Jacob Palis, Jr. Welington de Melo
Geometric Theory of Dynamical Systems An Introduction Translated by A. K. Manning
Jacob Palis, Jr. Welington de Melo
Geometric Theory of Dynamical Systems An Introduction Translated by A. K. Manning
With 114 Illustrations
Springer-Verlag New York Heidelberg
Berlin
Jacob Palis, Jr. Welington de Melo Instituto de Matematica Pura e Aplicada Estrada Dona Castorina 110 Jardim Botilnico 22460 Rio de Janeiro-RJ
A. K. Manning (Translator) Mathematics Institute University of Warwick Coventry CV4 7AL England
Brazil
AMS Subject Classifications (1980): 58-01, 58F09, 58FI0, 34C35, 34C40
Library of Congress Cataloging in Publication Data Palis Junior, Jacob. Geometric theory of dynamical systems. Bibliography: p. Includes index. 1. Global analysis (Mathematics) 2. Differentiable dynamical systems. 1. Melo, Welington de. II. Title. QA614.P2813 514' 74 81-23332 AACR2
© 1982 by Springer-Verlag New York Inc All rights reserved No part of this book may be translated or reproduced in any form without written permission from Spninxer-Verlag. 175 Fifth Aveeme New York, New York 10010, U S A Printed in the United States of America
987654321 ISBN 0-387-90668-1 Springer-Verlag New York Heidelberg Berlin ISBN 3-540-90668-1 Spnnger-Verlag Berlin Heidelberg New York
Acknowledgments
This book grew from courses and seminars taught at IMPA and several other
institutions both in Brazil and abroad, a first text being prepared for the Xth Brazilian Mathematical Colloquium. With several additions, it later became a book in the Brazilian mathematical collection Projeto Euclides, published in Portuguese. A number of improvements were again made for the present translation. We are most grateful to many colleagues and students who provided us with useful suggestions and, above all, encouragement for us to present these
introductory ideas on Geometric Dynamics. We are particularly thankful to Paulo Sad and, especially to Alcides Lins Neto, for writing part of a first set of notes, and to Anthony Manning for the translation into English.
Introduction
... eette etude qualitative (des equations diferentielles) aura par elle-meme un interet du
premier ordre ... HENRI POINCARE, 1881.
We present in this book a view of the Geometric Theory of Dynamical Systems, which is introductory and yet gives the reader an understanding of some of the basic ideas involved in two important topics: structural stability and genericity. This theory has been considered by many mathematicians starting with Poincare, Liapunov and Birkhoff. In recent years some of its general aims were established and it experienced considerable development. More than two decades passed between two important events: the work
of Andronov and Pontryagin (1937) introducing the basic concept of structural stability and the articles of Peixoto (1958-1962) proving the density of stable vector fields on surfaces. It was then that Smale enriched the theory substantially by defining as a main objective the search for generic and stable properties and by obtaining results and proposing problems of great relevance in this context. In this same period Hartman and Grobman showed that local stability is a generic property. Soon after this Kupka and Smale successfully attacked the problem for periodic orbits. We intend to give the reader the flavour of this theory by means of many
examples and by the systematic proof of the Hartman-Grobman and the
Stable Manifold Theorems (Chapter 2), the Kupka-Smale Theorem (Chapter 3) and Peixoto's Theorem (Chapter 4). Several of the proofs we give
vii
Introduction
are simpler than the original ones and are open to important generalizations. In Chapter 4, we also discuss basic examples of stable diffeomorphisms with infinitely many periodic orbits. We state general results on the structural
stability of dynamical systems and make some brief comments on other topics, like bifurcation theory. In the Appendix to Chapter 4, we present the important concept of rotation number and apply it to describe a beautiful example of a flow due to Cherry. Prerequisites for reading this book are only a basic course on Differential
Equations and another on Differentiable Manifolds the most relevant results of which are summarized in Chapter 1. In Chapter 2 little more is required than topics in Linear Algebra and the Implicit Function Theorem and Contraction Mapping Theorem in Banach Spaces. Chapter 3 is the
least elementary but certainly not the most difficult. There we make, systematic use of the Transversality Theorem. Formally Chapter 4 depends on Chapter 3 since we make use of the Kupka-Smale Theorem in the more elementary special case of two-dimensional surfaces. Many relevant results and varied lines of research arise from the theorems proved here. A brief (and incomplete) account of these results is presented in the last part of the text. We hope that this book will give the reader an initial perspective on the theory and make it easier for him to approach the literature. Rio de Janeiro, September 1981.
JACOB PALIS, JR. WELINGTON DE MELO
Contents
List of Symbols
xi
Chapter 1
Differentiable Manifolds and Vector Fields §0 Calculus in 6d" and Differentiable Manifolds §1 Vector Fields on Manifolds §2 The Topology of the Space of C' Maps §3 Transversality §4 Structural Stability
I 1
10 19
23 26
Chapter 2
Local Stability
39
§1 The Tubular Flow Theorem
39
§2 Linear Vector Fields
41
§3 Singularities and Hyperbolic Fixed Points §4 Local Stability §5 Local Classification §6 Invariant Manifolds §7 The A-lemma (inclination Lemma). Geometrical Proof of Local Stability
54 59 68 73 80
Chapter 3
The Kupka-Smale Theorem
91
§1 The Poincare Map §2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic §3 Transversality of the Invariant Manifolds
92 99 106 ix
X.
Contents
Chapter 4
Genericity and Stability of Morse-Smale Vector Fields
115
§1 Morse-Smale Vector Fields; Structural Stability §2 Density of Morse-Smale Vector Fields on Orientable Surfaces §3 Generalizations §4 General Comments on Structural Stability. Other Topics Appendix: Rotation Number and Cherry Flows
116 132 150 153 181
References
189
Index
195
List of Symbols
real line Euclidean n-space C" complex n-space C" differentiability class of mappings having n continuous derivatives infinitely differentiable C'° real analytic CO df(p), df, or Df(p) derivative of f at p (a/at)f, of/at partial derivative partial derivative with respect to the second variable D2 f (x, y) R
U8"
ntlf derivative off at p space of linear mappings L'(Rr; 98k) space of r-linear mappings norm composition of the mappings g and f gof empty set f1M restriction of map f to subset M U closure of set U TM, tangent space of M at p TM tangent bundle of M £(M) space of C' vector fields on M f. X vector field induced on the range of f b X X, diffeomorphism induced by flow of X at time t CV(p) orbit of p urlimit set of p W(p) a(p) a-limit set of p S" unit n-sphere d"f (p)
L(fd", 62"') II
U
0
xi
xii
List of Symbols
T2
two-dimensional torus gradient field off integral of f identity map of Al Riemannian metric inner product in the tangent space of p defined by Riemannian metric space of C' mappings C'-norm space of C' diffeomorphisms f is transversal to S orbit of X through p positive orbit of p derivative at t of map of interval n-dimensional torus space of linear operators on R" complex vector space of linear operators on C"
grad f
jf
idM >
C'(M, N) II1
II
1
Diff'(M) f T- S CIX(p)
pi+(p)
a'(t) T"
Y(l
)
2(C") Lk
LoL......L
Exp(L), e' exponential of L group of invertible linear operators of R" GL(R) space of hyperbolic linear isomorphisms of 08" H(R-) space of hyperbolic linear vector fields of fib" .*'(R") spectrum of L Sp(L) space of vector fields having all singularities simple 4o determinant of A det(A) hyperbolic space of vector fields having all singularities '1 space of diffeomorphisms having all fixed points elementary Go space of diffeomorphisms whose fixed points are all hyperbolic G1 space of continuous bounded maps from R to it Cb(Rm) dimension of M dim M stable manifold of p WS(p) unstable manifold of p W"(p) stable manifold of size (3 W,(p) unstable manifold of size j W;(p) local stable manifold W11o0) local unstable manifold W;«(0) 9F 12
dr(T) L1(X)
(XX)
M-S am
int A
space of vector fields in 91 whose closed orbits are all hyperbolic space of vector fields in 1.41 whose closed orbits of period :5 Tare all hyperbolic union of the a-limit sets of orbits of X union of the w-limit sets of orbits of X set of nonwandering points of X set of Morse-Smale vector fields
boundary of M interior of set A.
Chapter 1
Differentiable Manifolds and Vector Fields
This chapter establishes the concepts and basic facts needed for understanding later chapters.
First we set out some classical results from Calculus in R", Ordinary Differential Equations and Submanifolds of W. Next we define vector fields on manifolds and we apply the local results o, the Theory of Differential Equations in R" to this case. We introduce the qualitative study of vector
fields, with the concepts of a- and w-limit sets, and prove the important Poiricare-Bendixson Theorem. In Section 2 we define the C' topology on the set of differentiable maps between manifolds. We show that the set of C' maps with the C' topology is a separable Baire space and that the COO maps are dense in it. From this we obtain topologies with the same properties for the spaces of vector fields and diffeomorphisms. Section 3 is devoted to the Transversality Theorem, which we shall use frequently. We conclude the chapter by establishing the general aims of the Geometric or Qualitative Theory of Dynamical Systems. In particular we discuss the concepts of topological equivalence and structural stability for differential equations defined on submanifolds of R".
§0 Calculus in R" and Differentiable Manifolds In this section we shall state some. concepts and basic results from Calculus in R', Differential Equations and Differentiable Manifolds. The proofs of the facts set out here on Calculus in R" can be found in [46], [48] ; on Differential 1
2
1
Differentiable Manifolds and Vector Fields
Equations in the much recommended introductory texts [4], [417, [116] or the more advanced ones [33], [35] and also [47];on Differentiable Manifolds in [29], [38], [49]. Let f : U c R' --* R' boa map defined on the open subset U of W. We say that f is differentiable at a point p of U if there exists a linear transformation T : If8' -. l such that, for small v, f (p + v) = f (p) + T(v) + R(v) with
Jim.-o R(v)/llvll = 0. We say that the linear map T is the derivative off at p and write it as d f (p) or sometimes d f, or D f (p). The existence of the derivative
of f at p implies, in particular, that f is continuous at p. If f is differentiable at each point of U we have a map df: L(IWM, Pk) which to each p in U associates
the derivative off at p. Here L(R", R') denotes the vector space of linear maps from 68m to R" with the norm 11 T11 = sup{IITTu; Ilvil = 1). If df is continuous we say that f is of class C' in U. It is well known that f is C' if and only if the partial derivatives of the coordinate functions off, Of '/ax;: U -+ 68, exist and are continuous. The matrix of d f (p) with respect to the canonical bases of P' and 18" is [(af `; ax;)(p)]. Analogously we define df (p) as the
derivative of df at p. Thus d2f(p) belongs to the space L(ll'", L(P", P")), which is isomorphic to the space L2(R'"; 18") of bilinear maps from U8"' x I8'"
to R. The norm induced on L2(R; R') by this isomorphism is
IIBII =
sup{IIB(u, v)Il;hull = Ilvil = 1}. We say that f is of class C2 in U if d2f: U -> L2(lr; R") is continuous. By induction we define d'f (p) as the derivative
at p of d'- 'f. We have d' (p) e L'(Pr;.R ). where L' R; R) is the space of
r-linear maps with the norm
if C 1 = sup{ li C(vl, ... , v,)II ; IIv, II =
_
11v,11 = 1). Then we say that f is of class C' in U. if d'f : U -> L'(P; 08") is continuous. Finally, f is of class C°° in U if it is of class C' for all r 0. We remark that f is of class C' if and only if all the partial derivatives up to order r of the coordinate functions off exist and are continuous. Let U, V be open
sets in P' and f : U -+ V a surjective map of class- C'. We say that f is a diffeomorphism of class C' if there exists a map g: V - U of class C' such that g o f is the identity on U.
0.0 Proposition. Let U e P' be an open set and f": U -+ R' be a sequence pf maps of class C'. Suppose that f" converges pointwise to f : U - P" and that the sequence df" converges uniformly to g: U -+ LQr, P"). Then f is of class C' and df = g.
0.1 Proposition (Chain Rule). Let U e R' and V e P" be open sets. If f : U -+ P" is differentiable at p e U, f (U) e V and g: V -+ P' is differentiable
at q = f (p), then g o f : U -+ P" is differentiable at p and d(g o f)(p) _ dg(f(p)) o df(p) Corollary 1. 1f f and g are both of class C', then g o f is of class C.
-
Corollary 2. If f : U - P'' is differentiable at p e U and a: (-1, 1) -+ U is a curve such that a(0) = p and (d/dt)2(0) = v, then f o a is a curve which is differentiable at 0 and (d/dtx f ax)(0) = df(p)v. -
§0 Calculus in Si" and Differentiable Manifolds
3
0.2 Theorem (Inverse Function). Let f : U c R" - 1W" be of class C, r >- 1. If df(p): R" R'" is an isomorphism, then f is a local d feomorphism at p E U of class C'; that is, there exist neighbourhoods V c U of p and W c R' of
f(p)andaC'mapg:W--+Vsuch that gc f =1,andfeg=1w,where 1, denotes the identity map of V and I w the identity of W.
0.3 Theorem (Implicit Function). Let U c 1" x R" be an open set and f : U -. Q8' a C map, r >_ 1. Let zo = (xo, yo) E U and c = f(zo). Suppose that the partial derivative with respect to the second variable, DZ f(zo): 68" -- 1", is an isomorphism. Then there exist open sets V e II8" containing xo
and W c U containing zo such that, for each x e.V,, there exists a unique fi(x) E I8" with (x, S(x)) E W and f(x, g(x)) = c. The map : V -- R", defined in
this way, is
of class C' and its
derivative is
given by d(x)
[D2f(x, fi(x))]-' ° Dif(x, fi(x)). Remark. These theorems are also valid in Banach spaces.
0.4 Theorem (Local Form for Immersions). Let U c R' be open and f : U - 18""+" a C' map, r >_ 1. Suppose that, for some xo e U, the derivative df (xo): R"` -+ R'" +A is injective. Then there exist neighbourhoods V C U of xo,
W c 98" of the origin and Z c 118'"+" of f(xo) and a C' diffeomorphism h: Z V x W such that h o f (x) = (x, 0) for all x e V O
0.5 Theorem (Local Form for Submersions). Let U c Rm " be open and f : U -4 68" a Cr map, r > 1. Suppose that, for somv zo E U, the derivative df (zo) is surfeetive. Then there exist neighbourhoods Z c U of zo, W c 118" of c = f (zo) and V e 1W" of the origin and a C' diffeomorphism h: V x W -+ Z
such thatf oh(x,w)=wforall xaVand wEW Let f : U e R' --+ R' be a C' map, r 21. A point x e U is regular if df(x) is surjective; otherwise x is a critical point. A point c e R8" is a regular value if every x e f `(c) is a regular point; otherwise c is a critical value. A subset of I8" is residual if it contains a countable intersection of open dense subsets. By Baire's Theorem every residual subset of a8" is dense.
0.6 Theorem (Sard'[64]). If f : U c Ir -+ R" is of class C°° then the set of regular values off is residual in 1R".
We should remark here that if f -'(c) = 0 then c is a regular value. For the existence of a regular point x e U we need m -> n. If m < n all the points
of U are critical and, therefore, f(U) is "meagre" in R", that is R" - f(L') is residual. We are going now to state some basic results on differential equations. A
vector field on an open set U c 1W" is a map X: U -+ R. We shall only consider fields of class C, r >_ 1. An integral curve of X, through a point
4
1
Differentiable Manifolds and Vector Fields
p e U, is a differentiable map a: I -+ U, where I is an open interval containing
0, such that x(0) = p and Y(t) = X(a(t)) for all t c- 1. We say that a is a solution of the differential equation dx/dt = X(x) with initial condition x(0) = p. 0.7 Theorem (Existence and Uniqueness of Solutions). Let X be a vector field
of class C', r > 1, on an open set U c s' and let p c- U. Then there exists an integral curve of X, a: I --+ U, with a(0) = p. 1 f /3: J -+ U is another integral curve of X with /3(0) = p then a(t) = #(t) for all t e I n J.
A local flow of X at p e U is a map q : (-s, e) x VD
U, where V. is a
neighbourhood of p in U, such that, for each q c- Vi,, the map coq: (- E, e) --+ U, defined by cpq(t) = cp(t, q), is an integral curve through q: that is, (p(0, q) = q
and (c'/dr)cp(t, q) = X(gp(t, q)) for all Q. q) e ( - E, r) x V.
0.8 Theorem. Let X be a vector field of class C' in U, r > 1. For all p e U there exists ti 'local flow, (p: (-t;, E) x V. - U, which is of class C'. We also have
D, D2 rp(t, q) = DX((p(t, q)) D2 cp(t, q)
and D2 (p(0, q) is the identity map of R", where Dt and D2 denote the partial derivatives with respect to the first and second variables.
We can also consider vector fields that depend on a parameter and the dependence of their solutions on the parameter. Let E be a Banach space and
F: E x U -» R' a C' map, r > 1. For each e e E the map F,: U
R',
defined by F,(p) = F(e, p), is a vector field on U of class Cr. The following theorem shows that the solutions of this field F, depend continuously on the parameter e e E. 0.9 Theorem. For every e e E and p e U there exist neighbourhoods 11' of e in F, and V of p in U and a C' map (o: (-c, i.') x V x W -+ U such that (p(0, q, A) = q, D t cp(t, q, At) = F(2, (p(t, q A))
for ezerv (t, q. A) E(-E,, ,) x f x W Next we introduce the concept of differentiable manifold. To simplify the exposition we define manifolds as subsets of R'. At the end of this section we discuss the abstract definition. Let M be a subset of Euclidean space lk. We shall use the induced topology
on M: that is, A c M is open if there exists an open set A' c R' such that A = A' n M. We say that M e RI is a differentiable manifold of dimension m if, for each point p E M, there exists a neighbourhood U e M of p and a homeomorphism x: U -+ Uo, where Uo is an open subset of 1m, such that
40 Calculus in 88" and Differentiable Manifolds
5
Figure I
the inverse homeomorphism x ': Uo -+ U c Rk is an immersion of class C'. That is, for each u e U0. the derivative dx -'(u): .,T'" --+ Rk is injective. In this case we say that (x, U) is a local chart around p and U is a coordinate neighbourhood of p. If the homeomorphisms x - ' above are of class C' we say that M is a manifold of class Cr. What we have called a differentiable manifold
corresponds to one of class C. It follows from the Local Form for Immersions 0.4 that, if (x, U) is a local chart around p e M, then there exist neighbourhoods A of p in R', V of x(p) and W of the origin in Rk ° ` and a C"
diffeomorphism h: A - + V x W such that, for all q e A n M, we have h(q) = (x(q), ©). In particular, a local chart is the restriction of a C' map of an open subset of fk into R' (Figure I). From this remark we obtain the following proposition.
0.10 Proposition. Let x: U -+ R' and Y: V -+ ff"' he local charts in M. If x(I" n V) ti(U n V) U n V s 0 then the change o/ coordinates ' x is a C' diffeomorphism (Figure 2).
C.]
Figure 2
6
1
Differentiable Manifolds and Vector Fields
We shall now define differentiable maps between manifolds. Let M'" and N" be manifolds and f : Mm --p N" a map. We say that f is of class C' if, for each point p e M, there exist local charts x: U R' around p and y: V - R"
with f (U) c V such that y - f o x - ': x(U) -' y(V) is of class Cr. As the changes of coordinates are of class C`' this definition is independent of the choice of charts. Let us consider a differentiable curve a: (-E, E) -+ M c R' with a(0) = p. It is easy to see that a is differentiable according to the above definition if and only if a is differentiable as a curve in Rk. Hence there exists a tangent vector (da/dt)(0) = a'(0). The set of vectors tangent to all such curves a is called the
tangent space to M at p and denoted by TM.. Let us consider a local chart x: U R', x(p) = 0. It is easy to see that the image of the derivative dx-'(0) coincides with TMp. Thus TMp is a vector space of dimension m. Let f: M -* N be a differentiable map and v e TM, p e M. Consider a differentiable curve a: (-E, c) M with a(0) = p and a'(0) = v. Then
f - a: (-E, E) -+ N is a differentiable curve, so we can define df(p)v = (d/dtx f o ax0). This definition is independent of the curve a. The map df (p) : TMp -b TN Jip) is linear and is called the derivative off at p. As a differentiable manifold is locally an open subset of a Euclidean space
all the theorems from Calculus that we listed earlier extend to manifolds.
0.11 Proposition (Chain Rule). Let f : M N and g: N -+ P be maps of class C' between differentiable manifolds. Then g o f : M P is of class C and d(g ° f)(p) = dg(f (p)) ° df (p)
A map f : M -+ N is a C' diffeomorphism if it is of class C' and has an inverse f'' of the same class. In this case, for each p e M, df(p): TMp TN f(p) is an isomorphism whose inverse is df -'(f (p)). In particular, M and N have the same dimension. We say that f : M N is a local diffeomorphism
at p e M if there exist neighbourhoods U(p) c M and V(f (p)) c N such that the restriction off to U is a diffeomorphism onto V.
0.12 Proposition (Inverse Function). If f: M - N is of class C', r >_ 1, and df (p) Is an isomorphism for some p e M then f is a local diffeomorphism of class
C'atp. Now consider a subset S of a manifold M. S is a submanifold of class C' of M of dimension s if, for each p e S, there exist open sets U c M containing p, R'"_5 containing 0 and a diffeomorphism of V c 1W' containing 0 and W c
class C' p: U -+ V x W such that cp(S n U) = V x {0} (Figure 3). We remark that 68' is a differentiable manifold and that, if M c R is a manifold as defined above, then M is a submanifold of Rk. The submanifolds
of M c R' are those submanifolds of l
that are contained in M.
§0 Calculus in R" and Differentiable Manifolds
V
Figure 3 N'"+" be 0.13 Proposition (Local Form for Immersions). Lei f : Rfi° a map of class C, r z 1, and p e M a point for which df(p) is injective. Then there exist neighbourhoods U(p), V (f (p)), Uo(0) in I8m and V0(U) in 68" and C' Uo x V O such that 0 o f c cp-'(x) = d(eomorphisms tp: U -+ Uo and V/: V
(x, 0).-
A C' map f : M -+ N is an immersion if df(p) is injective for all p E M. An f(M) c N is a injective immersion f : M -+ N is an embedding if f : M homeomorphism, where f (M) has the induced topology. In this case f (M) is a submanifold of N. If f : M -- N is only an injective immersion we say that f(M),is an immersed submanifold. The examples of Figure 4 show submanifolds that are immersed but not embedded.
0.14 Proposition (Local Form for Submersions). Let f : M' +" - N" be a map of class C, r a 1, qnd p e M a point for which df (p) is surjective. There exist neighbourhoods U(p), V(f (p)), Uo(0) in U8' and Vo(0) in Wand diffeomorphisms q : U =+ Uo .x. Vo and t/i: V -+ Vo such that Vi o f o (p-'(x, y) = Y.
A point q e N is a regular value off: Mm - N", where fis aC'map with r > 1, if, for all p e M with f (p) = q, df(p) is surjective. It follows from the last proposition that f -'(q) is a C' submanifold of M of dimension m - n.
Figure 4
8
1
Differentiable Manifolds and Vector Fields
0.15 Proposition (Sard). Let f : M -+ N be a C" map. The set of regular values off is residual; in particular, it is dense in N. The proof of Proposition 0.15 follows from Sard's Theorem by taking local
charts and using the fact that every open cover of a manifold admits a countable subcover.
We remark that, if M is compact, the set of regular values off: M -+ N is open and dense in N. Let us consider a countable cover (U.) of a manifold M by open sets. We say that this cover is locally finite if, for every p e M, there exists a neighbourhood V of p which intersects only a finite number of elements of the
cover. A partition of unity subordinate to the cover {U"} is a countable collection {gyp"} of nonnegative real functions of class C°° such that:
(a) for each index n the support of tp" is contained in U,,; recall that the support of 9. is the closure of the set of points where qp, is positive; (b) >"cp"(p) = 1 for all p E M. . `
0.16 Proposition. Given a locally finite countable cover of M there exists a partition of unity subordinate to this cover. Corollary 1. Let K e M be a closed subset. There exists a map f : M --+ R of class C°° such that f -'(0) = K. Corollary 2. Let f : M -+ l he a Cr map where M e Rk is a closed manifold. Then there exists a C' map, ': R' - 18s such that I. I M = f.
It follows from this proposition that, given open sets U and V in M with U c V, there exists a C- real-valued function cp > 0 with tp = I on U and (p = 0 on M - V. Such a function is called a bump function. We shall now define the tangent bundle TM of a manifold Mm c R'. Put TM = {(p, v) a 68' x Rk; p e M, v c- TMP}. Give TM the induced topology as a subset of fl' x Rk; then the natural projection n: TM -+ M, n(p, v) = p, is continuous. Let us show that TM is a differentiable manifold and that it is actually of class C"'. Let x : U --+ R' be a local chart for M. We define the map
Tx: n-'(U) -+ 1l'" x Rm by Tx(p, v) = (x(p), dx(p)v)._ It is easy to see that (Tx, n-'(U)) is a local chart for TM and, therefore, TM e OBk x Rk is a manifold. Note that the expression for it using the local charts (Tx, n-'(U)) is simply the natural projection of Q8' x Ir on the first factor; thus it is C'. It is also easy to see that, if f : M -+ N is of class C'then df : TM --+ TN, df (p, v) = (f (p), df (p)v), is C'. As we remarked earlier there is an abstract definition of manifolds which, a priori, is more general than the one we have just presented. For this let M be a Hausdorff topological space with a countable basis. A local chart in M
is a pair (x, U), where U c M is open and x: U --+ Uo a I8' is a homeomorphism onto an open subset Uo of P'". We say that U is a parametrized
9
§0 Calculus in R" and Differentiable Manifolds
neighbourhood in M. If (x, U) and (y, V) are. local charts in M, with U n V # 0, the change of coordinates y o x-': x(U n V) --+ y(U n V) is a homeomorphism. A differentiable manifold of class Cr, r >- 1, is a topological
space together with a family of local charts such that (a) the parametrized neighbourhoods cover M and (b) the changes of coordinates are C diffeomorphisms. Such a family of local charts is called a Cr atlas for M. Using the local charts we can define differentiability of maps between these
manifolds just as we did before. In particular, a curve a: (- a, E)
M is
differentiable if x o a: (- E, E) --+ III'" is differentiable, where (x, U) is a local chart with a(-E, E) c U. The tangent vector to a at p = a(O) is defined as the set of differentiable curves f : (- E, e) - M such that P(0) = p and d(x o 9)(0) _ d(x o aXO). This definition does not depend on the choice of the local chart
(x, U). The tangent space to k at p, TM,,, is the set of tangent vectors to differentiable curves .passing through p. It follows that TM, has a natural m-dimensional vector space structure. If f : M - N is a differentiable map . between manifolds with f (p) = q, we define df(p): TM,, - TN, as the map which takes the tangent vector at p to the curve a: (-a, E) - M.to the tangent vector at q to the curve f o a: (- E, E) - N. It is easy to see that this definition does not depend on the choice of the curve a and that df(p) is a linear map. We say that f: M -+ N is an immersion if df(p) is injective for all p e M. An
embedding is an injective immersion f : M -+ N which has a continuous inverse f -' : f (M) c N -+ M. ' If f : M -+ ftk is an embedding of class C" then f (M) a 68k is a submanifold of R5 according to the definition given earlier.
The next theorem relates the abstract definition of manifold to that of submanifold of Euclidean space.
.
0.17 Theorem (Whitney). If M is a differentiable manifold of dimension m there exists a proper embedding f : M -+ R2" t .
Let M be a differentiable manifold and S c M a submanifold. A tubular neighbourhood of S is a pair (V, n) where V is a neighbourhood of S in M and n: V -+ S is a submersion-of class C°° such that n(p) = p for p e S.
0.18 Theorem. Every submanifold S c M has a tubular neighbourhood.
Lastly, every manifold of class C', r > 1, can be considered in anatural way as a manifold of class C. .0.19 Theorem (Whitney). Let M be a manifold of class Cr, r > 1. Then there such that f (M) is a closed C °° submanifold exists a C' embedd ing f : M -* 182m "such
of
By Theorem 0.17 this tesult is equivalent to the following: if d is a C atlas, on M there exists a C' atlas sat on M such that, if (x, U) e d and (x, U) e .zl with U n 0 96 0, then z o x-' and x o g` are of class C.
10
1
Differentiable Manifolds and Vector Fields
§1 Vector Fields on Manifolds We begin here the qualitative study of differential equations. As this study has
both local and global aspects the natural setting for it is a differentiable manifold. One of the first basic results which is global in character is the Poincare-Bendixson Theorem with which we shall close this section. Let M" c Rk be a differentiable manifold. A vector field of class C on M is a C' map X : M fflk which associates a vector X(p) a TM,, to each point p E M. This corresponds to a C' map X : M - TM such that xX is the identity on M where it is the natural projection from TM to M. We denote by £'(M) the set of C' vector fields on M.
An integral curve of X E X'(M) through a point p e M is a C'+' map a: I - M, where I is an interval containing 0, such that a(0) = p and a'(t) = X(a(t)) for all t e I. The image of an integral curve is called an orbit or trajectory.
If f : M - N is a C'+' diffeomorphism and X E I'(M) then Y = f.X, defined by Y(q) = df (pXX (p)) with q = f (p), is a Cr vector field on N, since
f. X = df o X o f -'. If a: I - M is an integral curve of X then f o a: 1 - N is an integral curve of Y. In particular, f takes trajectories of X onto trajectories of Y. Thus, if x: U -' Uo c ff8' is a local chart, then Y = x.X is a C' vector field in Uo ; we say that Y is the expression of X in the local chart (x, U). By these remarks the local theorems on existence, uniqueness and differentiability of solutions extend to vector fields on manifolds as in the following proposition.
1.1 Proposition. Let E be a Banach space and F: E x M - TM a C map, r > 1, such that iF(A, p) = p, where x: TM M is the natural projection. For every Ao E E and po E M there exist neighbourhoods W of Ao in E ana V
of po in M, a real numbers > 0 and a Cr map cp: (-s, e) x V x W - M such that V (O, p, A) = p and (a/at)rp(t, p, A) = F(A, (p(t, p, A)) for all t e (- C. e), p E V and A E W. Moreover, if a: (- e, e) -. M is an integral curve of the vector
field Fx = F(,, ) with a(0) = p then a =
p, 2).
1.2 Proposition. Let I, J be open intervals and let a: I -- M, /3: J -. M be integral curves of X E 1'(M), r z 1. If a(to) = f(to), for some to c- I n J, then a(t) = fl(t)for all t E I n J. Hence there exists an integral curve y: I U J M which coincides with a on 1 and with /3 on J.
PROOF. By the local uniqueness, if a(t,) = /3(t,) there exists e > 0 such that
a(() _ /3(t) for I t - tI I < s. Therefore the set II e I n J where a coincides with /3 is open. As the complement of Iis also open and .1 n J is connected we
have l=InJ.
II
§1 Vector Fields on Manifolds
1.3 Proposition. Let M be a compact manifold and X e X'(M). There exists on M a global C' flow for X. That is, there exists a C' map tp: I x M -+ M such that 4p(0, p) = p and (8/8t)cp(t, p) = X ((p(t, p)).
PROOF. Consider an arbitrary point p in M. We shall show that there exists an integral curve through p defined on the whole of R. Let (a, b) c f8 be the domain of an integral curve a: (a, b) M with 0 e (a, b) and a(0) = p. We say that (a, b) is maximal if for every interval J with the same property we have J c (a, b). We claim that, if (a, b) is maximal, then b = + oo. If this is not the case consider a sequence t,, - b, t E (a, b). As M is compact, we may converges to some q E M. suppose (by passing to a subsequence) that
Let tp: (- c, e) x VQ -. M be a local flow for X at q. Take no such that e) - M by y(t) = a(t) if e/2 and E Vy. Define y: (a, bt < t,,,0 and y(t) = (p(t t z t.0. It follows that V is an integral curve of X, which is a contradiction because (a, t,,0 + e) (a, b]. In the same way we may show that a = - oo and, therefore, there exists an integral curve a: UB --+ M with a(0) = p. By Proposition 1.2 this integral curve is unique. We define (p(t, p) = a(t). It is clear that (p(0, p) = p and (49/8t)fp(t, p) = X((p(t, p)). We claim that (p(t + s, p) = cp(t, ep(s, p)) for t, s e Ua and p e M. Indeed, let
ft(t) = (p(t + s, p) and y(t) = tp(t, (p(s, p)). We see that f and y are integral curves of X and Q(0) = y(0) = cp(s, p), which proves the claim. Lastly we show
that tp is of class C. Let p e M and 0: (-ep, ep) x Vp - M be a local flow for X, which is of class C' by Proposition 1.1. Also, by the uniqueness of solutions 0, is the restriction of tP to (-ep, e,,) x Vp. In particular, cp, = (p(t, ) is of class C' on V. for I t I < sp. By the compactness of M there exists e > 0 such that gyp, is of class C' on M for It I < e. Moreover, for any t e R, we can o (p,I is choose an integer n so that I t/n I < e and deduce that (p, = lp,J o of class C'. For any to e R and po e M, cp is C' on a neighbourhood of (to, po). For if It - toI < e,, and p e Vpo, then cp(t, p) = cp,, o (p(t - to, p) is Cr since cp,o and fp I (- epo, a..) x VD0 are Cr. That completes the proof. -
Corollary. Let X e X'(M) and let rp: f8 x M - M be the flow determined by X. For each t e R the map X,: M - M, X,(p) = tp(t, p), is a Cr difjeomorphism. Moreover, Xo = identity and X,+, = X, 0 X, jor all t, s e R. Let e X'(M) and let X,, t e U8, be the flow of X. The orbit of X through p E M is the set 0(p) = {X,(p); t e R}. If X(p) = 0 the orbit of p reduces to
p. In this case we say that p is a singularity of X. Otherwise, the map a: U8 -+ M.
a(t) = X,(p), is an immersion. If a is not injective there exists w > 0 such that a(w) = a(0) = p and a(t) # p for 0 < t < w. In this case the orbit of p is difleomorphic to the circle S' and we say that it is a closed orbit with period
to. If the orbit is not singular or closed it is called regular. Thus a regular orbit is the image of an injective immersion of the line. The w-limit set of a point p e M, w(p), is the set of those points q E M for which there exists a sequence t -* co with X,,,(p) -* q. Similarly, we define
12
I
Differentiable Manifolds and Vector Fields
PS
Figure 5
the a-limit set of p, a(p) _ {q e M; X'"(p) q for some sequence t -oo}. We note that the a-limit of p is the w-limit of p for the vector field - X. Also,
a(p) = w(p-) if p belongs to the orbit of p. Indeed, p = X,o(p) and so, if X,jp) -+ q where t -+ oo, then X, -,o(p-) -' q and t - to -+ oo. Thus we can define the co-limit of the orbit of p as co(p). Intuitively a(p) is where the orbit of p "is born" and w(p) is where it "dies".
