold of

t

class ~ invariant under (M, T, /). Denote = /IA (' E 1). Henceforth, we shall assume that A is exponentially stable (for definiteness, in the positive direction). Let N

= TM[A] I

N/: N -+ N (t

E

TA. The linear morphism

T/:

TM[A] -+ TM[A] induces a morphism

1). As usual, denote

O(Nf, x) w(Nf, x)

= - lim sup!t In H+CD

0(7]1 TA, x) w(7]1 TA, x)

= lim sup!, In t-++CD

IIN/INxll, IIN/"tIN

t

f (x)

II,

= lim sup !, In IIT/ I TxA", t-++CD

= - lim sup !, In H+ CD

117]' -t ITt f

ex E

All (X)

A).

In what follows, we shall assume that O(Nf, x)

< 0,

O(Nf, x) - w(7]1 TA, x)

<0

(x

E

A).

(1.1)

319 Let 1 denote the greatest integer satisfying the inequalities 1 ~ k and sup [n(J.if, x) - w(7]l TA, x)

+ I· max to,

o(7]l TA, x)}]

<

0.

(1.2)

xEA

According to Theorem V.4.6, the asymptotic phase defined in the neighbourhood of the submanifold A is of class C,If.. This means that there exist a neighbourhood U of the zero section Z(A) of the normal bundle (N, 1[, A), a neighbourhood V of A in M, and a tubular neighbourhood h: U .... V of class

C,If.

such that

1[

0h- I 0/ 0h = if IA)

•

1[

r+, h is a semidynamical system (i.e., a transformation semigroup) and It: (U, r+, h . . (A, r+, ftJ is a homomorphism. The semidynamical system (U, r+, h can be embedded into a dynamical system (N, T, h. for all 1 it 1. Denote .,

=

h- I . / •

h, then (U,

Thus, the investigation of (M, T, j) invariant submanifold

in the vicinity of

the asymptotically

A is reduced (via the asymptotic phase of class

C,If.)

stable

to that of

the extension 1[: (N, T, h . . (A, T, ftJ. So, our problem can be reformulated as follows. Let (N, It, A) be a vector bundle of class

C,If.,

1~ k

and

It: (N, T, j) .... (A, T, p)

be an extension of class

C,If.

(i.e.,

the mapping /: N .... N is a pt-morphism of class C,If. for every element 1 E 1). Further, we assume that the zero section Z(A) is invariant under (N, T, j). Denote D~/(v)

= DPr/INx)(v)

= Dv/(Z(x»

At(x)

= lim

n(A, x)

w(A, x)

= - lim

O(Tp, x)

w(Tp, x)

t .... + DO

= - lim

t .... + DO

= lim

t .... + DO

= It(V) ,

x

(v E N,

(x

sup

sup

A,

E

t In

-11

1

= 0,

P E

1, ... , k),

1),

IIAt(x) II ,

In IIA-t(pt(x» II ,

sup!1 In IITptIT;,cAII,

sup!1 In IITp-t IT

H+DO

The conditions (1.1) - (1.2) can be rewritten as

t

p (x)

All

(x

E

A).

320

n(A, x) < 0; n(A, x) - w(Tp, x)

+I

(1.3)

< 0

• max {O, n(7]1 TA, x)}

(x

A).

E

Let L denote the smallest positive integer satisfying L·n(A, x) - w(A, x)

+ I·max

{O, n(Tp, x)}

< 0

(x

E

(1.4)

A).

1.2.. Definition. We shall need the following generalization of jets to the case of C,Ie mappings. Our exposition is very brief. We leave the details to the reader. Let U c IRm and V c IR" be open subsets, (xo, yo> E U X V, and fi.: U x V ... IRd

= 1, 2) be mappings of class C,Ie such that It (xo, that It and 12 have contact of order [I, k] at (xo, yo> if

(i

IIDfDicti - fJ(x, y)1I

= O(lIy - YolIJc.l)

yo> = 12(Xo, yo>.

(0

:S

P

+ q :S

We shall say

l)

as x -+ Xo and y -+ Yo' In other words, this means that the I-jets of Ji and 12 at (xo, Yo) have vertical contact of order k - I. The corresponding equivalence classes are called jets of order [I, k]. Let zEN and W be a neighbourhood of z. Using charts on the manifold A and vector bundle charts on N, we may assume that W lW U x V, where U and V are open subsets of linear spaces, and pr.: W -+ U corresponds to '1[: N -+ A. This makes it possible to extend the above definition to the case of a C,Ie mapping F: W -+ N. If the chart W

lW

U x V

is fixed then j!,Ie(F), the jet of order [I, k] of F at

characterized by

o :s p + q :s k).

the

collection

partial derivatives

DfDiF(z)

«O:s p

Given a C,Ie section IT: W -+ 7N[W), we can define the jet c

in a similar way. The family of all jets of order P',1e • P',Ie(T:N):

of all

z, is completely

It can

:S

I,

= j!,Ie(IT)

of sections IT: N ... TN will be denoted by

[I, k]

be provided with a naturally defined vector bundle structure.

Clearly, p',l. P'(T:N) (see subsection A.14). The canonical projections ..l,I+1

----+ ..• ----+ r ... ----+ pi

JIl-TN

321

are vector bundle morphisms.

Let

rOfP, ... , rOy, rOy,l+t, ... , rOy,k

be the spaces

of continuous bounded sections of the corresponding vector bundles. The morphisms pI, induce projections '/tl' ... , '/tk of the above spaces of sections. Suppose that along with the extension '/t: (N, T, j) .... (A, T, p), there is another

... , pk

extension 'It: (N, T, g) .... (A, T, p) of class C,k such that /(x) = i(x) (x E Z(A) , t E 7). We shall say that (N, T, j) and (N, T, g) have contact of order (Q, 1, k) at wo E Z(A) if for some chart W 11$ U x V, w 11$ exo, 0) E U x V, and for all O::s p ::s I, o ::s p + q::s k, It I ::s 1, the relation

IIDfD1r/ex, y) - i(x, y)]11

= O(lIyllmax{Q-q,O»

(llyll .... 0)

(1.5)

holds uniformly on U.

1.3. Theorem. Let p, f, k, I and L satisfy the above conditions. If the system (N, T, g) has contact of order (Q, 1, k) with (N, T, j) at all points w E Z(A) and Q e L, then there exists an idKmorphism of class

C,k

defined in a neighbourhood of

Z(A) which co'!iugates f and g .

• We shall consider the case of cascades (i.e., T = Z). Let E > O. Denote t(x) .;. n(A, x) + E, ,,(x) = ",(A, x) - E, lItex) = n(Tp, x) + E, I/I(x) = ",(Tp, x) - E. If E is sufficiently small, then in virtue of (1.3), (1.4), we have by Lemma m.2.3 that sup lex)

<

xEA

0,

sup [I(X) - I/I(x)

+

I'max {O, lItex)}]

<

0,

xEA

(1.6) sup [L'I(x) - flex)

+

I·max {O, lIt(x)}]

<

O.

xEA

Besides, for all sufficiently large numbers n, the following estimates hold: exp [,,(x)n] ::s IIA"(x)1I ::s exp [t(x)n], (1.7) exp [I/I(x)n] ::s IITp"(X)1I ::s exp [1It(x)n]

(x

E

A).

We need to show that the p-morphisms f and g are conjugate to one another by some local morphism H of class C,k. We shall assume first that the estimates (1.7) hold for all integers n ~ 1. The general case will be considered at the end of the proof.

322 Because the problem in question is local, we may suppose without loss of generality that g is uniformly C,k. close to f, and f = g out of some neighbourhood of Z(A), In fact, for every E > 0 there exist a number c5 > 0, a vector bundle morphism Ao: N -+ N of class

C ,k.,

and p-morphisms

g:

N -+ N and

sup {II(A - Ao)INxll: x g(y)

= g(y)

111- glll,k.

(llyll s c5),

<

A}

E

1:

N -+ N such that

fly)

E,

= fly)

(llyll s c5),

< E, fly) = g(y) = AO(lI(y»

(llyll

it

E),

Below, we shall introduce a certain space of idJ\-morphisms H: N -+ N of class C,k. and shall prove that the operator H 1-+ f -\ 0 Hog has a globally attracting fixed point in this space. When performing this, we shall assume that the mappings H: N -+ N are so near to id: N -+ N that they can be represented in the form H = exp 0 ~, where ~: N -+ TN is a C,k. vector field and exp denotes the exponential mapping that corresponds to a fixed Riemannian metric on (TN, TN, N). Define a vector field'll: N -+ TN

from the equality

= exp

f -log

0

'II.

According to the hypothesis (1.5), the relation (llyll -+ 0)

(x

is true for all (x, y)

E

U

U, Y

E

X

V, 0

E

S

psi, 0 s p

+qs

V ~ W.

r°p-,k..

Let us introduce a special topology on function c5 m on the fibers of

lIm

For 0 s m s k, define a metric-like

in the following way. Assume

difference c1(y) - C(y), when considered in a chart Wor.

Uor.

III

the partial derivatives DfD'i[cr! - cr!l(y) , where psi, p representative of the jet c'(y) (i Put

Y The function

Ci m

k)

E

= 1,

Wor., llyn

2).

¢

Let {Wor.:

0, 0

S

CIt

X

+q

E.I}

psi, p + q

lImC I

= lIm C2,

Vor.,

is determined by

= m,

then the

and cr! is a

be a fixed atlas on N.

= m,

«

E

.I}.

differs from a metric only in that it can take infinite values.

Each element c

E rO[Il

may be identified with a section c: N -+ TN of the vector

323 bundle (TN, H

t-+

TN'

N).

Consider the operator 10 that corresponds to the operator

II • H • g and maps rOJP into itself. To be more precise, 10 is defined for all

sufficiently small elements c E rOJP by the rule:

=

c·(y)

For m

=

expi/ • I

-I •

exP•• c • g(y), a

= 1';(expi,1 • I

and h is an arbitrary

-I •

C"

= c·,

smooth

., m( = }y' eXPb-I ·1-1 • exp.·

=I

b

-I •

yEN.

g(y),

E r°P" (for m

:S

gCY),

b

=f

N)

TN'

is defined similarly: if c

h • g),

I) or ct, ~

a

=

TN'

N)

gCY),

b

yEN,

-I • gCY),

satisfying the condition

is a representative of the jet c(a)

and h is an arbitrary C,m section of (TN, C2

=

(TN,

section of

= c(a) (in other words, h m = I + 1, ... , k, the mapping 1m small, then Im(c) = c·, where

Let c l ,

g(y),

where

exP•• h • g), a

}":(h)

C.(y)

=

where

1, ... , I, define a mapping 1m from r O p" into itself as follows: if c is a

sufficiently small element, then Im(c) c·CY)

= c·,

lo(C)

E

For

p").

E r°p-,m is sufficiently

= 1-I • g(y),

YEN,

satisfying j!,m(h) = c(a).

E r°p-,m (for I <

m :s k), and Kmcl

=

Km~'

We shall prove that if the norms IIclllo and IIC1llo are sufficiently small, then there is a constant 7, 0 < 7 < 1, such that (m

At first consider the case m

= O.

= 0,

1, ... , k).

We have

. sup {lIDj -1(g(y»II' [lIg(y) II I lIyll]Q: YEN, lIyll .. O}

(l.8)

324

where Do is a constant depending on the mapping expo arbitrarily close to 1 whenever the norms 10

= exp [sup

IICllio

[Q·t(x) - ,,(x)

Obviously, Do can be taken

and 1I~lIo are sufficiently small. Denote

+ I'max

{O, lIt(x)}]].

(1.9)

xE/\

According to (1.6),

10

< 1. Because the morphism g is C,l( close to/, we may assume by

(1. 7) that IIDvf "1(g(y»11 is arbitrarily close to exp [- ,,(1f(y»]. For the same reason, IIg(y)1I I lIyll is close to exp [t(1f(y))]. As a result, we get

where

1

is any number satisfying exp {sup [Q·t(x) - ,,(x)]}

<

1

< 1

xEII.

(see (1.9».

Now let 1:s m :s k. Using the Composite Mapping Formula (see subsection A.6) and taking account of the equality 1fmCI

y

• lIyll" max{Q-q,O}:

:s

doDm sup

= 1fm~'

E

Wet.'

we obtain

lIyll'"

0, O:s p :s I, p

+

q

= m,

ex E ,,}

{IIDfDi[h~ • g(y) - h! • g(y)]11 IIg(y)u" max{Q-q,O}}

where

. do =

sup {IIDvfl(g(y»II. [lIg(y) II I

and Dm is a constant close to ct(o) in the chart Wet.