EXAMPLE 1. We shall consider the unit sphere S2 c l with centre at the origin and use the standard coordinates (x, y, z) in 68'. We call PN = (0, 0, 1) the north pole and ps = (0, 0, - 1) the south pole of S2. We define the vector
field X on S2 by X(x, y, z) _ (-xz, --yz, x2 + y2). It is clear that X' is of class C' and that the singularities of X are PN, ps. As X is tangent to the meridians of S2 and points upwards, w(p) = pN and a(p) = ps if p c- S2 {pN, Ps} (Figure 5).
EXAMPLE 2. Rational and irrational flows on the torus. Let rp: R2 -+ T' e 68' be given by
tp(u, v) = ((2 + cos 2nv) cos 27ru, (2 + cos 2irv) sin gnu, sin 2nv).
We see that cp is a local diffeomorphism and takes horizontal lines in l 2
to parallels of latitude in T2, vertical lines to meridians and the square [0, 1] x CO, I] onto T 2. Moreover, cp(u, v) = cp(u, vv") if and only if u - u = in
and v - v` = n for some integers m and n. For each a e P we consider the vector field in P2 given by X'(u, v) = (1, a). It is easy to see that Y' = cp. X' is well defined and is a C' vector field on T 2. The orbits of Y' are the images by cp of the orbits of Xa and these are the lines of slope a in P2. We shall sliow that, for a rational, every orbit of Y' is closed and, for a irrational, every orbit
of Y' is dense in V. For each c E P let-A, denote the straight line in P1 through (0, c) with slope a; A, = ((u, c + au); u e P}. As we have already observed, cp(A,) is an orbit of Y8. If a is rational this orbit is closed for each c E R. For if a = n/m then (m, c + (n/m)m) a A, and tp(m, c'+ n) = (p(0, c). Suppose now that a is irrational and c E R. We claim thatC = {c E R; cp(Ac) = cp(Ae)} is dense in R. It follows that Ucc A, is dense in P2 and, therefore,
,l Vector Fields on Manifolds
13
Figure 6
49(Az) = (p(IJCEc Ac) is dense in T2. To show that C is dense in ll it is enough to prove that G = {ma + n; m, n e 77} is dense in R, because c e C if and only
if c - c e G. As G is a subgroup of the additive group l we know that G is either dense or discrete. It remains, therefore, to show that G is not discrete. But for each m e Z, there exists n e 77 such that un = ma + n belongs to the interval [0, 1]. The sequence un has a cluster point and, as a is irrational, its terms are distinct. Thus G is dense. The vector field Y° above is called the rational or irrational field on T2 according as to whether a is rational or not. If x is rational the w-limit of any orbit is itself. If a is irrational, the w-limit of any orbit is the whole torus T2. EXAMPLE 3 (Gradient Vector Fields). Consider a manifold Mm c Ilk. At each
point p e M we take in TM, the inner product < >, induced by R. We denote the norm induced by this inner product by 11 lip or, simply, by p 11. If X and Y are C°° vector fields on M then the function g: M -' R. g(p) = <X(p), Y(p)> p is of class C. Let!: M - R be a C'-+.l map. For each p e M
there exists a unique vector X(p) e TMp such that dfv = <X(p), v>p for all v e TMp. This defines a vector field X which is of class C'. It is called the gradient off and written as X = grad f We shall now indicate some basic properties of gradient fields. Firstly, grad f (p) = 0 if and only if dfp = 0. Along nonsingular orbits of X = grad f we have f strictly increasing because dffX(p) = IIX(p))12. In particular grad f does not have closed orbits. Moreover, the w-limit of any orbit consists of singularities. For let us suppose that X (q) # 0 and q e w(p) for some p e M. Let S be the intersection off -1(f (q)) with a small neighbourhood of q. We see that S is a submanifold of dimension m - 1 orthogonal to X = grad f and, by the continuity of the flow, the orbit through any point near q intersects S. As q .e w(p) there exists a sequence p in the orbit of p converging to q. Thus the orbit of p intersects Sin more than one point (in fact, in infinitely many points) which is absurd sincefis increasing along orbits. On the other hand, it is clear that, if the to-limit of an orbit of a gradient vector field contains more than one singularity, it must contain infinitely many. We are going to.show that this can in fact occur.
14
I
Differentiable Manifolds and Vector Fields
Figure 7
Figure 8
Let f : R2 -+ R be defined by
f(rcos0,rsin0)=
et"'' "
if r < 1;
0,
ifr= 1;
e- tiv,-"sin(l/(r - 1) - 0), if r > 1. Let X = grad f. We have X(r cos 0, r sin 0) = 0 if and only if r = 0 or r =
1.
We are going to show that there exists an orbit of X whose u)-limit is the circle
C with centre at the origin and radius 1. Note that f '(0) = C u Et u E2, where E, and E2 are the spirals (Figure 7) defined by
E, = {(r cos 0, r sin 0); r = I + 1/(n + 0), -a < 0 < x-), E2 = {(r cos 0, r sin 0); r = 1 + 1/(2n + 0), -2n < 0 < oo}. Let us consider the region U = {(r cos A, r sin 0); 1 + 1/(271 + 0) < r < I + 1/(71 + 0), 0 > 01 and let I be the interval ((x, 0); 1 + 1/2n < x < I + 1/n). We shall show the existence of a point po E I whose positive orbit remains in the region U. Hence the cu-limit of po will be the circle C. In Figure 8 we draw some level curves of the function f on U.
The intersection of the level curve through a point p e 1 with U is a compact segment whose ends are in I. The length of this segment tends to infinity as p approaches the ends of 1. Let q e E,. As X(q) is orthogonal to E, and points out of U (because f is negative in U), we see that the negative orbit of q intersects one of the level curves through a point in the interior of 1. So the negative orbit of q intersects
1. Therefore, the set J = (p e I ; X,(p) e U for 0 < t < s and X,(p) a E,) is nonempty Moreover, given q e E,, there exists p e J such that the positive orbit of p contains q and the segment of the orbit between p and q is in.U. On the other hand, given q e E2, the negative orbit of q also intersects I so that
J#1
' Let po be the infiinum of J We claim that the positive orbit of po remains in U. For if this is not the case there exists a point q in the positive orbit of po s4ell at the segment of the orbit between po and q is contained in U and
k,(rj)` U for suffiicietitly small t > 0. Thus q e E, or q e E2 or q e I. If q e E, then each positive orbit through a point of J intersects E1 in a point of tbe.segtnent lrtweeil (I + 1/n, 0) and q. This is absurd because the negative
§1 Vector Fields on Manifolds
%
1 5
orbit through any point of E, intersects I and therefore J. If q e E2 or q e I then the positive orbit of each point near po leaves U without meeting E, which is absurd since po is the infimum of J. Thus the positive orbit of po is contained in U, which proves our claim. We remark that the vector field on S2 in Example.1 is the gradient of the height function that measures height above the plane tangent 'to the sphere S2 at ps. Other simple examples can be obtained by considering the function on a surface in 683 which measures the distance from its points to a plane. Some of these examples will be considered later. Next we shall discuss some general properties of w-limit sets. 1.4 Proposition. Let X e X'(M) where M is a compact manifold and let p e M. Then (a) o(p) # 0, (b) oXp) is closed, (c) co(p) is invariant by the flow of X, that is a(p) is a union of orbits of X, and (d) co(p) is connected.
PROOF. Let t -+ oo and p = X,,,(p). As M is compact p has a convergent subsequence whose limit belongs to.co(p). Thus w(p) # 0. Suppose now that q It co(p). Then it has a neighbourhood V(q) disjoint from {X,(p); t Z T) for some T > 0. This implies that the points of V(q) do not belong to (0(p) and so co(p) is closed. Next suppose that q a w(p) and 4 = Xs(q). Take tp -+ oo
with X,,,(p) -' q. Then X,,+5(p) = X,X,,,(p) converges to Xs(q) = 4 and so 4 e co(p). This shows that w(p) is invariant by the flow. Suppose that co(p) is not connected. Then we can choose open sets V, and V2 such that w(p) e
V,u V2,co(p)nV, #0,w(p)nV2#0and 17,nV2=0. The orbit of p accumulates on points of both V, and V2 so, given T
0, there exists t > T
such that X,(p) e M - (V, u V2) = K, say. Thus there exists a sequence t -+ op with X,,(p) a K. Passing to a subsequdnce, if necessary, we have X,.(p) - q for some q e K. But this implies that q e w(p) c V, u V2 which is absurd. 0 Remark. Clearly the properties above are also true for the a-limit set. On the
other hand, if the manifold were not compact we should have to restrict attention to an orbit contained in a compact set for positive time (or for negative time). Figure 9 shows an orbit of a vector field on 682 whose w-limit is not connected.
Figure 9
16
I
Differentiable Manifolds and Vector Fields
Figure 10
As we have already seen, the co-limit of an orbit of an irrational flow on the
torus T2 is the whole torus. There are more complex examples of vector fields on T2 with rather complicated co-limit sets as in Example 13 of Chapter 4. Meanwhile for the sphere S2 the situation is much simpler because of the
following topological fact: every continuous closed curve without selfintersections separates S2 into two regions that are homeomorphic to discs (the Jordan Curve Theorem). The structure of an w-limit set of a vector field
on S2 is described by the Poincar6-Bendixson Theorem whose proof we develop now through a sequence of lemmas. Let X C X'(S2), r z 1.
1.5 Lemma. Let E e S2 be an arc transversal to X. The positive orbit through a point p e S2, 0+(p), intersects E in a monotonic sequence; that is, if pi is the . ith intersection of 0+(p) with E, then pi a [pi_,, p;+ t] PROOF. Consider the piece of orbit from p,-, to pi together with the segment [pi_,, pi] c E. These make a closed curve which bounds a disc D and, as E
is transversal to X which points inwards into D, the positive orbit of pi is contained in D. Thus pi a [pi- 1, pi+t]. (] Corollary. The w-limit of a trajectory y intersects E in at most one point.
PROOF. Suppose that co(y) contains two points q, and q2 of E. Let p be the sequence of intersections, of y with 1r. Then there exist subsequences of p
converging to q, and q2, which leads, to a contradiction because of the monotonic property of p,,. 1.6 Lemma. If the co-limit set of a trajectory y does not contain singularities then w(y) is a closed orbit and the orbits through points close enough to a point p of y have the same closed orbit as their (o-limits.
PROOF. Let q e w(y). We show that the orbit of q is closed. Take x e w(q), which cannot, therefore, be a singularity. Consider a segment E transversal
17
§1 Vector Fields on Manifolds
Figure 11
to X containing x. By the previous lemma the positive orbit of q intersects E in a monotonic sequence q -' x. As q a co(p) the above corollary shows that q = x for all n. Thus the orbit of q is closed. By taking a transversal segment containing q we conclude as in Lemma 1.5 that co(y) reduces to the orbit of q. The proof of the second statement is immediate. 1.7 Lemma. Let pl and p2 be distinct singularities of the vector field contained in the w-limit of a point p e S2. Then there exists at most one orbit y c'w(p)
such that a(y) = Pi and w(y) = P2. PROOF. To get a contradiction suppose there exist two orbits y, y2 c co(p) such that a(yi) = p, and co(y) = p2 for i = 1, 2. The curve C1, consisting of the orbits y1, y2 and the points pl, P2, separates S2 into two discs, one of whichcontains pas shown in Figure 1z. Let E 1 and E2 be segments transversal to X through the points q1 a yl and q2 a y2 respectively. As yl, y2 (-- w(p) the positive orbit of p intersects E1 in a point a and later intersects E2 in a point b. Consider the curve C2 consisting of the arcs ab c O(p), bq2 C E2, q2 P2 C Y2, P2g1 c y1, qla c E1 and the point p2. We see that C2 separates S2 into two discs A and B. The positive orbit of the point b remains entirely in A which gives a contradiction since y1i y2 c uw(p).
Figure 12
1 Differentiable Manifolds and Vector Fields
18
1.8 Theorem (Poircare-Bendixson). Let X C X'(S2) be a vector field with a finite number of singularities. Take p e S2 and let w(p) be the a)-limit set of P. Then one of the following possibilities holds: (1) w(p) is a singularity; (2) w(p) is a closed orbit; (3) co(p) consists of singularities pl, ... , p and regular orbits such that if y e co(p) then a(y) = p, and w(y) = Pi-
PxooF. If w(p) does not contain a singularity then, by Lemma 1.6, w(p) is a closed orbit. If w(p) does not contain regular points then w(p) is a unique
singularity since X only has a finite number of singularities and w(p) is connected. Suppose, then, that w(p) contains regular points and singularities. Let y be a regular trajectory contained in o(p). We claim that w(y) is a singularity. If w(y) contain some regular point q take a segment E through q transversal to X. As y c o(p) the Corollary to Lemma 1.5 says that y intersects E in only one point. By Lemma 1.6 y is a closed trajectory and w(p) = y. This is absurd because w(p) contains singularities. Thus w(y) is a singularity. Similarly a(y) is a singularity.
In the following examples we illustrate some facts about this theorem. EXAMPLE. Let X be a vector field on S2 as in Figure 13. The north and south
poles are singularities and the equator is a closed orbit. The other orbits are born at a pole and die at the equator.
Let cp: S2 - t be a nonnegative C' function which vanishes precisely on the equator of the sphere. Consider the vector field Y = (p X. Each point of the equator is a singularity of Y and the w-limit set of a point p which is neither a pole nor on the equator is the whole of the equator. This example
shows that the Poincar6-Bendixson Theorem is not valid without the hypothesis of a finite number of singularities.
Ps
Figure 13
§2 The Topology of the Space of C Maps
19
Northern hemisphere Figure 14
Southern hemisphere
EXAMPLE. Let X be a vector field on SZ as in Figure 14. The vector field X
has two singularities ps and pN and one closed orbit y. The orbits in the northern hemisphere have PN as a-limit and y as w-limit. In the southern hemisphere we have the singularity ps which is the centre of a rose with infinitely many petals each bounded by an orbit which is born in, ps and dies in ps. In the interior of each petal the situation is as in Figure 15.
ep$
Figure 15
The other orbits in the southern hemisphere have y as a-limit and the edge of the rose as w-limit. Therefore the w-limit of an orbit can contain infinitely many regular orbits, which shows that Lemma 1.7 is not valid if
Pt ° P2
§2 The Topology of the Space of C Maps We introduce here a natural topology on the space '(M) of C vector fields on a compact manifold M. In this topology, two vector fields X, Y e X'(M) will be close if the vector fields and their derivatives up to order r are close at all points of M.
20
1
Differentiable Manifolds and Vector Fields
Let us consider first the space C'(M, OB) of Cr maps, 0 5 r < oo, defined
on a compact manifold M. We have a natural vector space structure on C'(M, OB'): (f + gXp) = f(p) + g(p), (AfXp) = Af(p) for f, g E C'(M, RS) and A E R. Let us take a finite cover of M by open sets V1,..., Vk such that each V, is contained in the domain of a local chart (xi, U,) with x,{U;) = B(2) B(1), where B(1) and B(2) are the balls of radii 1 and 2 with and centre at the origin in OBm. For f e C'(M, OBI) we writef' = f o x; ' : B(2) -' 0r. We define
If II, = max sup{IIf'(u)II, Ildf'(u)II,
.
.,
Ild'f'(u)ll ; u E B(1)}.
2.1 Proposition. II II, is a complete norm on C'(M, OBW). PROOF. It is immediate that II 11, is a norm on C'(M, OBI). It remains to prove That every Cauchy sequence converges. Let f" : M -' 1W be a Cauchy sequence
in the norm II II,. If p e M then f"(p) is a Cauchy sequence in R' and so converges. We putf(p) = lim f"(p). In particular, f;(u) - f'(u) for u a B(1) and i = 1, ... , k. On the other hand, for each u e B(1), df;,(u) is a Cauchy sequence in L(W", OBI) and so converges to a linear transformation V(u). We claim that the convergence df;, .- T' is uniform. For notice that
ildf"A) - T(u)II <- ildf'(u) - df'.(u)II +
T'(u)ll
Given e > 0, there exists no such that, if n, n' >- no, then II df;,(u) df (u)II < s/2 for all u e B(1). On the other hand, for each u e B(1) there exists n' >- no, which depends on u, such that Ildf',.(u) - T'(u)11 < e/2. Thus, T'(u)II < & for all u e B(1). By Proposition 0.0, for n z no, we have
f' is of class C' and df' = T. It follows thatf" -+ f in the norm II IIt. With the same argument we can show' by induction that f is of class C' and f" in the norm II 11,.
f
0
It is easy to see that the topology defined on C'(M, OBs) by the norm II II, does not depend on the cover Vi, ... , Vk of M. Next we describe some important properties of the space C'(M, 1) with the C' topology. A subset of a topological space is residual if it contains a countable inter.
section of open dense sets. A topological space is a Baire space if every residual subset is dense. As C'(M, OBI) is a complete metric space we immediately obtain the following proposition. 2.2 Proposition. C'(M, OBI) is a Baire space.
0
Let us show that C'(M, 1W) contains a countable dense subset. For f e C'(M, Or) consider f' = f o x; ' : B(2) c OB' - R. Note that the map
'f':
B(2)
B(2) x OBI x L(OBm, OBI) x
x L'(1W'; OBI) = E defined by
21
§2 The Topology of the Space of C Maps
ff'(u) = (u, f'u, df'(u), ... , d'f'(u)) is continuous. Thus J'(f') = j'f'(B(1)) is a compact subset of E. It is easy to see that, if 7' is a neighbourhood off in C'(M, l3'), there exists a neighbourhood W of J'(f) = J'(f') x . x Jr(fk)
in E XE X ... x E such that, if g e C'(M, R3) and J'g = J'g' x .
x
J'g' c-W, then g e W. 2.3 Proposition. C'(M, R) is separable; that is, it has a countable base of open sets. PROOF. As Ek = E x
x E is an open set in a Euclidean space, there exists
a countable base of open sets E1, . . . , E,, ... for the topology of Ek. Let i, ... , Ej,... be the collection of those open subsets of El that are finite unions of the E. Let d'j = {g e C'(M, W); J'(g) c 9j} for each j. It is clear that d j is open in C'(M, IB'). Let i ' be a neighbourhood off in C'(M, ff) and
W a neighbourhood of J'(f) such that g e #' if J(g) c W As J'(f) is compact there exists a finite cover of J(f) by open sets E, contained in W Let -`j be the union of these E; ; it is clear that J'(f) c E j c W. Therefore, 9,
tj, ...} is a
contains f and is contained in #'. This shows that countable base for the topology of C'(M, R).
Next we show that every C' map can be approximated in the C' topology by a C o' map.
2.4 Lemma. Let f : U e S8' - W be a C map with U an open set. Let K c U
be compact. Given e > 0 there exists a C° map g: Ilf 91jr<eonK.
QB"'.-+ R' such that
PROOF. Let us consider a bump function tp: R'" -+ F which takes the value 1 on K and is 0 outside a neighbourhood of K contained in U. Taking h = cpf
we have h = fin K and h = 0 outside U. As his of class Cand K is compact, there exists S > 0 such that sup{ IIdjh(u + v) - -djh(u)II ; u e K, Ilvll < b) < e,
where dj denotes the derivative of order j, f o r j = 1, ... , r. Let spa: l8' - R be a bump function such that rpb(v) = 0 if Ilvll > S and J rpa(v) dv = 1. We rpa(z - u)h(z) dz. It define g: R"' -+ l3' by g(u) = s cpa(v)h(u + v) dv follows that djg(u) = j'coo(v) djh(u + v) dv .
and. dig(u) _ (- l Y 1 djwa(z - u)h(z) dz.
I Differentiable Manifolds and Vectot Fields
22
From the second expression it follows that g is C'. On the other hand, from the first expression we have p
Ild'g(u) - d'h(u)ll
Jq,o(v)dih(u + v) dv - I gia(v) AM dv 5 pb(v)(d'h(u + v) - d'h(u)) dvlf < e
for u E K.
As It = f on K, g satisfies the required condition. 2.5 Proposition. The subset of C°° maps is dense in C'(M, Rs).
PROOF. Let (xi, U;), i = 1, ... , k be local charts with x,.(U,) = B(2) and M = U V where V = x7'(B(1)). Take a partition of unity {(pi: M -+ Ill;) subordinate to the cover {Y). Let f e C'(M, 18') and c > 0. By the previous
lemma, given b > 0, there exists
g" `: 11" - l
of class C°° such that
Il f` - g''ll, < Son B(1), where f' = f o xi '. Taking b sufficiently small we have IJtpi f --(p;§'- x;ll, < E/k. Thus, g = tpig" ° x; is of class C'° and
IJf -g11,= IIZ(cvi.f -(pig'0X) II,<E/k+...+E/k=E. Let us now consider a closed manifold N. By Whitney's Theorem, we may assume that N is a closed submanifold of 18, for some s > 0. As N is a closed subset of U8', C'(M, N) is closed in C'(M, Re). Therefore, C'(M, N), with the topology induced from C'(M. 18'), is a separable Baire space. Let N1 c Ri', N2 e 18'= be closed manifolds and 0: N1 N2 a C' map,
r<1 <m.Define
=cbo f.
2.6 Proposition. The map (D. is continuous.
PROOF. By Corollary 2 of Proposition 0.16 there exists a C' map, di: 18" W2, such that di = 0 on N1: Let (xi, U.), i = 1, ..., k, be local charts in M as before. It is easy to see that, given E > 0, there exists b > 0 such that, if II f o x; ' - g ° xi ' II, < b on B(1), then II(di ° f - di ° g) Xi
< E on B(1). Therefore, if IIf - gll, < b then we have 110 ° f & ° gll, < E, which shows the continuity of 4)..
1115 ° f
-
° gIl, =
Now let M and N be abstract manifolds with M compact. We can define a C' topology on C'(M, N). For this it is sufficient to embed N in a Euclidean space 182 (Theorem 0.17). The proposition above shows that this topology is independent of the embedding. 2.7 Proposition. The subset of maps of class C°° is dense in C'(M, N).
PROOF. Suppose N e R. Let V c 18' be a tubular neighbourhood of N and n: V - N the associated projection. Let f e C'(M, N); by Proposition 2.5 we can approximate f by a C°° map g: M -- W. Thus, n ° g: M -+ N is C°° and it o g approximates f because n. is continuous.
§3 Transversality
23
2.8 Proposition. The set Diff'(M) consisting of the C' diffeomorphisms of M is open in C'(M, M).
PROOF. Let f e Diff'(M). We can suppose that M c W. If p E M there exist,
by the Inverse Function Theorem, neighbourhoods V, of p in M and P of f in C'(M, M) such that, if g e p then gI Vp is a diffeomorphism. Let Vp,, ..., VD, be a finite subcover of M and put Y'' = .1 p,. If 8 is the 1
Lebesgue number of this cover then d(p, q) < b and p # q imply g(p) # g(q) for every g e y''. On the other hand, p = inf {d(f (p), f (q)); p, q e M and d(p, q) >_ b} is positive. By reducing 'V we can therefore suppose that if g Ef*' then g is injective. Since g is a local diffeomorphism it is a diffeomorphism. O
It follows from the previous proposition that Diff'(M) is a separable Baire space and that the subset of C°o diffeomorphisms is dense in it.
Finally let us consider the space X'(M) of C vector fields on a compact manifold M. Supposing that M c f8' we can easily see that X'(M) is a closed subspace of C'(M, Rs). Thus, X'(M) is a separable Baire space. Let us show that every vector field X E £'(M) can be approximated by a C°° vector field.
In fact, X can be approximated by a C' map Y: M -, TM. Let n: TM - M be the natural projection. As it. is continuous, it o Y is Cr close to it o X = idM By Proposition 2.8, rp = it o Y is a diffeomorphism. Let Z = Y o 9711; Z is a C°° vector field since cP is C°° and it o Z = idM. Moreover Z approximates X.
(See Exercise 15 at the end of the chapter.)
13 Transversality Lit S e N be a C' submanifold and let f : M - N be a C' map where k, r z 1. We say that f is transversal to Sat a point p e M if either f (p) 0 S or dfp(TMp) + TSf(p) = TN f(pl; that is, if the image of TMp by dfp contains a subspace of TNflp1 that is complementary to TS flp). We say that f is transversal to S, f S, if it is transversal to S at each point p e M. Note that if the dimension of M is less than the codimension of S then f is.transversal to S if and only if f (M) n S = 0. An interesting special case happens when f : M -+ N is a submersion; here f is transversal to every submanifold S c N. We define transversality between two submanifolds S1, S2 a N in the following way: S1 is transversal taS2 if the inclusion map is S1 - N is transversal to S2. We recall that every submanifold is locally the inverse image of a regular value. More precisely, if q e S c N" then there exist a neighbourhood Vq of q
in N and a diffeomorphism- cp: Vq -+ 68s x R` of class C such that
24
I
Differentiable Manifolds and Vector Fields
92-
0
Figure 16
cp(S n Vq) = 08' x {0}; thus, S n V. = (n, o (p)- '(0) where 7E2: W X R"S
R" is the natural projection. Now let f : M -+ N and jet U, c M be a neighbourhood of p with f(UP) c Vq where q = f(p). Consider the map n2 o cp a f I Up as in Figure 16.
The next proposition is immediate.
3.1 Proposition. The map f : M -+ N is Sransversal to S at p e f -'(S) if and only if 0 is a regular value of z2 o cp o f I Up for some neighbourhood UP as above.
Corollary. Let f e Ck(M, N") and let S' be a C' submanifold of N with k, r >_ 1.1f f is transversal to S then f -'(S) is either empty or a C'submanifold of codimension n - s, where I = min(k, r).
3.2 Proposition. If M is compact and S c 1V is closed then those maps in Ck(M, N) that are transversal to S form an open subset.
PROOF. Let f e C'(M, N) be transversal to S. For each q e S we take a neighbourhood V. and a diffeomorphism cpq: V, --+ R' x OB"-' such that cpq(S n Vq) = Og' x {0}. For each p a f - '(q) let us consider a neighbourhood
UP such that f c V. and the derivative of rte o cpq o f is surjective at every point of V P. There exists a neighbourhood D(f) e C5(M, N) such that the same happens for 7r2 ° cpq o g on UP for every g E D. Let UP1, ... , Up." be a finite subcover of the compact set f '(S) and put U = UP, V . U Up,,, and 71(f) _ ;, n ... n p",. It is clear that if g E V then g is transversal to Sat all points of U. As M - U is compact and f (M - U) n S = 0 we can reduce Y,' if necessary and obtain g(M - U) n S = 0 for all g e 11. Thus, every g E' is transversal to S, which proves the proposition.
We remark that, if S e N is not closed, the proposition remains true for closed subsets of S. That is, if S e S is closed in 11, then the set of maps f : M -+ N that are transversal to S on f '(s) is open in C(M, N).
LetA,M,Nbe manifoldsandletF:AxM -+ N beaC°°map. For A e A we denote by F,%: M - N the map defined by FZ(p) = F(.4, p)., Let S a' N be a C°° submanifold and let Ts e A be the set of points .1 such that' ) is transversal to S.
§3 Transversality
25
3.3 Proposition. If F: A x M -+ N is transversal to S e N then Ts is residual in A.
PROOF. Let n: A x M - A be the natural projection. Since F is transversal
to S, . = F- t(S) is a submanifold of A x M and ns - x13: S - A is a C' map. It is easy to see that Fx is transversal to S if and only if A is a regular value of ns. The proposition now follows from Sard's Theorem. Corollary 1. Let f : M - R" be of class C°° and let S e R" be a submanifold. The set of vectors v e R" such that f + v is transversal to S is residual.
PROOF. The map F: R" x M - R" defined by F(v, p) = f (p) + v is a submersion and, therefore, transversal to S. The corollary now follows immediately from the proposition. Corollary 2. If M is a compact manifold then the set Ts a Ck(M, R") of maps that are transversal to a closed submanifold S z R" is open and dense.
PROOF. The openness of Ts follows from Proposition 3.2 while the density comes from Corollary 1 and the density of Cm maps in Ck(M, R").
We emphasize that the manifolds A, M and N in Proposition 3.3 are not necessarily compact. Note too that in Proposition 3.3 and Corollary i we require the maps and the submanifold S to be of class C. This is because we use Sard's Theorem in the proof. 3.4 Theorem (Thom). Suppose that M is compact and that S c N is a closed submanifold. The set of maps f e C*(M, N) that are transversal to S is open and dense.
PROOF. As we have already shown the openness it remains to prove the density of maps transversal to S. Take f e CC(M, N). We have to show that,
for each p e M, there exist neighbourhoods U, of p in M and v of f in C*(M, N) such that the set of g e v that are transversal to S on U. is open and dense in p. In fact, if Up,, ... , U,,,, is a finite subcover of M and 'V(f) =
,n
r p,,, then the set of g e *' that are transversal to S at every
point of M is open and dense. In particular we can approximate f by a map transversal to S. Let us then construct these neighbourhoods Up and D. Let y: V -+ R be a local chart around f(p). We take a neighbourhood U. of p such that f (U,) e V and a neighbourhood r of f such that g(II,) e V for all g e p. The argument in the proof of Proposition 3.2 also shows that
the set of those g e n that are transversal to S on U. is open. In proving density we can start from a C°° map g e r because C°° maps are dense in +C"(M, N). It is easy to see that, if v e R" has small norm, there is g e v near g with y o g = y e g + v*on U. and g = g outside a neighbourhood of U,. By Corollary 1 of Proposition 3.3 there exists v e R" of small norm with y e g + v transversal to y(S) c R°. Thus g is transversal to Son U,,, which completes the proof.
26
1
Differentiable Manifolds and Vector Fields
Remark. In Theorem 3.4 we required S to be a C°° submanifold. However, the result remains true for S of class Cr, r > 1. In fact it suffices, in the proof above, to take a Cr local chart y such that y(S) is a C`° submanifold of I8" and then to approximate g by a map g that makes y o g of class C' on Up.
§4 Structural Stability The qualitative study of a differential equation consists of a geometric description of its orbit space. Thus it is natural to ask when do two orbit spaces have the same description, the same qualitative features; this means
establishing an equivalence relation between differential equations. An equivalence relation that captures the geometric structure of the orbits is what we shall define below as topological equivalence. Let X'(M) be the space of C' vector fields on a compact manifold M with
the C' topology, r >_ 1. Two vector fields X, Y E 3r'(M) are topologically equivalent if there exists a homeomorphism h: M - M which takes orbits of X to orbits of Y preserving their orientation; this last condition means that
if p E M and b > 0 there exists e > 0 such that, for 0 < t < b, hX,(p) = for some 0 < t' < e. We say that h is a topological equivalence between X and Y. Here we have defined an equivalence relation on X'(M). Another stronger relation is conjugacy of the flows of the vector fields. Two vector fields X and Y are conjugate if there exists a topological equivalence h that preserves the parameter t; that is, hX,(p) = Yh(p) for all p e M and
teR. The next proposition, whose proof is immediate, shows some of the qualitative features of the orbit space that must be the same for two equivalent vector fields.
4.1 Proposition. Let h be a topological equivalence between X, Y c X'(M). Then
(a) p e M is a singularity of X if and only if h(p) is a singularity of Y, (b) the orbit of p for the vector field X, Ox(p), is closed if and only if OY(h(p)) is closed,. (c) the image of the w-limit set of Ox(p) by h is the co-limit set of Or(h(p)) and similarly for the a-limit set. O
Figure 17
§4 Structural Stability
27
Figure 18
EXAMPLE 1. Let us consider the linear vector fields X and Yon R2 defined by
X(x, y) = (x, y) and Y(x, y) = (x + y, -x + y). The corresponding flows are X,(x, y) = e`(x, y) and Y,(x, y) = e`(x cos t + y sin t, - x sin t + y cos t). We shall construct a homeomorphism h of R2 conjugating X, and Y. As 0 is the only singularity of X and Y we must have h(O) = 0. It is easy to see that the unit circle S' is transversal to X and Y. Moreover, all the trajectories of X and Y except 0 intersect S'. We define h(p) = p for p e S'. If q e R2 - {0) there exists a unique t e R such that X,(q) = p e S'. We put h(q) = Y_,(p) = Y_,X,(q). It is immediate that It is continuous and has continuous inverse on 082 - {0}. The continuity of It and its inverse at 0 can be checked using the flows of X and Y. EXAMPLE 2. Let X and Y be linear vector fields on R2 whose matrices with respect to the standard basis are
X=
I
I),
Y=
101
-10)
These vector fields are not equivalent since all the orbits of Y are closed and this is, not true for X.
A vector field is structurally stable if the topological behaviour of its orbits does not change under small perturbations of the vector field. Formally we say that X e 3E'(M) is structurally stable if there exists a neighbourhood
I' of X in £'(M) such that every Y e y' is topologically equivalent to X.
Figure 19
28
1 Differentiable Manifolds and Vector Fields
Figure 20
The zero vector field on any manifold is obviously unstable. On the other hand the linear field. X(p) = p considered in Example 1 is structurally stable in the space of linear vector fields on 182. In order to motivate the necessary
conditions for structural stability that we shall introduce in l4ter chapters we present next some examples of unstable vector fields:
EXAMPLE 3. Let us consider a rational vector field on the torus T2, as in Example 2 of Section 1. This vector field is unstable in 1'(T 2). In fact all its
orbits are closed, whereas it can be approximated by an irrational vector field, which does not possess closed orbits. Actually, on (compact) manifolds
of dimension two every vector field with infinitely many closed orbits is unstable. This is because we can approximate it by a vector field with only a finite number of closed orbits, as we shall see in Chapter 4. EXAMPLE 4. Let it be a horizontal plane tangent to the torus T2 which is embedded in the usual way in 683 so that it meets T2 in a "parallel" or horizontal circle as in Figure 20. Let f : T2 - 68 be the function which to each point of T' associates its distance from it. We take X = grad f. The parallel of T2 contained in it is composed entirely of singularities of X. Now let X' = grad f', where fis distance from a plane n' obtained from it by a small rotation. As only four of the planes parallel to n' are tangent to T2 and each is only tangent at one point it follows that X' has only four singularities. Thus X is not equivalent to X' and so X is unstable. We shall show in Chapter 2 that every vector field with infinitely many singularities is unstable, since it can be approximated by another with a finite number of singularities.
Figure 21
29
§4 Structural Stability
Figure 22
EXAMPLE 5. We now describe a vector field on S2 which is unstable, even though it-is topologically equivalent to the north pole-south pole vector field (Example 1 of Section 1), which is stable. Let f : R -+ R be a CI map satisfying
the following conditions: f(t) > 0 for t # 0; f(t) = 1/t for t > 1; f(0) = df/dt(0) = . = d'f/dt'(0) = = 0. Consider the vector field on R2 .
defined by I(r cos 0, r'sin 0) = (rf (r) cos 0, rf(r) sin 0). The vector field 9 is radial and the origin is its only singularity, dX(0) = 0 and III(p)II 1 if IIPII z 1.