Itj

1.

lIyll]max(Q"q,O}

Recall that h; denotes a representative of the jet

Uet. )( Vet.' i.e., J":(h!.>

= ct(a)

(0

= g(y),

i

=

1, 2). Let us

325 estimate the value of do. First of all, observe that

IITg(Z(x»1I s exp [max {t(x) , 'lI(x)}] s exp ['lI(x)], IIDJ{Z(x»1I s exp [t(x)]

(x e A)

by (1.6) and (1.7). Because the morphism g can be chosen as C,k close to f as we like, and jlZ(A) = gIZ(A), we conclude from (1.6) and (1.9) that do can be estimated from above by any number greater than fo (recall that fo < 1). The case I < m s k can be examined in a similar way. Thus, the estimate (1.8) is established for all numbers m. Applying Theorem A.2S, we obtain by induction that the operator H

'r--+

rl

0

Hog has a

unique ftxed point Ho in our special space of C,k morphisms H: N -+ N. Clearly,

f

0

H

= Hog.

Thus, the statement is proved under the additional assumption that (1.7) holds for all n it 1. Assume now that (1.7) is satisfted only for n it no. By the previous 1-+ jnO 0 Hog no has a unique ftxed point Ho. Then g according to Lemma A.30.

considerations, the operator H

f

0

Ho = Ho

0

1.4. Theorem. Let N, p, f and k be the same as before. Assume that o(A, x) (x e A). Let L denote the smallest positive integer such that L'O(A, x) - w(A, x)

< 0

(x e A).

<0

(1.10)

Let (N, T, g) be a dynamical system of class cf which has contact of order (Q, k, k) at all points y e ZeAl with (N, T, /). If Q it L then there exists a number I: > 0 such that whenever O(Tp, x) < I: and w(Tp, x) > - 1:, then (N, T, g) is cf conjugate to (N, T, /) in a small enough neighbourhood of ZeAl. ~ This statement follows directly from Theorem 1.3. Indeed, if the number I: > 0 is sufftciently small, then one can take 1 = k in (1.3). Note that (1.10) implies (1.4).

1.5. Remark. Suppose the hypotheses of Theorem 1.4 are fulftlled. Assume in addition

f e c!Hk, and the bundle (N, n, A) is trivial (i.e., N ~ A x IRn ). Then the mapping f. N -+ N can be written as ft..x, y) = (p(x), fix, y» (x e A, y e IRn ). that T = I,

Clearly, fix, 0)

==

O.

According to Taylor's formula,

326 f(x, y)

= a1(x)y + ... + aQ(x)yQ +

r(X, y),

where aJ(x)

1 D'J.j f(x, = 7T lIr(x, y)1I

Since /

E

cP+ lt ,

we have aJ

Clearly, g: N -+ N is are

CC

E

CC

CC smooth.

(j

(j

0)

= 1,

... , Q),

= o(lIyIlQ).

= 1,

... , Q). Set

It follows from Theorem 1.4 that the mappings/and g

conjugate in the vicinity of the zero section (i.e., for small

natural to say that the g is the generalized polynomial / near the submanifold Z(A)

ItS

CC

It seems

lIyll).

normal form of the p-morphism

A. When A is a fixed point, then the hypothesis

can be replaced by the weaker condition / E cP because aJ = const (j in this case. A similar statement holds for vector fields, as well.

=

f

E

cP+ 1t

1, ... , Q)

1.6. Proof of Theorem ll.3.17. We need to apply Theorem 1.4 to the case where A is replaced by the center manifold

we = g:

(see subsections 11.3.9 and 11.3.12). However,

previously we must compactify the space g: and extend the dynamical system, as it was done in subsection V.4.9.

1.7. Example. Consider the following system of differential equations

g!j = - (cose ~

+

4)u

+ i,

= -2y,

~ =

i sine

(1.11) (u, y

and the corresponding phase flow

E

IR;

e

E

SI

(IR'J. x SI, IR, J).

Ii

1R/2nZ).

The sub manifold A

e E SI} ItS SI is invariant and exponentially stable in Linearizing the given system in the direction normal to

the A,

=

{(O, 0, e):

positive direction. we obtain the flow

327 (1R2 x S, IR, Nf) determined by the equations

~ = - (oose + 4)x, ~ = - 2y,

de

1. sme.

at = 2

(1.12)

It is easy to check that

= w(7]1 TA,

g(7]1 TA, 0) g(7]1 TA, b)

= w(7]1 TA, g(Nt. b)

b)

= 112,

0)

= - 112

=-2

(b

E

b

A,

¢

0),

(b e A),

weN/, 0) = - 5, weN/, b) = - 3

(b e A, b

¢

0).

We see from these relations that the submanifold A is normally 3-stable, i.e., g(Nt. b)

+

3w(7]1 TA, b) < 0

(b e A).

By Theorem V.2.2, every system of differential equations of class sufficiently

C

close to (1.11) has a

C

C

on

1R2 x S

smooth invariant submanifold A' situated near

A. Note that

= 2,

f

and

Nf have

we see from (1.2) that I

oontact of order (2, 2, 2) at each point of A. Since I :s k

= 2.

For the point b

= 0 e St,

+ 5 + 2 max {O, 112} < 0, hence, L > 3. But Theorem C linearizability of f near A because Q = 2 < L. Let us show that system (1.11) can be C linearized at - 2L

the condition (1.4) gives 1.3 does not guarantee

A,

but admits no

C

linearization. Observe first that (1.11) cannot be linearized in the normal direction by a transformation which is analytic in x and y. Suppose, to the contrary, that there exists a function

(1.13) m+n~

such that the change of variables u The coefficients Omn(9) are assumed

= x + h transfers system (1.11) to system (1.12). to be C smooth, and the series (1.13) is convergent

328

for all U

e

E

=X + h

Sl and for sufficiently small Ixl and Iyl. Differentiating the equality along the solutions of (1.11) and (1.12), we get

- (cose

+

4)h

1

+ Y = - (cose +

ah ah 4)x ax - 2y ay

+

1. ah 2 sme ae'

(1.14)

hence

(1.15)

Equating the coefficients of

l

in both parts of (1.15), we obtain

sine· a~z(e)

= - 2 cose· tlo2(e) + 2.

(1.16)

Now let us show that the differential equation (1.16) has no solution. Indeed, according to (1.11), e(t)

= 2 arctg(tg(eol2)

etll)

C

smooth 2n-periodic

(eo;l: kn).

Consequently, the function aoz(e(t)) satisfies the equation daol CIt

=

daoz(e)

de

.

de

(2

at = -

cose· a Ol

+

+

1.

1 2) 2 '

or

d~~z = _cos[2

arctg(tg(eol2) etll)]aoz

(1.17)

The graph of the function rp(t) = - cos[2 arctg(tg(eol2) etll)] looks like Figure 1.1. = rp(t)x and Consider the linear extension generated by the differential equation look at the corresponding phase portrait (see Figure 1.2). It is easily seen that equation (1.17) has no non-trivial bounded solution. Therefore equation (1.16) does not have bounded solutions.

x

Similarly, we can show that (1.11) admits no contrariwise, that there exists a

C'

C'

transformation u

smooth linearization. For suppose,

=x +

h(x, y, e)

which brings

329

------------t

-1

-------------

Figure 1.1

Figure 1.2

(1.11) to (1.12), then (1.14) holds. Differentiating the equality (1.14) twice with respect to y and letting x = 0, y = 0, we obtain

or sine.!..c alh) ae ayl

= _2

cose. alh ayl

+

4.

Thus, the function (alh I al)(o, 0, e) satisfies a differential equation which is much the same as (1.16) and has no periodic in e solutions for reasons given above. This means that alh(O, 0, e) I

al

is not

C

C, no C

smooth, hence h is not of class

contradicting our assumption. The reasons for which the system (1.11) admits linearization cannot be abolished by means of small perturbations because they are conditioned by intrinsic properties of linear extensions with no non-trivial bounded motion, and the set of all such extensions is open (see Bronstein [4, Proposition 8.16]). Consequently, not only the system (1.11), but also each sufficiently system of class C is not C linearizable near the corresponding manifold. Write down the characteristic equations

C

C

close

smooth invariant

330

dh = dx = ..EL = d~ - (cose + 4)h + y2 - (cose + 4)x - 2y ~

(1.18)

~e

M

From the last two equalities, we get two integrals of (1.18): ( I - oose ) 2 Y 1 + cose

= CI'

5 x-(1 - oose)(1 + cose)3

~

C2'

Using the equation

dh _ cose + 4 h _ l' 2y 2'

ay -

we obtain the general solution of (1.14):

b:

=

h(x, y, 9)

1 + 2 S9 + 4(1 + cose)3 . (1 - oose)5

~(x

(1 - ooS9?: (1 - oose)2) (1 + cos9)a, Y 1 + oose '

where ~ = ~(u, v) is an arbitrary function. Put ~1 = ~ + \14(1 + ~rl. Instead of v4(1 + ~rl one could take any smooth function ~o(v) satisfying ~o(O) = 0 and lim V-2~O(V)

~oo

hex

=

' y,

1. Then we write

= h (y

9)

0

,

e) + 4(1 + oose)3 (1 - oose)S

.

~rx I:

(1 - coset

(1 + COSe)3' y

(1 - cose)2), 1 + cose

where

l(l -

2l [(1 ho(y, 9)

+cose) 3 cos e)3] (1 + co s e)4 + y2(l - ooS9)4

=

o

if

if

(y, e)

(y, e)

;*

= (0,

(0, n);

n).

Further, we have

BhO By

= 4y[(l Bho B9

+ oose) 7 - 2/( 1 + cose)4 (1 - cose)3 [(l + oose)4 + y 2(1 - oose)4]2

= 2/sine[(I+cose)6

/(1 - cose) 7J

l(l - oose)6 J

- 2y2(7 - oos 2e) + [( 1 + cose)4 + y2(1 - cose) 4]2

331

These derivatives exist everywhere except the point y

= 0,

e

= 71:,

and

. aho . aho 11m ay = 11m 89 = 0.

y-+ 0 e-+n

y-+ 0

e-+n

For (y, e) - (0, 71:), we obtain

therefore a2ho I al -+ co as e -+ 'It and y = 0, consequently, ho is C but not C smooth. Suppose that a suitable choice of ~1 ensures the existence and continuity of the derivative alh I ai. Let x :3

....!..'2. ay'2.

=

°

and e

1

E

(0, 'It). According to this supposition,

1

~ (x ( - cose) y ( - cose) l ) 1 (1 + cose)3' 1 + cose $

I

x-O,y-O

'

°

=

consequently, :3 (a 2 I a;)~I(u, v) Iu-O,v-O A. Because h(x, y, e) = acyl) as y -+ and the variable y is proportional to the second argument of ~1' the function h(O, y, e) can be represented in the form

h(O, y, e)

= ho(y,

e)

+ 2Ail

sinle (1.19)

+

4(1 + cose)3 (1 - cose)$

.

Let k be an arbitrary number. Put y

oC l(

1 - cose)4) 1 + cose

= k sine.

(e

E

(0, 'It».

Because h is continuous at (0, 0, 0), by

taking the limit as e -+ 0, we deduce from (1.19) that h(O, 0, 0) hence A = 0. Therefore, we get the equality

= ho(O,

0)

+ 2AK,

332

which contradicts the properties of ho established above. Thus, the systems (1.11) and (1.12) there is no

C

C

are

conjugate near the submanifold A, but

conjugacy.

§ 2. Polynomial Normal Forms (the Saddle Case) In this section, we investigate conditions sufficient for smooth conjugacy of two

dynamical systems near their common smooth compact normally hyperbolic invariant submanifold. 2.1. Standing assumptions. Let M be a smooth manifold, k (M, T, f) be a dynamical system of class

sub manifold of class

C

(0, 1) and c

>

According to Theorem V.4.4,

Fix some

= I, and

f.

Suppose that A is normally k-hyperbolic,

E

A; t

l!:

0; r

7M[A]

=r

&l

TA

&l

Jt'

and numbers

w: =~,

C" Riemannian

= 0,

1, ... , k).

= W(f), and ~ = ~(f) are, in fact, c!< smooth

bEA

A, and Tb

or T

0 such that

By Theorem V.2.2, the sets A, W sub manifolds and

E

= IR

Let A c: M be a compact (boundaryless)

invariant under

(b

b

1, T

c!<.

i.e., there exist a 1J-invariant decomposition ~ E

l!:

bEA

w: and ~

Tb ~

=~

are (b

metric on (TM,

E

c!<

sub manifolds depending continuously on

A).

TM'

M). Let d denote the corresponding metric

333 on M and dr denote the metric on the space f (M, M) of all r-jets of mappings from M into itself. 2.2. Dermition. Let (M, T, g) be a dynamical system of class

= /(x) (x

E

A,

t

7). Let Q

E

it

k.

c!

such that i(x)

We say that (M, T, g) and (M, T, f) have contact

(Q, k) at x E A, whenever j!if) = j!(l) for It I :s 1 and, moreover, the following relations hold for r = 0, 1, ... , k and It I :s 1:

0/ order

Since A is compact, this property does not, in fact, depend on the choice of metrics d and dr and can be reformulated as follows: ,kU -t )xV

as y -+ x, y

E

M.

0

gt)

'k('d) =)x 1

(It I :s 1),

If / and g have contact of order (Q, k) at all points

there exist a neighbourhood U of A and a number D

(y

If the

neighbourhood

U

E

U;

It I :s 1; r

= 0,

x

E

> 0 such that

(2.1) 1, ... , k).

is sufficiently small, then there exists a vector field

i

A, then

vt :

U

-+ TM[U] of class c! such that / -t 0 = exp 0 where exp is the exponential mapping that corresponds to the fixed Riemannian metric on (TM, "CM' M). It therefore follows that (2.1) can be rewritten as

vt ,

where 0 denotes the zero section of the tangent bundle

TM[U], and II· IIr is a norm

defined on the vector bundle P(TM[U]) of r-jets of sections 17': U -+ 7M[U]. The mapping Exp: TM[A] -+ A x M defined by Exp(~) = ("CM(~)' ex~) (~E TM[A]) is a local ~ diffeomorphism from a neighbourhood of the zero section Z(A) into A x M.

334 Because 1M[A]

=r

Ell

TA EIl~, we see that explr

neighbourhood of the zero section of the bundle E in M. In other words, explr a

C<

8ubmanifold, there exists a

C<

to a

C<

C<

vector

structure. Consequently, H of class

c!

~ is a

C<

II

Ell

~ onto a neighbourhood of A

tubular neighbourhood for A uniquely determined up

bundle

exp I r

r

~ maps a sufficiently small

tubular neighbourhood of A in M. Since A is

vector bundle isomorphism. Further, (r

unique structure of a

c!,

Ell

Ii

Ell

Ell

Ell

which

~, is

TM'

c!

A) can be provided with a

equivalent

to

the

initial

~ can be regarded as a tubular neighbourhood

and ~ of E

(bearing in mind that the subbundles r

==

r

Ell

~ are of class

as before).

2.3. Theorem. Let (M, T, /) and A satisfY the above conditions. There exists a number Q, Q I: k, such that if (M, T, g) is a dynamical system of class C< having contact of order (Q, k) with f at every point X e A, then one can find a small enough

neighbourhood U of A in M and a i.e., t o/(x) ~

=l

0

C< diffeomorphism

t: U .. M which conjugates f and g,

t(x) (t e T, x e Un f -t(U».

Let us first prove the following proposition.

2.4. Lemma. For every number K, K > k, one can pick a number Q > K

+k

such that

if the dynamical systems f and g have contact of order (Q, k) at A, then there exists a

C<

local hog

0

diffeomorphism h: U .. M satisfYing the following condition: the systems f and h- I

have contact of order (K, k) along W(f) n U (this means, in particular,

that f and hog ~ H -I

0

h-I agree on W(f).

The proof will be given for cascades. Replacing 0

f

0

by H

-I

0

f

0

H and g by

H, we may assume that f and g are defined in the vicinity of the zero section

of the vector bundle E

c!

f

close to r

==

r

Ell

~. Let X and Y be

C< vector subbundles of E

sufficiently

and ~,respectively_ It is seen from the proof of Theorem V.2.2 that

= graph(cps), WU(f) = graph(cpu), id...-morphisms of class C< in the category :BUll.. A

W(f) and WU(f) can be represented in the form W(f)

where CPs: X .. Y, CPu: Y .. X are similar statement holds for

W(g)

and

WU(g),

because the hypothesis j!(f)

= j!(g)

335 A) implies that g is arbitrarily ~ close to / when restricted to a small enough neighbourhood U of A. Replacing/and g, if necessary, by some diffeomorphisms which are (x

E

~ conjugate to / W(f)

= W(g) = X,

and g, respectively, we may (and do) henceforth assume that

WU(f)

= WU(g) = Y.

Although

bEA

bEA

for the (modified) diffeomorphism f, we point out that

Tb ~(f)

is not necessarily

equal to Xb and, similarly, Tb Wt!(f) may fail to coincide with Yb • Let 'ltlo denote the projections that correspond to the direct sum decomposition 7E[1\] • Y. Clearly, the mapping 'lt3 •

1)'1 Yb

'lt l •

is arbitrarily close to

1)'1Xb:

Xb

'lt2'

and

= X.

'lt3

Til.

-+ Xf(b) is arbitrarily close to 1)'1~ and

1)'1~ whenever X and Yare sufficiently close to

r

and yB, respectively. Let L: E -+ E be a ~ vector bundle morphism sufficiently ~-I close to ('It I • 1)'1X) e ('lt3 • 1)'11'). The mapping L-I • / is arbitrarily close to the identity mapping in a small enough neighbourhood V of the zero section of E. Hence, there exists a vector field (: V -+ 1E[A] of class

~

satisfying £"1 • /

=

exp • (.

Let ~: E -+ TE be a ~ vector field wich agrees with ~ in a smaller neighbourhood VI c V and vanishes out of V (such a vector field ~ can be built by multiplying ( by a Use the same scalar function). Define a mapping J: E -+ E by J = L • exp~. construction to obtain E -+ E. Clearly, the vector subbundles X and Y are invariant > 0 such that ](z) = g(z) = Lz under both J and g. Moreover, there exists a number for IIzll it ro. By making the neighbourhoods smaller, we can assume to be arbitrarily close to J. Note that J and / agree near A. In what follows, we shall omit the tildes placed over / and g.

g:

'0

g

Assume that the Riemannian metric on E is of class ~ and satisfies the condition: if X <& Y Ii E, then d(z, A) = (IIXII 2 + lIyIl2)1/2. It is meant here that A is identified with the zero section Z(A) c E. We also assume that d(z, X) = lIyll and d(z, 1') = IIxli. Because we think of the mappings / and g as defined on E, there exists a

z,. (x, y)

E

vector field 0': E -+ TE satisfying /.1 • g = exp • 0'. According to our hypothesis, / and g have contact of order (Q, k) at each point of 1\. Therefore

336

1I1J'~(II')-J~(0)lIrsD'[~IXIl2+IIYIl2"]Q-r _ _

( ZE E ;

r

= 01 , , ... , k).

Let us show that the vector field 11': E -+ TE can be represented as II' where both 11'1 and 11'2 are C- vector fields satisfying II'I(Z) IIzll = 0 or IIzll it ro' and, moreover,

(2.3)

= 11'1

= 1I'2(Z) = 0

+ 11'2,

whenever

(2.4) (Z

(x, y)

E

E

V; r

= 0,

1, ... , k).

Indeed, the inequality (2.3) implies that II' can be written in the form

cr(z) where cp is a

C-

= (IIXII

+ IIYII)Qcp(z),

function defined on the set E \

ZeAl

and such that all its derivatives

are bounded. Expanding (lixil + lIyll)Q according to the binomial formula, we get

where ul and VI are polynomials in IIxll and lIyll of order greater than consequently, UI and VI are crl(Z) crl

+

= U(z)cp(z) , cr2

= cr.

crl(Z)

C-

smooth. Denote u(z) = lIyIlKUI(Z),

= v(z)cp(z).

k and,

v(z) = IIXIIQ·KVI(Z)'

Then crl and 11'2 are vector fields of class

C-

and

Besides that,

11j~(V) - J~(O)lIr s D lIyll(Q'K)-r (z

E

E;

r

= 0,

(2.5)

1, ... , k).

One can easily show by induction that

IIJ~(cp) - J~(O)lIr

s DC

~IXII2

+ lIyll2 ')-r

(r

= 0,

1, ... , k).

(2.6)

337 The required inequalities (2.4) follow from (2.5) and (2.6). Denote g\ = f • exp • crt. Clearly. g\ is a local diffeomorphism of class establish the following estimates:

C<.

Let us

(2.7)

dr(J~(gi\ • g). J~(id» ~ D

IIXII(Q-K)-r

(r

= O.

1..... k).

Using Taylor's formula. it is not hard to show that the mapping expo (0'\

=exp(O'\

+ 0':0 can be expressed in the form exp(O'\ + cr:0 = expO'\ • exp(O'z where P(O'\. 0':0: E -+ TE is a vector field satisfying the inequality

+

+

0':0

P(O'I' 0':0).

c = const (see Moser [3]). Since g = f. exp(O'\ + 0':0 and g\ = f. expO'\> we obtain g = g\ • exp(O'z

+

P(O'\. 0':0). Denote 0'3 = O'z

+ P(O'\.

0':0. Then gil

0

g = eXpO'3 and

(2.8) The promised estimates (2.7) follow from (2.4) and (2.8).

C<

To finish the proof. we need only show that the diffeomorphisms g and g\ are conjugate with one another. Consider the Lyapunov exponential rates (see III.2.1):

n(T/, b)

=

lim s up ~ In 117]1 T~II. n

n(7]1XS. b)

=

-+ +00

lim s up n

-+ +00

~ In lIiflX:1I

(b

E

A).

According to our hypothesis. there is a number CXo < 0 such that n(7]1 XS. b) (b E A). Therefore. given a number K. one can find an integer Q = Q(K) > K satisfying

sup{- w(T/, b) + r·n(T/, b) + [(Q - K) - r]·c(7J1X'". b)} < 0

bEll.

:S

CXo

+k

338 (r = 0, 1, ... , k).

Applying Lemma m.2.2, choose a integer v so that IITr" IT" Ell .1I1'j IEbl(' 1I..,.p IX'::bII (Q.K).r f

'J

(b)

(b e A;

:!S

(2.9)

114

r = 0, I, ... , k).

Denote

(z e E \ y),

1 i m sup lilt, w w

e

0

-+ z

gn(W)1I I IIlt,(w)1I

(z

E

Y);

E\Y

Bn(z) = IITgn (z)1I

(z e E).

Since the subbundle X is close to r and the mapping II Xb is close to D(j1 Xb)(Ob) (b e B) in the vicinity of the zero section, we conclude that IlflXbli is close to

1I1]l~II. Further, the mappings g and gl are assumed to be arbitrarily c!' close to f (at least, in a sufficiently small neighbourhood of the zero section). Therefore, we obtain (2.10)

For simplicity of notation, assume v Let I'"

== I"'(-r E )

=

1.

be the vector bundle of m-jets of sections IT: T -+ TE. Let r°P"

denote the Banach space of all continuous bounded sections of I'" (m = 0, I, ... , k). Let

r,m-,: I'" -+ I"'-l

a function

Pm

and ltm : rOpn -+ rOpn-l denote the canonical projections. Defme

on the fibers of ltm as follows: if ltmCl = ltmc? then

339

(note that

Pm

can take infinite values).

Let b:

JIl

-+ TE denote the projection On the

=b

target of the jet. Each element c E rOJll can be identified with a section Co E -+ TE of the bundle (TE,

TE,

For

m

= (z,

expi,l

0

element of rOpn with

and h

E

0

eXPa

gl·1

0

0

eXPa

r'"(71i.) is an arbitrary

=

E rOJll by the rule: to(c)

= g(z) ,

g(z» , a

0

small enough, then tm(c)

IICllo

-I = J""( z eXPb

satisfying J":(h)

Co

0

C

b

= gil

c*,

=

0

g(z) ,

where

z E E.

define an operator from rOpn into itself as follows: if

= I, ... , k,

C.(Z)

gil

0

hog), a

= c·,

= g(z) ,

c is an

where

b

= gi I

0

ze E

g(z) ,

C" smooth section of the vector bundle

(TE,

1["/

• Consider first the case

In

=

O. Assuming ...

0 , 1, ... , k).

(2.11)

= I, We obtain by virtue of (2.10) that

Po(toe l , toC'-) = sup {lIt..rI(Z) - t..r2(z) II II II'~Q K) II'" II'" x·: -I

lIexPb

. II II-(Q-K)·

. z

x

-I

0

gl

- ( = x,

:s sup {Dodl(z) IIbcl

. sup {[lIl[l(g(z»1I /

0

eXPa

bel

Q

y) e E,

II~II

g(z) -

h
0

) z;: (t, YEll,

gez) _ eXPb' l

0

0

g'll

o

* O} exp.

g(z)1I 11'lf1(g(Z»II'(Q-1C)}

: z-

'" }

Q

IIxll

* O}

111[1(Z)II]Q-/r

:s Do sup {dl(z)AfK(z)} po(el ,....2. zEE

E)

= l[rnc. If the norms IICllio and litho

em --

{

TE,

c(a).