Let n: S2 - {PN} -+ R2 be the stereographic projection as shown in Figure 22. We define the vector field X on S2 by X(p) = d7r }I(7t(p)) if p # pN and X(pN) = 0. We see that X is a L°° vector field and has two singularities PN and ps. Note that the identity map is a topological equivalence between X and the north pole-south pole vector field of Example 1, Section 1. Let us show that there exists a vector field C' close to X having a closed orbit. Let 7 be the vector field on 182 defined by V'(r cos 0, r sin 0) = (rl(r) cos 0 + rg(r) sin 0, -rg(r) cos 0 + rl(r) sin 0), where 1, g: R -* R are C° maps with graphs as in Figure 23. 1(0) = 1(a) = 0,
g(0) = g(t) = 0, if t z c g(t) > 0,
1(t) = 1/t,
if t Z 1
1(t)<0,
if0
g'(a) = 0,
1(t) > 0,
if t > a
g(a) = b.
Figure 23
if 0 < t < c
30
I
Differentiable Manifolds and Vector Fields
Figure 24
The circle S. with centre at the origin and radius a is a closed orbit of Y since
: is tangent to S. at each of its points. Outside the disc of radius 1, V = X. Thus V defines a vector field Y on S2 which has the north and south poles as
"attracting" singularities, and a "repelling" closed orbit y = 7t-'(Sa). If we choose l to be C' close to f and g C' close to the zero function then Y will be C' close to X. As X is not topologically equivalent to Y, X is not structurally stable in X'(S2). We emphasize that, out of these examples, only the vector field of Example 5 is equivalent to a stable vector field. The basic reason for the instability in
this case is that the derivative of the vector field at the singularity ps is degenerate. It is, in general, a delicate problem to prove the stability of a vector field. Many examples will be given in Chapter 4. We shall next analyse the stability of vector fields on S'. This is a very simple case but it offers an insight into the general aims of the Theory of Dynamical Systems.
Let X° be one of the two unit vector fields on S. Any X e 1'(S') can be written in a unique way as X(p) = f(p)X°(p), p e S', with f e C'(S', R). It is clear that X(p) = 0 if and only if f (p) = 0. As we have already remarked, given any compact set K c S' there exists f e C'(S', fl) with f '(0) = K. Thus K is the set of singularities of X = fX°. As topological equivalence preserves singularities we have at least as many equivalence classes of vector fields as homeomorphism classes of compact subsets of S'. This shows that it is impossible to describe and classify the orbit structures of all the vector fields on S'. It is, therefore, natural to restrict ourselves to a residual subset of 3`'(S'), or preferably to an open dense one. " A singularity p of X E X'(S') is nondegenerate (or hyperbolic) if dX(p) # 0, that is df (p) 0 0 where X = f X °. If df (p) < 0 then p is a sink (or attracting singularity) and if df(p) > 0 then p is a source (or repelling singularity). Let G c X'(S') be the subset consisting of those vector fields whose singularities are all hyperbolic; as these singularities are isolated it follows that the number of them is finite (possibly zero!). We claim that G is open and dense. In fact,
let X = f X' and let 1: S'
S' x If8 be defined by .'(p) = (R f (p)). It is clear that X e G if and only if f is transversal to S' x {0}. But the set of f e C'(S', J) such that f is transversal to S' x {0) is open. This set is also dense, since for any f the set of v c R such that f + v is transversal to S' x {0} is residual and so we can choose v small with (.f + v)X° e G.
§4 Structural Stability
31
If X e G and X = f X1 we see from the graph off : S' - R that the sinks and sources of X must alternate round S'. In particular, the number of singularities is even. From this it follows that if X, ' Y e G have the same number of singularities then X and Y are topologically conjugate. In fact,
let a, b,, a2, b2, ... , a b, be the sinks and sources of X in order on -S'. Similarly, let a,, ba2, b2, ... , as, b; be the sinks and sources of Y in order
on S'. Define h(a,) = a; and h(b,) = b;. Choose points p, a (a,, b,), q, e If (b a,+ t) and p; (a', bi), qi e (bi, a;, 1). Define h(pi) = pi and h(q,) = p e (a,, b,) there exists a unique t e R such that X,(p) = pi; define h(p) _ Y_,(p;) = Y_,hX,(p) and proceed similarly for points of (b,, a,+ 1). It is now clear that h is a homeomorphism that conjugates the flows of X and Y If X does not have singularities the only orbit of X is the whole of V. If
X = f X° then either f > 0 or f < 0 on S. If f > 0 the identity is a topological equivalence between X and V. If f < 0 we take an orientationreversing homeomorphism as our topological equivalence. Finally we claim that if X e 3:'(S') is stable then X e G. First we remark that the number of
singularities of X is finite. This is because G is dense and so X must be equivalent to a vector field Y E G near X. We leave it to the reader to show that these singularities are hyperbolic; if not, we can perturb X in a way that increases the number of singularities contradicting the hypothesis that X is stable. Thus X E 1'(S) is stable if and only if X e G. Therefore, the structurally stable vector fields in X'(S') form an open dense set and, as we have seen, it is possible to classify them. The development of the geometric theory of differential equations led naturally to a parallel study of diffeomorphisms. We next introduce some basic concepts in the study of the orbit structure of a diffeomorphism. Let f e Diff'(M). The orbit of a point p e M is the set O(p) _ { f"(p); n c- Z). When O(p) is finite we say that p is periodic and the least integer n > 0 such that f"(p) = p is called the period of p. If f(p) = p we say that p is a fixed point. A point q belongs to the co-limit set of p, co(p), when there exists a sequence of integers ni -+ co such that f"'(p) -- q. If x e O(p) then (O(x) = co(p). Also a(p) is nonempty, closed and invariant. Invariant means that w(p) is a union of orbits off. For p periodic w(p) = O(p); thus w(p) is not connected if the period of p is greater than one. Analogously we define the a-limit set of p, a(p), as the cv-limit set of p for f ''. The above properties of w(p) hold for a(p) too. Equivalence of the orbit structures of two diffeomorphisms is expressed by conjugacy. A conjugacy between f, g e Diff'(M) is a homeomorphism
h: M - M such that h o f =g-h. It follows that h o f" = g" o h for any integer n and 'so h(O f(p)) = OB(q) if q = h(p). That is, h takes orbits off onto orbits of g and, in particular, it takes periodic points onto periodic points of the same period. Also h((of(p)) = wo(q) and h(a f(p)) = a,,(q).
EXAMPLE 6. Let us consider two linear contractions in l, .1(x) = Ix and g(x) = Jx. We shall show that f and g are conjugate. Take two points with coordinates a > 0, b < 0 and consider the intervals [1(a), a], [b, f (b)] and
32
I
Differentiable Manifolds and Vector Fields
[g(a), a), [b, g(b)]. Define a homeomorphism h: [f (a), a] u [b, f (b)] -[g(a), a] u [b, g(b)] such that h(a) = a, h(b) = b, h(f(a)) = g(a) and h(f(b)) = g(b). For each x e 18, x 0, there exists an integer n such that f"(x) e [ f (a), a] u [b, f (b)]. We put h(x) = g - "hf "(x) and h(0) = 0. It is easy to see that It is well defined and is a conjugacy_ between f and g. On the
other hand, the contractions f (x) = Ix and g(x) = -Jx are not conjugate. By this argument we can also see that two contractions of R are conjugate if and only if they both preserve or both reverse the orientation of R. EXAMPLE 7. The linear transformations of 6$2, f (x, y) = (+x, 2y) and g(x, y) = (ix, 4y) are conjugate. We construct, as in Example b, a conjugacy h t between
f I C8 x {0) and g I R x (0) and a conjugacy h2 between f I {0} x ll and g 110) x R. The conjugacy between f and g is given by h(x, y) = (h t (x), h2(y)). EXAMPLE 8. The linear transformations Xt, Yt induced at time 1 by the vector fields X, Y of Example 2 are not conjugate. It is sufficient to observe that Yt leaves invariant a family of concentric circles and Xt does not.
Conjugacy gives rise naturally to the concept of structural stability for diffeomorphisms. Thus f e Diff'(M) is structurally stable if there exists a neighbourhood y'' off in Diff'(M) such that any g E V is conjugate to f. The identity map is obviously unstable. Also, the diffeomorphisms induced at time t = 1 by the vector fields X in Examples 3,4 and 5 of this section are unstable.
EXAMPLE 9. Let us take a vector field X e 3E'(St) which is stable and has singularities. As we have already seen, X has an even number of singularities
alternately sinks and sources al, bl, a2, b2, ... , a b,. Choose points pi a (a,, bi) and q, E (b,, a,+t). Now consider the diffeomorphism f = Xt
Figure 25
33
§4 Structural Stability
Figure 26
induced by X at-time t = 1. We shall prove that f e biff'(S') is structurally stable. We know that f is a contraction on [q,_ 1, p,] with fixed point a, and f is an expansion on [pi, q,] with fixed point b1. If g is C close to`f then g is a contraction in [q1_ 1, p,] with a single fixed point a, close to a1. In addition, g is an expansion on [p,, q,] with a single fixed point 5, near b,. We put
h(al) = a,, h(b) = $,, h(p) = p h(q,) = qt, h(f(P)) = g(Pt) and h(f(q,)) = any homeomorphism from the interval [p,, f(p,)] to [Pi' g(p)] and from [q,, f(q,)] to [q,, g(q,)] and then we extend h to [a,, b,] and [b,_ 1, a1] as in Example 6. We obtain a conjugacy between f and g, which shows that f is structurally stable. g(q,). We define h
We emphasize that the stable diffeomorphisms in Difl'(S') form an open dense subset. The proof of this result is much more elaborate and will be done in Section 4 of Chapter 4. We also remark that, in the example above, we started from a stable vector field in X'(S') and showed that the diffeomorphism it induced at time t = 1 was stable in Diff'(S1). The next example shows that this is not always the case. EXAMPLE 10. Consider the unit vector field X° on S1. Then S' is a closed orbit
of X° of period 2n. The diffeomorphismf = X° induced at time t = 1 is an irrational rotation. The orbit 0 f(p) is dense for every point *p a S1. To see that f is unstable we approximate f by g = X° with t near one and t/2>t rational. Every orbit Oo(p) is periodic and so f is not conjugate to g. We now show why we defined a conjugacy to be a homeomorphism rather
than a diffeomorphism. Let us consider again the diffeomorphism f in Example 9. As we saw f is structurally stable: if g is C close to f then there exists a homeomorphism h of S' such that It o f = g c It. To construct It we note first that, for each sink a, off, we have a sink a; of g close to a,. The same' is true for the sources. We put h(a) = a'. It is easy to see that we can choose g close to f such that a; = a, and g'(a,) # f'(a,). Now suppose that h is, in
I Differentiable Manifolds and Vector Fields
34
Figure 27
fact, a diffeomorphism. Then we have h(a) = a; and h'(a;) f(a) = g'(a) h'(a;) which -implies that f '(a) = g'(a,) contrary to our hypothesis. Thus, if we required the conjugacy to.be a diffeomorphism, f would not be stable in Diff'(S 1). Similarly we can show that no f e Diff'(M) that has a fixed or periodic point would be stable. This shows that we ought not to impose the condition of being differentiable on a conjugacy. The same idea applies to topological equivalence between vector fields. Although the proof is more complicated it is also true that no vector field with a singularity or a closed orbit would be stable if we required the equivalence to be differentiable. See Exercise 13 of Chapter 2 and also Exercise 5 of Chapter 3.
ExBxcis>^s
1. Show that every C' vector field on the sphere S2 has at least one singularity.
2. Two vector fields X, Y e £'(M) commute if X,(Y,(p)) = Y,(X,(p)) for all p e M and s, t e R. Show that if X, Y e X'(S2) commute then X and Y have a singularity in common (E. Lima). 3. Let X = (P, Q) be a vector field on P2 where P and Q are polynomials of degree two. Let y be a closed orbit of X and D c P2 the disc bounded by y. Show that X has a unique singularity in D. 4. Let X e 3i'(M2) and let F 'c- M2 be a region homeomorphic to the cylinder such that X,(F) e F for all t - 0. Suppose that X has a finite number of singularities in F. Show that the co-limit of the orbit of a point p e F either is a closed orbit or consists of singularities and regular orbits whose co- and a-limits are singularities.
5. Let F e M2 be a region homeomorphic to a Mobius band and let X e £'(M2) be
a vector field such that X,(F)c F for all t z!t 0. If X has a finite number of singularities in F then the o)-limit of the orbit of a point p e F either is a closed orbit or consists of singularities and regular orbits whose to- and a-limits are singularities.
35
Exercises
6. Let y be an isolated closed orbit of a vector field X e £'(M2). Show that there exists a neighbourhood Y of y such that, for p e V, either a(p) = y or w(p) = y. 7. A closed orbit y of X e 1'(M2) is a i attractor if there exists a neighbourhood V of y such that X,(p) e V for all t z 0 and w(p) = y for all p e V. Show that, if X has a closed orbit that is an attractor, then every vector field Y sufficiently near X also has a closed orbit.
8. Let X be a C' vector field on the projective plane. Show that, if X has a finite number of singularities, then the w-limit of an orbit either is a closed orbit or consists of singularities and regular orbits whose w- and a-limits are singularities.
9. Let X be a vector field on the torus T2 which generates an irrational flow X,. Show that, given n e N and e > 0, there exists a vector field Y with exactly n closed orbits such that II Y - X II, < e.
10. A cycle of a vector field X e £'(M) is a sequence of singularities pl,... , pj, Pj+ i = Pi and regular orbits y, ... , y j such that a(y;) = pi and w(y) = p,+,. Let X e X'(S2), r 1, satisfy the following properties: (1) X has a finite number of singularities; (2) if p e S2 is a singularity of X then either p is a repelling singularity or the set of orbits y with a(y) = p is finite. Show that, for any orbit y, (a) if w(y) contains more than one singularity then w(y) contains a cycle; (b) if p, and P2 are singularities contained in w(y) then there exists a cycle which
contains p, and p2. I I.A. Let G e R" be an additive subgroup. Show that, if G is closed, then it is isomorphic
to Rk x Z' for some k and I with k + 1:5 n. Hint. (a) Show that, if G is discrete, then it is isomorphic to Z'; that is, there exist vectors v,, ... , v, a R" such that G = {Y,=, n; v,; n, a Z}. (b) Show that, if G. is not discrete, then it contains a line through the origin. (c) Let E e R" be the subspace of largest dimension contained in G. Let El be the orthogonal complement of E. Show that G = E ® (El n G) and that El n G is a discrete subgroup of El.
11.B. Let a = (a,, ... , a") a R. Let G = (la + m; l e Z and m e Z). Suppose that the
coordinates of a are independent over the integers; that
is, if <m, a>
;,"s,mia, = 0 with m e Z" then m = 0. Show that G is dense in R".
Hint. (a) n: R" -. R"-' be the projection
Let
6 be the closure of G. Suppose, by induction, that n(C) = R"-'. (b) Let E e R" be the subspace of largest dimension contained in C. Then either the dimension of E is n - 1 or it is n and E contains the vector a. (c) Let E1j = {x; xk = 0 if k s i, j}. Show that E,j n E is a straight line in E,j with rational slope. Deduce that E contains n - 1 linearly independent vectors with integer coordinates. The vector product of these vectors is a vector with integer coordinates that is perpendicular to E.
11.C. Find an example of a vector field of class C' on the torus r = S' x . . . x S' such that all its orbits are dense in T.
36
I
Differentiable Manifolds and Vector Fields .
Figure 28
12. Let X' be a C°° vector field defined on a neighbourhood of a disc'D; c 682 for i = 1, 2\ Suppose that X.' is transversal to the boundary C; of Di and that X' points out o7 D, while X2 points into D2. Show that there exists a C° vector field X on S2 and embeddings h,: Di -. S2 such that: (1) hi(D,) n h2(D2) = 0; (2) dh,{ p) X(p) - X(h,(p)) for all p e D,; (3) if p e h,(C,) then the arlimit of p is contained in h2(D2). 13. Let X', X2 be C° vector fields on manifolds M,, M2 of the same dimension. Let D; c Mi be discs such that X' is transversal to the boundary Ci of D;, i = 1, 2, with X' pointing out of D, and X2 pointing into D2. Show that there exist a C' vector field X on a manifold M and embeddings h,: M, - D; -+ M such that:
(1) hi(M, - D1) n h2(M2 - D2) = 0; (2) dh;(p) Xr(p) = X(h1p)) (3) if p E h,(C,) then the a-limit of p is contained in h2(M2 - D2)14. Let X be tote parallel field a/at on the cylinder S' x [0, 1]. Let M be the quotient
space of S' x [0, 1] by the equivalence relation that identifies S' x {0} with S' x {1} by an irrational rotation R: S' x {0} -. S' x (1).Letn:S' x [0, 1)-+M be the quotient map. Show that (a) there exists a manifold structure on M such that it is a local diffeomorphism and n.X is a C°° vector field on M; (b) there exists a diffeomorphism h: M -- T2 such that if Y ,h.X then Y, is an irrational flow. 15. Let M, N and P be manifolds with M and N compact. Show that
(a) the map comp: C'(M, N) x C(N, P) - C'(M, P), comp(f, g) = g o f, continuous; (b) the map is Diff'(M)
is
Diff'(M), i(f) = f -' is continuous.
16. Let M and N be manifolds, with M compact, and let S e M x N be a submanifold.
Consider the set Ts = {f e C'(M, N); graph(f) is transversal to S}, where graph(f) _ {(p, j(p)); p e M}. Show that Ts is residual in C'(M, N). 17. For each f e C(68", 68"') consider the map
j'f : 68" - 68" x R' x L(RR, R) x ... x L,(R"; 68) x r+ (x, f (x), df (x), ... , d'f (x)).
37
Exercises
Let E be the Euclidean space R" x R" x L(P", R") x
. x L;(R; R). For
each open set U c E define the subset
..rtr(U) _ (f e C'(08", R);ff(R) c U). (a) Show that the sets ..K(U) form a base for a topology on C'(08", l8) (the Whitney topology). (b) Show that C'(R", l8) with the Whitney topology is a Baire space. (c) Show that the C°° maps form a dense subset in C'(R", l8"). (d) Show that, if k < r, the map C(08", R")
C''
(1", Q8" x C8"' x L(R", R) x ... x L;(R"; R))
f -If is continuous.
(e) Let S c l8" x R' x L(08", E") be a submanifold. Consider the set Ts = (f e C(R", l8"); j'f f S), where r >_ 2. Show that Ts is residual. 18. Let X° a X'(S') be a vector field without singularities. Let E, be the set of vector
fields X = fX° e 1'(S') such that the singularities of X are all nondegenerate except one at which the second derivative off is nonzero. Let 11.1 be the set of vector fields X = f X' such that the singularities are all nondegenerate except two at which the second derivative off is nonzero. Let 1;,,, be the set of vector fields X = f X' such that the singularities of X are nondegenerate except one at which the second derivative off is zero but the third derivative is nonzero. (a) Show that 1, is a codimension 1 submanifold of the Banach space X'(S') and that E, is open and dense in 1'(S') - G, where G consi$ts of the structurally stable vector fields as in Section 4. (b) Show that 12 = 11.1 v 11.2 is a codimension 2 submanifold of 1'(S') and that 12 is open and dense in X'(S') - (G v 11). (c) Describe all the equivalence classes in a neighbourhood of a vector field in 11 and of one in 12. Remark. Sotomayor [114] considered conditions like these in the context of Bifurcation Theory.
19. (a) Show that, if g: R - R is a C' diffeomorphism that commutes with f : R - R given by f(x) = Ax where 0 < . < 1, then g is linear. (b) Show that, if g: R2 - R2 is a C' diffeomorphism that commutes with a linear contraction whose eigenvalues are complex, then g is linear. (c) Show, however, that there does exist a nonlinear C' diffeomorphism that commutes with a linear contraction. 20. Poincare Compactifrcation. Consider the sphere S2 = {y a R3; 3=, y; = 1) and
the,plane P = {y a 183; y3 = 1} tangent to the sphere at the north pole. Let
Ui={yeS2;y1>0}and Vi ={yeS2;yi<0}. Let n3:P-'U3,a3:P-.V3be the central projections, that is, n3(x) and rz3(x) are the intersections of the line joining x to the origin with U3 and V3. Let L be a linear vector field on R2. Consider the fields X' = (n3).L. on U3 and X2 = (k3).L on V3. (a) Show that X' and X2 extend to a C°" vector field X, which we call n(L), on S2 and that the equator is invariant for X.
I
38
Differentiable Manifolds and Vector Fields
(b) Describe the orbits of the fields n(L), i = 1,2,3,4, where the L' are represented, with respect to the standard basis, by the matrices
(Al \0
O/'
\1
O/'
pO
(c) Show that, if L' and L2 = AL'A - ' are linear vector fields and A is a linear isomorphism, then a(L`) and n(L2) are topologically equivalent. (d) Show that, if L is a linear vector field which has two equal eigenvalues or an eigenvalue with real part zero, then n(L) is not structurally stable in X '(S2). Hint. Use the local charts
Remark. The vector fields that are structurally stable among those induced on the sphere S" by linear vector fields. were characterized by G. Palis in [73].
21. An orbit y of X a 3:'(M"), r >_.1, is said to be a)-recurrent if y c w(y). Let M be a
compact manifold and y an w-recurrent orbit for X E X'(M). If f = X,=, and x e y show that x is w-recurrent, that is x e w f(x).
Remark. The Birkhoff centre C(X) of X E 3:'(M) is defined as the closure of the set of those orbits that are both w- and a-recurrent. The same definition works for J 'c Diff'(M). This exercise shows that C(X) = C(f) when f is the time 1 diffeomorphism of X.
Chapter 2
Local Stability
In this chapter we shall analyse the local topological behaviour of the orbits of vector fields. We shall show that, for vector fields belonging to an open dense subset of the space 1'(M), we can describe the behaviour of the trajectories in a neighbourhood of each point of the manifold. Moreover, the local structure of the orbits does not change for small perturbations of the field. A complete classification via topological conjugacy is then provided. Such a local question is considered in two parts: near a regular point and near a singularity. The first part, much simpler, is dealt with in Section 1. The second part is developed in Sections 2 through 5. Section 2 is devoted to linear vector fields and isomorphisms for which the notion of hyperbolicity is introduced. In Section 3 this notion is extended to singularities of nonlinear vector.fields and fixed points of diffeomorphisms. Local stability for a hyperbolic singularity or a hyperbolic fixed point is proved in Section 4. Finally,
in Section 5 we present the local topological classification. Section 6 is dedicated to another important result, the Stable Manifold Theorem. Much related to it is the A-lemma (Inclination Lemma) that is considered in Section 7, from which we obtain several relevant applications and a hew proof of the local stability,
§1 The Tubular Flow Theorem Definition. Let X, Y e X'(M) and p, q e M. We say that X and Y are topologically equivalent at p and q respectively if there exist neighbourhoods Vp and Wq and a homeomorphism h : Vp - W,,, with h(p) = q, which takes orbits of X to orbits of Y preserving their orientation. 39
40
2 Local Stability
Figure 1
EXAMPLE. Consider the vector fields X and Y on S2 given in Figure 1. X and
Y are not equivalent at PN, PN since each neighbourhood of P;, contains closed orbits of Y but there are no closed orbits of X near P. Definition. Let X e .'(M) and p e M. We say that X is locally stable at p if for any given neighbourhood U(p) c M there exists a neighbourhood .A' of X in £'(M) such that, for each Y e,* x, X at p is topologically equivalent to Y at q for some q e U. The next theorem describes the local behaviour of the orbits in a neighbourhood of a regular point. 1.1 Theorem (Tubular Flow). Let X e X'(M) and let p e M be a regular point
of X. Let C = {(xl,..., x'") e R'; Jx`J < 1) and let Xc be the vector field on C defined by Xc(x) = (1, 0, ... , 0). Then there exists a C' difteomorphism h: V, - C, for some neighbourhood V, of p in M, taking trajectories of X to trajectories of X.
PROOF. Let x: U - U0 c R"' be a local chart around p with x(p) = 0. Let x. X be the C' vector field induced by X on U0. As X(p) # 0 we have x. X(0) 0 0. Let W : [ - r, T] x Vo -+ U0 be the local flow of x. X and put H = {co e R'; <w, x.X(0)> = 0), which is a subspace isomorphic to R'"-'.
Let 0: [ - T, r] x S -' U0 be the restriction of tp to [ -,r, T] x S where S = H n V0. Take a basis {el, e2, ... , e,,,} of R x H R' where e, _ (1, 0, ... , 0) and e2, ... , e,,, c {0) x H. It follows that DO (O, 0)e, = x. X(O)
(by the definition of local flow)
Dtfi(0, 0)e1 = ej,
j = 2, ... , m,
since s'(0, y) = y for all y e S. Thus, Dtf(0, 0): R x H -- R' is an isomorphism. By the Inverse Function Theorem, t/i is a difleomorphism of a neighbourhood of (0, 0) in [ -,r, T] x S onto a neighbourhood of 0 in R'". Therefore, ifs > 0 is small enough, Ct = {(t, x) e R x H; Jti < e} and 11 x1J < e and : Ct -+ U0 is the restriction of
41
§2 Linear Vector Fields
Hx (0)
H x (- r l
Hx (r)
VOx{"'r .
Figure 2
q to C,, then
is a C' diffeomorphism onto its image which is open in U0. Moreover, takes orbits of the parallel field Xc. in Cr to orbits, of x. X. Let us consider the C°° diffeomorphism f : C -+ C,, f (y) = cy and define h-' =
x-' if : C -+ M. Then h: x-' (Q - C is a C' diffeomorphism which satisfies the conditions in the theorem.
Remark. The diffeomorphism h-': CL -' M defined by h-' = x-' takes orbits of the unit parallel field Xc. to orbits of the field X preserving the parameter t. Corollary 1. If X, Y E £'(M) and p, q e M are regular points of X and Y, respectively, then X is equivalent to Y at p and q.
Corollary 2. If X e X'(M) and p e M is a regular point of X then X is locally stable at p.
§2 Linear Vector Fields Let 2'(R") be the vector space of linear maps from R" to R" with the usual norm: IILII = Sup{IILvlI; IIvUI = 1).
First we recall some basic results from linear algebra. If L e 2(R") and k is a positive integer we write Lk for the linear map L c o L. It is easy to show, by induction, that IILkll < IIL11". Let us consider the sequence of linear maps E," = Em= ° Ilk! Lk where L° means the identity map. 2.1 Lemma. The sequence E. converges.
Pttooi. The sequence of real numbers Sm = Em=0 1/k! IlLllk is a Cauchy sequence that converges to ell 11. On the other hand, m+m'
IIEm+m' - Emil
k=m+t k
Lk s
m+m' ,
km+t k
IILIIk = I1S1+m' - S.H.
2 Local Stability
42
This shows that {Em} is a Cauchy sequence. As x(R") is a complete metric space it follows that the sequence {Em} converges.
Definition. The map Exp: £°(R") - .2'(18") defined by Exp(L) = e' _ E. 0 1/k! L' is called the exponential map.
2.2 Lemma. Let a: 01 -+ 2'(18") be defined by a(t) = e`L. Then a is differentiable and a'(t) = L e`L.
PROOF. Let am(t) = I + tL + (t2/2!)L 2 + - - - + (t'"/m!)Lm. It is clear that am is differentiable and -1
a;,,(t)=L+tL2+...+ (m - 1)! Lm=Lam- 1(t) As am_ 1(t) converges uniformly to e`L on each bounded subset of 18, it follows
a is differentiable and a'(t) = LeL. '
that
2.3 Proposition. Let L be a linear vector field on W. Then the map cp: 18 x 11" -+ R" defined by cp(t, x) = e`Lx is the flow of the field L.
PROOF. As the map .P(l ") x18" .- R", (L, x) -+ Lx is bilinear and the map t --+ e`L is differentiable it follows by the chain rule that cp is differentiable.
Moreover, a/at ip(t, x) = Lip(t, x) by Lemma 2.2. As ip(0, x) = x for all x e R" the proposition is proved.
Let C" be the set of n-tuples of complex numbers with the usual vector space structure. An element of C" can be written in the form u + iv with u, v e R". If a + ib e C then (a + ib)(u + iv). = (au - bv) + i(av + bu). Let 2(C") denote the complex vector space of linear maps from C" to C" with the usual norm; IILII = sup{IlLvll ; v e C" and IILII = 1}. If L e x(18") we can
define a map L: C" - C" by i?,(u + iv) = L(u) + iL(v). It is easy to see that L is C-linear; that is, L e .'(C"). Let Exp: 2(C") -i. 2'(C") be the exponential map, which is defined in the same way as in the real case. Let W:. '(R") -+ 2(C") be the map which associates to each operator L its complexification L defined above. The proposition below follows directly from the definitions. 2.4 Proposition. The map 'g: 2'(R") -+ £(C") satisfies thefollowing properties: (1)
T) = BB(L) + (T), le(aL) = aW(L);
(2) W(LT) _ `B(L)`B(T); (3) W(Exp L) = Exp 1'(L); (4) 11W(L)II = IILII,
for any L, T e.'(R") and at e R.
43
§2 Linear Vector Fields
EXAMPLE. Let L e 2(R2) and let {et, e2} be a basis for 082 with respect to
which the matrix of L has the form
/a i
f
.
a
The matrix of L = 6(L) in 0
the basis {et + ie2, e, - ie2) for C2 is 0
where A = a + ifi and
.1
0
eA
ezj. On the other
and 11 = a - if. Thus, the matrix of e' in this basis is 0
hand, le(eL)(e, + ie2) = e'e, + iete2 = e''(e, + ie2) = ez(e, + ie2). As e' = e°(cos + i sin fi) it follows that eLe, = e°(cos fi e, - sin Q e2) and eLe2 = e'(sin /3 et + cos /3 e2). Therefore, the matrix of e' in the basis {e,, e2} is e°
cos fl
-sin
sin fl. cos NJ
2.5 Theorem (Real Canonical Form). If L E 2(08") there exists a basis for R" with respect to which the matrix of L has the form A,
0 A,
B,
O B,
where A;
0 i = 1,...,r,
A; =
C.
Bi =
O
Ci=(
faii
0 Ci
I
aii
A;ER
1
j = 1,...,s
'Ci
1=
1
(0
0 ,
i
1
The submatrices A,__ , A B ,, ... , B, are determined uniquely except for their order.
2 Local Stability
44
Corollary. Let L e Y(R"). Given E > 0 !here exists a basis for U8" with respect to which the matrix of L has the form
0 A,
with
A.=
B;=
Aj
O
a1
lji
- Ri
ai
0 D
o
ti;
O
e
0
ai
0
C
-/ii
2.6 Lemma. If A, B e.'(118") satisfy AB = BA then
eA+,B
ai
= eaea
PxooF. Let I + 4 + + (tm%m!)Am. As AB = BA we have AkB BAk and so Sm(t)B = BSm(t). Since Sm(t) > e1A we have e`AB = Be".
Let x e R" and consider the curves x, /3: R -+ 18", a(t) = etA+elx, /3(t) _ e`Ae`Bx. By Lemma 2.2 we have Y(t) =.(A + B)e`tA-Blx = (A + B)a(t) and Ae`"e`Bx + e`ABe`x = Ae`AetBx + Be`AetBx = (A + B)/3(t) using e`AB = BeA. Therefore a and $ are integral curves of the linear vector field A + B and satisfy the same initial condition a(0) = /i(0) = x. By the uniqueness theorem we have a(t) = /3(t) for all t. In particular, eA+Bx = eAeBx. As this holds for all x e 08" it follows that eA+B = eAeB If CE x(R) then the spectrum of L, that is, the set of eigenvalues of L, is called the complex spectrum of L and coincides with the set of roots of the characteristic polynomiaa of L. The Jordan canonical form of the complexified
operator L is represented by At
O ,
O Ar
O
where A,= O
1
A;
and the A; are the eigenvalues of L.
We remark that a triangular complex matrix has its diagonal entries as eigenvalues with multiplicity equal to the number of times they appear.
§2 Linear Vector Fields
45
2.7 Proposition. If L e 2'(R") and A is an eigenaalae of L then e`_ is an eigenvalue of e' with the same multiplicity. PROOF. Consider an m x m matrix
A
1
where A e C. We have A = D + N where
O D =
and
O 2 It is easy to see that M" = 0 and that ND = DN. By Lemma 2.6 we have
`/(m-I)!sinceN"=0
e`=eDeN.But fork >_ m. Thus
..:
1/(m - 1)!
2'
1
1
Now e" = e D e N = ezeN
Therefore e" is a triangular matrix with all its diagonal elements equal toe` and so e` is an eigenvalue of e" with multiplicity m. Now let L e 2(R ). By the Real Canonical Form Theorem the matrix of L, with respect to a certain basis of /3 of C", has the form .
A=
,
O
0
AO ;
Al AO
with
A,=
r
1
It is easy to see that (Ak,.
,
A': =
0
,
0
A,
A;
2 Local Stability
46
for all k ,E N and, therefore,
eO PA =
O
eA,
This shows that the eigenvalues of e" are exactly e''-, ..., ez, where At......k, are the eigenvalues of A. But et = eL is represented with respect to the basis 1 of C" by the matrix e" which shows that e"', .. , ea, are the eigenvalues of the complexification of eL.
Definition. A linear vector field L e.(R") is hyperbolic if the spectrum of L is disjoint from the imaginary axis. The number of eigenvalues of L with negative real part is called the index of L. Note that a hyperbolic linear vector field has only one singularity which is the origin.
2.8 Proposition. If L e 2'(R") is a hyperbolic vector field then there exists a unique decomposition (called a "splitting") of Was a direct sum 68" = E' ® E", where E' and E" are invariant subs paces for L and for the flow defined by L such
that the eigenvalues of L' = L! E' have negative real part and the eigenvalues of L" = LSE" have positive real part. PROOF. Let et, ... , e" be a basis for R" for which the matrix of L is in the real
canonical form. For an appropriate order of the elements of this basis the matrix of L has the form At A,
O
B,
Ct
.CM
O
Dt
D.] where
O
A;
A. =
1
A,
O
1
,
A;
with
a; < 0,
47
§2 Linear Vector Fields
O
M;
B. =
I
M,
O
1
aJ ,
with M. _
f,
- fl aj)
i
I
0,.
(0 1)
M;
and a; <0 Ck=
V
O
k
with Ak>0
1
1
CO
2k
and
D, =
I
\O
a1
HI,
I
with
MI =
Mi
(1't
011)
x,
Let Es be the subspace generated by el, ..., e,, where e...... e., correspond to the invariant subspaces associated to A,, ... , AS., B1, ..., B3... Let E" be the subspace generated by es+,, ... , e". It is clear that E' and E" are invariant for L and that the matrix of L', for the basis {e,, . , . , es}, is IA,
Bs
{
while the matrix of L", for the basis.{e,+,, ... , e"}, is
D,
O D"! which shows the existence of the required decomposition. Uniqueness is O immediate. Let L e .(R") be a hyperbolic vector field. If L, denotes the flow of L then L, = et and, as L does not have an eigenvalue on the imaginary axis, it follows from Proposition 2.7 that L1 does not have an eigenvalue on the unit circle S'. This suggests the following definition.