2.5. Proposition. Let c l , e2 e r°p" and are sufficiently small, then

= sup

c:

Let to denote the operator from raP> into itself

E).

defined for all sufficiently small elements C·(z)

0

:s

)

(.t. y

1

n

3' Vo

•

eE

'!lxlI

p fe l ....2.

o\:, '" },

* O}

2

0

be

0

g(z)1I

340 where Do is a constant depending only on the exponential mapping. We may assume that

Do :s 2 whenever we consider the restriction of exp to a small enough neighbourhood of the zero section.

if IIclli o and IIc?ll o are sufficiently small and the mapping

Therefore,

= O.

gl is sufficiently close to g, we have (2.11) for m

Now let m

i!:

1. Then

= sup { I!J'"'I z\exPb-I . IIx II-(Q-K) +m:

gl-I

0

Z

E

0

exp.

0

(X, y) e E,

h ) - ] nt{ log Z\exPb-I

0

gl-I

exp.

0

0

J. ) 0 g II

'"2

IIXIl '" O}

..m Q-K-nl(} I 2 :s Dm sup {dl(Z)D I (Z)A1 Z) Pm(C, C) zEE

If IIcllio and IIC2 110 are small enough, we may assume that Dm:s 2, thus proving (2.11). of mappings hn : E -+ E by letting

Let us inductively define a sequence {hn }

= id, hn+l = gil 0 hn 0 g. Because gl can be chosen arbitrarily close to g, we see that all the mappings hn are so close to id: E -+ E that they can be represented in the

ho

form hn = exp

0

~n'

~n+I(Z) = exp;:

0

where ~n is a vector field on E. Thus gil

0

exp.

0

~n

0

g(z) ,

a

= g(Z) ,

b = gil

0

g(z) , Z E E.

Taking into account Proposition 2.5 and Theorem A.25, we see that the sequence {~n} is

C"

convergent. Hence, the sequence {hn } is also

Clearly, h

C"(E, E)

and

h

diffeomorphisms gl and g are

C"

E

= gil

0

hog,

C"

convergent. Denote h = lim hn ·

i.e.,

g)

0

h

= hog.

Thus,

the

conjugate to one another.

To finish the proof of Theorem 2.3, it remains to show that the diffeomorphisms f and

341 gl are also and

d'-

conjugate. Let K denote the smallest positive integer such that K

+

sup {- w(Tf- I , b)

r . C(Tf· I, b)

+

>

k

< 0

(K - r) C(Tf-IIX", b)}

bEJ\

for r

= 0,

1, ... , k. Put Q

= Q(K).

Apply Lemma 2.4 to the diffeomorphisms r l and

gil, h being replaced by WS(f). We get that f

-I

and gil are

d'-

conjugate to one another

in the vicinity of WS(f) and, consequently, in the neighbourhood of h. This would finish the proof of Theorem 2.3, but unfortunately Lemma 2.4 is not directly applicable because X is not compact. To get over this difficulty, we compactify the space X by attaching the projective space PXb to each fiber Xb (b E B) and then extend the mappings f and glover X· x Y in a natural way by taking into account the fact that j{z) for 111.11 ~ roo

= gl(z) = Lz

2.6. Remark. It is seen from the proof of Theorem 2.3 that the number Q can be determined as follows. First choose the smallest integer K satisfying K > k and ~~ {- w(Tf- I , b)

+ r C(Tf-I,

b)

+

(K - r) C(Tf-IIX", b)}

< 0

(2.12)

(r = 0, 1, ... , k),

then pick out the smallest number Q such that Q sup {- w(T/, b)

+

r C(Tj, b)

+

> K + k and

[(Q - K) - r)]

bEJ\

(r

= 0,

ccmr, b)}

<

°

(2.13)

1, ... , k).

2.7. Proof of Theorem n.3.3. Apply Theorem 2.3 to the case where h is a rest point. the notation introduced in subsection 1l.3.2. It is easy to verify that the relations (2.12) and (2.13) become

Keep

Jim '),.l

Hence

+

+

k'),.l -

(K - k)Ji\

< 0,

/qJ.m - [(Q - K) - k]'),.\

< O.

342

consequently,

K

=

[Jlm JlI

+ k (~l + 1)] + 1 > k, ~I

where [a] denotes the integral part of a. Similarly,

Q _K

= [~l + ~l

k ( Jlm + 1)] + 1 Jll

> k.

Finally,

Q

= [~l~l

+ k (

Jlm JlI

+ 1 )] +

[Jlm Jll

+ k (~l + 1 )] + 2. ~l

1.S. Proof of Theorem ll.3.16. Apply Theorem 2.3 with A replaced by the center manifold W' II~. More precisely, first compactify ~ likewise in subsection V.4.9 and then apply Theorem 2.3 to the zero section of the corresponding vector bundle.

Bibliographical Notes and Remarks to Chapter VI The problem on smooth linearization of a dynamical system in the vicinity of an arbitrary smooth invariant sub manifold was considered by Robinson [1] and Sell [1, 2]. Samovol [3] has investigated this problem for systems of differential equations near a quasi-periodic torus assuming that the linearized system is reducible to constant coefficients. Samoilenko [2] has proved several theorems on C linearization in the vicinity of an exponentially stable torus without any additional assumptions on reducibility. Theorem 1.3 generalizes the results obtained by Sternberg [1], Samoilenko [2] and Sell [1, 2] concerning exponentially stable sub manifolds. Example 1.6 is taken from the paper of Bronstein and Kopanskii [I]. Theorem 2.3 is a strengthening of a theorem due to Robinson [I], which, in tum, is a generalization of results obtained by Sternberg [2], Chen [1] and Takens [I].

343

APPENDIX SOME FACTS FROM GLOBAL ANALYSIS

We present here some definitions and facts from differential calculus and the theory of smooth manifolds, as well as several more special results repeatedly used in the course of the book (in particular, the fiber contraction principle and the smooth invariant section theorem).

Differential calculus A.1. Notation. Let z+ denote the set of all non-negative integers. Let n be a positive integer. A multiindex is an element of z~. For ot e Z~ denote + otn . Define ot(}) = sup {k e z+:

1::5 k::5 n, j > ot l

+ ... +

(in other words, to obtain the sequence {ot(})} , write

otk_l}

1 Cli

loti = ot,

+ ...

(l:s j :s loti)

times, then write

2

«,.

times, etc.) Let E I , ••• , Em F be Banach spaces. By It" we shall denote the direct sum of the family of spaces Ea.(,}) for 1::5 j ::5 IClI. Let La.(EI' ••• , En; F) denote the Banach space of all continuous symmetric IIX I-linear mappings f; from It" into F endowed with the norm

Let Pt be the canonical projection from E

==

EI

$

••• $

En onto E t • Given

(l

e Z~, let

Pot. denote the mapping x ~ (Pa.(,})(x» from E into It". The mapping f E -+ F is said to be a multihomogeneous polynomial of multi-degree (l, if there exists an element (l e La.(EI' .•• , En; F) such that f = u 0 Pot.. The image of the space LOt.(E" ••• , En; F) under the linear map u ~ u 0 POI. is denoted by Pa.(E)o ••. , En; F). The value of the

344 polynomial I

E

POI.(E\t ... , En; F)

at the element (x\t ... , xn)

•

01.

E

EI

81 ... 81

En

will

•

OI.n

01.1

be wntten as ft.xl' ... , xn) - I ' x - I . XI ..... xn • Whenever the dlrect sum decomposition E = EI 81 ••• 81 En is fixed, we shall write POI.(E; F) instead of POI.(EI , ... , En; F).

= XI 81 ... 81 Xm ,

Let X, Y and Z be Banach spaces, X

fl

E

Z:.

Y = YI 81

... 81

Yn ,

Ot E

Z':',

A.2. Lemma. There exist canonical Isomorphisms

7

A.4. (i

there are polynomials I~.a

A.3. Lemma. Given IE POI.(X; Z), such thai

E

p~.a(X,

X; Z),

+ a = Ot,

=

f(x

+ y) =

L I~a

Lemma. If IE

P/3(Y; Z),

Ott

1, ... , n),

then

the /unction

E

• x~ya

z,:"

Ut

(X, y

= (Ot~,

E

X).

... , Ot7) and

h: X ... Z defined lTy

h(x)

gt

= j{gl(x),

E

POI.t(X; Z)

... , gn (x» ,

n

belongs to the space P~(X; Z), where 7J

=

LfltOt{

(j

= 1,

... , m).

t-I

A.S. Notation. Let E and F be Banach spaces and k be a positive integer. By Lk(E; F) we shall denote the Banach space of all continuous k-linear symmetric mappings from Ff into F. Given (E LJc(E; F), the mapping f. E ... F defined by f(x) = (x, ... , x) is called a homogeneous polynomial of (full) degree k on E with values in F. The set of all such polynomials is denoted by Pk(E; F). For k = 0 define Po(E; F) = F. Denote pk(E; F)

=

k 81

P,;(E; F), ac(E; F)

=

k 81

P.(E; F) .

• -0

Let

E

= EI

81 ... II

En.

Clearly,

PJc(E; F) is the direct sum of the spaces

345 Pa.(E\ • •..• En; I') for all Il

E

Z~.

IIlI

= k.

f E C(U. /) and l:s k :S r. denotes the k-th iterated derivative of f at the point x. Recall. that

Let U be an open subset of a Banach space E. x Dk.~)

As usual.

Dk.~) is an element of Lk.(E; 1'). In particular. D~)

= E\

Let E

• •.. • En and Il

E

U.

E

L(E; 1').

E

z~. The iterated panial derivative

is denoted by Da.~). Clearly. D"1Cx)

E

La.(E; 1').

A.c). Composite mapping formula. Let E. F and G be Banach spaces; U c E and V c F be open subsets; g

E

f

C(U. V);

Dk.(f. g)(x)

E

C(V. G);

l:s k

L LI1'k.W Dqj(g(X»

=

:S

r;

X E

U. Then

. D'\g(x) ..• D'qg(x).

\:Sq:Sk. ,

where the inner summation is over all such collections i of positive integers it • ...• iq that i\ + ... + iq = k. and the coefficients 11'k.(I) are certain positive integers. In particular. for k = I we have the chain rule DC!. g)(x)

Assume. additionally. that E fJ

E

Z:'.

l:s fJ

:S

Ij! I

I!:

E\ •.•.• Em. F

. Dg(x).

=

F\ •...• Fn. 8

=

(g\O •••• 8n).

.Ie l.

E

r. Then

Here T = (T\ ••..• Tn) moreover.

=

= Dj(g(x»

E

j = {j.k. .:

n Z+;

1 (1:5 k :5 n. 1:s

S :S

1:s k:5 n.

1:5

S

} :5 Tk..

m Z+

d an.

Tk.). The summation extends on all such pairs

n

(T.l) that

1:s ITI:s IfJl

and

LC!~

+ ...

+iZ:k.)

= fJ.

Thecoefficients

e(T.l)

take positive values. A. 7. The differentiation formula for a polynomial. Let

Il E

z:'.

E = E\ •...• Em

346

If tI e

and f e Pa.(EI, ... , Em; F). •.. , m), then

Z':'

and tI:s

CIt

(i.e.,

tlt:s a.t for all i = 1, (A.1)

where 1 is the polylinear mapping that corresponds to the polynomial f and e(a., (3) are

> a. p for some number p e {I •...• m}. then iff ==

definite positive integers. If 13 p

o.

A.S. The differentiation formula for a polynomial with variable coefficients. Let E and F be Banach spaces, a. e

z,:" E

of class C. Define A: E -+ F 13 e

z,:"

= EI

ED •••

by A(X)

=

CI)

Em' and ;\: E -+ Pa.(E; F) be a mapping

[;\(x)](x) == ;\(x) . xa.. For each multiindex

I:s 1131 :s r, the following formula is valid:

r

Df3 A(X)·hf3 =

(A.2)

d(CIt. 13. JL)(if-Ji;\(x) . hf3 -Ji ) . xa.-Ji . hJi.

",:Sf3

",:Sa.

• Define a function f E

$

E -+ F by f(u, v)

= ;\(u)(v)

==

;\(u)· va.. Then A(X)

= j{x,

x).

Clearly,

vaA(X)

• h(3

r

(A.3)

C(JL, v) Ii;D'U(x, x) . hVhJi ,

Ji+v=/3 where

JL,

v e z':' and c(JL, v) are positive integers. Since f(u, v) = ;\(u) . va., we have

D"J(u, v) . h~ therefore for

JL:S

= (Dv;\(u)

. h~)(v) == (Dv;\(u) . h~)va..

a. Ii;(D"J(u. v) . h~)(u, v) . h~ = Ii;[(Dv;\(u) . h~) . vj . h~ (A.4)

= ~(Dv;\(u)·h~)(v)·h~ = e(a., 1J.)(Dv;\(u)·h~)·va.-flh~

by (A.l). If there exists a number

k e {l, ... , m}

such that

JLk

>

Cltk'

then

Ii;D'U{u. v) == O. Substituting (A.4) with u = v = x, hi = ~ = h in (A.3), we obtain the equality

347

ifA(x)

. hf3 =

which is much the same as (A.2).