48
2 Local Stability
Definition. A linear isomorphism A e GL(R") is hyperbolic if the spectrum of A is disjoint from the unit circle S` c C. In particular, the diffeomorphism induced at time 1 by the flow of a hyperbolic linear vector field is a hyperbolic isomorphism. 2.9 Proposition. If A e GL(P") is a hyperbolic isomorphism then there exists a unique decomposition U8" = E' (B E" such that E5 and E" are invariant for A and the eigencalues of A' = A J E' and A" = A J E' are the eigenvalues of A of modulus less'than 1 and greater than I respectively.
0
PRooF. Similar to the proof of Proposition 2.8.
2.10 Proposition. If A e GL(R") is a hyperbolic isomorphism then there exists
a norm 11.11, on R" such that IIA'11, < 1 and II(A")'II < 1, that is AS is a contraction and A" is an expansion.
PR(X)F. Consider the canonical form for A' = A I E',
O
where
IA;I
and
.
B; _ 1
0
xj+f3 <1.
1
0
1
0
0
1
a;
ft;
-f3
x;
49
§2 Linear Vector Fields
For each t e 88 consider the matrices
O
Ai
Ai(t) =
t
'
(0.-
,
t_-A,
O
B,{t) =
t
0
0
t
0
t
0
0
t
and
M(t) =
BI(t)
0
8,:.(t)
We claim that there exists b > 0 with the following property _ if e < b and et,... , e, is an orthonormal basis of E' and A is a linear transformation of E' whose matrix in this basis is M(e) then II All < 1. In fact, let A(t) be the linear transformation of E' whose matrix in the basis et, ... , e, is M(t). It is easy to see that IIA(0)II = max{ I A, I, (a; + fl)}. ,Thus IIA(0)11 < 1. As the composition t --, A(t) --+ IIA(t)II is continuous there exists E > 0 such that 1IA(t)II < 1 for 0 < t < b, which proves the claim.. Now let b > 0 be as above. By the Corollary to Theorem 2.5 there exists a basis ei, ..., e, of Pin which the matrix of A' is M(e). We define a new inner
product on E' by <ei, e;)t = bi; where bi; = I if i = j and 0 if i # j. Let 1 1-01 be the norm associated to < , >,; As the basis is orthonormal in the new metric the claim implies IIA'II, < 1. Similarly we change the norm on E" so that II(A")-'lj, < 1. We define a normon UF" by Ilvll, = max{llvtll,, Ilv"II,}, where v' and v" are the components of v in E' and E respectively. It 1s clear that this norm satisfies the conditions in the proposition. 0 Corollary. If L is a hyperbolic linear vector field with flow L, and R" = E' ® E" is the splitting of Proposition 2.8 then L,(x)`eonverges to the origin if x e E'
and tw.+ooor ifxeEMand t-: -oo.
2 Local Stability
50
PROOF.. Let x E Es. It is sufficient to show that L"(x) -+ 0 where n e N and cc. In fact, if t e [0, 1] we have, by the continuity of L that, given e > 0, it there exists b, > 0 such that IIL,(y)II < s for Ilyll < b,. As [0, 1] is compact
there exists b > 0 such that IIL,(y)II < e for llyll < b and all t e [0, 1]. If L"(x) -. 0 as n - oc there exists no a N such that lIL"(x)Il < b if n z no. If
t > no then t = n + s for some n >- no and s e [0, 1]. Thus IIL,(x)II = IILSL"(x)ll < E. So it is sufficient to show that L"(x) = L;(x) tends to 0. By the proposition above there exists a metric on E' in which L, is a contraction, that is IILt iI < 1. Then IILixll < IJLillllxll <- IIL1II"IlxlI. As IIL, II" -. 0 we
have IILixIi - 0 as required. The second part of the corollary is proved similarly.
2.11 Proposition. The set H(P") of hyperbolic isomorphisms of P." is open and dense in GL(P"). PROOF. (a) Openness. Let A e H({."). Let us show that there exists b > 0 such
that, if IIA - BII < b, then B e H(R"). Let A e S'. As A is not an eigenvalue of A, det(A - Al) : 0 where I is the identity of C". Now, det: 2'(C") -* C is a continuous map so there exist 6,, > 0 and a neighbourhood VV of A in C such that, if 11B - A II < b,, and p e V;,, then det(B - p1) 0 0. Let V,,,, ... , V,, be a finite subcover of the cover { V2 ; A e V) of S'. Put b = min {b;,,, ... , b2,"}.
If IIB - All < b and p e S' then p e VAJ for some j, and, therefore, det(B - pI) 0 0. Thus, B e H(R") as required. (b) Density. Let A e GL(P.") and let A...... A. be its eigenvalues. It is easy to see that, if p e P., the eigenvalues of A + p1 are A, + p, ... , A" + p. Let A;,, ..., At, be the eigenvalues of A which do not belong to S'. Consider the following numbers:
b, = b2 = min{Il -
IA,II},
63 = min { I a I ; a + iQ is an eigenvalue of A with a2 + #2 = 1 and a
01.
It is clear that b, > 01 b2 > 0 and b3 > 0. If 0 < p < min (61, 62, 63) and A
is an eigenvalue of A then A + p 0 S' and so B = A + µl is hyperbolic. Given e > Owe take p < e and u < min{b,, b2 , b3} and then B is by perbolic and
IIB - All = Ilplll < s. This shows that H(P") is dense in GL(P"). 2.12 Proposition. The set .,Y(P.") of hyperbolic linear vector fields on 1B" is open and dense in 2(P.").
PROOF. (a) Openness., The map Exp: 2'(P.") - GL(P.") is continuous. By Proposition 2.7 we have .°(R") = Exp-'(H(R")). As H(P.") is open it follows that .-E°(P.") is open too.
(b) Density. Let L e 2'(P."). Let b, = min{ lal; a + iJi is an eigenvalue of L and a # 0). Given e > 0 we take b < min (E, b,). It is easy to see that the vector field T = L +& is hyperbolic acid I T - L 11 < e.
51
§2 Linear Vector Fields
Our next aim is to give a necessary and sufficient condition for two hyperbolic linear vector fields to be topologically equivalent.
2.13 Lemma. Let L be a hyperbolic linear vector field on R" with index' n. There exists a norm II.11 on R such that, if S"-' = {v e R"; JJvJJ = 1), then the vector L(x) at the point x is transversal to S"-' for all x e S"-1. PROOF. Let us consider a basis et, ... , e" of II8" for which the matrix of L is
0
At(1) As (1)
A=
B,(1) Bs.(1)
0 with Ai(l) and Bj(l) as in Proposition 2.8. Let L be a linear vector field on R" whose matrix in an orthonormal basis is
0
(A1(o)
A,(0)
A=
Bt(0) 1
Q
BA0)
It is easy to see that L is transversal to S' '. Since S"-' is compact, if e > 0 is sufficiently small the field E, whose matrix in this orthonormal basis
0
JA t(e)
B1(e)
0 is transversal to S"-'. On the other hand, by the corollary to Theorem 2.5, there exists a basis of R" in which the matrix of L is A. We define an inner product on R" making this basis orthonormal and then, by the argument above, L is transversal to the unit sphere in this norm. 0 2.14 Proposition. If L and T are linear vector fields on R" of index n then there exists a homeomorphism h: R" -+ R" such that hL, =. T h for all t E R.
PROOF. Let 11-11, and 11-112 be norms on R' such that the spheres S,-' _ {v E R"; I I v 1 1 i = 1} and SZ ' = {v a R"; II012 = 1) are transversal to the vector fields L and T respectively. If x e R^ - {0} then, by the corollary to
2 Local Stability
52
Proposition 2.10, we have lira,- L,(x) = 0 and lime..,, IIL_,(x)II = co so. that 0L(x) does meet S'-'. As L is transversal to S,-' it follows that OL(x) meets S;-' in a unique point. Let h: S;-' - S2-' be any homeomorphism (for example, we can put h(x) = x/IIxI12). We shall extend h to R". Define h(O) = 0. If x e 1R" - {0} 'there exists a unique to e R such that L_,o(x) e S"-'.Put h(x) = T"h(L_,"(x)).
It is easy to see that hL, = T,h for all t e 98 and that h has an inverse. It remains to show that h is continuous. Let x e R" - {0} and let (xm) be a sequence converging to x. Take t, e R a S, - ' and to a l such that L_,,,(x) E Sr'. As the flow such that is continuous it follows that tin -' to and L_,,"(xm) -+ L_,o(x). Thus h(xm) = TJhL_,_(xm) converges to T,,0hL_,,,(x) = h(x) which shows the continuity of hat x. We now show that his continuous at the origin. From Proposition 2.10 and the compactness of S"2-' it follows that, given e > 0, there exists t, > 0 such that II T,(y)II < E for all t > is and all y e S"2-'. On the other hand, as
40) = 0, there exists S > 0 such that if IIx1I < S and L_,(x) a S, - ' then t > t,. Therefore, IIh(x)II < e if IIxII < S, which shows the continuity of h. 'Similarly we can show that h'.' is continuous. 2.15 Proposition. Let L and T be hyperbolic linear vector fields. Then L and T are topologically conjugate if and only if they have the same index. PROOF. Suppose that L and T have the same index. Let E', Es be the stable subspaces of L and T respectively. Then dim E' = dim E". By Proposition
2.14 there exists a homeomorphism h5: E' -+ E'' conjugating L' and T'; that is, h,L; = T-h, for all t e R. Similarly, there exists a homeomorphism h": E' -, E"' conjugating L' and T". We define h: E' $ E" - E" $ E" by h(x" + x") = h,(x') + h"(x"). It is easy to see that h is a homeomorphism and conjugates L, and T,. Conversely, let h be a topological equivalence between
L and T As 0 is the only singularity of L and T we must have h(0) = 0. If x e E' we have to(x) = 0. As a topological equivalence preserves the to-limit
of orbits, we have co(h(x)) = h(co(x)) = 0. Therefore h(x) a E' so that h(E') c E. Similarly, h-'(E") c E. Hence h I E' is a homeomorphism between E' and ES . By the Theorem of Invariance of Domain, from Topology,
it follows that dim E' = dim E', which proves the proposition.
We next intend to show that the eigenvalues of an operator depend continuously on the operator. By that we mean the following. For L e 3(R"), let )..1, a.2, ..., A be its eigenvalues with multiplicity m,, m2, .... mk, respectively. We consider balls B,(A;) of radius a and center A;, 1 < i < k, so that they are all pairwise disjoint. We want to show that given e > 0 there exists
6 > 0 such that if T e .'(Or) and II T - LII < b, then T has precisely mt eigenvalues iti B,(Ai) counting their multiplicities, for all 1 < i 5 k. For L e . '(6R") let Sp(L) denote the spectrum of L, the set of its eigen-
values. The lemma below shows' that Sp(L) cannot explode for a small perturbation of L.
53
§2 Linear Vector Fields
2.16 Lemma. Let Lc- 2'(R"). Given e > 0 there exists b > 0 such that, if T e .5'(R') and u T - LII < b, then for edch'A' e Sp(T) there exists A a Sp(L)
withIA-A'I.<e. PROOF. If A a Sp(L) then A is an eigenvalue of the complexified operator L, so that I A I 5 IILII = IILII. Thus, if 11T - LII < 1, the spectrum of T is con-
tained in the interior of the disc D with centre at the origin of C andradius 1 + IILII: Let V. be the union of the balls of radius a with centre the elements
of Sp(L). If p e D - V, then det(L - µl) # 0. By continuity of the determinant there exist a neighbourhood U. of p in C and b > 0 such that, if 11T - LII < b and p' a U,,, then det(D - u'l) # 0, so that p' ¢ Sp(?`). By the compactness of D - V we deduce that there exists b > 0 such that, if 11T - LII < b and p e D - V, then det(T - 111) # 0. As Sp(T) c D it follows that'S,p(T) c V,, which proves the lemma. 0 .
If the eigenvalues of Lure all distinct it follows from Lemma 2.16 that they
change continuously with the operator. Let A be an eigenvalue of L of multiplicity m and let E(L, A) a C" be the kernel of (L - Al)". Then E(L, A) is a subspace of dimension m. Moreover, if k Z in, the kernel of (L - Al)k is E(L, 1). ' 2.17 Lemma. If A is an eigenvalue of L E 2'(R") of multiplicity in then there exist eo > 0 and b > 0 such that, if 11T - LII < b, the sum of the multiplicities of the eigenvalues of T contained in the ball of radius eo and centre A is at most in.
PROOF. To get a contradiction suppose for all e > 0 and b > 0 there exists
T -2(R") with IIT - LII <,b such that the number of eigenvalues of T, counted with multiplicity, in the ball of radius a and centre A is greater than m. Thus there exists an m' > m and a sequence of operators Lk -+ L such that -Ak,+... , A,,_, are eigenvalues of Lk which converge to A. Let Ek be the kernel of (L'k -' Ak,1) o o (L - tk_,1). We may suppose that the dimension of Ek is m'. Let e' ,, ... , ek be an orthonormal basis of Ek. As II ej II = 1 and the unit sphere in C" is compact we can suppose (by taking a subsequence if necessary) that ek e . The vectors e,, ... , e",, are clearly orthonormal and so span a o (Lk - 2k, I) subspace,eof dimension m,'. As the operator (Lk - Ak,I) o converges to (L - ail)'"' we see, by continuity, that the kernel of (L - Al)"'' contains L which is absurd since it has dimension m < m'. 0
2.18 Proposition. The eigenvalues of an operator L e 2'(O ") depend continuously on L.
PROOF. Let A...... A. be the distinct eigenvalues of L with multiplicity n,; ... , nk. By Lemma 2.16,' given e > 0, there exists b > 0 such that, if IIT - Ln < b, then the eigenvalues of T are contained in balls of radius e and centre at the points A. It remains tb show that the sum of the multiplicities of the eigenvalues of T contained in the ball of centre A is exactly
2 Local Stability
54
n. By Lemma 2.17, taking e < to if necessary, this sum is less than or equal'to nj. If for some j this sum is strictly less than nj then the sum of the multiplicities of all the eigenvalues of T would be strictly less than D=1 nj = n, which is absurd.
.
Corollary.. If L E .P(R") is a hyperbolic vector-field then there exists a neigh-. bourhood V c .(p8") of L such that all T e V have the same index as L.
Another corollary of this proposition is that the roots of a polynomial vary continuously with its coefficients.
2.19 Proposition. A hyperbolic linear vector field is structurally stable in the space of linear vector fields.
:
PxooF. This follows immediately from the corollary above and Proposition 2.15.
2.20 Proposition. Let L be a structurally stable linear vector field. Then.L is hyperbolic.
PROOF. Let L be a nDnhyperbolic linear vector field and b = min{ l a i; a + i f
is an eigenvalue of L and a 0 0). If 0 < t < b then L + tI and L - tI are hyperbolic vector fields with different indices and so cannot be topologically equivalent. This shows that, in any neighbourhood of L, there exist two'vector
fields that are not topologically equivalent. Thus, L is not structurally
-0
stable.
To summarize the results above we may say that a linear vector field is structurally stable in the space of linear fields if and only if it is hyperbolic. Moreover, the structurally stable linear fields form an open dense subset of
§3 Singularities and Hyperbolic Fixed Points In this section we are going to define a subsetIi c 2''(M) such that every X e I has a local orbit structure that is stable and simple enough for us to be able to classify it.
The example below shows that a vector field can have an extremely complicated orbit structure near a singularity. EXAMPLE. If L is a linear vector field on 682 with a zero eigenvalue there exists
a one-dimensional subspace V e R2 consisting of singularities of L. Let L be the vector field whose matrix in the standard basis is I 0 0
case V = {(t, 0); t e 68}.
-0I. In this 1
§3 Singularities and Hyperbolic Fixed Points
55
Figure 3
We shall show that arbitrarily close to L there exists an uncountable set ' of vector fields such that, whenever X, Z e.,K, X is not locally equivalent to Z at the origin.
Let Y be the constant vector field Y = (1; 0). Let K c R be a compact subset. It is known that, there exists a bounded C°° map p: R -+ R that vanishes exactly on K. We can also suppose that the first r derivatives of p are bounded. Given e > 0 we choose n e tJ such that 11(1/n)pII, < e. Let Z = L + (1/n)pY. Then we have IIZ - LII, < E and the set of singularities of Z is K. Thus, if K and Kare two nonhomeomorphic compact sets, we deduce that the vector fields Z and Z', constructed as above, are not topologically equivalent. Hence there are at least as many equivalence classes of vector fields as there are homeomorphism classes of compact subsets of R.
I
Figure 4
This example motivates the next definition. Definition. We say that p e* M is a simple singularity of a vector field X e I'(M)
if DX,: TM, -- TM, does not have zero as an eigenvalue. 3.1 Proposition. Let X e X'(M) and suppose that p e M is a simple singularity
of X. Then there exist neighbourhoods .N'(X) c X"(*, -Up e M of X and p respectively and a continuous function p:.V(X) -+ U. which to each vector field Y e .N'(X) associates the unique singularity of Y in Up. In particular, a simple singularity is isolated.
PROOF. We shall use the Implicit Function Theorem for Banach spaces. As
the problem is a local one we can suppose, by using a local chart, that M = R, p = 0 and that X is a vector field in Xr(D"), where D' = {x a 08"`; IIxN S 1}. We have that X' = X'(D") is a Banach space and the map
56
2 Local stability
rp : Dm x I' -.' IR' given by tp(x, Y). = Y(x) is of class C. We have p(0, X) = 0 and, by hypothesis, D1gp(0, X) = DX(0): i8m -. RTM is an isomorphism. By the
Implicit Function Theorem there exist neighbourhoods U of 0 and :N' of X and a unique- function p: A' -+ U of class C' such that (p(p(Y), Y) = 0. Thus, for x e U, Y(x) = 0 if and only if x = p(Y). Moreover, as DX(0) is ala isomorphism and the set of isomorphisms is open, we can suppose; by shrinking N' and U if necessary, that DY(p(Y)) is an isomorphism so that p(Y) is a simple singularity of Y
Next we shall characterize a simple singularity of vector field X in M in terms of transversality. For this, let us consider the tangent bundle TM = ((p, v); p e M, v e TMp) and let Mo = {(p, 0); p e M} be the zero section. Mo is a submanifold of TM diffeomorphic to M and a vector field X e X'(M) can be thought of as a C' map from 'M to TM which we shall denote by the same letter X. Therefore p is a singularity of X if and only if X(p) a Mo.
3.2 Proposition. Let X be a C' vector field (r z 1) on a manifold M and let po e M be a singularity of X. Then po is a simple singularity of X. if and only if the map p r+ (p, X(p)) from M to TM is transversal to the zero section Mo at Po-
PROOF. Let x: U -+ R."'
be a local chart with x(po) = 0. Let T U =
{(p, v) e TM; p e U}. The map Tx: TU -,. R' x R" defined by Tx(p, v) = (x(p), Dxp(v)) is a local chart for TM. Consider the following diagram
TU
U
Rm x Rm
=
Q8m,
where n2 is the projection 7E2(x, y) = y.
Put h = n2 TxX. Then X is transversal to Mo at po if and only if po is a regular point of h, that is, if dh(po): TMp -+ 48'" is an isomorphism. On the other hand, dh(po) = Dx(po)DX(po). Thus, Dh(po) is an isomorphism if and only ;if DX(po) is an isomorphism, which proves the proposition.
Let go c 3E'(M) be the set of vector fields whose singularities are all simple; that is, go = {X e X'(M); X: M -+ TM is transversal to Mo}. As a simple singularity is isolated and M is compact it follows that any X E 9o has only a finite number of singularities. 3.3 Proposition 10 is open and dense in X'(M). PROOF. (a) Openness. As the set of Cr maps from M to TM that are transversal
to M. is open we conclude that Wo is open.
(b) Density. Let X e X'(M). By Thom's Transversality Theorem there exist maps Y: M -+ TM transversal to Mo arbitrarily close to X. It may happen, however, that Y is not a vector field since we might have n(Y(p)) 0 p
57
§3 Singularities and Hyperbolic Fixed Points
for some p e M where a: TM -' M denotes the map (p, v) r-* p. But aX = idM
and, if Y is close enough to X, then cp = nY is close to idM so that it is a diffeomorphism since the set of diffeomorphisms of M is open in C'(M, M).
Now Z = Y(p-' is a vector field on M because nZ = xY(p-'
99-' =
iEtm. As Y is transversal to Mo and cp-' is a diffeomorphism it follows that Z is transversal to Mo. If Y is close to X then so is Z. EXAMPLE. In this example we shall consider a linear vector field on R2 with a
simple singularity and show that a nonlinear perturbation can give a vector field with an extremely complicated orbit structure. Consider the vector field given by
L=
0
1
a (-1).
Let P: IR -i (f8 be a C°° function such that p(O) = 0 and ptk1(0) = 0 for all k e N. Let X be the vector field on 1R2 defined by X(x, y) _ (y + p(r2)x, -x + p(re)y) where r2 = x2 + y2. It is easy to see that X is C°° and that (-0
101
DX(0, 0)
Thus, (0, 0) is a simple singularity of X. Let K be a compact subset of R containing 0. We can choose p so that p(K) = 0 and p is never zero on I = K where I = (- S, b) is an interval containing K. Given E > 0 and r > 0 we can choose p so that IIX = Lii, < e If rp a iR+ has p(ro) = 0 then the vector field X is tangent to the circle of radius ro so that this circle is a closed orbit for X. On the other hand, if (a, b) a R is an interval such that p(t2) > 0 for t e (a, b) and p(a2) = p(b2) = 0, then the orbits of X through points of the annulus Db - D. = {z a IfBe; a < Ilzll < b} are not closed and, in fact, they are spirals whose o.)-limit set is the circle of radius b. This follows from the observation that <(x, y), X(x, y)) = r2p(r2), so that the radial component of X is rp(r2) which is positive for r e (at b), Similarly, in an interval (a, b) where
p(a2) ='p(b2) = 0 and p(te) < 0 for t e (a, b), the orbits in the annulus Db - D, will have the circle of radius b as a-limit set and the circle of radius a as w-limit set. See Figure 5.-
p<0
'0 > 0 Figure 5
58
2 Local Stability
This construction gives a vector field X arbitrarily close to L whose closed orbits intersect the x-axis precisely at the points of the compact set Jr a R'; r2 a K) which is homeomorphic to K. If K and K' are two nonhomeomorphic compact sets and X and X' are vector fields associated to them by the above construction it follows that X and X' are not topologically equivalent. Definition. Let X e F (M) and let p e M be a singularity of X. We say that p is a hyperbolic singularity if DXP: TMP - TMP is a hyperbolic linear vector field, that is, DXP has no eigenvalue on the imaginary axis.
Let 9, c 3;'(M) be the set of those vector fields whose singularities are all hyperbolic. It is clear that 1, c %. 3.4 Theorem. 9, is open and dense in F(M). PROOF. As 9o is open and dense in 3:'(M) and 9, c 1o, it is sufficient to show
that 91 is open and dense in go. Let X e °o and let po, ... , pk e M be the singularities of X. By Proposition 3.1 there exist neighbourhoods .K(X) of X and U1,..., Uk of p,, ... , pk, respectively, and continuous functions pj:.A - U,, j = 1, ..., k such that p,{Y) is the only singularity of Y in U,.
We can suppose that these neighbourhoods are pairwise disjoint. If U j, p e M - Uj=, U, then X(p) # 0 and, by the compactness of M there exists 6 > 0 such that 11X(p)jI > 6 for all p e M - U U,. Therefore, shrinking .K if necessary, we can suppose that any Y e X. does not have a
singularity in M - U Uj. Let us suppose that X e 9,. As DXP, is a hyperbolic linear vector field and such vector fields form an open set we deduce from the continuity of the maps p1, shrinking A' if necessary, that DY,,,(r) will be a hyperbolic linear vector field for all Y e X. Thus A" c g which proves the openness of T,. Now suppose that X e `$o. We shall show that there exists Y e 9, n A' arbitrarily close to X. Note that, if u > 0 is small enough, then DXP, + ul is a hyperbolic linear vector field on TM,,, for all j = 1, ... , k. It will therefore
suffice to show that, given a neighbourhood X, c .N' of X there exists Y e .N; such that Y(p) = O and DY,,, = DXP, + ul. Let V e Uj be a neighbourhood of p, and x': V -' B(3) c III' be a local chart with x'(p) = 0 where B(3) is the ball with radius 3 and centre at the origin. Let gyp: R' -* R be a positive C°°
function such that q (B(1)) = 1 and rp(lr - B(2)) = 0. Let x: X denote the expression of the vector field X in the local chart x'; that is, x:X(q) = Dx'((x')-'(q))X((x')-'(q)). Then we define Y(p) = X(p) if p e M - U; V; and Y(p) = D(x')-'(x'(p))(xdX(x'(p)) + ucp(x'(p))x'(p)) if p e V'. It is easy to see that Y is a C' vector field, that Y(p) = 0 and that DYP, = DXP, + W. Moreover, by taking u small enough we have Y e .K,, which completes the proof.
Next we shall extend these results to diffeomorphisms of a compact manifold M. We shall omit the proofs of the propositions and the theorem as they are analogous to those just given for vector fields.
§4 Local Stability
59
Definition. Let p e M be a fixed point of the diffeomorphism f e Diff'(M).
We say that p is an elementary fixed point if 1 is not an eigenvalue of Dfp: TMp - TMp. 3.5 Proposition. Let f e Diff'(M) and suppose that p is an elementary fixed point off. There exist neighbourhoods X off in Diff'(M) and U of p and a continuous map p: X -+ U which, to each g e X, associates the unique fixed point ofg in U and this fixed point is elementary. In particular, an elementary fixed point is isolated.
Let A denote the diagonal {(p, p) e M x M; p e M}, which is a submani-
fold of M x M of dimension m. If f e Diff'(M) we consider the map f: M M x M given byJ(p) = (p, f (p)), whose image is the graph off. 3.6 Proposition. Let f e Diff'(M) and let p e M be a fixed point off. Then p is an elementary fixed point if and only ifJis transversal to A at p. Let Go c Diff'(M) be the set of diffeomorphisms whose fixed points are all elementary. Thus, f e Go if and only if I is transversal to A. By using Thom's Transversality Theorem we obtain the following proposition. 3.7 Proposition. Go is open and dense in Diff (M).
Definition. Let p e M be a fixed point off e Diff'(M). We say that p is a hyperbolic fixed point if Dfp: TMp - TMp is a hyperbolic isomorphism, that is, if Dfp has no eigenvalue of modulus 1.
Let Gt c Diff'(M) be the set of diffeomorphisms whose fixed points are all hyperbolic.
3.8 Theorem. Gt is open and dense in Diff (M).
In the next section we shall show that a diffeomorphismf e Gt is locally stable.
§4 Local Stability In this section we shall prove a theorem due to Hartman and Grobman according to which a diffeomorphismf is locally conjugate to its linear part at a hyperbolic fixed point. Analogously, a vector field X is locally equivalent to its linear part at a hyperbolic singularity. As a consequence we shall have local. stability at a hyperbolic fixed point and at a hyperbolic singularity. The proof we shall present is also valid in Banach spaces [25], [36], [74], [90]. Other generalizations and references can be found in [80].
60
2 Local Stability
4.1 Theorem. Let f E Diff'(M) and let p e M be a hyperbolic fixed point off. Let A = Dff: TMP -+ TMP. Then there exist neighbourhoods V(p) c M and U(0) TMP and a homeomorphism h: U V such that
hA = fh. Remark. As this is g local problem we can, by using a local chart, suppose
that f : R' -. R' is a diffeomorphism with 0 as a hyperbolic fixed point. $efore proving Theorem 4.1 we shall need a few lemmas.
4.2 Lemma. Let E be a Banach space, suppose that L e £(E, E) satisfies IIL11 < a < 1 and that G e .'(E, E) is an isomorphism with 11 G ' II < a < 1. Then
(a) I + L is an isomorphism and 11(1 + L)-'II S 1/(1 - a), (b) I + G is an isomorphism and 11(1 + G)-'I1 S a/(1 - a). PROOF OF LEMMA 4.2. (a) Given y E E, define u: E -+ E by u(x) = y - L(x). Then u(x,) - u(x2) = L(x2 - x1). Thus, IIu(xt) - u(x2)II < allx, - x211 so
'that u is a contraction. Hence, u has a unique fixed point x e E; that is,
x = u(x) = y - Lx. Therefore, there exists a unique x e E such that (L + 1)x
y; that is, I + L is a bijection. Let y e E have Ilyll = 1 and take
xeEsuch that (I+'L)-'y=x.Asx+Lx=y,wehave Ilxll -allxll <1 so that Ilxll <- 1/(1 - a). Thus, II(I + L)-'II < 1/(1 - a). (b) First note that I + G = G(I + G '). As I1G-' 11 <'a < 1, the first part
of the lemma says that I + G-' (I + G-')-'G-' and, therefore,
is
invertible. Thus (I + G)-' _
II(I + G)-'11 <- 11(1 + G-t)-111 JIG- '11 < 1a l .a =
a
a,
which proves the lemma. As A = Dfo is a hyperbolic isomorphism, there exists an invariant splitting U8' = E' ED E" and a norm II.11 on l8' in which II As 11 < a < L.
where
A' = A I Es: E' -. E',
JI(A")-' II < a < 1,
where
A" = AI E E" - E".
Let Cb(18'") be the Banach space of bounded continuous maps from R'" to Rm with the uniform norm: 1lull = sup(11u(x)11; x e 08'"}. As lf8' = E' ®.E"
we have a decomposition Cb(W") = Cb(R", E') ® C6(Rm, E) where u = u' + u" with u' = n, o u and u" = a" o u obtained from the natural projections
its: E'®E"-+E'and it.:E'®El PE". 4.3 Lemma. There exists t > 0 such that, if cp1, 92 e C6(ll ") have Lipschitz constant less than or equal to E, then A + (p1 and A + cp2 are conjugate.
§4 Local Stability
61
PROOF. We must find a homeomorphism h: R" -+ R' that satisfies the equation h(A + (p,) = (A +. (p2)h.
(1)
Let us try a solution of the form h = I + u with u e Cb(R'). Then we need
(I+uXA+,rpl)=(A+q,2)(1+u)
(2)
or
A+Qp, +u(A+(pl)=A,+Au+(p2(I +u) or equivalently, A
u-u(A +(pl)=(pt -4)2( 1 +u).
(3)
We shall show that there exists a unique u e Cb(Rm) that satisfies equation (3).
Consider the linear operator Y: Cb(Rm) --+ Cb(or),
2(u)=Au-u(A+(p,). We claim that 2' is invertible and that II2-`II < IIA-'111(1 - a). In fact, where -2*: Cb(Rm) - Cb(R'") is given by 2*(u) u - A - 'u(A + gyp,) and A: Cb(Rm) -+ Cb(R"') by A(u) = A u. As A is
2 = A2 *
invertible, we must show that 2* is invertible for then 2-' = 2*-'A-'. We remark that Cb(R'", E') and Cb(Rm, E") are invariant under 2* since E' and E" are invariant under A -'. Thus we can write 2* = Y*' ® 2*" where Y*' = 2* I Cb(Rm, E') and 9*" 2* I CA R-, E"). It is easy to see that if z is small enough then A + !p, is a homeomorphism and so the operator u't--+A-'u'(A + (p 1) is invertible and its inverse A'u'(A + gyp,)-' is a contraction with norm bounded by the number a < 1 involved in the hyper-
bolicity of A. By part (b) of Lemma 4.2, F*' is invertible and II-' II < a/(1 - a). From part (a) of Lemma 4.2 we also conclude that 2'*" is invertible
and II(2*")-'li <- 1/(1 - a) = max{l/(1 - a), a/(1 - a)}. Therefore, 2' is invertible and
II2'-'II =IIA* to-'II 5 IIA-'11/(1 - a), which proves our claim. Now consider the map p: C'(R"')-+ Ce(Rm),
µ(u) = 2-191
4)2(1 + u)].
We have
llµ(ut) - U(u2)II = 112-'Cw2(I + u2) - 4'2(1 + ut)]ll < IlA-'1l(1 - a)-'eIIu2 - U111If a is small enough to make e11 A - `11(1 - a) - ' < I then k is a contraction and so has a unique fixed point u, say, in Cb(R"'). As u e Cb(R"') is a solution
2 Local Stability
62
of (3) if and only if it is a fixed point of k we conclude that (3) has a unique solution in Cb(Rm'). It remains to prove that I + u is a homeomorphism. For this we first note that the method used above also shows that the equation
(A + cpt)(1 + v) = (I + v)(A + c02) also has a unique solution v e Cb(Rm). We claim that
(I + u)(1 + v) = (I + v)(I + u) = 1. In fact,
(I + u)(I + v)(A + c02) = (1 + u)(A + cpt)(I + v) = (A + c02)(1 + u)(1 + v).
On the other hand, as (I + u)(I + v) = I + v + u(1 + v) with w = v + u(1 + v) a Cb(1m) and I(A + c'2) = (A + (p2)1, the uniqueness of the solution of the equation (I + w)(A + (p2) = (A + q,2XI + w) shows that
(I+u)(I+v)=I. Similarly we have (I + v)(1 + u) = I. This shows that I + u is a homeomorphism which conjugates A + cp, and A + cp2 and that proves the lemma.
4.4 Lemma. Given E > 0 there exists a neighbourhood U of 0 and an extension off I U to R' of the form A + cp where cp e Cg(l m) has Lipschitz constant at most E.
PROOF. Let a: R
R be a C°° function with the following properties:
a(t)=0,
ift
- 1;
a(t) = I, , if t < 1;
la'(t)I2. Let f = A + , with 0(0) = 0 and Doo = 0. Let B, be a ball with centre at the origin and radius e such that Ii1N' ll < E/2K for x e Be. Define (p: R' -' O' by c0(x) = x(Ilxll/e)0(x). It is clear that cp(x) = 0 if Ilxll > e. Let us show that cp satisfies the conditions in the lemma. In fact, since cp(x) = O(x) for Ilx II < e/2 we see that A + cp is an extension off I Be/2.On the other hand, we have if IIcP(x,) - tp(x2)II = II[a(IIx,11/e)0(x,) - «(11x211/e)/(x2)]II = 11[«(11x,11/e) - 411x211/e)]4(x0
+ a(IIx21I/e)[0(x,) - 'P(x2)1II < (KIIx, - x211/e)(e/2K)IIx,II + (E/2K)llx, - X211 <- ellx, - x211 If x, a Be and X2 0 Be we have 119(x,) - w(x2)11 5 e/211x, - x211 5 Ellx, - x211 and, if x X2 0 Be, we have IIQP(x,) - (P(x2)II = 0 < 811X, - x211
Thus, q has Lipschitz constant at most s, which proves the lemma.