Jets of Banach space mappings A.9. Dermition. Let E and F be Banach spaces, U be an open subset of E, a E U, and k be a positive integer. Let I and g belong to C(U, F), and j{a) = g(a). One says that

I and g have contact olorder k at the point a if x -+ a.

iff

as

C(U, F) and O:s k :s r. The mappings I and g have contact DSj{a) = DSg(a) lor s = 0, I, ... , k.

A.10. Lemma. Let j, g 01 order k at a

I!f{x) - g(x)n = o(nx - an k )

E

A.11. Lemma. Suppose the hypotheses 01 the preceding lemma are fulfilled. Denote

j(x) = j{x that

.f

+ a) (x

E

U - a).

There exists exactly one polynomial Pk

E

r(E; F) such

and Pk have contact olorder k at the origin. In lact, k

Pk(h)

L -{ D1f..a)

=

. h(.

I.

(=0

A.12. Definiion. Let a

E

E,

b

E

F,

and k be a positive integer.

Assume that

= g(a) = b.

The functions I and g are said to be equivalent, if they have a contact of order k at the point a. Clearly, this is indeed an equivalence

j, g

E

CC(E, F) and j{a)

=

,relation. The equivalence class containing I is denoted by l" j!W and called the k-jet of the function I at the point a. We also say that a is the source and b is the

target of the jet. We write a

= s(l)

and b

=

b(/c). The family of all k-jets

l"

with

s(/') = a and b(l') = b is denoted by ~(E, F)b' Let U and V be some open subsets of E and F, respectively. Denote

348

r(U,

U UJ!(E, F)b'

V) =

aeu bey

According to Lemma A.II, there is a one-to-one correspondence r(U, V)

(A.S)

U X V x rI(E, F).


Jets (the general case) A.13. Defmition. Let X and Y be

C

smooth manifolds and X e X. Let 1 and g be

continuous mappings defined in the vicinity of x and taking values in Y. We say that 1 and g have contact

01 order k

at

the point x if

~) =

g(x) and there exist charts

(U, fI, E) on X at x and (V,!/I, F) on Y at j(x) == g(x) such that the mappings !/I

0/ 0

,,-1

and !/I

0

g

0

fI- 1

defined in some neighbourhood of the point ,,(x) in E with

values in F have contact of order k at ,,(x). It is not hard to show that this definition does not depend on the choice of charts (U, ", E) and (V, !/I, F). Just like before, we define the notion of a

k-jet j = J~(f)

with

b(j.Jc) = j(x). The set of all k-jets of mappings from X into Y is denoted by can be provided with a natural structure of a and A.12). Let j e

reX,

y),

s(j)

= x,

and

does not depend on the choice of mapping ,J'.l:

reX,

Y) -+

i(x,

I be an

1

C-

k

manifold

integer,

O:s

reX,

and Y). It

(by using subsections A.S

I

<

k.

Clearly, the jet j~(f)

e j, and it is therefore denoted by

Y) determines a bundle of class

this bundle are diffeomorphic to a definite Banach space.

=x

s(j)

C-

k •

But whenever

,J'.l(j). The The fibers of

k

~

2,

the

bundle ,J'.l is 1IOt a vector bundle. This fact is often referred to as the 1IOn-invariance

01 higher derivatives.

=

The bundle r 1•O: i(x, Y) -+ ~(X, Y) X x Y can be provided with a canonical structure of a vector bundle. In fact, let L(Tx; Ty) denote the naturally defined vector bundle with base X x Y and fiber L(T;xX; TyY) at (x, Y) E X X Y. There is a canonical idxxy-isomorphism T:i(X, Y) -+ L(Tx; Ty) of class C- 1• Namely, if j E i(X, Y) and f E j, then 1J{s(j) .. T(j) depends only on j and belongs to the space L(T;xX; TyY),

349

x

where

= S(}),

y

= b(}). Jets of vector bundle sections

C,

A.14. Definition. Let (E, n, X) be a vector bundle of class

J!(X, y), 0 ~ k ~ r.

j

E

j

= j!(f)

and f

is a

C

x

E

X

and

One says that j is a k-jet of a section of (E, n, X), section defined on a neighbourhood of x.

Equivalently,

= j!(idx)'

jb(J)(n) • j

r(E, n, X)

The set of all such k-jets is denoted by

is

It can be naturally

r(E).

provided with the structure of a C-le vector bundle over X. Indeed, let Co be a vector bundle chart on E and c 1 there is a one-to-one mapping

9

= (U,

= (U,

!p,

Fa)

1/1, F 1) be a chart on the manifold X. Then

from r(E) I U onto .rc(I/I(U), Fa)

which determines a vector bundle chart d

if

= (U,

9, G)

AI

I/I(U) x Fo x W(F\O Fa),

on r(E) with

le

G

= Fo

X

W(F1, Fa>

=

nPm(F

1;

Fa>.

m-O

A.l!. Lemma. Let EI> ... , En and F be vector bundles of class u: EI

III ... III

En -+ F be a polylinear morphism of class

one polylinear morphism r(u): r(E1)

III ... III

C with base X. Let

C. There exists one and only

r(En) -+ r(E)

such that

for every open subset U of X and every set of C smooth sections s,: U -+ E, I U

(i = 1•

.... n).

A.16. Lemma. Let I

E

Z. 0 ~ I

< k ~ r. The mapping

surjective vector bundle morphism of class

~": r(E) -+ r(E) is a

cole. Its kernel. Hle"(E), consists of all

jets j!(f) of sections f. X -+ E having contact of order I with the zero section at the point x

E

PleCIX, E).

X.

There is a canonical vector bundle isomorphism between Hle,le-I(E) and

350

Thus, r(E) is endowed with the natural structure of a filtered (but not a graded) vector bundle. A.17. Notation. Let (E, n:, X) be a vector bundle of class C, X be a compact manifold and O:s k :s r. Provide the vector bundle r(E) with a Riemannian metric. Let rX(E) denote the Banach space of all

c!

smooth sections cr: X ~ E of (E, n:, X) equipped

with the norm II· II x' where IIcrll X =

sup {11j~(cr)lI:

X

e X}.

There exists a canonical embedding of rJt(E) into rO(r(E); namely, to every section cr

e rX(E), we put in correspondence the section iT e rO(r(E) Given an element ~ e r(E), a = s(I;).

(x e X). cr(a)

=

defined by iT(x) = j~(II")

there is a section cr e rJt(E) satisfying

~,where

A.IS. Vector bundle structure on pl(E, n:, X). As before, let (E, n:, X) be a vector bundle of class C. Recall that the subbundle VE = {v e TE: Tn:(v) = O} of the tangent bundle (TE, TE , E) is called vertical. There is a canonical vector bundle isomorphism i 1: (VE, TE, E) ~ (E 61 E, pr.. E). Fix some C· I smooth vector subbundle HE complementary to VE in TE and name it horizontal. Usually, we shall assume that HEIZ(x) = T(Z(x» , where Z: X ~ E is the zero section. The mapping i2 : (HE, TE' E) ~ (E 61 TX, pr.. E), i 2(v) = (TE(V) , Tn:(v» , is also a vector bundle isomorphism. For 11 e TE, we set lIHE = ;21(T E(lI) , Tn:(lI», where i2l: E 61 TX ~ HE is the inverse

e VE, and i HE : (TE, TE , E) ~ (E 61 TX 6) E, prl' E), where i l (lI - TlHE»' is a (non-canonical) isomorphism in the category V:B E • Observe that i HE : (TE, n: 0 T E , X) ~ E 61 TX $ E is an isomorphism in the category :Bun.x '

morphism. Clearly, (11 iHE(TI)

TlHE)

= (T E(lI) , Tn:(lI), pr2

0

Let L(TX, TE; n:) denote the vector bundle with base E, whose fiber over the point y e E consists of all linear mappings h e L(T.,.(Yr, TyE) satisfying Tn: 0 h = id. If and cr e j, then T(j) = Tcr(s{J) actually depends only on j and belongs to L(TxX, TyE), where x = s{J) and y = b{J). Thus, we have established a one-to-one

j e r(E)

351

correspondence T: pl(E) -+ L(TX, TE; If), which is a vector bundle isomorphism. If h e L(T7f ey)X, TyE),

then prl

L(TX, TE; If) -+ E is surjective and prl It is easily seen that

0

0

iHE iHE

h(T7f(YYQ

0

0

= id.

h

= {y}.

Put i;'E

i;'E: L(TX, TE; If) -+ E • L(TX; E)

category V:B E• Thus, i;'E isomorphism.

0

Therefore prl

= (prl

0

i HE , pr3

is an isomorphism

0

0

i HE : i HE).

in the

T: pl(E) -+ E • L(TX, E) is a (non-canonical) vector bundle

Let if, h) be a C automorphism of (E, If, X) in the category :Bun.. Fix some horizontal subbundle HE c: TE. Define a mapping Dvf E • E -+ E from the equality il

Let

71

(7]1 VE)

0

0

ijl(U, w)

= (ltu),

Dvf{u, w» ;;; (ltu), Dvj{u) . w).

e TE. Then

Define a mapping C: E • TX -+ E by

Observe that C is linear in the second argument. Thus, we get iHE

where CIO(u)v CIO(u) iii O.

0

11

=

0

iH~(U, v, w)

C(u, v)

= (f(u),

«u, v, w) e E • TX. E).

Given a section cr: X -+ E, set cr· pIoo: pl(E) -+ pl(E)

Th(v) , ClO (u)v

as follows: if j

=f E

0

cr

pl(E),

0

+ Dvf{u)w),

If HE is

(A.6) 7]-invariant, then

hoi. This allows to define a mapping j

= j~(cr),

then

ploo(})

= j~(X)(cr·).

The mapping pi (f): pl(E) -+ pI(E) induces a mapping 1'100: E. L(TX; E) -+ E. L(TX; E) via the isomorphism

J.lI - i;'E

0

T: pl(E) -+ L(TX; E). From (A.6) we deduce

A.19. Lemma. 'The mapping 1'100 can be written as follows:

352

Thus, (ploo, /) is an affine morphism of the vector bundle (p'(E) , b, E) onto itself. Clearly, (P'oo, h) is an automorphism of (p'(E), 5, X) in the category :Bun.. Whenever the horizontal subbundle HE is 71-invariant, (pi 00, morphism (because CIO = 0).

Morphisms of the vector bundle P(E,

'It,

/) is a vector bundle

X)

A.20. Notation. Let LIt(E) = E • L(TX; E) •••.• LIt(TX, E) (k 2: 1). Choose some horizontal subbundles HE, HL(TX; E), ... , HLIt_I(TX; E) of the tangent bundles TE, 7I.(TX; E), ... , 7I.1t_I (TX; E), respectively. Proceeding by induction, it is not hard to show that for each number k 2: 1 there exists a vector bundle idx-isomorphism "'It of (r(E), 5, X) onto LIt(E) which depends on the choice of HE, HL(TX; E), ... , HLIt_I(TX;

E).

Let if, h) be a r(E)

c!

automorphism of (E,

'It,

X) in the category :Bun. and roo: r(E) -+

be the canonical mapping induced by if, h). Denote

A.21. Lemma. The mapping

«y, where CmO(y)

E

~I,

... ,

Lm(TX; E) (1

rOO:

~It)

E

rOO = "'itl

0

roo • "'It·

LIt(E) -+ LIt(E) can be expressed as follows:

E • L(TX; E) . . . . . LIt(TX;

< m s k),

Cm(y)

E

E»,

L(L(TX; E), ... , Lm_I(TX; E), Lm(TX;

E»

(1 s m s k). Thus, (roo, /) is an qffine morphism of the vector bundle (pIt(E) , b, E) onto itself. Moreover, its linear part is triangular.

353

The morphism if, h) IIc(If.)