§4 Local Stability
63
PROOF OF THEOREM 4.1. Let e > 0 be as in Lemma 4.3. Let A + 9 be an extension off I U(O) where U(O) is a neighbourhood of 0 and W has Lipschitz constant at most e. By Lemma 4.3 there exists a homeomorphism h: R'° -, R"'
such that hA = (A + cp)h. Thus hA = f It on a neighbourhood of 0 as required.
Remarks. (1) The conjugacy between A + tpt and A + cps in Lemma 4.3 is unique among maps having a finite distance from the identity. However, there are infinitely many conjugacies that do not satisfy this last condition. For example, consider a contraction gyp: R -, R with (p(0) = 0. Suppose that p(l) = a > 0 and (p(- 1) = b < 0. We pick any homeomorphism
h:[-1,b]u [a,1]-+[-1,b]u[a,1]with h(-1)= - 1, h(b) = b, h(a) = a, h(1).= 1. We extend h to the whole real line putting h(0).= 0 and h(x) = qp"hip-"(x) where n e Z is chosen so that (p -'(x) e [-1, b] u [a, 1]. In this way we find as many solutions of the equation hcp = (ph as there are homeo-
morphisms from [-1, b] u [a, 1] to itself. (2) Even if we require the conjugacy between A and f to be near the identity it is not unique because it depends on the extension A + cp off to the whole space R'.
Next we shall show from Theorem 4.1 that a hyperbolic linear isomorphism is locally stable in .'(R'"). In fact, as we shall see.in the next section,
a hyperbolic isomorphism is globally stable in 2(R'"). 4.5 Proposition. Let A be a hyperbolic isomorphism. There exists S > 0 such that, if B e £°(R'°) and JIB - All < 6. then B is locally conjugate to A. PROOF. We shall look for a homeomorphism h: R'° -+ R'° that conjugates A and an extension of B I U, where U is a neighbourhood of the origin. Let a: R -+ R be a C°° map with the properties:
2(t) = 1, if It 1 < 1;
a(t) = 0, if ItI >- 2;
a(R) c [0, 1]. We can write B as A + ($ - A). Let gyp: R' --* R' be defined by qp(x) _ a(IIxlI) . (B - AXx). Then we have cp I B, = (B - A)I Bt, where Bt is the ball with centre 0 and radius 1, and lp(x) = 0 if Ilxll ? 2. For all x e R'", IIDgpxll 5 K II B - A II + II B - A IJ, where K = sup { I a'(t) I ; t e R) is greater than 1. Given e > 0 we take b < e/2K so that JI Dco I } < e. It follows that (p has
Lipschitz constant less than e. By Lemma 4.3 there exists a homeomorphism
h: R' -+ R' such that hA = (A + (p)h.
Since A + cp is an extension of B, h is a local conjugacy between A and B.
2 Local Stability
64
4.6 Theorem. Let f e Diff'(M), r > 1, and let p e M be a hyperbolic fixed point off Then f is locally stable at p. PROOF. By Proposition 3.5 there exist neighbourhoods ,17(f ) and W(p) and
a continuous map p: .N'(f) -i W which associates to each g e .A (f) the unique fixed point of g in W. p(g), and this fixed point is hyperbolic. If we take
a small enough neighbourhood X(f) c ./V(f) we shall have Dfp near Dgp(e) and so, by Proposition 4.5, these linear isomorphisms are locally conjugate. As f is locally conjugate to Dfp and g is locally conjugate to Dgp(o) it follows, by transitivity, that f is locally conjugate to g.
We shall now extend these results to vector fields. Let V be a neighbourhood of 0 in R' and let X : V --. R"' he a C' vector field, r > I. We recall that 0 is a hyperbolic singularity of X if L = DXo is a hyperbolic linear vector field. We shall show that, if 0 is a hyperbolic singularity of X then the orbits of X in- a neighbourhood of 0 have the same topological behaviour as the orbits of the linear vector field L. For this we shall need some lemmas.
4.7 Lemma (Gronwall's Inequality). Let u, v: [a, b] -. f{B be continuous nonnegative Junctions that, for some x ? 0, satish u(t) < a. +
Ja
u(s)v(s) ds b t c [a, b].
Then
u(t) < a expIf." v(s) ds]. PROOF. Let co: [a, b] -+ R be the map w(t) = x + Jou(s)v(s) ds. Suppose first
that a>0.We have w(a)=x and w(1)
- a>0for all Ie[a,b].As (Or)=
v(t)u(t) < v(t)co(t), we have w'(t),4u(t) < C(t). Integrating from a to t we obtain
w(t)/a < exp[f."v(s)
exp[ft
v(s) ds].
u(t) S w(t) < or a
If a = 0 the previous case implies that, for all a, > 0, u(t) < a, exp[J'v(s) ds]. Thus, u(t) = 0 and the inequality remains true. 4.8 Lemma. Let Y: ff8m -. R' be a C' vector field with Y(0) = 0 that satisfies a Lipschitz condition with constant K. Then the flow of Y is defined on UI x Il8"' and I'; Y,(x) - Y,(y)ll < e"' 111x - y11 for all x, y e R'".
65
§4 Local Stability
PROOF. Let x e R. To get a contradiction suppose the maximal interval of the integral curve of Y through the point x is (a, b) with b < oo. Let (p: (a, b) -+ 68' be the integral curve through the point x. We have cp(t) = x + f Y((p(s)) ds. ` 0
Therefore, if t z 0, il(p(t)II !g IIxll +
f II Y((p(s))II ds s Ilxll + f KII(P(s)II ds. 0
0
From Gronwall's inequality we obtain II(n(t)II < ex'`'llxll < eKbpx!I
if t > 0.
Let t" -+ b and consider the sequence {(p(t")j whose terms all belong to the closed ball with centre 0 and radius M = e"bIIxII Since
f
rm
(p(t") - (P(tm) =
Y((P(s)) ds,
we have that i14'\t") - (p(tm) II < KM I t" - Q.
Thus, (p(t") is a Cauchy sequence and so it converges to a point y e R". The local flow of Y around y enables us to extend the integral curve (p to the right
of b contrary to our initial hypothesis. Thus, the flow of Y is defined on CB x Rm. Since
Y(x) - Y,(Y) = x - y + $[Y(c(x)) - Y(Y(Y')] ds, 0
we deduce that, for t z 0, II Y(x) - Y(Y)II S 11x - y11 +
fKIl1(x) - Y,(Y)II ds.
From Gronwall's inequality we have II Y(x) - Y(y)ll <- e"I"IIx - YII
if t z 0.
For t < 0 we obtain the same expression by applying this argument to th vector field - Y. 4.9 Lemma. Let X : V -. Rm be a C' vector field with X(0) = 0. Let L = DXO. Given e > 0 there exists a C' vector field Y: Rm -+ 08m with the following properties:
(1) the field Y is Lipschitz with some Lipschitz constant K so that-the flow induced by Y is defined on 18 x QBm;
(2) Y = L outside a ball B,;
2 Local stability
66
(3) there exists an open set U c V containing 0 such that Y = X on U; (4) if Y, = L, + ip, there exists M > 0 such that IItP,ll 5 M for all t e [-2, 2]
and rp, has Lipschitz constant at most B. Moreover, D(V,)o = 0 or, equivalently, D(Y,)o = eL = L1.
PROOF. As L = DXo we have X = L + 0 where 0: V -, R' is a C' map that
satisfies 0(0) = 0 and Da/ro = 0. Let a: R -+ R be a C°° map such that a(R) c [0, 1], a(t) = 1 if Itl < 1/2 and a(t) = 0 if t z 1. Let t : R' - R' be defined by ip(x) = a(IIxI]) i/i(x) if x e V and O(x) = 0 if x e R" - V. Given S > 0 we can choose l > 0 so that the map ip is C' and has Lipschitz constant at most b. It is clear that ip = 0 on But and w = 0 outside B,. Let Y: R' R' be the vector field defined by Y = L + Co. Again it is clear that Y = X on B1/2, Y = L outside B, and Y satisfies (1). It remains to show that condition (4) holds. In fact, IIY,(x) - Y,(y)II 5 e2Kllx - yll for
t e [ - 2, 2] by Lemma 4.8. Put ip, = Y - L. That there exists M such that II cot II < M for t e [ - 2, 2] follows from the fact that Y, and L, are bounded on B(0, 1) for t e [ - 2, 2] and Y = L outside B(0, 1). We also have
w,(x) - cot(y) = f 1[O(Y,(x))
- O(Y.(y))] ds + f
- ip.(y)) ds.
o
0
F rom Gronwall's inequality we now conclude that
I!cv,(x) - wdy)II S 2e2Kbe2uL,i Ilx - yj d t e C-2,2]. Provided b is small enough, condition (4) will be satisfied. Finally, let us see that D((p,)o = 0. We want to show that given p > 0 there exists r > 0 such that II co l(x) II 5 p lI x II for II x II 5 r. Since (Dip)o = 0, we can choose r so that IIO(z)II <- gllzli for llzll < r with q < pe-Ke-11L11. As in the expression above, we have
ipds f(Y.(x)) + fo L(ip.(x)) ds.
qp,(x) = 0
Using the fact that II Y,(x)II 5 eKIlxII for 0:5 s 5 1, from Gronwall's inequality we get IIcoi(x)II 5 ,1eKUxIIeIILu < pIIxll
This completes the proof of Lemma 4.9.
4.10 Theorem. Let X : V -+ R' be a C vector field and 0 a hyperbolic singularity of X. Put L = DXo. Then X is locally equivalent to L at 0.
PROOF. Let Y: R' - R' be a C' vector field as in Lemma 4.9. As Y = X on the neighbourhood U of 0, the identity map on U takes X-orbits to Y-orbits. Therefore, Y is locally equivalent to X It remains now to prove that Y is locally conjugate to L. In fact we shall show that there exists a homeomorphism of Rm that conjugates the flows Y, and L.
67
§4 Local Stabi
By Lemma 4.9, Y, = L, + tp, and, since DYo = L, we have D(Y,), e` = L,. Thus, the diffeomorphism Y, = Lt + gyp, has the origin as a hyperbolic fixed point and gyp, has Lipschitz constant at most F. By Lemma 4.3
there exists a unique homeomorphism h: R' --+ R''eat a finite distance from the identity (that is, h = I + u with u e Cb(R"')) that satisfies hY, = L, h. We define H: R' -+ R' by
H = fL5hzdt. 0
It is clear that H is a continuous map and condition (4) of Lemma 4.9 ensures that H is at a finite distance from the identity. We now show that
L,H=HY, forallseR. For this it is sufficient to considers a
We have t
/ r1
L_,HY = L_,I By
J
dt.
L_,hY, dt/// 1Y, = f o
putting u,= t + s - I we obtain 1
S
JL_($+)hY+$dt = 0
L-
J 1+s
=f
1hY"+, du
1+s
We put v = u + 1 in the first term and,v = u in the second, use L_,hY1 = h and deduce that
L_,hY, _
fL_,hYpdv + 5 .o
H.
This shows that H is a continuous map at a finite distance from the identity and it conjugates the flows L, and Y. It remains to show that H is a homeo-
morphism. In fact, as L,H = HY1 and L,h = hY,, the uniqueness of the solution of this equation shows that H = h, which proves the theorem. 4.11 Theorem. Let X e 1'(M) and let p e M be a hyperbolic singularity of X. Then X is locally stable at p.
PROOF. We have seen that there exist neighbourhood .K(X) in 3:'(M) and U of p in M such that every vector field Y e .#(X) has a unique hyperbolic singularity py in U. Moreover, for .,V(X) small enough, py has the same index as p. Thus, the linear vector fields DXP and DY, . are topologically equivalent.
As X is locally equivalent to DXP and Y is locally equivalent to DY,, this proves the theorem. -
2 Local Stability
6S
5 Local Classification We have seen that the subset 'y',(M) c X'(M) consisting of those vector fields hose singularities are all hyperbolic is open and dense in X'(M). Moreover, i iiese vector fields are locally stable at each point of M. We shall now describe the possible types of local topological behaviour for vector fields in IN,.
Let us consider the linear fields Lo, Lt, ... , L. on i8' which are represented, with respect to the standard basis, by the matrices:
0
)
0
11
1.,=
.
I
O
1,
Lt=
-1
O
-1
L,"=
1
O
O
1
If L is a hyperbolic linear vector field on R' then L is conjugate to Li, where i is the index of L. R' be the constant vector field given by C(x) = (1, 0, ... , 0). Let C: GRm
We may collect the results proven in previous sections into the following theorem. 5.1 Theorem. Let X E S, (M) and choose p E M. (a) 1 f p is a regular point of X then X is locally equivalent at p to the constant vector field Cat 0.
(b) lip is a singularity of X then X is locally equivalent to L; where i is the index of p.
PROOF. (a) This comes from the Tubular Flow Theorem.
(b) Apply the Grobman-Hartman Theorem and Proposition 2.15. Now let us consider the space Diff'(M) of C' diffeomorphisms of M. We
saw that the set G,(M) c Diff'(M) consisting of those diffeomorphisms whose fixed points are all hyperbolic is open and dense in Diff'(M). We shall now describe the possible types of topological behaviour of the orbits of a diffeomorphism in G, near its fixed points. By the theorem of Hartman and Grobman it is sufficient to classify the hyperbolic linear isomorphisms.
5.2 Proposition. Let A e £(G8') be a hyperbolic linear isomorphism. There exists c > 0 such that, if B E Y(R') satisfies 1IA - Bhj < c, then B is conjugate to A.
PROOF. By Proposition 4.5, B is locally conjugate to A; that is, there exists a homeomorphism h: V(0) U(0) such that hA = Bh. Let Es and E" be the
stable and unstable subspaces of A and let E" and E" be the corresponding subspaces for B. Let_ V' = V(0) n E', V" = V(0) n E", U' = U(0) n Ef and
U = U(0) n E"'. By the continuity of h we have h(V') = U' and h(V") _ U" . We shall define a homeomorphism h': E'
E' conjugating AS = .4 I E'
69
§5 Local Classification
and B' = B I E". If x c V', which is a neighbourhood of the origin in E', we
put h'(x) = h(x) e E. If x e E' - V' then, as A"(x) -+ 0 as n -+ oe, there exists r e fkl such that A'(x) E V'. We put h'(x) = B-'hA'(x). As h conjugates A and B in V' we see immediately that h' does not depend on the choice of r. It is also easy to check that h' is a homeomorphism and conjugates A' and B. Similarly we define a homeomorphism h": E" E"' conjugating A and B. Then we can define h: E' ® E" -+ E' ® E" by h(x' + x") = h'(x') + h"(x"). It is clear that h is a homeomorphism that conjugates A and B.
We leave it to the reader to prove the next proposition. 5.3 Proposition. Let A and B be hyperbolic isomorphisms of Q8'". Let E' and E"
be the stable and unstable subspaces of A and E'', E the corresponding subspaces for B. Then A and B are conjugate if and only if AS = A I E' is conjugate
to B' = BCE" and A" = ASE" is conjugate to B" = BCE". We want to give a necessary and sufficient condition for two hyperbolic isomorphisms to be conjugate. 5.4 Proposition. Let A, and A2 be the isomorphisms of 68m which are represented, with respect to the standard basis, by the matrices:
2
'Ti
O
z/
\ O
3/
Let A be a hyperbolic isomorphism of index m. If A preserves the orientation,
i.e. det(A) > 0, then A is conjugate to A,. If A reverses the orientation, i.e. det(A) < 0, then A is conjugate to A2.
PROOF. By local stability A is conjugate to each isomorphism in some neighbourhood of A. Thus, we can simplify the argument by assuming that A is diagonalizable. Let {v,, ..., be a basis for Rm in which the matrix j, that represents A, is in real canonical form:
A3
A'= B,
0
B5..;
70
2 Local Stability
where -1 <.l;<_0,0
- A a r/
withf; -A 0 anda?+(32<1. We remark that two hyperbolic isomorphisms, Bo and B,, are conjugate
if they are in the same connected component of the space of hyperbolic isomorphisms, which is open in r(Rm). In fact let a: [0, 1] -+ GL(08') be a continuous curve such that a(t) is hyperbolic for all t E [0, 1] and a(0) = Bo, a(1) = B,. By local stability, for each t e [0, 1], a(t) has an open neighbourhood V c GL(68m) such that every isomorphism in V, is conjugate to a(t).
As a([0, 1]) is compact, there exist t, = 0 < t2 < < t,_, < tk = 1 such that V, u - u Vk =, a([0, 1]). As V, r V,,, # 0 we see that a(t;) is conjugate to a(t;+ ); thus, Bo is conjugate to B,. -
By this remark it is sufficient to find a continuous path a: [0, 1] -+ GL(RM)
through hyperbolic isomorphisms such that the matrix of a(0), in the basis U ... , tm, is A and the matrix of a(1) is A, or A'2. This is because the iso-
morphisms A, and A? are similar (and therefore conjugate) to the isomorphisms represented, respectively, by A, and A2 in the basis {v...... v",}.
First we construct a continuous path a,: [0, 1] -+ GL(lm) such that a,(0) = A, Y, (t) is hyperbolic for all t e [0, 1] andO -1
2 NS 1
Afterwards we shall construct a continuous path a2: [0, 1] - GL(lm) such that a2(0) = a,(1) and a2(1) = A, or A2. We put
0
A,(t)
,
(t)
B1(t)
O
B: (t)
µ,(t) _ (1 - t)µ; + (#)t, cos((, t)
sin(w, t)
-sin(wrt)
cos(rvit)
-
a;
;
tg,
a.,
if tE[0, ]
71
§5 Local Classification
and
B,{t)
0
((fX2t - 1) + 2(1 - t) (a? + f?)
0
(IX2t - 1) + 2(1
+ J3Zll
if t e [;, 1], where ,;os(zco,) = a;/
(a + fl)
and
sin(ZCO)
:t is easy to see that -1 < 2;(t) < 0, 0 < p1(t) < 1 for all t e [0, 1) and that the eigenvalues of B;(t) have modulus less than 1 fort a [0, 1]. This implies that al(t) is hyperbolic for all t e [0, 1]. As a, is continuous we deduce that a1(0) = A is conjugate to a,(1) whose matrix is I.
I\
0
Now we construct the curve a2. Suppose that A reverses the orientation, that is, det(A) < 0. As a, is continuous and x,(t) is an isomorphism for all
t e [0, 1], we have det(a,(t)) < 0 for all t e [0, 1] and, in particular, det(a1(1)) < 0. Thus, in this case the number s' of negative entries on. the diagonal of a1(1) is odd. We put
I 2
where
sin(1rt)
cos(irt)
C,{r)
sin(nt)
Them we have
q0)
=
(-1 0
-
cos(rtt)
,
,
0
2
_2 0
/1. 0
C/i) = I
i).
Moreover, the eigenvalues of C;(t) are of the form - Ze" 1t and so have modulus less than 1 for all t e [0, 1]. As a2(0) = o(J1) and a2(1) is represented by the
matrix A2, we have A conjugate to A2. Similarly, if det(A) > 0 then A is conjugate to A1. Using degree theory, as presented in [64] or [38], we can prove that AI is not conjugate to A2. 1'h fact, if h is a homeomorphism conjugating AI and A2 we have: deg h = (deg hXdeg AI) = deg(h.41) = deg(A2 h) = (deg A2Xdeg h) = -deg It and this is a contradiction because
degh=±1.
0
2 Local Stability
72
Remark. If A is a hyperbolic isomorphism of index 0 then A is conjugate to one of the following isomorphisms:
A,=
-O2
C
2
or A2=
2
O 2
2 O 2 according as det(A) > 0 or det(A) < 0, respectively. The proof of this is
entirely analogous to that of Proposition 5.4.
We shall now classify hyperbolic fixed points of diffeomorphisms using the equivalence relation of local conjugacy. 5.5 Theorem. Let f e DiW(M) and suppose that p e M is a hyperbolic fixed point o f f . Then f is locally conjugate at p to one of the following linear isomorphism. A o f Q8', where m = dim M. The isomorphisms A , 0 < i < in and 1 < j < 4, have index i and are represented in the standard basis of Rm by the matrices
A(' = 44 =
C
C
11
C
-2
A 4
-2 2 2
O Al = Aa m rM
i1
O
§6 Invariant Manifolds
73
PROOF. This follows immediately from the Theorem of Hartmanand Grobman, Proposition 5.3 and Proposition 5.4.
§6 Invariant Manifolds Let f e Diff'(M) and suppose that p e M is a hyperbolic fixed point off. The set Ws(p) of points in M that have p as co-limit is called the stable manifold of p and the set W"(p) of points that have p as a-limit is called the unstable manifold of p. It is clear that W'(p) and W"(p) are invariant by f. Using the hyperbolicity of p we shall ,describe in this section the structure of these sets and we shall analyse the way they change under perturbations of the difieomorphism f. Analogous definitions and results are valid for singularities of vector fields.
EXAMPLE 1. If A e GL(Rm) is a hyperbolic isomorphism there is an invariant splitting 08"' = E' ED E" such that, for q e E', A"(q) -+ 0 as n -+ oc and, for q e E", A-"(q) -+ 0 as n - x. Moreover, for any other q, 1IA"(q)11 x both as n -+ oo and as n -+ - oo. Thus, W'(0) = E' and W"(0) = E". Let us suppose that M c Rk'and let d be the metric induced on M from R'. For 13 > 0 we shall write Bs c M for the ball with centre p and radius P. Definition. The sets
Ws (p) = {q E Bp; f"(q) e B., V n > 0},
Wp"(p) = {q a Bp;.f-"(q)c- B,,V n > 01 are called the local stable and unstable manifolds, of size 13, of the point p.
We recall that a topological immersion of It' in M is a continuous map F: R -s M such that.$very point x e R' has a neighbourhood V with the following property: the restriction of F to V, F I V. is a homeomorphism onto
its image. In this case we say that F(1t') c M is an immersed topological submanifold of dimension s. A topological embedding of R' in M is an injective topological immersion which is a homeomorphism onto its image. 6.1 Proposition. If fi > 0 is sufficiently small we have:
(1) Ws,(p) c W'(p) and W"p(p) c Wu(p); that is, those points in a neighbourhood of p whose positive (respectively negative) orbit remains in the neighbourhood have p as w-limit (respectively a-limit); (2) W p(p) (respectively W"(p)) is an embedded topological disc in M whose dimension is that of the stable (respectively unstable) subspace of A = Dfo;
(3). W'(p) = U. a o f "(W'p(p)) and W"(p) = U., o f"(W'(p)). Hence there E" -s M) whose image is W'(p) (respectively W"(p)), where E' and E" are the stable and exists an injective topological immersion cp,: E' -s M ((p unstable subspaces of A = DfD.
74
2 Local Stability
PROOF. (1) and (2): By the Grobman-Hartman Theorem there exists a neighbourhood U of 0 in TMv and a homeomorphism h: B0 -+ U which conjugates f and the isomorphism A. As A is a hyperbolic isomorphism it follows that if x e U has A"(x) e U for all n >_ 0 then x e E' and so A"(x) --+ 0 when n--+ cc. Let q e W'(p). As f"(q) a B for n >_ 0, and hf"(q) = A"h(q)
we have A"h(q) e U for n 2! 0 so that A"h(q) -+ 0. Thus, f"(q) = h-'A"h(q)
converges to p - h-'(0) which shows that W'(p) c W'(p). Moreover, h-'(E' n U) = W'(p) which proves part (2). Similarly, W'(p) c W"(p) and W' p) = h-1(U n E"). (3) As W'(p) is invariant by f and W'(p) W'(p), we have f - "(W'(p)) c W'(p) for all n so that U-10 f -"(WR(p)) c W'(p). On the other hand, if q e W'(p) then lim"-. f"(q) = p so there exists no e N such that f"(q) e Bs for all n >_ no. Thus, f"0(q) e W'(p) and so q E f `0 W,(p). Similarly, we may show that W"(p) = U"Zo f"Wp(p). We shall now define'a map tp,: E' M whose image is W'(p). If x E E' there exists no e N such that A"°(x) e U where
U is the neighbourhood of 0 considered above. We define cp,(x) = f -"°h-'A"°(x). As h - ' conjugates A and f, it follows that cp, is well defined, that is, it does not depend on the choice of no. It is easy to see that cp, is an injective topological immersion and that cp,(E') = W'(p). Similarly, we may construct an injective topological immersion cp": E" -+ M whose image is W"(p). O
Remarks. (1) If p c M is a fixed point off then the stable manifold of p for f coincides with the unstable manifold of p for f -'. This duality permits us to translate each prope ty of the stable manifold into a property of the unstable manifold. (2) Although the local stable manifold is an embedded topological disc, the global stable manifold may not be an embedded submanifold of M as Example 2 below shows. (3) It is important to stress that the Grobman-Hartman Theorem only provides W'(p) with the structure of a topological submanifold, as we saw
in Proposition 6.1. However, the next theorem is independent of the Grobman-Hartman Theorem and shows that W'(p) is in fact a differentiable immersed submanifold of the same class as the diffeomorphism. We presented Proposition 6.1 as motivation for the train result of this section. EXAMPLE 2. Let f : S2 -+ SZ be the diffeom *phism induced at time 1 by the flow of the vector field X whose orbit structure is as follows: the north pole pN, is the only singularity in the northern hemisphere; the south pole ps is a
saddle whose stable and unstable manifolds form a "figure eight" that encircles two other singularities. See Figure 6. In this example the stable manifold of ps is not an embedded submanifold of S2. EXAMPLE 3. Let f = Y1 where Y is the vector field on S2 whose orbit structure
is shown in Figure 7. In this example W'(ps) and W"(ps) are embedded submanifolds of S2.
§6 Invariant Manifolds
75
Northern hemisphere
Southern hemisphere
Figure 6
Definition. Let S and S' be C' submanifolds of M and let e > 0. We say that S and S' are e C'-close if there exists a C' diffeomorphism h: S -+ S' c M such that i'h is a-close to i in the C' topology. Here i: S - M and i': S' -' M denote the inclusions.
-6.2 Theorem (The Stable Manifold Theorem). Let f e Diff'(M), let p be a hyperbolic fixed point off and E' the stable subspace of A = Dfp. Then: (1) W'(p) is a C' injectively immersed manifold in M and the tangent space to W'(p) at the point p is E';
(2) Let D e W'(p) be an embedded disc containing the point p. Consider a neighbourhood X c Diff'(M) such that each g e .N° has a .unique hyperbolic fixed point po contained in a certain neighbourhood U of p. Then, given e > 0, there exists a neighbourhood 4?' e .N' off such that, for each g e Y, there exists a disc D. c W'(po) that is c C'-close to D. We are going to present a proof of this theorem using the implicit function theorem in Banach spaces. The proof is due to M. Irwin [43]. We base our presentation on a set of notes by J. Franks.
Northern hemisphere
Southern hemisphere Figure 7
2 Local Stability
76
We shall prove that the local stable manifold, W' p), is the graph of a C' map and that the points of Wo(p) have p as their w-limit. Thus, the global
stable manifold is of class C", since W$(p) = U"zoj -"W'(p) We can therefore restrict ourselves to the case where f is a diffeomorphism defined on a neighbourhood V of 0 in ff8m, with 0 as a hyperbolic fixed point.
Let A = Df(0). Let us consider the A-invariant splitting Rm = E' ® E° 11 % on E', E" such that IIA'11, < a < 1 and !I(A")'II" < a < 1. and norms II For Q > 0 we write On Rm we use the norm !ix, Q+ x"44 = max{11x3!!:,
Bp for the open ball with centre 0 and radius f and we put B' = B0 n E', B" = Bw n E". We choose f so that, in Ba, we can write
.4 + ,
(D(0) = 0,
!!D(D!! < E
for some t;, 0 < r(D< 2(a- ' - 1). We shall also use the notation f = (f', j"), A = (A', A") and
_ (c', (D"). To prove Theorem 6.2 we need the following
lemma.
6.3 Lemma. If z = (z x") and z' for all n > 0 then z = z'.
(x,, x;,) satisfy f"(z) e Bp andf"(z') a B0
PROOF. Consider two points y = (y y") and
(y;. y;,) in B, such that
iIY, - Ysll <_ IIY" - Yull We claim that 11f"(y) - f"(Y')11 >- (a - ' - E)IIY" - y. 11 and
VJ'(Y) - f"(Y')11 s of"(Y). -. f (Y,) ii
In fact, f"(y) - f°(1') = A"(!°) - A"(Y') +'I"(Y) - (D"(),'). By the Mean Value Theorem we have Irf"(Y) - f '(Y')II ? a-' I!Y" - y.11 - elly - yli. As IIY' - y ll = fIv" - v;,11 we deduce that
f"(v )iI > (a-t - e)IIY" - Ywll. Similarly.
Ilf'(Y) - f'(o')il < ally, - Ysll + EI!y - Y':I. As IIY = Y'll = 11y. - v;,il ? ilv, - Y;II we obtain
I.f'(Y) - f'(Y')I! (a + E)IIY" - Y"11 s lif"(v) -- f"(v')il. From these inequalities we conclude that
iIf(y) - f(Y')II 2t (a-' - r)VY - y p. Now consider the original points z, z' and put y = f"(z), y = f"(z') for n > 0. From the above argument we obtain
of"(-) - f"(z')II z
(a-t - E)"Ilz - z11.
§6 Invariant Manifolds
77
As a - ' -'r; > 1 we conclude that z z'. If not, the distance between f"(z) and f"(z') would tend to infinity with n contradicting the fact that f"(z) and f"(z') belong to BO for all n >- 0.
PROOF OF THEOREM 6.2. We want to show that the set of points whose positive orbits remain in a neighbourhood of 0 form a C''submanifold. We shall also show that this set coincides with the set of points whose positive orbits converge to 0. First let us motivate the proof. Let K be the space of sequences y(n), n >_ 0, in Rm such that y(n) -+ 0, with the norm IIYII = sup"Ily(n)JI. Let G be the subset of K defined by G = (y e K; y(n) e B, for n >_ 0). Suppose, for some z E BR, that y e G and y(n) = f"(z). Then
r(n) = A(A + (D)"-'(=) + t(y(n - 1)) = A2(A + (D)n-2(z) + A4(y(n - 2)) + F(y(n - 1)) n-I
A"(z) + I A "
° 1 (:'(i)) 10 The second component of (n) has the expression
l
n-1
(A')-'
`4'"(y(1)) +.
As A" is an expansion we conclude that Y;_o (Au)- '
converges to
- Z1.
This motivates us to define a map F: B' x G -+ K by (A')"
F(x, )(n) = y(n) - CA-)"(x) +
(Au)n
I
We shall show that F(. ,-) c K and then that for each x there exists , e G such that F(x.,,) = 0. Choose h > 0 such that I4D(_)I; < b for z E By. As 0 < a < 1, Y-0 a' is
bounded and converges to (I - a) "'. Firstly, notice that y(n) -+ 0 and (As)"(x) -» 0 since
< a". Also <_ (1 - a)
sup III"(y(i))I$. iZn
Thus, given c > 0, we choose n large enough so that II(D"(y(i))II <(1 - a). for i > n. Now the second component of F(x, y) tends to 0. Look at the first component. For 0 <- in n we have n-1
Y (A')"- 1 -ic s(y(i)) 5 (1 1=0
a)7 1(j---h + (1 - a)-' sup jIV(y(i))Ij. Lam
78
2 Local Stability
Given e > 0 we make m large enough for the second term to be less than a/2. Then we can choose n large enough for the first term to be less than a/2. Thus, F(x, y) E K. Now we shall use the Implicit Function Theorem. If we fix y c- G the map
x -i F(x, y) from BQ to K is affine and continuous. In particular it is of class C''. We shall show that, for u E K, n- 1
D2 F(x, y)(uXn) = u(n) - Y (A5)"- 1 -'Dt1(y(i)Xu(i)), i=o Y(Au)n-
1 -'D(p"(y(i))(u(i))
i=n
).
/
To simplify the expressions below we write A for the right-hand side of this equation. We want to prove that, given S > 0. II F(x, y + uXn) - F(x, y)(n) - A II
<- d II u II
for small (lull. The left-hand side of this is less than or equal to n-1
Y (As)n-1- i [ V(y(i) + u(i)) - D'(y(i)) - DV(y(i)Xu(i))]
i-0
X
Y (A")n -1- i [(D"(y(i) + u(i)) - (D"(y(i)) - D(D"(y(i))(u(i))1 ,=n
Since DO is continuous on the closure of Ba, it is uniformly continuous on B. Thus, given 6'> 0, there exists p > 0 such that jID(D(z + u) - D(D(z)Il < 6' for z E Ba and (lull so small that z + u c- Bb and IIuII < p. Applying the Mean Value Theorem to (U(z + u) - (D(z), we get
IID(z + u) - 1(z) - DD(z)ull <- b'Ilull.
We conclude that each term in the above expression is smaller than (1 - a)-'S' Ilu ll. So it is enough to consider 2(1 - a)-'(Y < S. This shows that A = D2F(x, y)(u)(n). It is easy to check directly that D2F is continuous.
We conclude that F is of class C' since D,F and D2F are continuous. Moreover we can check directly that D2 F(0, 0) is the identity I on K. By the
Implicit Function Theorem there exists a ball B; in ES and a C' map (p: B; -+ G such that 9(0) =(0 and F(x, (p(x)) = 0 for all x E B;. We have X,
W(xx0) = I
- Y (A")
-i(P"(9(x)(0))
i=0
Then we define h: B; E' as the second component of this expression, so that (p(x)(0) = (x, h(x)). The map h is of class C' since (p is C' and the map 0: G - ' taking,,, to y(0) is continuous linear. We remark that h = n2 o 0 - (p where 7[2: 68'
E" is the natural projection. As F(x, (p(x))(n) = 0 for all
n z 0 we have n- 1
(p(xxn) _ ((A3)1(x) +
i=0
ap
(As)"
1
i(D'((p(xxi)),
- i=n (A")"
1
1 J.
§6 Invariant Manifolds
79
From this it follows that cp(x)(n + 1) = (A + D)cp(xxn) = f(cp(x)(n)). By induction we have cp(xxn) = f"(x, h(x)) for all n > 0. This shows that if z e graph(h) then f"(z) tends to 0 as n -+ oo. On the other hand, if z and its positive orbit remain in a small neighbourhood of 0 then Lemma 6.3 shows that z e graph(h). Thus f (graph(h)) c graph(h) and graph(h) represents the local stable manifold of 0 for f. It remains to prove that Dh(O) = 0 to conclude that the local stable manifold is tangent to E' at 0. As F(x, (p(x)) = 0 we have D1F(x, (p(x)) + D2 F(x, (p(x))Dcp(x) = 0. From this it follows that Dcp(0) = - D1F(0, 0) since D2 F(0, 0) _ I. If v e E'then Dcp(O)(v) - D 1F(0, OXv) = -u where u(n) = (- (A')"v, 0) for n >- 0. Ash = n2 o O o cp we have Dh(O)(v) = n2 o 0 o Dcp(0)v since n2 and 0 are linear. Thus Dh(0)v = ir2((A')°v, 0) = 0 as we wanted to prove.
We proved above that the stable manifold is of class C'. In fact it is C iff is C. This comes from the fact that F above is C; we saw that D1F is C'-' and it can be checked directly that D2 F is Cr-1. The continuous dependence of the local stable manifold on the perturbation is proved in the same way
as above. It is enough to make the map F depend also. on the bounded C' perturbation F = F(x, y, (D). Then one checks that D3F is Cr-1 so that F is C'. The rest of the argument is as before. We just notice that the fixed point for A + 0 may vary (continuously) with 0. But this is not very relevant for the proof above: with a C' diffeomorphism we can translate the fixed point back into the origin. It is interesting to remark that the proof works as well for maps: A' may have some eigenvalues equal to zero. However, in this way we can only show the existence of the stable manifold. The unstable manifold also exists as we
can show using another proof, the so called graph transform, etplained below.
Another interesting remark is that the proof above is valid in Banach spaces : we only add as an assumption that the local map f and its derivatives are uniformly continuous in B5. Now let us consider a vector field X e £'(M) and let p e M be a hyperbolic singularity of X. The stable manifold of p for the vector field X, W'(p, X), is the set of points of M whose co-limit is p. Let f = X, be the diffeomorphism induced at time t = 1. As we have seen, p is a hyperbolic fixed point off. If
W'(p, f) denotes the stable manifold of p for f then W'(p, f) = W(p, X). In fact, if x e W'(p, X), that is, if X,(x) -+ p as t - oo, then clearly X"(x) = f"(x) - p as n oo. Thus, W'(p, X) c W'(p,f). On the other hand, let U be any neighbourhood of p. As X(p) = 0 there exists a neighbourhood V of p such that X,(V) c U for 0 < t < 1. If x e W'(p, f) then there exists no a N such that f"(x) e V for n >- n°. Thus X1(x) e U for t z n°. This shows that
X,(x)-,past-+ oo. Thus W'(p,f) c W'(p, X). We shall also present a sketch of another more geomettical proof of Theorem 6.2 that can be found in [39] and [77]. We shall omit the details of this second proof since they are technically more complicated. We emphasizg,
80
2 Local Stability
Figure 8
however, that this proof lends itself to important generalizations as in [40].
Let . c C'(Bs, B") be the set of C' maps whose Lipschitz constant is at most 1. Then F is a closed subset of C'(B', B") and hence a Baire space. If x e .f then the graph of a is a C' submanifold of W. As A contracts vectors in ES and expands vectors in E" it follows that A-t(graph(a)) n B. is the graph of a C' map from BO to B" which we denote by FA(a). In fact, FA(xXXS) _ (A")- `ot(Asxs). As I'A(a) also has Lipschitz constant at most 1 we have defined
a map r,: .-+ .F called the graph transform associated to A. It is easy to see that, if a E .IF, the sequence tr" (ot)} converges to the zero map whose graph is the local stable manifold of A. As A is the derivative off at the point 0 it is reasonable to expect that we can, by taking fi small enough, define a'
graph transform Ft-: F -+ f associated to f If F f has an attracting fixed point x f e C'(BQ, Bu), that is, if F1(a1) = a f and r' "(x) x f for any a e ,F, then f (graph(a f)) = graph(a f) and graph(a f) is the set of points of B'6 whose positive orbits for f remain in BQ. Thus graph(a f) = W'(0). In other words the proof of the theorem can be done in the following steps:
(1) r f is well defined and has an attracting fixed point U f E C'(B', Be);
(2) if g is near f in the C' topology then F9 is well defined and has an attracting fixed point, which varies continuously with g.
§7 The A-lemma (Inclination Lemma). Geometrical Proof of Local Stability In this section we shall discuss a local fact which is relevant to several results in Dynamical Systems and explain some of them [75]. In particular, we shall
give another more geometrical proof of the Grobman-Hartman Theorem. Again we observe that the same result (and the proof) is valid in Banach spaces.
Let f be a C' diffeomorphism of a neighbourhood V in R' with 0 as a hyperbolic fixed point. Consider the hyperbolic isomorphism A = Df (0)
§7 The A-lemma (Inclination Lemma). Geometrical Proof of Locai Stability
81
E" R u (0) toe wls
oc (0)
Es
N Figure 9
and the invariant splitting OB' = E5 E) E". In Section 6 we saw that the local
stable manifold of the fixed point 0, W,o(0), is the graph of a C' map cp,: B'
E" with 9,(0) = 0 and Dcp,O) = 0. Here B,3,(0) c E' denotes the ball
with centre 0 and radius P. In the same way the local unstable manifold, W".(0), is the graph of a C' map (p.: B; - E' with p"(0) = 0 and Dcp"(0) = 0. Let us consider the map cp: BBe Q BB -+ E' $ E",
(AX. , X") = (x, - cP"(x"),
X. - (Ps(xs))-
It is clear that tp is Cr and that Dcp(0) is the identity. Thus cp is a diffeomorphism
when restricted to some neighbourhood of 0 in Rm. Let us consider the diffeomorphism! = cp. f = cpf cp t. Thenjis a diffeomorphism of a neighbourhood of the origin with J(0) = 0 and DJ'(0) = A. Moreover, the local stable manifold off is a neighbourhood of the origin in E' while the local unstable manifold is a neighbourhood of the origin in E". In other words, we can always assume that the local stable (respectively unstable) manifold of a hyperbolic fixed point of a diffeomorphismf is a neighbourhood of the fixed point in the stable (respectively unstable) subspace of the linear part off. Let 11.11 be a norm on I' such that IIA'11 < a < I and jj(A")- t jI < a < 1
where A' and A" are the restrictions of A to Es and E" respectively. If P: B` -+ E' is the restriction off to Wp(0) c B' c E' then Df'(0) = A'. We have that f' is a contraction for small enough since A' is a contraction. Therefore, if B' c B' is an open ball with centre at the origin then f (8B') c B'
where 8B' = B' - B' is the boundary of R. The annulus G'(0) = B' - f (B,) is called a fundamental domain for the stable manifold of 0. It is clear that a&(O) = 8B u f (8B5)
Figure 10
82
2. Local Stability
If x e W'(0) - {O} then the orbit of x has at least one and at most two points in the fundamental domain G'(0), that is, U"ea f"(G'(0)) = W'(0) - {0} and, if x e int G'(0), f"(x) 0 G'(0) for all n e 7L - {0}. Any neighbourhood N'(0) of G'(0) that is disjoint from W;.,(0) is called a fundamental neighbourhood for the stable manifold of 0. Similarly, we define a fundamental domain G"(0) and a fundamental neighbourhood N"(0) for the unstable manifold of 0.
Let B' c E' be a ball contained in Wioc(0), B" c E" a ball contained in W;",(0) and V = B' x Bu. Consider a point q e W{.,(0) and a disc D" of dimension u = dim E" transversal to W1.(0) at q.
7.1 Lemma (The A-lemma). Let V = B' x B", let q e W'(0) - {0) and let D' be as above. Let D." be the connected component off "(D) n V to which f"(q)
belongs. Given c > 0, there exists nc a N such that, if n > no, then D.' is e C'-close to B".
eighbouthood Vof 0 is given by
PROOF. The expression for f in
f (x3, x") = (A'x5 + *p,(x,, x"), A'x. + 9N(x3, x")) where
(Df)o = (A', A"), 11A311
a<1,
x, e B',
x. c- B"
Ji(A")-'f1 5a<1,
.
a4s
a(Pu
ax" B.
ax,
B.
=0.
As app;/8x,{0, 0) = 0 for i, j = s, u, the continuity of these partial derivatives
implies that there exists k with al = a + k < 1, 0 < k < 1, b = (a-' - k) > 1, k < (b - 1)Z/4 and there exists V' c V such that k Z max V.
I, axj
Figure 11
i, j = s, U.
§7 The A.lemma (Inclination Lemma). Geometrical Proof of Local Stability
83
We can suppose that q e V' and B" c V'. Let vo be any unit vector in (TD"),. We can write vo = (vo, vo) in the product V = B° x B". Let ao be the slope of vo, 1o = Ilvoll/Ilvoll, with 11v0" 11 :0 0 since D" is transversal to B3 at q. Consider qt = .f (q),
vt = D.fq(vo)
q2 = .f 2(q),
v2 = D.fq,(vt)
qn = f "(q),
V. =
ForgeaB5, A' + agqlax3(q)
aq/ax"(q)
O
A" + a4p/ax"(q.)
D.f"(vo) =
vo " vo)
(A5vo + acp,/ax3(q)vo +
A+ acp"/ax"(q)vo Thus
'-
II v, II II VI II
=
II A3vo + ag,/ax,(g)vo + acp,/ax"(q)vo II IIA"vo + a(p,Jax"(q)vo I
The numerator is bounded above by I1Asv0' l1 + II 0(p,10x,(q)v°o11 + 11 aQ,,/ax"(q)v°oll 5 allvsoll + kllvoll + kllvoll.
The denominator is bounded below by IIA"volt - Iiatp"/ax"(q)voll ? a-'110011 - kllvoll
Hence .1
'
2 A
b
Ilvoll
At + k S
A0
C`
1
h
IIviII IItnII
A0
b2
b
1 + k t2 b;>
A'0
k
b^+b--1.
As 1o/b" - 0 as n oc and k/(b - 1) < (b - 1)/4, there exists no a V such that, for any n > no we have A,, < (b - 1)/4. Let 0 < kl < min(e, k). As a(p,iax" I B. = 0 and B" is compact, there exists b < s such that, for Vt = SB' x B" c V. we have max 1.,
'(D, II 0 x"
< k1.
2 Local Stability
84
Here SB' means the ball whose radius is b times the radius of B'. As vo can be chosen so that A0 is the maximum possible slope of unit vectors in (TD")4, there exists no such that, for n >- no, all the nonzero vectors of (TD")q, have slope A. < (b - 1)/4 and qn0 E V,. Then, by the continuity of the tangent plane of D"0, there exists a disc D" embedded in D"0 with centre qn0 and such that the slope a of any unit vector in (TD")p, p e D" satisfies
d<(b-1)%2. Let v c- (TD")D for p e D". In the decomposition v = (v', v"), v has slope Ano = II u' II / II u" II. Let us calculate the slope of the iterates of v. We have
=
Dfp(v)
A'v' + ap,!ax,(p)v' + acp,/ax"(p)v" (atp"%ax,(p)u'
+ A"v" + acp,
so
IIA't + a(p,laxs(P)v' + a(P"/ax"(P)u"II Ilagp,/ax,(p)v' + A"v" + acp,jax"(p)v"II
whose numerator is less than a11011 + k II 011 + k, 11t;'11 and whose denominato
is greater than ;IA"v"II - Ilaw.'ax"(P)t'"II - !a(p,jax,(p)u'p > a-lllv"Il - klly"ll - kllu'p. Hence aAn°+kAn0+k,
a-t -k-k2n0
'ln°+k, b-k2,0
;.o+k, b-2k(b- 1)-b--tz-(h- 1) ,ln,, +.k,
An°+k, 2(b+ 1)
Let h, = 2(b + 1), b, > 1. Then ;n+no 5 An0/bi + k,/(b, - 1). There exists ,n" such that, for n >- n, i
An+nn < f.
I+
b1
-
I
As we could have considered u such that An. was the maximum possible slope
of unit tangent vectors to D", we see that, for n >- n, any nonzero vector ta'igent to f"(D") n V, has slope less than s[1 + (b, - 1)-t]. Thus, given t > 0, there exists n such that, for n >- n, all the nonzero tangent vectors to f"(D") n V, have slope less than F. Let us compare the norm of a vector tangent to f "(D") n V, with that of its image by Df : (vn, v") - DJ (tn, vn) _ (vn+ `IILn+1111 + lt'"+,IIZ ilu"IIZ
v+
.tII liu"11
1 + A.2+t
1+
.
,1z
85
§7 The A-lemma (Inclination Lemma). Geometrical Proof of Local Stability
From the expressions for v,",, and vu we conclude that
_
Il4+111>a'-kk.?. "
IL"II
As the slopes A,, and A. are arbitrarily small, we see that the norms of the iterates of nonzero vectors tangent to f"(D") n V, are growing by a ratio that approaches b = a - ' - k > 1. Hence the diameter of f"(I3") n V, increases, and this, together with the fact that its tangent spaces have uniformly small slope, implies that there exists n such that, for n > n, f"(f5") n V, is C' close to B" via the canonical projection onto B". This completes the proof of the A-lemma. U Remarks. (1) The A-lemma can be stated for a family of discs transversal to W'(0) provided that this family is continuous in the C' topology. Thus, let F: G'(0) -+ C'(B", M) be a continuous map which associates to each point q of the fundamental domain G'(0) a disc Dq = F(q)B" transversal to B'. Let
U = B' x B" as above. Then, given e > 0, there exists no E N such that f"(D) n U is a disc a C'-close to B" for any q E G'(0) and n >_ no. (2) Although this is not necessary for the majority of applications, these discs can be proven to be C' close, r >- 1, if F is a continuous family of C' discs; that is, if we have a continuous map F: G'(0) C'(B", M). (3) The following fact is an immediate consequence of the A-lemma. Suppose D' is a small s-dimensional disc transversal to W"(0). Then, there exists`no > 0 and a sequence of points z" a D", n >_ no, such that f"(z") e V.
Corollary 1. Let p,, P2, p3 e M be hyperbolic fixed points off e Diff'(M). If W"(p,) has a point of transversal intersection with W'(p2) and W"(p2) has a point of transversal intersection with W3(p3) then W"(p,) has a point of transversal intersection with W'(P3).
Figure 12
86
2 Local Stability
Figure 13
PROOF. Let q be a point of transversal intersection of W"(p2) and W'(p3). We consider a closed disc D c W"(p2) containing p2 and q. As D has a point of transversal intersection with WI(p3) it follows that there exists e > 0 such that if D is a disc e C'-close to D then D also has a point of transversal intersection with W'(p3). Now let q2 be a point of transversal intersection of
W"(p,) with W(p2) and D" c W"(p,) a disc containing q2 of the same dimension as W"(p2). By the A-lemma there exists an integer no such that f"°(D") contains a disc D that is s C'-close to D. Thus there exists a point g e D n W'(p3). As W"(p,) is invariant we have f"°(D") c W"(p,) so that 4 is a point of transversal intersection of W"(p,) and W'(p3).
Corollary 2. Let p e M be a hyperbolic fixed point off e Diff'(M) and let N'(p) be a fundamental neighbourhood of W'(p). Then Un2of "(N'(p)) U - Wj°,(p) for some neighbourhood U of p.
PROOF. First, notice that the iterates by f of a fundamental domain G'(p) c
N'(p)cover W'(p) - {p};that is, U"Eaf"(G'(p)) = W'(p) - {p}. Moreover, by the A-lemma, every point in a certain neighbourhood U of p that is not in W;°,(p) belongs to some iterate of a section that is transversal to G'(p) and contained in N'(p). We shall now prove the A-lemma for vector fields. Let p e M be a hyperbolic singularity for X e Xr(M). Let W(,,,(p) and W,,,Jp) be the local stable and unstable manifolds of the point p. Let B' be a disc embedded in W;.(p) such that aB' is transversal to the field X in W'(p). The sphere 9'(p) = OW is called a fundamental domain for W'(p). It is easy to see that, if x e W'(p) {p}, the orbit of x intersects 9'(p) in only one point. Similarly, we can define a fundamental domain 9"(p) for W"(p).
§7 The A-lemma (Inclination Lemma). Geometrical Proof of Local Stability
87
Let D" be a disc transversal to W;"c(p) that contains a point q e W',a,(p)
and has dim D" = dim Wlo,(p). Let K be the compact set K
{X,(q); t e [0, 1] } and, for each point X,(q) E K, consider the disc D"(X,(q)) = X,(D")
which contains the point X,(q) and is transversal to W' (p) since X, is a diffeomorphism and W" ,(p) is invariant for X. Let f = X1 be the diffeomorphism induced at time 1. Then p is a hyperbolic fixed point for f and the stable and unstable manifolds of p for the diffeomorphism f coincide with the stable and unstable manifolds of p for the vector field X. Let B' be a disc embedded in containing p, B" a disc embedded in W;.(p) containing
p and V = B' x B" a neighbourhood of p. By the A-lemma for diffeomorphisms, given e > 0, there exists no a N such that, if n > no, D."(x) is c C'-close to B", where Du(x) is the connected component of f"(D"(x)) n V that contains f"(x) and x E K. This proves the following lemma.
7.2 Lemma. Given e > 0 there exists to > 0 such that, if t > to and D, is the connected component of X,(D") n V that contains X,(q), then D, is c C'-close to B".
D
Now we present another, more geometric, proof of the GrobmanHartman Theorem. We shall use the A-lemma and the Stable Manifold Theorem, whose proof is independent of the Grobman-Hartman Theorem as we have already remarked. The proof we present is for flows but similar arguments work as well for diffeomorphisms. 7.3 Lemma. Let p e M be a hyperbolic singularity of a vector field X C- N'(M). There exists a neighbourhood U of p and a continuous map n,: U -, B where B, = U n W,,,(p) is a disc containing p, with the following properties:
(1) 7r- '(p) = B. = U n WI.(p) is a disc containing p; (2) for each x e B as '(x) is a C' submanifold of M transversal to W'o'(p) at the point x; (3) n, is of class C' except possibly at the points of B"; (4) the fibration defined by it, is invariant for the flow of X, that is, if t >_ 0 then X1(na 1(x)) n" I(X1(x))
PxooF. We can suppose, by using a local chart, that X is a vector field on a
neighbourhood V of the origin in R' = E' ® E" with 0 as a hyperbolic singularity. We can also suppose that W" (0) is an open subset of E' contain-
ing 0 and that W' (0) is an open subset of E" containing 0. Let 5'(0) be a fundamental domain for W'(0). 9'(0) is a sphere contained in E' and transversal to the vector field X on E'. Let B" c E" be a disc containing 0. If we take B' small enough the cylinder "(0) x B" is transversal to the vector field. In W"(0) x B" we have a C' map n,: 9'(0) x B" - W'l,,(0) which is the projection on the first factor. The fibres 7r, -'(x) through points x e 9'(0) are
88
2 Local Stability
W"(0)
Figure 14
transversal to W, o,(0). By Corollary 2 of the A-lemma U, z o X,( x B") U - E" where U is a neighbourhood of p. If x e U n E" we define ir,(x) = p. If x e U - E" then there exists t > 0 such that X _,(x) e x B". Then we define n,(x) = X, n, X _,(x). It is clear that n, is of class Cr in U - E". The continuity of n, at points of E" follows from the A-lemma. discs
Using the fibres constructed above we can prove the local stability of a
hyperbolic singularity. In fact, let p e M be a hyperbolic singularity of X e .'(M). Let N be a neighbourhood of X such that any Y E N has a singularity py near to p and of the same index. Define homeomorphisms that conjugate the flows of X h': Wio"(p) ' Wio"(py), h": Wio,(p) -' and Y by first defining them on fundamental domains and then extending them as in Proposition 2.14 to Wl.(p) and W;o,(p) using the flows of X and }: Consider fibrations Its : Up -u Wloc(p), nu : Up W"'(p), ns : VPr 10 4170 (py), irt, V,, -. WioJpy) as in Lemma 7.3. If q e UP define h(q) = q where 4 is such that it (q") = hs(n, q) and 7r'(q") _ h"(r.5 q). It is easy to see that h is a homeomorphism that conjugates the flows of X and Y. We remark that the fibrations considered above define continuous coordinate systems in which the flows are expressed as products and consequently the homeomorphism h is the product of h' and h". A proof of the Grobman-Hartman Theorem for diffeomorphisms using the A-lemma is in [75]. The constructions are similar but more elaborate. :
EXERCISES
1. Show that a linear vector field L is hyperbolic if and only if the 0)-limit of each orbit is either the origin or empty.
2. Show that there exists a linear vector field L on F and an orbit y of L such that the w-limit of y contains y but y is neither singular nor a closed orbit.
89
Exer.ises
3. We say that a linear isomorphism A- Ca" --. tR' embeds in a Hcnt if th, ;
;a .I -'cctur
field X generating a flow X, with A = X,. Characteriic. b) their .unmic.;l forms. the hyperbolic linear isomorphisms that embed in flow. 4. Let,/': M - R be of class C', r > 2, and let X = grad /. Show that P E Al is .t hyperbolic singularity of X if and only if (if (p) = 0 and 121 (p) is a nondegenerate bilinear form.
5. Let X = grad J where f : M -. R is of class C'" Show that if p e 11 is a singularity of X then the eigenvalues of dX,, are real. 6. Give an example of a vector field X E X'(S2) such that .Y E q, and X,:_ , ¢ G,, that
is, X,has a non hyperbolic fixed point.
7. Let X = grad f where f : M -- f8 is of class C', r >t I. Show that ,\. e :". if and
onlyifX,_,eG,. 8. We say that a C' function f :.M - R, r > 2,.is a Morse fiax rion if grad f E :4:, that is, if the singularities of grad f are all hyperbolic. Show that the set of ?Morse functions
is open and dense in C'(M).
Hint. Let cp: R" -+ R be of class C'. Show that 0 is a regular value of the map 4>: U8" x L(Ud", l) -+ L(l8", 98) defined by D(x, A) = drp(x) + A.
9. Let X and Y be C' vector fields on R'". Suppose that 0 is an attracting hyperbolic singularity for X and Y. Show that there exists a homeomorphism h of a neighbourhood of the origin which conjugates the diffeomorphisms X,.., and Y,but does not take orbits ofX to orbits of Y. 10. Let p e M be a hyperbolic fixed point of a diffeomorphism f Let (p.1, be a sequence of
periodic points off with p" # p and p, - p. Show that there exists a sequence of periodic points off that converge to a point other than p on the unstable manifold of p.
it. Show that if f e Diff'(M), r >- 1, is structurally stable then all the fixed points off are hyperbolic.
12. Let p e M be a hyperbolic periodic point off e Diff'(M). Show that given n E ' there exists a neighbourhood V of p such that any periodic point of / in I period greater than n.
has
13. Let 0 e R" be a hyperbolic singularity for the vector fields X and Y e 3:'(R"). Show that if there exists a C' diffeomorphism, f : R" -+ IR", taking orbits of X to orbits of Y then the eigenvalues of L = DX(0) are proportional to those off = DY(0).
Hint. Under the above hypothesis there exists a function A: R" - R such that Df (x) X(x) = A(x) Y(f (x)). Show that, for each v # 0, there exists ,(v) _ lim,..°A(rv) and that Df(0) Lv = l(v)ILDf(0)
v.
14. Let f : R" 9d" be a C' diffeomorphism with f(0) = 0. Let E" and E° be invariant subspaces for f such that ll8" = E° ® E" and write f": E" -. E" and f °: E° . E° for
the restrictions off to E" and E°, respectively. Suppose that the eigenvalues of df"(0) have absolute value > 1 and that the eigenvalues of df°(0) have absolute
2 Local Stability
90
value ,:5 1. Let D" c E" be a disc containing 0 and P a disc transversal tt E° containing a point q e E° such thatf"(q) .- 0 as n co. Show that there exists a neighbourhood V of 0 in R" such that, for all t > 0, there exists no a t01 satisfying the following property: if n >- no then f"(D) contains a disc E C'-close to V n D.
15. Let f : R2 -+ RI be a C' diffeomorphism, f (x, y) _ (f,(x, y), f2(x, y)), with the following properties:
(1) f,(O,y) = Ofor all ye R; (2) f2(x, 0) = 0 for all x e R; (3) af2/ay(O, 0) > 1;
(4) if a(x) = f1(x, 0) then a'(0) = 1, a"(0) = 0 and a(0) < 0. Show that there exists a > 0 and a neighbourhood V of (0, 0) such that, given c > 0 and a segment D transversal to the axis x = 0 through a point (0, y) e V, there exists no e N such that if n > no then f -"(D) contains a disc E C1-close to the interval
((x,0);-aSx5a).
16. Show that the diffeomorphismf in the previous exercise is locally conjugate to the diffeomorphism
g(x, y) = (x - x', 2y).
17. Let f : R2 - R2 be a C2 Qiffeomorphism, f(x, y) = (f, (x, y), f2(x, y)) with the following properties:
(1) f1(O,y)=OforallyeR; (2) f2(x, 0) = O for all x e R; (3) aft/ay(O, 0) > 1; (4) if a(x) = f1(x, 0) then a'(O) = 1 and a"(0) # 0. Show that f is locally conjugate to the diffeomorphism g(x, y) = (x + x2, 2y).
Chapter 3
The Kupka-Smale Theorem
Let M be a compact manifold of dimension m and X'(M) the space of Cr vector fields on M, r >_ 1, with a Cr norm. In Chapter 2 we showed that the set 11 c X'(M), consisting of fields whose singularities are hyperbolic, is open and dense in X'(M). This is an example of a generic property, i.e. a property that is satisfied by almost all vector fields. In this-chapter we shall analyse other generic properties in 1'(M). The original proof of the results dealt with here can be found in [44], [82] an* 407]. First 'we introduce the. concept of hyperbo ficity for closed orbits. As in the case of singularities a hyperbolic closed orbit y persists 'under small perturbations of the original vector field. Moreover, the structure of the trajectories of the field is very simple and is stable under small perturbations. In particular, the set of points which has y as w-limit (a-limit) is a differentiable
manifold called the stable (unstable) manifold of y. In a sense that will be made precise in the text, compact parts of these manifolds change only a little when we change the field a little. Let us consider two hyperbolic singularities Ql and v2. If the stable manifold of al intersects the unstable manifold of a2 then al and a2 are related by the existence of orbits which are born in v2 and die in a,. If the intersection is transversal then a small perturbation of the field will have hyperbolic singularities that are related in the same manner. Analogous concepts and properties are valid for closed orbits as we shall see later. We shall show here that all these properties hold for the fields in a residual subset of X'(M). At the end of the chapter we shall establish similar properties for Diff'(M). 91
91
The Kupka Smale Theorem
§1 The Poincare Map In the previous chapter we described the topological behaviour of the orbits of a vector field in the neighbourhood of a hyperbolic singularity. Now we are going to make an analogous study for closed orbits. As in the case of singularities we need to restrict ourselves to a subset of the space of sector fields in order to obtain a simple description of the orbit structure in neighbourhoods of the closed orbits.
Let y be a closed orbit of a vector field X c '(M). Through a point we consider a section E transversal to the field X. The orbit through xo returns to intersect E at time r, where r is the period
xo e
of y. By the continuity of the flow of X the orbit through a point x e Y sufficiently close to xu also returns to intersect Y. at a time near to r. Thus if
V c E is a sufficiently small neighbourhood of xo we can define a map P: V -+ E which to each point .x E V associates P(x), the first point where the orbit of x returns to intersect E. This map is called the Poincare map associated to the orbit (and the section E). Knowledge of this map permits us to give a
description of the orbits in a neighbourhood of ;. Thus, if x E V is a fixed point of P then the orbit of x is closed and its period is approximately equal to the period of if x is near to x,,. In the same way if x is a periodic point of P of period k, i.e. P(x) e l' P-(x) C V, ... , P'(-x) = x, then the orbit through x is periodic with period approximately equal to kr. ii' Pk(x) is defined for all k > 0 the positive orbit through x will be contained
in a neighbourhood of y and if, in addition, Pk(x) xo as k -+ x, then the u,-limit of the orbit of x is y. We can also detect the orbits which have y as otlimit using the inverse of P, which is the Poincare map associated to the field -- X.
From the continuity of the flows of X and - X it follows that P is a homeomorphism from a neighbourhood of xO in E into E. Later we shall show, using
the differentiability of the flow via the Tubular Flow Theorem, that P is in fact a local diffeomorphism of the same class as the field. We shall then be able to use the derivative of P at xo to describe the orbit structure in the neighbourhood of Y. For that we shall need some preliminary results. A tubular flow for X e X'(M) is a pair (F, f) where F is an open set in M
and f is a C' diffeomorphism of F onto the cube I' = I x I'- t = {(x, y) C
Figure 1
§1 The Poincare Map
93
Figure 2
R x U"; I x I < I and l y' I < 1, i = 1, ... , m - 1} which takes the trajectories of X in F to the straight lines I x {y} c I x I". If f*X denotes the field in IT induced by f and X, i.e. f,X(x, y) = Df f-:(x, y) X(f -1(x, y)), then f*X is parallel to the constant field (x, y) - (1, 0). The open set F is called a flow box for the field X. In the previous chapter we saw that, if p e M is a regular point of X then there exists a flow box containing p (Tubular Flow Theorem). 1.1 Proposition (Long Tubular Flow). Let y c M be an arc of a trajectory of X that is compact and not closed. Then there exists a tubular flow (F, f) of X such that F _D y.
PROOF. Let a: [-e, a + e] -+ M be an integral curve of X such that a([0, a]) = y and a(t) # a(t') if t # t'. Let us consider the compact set y' = a([ - e, a + e]). As the points of y are regular there exists, by the Tubular Flow Theorem, a cover of y by flow boxes. Let b be the Lebesgue number of this cover. We take a finite cover {F1, . . . , F,r} of y by flow boxes of diameter less than 6/2. By construction it follows that, if F1 n Ff 0 0, then F; u F. is contained in some flow box of X. Using this property we can reorder the F,, reducing them in size if necessary, so that each F; intersects only F; and
F,+1 Let
be such that pi =a(t,)eF,n5' and
let us write Id - 1 for {(0, y) e I x 1''-1; 1yi1 < d, j = 1, ... , m - 1}. Let (F,, f) be the tubular flows corresponding to the flow boxes above. It is clear that E1 = f , 1(1d - 1) is a section transversal to X because fl is a local diffeomorphism and po E E. If E, = X,,_,,(E,) it follows that E, is a section transversal to X which contains the point p,. If d is sufficiently small E, C F,. For each p e y' we take t e [0, a + 2e] such that p = X,(p,) and consider
the section EP = X,(E,). Using the Tubular Flow Theorem we have EP n Eq = 0 if p # q and also that F = UPE r EP is a neighbourhood of y.
Figure 3
3 The Kupka Smale Theorem
Figure 4
In this neighbourhood we have a C fibration whose fibre over the point " is E.,. i.e. the projection n, : F -+ y, which associates to each z E F the point p such that : e Ep, is a Cr map. We have another Cr projection defined on F, 1,, which associates to each point z E F the intersection of the orbit rz2 : F
of'-- with Z. More precisely, if z e Ep and p = X,(p,) then n2(z) = X _,(z).
Let us consider two diffeomorphisms g,: y - [-1, 1] and g2: E1 -+ I--'Then we define f : F -+ I x I'-1 by f (z) = (glnl(z), g2 n2(z)). It is clear that (F, f) is a tubular flow which contains y.
Remark. The diffeornorphism f obtained above takes orbits of X in F to orbits of the constant field C: I x I' -1 -+ R", C(x, y) = (1, 0). In general f does not preserve the parameter t, i.e.f.X is not equal to the field C. However,
we can find a neighbourhood of y, P c F, and a diffeomorphism J: P -+ (-b, h) x I", where b > 0, such thatf.X is the constant field. In fact, take p e y and b > 0 such that y = U..(-b.b)X,(p) c F. Let E, c F be a section transversal to X through the point p small enough for = U1E(-b,b)X,(1;d to be contained in F. IL z e F and X _,(z) e Ep then put j(z) = (t, hX _,(z)) I' -' is a diffeomorphism. It is easy to see that j`is a C diffeomorphism and that.j:X is the constant field. where h: Ep
1.2 Proposition. Let y be a closed orbit of a vector field X e £'(M) and let E be a section transversal to X through a point p e y. If PE: U e E -+ E is the Poincare map then PE is a C' diffeomorphism from a neighbourhood V of p in E onto an open set in E.
Figure 5
§1 The Poincare Map
95
PROOF. Let (F,, f,) be a tubular flow containing p and let (F2, f2) be a long tubular flow such that y c F, u F2 as in Figure 5. Let E, and E2 be the components of the boundary of F2 which are trans= f2 '({-1} x I'°') and E2 = f2'((l} x I'"-').Write versal to X, i.e. rtl : V c E -+ E,, 712: E, -+ EZ and n3: E2 -+ E for the projections along the
trajectories of X, where V is a small neighbourhood of p in E. PE _ n3 o n2 o 1r,. It is easy to see from the Tubular Flow Theorem that it,. Jr2 and 7E3 are maps of class Cr. Thus PE is C. As P has an inverse of class C. which is the Poincare map corresponding to the field - X, it follows that PE is a C' diffeomorphism from V onto an open set in E, which finishes the proof. U
Let E, and E2 be sections transversal to X through points p, and p2 of a closed orbit y as in Figure 6. E2 be the map which associates to each q e E, the first point Let h: E, in -which the orbit of q intersects E2. By the Tubular Flow Theorem h is a Cr diffeomorphism. If PE, and. PE, are the Poincare maps for the sections E, and E2 respectively we have PEZ = h o PE o h-'. So DP,,(p2) _
Dh(p,) o DPE,(p,) o Dh-'(p2) and therefore DPE,(p2) has the same eigenvalues as DPE,(p,). This shows that the next definition depends only on the field and not on the section E. Definition. Let p r_ ', where y is a closed orbit of X. Let E be a section transversal to X through the point p. We say that y is a hyperbolic clr,sed orbit of X
if p is a hyperbolic fixed point of the Poincare map P: V c E
E.
Remark. As the flow of a vector field depends continuously on the field the Poincare map also depends continuously on the field. More piccisely. let P.r: V c E E be the Poincare map of X. Let Y. he a neighaourhood of X in :V(M) such that, for all Y in Y', E is still a section transvers< l to Y and the orbit of Y through each point of V still returns to intersect E. Then the map
i!'' -. ("(I', E) which associates to each Y E Y' its Poincare map P, continuou,.
Figure 6
is
3 The Kupka-Smale Theorem
96
From this remark we conclude that, if y is a hyperbolic closed orbit of the vector field X, there exists a neighbourhood Y 'of X in X'(M) such that every Y E Y' has a hyperbolic closed orbit y, close to ; . This is because an analogous property holds for hyperbolic fixed points of diffeomorphisms as we saw in the last chapter. In what follows we shall show that, if y is a hyperbolic closed orbit of a vector field X e X'(M), then X is locally stable at y. That is, for each vector field Y belonging to a neighbourhood w° of X, there exists a homeomorphism It: V - V' where V is a neighbourhood of y, taking orbits of X to orbits of Y As we have already remarked, we cannot require the homeomorphism h to conjugate the flows X, and Y because this would imply. among other things, that the closed orbit -,-y c V' has the same period as ;'. It is clear that there exist vector fields Y arbitrarily close to X such that has period different
it is enough to take Y = (1 + n- `)X with n large enough. Let E he a section transversal to the field X through the point p e y. We sav that E is an invariant section if there exists a neighbourhood U C- Y. of p such that X,,,(U) c E where w is the period of The next lemma shows that we can reparametrize the vector field X in such a way as to make a given
from
:
transversal section invariant. The proof is quite technical although the result is intuit]Nely clear.