=I ·

If. • (Th-I)m

induces mappings lie: Lm(TX.; E) -+ Lm(TX; E) (m

HL(TX.; E), ... , HLIe_I(TX.; E)

defined

by

= 1,

... , k). Whenever the horizontal subbundles HE, are invariant under TE, TL(TX.; E), ... , TLIe_I(TX.; E),

respectively, then all the functions Cm,o, Cm (1:11 m :II k) vanish. Therefore a vector bundle morphism in this case. The material presented in A.9 - A.21 is adapted from Bourbaki [1].

r(/)

is

Smooth invariant sections A.22. Lemma. Let (X, d) be a complete metric space and t be the set 01 all contractions f. X -+ X endowed with the topology 01 uniform convergence. Let xf denote

the fixed point 01 the mapping lEt. continuous . ~

Then the mapping :>.: t -+ X, :>.(/)

= xf'

is

If I and g belong to t, then d(Xf' Xg)

= d(f(xf ) , g(Xg» :II

d(f(xf ), j{xg»

+ d(f(Xg) ,

g(xll»

therefore d(xf , xll )

:II

dC/(xg ): ~Il» 1 - LIp

:II

[1 - Lip(f)]-I sup d(f(x) , g(x». xEX

Consequently, if g tends to I in the cD-topology, then d(xf' xll) --+ O. A.23. Dermition. Let E and X be topological spaces, and p: E -+ X be a continuous

surjective mapping. Denote p-I(x) = Ex (x E X). Suppose that each fiber Ex is provided with the srtucture of a Banach space so that the mappings s: E • E -+ E, sea, b) = a + b; «: IR x E -+ E, «(k, a) = lea and II' II: E -+ IR are continuous. Then (E, p, X) is called a Banach bundle.

354

A.14. Theorem. Let (E, p, X) be a Banach bundle, h be a homeomorphismJrom X onto itself, and f E -+ E be a continuous mapping which covers h, i.e., p 1= h p. Assume that fI Ex is Lipschitlian lor all x E X and D

k

=Lip(f) = sup Lip(fl Ex)

< 1,

xEX

sup I!f{Ox) II xEX

<

D

(A.?)

ID,

where Ox denotes the zero element 01 the Banach space Ex. Then there exists a unique bou1lfkd I-invariant section rrf: X -+ E, and rrf is continuous. Besicks that, rrf depends continuously on I with respect 10 the C'-topology .

=

• That rr f is invariant means Jf..rr f(X»

rr f(X)' Let r.b denote the Banach space of all

bounded sections rr: X -+ E equipped with the norm IIrrll mapping

I

induces a transformation /,: r.b -+ r.b

fact, it suffices to show that

II/,(rr) II

<

ID.

= sup {lIrr(x)II:

defined by /,(rr) =

X

E Xl

I

<

ID.

The

D rr D h-I. In

We have

sup IlfD rr D h-I(x)1I xEX

sup Ilf,(rr)(x) II xEX :s

S up Ill' D rr(x) - Jf..Ox) II xEX

:s k sup IIrr(x) II xEX

Clearly, rr is I-invariant iff I,(rr)

=

+ sup I!f{Ox)1I xEX

+ sup I!f{0x) II <

ID.

xEX

rr. Let d denote the metric on r.b that corresponds

to the above norm. It then follows from (A.7) that d((,(rr l), /,(rr.J)

=

sup Ill' D rr 1 D h-I(x) - I D rr1 D h-l(x)1I :s k d(rrl' rr.J. xEX

Consequently, /, has a fixed point, rrf'

Since the subset r.0 of all continuous bounded

sections is I,-invariant and closed in r.b , we get rrf E r.0 , as required. Provide the set • of all Lipschitz h-morphisms f E -+ E satisfying the condition Lip(f)

< 1

with the uniform convergence topology. Further, provide the set {f,: IE.}

with the induced C'-topology. According to Lemma A.22, the mapping continuous.

I

t---+

rrf

is

355 Theorem A.24 is taken from the book by Hirsch, Pugh and Shub [1].

Fiber contraction principle A.2S. Theorem. (Hirsch and Pugh [1]). Let (E. p, X) be a Banach vector bundle and h be a homeomorphism from X onto X having a globally attracting fixed point a E X. Let f. E -+ E be a continuous mapping which covers k and satisfies the conditions (A.7). Let b f denote the fixed point of /lEa. Then bf is a globally attracting fixed point of the transformation f. E -+ E• • We must show that

lim /,(v) n-+OII

= bf

(v

E

E).

According to Theorem A.24, there exists a continuous f-invariant bounded section vf: X -+ E such that for each bounded section 11'f as n -+ 011. Let vEE. Define a section

v(y)

Denote x

= p(v).

get 1!f(V) -

={

V,

X -+ E the sequence {f,(v)}

11'

as follows:

if Y

tends to

= p(v),

0, if y;l: p(v).

Because d(f,(v), vf) -+

11' f(h"(x»

11':

°

as n -+

II -+ O. Since h"(x) -+ a and

IT'

+011

and t;(v)(hn(x»

= /,(v),

f is continuous, we have

IT'

f(hn(x»

/' -+ vf(a) as n -+ 011. It remains to show that vf(a) To this end, define a section s: X -+ E by the rule:

-+ vf(a), consequently,

s(y)

== {

bf' if Y

0, From h(a)

= a we infer

= a,

if y;l: a.

we

= bf .

356 Since /;(s) -+ IT" we obtain IT,(a) = b,.

A.26. Remark. In particular, assume that X is a complete metric space and h is a contraction. Then a = ah depends continuously on h in the uniform cfl-topology (see Lemma A.22). According to the same lemma, the section IT, depends continuously onfwith respect to the uniform cfl-topology. Since IT, = IT,(a~, continuous in f.

we conclude

that

b, is

The inverse of a Lipschitz morphism A.27. Lemma. Let B be a compact space, (X, p, B) be a vector bundle and IT: B -+ B be a homeomorphism. Further, let (L, IT) be a vector bundle isomorphism of (X, p, B) into itself and (rp, IT) be a morphism in the category :BUll.. If rp satisfies the Lipschitz condition and Lip(rp)

<

ilL-lift, then (L

+

rp, IT) is an isomorphism from

(X, p, B)

into itself and

~

See Hirsch and Pugh [1] or Bronstein [4, Theorem 9.3].

Complementary invariant vector subbundle A.28. Lemma. Let X be a paracompact space, (V, p, X) be a vector bundle, and (~, h) be a vector bundle automorphism of (V, p, X). Let VI be a ~-invarianl vector subbundle of (V, p, X), and V1 be some complementary vector subbundle. Fix a Riemannian metric on (V, p, X) so that VI 1 V1 • Let Pt : V -+ Vt (i = I, 2) denote the projectors that correspond to the splitting V = VI III V1 • Denote VIX = VI (x EX).

n

p-I(x), Vlx = V1 n p-l(x)

If k

E

sup (11;\ 1VIX" IIP1 leEK

0

;\-1

1V1,h(X)")

< 1

(A.S)

357 and

sup II PI

0

AI V2,xII

< "',

(A.9)

xex

then there exists exactly one invariant vector subbundle complementary to VI' and it can be represented as the graph 0/ a certain bounded vector bundle morphism from VI into V2. ~

There is a one-to-one correspondence between vector subbundles complementary to VI

in V and continuous sections of the vector bundle L(V2; VI)' The mapping induces a transformation A.: L(V2; VI) -+ L(V2 ; VI) defmed by the formula

where Ax = AI Vx , Ix = id: V2 ,x -+ V2 ,x' Clearly. A.(rp) e L(V2 ,h(X); VI ,h(X» covers h. Since A(VI) = VI and P2 1VI = 0, we have

A: V -+ V

,

i.e., A.

Therefore

Since VI is A-invariant,

P2 0 A • P2 = P2 • A, consequently,

[P2 0 AI V2 ,Xr l = P 2 •

A·II V2 ,h(X) • Set E = L(V2 ; VI)' / = A. and apply Theorem A.24. It follows from (A.8) that A. is uniformly fiber contracting, and (A.9) implies sup {1lf(OX)II: x e X} < "'. Thus, the hypotheses of 'Theorem A.24 are fulfilled. Let rrf: X -+ L(V2 ; VI) be the invariant section. Then {(v, ITf(x){v»: x e X, v e V2 ,x} is the required invariant vector subbundle. Remark. Whenever X is compact, the condition (A.9) is automatically fulfilled.

358 The norm of a composition operator

A.29. Lemma. Let E and F be Banach spaces and E = EI (I ... ® En' Let A: F -+ F and B t : Et -+ Et (i = 1, ... , n) be bounded linear operators. Let L(EI' ... , En; F) denote the space of all polylinear mappings with the usual norm, and B = diag [B I, ... , Bn]· Define a linear operator C: L(EI' ... , En; F) -+ L(EI, ... , En; F) by C(I;) = A 0 I; 0 B. Then IICII

= IIAII

• "BI" ..... "Bn ".

IIBII ~ O. Let E = EI ® ... ® En be the tensor product of E I, •.. , En and B: E -+ E denote the canonical linear mapping induced by B: E -+ E. Then liB II = "B I"'" IIB n II. To each element I; E L(EI' ... , En; F) we put in correspondence the canonically defined linear operator ~ E L(E; F). Recall that II~II = 111;11. For each c > 0 there exist elements Vo E E and Uo E F such that "Vo" = 1, IIUoll = 1, IIBvoll it (1 - c) liB II , and IIAuoll ~ (1 - c)IIAIi. By the Hahn-Banach theorem, one can find a linear functional f. E -+ IR such that IIjlI = 1 and I!f{Bvo) II = IIBvoli. Define a linear mapping ~o E L(E; F) by ~o(w) = ftw)Uo (w E E). Then ~o (Bvo) = IIBvollUo and lI~oll = 1. Therefore ~ Without loss of generality assume

IICII = sup {IIA ~ BII:

=

II~II = I}

sup {IIA ~o B VII

it

it

IIA ~o BII

IIA ~o 11 Voll

IIYII~I

On the other hand, lIell = sup {IIA I; BII:

II~II =

I}

Since c is an arbitrary positive number, we get the desired result.

359

Common fixed point A.30. Lemma. Let M be a set and F be a commutative family of transformations !p: M -+ M. Assume that some transformation !Po e F has exactly one fixed point mo e M. 1.7um !perno) = mo for all !P e F.

In other words, mo is a common fixed point of the

family F . • Indeed, !perno) = mo implies !p(mJ = !p(!po(mJ) = !po(!p(mJ) for all !P e F. Hence, for each element !P e F, the point !p(mo) is fixed under !Po, consequently, !p(mo) = mo because rno is the unique fixed point of !Po' Thus, !perno) = mo for all !P e F.

Smooth invariant section theorem

e

A.31. Preliminaries. Let X be a fmite dimensional

e, and

a vector bundle of class

f.

E -+ E be a

C<-

I:s k :s r. Let h: X -+ X be a

mapping which covers h, i.e .• p

0

f

Xl be

smooth manifold, (E, p,

C<-

= hop.

diffeomorphism and

Xo be an open

Let

subset of X such that h·I(XJ c: Xo and Xo is compact. Denote Eo = EIXo. Equip (E, P. X) and (TX, T X ' X) with some Riemannian metrics. For Xo e X set IXX'

fJX'

=

= IIT.fl TE.I

l11.7il(X)1I = sup {IITh·I(V)II:

11 (X')

II

ve

= sup {1I1J{W)II: W e T,fi.1

TxX.

11 (X')

IIvll:s I};

, veE. 1

11 (X')

,

IIwlI:S I}.

The morphism f. E -+ E is called k-contracting over Xo if there exist numbers c j.! e (0, 1) such that sup {IITh·n(X)lI s II if I TE

-n II: 11 (X')

> 0,

x eX} :s C j.!n (n = 1, 2, ... )

for s = 0, 1, ... , k. It is called strongly k-contracting over Xc, if ;\s

=

for s = 0, 1, ... , k.

(A. 10)

360

A.31. Remark. The following assertions are pairwise equivalent: (1) f. E -+ E

is

Xo; (2) for some number n > 0, the morphism /' is strongly over Xo; (3) /' is strongly k-contracting over Xo for all sufficiently

k-contracting over k-contracting large numbers n.

A.33. Theorem (Hirsch, Pugh and Shub [l]). Let f. E -+ E be k-contracting over

Xo.

Then there exists a CC smooth finvariant section rI'f: Xo -+ Eo. Moreover, rI'f is the globally attracting fixed point 0/ the trans/ormation Ji: r:(EO> -+ r:(EO>. If g: E -+ E is another CC morphism which is also k-contracting over Xo and g tends to / in the

C-

uniform

topology, then

rI'.

-+

rl'f

in r:(EO> .

Xo,

by Remark A.32 there is a number n such that /'

• Since / is k-contracting over is strongly k-contracting over

Xo.

We shall show that there exists a unique

CC

smooth

/,-invariant section rl'f: Xo -+ Eo. By Lemma A.30, this section will be finvariant, as well. Therefore, without loss of generality we may (and do) assume that / is strongly k-contracting over Xo. It follows from (A. 10) that ~o = sup {~x: X E Xo} < 1, hence, LipOO • sup {Lip(flEx): x E Xo} s ~o < 1. According to Theorem A.24, we have a continuous section rl'f: Xo -+ Eo which is I-invariant and, moreover, coincides with the fixed point of the contraction rl'f E

/,: r~(Eo) -+ r~(Eo).

Our goal is to prove that

Jc

rb(Eo).