1.3 Lemma. Let X e £'(M) and let he a hyperbolic closed orbit of X with period w. Let E be a section transversal to X through a point p e y. Then there exi.,t s a, Ontinuous map; : Y
X'(M), where Y is a neighbourhood of X, such
that:
(a) µO) = t
, Y, where py.:.M .-+ F is a positive di/ferentiabie function that takes the value I outside a neighhourh,' d of 'o point of ; . .
(b) there exists a neighbourhood if c E of p such that
c E where
y* = p( Y) That is, E is an invariant section of Y* = P(Y)for all Y e y
PR(x)t. Let E' X with 0 < to < r,. We have X,,,(p) = p' c- E'. Let us consider the C map, z: Y. -+ lk, that associates to each y e Y. the least positive time z(}) for which a neighbourhood of p in E, then
E Y-'. Notice that if x(y) = to, for all Y in is invariant.
let L' C 2: be neighbourhoods of p with i' U. Using a bump fun.tion that takes the value I on U and 0 outside C. we define a C' map,
Figure 7
§1 The Poincare Map
97
P: E -+ R, that is equal to a on U and is constant at the value to outside V. With the same bump function we define for each vector field Y in a neighbourhood of X a Cr function, fly: E -+ R, which coincides with the function ay on U and is equal to to outside f3. Here ay: Y. -+ f8 is the function that associates to each y e E the least positive time av(y) for which YY,,(yl(y) E EY where EY =
Next we construct the desired reparametrization. Let G:,V x Y. x tR - R be a C' map satisfying the following condition: for each Y E P and each y E E, Gy, y(t) = G(Y, y, t) is a polynomial in t of degree 2r + 3 whose coefficients are determined by Gy,y(to) = NF(y);
Gy, y(0) = 0, dGY
d y., (to) ^ 1;
_
y
1,
dt
(0)
dt" y (0) = 0,
dk
Gr' T. (to) = 0,
We,have Gy,y(t) - t + a,ti+2 + __
a' I
:
+.a,+2t2'+3 where
flr(y)Alj - toA1j detA r+2 to
A= (r + 2)t 7' (r + 2)!to
k=2, ..,r+1.
J=1,
r+2,
tor+3
to2r+3
(r,+ 3)to+2
(2r + 3)to +2
,
(r + 3)! 2!
( 2 r+3 ) !+
to
(r+2)!
and A,; is the cofactor of the entry a,; = t'o+1+ 1 in A. It now follows that Hr(y, t) = dG1,y/dt(t) satisfies the following conditions:
(a) Hy(y, 0) = Hy(y, to) = I for y e E; (b) Hy(y, t) = I for y 0 f3; (t;) DkHy(y, 0) = DkHy(y, to) = 0 for all y e Y. and k = 1, ... , r. Condition (c) follows easily from the previous equalities.
Thus H I E x [0, to] extends to a C' map, H: I x R -. R, with H = 1 outside E x [0, to]. For small *and 0, the map GY, y: CO, to] -+ [0, )3y(y)] is a diffeomorphism M,
since Gy,y is the identity for y # 0. Also, the map tpy: I x [0, to]
defined by rpy(y, t) = Y,.(y) where t* = Gy,y(t), is a C' diffeomorphism. Let W be rpy(E x [0, to]) c M and let us define py: M 08 to be equal to 1 outside W and to be, on W, the composition of the maps d6,, q '_-' (y, t) F'--'
dGy,y
dt
(t) = Hr(y, t).
3 The Kupka-Smale Theorem
98
By the construction above pr is of class C. Let us consider the vector field Y* = Rr Y. We claim that Y,(y) e E; for any y e E. In fact, let y e U and let 1': [0, py(y)] M be the integral curve of Y through the point y. Thus, fi(t) = Y,(y) and >'(f,.(y)) e E,. Now let b*: [0, to] M be the map i/i* _ 0 ° G. We check that /i* is the integral curve of Y* through y. We
have *
do (t) = d (G'(t)) =
dGY_> dt
r
dt
(t)
(t) - Y(O ° Gr.r(t)) = Py(o*(t)) Y(O*(t)) = Y*W(t))
Thus. Y *(Y) = 0*(to) ='(fir(Y)) E Er We complete the proof by defining µ(Y) = Y*.
Remark. Let X e >`'(M), po e M and let E be a section transversal to X containing po. Let E' = X,,(E) and pl = X,,(po). From the proof of the lemma it follows that we can reparametrize all the vector fields near X so that they take the section E to the section E' in time tl. Such a reparametriza-
tion can be concentrated in a neighbourhood of a point p = X,(po) for
0 < t
a point p E y. Then p is a hyperbolic fixed point of the Poincare map P. For Y close to X the Poincare map Py is close to Px. Thus, by the local stability of a hyperbolic fixed point. there exist neighbourhoods Y` of X and U c E
of p with the following property: for each Y e Y'' we can find a homeomorphism hr. from U to a neighbourhood of p in E conjugating Px and Pr, that is h1l',K = P1hz. Let us extend hr to a neighbourhood of y. Let ,u: x
3:'(M) be the map obtained in Lemma 1.3. Let V be a neighbourhood
of y such that, if y e V, there exists 0 5 t 5 to with X; (y) e U where X* _ Ic(X ). Then we define hy(y) = Y*_,h1X *(y). It is easy to see that, if V is small
enough, hr is well defined, is a homeomorphism and conjugates the flows X* and Y. As X* and Y* have the same orbits as X and Y, respectively, we conclude that by takes orbits of X to orbits of Y. If y is a hyperbolic closed orbit of a vector field X e X'(M) we define the stable and unstable manifolds of y by W5(y) = {y e M; co(y) = Y},
W°(Y) = {y e M; a(Y) = Y}.
§2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic
99
There exists a neighbourhood V of y such that if X,(q) a V, for all t >- 0, then q e W'(y}. This follows from an analogous property of the. Poincare map of y. Let us consider the sets
W,, (y) = (y e V ; X,(y) e V for all t > 01 W"v(y) = {y e V; X,(y) e V for all t < 0}. We have W'(Y) = UnENX-nW'v(Y) and W"(y) = UhsNX.W"v(Y) 1.5 Proposition. Let y be a hyperbolic closed orbit of a vector field X e X'(M). If V is a small neighbourhood of y then W'v(y) and W i,(y) are C' submanifolds of M, W),(y) is transversal to My(y) and W;,( y) n Wj,(y) = y.
PROOF. Let E be a section transversal to X through a point p e y. If U is a small neighbourhood of p in E we denote by W'v(p) and W,(p) the stable and unstable manifolds, respectively, of the Poincare map Px. Asp is a hyperbolic fixed point of P. we see that W' (p) and W"v(p) are C'-submanifolds which are transversal to each other in E and W' (p.) n p}. Therefore, if co is the period of y then UsEto.2w X,(W'v(p)) and UtE(0,2W)X_,(Wuu(p)) are Cr submanfolds which intersect each other transversally along y. If V is a small neighbourhood of y then W.(y) and W;,(y) are open neighbourhoods of y in and in U,E(o, 2w) X _,(WU(p)) respectively. This proves the U:E(o, 2.) proposition. Corollary. W4(y) and W"(y) are immersed submanifolds of M of class C'.
We leave it to the reader to prove the next statement. Let y be a hyperbolic closed orbit of a vector field X e JE'(M). Show that there exists a neighbourhood r of X and, for each Y e Y', a neighbourhood Wye W'(yY) of yY such that the map Y F- WY is continuous. That is given e > 0 and Yo E Y,, there exists S > 0 such that, if 11Y - YouI < a, then W, is a C'-close to Wyo.
§2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic In the last chapter we showed that the set 11 c X'(M), consisting of vector fields whose singularities are hyperbolic, is open and dense in X'(M). In this section we shall show that the set 912 c- 41, of vector fields in W1 whose closed orbits are hyperbolic, is residual. For this, it is enough to show that, if T > 0 is any integer, then the set X(T) c 91 of those vector fields whose closed orbits of period 5 T are hyperbolic is open and dense. As q12 = nrZ 1X(T) it will then follow that 912 is residual.
100
3 The Kupka-Smale Theorem
2.1 Lemma. Let p e M be a hyperbolic singularity of X e X'(M). Given T > 0 there exist neighbourhoods U c M of p and It c X'(M) of X and a continuous map p: V -+ U such that (i) if Y e c! then p(Y) is the unique singularity of Y in U and it is hyperbolic; (ii) every closed orbit of a field Y E V that passes through U has period > T. PROOF. Part (i) was proved in the last chapter. By the Grobman-Hartman Theorem there exists a neighbourhood V of p such that there is no closed orbit of Y e 1'l entirely contained in V. As X (p) = 0 we can find a neighbourhood U of p contained in V such that, if q e U, then
X,(q) e V for .t S 2T Shrinking °h if necessary we shall have Y,(q) e V for t < T if q e U and Y E W. As Y does not have a closed orbit entirely contained in V. the lemma is proved.
2.2 Lemma. Let T > 0 and let y be a hyperbolic closed orbit of a vector field X e X'(M). Then there exist neighbourhoods U c M of y and W c: r(M) of X such that :
(i) each Y e V has a hyperbolic closed orbit y, c U and each closed orbit of Y distinct from yr that passes through U has period greater than T ; (ii) the orbit yr depends continuously on Y. PROOF. Take a transversal section E through a point p of y and let Px: V - E be the Poincarb map associated to X. Let r be the period of y and n a positive integer such that nr > 2T For sufficiently small V c T. we have Px defined on V. As the Poincar@ map depends continuously on the vector field there exists a neighbourhood W of X such that, if Y e W, My is defined on V. As p is a hyperbolic fixed point of Pa., there exists, for possibly smaller,& and V, a
continuous map p: 9! -. V that associates to each Y e °1! the unique fixed point p(Y) of P1 in V and this fixed point is hyperbolic. If yy is the orbit of Y
through p(Y) it follows that y is a hyperbolic closed orbit and yy clearly depends continuously on Y. By the Grobman-Hartman Theorem for diffeomorphisms and by the continuous dependence of the Poincare map on the vector field there exists a neighbourhood 17 c V of p and a neighbourhood ft of X such that, for all Y e W and all q E V, Py(q) e V for k = 1, ... , n. Therefore, for possibly smaller Rl, every closed orbit of Y e,&, other than y,, through a point q e V has period > T, because q is not a periodic point of Pr of period less than or equal ton. It is now sufficient to take U = U,. [o,T+LjX f" where t > 0 is small enough. Corollary. Let X e 512, that is, all the singularities and closed orbits of X are hyperbolic. Given T > 0 there are only a finite number of closed orbits of period < T. In particular, X has at most a countable number of closed orbits. PROOF. Suppose, if possible, that X has an infinite number of closed orbits of
period < 7' and let y. be a sequence of them with y # y,,, if n # n'. Take As M is compact we can suppose, by passing to a subsequence if p,, e
§2 Genericity of'Vector Fields Whose Closed Orbits Are Hyperbolic
101
necessary, that p" converges to some point p. Thus the orbit of p is in the closure of the set which consists of an infinite number of closed orbits of period < T. By Lemma 2.1, p cannot be a singularity and, by Lemma 2.2, the orbit of p cannot be closed. Thus, the orbit of p is regular. Set p' = X _(p) and p" = X,{ p). Let F be a flow box containing the arc p'p" of the orbit of p. If p" is close enough to p then X,(p") E F for t c- [ - 'IT, #T] so that the orbit of p" has period > T which is absurd. 2.3 Lemma. Let X E X'(M) and let K c M be a compact set such that X has no singularities in K and the closed orbits of X through points of K have period > T. Then there exists a neighbourhood V a X'(M) of X such that each Y E'Pl has no singularities in K and the closed orbits of Y through points of K have period > T
PROOF. As K is compact and X # 0 in K there exists a neighbourhood ..Y c X'(M) of X such that each vector field Y e .N' has no singularities in K. Take p e K. As the orbit of X through p is either regular or has period greater than T, it follows that there exists e > 0 and a neighbourhood U. of p such that, for q c- Up, we have X,(q) f Up for all t e [e, T+ e]. As the flow
depends continuously on the vector field, there exists a neighbourhood VP c .At of X such that the same property holds for all Y e'Plp. In particular, each closed orbit of Y E 9lp through a point of Up has period greater than T Let U,,,..., U p,, be a finite cover of the compact set K and let 'P1 = nk- t qlp,.
It is clear that each closed orbit of Y E V through a point of K has period greater than T Let X be a C°° vector field on M, y a closed orbit of X and E a transversal section through a point p e y. Let 'P c 3E'(M) be a neighbourhood of X and let V c Y. be a neighbourhood of p such that, for all Y e all, the Poincare map of Y is defined on V.
2.4 Lemma. In the above conditions there exists a neighbourhood U e M of y with the following property: given e > 0 there is a C°° vector field Y E 4l, with II Y - X 11, < E, such that P,, has only a finite number of fixed points in Y. e U and they are all elementary.
PROOF. Let (F, f) be a tubular flow with centre p such thatf -'({0} x I'"-') _
E n F and also f.X is the unit vector field Con [-b, b] x I". Let C be a C' vector field defined on f (F) c 11' such that C is transversal to {-b) x I' -' and {b} x I' -' and each orbit of C through a point of {-b} x I` meets {h} x I". Then we can define a map Le: {-b) x {b} x I"'which associates to each point of { -b} x I' -' the intersection of its orbit with {b} x 1"'-'. By the Tubular Flow Theorem Le is a diffeomorphism. Let
A = [-b, -1b] x 1'"-' v [4b b] x I'"-' v [-sb,+jb] x (I'"-t - I3/4')
102
3 The Kupka-Smale Theorem
!nt-1 '/4
I -b
r
-b
0
b
T
b
Figure 8
We claim that, givens > 0, there exists s, > 0 such that, for all v e I8' with lull < e, we can choose the C`° vector field C such that IIC - Cll, < s on
[-b, b] x 1", C = C on A and Le(-b, y) = (b, y + v) if y e I1,4'. In fact, let 0: [-b, b] -+ R+ be a C°° function such that 0(t) = 0 if t e [-b, --fb] u [Zb, b] and fi(t) > 0 if t e (--b, b). Take cp: I--' R+ such that (p(y) = 0 if Ilyll > a and cp(y) = 1 if IIyII < #. We define C(x, y) = (1, pcp(y)c/i(x)v) and look for a real number p for which C has the required properties. The differential equation associated with C can be written as
= W(Y)/(x)v. P
The solution of this equation with initial conditions x(0) = -b, y(0) = Yo e 11 /4 can be written as x(t) = t - b, y(t) = yo + (1' p(p(y(s))'(s - b) ds)v. Let us take 1/p = fo'i/i(s - b) ds. It is easy to see that, if jlvll is small enough, we shall have IIY(t)II < I for all t E [0, 2b]. Thus,
Yo + (400(s - b) ds)v. Thus L,-(-b, yo) = (b, yo + v), which proves our claim.
Let Y be the vector field on M which is equal to X outside f -'([-b, b] x 1"`-') and equal to (f -').C on f -'([-b, b] x I"'-'). It is clear that Y is C°° and II Y - X11, < s. We write E, for the transversal section
defined by f -'({b} x 1'-') and let V e E, be a small neighbourhood of p, = f -'(b, 0) on which the Poincare maps Px and Pt. are defined. The expression for Py in the local chart f l E, is Py(b, y) = Px(b, y) + v if
V c f -'({b} x 11,4) and y e 1114.
By Proposition 3.3 of Chapter 1 we can choose v e R", Ilvll < e,, such that the map y F- ()-, Py(y)) is transversal to the diagonal in O "-' x 118"'-' With this choice the fixed points of Py in V are all elementary which proves the lemma. 0
§2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic
103
2.5 Lemma. Let y be a closed orbit of a C°° vector field X. Given e > 0, there exists a C" vector field Y on M such that 11 Y - X11, < e and y is a hyperbolic closed orbit of Y.
PROOF. Let (F, f) be a tubular flow with centre a point of y such that f. X is
the unit field C on [-b, b] x I"'-'. We shall show that, given e > 0, there exists et > 0 such that, for any 0 < b < el, we can find a C°° vector field
Csuchthat IIC - CII, < eon[-b, b] x 1'-1,C = Con A and Le(-b, y) = (b, (1 + S)y) if y e I1J4t. Here we are using the same notation as in the proof of Lemma 2.4.
In fact, let 0: [-b, b] -- 68+ be a C' function such that fi(t) = 0 if t e [ -b, -1b] v [4b, b] and 0(t) > 0 if t e (-ib, sb). Take tp: I'-' R' such that cp(y) = 0 if IIYII >- a, ro(Y) = 1 if IIYII <- # and (p(y) > 0 if I < Ilyll < 1. We define C(t, y) _ (1, pcp(y)0(t)y) and look for p for which C satisfies the required conditions. It is immediate from the definition that C = C on A. The differential equation associated with C can be written as dx
dt dy
dt
_ 1'
= P.P(Y)4/(x)Y
Let yo a I'-' satisfy 11yoll 51. We have 4p(yo) = 1 and fp(y) = I in a neighbourhood of yo. The solution of the above equation with initial conditions x(0) = -b, y(O) = yo can be written as x(t) = t - b, y(t) = Yo + Jop(p(y(s))bi(s - b)y(s) ds. By the continuity of y(s), there exists 1 > 0 such that rp(y(s)) = 1 for all s e [0, -Zb + 1]. Hence y(t) = yo x exp(p Jo t'(s - b) ds) in [0, -lb + 1] as we can check by differentiating. Let
µ(t) = exp(p$o 0(s - b) ds). Then µ(0) = 1. Now, for 0 < p < e = log 2/ Job ri(s - b) ds we have u an increasing function, 0 < p(2b) < 2 and BY(s)II 5
211Yo11 <- #. Therefore, (p(y(s)) = 1 for all s e [0, 2b] so that y(s) = u(s)yo and
Le(-b, yo) = (b, p(2b)yo). It is easy to see that we can choose p to make
92b) = 1 + S where 0 < S < ei, so that Le(-b, yo) = (b, (1 + S)yo). Moreover, by taking p small enough we clearly get 11 C - CII, < e.
Let Y be the vector field on M which is equal to X outside f''([-b, b] x I'"-') and to (f -').C on f -'([--b, b] x 110-1). It is clear that Y is C°°, BY - X 11, < e and y is still a closed orbit of Y. We show that,
for b small enough, y is a hyperbolic closed orbit of Y. In fact, put 1 = f ''({b} x 1m'). The expression for the Poincarb map in the local chart
f11is Pr(b, y) _ (1 + b)Px(b, y)
if IIYII < J.
Thus, D(PY)(b,o) = (1 + S)D(Px)(b, o) For small b > 0 the eigenvalues of D(Pr)(b, 0) will have absolute value different from 1.
104
3 The Kupka-Smale Theorem
Remark. In Lemmas 2.4 and 2.5 we started from a vector field X of class C"°
and made the perturbation in the C' topology. The proofs of these lemmas cannot be made as above if X is of class C. This is because the tubular flow (F, f) is then of class C' and the vector field (f -'). t is only C'-'. However, the reader can prove that givens > 0 there exists a C' vector field 9 which is EC' near X and has a closed orbit y near y. We shall return to this question at the end of this chapter. 2.6 Theorem. The set 912, which consists of the vector fields whose critical elements (that is, singularities and closed orbits) are hyperbolic, is residual (and therefore dense) in '(M).
PROOF. Take T > 0 and consider the set X(T) _ {X E 9,; the closed orbits of X with period < T are hyperbolic}. We shall show that 3`(T) is open and 3£(n). dense in 3:'(M) and then T12 will be residual since it is equal to Part 1. X(T) is open in 3£'(M). Let X e 3:(T). By the corollary of Lemma 2.2, X has only a finite number of closed orbits of period < T. Choose p e M. We have three cases to consider:
(a) p is a singularity of X ; (b) 0(p) is regular or closed with period > T ; (c) (9(p) is closed with period
In case (a) there exist, by Lemma 2.1, neighbourhoods Up of p in M and XP of X in 3f'(M) such that every vector field Y E v has only one singularity p(Y) e Up, which is hyperbolic, and any closed orbit of Y that intersects Up has period > T. In case (b) there exists, by the Tubular Flow Theorem, a neighbourhood
Up of p in M such that any closed orbit of X that intersects Up has period > T and X has no singularities in Up. By Lemma 2.3, there exists a neigh-
bourhood v of X in 3v'(M) such that every vector field Y c r has no singularities in Up and the closed orbits of Y through points of Up have period > T. In case (c) there exist, by Lemma 2.2, neighbourhoods Up of 0(p) in M and AP of X in 3:'(M) such that any Y e X. has only one closed orbit in Up, yy, it is hyperbolic and all other closed orbits of Y that intersect Up have period > T. Moreover Y has no singularities in Up.
Now {Up; p e M) is an open cover of M. Choose a finite subcover U1, ... , U,, let .A , ... , A/1' be the corresponding neighbourhoods of X in £'(M) obtained in cases (a), (b) and (c) and put 9l = Ac o . n .. It is now easy to see that any vector field Y e all has its closed orbits of period 5 T near the corresponding closed orbits of X and they are still hyperbolic. Also Y has the same number of singularities as X (again still hyperbolic), which proves the first part of the theorem.
Part 2. X(T) is dense in £'(M). It is sufficient to prove that 3£(T) is dense in 91 so choose X e IJ1.
§2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic
105
Claim 1. There exists r > 0 such that every closed orbit of X has period > r. Suppose, if possible, that there exists a sequence of distinct closed orbits and that the sequence of their periods decreases and tends to 0. Take a sequence of points p,, e y,,. We can suppose, by passing to a subsequence if necessary, that p converges to some p E M. The point p must be a singularity of X for, if not, there would be a flow box containing p and the closed orbits intersecting this box could not have arbitrarily small period. Since X e 1,, p is a hyperbolic singularity. By Lemma 2.1, there exists a neighbourhood U of p in M such that every closed orbit of X that intersects U has period greater than one and this contradiction proves Claim 1. Now consider the set F = F(r, 3r/2) = (p e M; 0(p) is closed with period
t and r < t < 3r/2). Claim 2. I is compact.
It is enough to prove that r is closed. Let p be a sequence in r with p - p. As noted above p cannot be a singularity of X. If the orbit of p is regular or closed with period greater than 3r/2 then any closed orbit of X through a point near p has period greater than 3r/2, which is a contradiction. Thus, 0(p) is closed with period between T. and 3z/2 which proves the claim. Given e > 0, we want to find Y e 1(T) with IIX - Y11, < r to conclude the proof. First, we outline the construction of Y. Express T as nr + q where 0 < q < 21T.
Initially, we approximate X by a C' vector field X with
IIX - Xlir < E/2n. Next, we approximate X by a C°° field Y, E 3`(3x/2) such
that 119 - Y, IIr < e/2n. The next step is to approximate Y, by a C`° field Y2 e X(2T) with 11 Yi - Y211, < e/n. We carry out the process used in approxi-
mating Y, by Y2 n - I times and obtain C°° fields Y,, Y2, ... , Y with Y; e 1( -jr + r) and 11 Y;+t - Y;JI, < c/n. Then putting Y = Y we have Ye1(T)and 11Y-X11,
Let p,_., p, be the singularities of X. By Lemma 2.1 there exist neighbourhoods U,..... U, of p,, ... , p, and .i4 c T, of X in 3E(M) such that, for all Y e .4 , the closed orbits of Y that intersect U = U, u u U, have period greater than T Moreover, we can suppose that any Y E .N, has no singularities in M - U. From now on we restrict ourselves to fields in .1;. Let y be a closed orbit of 9 in F and Z. a transversal section through p e y. Let us consider neighbourhoods V;, c EY of p and .A' of k such that Py is defined on Vv for all Y e X, and the positive orbit of Y through a point of YY first intersects EY at a time t > 3r/4. We also consider a neighbourhood
W, of,y such that the positive orbit of Y through any point of W, intersects EY at least twice.
The open sets W. for closed orbits y c r cover the compact set 1. Let W,, ... , Wk be a finite subcover and let .N;,, ... , , ' be the corresponding neighbourhoods of X. Put A12 = ., n ... c .f Y4 and W = W, u u Wk.
Now consider the compact set K = M - (U u W). Since k has no singularities in K and every closed orbit of 9 through points of K has period greater than 3r/2, there exists, by Lemma 2.3, a neighbourhood c X2
3 The Kupka-Smale Theorem
106
of 2 such that the closed orbits of any Y e. through points of K have period greater than 3r/2. From now on we restrict attention to fields in .. For each j = 1, ... , k and for each Y C- X3 consider the Poincarb map Pjr.: Vj -+ Ej. By Lemma 2.4, we can approximate 9 by a C°° field P, such that P,f, only has elementary fixed points in W, n E,. As we saw in the last chapter, every field close enough to £, has the same property. Thus we can approximate ;£, by a field 72 such that P,;, and Peg, only have elementary fixed points in W, n E, and W2 n E2. By repeating this argument we obtain a field f x arbitrarily close to 9 for which the Poincar6 maps Pj;.,, only have elementary fixed points in Wj n Y j for j = i, ... , k. Let'p1...... , be the closed orbits of £ corresponding to the fixed points of these Poincare maps. The other closed orbits of Irk have period >3T/2. Moreover, there exists a neighbourhood .N'(4,,) such that each field Y E .K(?,) has its closed orbits of period < 3r/2 elementary and they are near µ ... , Jul. Using Lemma 2.5 repeatedly we approximate ? by a C°° field Y, whose closed orbits of length less than or equal to 3r/2 are the same as those of 7, and they are hyperbolic for Y1. Thus, we have Y, a X(3t/2).
As 1(3t/2) is open, there exists a neighbourhood .N4 of Y, contained in 3:(3x/2). Consider neighbourhoods U.,,,,._ Us+, of the closed orbits of Y, of period less than or equal to 3t/2 as in Lemma 2.2. Put U = U; ±i U;. Thus every closed orbit of a field near Y, through a point of U is hyperbolic or has period greater than 3r/2. Approximating Y, by Y2 e X(2r). From the compactness of F = r(3t/2, 2t) and Lemmas 2.4 and 2.5 as before
we obtain a neighbourhood W of I such that Y, can be approximated by a field i '2 whose closed orbits through points of W are either hyperbolic or have period greater than 2t. As the closed orbits of Y, in the compact set K = M - (U u W) have period greater _than 2r it follows that we can choose Y2 to be a C°° field in 3:(2r). Similarly we can obtain Y3, ..., Y. and this O completes the proof of density.
§3 Transversality of the Invariant Manifolds In this section we shall complete the proof of the Kupka-Smale Theorem. We say that a vector field X e 1'(M) is Kupka-Smale if it satisfies the following properties: (a) the critical elements of X (the singularities and closed orbits) are hyperbolic, that is, X e 112;
(b) if Q, and a2 are critical elements of X then the invariant manifolds W3(al) and Wu(a2) are transversal.
We write 9123 or K-S for the set of Kupka-Smale vector fields.
§3 Traoeversality of the Invariant Manifolds
107
3.1 Theorem (Kupka-Smale). K-S is residual in E'(M). We have already shown that T12 is residual in 3:'(M) so it only remains to
show that K-S is residual in 1t 2. We shall divide the proof of this fact into several lemmas.
To simplify the notation we make the convention that a singularity of X e YJ(M) is a critical element of period zero. Thus, if T > 0 and X e X(T) then X has only a finite number of critical elements of period 5 T and they are hyperbolic. Let X(T) be the set of=vector fields X E X(T) for which W'(at) is transversal to W"(a2) whenever at and a2 are critical elements of period f,- T.
3.2 Lemma. If X(T) is residual in 3:'(M) for all T Z 0 then K-S is residual. PROOF. As K-S = residual.
X(n) and each t(n) is residual it follows that K-S is
3.3 Lemma. Let E be a separable Baire space and F c E a dense subset. A subset U c E is residual if and only if each x e F has a neighbourhood Vx such
that U n V, is residual in V.
PROOF. Let J,,...,
be a countable cover of F such that U n VS, is residual in Vs, for all i. Then U n Vs, a nj t Uij where U,j is open and dense
in V. Let V = U = t V., and W j = Uij v (V - Vx). Then V and Wj are open and dense. It is easy to see that U contains n, t n t Wj. Hence U is residual. The reciprocal implication is trivial. Corollary. If for all T Z 0 and for all X E X(T), there exists a neighbourhood .N' of X such that X(T) is residual in .N' then K-S is residual.
PROOF. This follows from Lemmas 3.2 and 3.3 and the density of 3:(T) in 3:'(M).
Let X e X(T) and let at, ... , a, be the critical elements of X with period 5 T. For each i let us take compact neighbourhoods W,(a,) and Wo(ai) of a, in W'(ai) and W(a,), respectively, such that the boundaries of W,(ai) and Wo(a) are fundamental domains for W'(a) and W'(a,). Let E; be a codimension I submanifold of M that is transversal to the vector field X and to the local stable manifold of ai which it meets in the fundamental domain aWo(a1), see Figure 9. For each Y in a small enough neighbourhood .K of X, Y is transversal to each E; and the critical elements of Y of period less than or equal to T are hyperbolic and are near the corresponding critical elements of X. Thus, if ai(Y),... , a,(Y) are the critical elements of Y of period less than or equal to T then there exists a compact neighbourhood Wo(a,, Y) of a,( Y) in W'(a,, Y)
whose boundary is the intersection of Ei with Wo(ai, Y). By the Stable
,
3 The Kupka-Smale Theorem
108
rW fo(a;)
Figure 9
Manifold Theorem the map Y H W'(ai, Y) is continuous; that is, given Yo e.' and e > 0 there exists S > 0 such that if 11Y - Y0JI, < S then W'(rr;, Y) is E C'-close to Wo(ai, YO). Similarly, for each Y E .N and each i = I, ... , s, we construct a compact neighbourhood W"(a;, Y) of ai(Y) in W"(a;, Y) so that the map Y F-+ W"(aj, Y) is continuous.
For each positive integer n we define W.(aj, Y) = Y)) and WR(a;, Y) = Y"(Wo(aj, Y)). It is clear that the maps Y F-+ Y) and Y F-+ Y) are continuous since Y. and Y_ are diffeomorphisms that depend continuously on Y. Moreover, W'(aj, Y) and W'(ai, Y) are compact submanifolds with boundary, W'(aj, Y) = Wl(ai, Y) and W°(aj, Y) _ U.2 o WR(aj, Y). Let be the set of vector fields Y e A' such that W'(aj, Y) is transversal to W"(aj, Y) for all i and j. 3.4 Lemma. Let X e X(T) and let X be a neighbourhood of X as above.1f,for all n e N, 1,(T) is open and dense in ..Y then X(T) n A' is residual in X.
PROOF. If suffices to observe that X(T) n .N' = l. . , X(T).
0
3.5 Lemma. Let X e X(T) and let ,N' be a neighbourhood of X as above. Then, for all n e N, X"(T) is open and dense in .ti'.
PRooF. Let a ... , a, be the critical elements of X of period less than or equal to'T. We write .T,,, j, (T) for the set of vector fields Y e ..V' such that Y) is transversal to W.u(aj, Y). It is clear that n;,;=, I,,, ;_,{T). Thus it is enough to show that each T) is open and dense in X. Part 1. Openness of T. ,j{T). Let . e X,,,;,{T). As Y f-+ W.I(a;, Y) and Y F-4
and the maps Y) are continuous, it follows that there
9) is transversal to W.u(aj,
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109
exists a neighbourhood .N;; of $ in .K such that, for all Y e.Alj, W,(a;, Y) is transversal to W,"(a;, Y). Therefore .; c X ;,AT), which proves this set is open.
Part 2. Density of I,,,;,AT).
Let 2 e X(T) n .K We shall show that there exists a neighbourhood N of 2 in At such that X,,,;,AT) n J.' is open and dense in At In particular, 9 can be approximated arbitrarily well by elements of 1,,,1,XT). Let us consider the compact set K = W;(a,, 2) n WR(ap 2). If x c- K there exist a tubular flow (Fx,fx.x) containing x and a positive real number bx such that ff,x(Fx) [-bx, bx] x I'and the vector field (f2
coincides with the unit field on [-b5, bx] x I". Let Ax c F,, be an open neighbourhood of x such that the closure of A. is contained in the interior of
(ft,x)-'([-bx, bx] x 1By shrinking Fx if necessary we can suppose that Wzn(a,, 9) n Fx and 2) n F,, have only one connected component each. Let A 1, ... , Al be a finite cover of K by such open sets and let us write (Fk,ft.k) for the corresponding tubular flows. Then
(fX.k)-'([-bk, bk] x 11,41) contains At. As the maps Yt--kW°(0j, Y), YF--+ Wn(a;, Y) are continuous, there exists a neighbourhood .47 of 9 in At such that W'.(a,, Y) n Y) c Uk-1 Ak for all YeA7. By shrinking A7 if necessary we can even suppose that, for each Ye AP and k = 1, ..., 1, there exists a tubular flow (Fy,,,fr,k) for Y with Fr,k Ak,fr,k(Fy,k)
[-bk, bk] x I'` and such that the interior of (fr.k)-'([-bk, bk] x Ii', ) contains the closure of Ak. This is possible because the flow depends continuously on the vector field. Now let Xk be the set of those vector fields Y e 3'
such that W' (a,, Y) is transversal to W."(aj, Y) at all points of .4k. Clearly 1k is an open subset of 3 and it will be sufficient for our purposes to show
that Ik is dense in 3 since X",,,,{T) n f'= nk-1 Z. We shall now show that YEk is dense in 3 Let us take a C'° vector field Y e 3. We denote by S+(Y) the intersection of fr,k(W'(a,, Y) n Fy,k) with {bk} x' I"'-' and by U+(Y) the intersection of fr,k(W."(aj, Y) n Ff,k) with {bk} x I'"-'. It is easy to see that if S+(Y) is transversal to U+(Y) in (bk) x I'i4' then W.'(a,, Y) is transversal to W;(aj, Y) in Ak. See Figure 10.