Let

°

s m s k. The mapping pnoo: pn(E) -+ pn(E) induces a transformation

r~(pn(E) -+ r~(pn(E)

induction that

J100 (m

rI'f

E

defined by

r';;(EO>

J1OO(rI')

= pnoo

0

rI'

0

(Th·I)m.

J1OO:

Let us show by

and that }'"'(rrf) is an attracting fixed point of the mapping

=

= 0,

°

1, ... , k). The validity of this assertion for m was established above. Suppose that it holds for m = 0, ... , I - 1, where I - 1 < k. Let us show that it is true for m = I. The vector bundles TE, TL(TX; E), ... , 7Z.Jc_I(7X; E) are of class C-I. Choose C- I smooth horizontal subbundles HE, HL(TX; E), ... , HL Jc _I (7X; E), then the vector bundle isomorphisms JIm: (pn(E),

C-

Jc

(m

= 0,

5,

X) -+ Lm(E) ...

E. L(7X; E)

1, ... , k). The mapping roo. JIm

0

81 ... 81

pnoo

0

Lm(TX; E) are of class

JI~I is described in Lemma

361

"'m (m = 1, ... , k).

A.21. Identify p"(E) with Lm(E) via the isomorphism

= p-I(E)

• L,(TX; E), and the mapping

P"oo can "e expressed in the form

where 11,_1 = (y, ~I, ... , ~'-I) E L'_I(E) we have rO(P(E»

• L,(TX; E),

peE) -+ p-I(E)

induces

Then r(B:)

(see Lemma A.21).

= rO(p-I(E»

Since

• rO(L,(TX; E».

r!,'-I: rO(p(E» -+ rO(pl-I(E»,

peE) = p-I(E)

The mapping

where

r,,'-I:

r!,'-I(~, ",) = ~

(~ E rO(p-I(E», '" E rO(L,(TX; E»). The mapping ~oo can be written in the form

where tr

= r!-I,O(~),

E rO(P(Eo»

and cp~(~) E rO(L,(1Xo; Eo» depends only on~.

be such that r!,l-l(lJl)

where "'., IJ2

E

= r!,l-l(lJ~ = ~.

Then lJl

=

Let 111' 112

(~, "'I)'

lJ2

=

(~, ~,

rO(L,(TXo; Eo» , Therefore

= sup

IIDvj{tr

0

h-I(x»

0

("'I - ~

0

(Th-I(x»'n

xEXO

According to the inductive assumption,

trf

E

~-Ioo.

globally attracting fixed point of

coincides with the

r~(L,(1Xo; ErJf) such that 1If

E

is the globally attracting fixed point of

F,oo,

c!

and /-1 (trf)

By the fiber contraction principle (see

Theorem A.2S) there exists a section "'f

the uniform

r~-I(Eo)

E

and lJf depends continuously on

topology (see Remark A.26). It remains to prove that lJf

this end, take on arbitrary section

tro

{;'(f,(tro»}n-I,2,....

=

Since

attracting fixed point of

/(f,(tro» F.,(f) ,

E

OJ.-I (trf)'

r'(Eo)

"'f)

Poo

= /(trf)'

in To

and consider the iterative sequence

[~(f)t(l(tro»

and

1If

we conclude that the sequence

is

the

globally

°

U"~(tr v, »} n~I,2, ...

362

converges unifornily to

lIf'

uf e r~(Eo) and /(Uf) =

Because t;(uo) ~ uf in the uniform

c!

topology, we get

lIf'

Whitney extension theorem A.34. Theorem (see Abraham and Robbin [1] or Hormander [1]) Let A be a compact

subset of IR". Assume that for each multiindex

a e z~,

Ia I

s

continuous functions uOl.: A ~ LOI.(IR" , IR"). For every integer I, jUnction Uz: A x A ~ IR as follows:

rl.. UOl.(x) -

Uz(x, y) =

IOI.I-z

rl.. UOl.+/3O') (x - y) /3 / f3! II

k,

we

are

0 sis k,

IIx - yll

given

define a

Z-k

IOI.I-z 1/31 :5k-l

for x, yeA with x

* y;

Uz(x, x) = 0 for x

e A.

If all the jUnctions Uz, 0

are continuous, then one can find a jUnction v e = UOI.(x) for all x e A and lal s k.

s k,

eft (IR",

s

I

IRm) such that DOI.v(x)

Generalization of the smooth invariant section theorem A.35. Defmition. Let E, X, f and h have the same meaning as in subsection A.31. Assume additionally that X is a compact manifold. Let A be a closed subset of X satisfying h- 1(A) c A. The morphism f. E ~ E is said to be k-contracting over A if there exist such numbers c > 0 and JI. e (0, 1) that sup 1111("(x)lIS' n7/'l1E

1I:s c JI." (n = 1,2, ... )

-n

h

xeA

(x)

for s = 0, 1, ... , k. A section u: A -+ EIA is said to be in X and a

eft

eft

smooth if there exist a neighbourhood U of A

smooth section u.: U ~ EI U such that u.(x)

A.36. Theorem. If the morphism

=

u(x)

for x e A.

f. E -+ E is k-contracting over the set A, then there

363

exists a /-invariant section 11'f: A

-+ E 01 class

cf .

• It is well-known (see Husemoller [1]) that for the vector bundle (E, p, X) there

exists a vector bundle (Eo, no, X) such that

E

$

Eo

is trivial. Define a k-contracting

vector bundle h-morphism 10: Eo -+ Eo of class cf. Then 1 $ fa is k-contracting over Hence it follows that without loss of generality we may assume the vector bundle (E, n, X) to be trivial. Repeating the arguments used in the proof of Theorem A.33, one can show the existence

A.

of a section

"1

e rO(r(E) IA)

which

is

the globally

mapping ~(f). Let 11'0 e rl«E) and ITn = j

(j~n}n=I.2.... converges to as ITn(x)

=

(x, u(n) (x» ,

"1:

0

ITo

0

attracting

fixed

point of the

h-n (n = 1, 2, ... ). Then, clearly,

A -+ E uniformly on E.

u(n) maps A into IRd , where d

The section ITn can be written

= dim

Ex, x e X. Denote

and set

Lu~n)(x) -

u~n)(x, y) = II

L u~~k(y) (x - y)~ I /3! II

IIX _ yll/-I<

1",1 =1

led -I

1~I:Sk-1

for x, yeA, x.;. y, and

dzn)(x, x) = 0

(n = 1, 2, ... ). Because 1 is k-contracting

over A, we deduce that the sequence of continuous functions U~n): A x A -+ R converges uniformly to the continuous function U I : A x A -+ IR (I = 0, 1, ... , k) that corresponds to the section of functions u""

"1:

A -+ r(E) IA.

This means that UI corresponds to the collection

10:1:s k, where

(x e A).

According to the Whitney extension theorem, the section 11'0 iA is

cf smooth.

364

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R. Abraham and J. Robbin

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w.

Altherr

[1] An algorithm for enumerating all vertices of a convex polyhedron, Computing, 15 (1975), no.3, 181-193. D. V. Anosov [1] A higher-dimensional analogue of a theorem of Hadamard (Russian), Nauchn. Doklady Vysshei Shkoly, Fiz.-Mat. Nauki, 1959, no.l, 3-12. [2] Geodesic flows on closed Riemannian manifolds with negative curvature, Amer. Math. Soc.: Providence, R.I., 1969 [Russian original, 1967]. [3] On a class of invariant sets of smooth dynamical systems (Russian), Proc. of the 5th international conf. on nonlinear oscillations, 1969, Naukova Dumka: Kiev, 2 (1970), 39-45. V.I. Arnold [1] Small denominators. I. Mapping the circle onto itself, Amer. lations, Series 2, 46 (1965), 213-284 [Russian original, 1961]. [2] Small denominators. II. Proof of a theorem of A.N. Kolmogorov conditionally periodic motions under a small perturbation Russian Math. Surveys, 18 (1963), no.5, 9-36. [3] Small denominators and problems of stability of motion in mechanics, Russian Math. Surveys, 18 (1963), no.6, 85-193. [4] Geometrical Methods in the Theory of Ordinary Differential Verlag: New York, 1983 [Russian original, 1978].

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tekhn. i mat. nauk, 1983, no. 3, 16-20. Non-autonomous dynamical systems, Shtiintsa: Kishinev, 1984 [Russian]. A theorem on decomposition of weakly non-linear extensions (Russian), Differents. uravneniya i dinamicheskie systemy, Shtiintsa: Kishinev, (1985), 25-31. Necessary conditions for persistence of smooth invariant sub manifolds (Russian), Differents. uravneniya, 22 (1986), no. 4, 573-581. Jet transversality and the existence of a Green-Samoilenko function (Russian), Geometr. teoriya differents. uravnenii. Shtiintsa: Kishinev, 1990, 41-49.

I.U. Bronstein and V.P. Burdaev [1] Invariant manifolds of weakly non-linear extensions of dynamical systems, Institut matematiki AN MSSR: Kishinev, 1983 [Russian].

Preprint,

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d'un systeme d'equations differentielles dans Ie voisinage des valeurs singulieres, Bull. Soc. Math. France, 40 (1912), 324-383.

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Program., 12 (1977), no. 1, 81-96. [2] An improved vertex enumeration algorithm, Europ. 1. Oper. Res., 9 (1982), no. 4, 359-368. M.E. Dyer [1] The complexity of vertex enumeration methods, Math. Oper. Res., 8 (1983), no. 3, 381-402.

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les equations

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a

coefficients periodiques, Ann. Ecole

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128 (1959), no. 5, 880-881.

.

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des equations differentielles,

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M. W. Hirsch, C.C. Pugh and M. Shub [1] Invariant manifolds,

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Berlin-

370

L. Hormander 1. Distribution Theory and Fourier Analysis, Grundlehren der math. Wiss., 256, Springer-Verlag: Berlin Heidelberg - New York - Tokyo, 1983.

[1] The Analysis of Linear Partial Differential Operators.

D. Husemoller [1] Fibre bundles, McGraw - Hill: New York, 1966.

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371 A. KeUey [1] The stable, center-stable, center, center-unstable and unstable manifolds, J. Equat., 3 (1967), no. 4, 546-570.

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Ukraine Mal. Zh., 34 (1982), no. 1, 43-49. On relations between quadratic forms and Green's functions of linear extensions of dynamical systems on the torus (Russian), ibid., 36 (1984), no. 2, 258-262. Green's function for bounded solutions of linear systems of differential equations (Russian), Differents. uravneniya, 20 (1984), no. 4, 540-577. Weakly regular linear systems of differential equations, Ukraine Mat. Zh., 37 (1985), no. 4, 501-506. Bounded solutions of systems of linear differential equations, ibid., 39 (1987), no. 6, 727-732.

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372 M. Kurata [1] Hartman's theorem for hyperbolic sets, Nagoya Math. J., 67 (1977), 41-52.

W. T. Kyner [1] Invariant manifolds, Rend. Cire. Math. Palenno, Ser. 2, 10 (1961), F. I, 98-110. N.N. Ladis

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Lang

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and

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R. Maii~ [1] Qusi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370. [2] Persistent manifolds are normally hyperbolic, Trans. Amer. Math. Soc., 246 (1978), 261-283.

J. Mather [1] Characterization of Anosov diffeomorphisms, Indag. Math., 30 (1968), no. 5, 479-483.

J. McCarthy [1] Stability of invariant manifolds, Bull. Amer. Math. Soc., 61 (1955), no. 2, 149-150.

373 Yu.A. MitropolskU and V.L. Kulik [1] Bounded solutions of non-linear systems of differential equations (Russian), Ukrain.

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application

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ibid., 37 (1985), no. 3, 306-317. Yu.A. Mitropolskii and O.B. Lykova [1] Integral manifolds in non-linear mechanics, Nauka: Moscow, 1973 [Russian].

Yu.A. Mitropolskii, A.M. Samoilenko, and V.L. Kulik [1] Application of quadratic forms to the study of systems of linear differential equations (Russian), Differents. uravneniya, 21 (1985), no. 5, 776-788. [2] The investigation of dichotomy of linear systems of differential equations l1y the aid of Lyapunov functions, Naukova Dumka: Kiev, 1990 [Russian].

J. Moser [1] The analytic invariants of an area preserving mapping near a hyperbolic point, Comm. Pure Appl. Math., 19 (1956), no. 4, 673-692. [2] A rapidly convergent iteration method and non-linear partial differential equations. 1, Ann. Scuola Norm. Sup. Pisa, Ser. 3, 20 (1966), 265-315. [3] On a theorem of D. Anosov, J. DijJ. EqUal., S (1969), no. 3, 411-440.

Yu.l. Neimark [1] On the existence and roughness of invariant manifolds of mappings (Russian), Izv. VUZ, Radiojizika, 10 (1967), no. 3, 311-320. [2] Integral manifolds of differential equations (Russian), ibid., 321-334. [3] The method of point transformations in the theory of non-linear oscillations, Nauka: Moscow, 1972 [Russian].

V. V. Nemytskii and V. V. Stepanov [1] Qualitative theory of differential equations, N. 1., 1960 [Russian original, 1947, 1952].

Princeton Univ.

Press:

Princeton,

374 Z. Nitecki [1] Differentiable dynamics, MIT Press: Cambridge, Mass., 1971.

G.S. Osipenko [1] The behaviour of solutions of differential equations near invariant manifold~ (Russian), Differents. uravneniya, 15 (1979), no. 2, 262-271. [2] Perturbation of dynamical systems near invariant manifolds. I, II (Russian), ibid., 15 (1979), no. 11, 1961-1969; 16 (1980), no. 4, 620-628. [3] Perturbation of invariant manifolds. I - IV (Russian), ibid., 21 (1985), no. 4, 615623; 21 (1985), no. 8, 1337-1344; 23 (1987), no. 5, 818-825; 24 (1988), no. 6, 987993.

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Math. Anal. Appl., 41

(1973), 753-758.

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Stabilitiit und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Zeitschrijt, 29 (1928), no. 1, 129-160. [2] fiber Stabilitiit und asymptotisches Verhalten der LOsungen eines Systems endlicher Differezengleichungen, J. reine und angew. Math., 161 (1929), no. 1, 41-64. [3] Die Stabilitiitsfrage bei Differential-gleichungs systemen, Math. Zeitschrijt, 32 (1930), no. 5, 703-728. V.A. Pliss [1] The

reduction principle in the theory of stability of motion (Russian), [zv. AN SSSR, ser. matem., 28 (1964), no. 6, 1297-1324. 374

375

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[Russian].

H. Poincari [1] Sur

les proprietes des fonctions d~nies par les &juations aux diJjerences panielles. These. Gauthier-Villars: Paris, 1879. [2] On curves defined by differential equations, OGtz G1'ITL: Moscow, Leningrad, 1947 [Russian].

L.S. Pontryagin [1] Ordinary differential equations, Nauka: Moscow, 1974 [Russian].

c.

Pugh and M. Shub

[1] Linearization of normally hyperbolic diffeomorphisms, Invent. Math., 10 (1970), no. 3, 187-198. L.E. ReizinS [1] Local equivalence of differential equations, Zinatne: Riga, 1971 [Russian]. A.A. Reynfeld [1] Topological equivalence of differential equations with decomposed principal part (Russian), Differents. uravneniya, 9 (1973), no. 3, 465-468. [2] Reduction theorem (Russian), ibid., 10 (1974), no. 5, 838-843. [3] The reduction theorem for periodic orbits (Russian), ibid., 11 (1975), no. 10, 18111818. [4] The dynamical equivalence of the full and the reduced equations (Russian), LaN. mat. ezhegodnik, 19 (1976), 222-232. [5] Homeomorphism of dynamical systems in the vicinity of a stable invariant manifold (Russian), Differents. uravneniya, 19 (1983), no. 12, 2056-2065. [6] Differential equations in the neighbourhood of a stable invariant manifold in a Banach space (Russian), ibid., II (1985), no. 12, 2068-2071. [7] Dynamical equivalence in the vicinity of an asymptotically stable manifold in a Banach space (Russian), LaN. mat. ezhegodnik, 30 (1986), 76-91.

376 J.W. Robbin [1] A structural stability theorem, Annals of Math., 94 (1971), no. 3, 447-493. R.C. Robinson [1] Differentiable conjugacy near compact invariant manifolds, Bol. Soc. Brasil. Math.,

2 (1971), no. 1, 33-44. [2] Structural stability of vector fields, Annals of Math., 99 (1974), no. 1, 154-175. M.R. Rychlik [1] Lorenz attractors through Sil'nikov-type bifurcation. Part I, Ergodic Theory and Dynamical Systems, 30 (1990), no. 4, 793-821.

R..J. Sacker [1] A new approach to the perturbation theory of invariant surfaces, Comm. Pure Appl.

Math., 18 (1965), no. 4, 717-732. [2] A perturbation theorem for invariant manifolds and HOlder continuity, J. Math. Mech., 18 (1969), no. 8, 705-761.

R..J. Sacker and G.R. Sell [1] Existence of dichotomies and invariant splittings for linear differential systems. I - ill, J. Diff. Equat., 15 (1974), no. 3, 429-458; 22 (1976), no. 2, 478-496;

22 (1976), no. 2, 497-522. V.S. Samovol [1] On linearization of systems of differential equations in the vicinity of a singular point (Russian), Doklady AN SSSR, 206 (1972), no. 3, 542-548. [2] On linearization of an autonomous system on the plane in the vicinity of a singular point (Russian), Vestnik MGU, Ser. matem., mech., 1973, no. 1, 39-45. [3] Linearization of systems of differential equations in the neighbourhood of toroidal manifolds (Russian), Trudy Mosk. mat. obschestva, 38 (1979), 187-219. [4] Equivalence of systems of differential equations in the vicinity of a singular point (Russian), ibid., 44 (1982), 213-234.

377

[5] Linearization of an autonomous system in the neighbourhood of a hyperbolic singular point (Russian), Differents. uravneniya, 23 (1987), no. 6, 1098-1099. [6] On smooth linearization of systems of differential equations in the neighbourhood of a saddle singular point (Russian), Uspekhi mar. naulc, 43 (1988), no. 4, 223-224. [7] On some conditions for smooth linearization of an autonomous system in the vicinity of a singular point (Russian), lzv. AN Kazakh. SSR, ser. ./i1..-mar. naulc, 1988, no. 3, 41-44. [8] Linearization of a system of ordinary differential equations in the vicinity of a singular point (Russian), Doklady AN Ukrain. SSR, ser. A, jiz.-mar. tekhn. nauki, 1989, no. I, 30-33. [9] A necessary and sufficient condition for smooth linearization of an autonomous system on the plane in the neighbourhood of a singular point (Russian), Mat. :zametki, 46 (1989), no. I, 67-77. [10] A criterion for C smooth linearization of an autonomous system in the vicinity of a non-degenerate singular point (Russian), ibid., 49 (1991), no. 3, 91-96.

A.M. Samoilenko [1] On

the persistence of an invariant torus under perturbations (Russian), 11.v. AN SSSR, ser. mar., 34 (1970), no. 5, 1219-1240. [2] On the reduction of a dynamical system in the neighbourhood of an invariant torus to the canonical form (Russian), ibid., 36 (1972), no. 1, 209-233. [3] Necessary conditions for the existence of invariant tori of linear extensions of dynamical systems on a torus (Russian), Differents. uravneniya, 16 (1980), no. 8, 1427-1437. [4] Green's function of a linear extension of a dynamical system on a torus, the conditions ensuring its uniqueness and properties following from these conditions (Russian), Ukrain. Mat. Zh., 32 (1980), no. 6, 791-797. [6] Elements of the marhematical theory of oscillations with many frequencies, Nauka: Moscow, 1987 [Russian].

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378 [2] Exponential dichotomy of an invariant torus of a dynamical system (Russian), Differents. uravneniya, 15 (1979), no. 8, 1434-1443. [3] On splitting of linearized systems of differential equations (Russian), Ukrain. Mat. Zh., 34 (1982), no. 5, 587-593. [4] Splitting of linear extensions of dynamical systems on a torus (Russian), Doklady AN Ukrain. SSR, ser. A, ftz.-mat. tekhn. nauki, 1984, no. 12, 23-27.

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1283-1295. [3] Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), no. 5, 10351091.

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Noordhoff: Leiden, 1975 [Russian original,

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379 S. Sternberg [1] Local contractions and a theorem of Poincare Amer. J. Math., 79 (1951), 809-824. [2] On the structure of local homeomorphisms of Euclidean n-space. n, ibid., 80 (1958), 623-631.

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systems (Russian), Priblim. metody analiza nelineinykh kolebanii, Naukova Dumka: KJev, 1984, 128-133. [2] Necessary condition for the existence of an invariant manifold of a linear extension of a dynamical system on a compact manifold (Russian), Ukrain. Mat. Zh., 36 (1984), no. 3, 390-393. [3] A criterion for rough diagonalizability of linear extensions of compact flows (Russian), ibid., 37 (1985), no. 4, 523-528. E.M. Vaisbord [1] On equivalence of systems of differential equations in the vicinity of a singUlar point (Russian), Nauchn. Doklady Vysshei Shko/y, Fiz.-Mat. Nauki, 1958, no. 1, 37-42

380

SUBJECT INDEX

Adapted Riemannian metric

m.2.6

Affme extension

m.S.1

Asymptotic phase

1.4.6;

Attractor

m.1.1

Bounded motion

m.3.1

Cascade

1.1.8

Chain recurrent point

m.1.1

Cocycle

m.2.6

Contact of order k - - - - - - - - (Q, k) - - - - - - - - [I, k] - - .... - - - - - (Q, I, k)

A.9 11.3.1, VI.2.2 VI. 1.2 VI. 1.2

k-contracting morphism

A.31

Demiperiodic pseudochart

1.3.6

T-divisible polynomial

11.6.1

Dual linear extension

m.1.8

Dual vector bundle

m.L7

381

Dynamical system

m.1.1

Exponential dichotomy

m.2.6

Exponential separation

m.2.4

Exponential splitting

m.2.6

Filtraction

III. 1.3

Floquet-Lyapunov theorem

1.3.9

Flow

1.1.6

Green-Samoilenko function

III.5.6

Grobman-Hartman theorem

1.2.5; 1.2.11; 1.3.13; IV.3.5

Hadamard-Bohl-Perron theorem Homomorphism of dynamical systems Hyperbolic fixed point

I

1.4.3; 1.4.10; V.2.2; V.4.7

m.l.s 1.2.2

Hyperbolic linear extension

m.2.6

k-hyperbolic linear extension

m.6.1

Hyperbolic periodic orbit

1.3.10

Hyperbolic singular point

1.2.9

Jet of a mapping Jet of a vector bundle section k-jet hyperbolicity condition

A.12, A.13 A.14 m.6.1

382

k-jet transversality condition

Linear extension Local C conjugacy

Locally invariant sub manifold Lyapunov exponent (number) Lyapunov function Lyapunov metric

Manifold (stable, unstable, center, center-stable, center-unstable) Morse collection Morse set Nodal singular point Non-homogeneous linear extension Non-trivial motion Normal k-hyperbolicity Normal linear extension

CC-persistence Pullback of a linear extension Quotient linear extension

m.6.1 m.1.8 1.1.10, 1.1.14 1.4.2 m.2.1 m.1.l m.2.6 1.4.4, 1.4.10

m.1.3 m.1.3 11.2.13

m.5.1 m.3.1 V.2.1 V.1.3

V.1.3 m.1.8 111.3.6

383 Regular linear extension

m.S.3

Repeller

m.I.I

Resonant normal form of a jet

n.2.9; n.2.17

Riemannian metric

m.I.7

Saddle singular point

n.2.13

Smale suspension

1.1.8

Smooth linear extension

m.6.1

Tangent linear extension

m.1.S; V.1.3

Topological transformation group

m.1.1

Transversality condition

m.3.3

Twisted (untwisted) periodic orbit

1.3.2

Uniformly weakly regular linear extension

m.5.6

Vertical contact of order Q

Il.3.14

Vertical contact of order (Q, k)

n.3.17

Weakly regular linear extension

IlLS. 3

/'

Whitney sum of linear extensions

m.1.7

LIST OF SYMBOLS

Dif~(M)

1.1.9

C(n, b), "'(11:, b)

Homeo,,(M)

1.1.9

JtI,

W"", WU, W', W'., W'u

1.4.4

W"",WU,~,W:

V.2.1

Dift:;:(E)

11.1.4

L.,CEII ... , En; F)

A.l

r~(E)

11.2.1

PIJI.(EIo

A.l

E't'~

11.3.1

LIe(E, F), PIe(E, Ii)

Qo(k)

11.3.3

jt(X, 1')

A.13

QI

11.3.4

r(E)

A.14

11.4.2; 11.7.4 11.4.5; 11.7.2 IT.S.2 11.5.3 11.5.6 11.5.7

~(E)

A.17

S(k) MS(k)

A(k, Ir) S(k)

Ao(k, Ir) So(k) AI(k, Ir) SI(k) [I(k) [2(k)

trek) C(k)

11.7.13 II.7.1S

SoCk) 1l(f)

11.5.15 n.5.16 11.7.9 11.7.10 11.7.13

III.l.l; III.1.3

i.

fJ, WU(A, fJ

:Bun

m.1.1 m.1.7 III. 1.7

I-

III.1.7

W(A, V2J

~

"'1

m.2.1 m.3.1

En; F)

A.5