Figure 10
110
3 The Kupka-Smale Theorem
By the same argument as in the proof of Lemma 2.4, given e > 0 and v E R` with Ilvll small enough, we find a C°° vector field V on M with II Y - VII, < e such that
(a) V = Y outsidefY,k([-bk, bk] x I"),
(b) Lk(-b,y)=(b,y+v)forallye11j4'. Here b = bk and LY is the map from {-b} x 1ei-1 to (b) x 1" which associates to each (-b, y) the point of intersection with {b} x /' of the orbit of (f,,k),, V through the point (-b, y). On the other hand, by Proposition 3.3 of Chapter 1, we can choose v small so that S+(Y) will be trans-
versal to U+(Y) + v. Then fY,k(S+(Y)) is the intersection of WZ"(a;, V) with the transversal section f Y ),({bk} -x I") and f Y,k(U+(Y) + v) is the x lira'). Thus, we intersection of WZ"(a;, Y) with the section conclude that these two submanifolds are transversal in f -'([-bk, bj x 1,J41). Consequently W'(ai, V) is transversal to W," (e-, V) on Ak. This shows
that Velk. Thus any C°° field in .A%' can be approximated by a field in Xk. As any field in 47 can be approximated by a C°° field it follows that 1k is dense in .N' which completes the proof of the lemma.
Theorem 3.1 is now an immediate consequence of Lemmas 3.2, 3.3, 3.4 and 3.5.
Remark. It is important to note that K-S is not open in X'(M). In fact, consider an irrational flow X, on the torus T2. The vector field X e K-S since it has no critical elements. However, X can be approximated by vector fields Y for which Y is a rational flow. All the orbits of Y are closed and nonhyperbolic. Thus Y K-S. Now we shall state the Kupka-Smale theorem for diffeomorphisms. The proof is similar to the case of vector fields and will be left as an exercise for the
reader. We shall, however, present two separate sketches of proofs. A diffeomorphism f e Diff'(M) is said to be Kupka-Smale if
(a) the periodic points off are hyperbolic, and (b) if p and q are periodic points of f then W'(p) is transversal to W"(q). We shall denote the set of Kupka-Smale diffeomorphisms by K-S too.
3.6 Theorem K-S is residual in Diff'(M). SKETCH OF PROOF. (1) If f e Diff'(M) and k e N we write fk: M - M X M for the map given by f k(p) = (p, f k(p)). If p is a periodic point of f of period k then fk(p) = (p, p) which belongs to the diagonal A c M x M. Such a point
p is elementary if and only if f' is transversal to A at p. Let n e N and let :a" be the set of diffeomorphisms f e Diff'(M) such that f k is transversal to A for k = 1, ... , n. Then V" is open and dense.
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111
(2) Let f e 2" and let p be a periodic point off of period less than or equal to n. It is easy to approximate f by g c -2n so that g = f outside a neighbourhood of p and p is a hyperbolic periodic point of g. Thus the set :)" a V of those diffeomorphisms whose periodic points of period k = 1, ... , n are all hyperbolic is open and dense.
(3) Let s" be the set of those diffeomorphisms in k whose stable and unstable manifolds of periodic points of period up to n are pairwise transversal. r is residual. As K-S = fl,, v it follows that K-S is residual. O Another proof of the Kupka-Smale Theorem for diffeomorphisms can be obtained from the corresponding theorem for vector fields. For this we shall need a construction that enables us to relate diffeomorphisms on a manifold M with vector fields on a manifold ft of dimension one higher. This construction is called the suspension of a diffeomorphism. Let X e X'(M) and let I c M be a compact submanifold of codimension 1. We say that l is a global transversal section for X if (a) X is transversal to !, and (b) the positive orbit of X through each point of T. returns to intersect again. WE is a global transversal section for X e X(ft) then the flow of X induces a diffeomorphism f : ! -- ! which associates to each point p" e I the point l(p3) where the positive orbit of p first intersects 1. The diffeomorphism f is called the Poincare map associated with l. It is easy to see that if X e X'(M) admits a global transversal section !
then the saturation of T. by the flow of X coincides with M, that is, U,Ea X,(E) = M. In particular, X has no singularities. Lastly, let us remark that the orbit structure of X,is determined by the orbit structure of the Poincare map land vice-versa. In effect, the following facts are immediate:
(i) p e ! is a periodic point off if and only if Ox(p) is closed; (ii) P e ! is a hyperbolic periodic point of f if and only if Ox(p7 is a hyperbolic
closed orbit; (iii) if pt and P2 are hyperbolic periodic points off then W'(pt) is transversal to W"(P2) if and only if W'(OX(Pj)) is transversal to W"(Ox(p"2)); (iv) q e cv(p) if and only if OX(q") c ca(OX(p)).
The next proposition shows that every diffeomorphism is the Poincare map associated with a global transversal section of some vector field. 3.7 Proposition. Let f e Diff'(M) where M is a compact manifold. Then there exist a manifold M, a vector field X e X'- t(M) admitting a global transversal section ! and a C' diffeomorphism h : M -+ Z that conjugates f and the Poincare
mapl: - E.
PaooF. Consider the following equivalence relation on M x R:
(p,s)-(q,t)c>s-t=nEZ and q= f"(p).
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3 The Kupka-Smale Theorem
Let M be the quotient space M x R/-r and n: M x R -+ M the natural projection. Let t c M denote the image of M x {O} by it. For each to a R the restriction of it to M x (to, to + 1) is a one-one correspondence between
M x (to, to + 1) and M - zt(M x (to)). Moreover, x(p, 1) = x(f(p), 0). On M we use the topology induced by it, that is, A c M is open if and only if
n-'(A) is open. We shall show that M has a natural differentiable manifold structure and that n is a C' local diffeomorphism. Let xi : U, - Uo c R m, i = 1, ... , s be local charts on M such that U7.' U1 = M. Then 0, = n(U, x (-4, 4)) and
(, = n(Ui x (J, -)) are open in 1N. We define x,: 0, -+ Uo x (-4, 4) and Y1: D,' -+ Uo x (, a) by z,(n(p, t)) = (x,(p), t) and Y,(n(p, t)) _ (xXp), t). Clearly z, and y, are homeomorphisms. We claim that {(x,, U,), i = 1, ... , s} is a Cr atlas on M. In fact, z,z j '(u, t) = (x,xi '(u), t), y,y; '(u, t) = (x,xi '(u), t)andz,yj '(u, t) = (x, fxj '(u), t - l)areC'diffeomorphisms, which proves the claim. In fact we can consider M as a C°° manifold since, by Theorem 0.19 of Chapter 1, there is a C'° manifold structure on M such that x, and y, are C diffeomorphisms. As z, o n o (x(-' x id) is the identity on Uo x (-I, #) and y, o n o (x;` x id) is the identity on Uo x (1, 1) it follows that it is a C' local diffeomorphism. Let 0/a1 be the unit vector field on M x IB whose orbits are the lines {p} x I8, p e M. Let X(n(p, t)) = dn(p, t) ((3/dt(p, t)). It is easy to see that X(n(p, t)) =
X(n(f(p), t - 1)). Thus X is a C'-' vector field on 1N. The field X is transversal to E and its orbit through the point p = n(p, t) is n({p} x R). Thus the positive orbit of X through a point p = n(p, 0) e T. returns to inter-
sect t for the first time again at the point q = n(p, 1) = n(f (p), 0). The Poincare map associated to E, f: E -- E is, therefore, defined by f (n(p, 0)) = n(f (p), 0). The map h: M E given by h(p) = n(p, 0) is a C' diffeomorphism and l o h = h o f which completes the proof. O Remark. By Proposition 3.7 the suspension of a C'diffeomorphism is a C'-' vector field. The above construction is modified in [80] to give a C' vector field as the suspension of a C' diffeomorphism but we shall not make use of this fact.
We shall now show the density of the Kupka-Smale diffeomorphisms using the method of suspension. Takefo E Diff'(M). First we approximate fo by a C°° diffeomorphism f. In order to approximate f by a Kupka-Smale diffeomorphism we consider the C°° vector field X on ft obtained by suspendingf and let h: M -+ E be the diffeomorphism which conjugates f with the Poincare map': E of X. We approximate X in the C' topology by a C°° Kupka-Smale vector field Y and write g": E -+t for the Poincare map of Y. As g is a Kupka-Smale diffeomorphism C' close to f it follows that g = h-' o j- h is a Kupka-Smale diffeomorphism C' close to f. 0
Exercises
113
ExPatclsEs
1. Let X, be the flow generated by a vector field X e 3:'(M). Let y be a hyperbolic closed orbit of X of period I Consider the diffeomorphism f = X,t and a point PE Y.
(a) Show that T,M is the direct sum of two subspaces H and Ho, each invariant for df, such that Ho has dimension 1 and contains the vector X(p).
(b) Show that if S c M is a submanifold whose tangent space at p is H and a: U e S -i S is the Poincar6 map associated toy then da(p) = df(p)IH. 2. Show that, if X is a Kupka-Smale vector field on S2 then the w-limit of any orbit is a critical element.
3. Show that the set of Kupka-Smale vector fields is open in 3:'(S`). 4. Show that the set of Kupka-Smale vector fields is open in X(S2). 5. Let y and i be closed orbits of vector fields X and 9, respectively. Show that if there exists a homeomorphism h from a neighbourhood of y to a neighbourhood of y' taking orbits of X to orbits of 9 and preserving the orientation of the orbits then the
Poincare maps associated to these closed orbits are conjugate. If h is a diffeomorphism then the Poincare maps are conjugate by a diffeomorphism. 6. Show that any Kupka-Smale vector field on a compact manifold of dimension two has a finite number of closed orbits. Hint. Any compact manifold of dimension two is diffeomorphic, for some n, to the sphere with n handles, if it is orientable, and to the projective plane or the Klein bottle with n handles, if it is nonorientable. The Klein bottle is diffeomorphic to the projective plane with one cross-cap. 7. Show that a structurally stable vector field on a compact manifold of dimension two is Kupka-Smale. 8. Sketch a vector field on S2 with infinitely many hyperbolic critical elements. 9. Let C c S2 be a circle. Consider the set 3:c a X'(S2) of vector fields that are tangent to C.
(a) Shr,w that arc n K-S is not dense in X. (b) Consider the set KSc c 3:c of vector fields with the following properties: (i) the singularities and closed orbits are hyperbolic; (ii) if y is an orbit such that x(y) and w(y) are saddles then y c C. Show that KSc is open and dense in X.
10. Let Grad'(M) a 3:'(M) be the set of gradient vector fields on M, that
is,
X e Grad'(M) if and only if there exist f e C"'(M) and a Riemannian metric g such that X = grad fin the metric W. Show that the set of Kupka-Smale vector fields is residual in Grad'(M). Hint. Show that, if X = grad f in the metric g, F is a flow box for X and Y is a vector field near X that coincides with X on M - F, then there exists a metric g" such that Y is the gradient off in the metric J. Also use Exercise 8 from Chapter 2.
11. (a) Show that, if X e 3r'(M) is a Kupka-Smale vector field which has a global transversal section, then the associated Poincare map is a Kupka-Smale diffeomorphism.
3 The Kupka-Smale Theorem
114
(b) Show that the suspension of a Kupka-Smale diffeomorphism is a Kupka-Smale vector field.
12. (a) Show that if f e Diff'(M), r Z 1, is structurally stable then its periodic orbits are hyperbolic. (b) Show that if f e Diff'(M) is structurally stable then f is a Kupka-Smale diffeomorphism. Notice that in [98], C. Robinson proved this result for f e Diff'(M),
rz2.
(c) Show that a structurally stable vector field'X a 3:'(M) is Kupka-Smale.
Remark. It is an open problem to show that a structurally stable vector field X e 3:'(M), r z 3, has its closed orbits hyperbolic. 13. Let X e 3r'(M) be a vector field with a closed orbit. Let X, be the flow generated by X. Show that the diffeomorphism X, is not structurally stable.
14. Show that on any compact manifold M", n Z 1, the set of Kupka-Smale diffeomorphisms is not open in Diff'(M), r Z 1. Hint. Show that any manifold has a Kupka-Smale vector field with closed orbits. 15. Show that on any compact manifold M", n 2t 3,.the set of Kupka-Smale vector fields is not open in X'(M), r z 1.
Chapter 4
Genericity and Stability of Morse-Smale Vector Fields
As we have emphasized before, the central objective of the Theory of Dynamical Systems is the description of the orbit structures of the vector fields on a differentiable manifold. There exist, however, fields with extremely complicated orbit structures as the example in Section 3 of Chapter 2 shows.
Thus the strategy this programme must adopt is to restrict the study to a subset of the space of vector fields. It is desirable that this subset should be open and dense (or as large as possible) and that its elements should be structurally stable with simple enough orbit structures for us to be able to classify them. As far as the local aspect is concerned this problem is completely solved as we saw in Chapter 2.
In this chapter, in Sections 1 and 2, we show that the global aspect of the above programme can be achieved on compact manifolds of dimension two. This result, due to Peixoto [81], [85], is one of thu, early landmarks in the recent development of the theory. Besides the previous fundamental work of Andronov-Pontryagin, followed by that of De Baggis, on the disc D2 or the sphere S2 (see [5], [47]), we also mention that Pliss [87] obtained the same result as Peixoto's for vector fields without singularities on the torus T2.
In higher dimensions the structurally stable fields are still plentiful, but they are not dense. There exist richer and more complicated phenomena that
persist under small perturbations of the original field. Even for the stable fields, the orbit structures of the limit sets are not always completely understood and their description is still an active area of research. These facts are discussed in Sections 3 and 4.
In this context we should emphasize again the importance of studying generic properties, that is, properties satisfied by almost all (a Baire subset of) vector fields. This was the case with the Kupka-Smale Theorem in the last chapter. 115
1 16
4 Genericity and Stability of Morse -Smale Vector F'elds
Finally we remark that the above programme can be posed for certain subsets of the space of vector fields of special interest. One of the most relevant examples is the gradient vector fields on a compact manifold. In this case the structurally stable vector fields form an open dense subset [79], [106]. We shall indicate in Section 3 some of the basic properties of their orbit structures. In Section 4 we present a collection of general results on structural stability. In particular, we exhibit structurally stable systems with infinitely many periodic orbits.
§1 Morse-Smale Vector Fields; Structural Stability Here we define a class of vector fields which play an important role in the Theory of Dynamical Systems. This class, called Morse--Smale systems,
forms a nonempty open subset and its elements are structurally stable. Although these results hold on compact manifolds of any dimension [75], [79], we shall only study in this chapter the case of dimension two, where the class is also dense.
We begin the section by defining Morse-Smale vector fields. Then we prove that a Morse-Smale field on M2 is structurally stable. The proof we give was introduced in [75], is different from the original one [85] and can be generalized to higher dimensions. Before presenting the definition of Morse-Smale vector fields formally we shall motivate it by means of some examples. As our aim is to find a class of structurally stable fields we must certainly require the singularities and closed orbits to be hyperbolic. Without this requirement the fields would not even be locally structurally stable. We also emphasize that the intersections of the stable and unstable manifolds of critical elements (singularities and closed orbits) have to be preserved by any topological equivalence. Therefore,
it is natural to require these intersections to k e transversal since this will guarantee that they persist under small perturbations of the vector field. EXAMPLE 1. Consider the torus T2 e R3 and let X = grad h where h is the height function of points of T2 above the horizontal plane in Figure 1. This vectorfield has four singularities pt, p2, p3, p4 where pt is a sink, P2 and p3 are saddles and p4 is a source. The stable manifold of p2 intersects the unstable manifold of p3 nontransversally. As in Chapter 1 we can destroy this intersection with a small perturbation of the field X. The resulting field Y will therefore not be equivalent to X. The discussion so far leads us to define Morse-Smale fields as a subset of the Kupka-Smale fields. At this stage it is fundamental to notice that saying a vector field is Kupka-Smale gives no information about the a- and cu-limit sets of a general orbit. As a topological equivalence between vector fields preserves the a- and cu-limit sets of corresponding orbits we ought to ina22ose some specific conditions on these limit sets.
§1 Morse Smale Vector Fields; Structural Stability
117
Pt
Pa
Figure I
EXAMPLE. 2. Consider the vector field X that induces an irrational flow on the
torus V. The a- and (,)-limit of any orbit of X is the whole torus T2. In particular X does not have singularities or closed orbits. Consequently X is a Kupka-Smale field. However X is not structurally stable because it can be approximated by it field Y that induces a rational flow (see Section 4 of Chapter 1). All the orbits of Y are closed so this is a radical change in the wand o)-limit sets.
Before defining the set of Morse-Smale fields we still need some new concepts and notation.
Let X e X'(M). Consider the sets L,(X) = (p e M. p e a(q) for some q e M) and L4,(X) = { p e M. p e w(q) for some q e M}. These sets are invariant by the flow generated by X and the orbit of any point is "born" in L, and "dies" in L,,,.
Figure 2
4 Genericity and Stability of More-Smale Vector Fields
1 18
P2
Figure 3
Definition. Let X e 3:'(M). We say that p E M is a wandering point for X if there exists a neighbourhood V of p and a number to > 0 such that X,(V) n V 0 for It I > to. Otherwise we say that p is nonwandering.
We write Q(X) for the set of nonwandering points of X. The following properties follow immediately from this definition: (a) S2(X) is compact and invariant by the flow X,;
(b) f4X) - LQ(X) u L.(X ). In particular, S)(X) contains the critical elements of X;
(c) if X, Y e X(M) and h: M - M is a topological equivalence between X and Y then h(fl(X)) = il(Y). The next example shows that, in general, 0 contains L,, u L,,, strictly. EXAMPLE 3. Consider a C' vector field X on S2 with two singularities pt and
p2 such that all the other orbits are closed, see Figure 3. We now multiply the field X by a nonnegative C°° function cp: S2 - ff that only vanishes at one point p where p is distinct from pt and p, Let Y = cpX. The field Y is C°°, has three singularities Pt, p2 and p and all other orbits are closed except for one orbit y with a(y) = co(y) = p. No point x of y belongs to L,, L L. but x e D since it is accumulated by closed orbits of Y. See Figure 4. Definition. Let M be a compact manifold of dimension n and let X E 3:'(M). We say that X is a Morse-Smale vector field if; P1
Figure 4
§1 Morse-Smale Vector Fields; Structural Stability
119
Ps
Figure 5
(1) X has a finite number of critical elements (singularities and closed orbits) all of which are hyperbolic;
(2) if a, and a2 are critical elements of X then WS(at) is transversal to W°(a2);
(3) SZ(X) is equal to the union of the critical elements of X. Next we give some examples of Morse-Smale fields. Examples 4, 5, 6 and 7 are on S2, and Examples 8 and 9 are on T2. A critical element is called an attractor or repellor if its index is the maximum possible or zero respectively. Otherwise it is called a saddle. EXAMPLE 4. Any vector field on S2 with the following characteristics is called
a north pole-south pole field : pn, ps are hyperbolic singularities; pN is an attractor; ps is a repellor;
if x e S2 - {p,,, ps) then W(x) = pN and a(x) = ps. EXAMPLE 5. pN, ps are hyperbolic repelling singularities, y is a hyperbolic
attracting closed orbit and if X e S2 - {PN, Ps} - y then w(x) = y and «(x) = PN or Ps
Figure 6
120
4 Gcnericity and Stability of Morse-Smale Vector Fields
r2
P2
Figure 7
t.e 6. p,, p2 are hyperbolic attractors, r,, r2 are hyperbolic repellors, s, s, are hyperbolic saddles and (2(X) _ {p,. P2, r,, r2, s,, s2}. F:xAMri.l: 7. r,, r2, r3 are hyperbolic repellors, s is a hyperbolic saddle, y,, 72
are hyperbolic attracting closed orbits and Q(X) = {r,, r2, r3, s} tJ y, U y2. The orbits oft his field on the cylinder bounded by y, and y2 areas in Figure 9.
::xAMPLh 8. p is a hyperbolic attractor, r is a hyperbolic repellor, s,, s2 are hyperbolic saddles and S2(X) = (p, r, s s2}. See Figure 10. 9. T, is an attracting hyperbolic closed orbit, y2 is a repelling hyperholic closed orbit and 12(X) = y, u y2. See Figure 11. ExAMPI i
We shall write M-S for the set of Morse-Smale fields. The following proposition gives a simpler characterization of Morse-Smale fields on twodimensional manifolds. A saddle-connection is an orbit whose a- and o--limits are saddles.
r2
Figure 8
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121
I
Figure 9
Figure 10
Figure 11
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4 Genericity and Stability of Morse-Smale Vector Fields
1.1 Proposition. Let M be a compact manifold of dimension two. A vector field
X e £(M) is Morse-Smale if and only if: (a) X has a finite number of critical elements, all hyperbolic; (b) there are no saddle-connections; and
(c) each orbit has a unique critical element as its x-limit and has a unique critical element as its w-limit.
PROOF. Clearly, if X e M-S then X satisfies conditions (a), (b) and (c) above. Let us show the converse. Take X C Tr(M) satisfying (a), (b) and (c). As the
stable manifold of a sink and the unstable manifold of a source are twodimensional, the transversality condition can only be broken by the stable and unstable manifolds of saddles. This does not happen because there are no saddle-connections. Thus, it is sufficient to prove that Q(X) consists of the critical elements. First let us show that the stable manifold of a sink consists of wandering points except for the sink itself. Suppose that the sink is a singularity p. As
we have already seen there exists a disc D c W'(p) containing p, whose boundary is a circle C transversal to X. As Ws(p) - {p} = U,EttX,(C) and the set of wandering points is invariant, it is enough to show that the points of C are wandering. Consider the discs D, = X, (D) contained in the interior of D and D_ 1 = X_ 1(D) whose interior contains D. Take x e C. Let V be a
neighbourhood of x disjoint from D, and M - D_,. Then X,(V) n V = 0 for It I > 2. This proves that x is wandering. Now suppose that the sink is a closed orbit y. In this case there also exists a neighbourhood U of y whose boundary S is transversal to X. If y is an orientable curve then U is homeomorphic to an annulus and S is the disjoint union of two circles. If y is not
orientable then U is homeomorphic to a Mobius band and S is a circle. Moreover, W3(y) - y = U,, R X,(S). By the same argument as before S, and
therefore W'(y) - y too, consists of wandering points. In an entirely. analogous manner we can show that W"(Q) - a consists of wandering points when o is a repelling critical element. Finally, if x e M is not a singularity and does not belong to a closed orbit then either rn(x) or x(x) is a repellor, since there are no saddle-connections. Thus, x is wandering, which completes the 0 proof.
Figure 12
§1 Morse-Smale Vector Fields; Structgral Stability
123
S
Figure 13
EXAMPLE 10. Polar fields on the pretzel.
We present some examples of polar Morse-Smale vector fields on the pretzel, that is, vector fields without closed orbits and with just one source and one sink. The pretzel (or the "torus with two holes" or sphere with two handles) can be represented [59] by an octagon with its edges identified in pairs in the following way: (1) two edges to be identified do not have a vertex in common; (2) the diffeomorphism that identifies the two edges reverses orientation; (3) the vertices are all identified to one point. In Figure 12 we represent the pretzel in two ways. Given a representation of the pretzel as an octagon we can construct a polar Morse-Smale field by putting a source at the centre of the octagon, a saddle at the mid-point of each edge and one sink at the vertices. In Figure 13
we sketch the Morse-Smale fields on the pretzel corresponding to its representations above. Conversely, let X be a polar Morse-Smale field on the pretzel. By cutting the pretzel along the unstable manifolds of the saddles we obtain a field like the one described above.
Similar constructions enable us to exhibit polar Morse-Smale fields on any compact manifold of dimension two. Definition. Given a Morse-Smale field X we define the phase diagram r of X to be the set of critical elements of X with the following partial order:
al, a2 E r, at < 0-2 if W°(at) n Ws(a2) o 0. That is, there exists an orbit which is born in al and dies in a2. F
The relation S is a partial order since there are no saddle connections, We may also remark that, as dim M = 2, the phase diagram of any MorseSmale field has, at most, three levels.
4 Genericity and Stability of Morse-Smale Vector Fields
124
Figure 14
EXAMPLES. The phase diagrams for Examples 4 to 9 of this section are shown in Figure 14.
Definition. Let X, Jk E M-S and let r, f' be their phase diagrams. We say that r' and r are isomorphic if there exists a one to one correspondence h: r -> r such that (a) x e V is a singularity if and only if h(x) e 1' is a singularity; (b) for x1, x2 e V, x1 5 x2 if and only if h(x1) 5 h(x2).
We are going to show that small perturbations of a Morse-Smale field give rise to Morse-Smale fields with isomorphic phase diagrams. For this we shall use the concept of a filtration associated to a Morse-Smale field. We observe that such a concept can be usefully applied to a more general kind of vector field [14], [109].
Definition. Let X E M-S. A filtration for X is a sequence MO = 0, c Mk = M of compact submanifolds Mi (with boundary M1 C M2 c for 0 < i < k) such that: (a) X is transversal to the boundary of MI and X,(M1) a Interior MI for
t > 0; (b) in M1+ 1 - M1, the maximal invariant set of the flow X, is just one critical element 01 + 1, that is, f,, R X,(Mj+ 1 - int MI) = al+ t.
1.2 Lemma. Let X e X'(M2) be a Morse-Smale field. Then there exists a filtration for X.
PROOF. Let al, 472i ..., aj be the attractors of X. Let us take disjoint neighbourhoods V1, V2, ... , Vj with boundaries transversal to X as in the proof of Proposition I.I. We define M 1 = V1, M2 = M1 v V2,..., Mj - Mj_ 1 v V1.
§1 Morse-Smale Vector Fields; Structural Stability
125
i
Figure 15
Let aj+ 1, ._a, a, be the saddles of X. Let us consider aj+ t and the components of W °(v j+ t) - a j+ t (whioh intersect aM j transversally). In a neighbourhood of a,. t we construct two sections, St and S2, transversal to W5(aj+t) - Uj+ t. If this neighbourhood is small enough then the trajectories of X through the end points of St and S2 cut aMj transversally. Near these arcs of trajectories we construct curves ct, c2, c3 and c4 transversal to X joining the end-points of St and S2 to aM j. We can construct these curves to be tangent to the sub-
manifolds S1, S2 and 3Mj, by using tubular flows containing these arcs of trajectories as indicated in Figure 16. Let Vj+ t be the region containing Qj+ t and bounded by S1, S21 cl, c2,
c3, c4 and part of aM j. Put Mj+ I = M; U Vj+ t- It is easy to check that n, . R X r(V1+ 1) = aj+ t and that Mj+ t satisfies the required conditions. We
repeat the construction for each saddle and thus obtain the sequence of submanifolds 0 = MO c M, c -- c Mj+ t c ... c M,. Finally, let 03 + t, ... , ak be the sources of X. We consider neighbourhoods V,,,, Vi+2, ... , Vk of these sources with their boundaries transversal to X, as in the case of the sinks. Then we define
M,+ I = M - (int V,+2 v . . . v int Vk),
M,+2 = M - (int V,+3v...vint Vk), 0Mj
S, Figure 16
126
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 17
and so on up to .Mk = Al. It is easy to check that 0 = Mo c M, c is a filtration for X.
c Mk
p
The next two theorems are true in higher dimensions [75], [79]. We have adapted their proofs to the much simpler two-dimensional case.
1.3 Theorem. Let X E M-S. Then there exists a neighbourhood & of X in X'(M2) such that if YE V then Ye M--S and its phase diagram is isomorphic to that of X. PROOF. As the critical elements of X are hyperbolic, for each a; a Q(X) there
exist neighbourhoods ,)11; c ?'(M2) of X and U; of a; such that any Y e ?l;
has a unique critical element a;(Y) c U;. Moreover, by the GrobmanHartman Theorem we can, by shrinking the neighbourhoods ill;, U; if necessary, suppose that a;(Y) is the unique set invariant under the flow Y
that is entirely contained in U. Put '"[l =null;. Now let us consider a filtration 0 = M0 c 1111 c M2 e c Mk T Al for X. We shall show
that, for W small enough, 0 = M0 c All c
c Mk = Al is also a
filtration for any Y e 1. This will imply that i2(Y) consists of the critical elements a;(Y) defined above. First, we remark that, as X is transversal to the compact set O AI,, so is any Y near X. Now shrink the neighbourhoods U, of until U; c Mi - M;_,. As nreR X,(.M; - int M;_,) = a;, there exists T > 0 such that nT _TX,(M; - int A1;_,) c U;. The same fact holds for Y e ill if 4l is small enough. As n,ER Y,,(U;) = a;(Y), it follows that
n..Y(M;-intM;-,)= e;(Y). Hence 0=M0
All
is a filtration for all Y e V. We claim that fl(Y) n (Mi -- int M;_,) = a,,{Y).
In fact, any orbit 7 distinct from a;(Y) through a point of M. - int M;_, must intersect dM1 or dM, - , (or both). This is because the only orbit entirely
127
§1 Morse-Smale Vector Fields; Structural Stability
Figure 18
contained in M; - int M;_ I is ai(Y). As Y,(Mi_ 1) c M;_ I for t > 0 and Y,(M - Mi) c M - M; for t < 0, y is a wandering orbit. Thus, a;(Y) is the only orbit of fl(Y) in Mi - int M;_ 1. Therefore, Q(Y) = Uja{Y) and ai(Y) is hyperbolic for each i. In order to conclude that Y c- 0& is a Morse-Smale field it is enough to show that, for small V, there are no saddle-connections. Thus, let aj = aj(X)
be a saddle and suppose that one component y of W"(aj) - a; has the sink a = a(X) as its w-limit. Let V be a neighbourhood of a as in the construction of the filtration. As compact parts of W"(ai(Y)) are near W"(ai), one com-
ponent of W"(aj(Y)) - ai(Y) also intersects aV transversally. Thus, its w-limit is a(Y). The same reasoning applies to all the components of stable and unstable manifolds of saddles. Thus, for small enough ', if Y e ,& then Y e M-S and the above correspondence a,{X) t- a,{Y) is an isomorphism of phase diagrams. This proves the theorem. 0
1.4 Theorem. If X e X'(M2) is a Morse-Smale field then X is structurally stable.
PROOF. By the previous theorem we know there exists a neighbourhood
* c X'(M2) of X such that if Y e,& then Y e M-S and there exists an isomorphism a,{X) i-+ ai(Y) of phase diagrams.
Part 1. Let us suppose initially that X has no closed orbits. Consider a sink a of X and the corresponding sink a(Y) with Y e V. Let V be a disc in
W'(a) containing a as before. That is, aV is transversal to X and to all Y e W. Also a(Y) e V c Ws(a(Y)). Let al, a2, ... be the saddles of X such V
Figure 19
128
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 20
that a, < a. Let pt, P2, . .. be the points at which the unstable separatrices of the saddles at (that is, the components of W"(a,) - a,) intersect c3V. Let p ,(Y), p2(Y), ... be the corresponding points for Y. For each a, let us consider
sections Si, 3, transversal to the stable separatrices of a; through points q,, q, as in Figure 20. Saturating Si, ', by the flow X, we obtain a tubular family for W"(a,) as in Section 7 of Chapter 2. The fibres of this family are X,(S,) and X,(3) for each t E 68 and also W"(a,). The projection n;, which associates the point f n W'(a,) to each fibre f, is continuous. Moreover nt is a homeomorphism from a neighbourhood I, of p, in al/ to a neighbourhood of a, in W'(a,). We make the same construction for the field Y. Now we begin to define the topological equivalence h between X and Y. We put h(a) = a(Y), h(a) = c,(Y), h(p) = p,(Y), h(q) = q,(Y) and h(4i) = 4;(Y). We extend h to W'((T,) by the equation hX,(q,) = Y,h(q;) = Y,q,(Y), U,(4,) = Y,q,(Y). Now we define h on 1i in the following way: for x e 1;, hx = [n,(Y)]- thn,x. In this way h has been defined on a finite number of disjoint intervals I. in 11V. Notice that if the neighbourhood ,W of X is small then It I I, is near the identity. Thus, we can extend h to the whole circle SV. We repeat the same construction for all the sinks. Finally, we define h on the whole of M2 by the equation hX,z =
Figure 21
129
§1 Morsc-Smale Vector Fields; Structural Stability
h
iri (x)
Figure 22
Y,hz. It is easy to see from the construction that h has an inverse h-', which can be defined exactly as h was by interchanging the roles of X and Y. Thus it remains to prove the continuity of h. This is obvious at the sinks and sources and on the stable manifolds of the sinks. We shall analyse the
case of the stable manifolds of the saddles. Take x e W'(ai) where a, is a saddle. Recall that h takes fibres of the tubular family for a, to fibres of the tubular family for a,(Y), that is, n,(Y)hz = hn,(X)z. Consider any sequence x -+ x. We want to show that hx5 - hx. By the above remark, the fibre through hx converges to the fibre through hx. That is, n,(Y)hx -' hx. It remains to prove that hx converges to W'(a,(Y)). For this we construct tubular families for W'(a,) and W'(a,(Y))., This is done by starting from segments Ti, 1,(Y) in aV and saturating them by the flows X, and Y. As h(l,) = 1,(Y), we see that h takes fibres of the tubular family of W'(ai) to fibres of the tubular family of W'(a,(Y)). Therefore, if n, and n,(Y) are the respective projections onto W"(ai) and W"(a,(Y)) then hit,(z) = iri(Y)h(z). As n, is continuous and x x E W'(ai) we see that irix ir,x = a,. Also, the restriction of h to W"(a,) is continuous, so that h(a) = a,(Y).
Figure 23
130
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 24
ai(Y). On the other hand, and therefore This shows that converges to the stable manifold of a,{ Y), hence h(x) This completes the proof in the case in which X has no closed orbit.
Part 2. Now suppose that X does have closed orbits. These closed orbits must be attractors or repellors because they are hyperbolic and dim M = 2. As we have already remarked, there exist fields Y arbitrarily close to X such that the flows X, and Y, are not conjugate. For this it is enough to alter the period of one of the closed orbits of X by a small perturbation. We shall avoid this difficulty by defining a conjugacy h between flows 9, and V, that are reparametrizations of X, and Y. As the orbits of X, and X, are the same and so are the orbits of Y, and 1 it follows that h will be an equivalence between tlae fields X and Y.
Using Lemma 1.3 of Chapter 3 we can suppose right from the beginning
that the closed orbits of X and Y all have the same period t and admit invariant transversal sections. To simplify the exposition we shall consider two subcases.
(2.a) Consider, first, the case in which all the closed orbits are attractors. We shall try to imitate the construction of the conjugacy made in the case
where there were no closed orbits. Around each attracting singularity ai, oi(Y) we consider a circle Ci transversal to X and Y. For each closed orbit
o,, a,(Y) we take an invariant transversal section E; and fundamental domains I;, I,(Y) in Ej for the associated Poincare maps. As before we construct unstable tubular families associated to the saddles ak, ak(Y) of X and Y: we take sections Sk and 3k transversal to W'(ok) and W'(at(Y)) and use the families X,(Sk), X,(3k) and Y(Sk), Y(3k). The homeomorphism we want to construct will have to take each fibre of the tubular family of Qk to a fibre of the tubular family of a ,(Y). Moreover, it will preserve the transversal circles C; and the transversal sections E. Thus, by defining a conjugacy between X I W(ok) and Y I W'(uk(Y)) for each saddle ak. we shall induce a homeomorphism h on a finite number of subintervals of C; and I . These subintervals contain the intersections of the unstable manifolds of the saddles
§1 Morse-Smale Vector Fields; Structural Stability
131
with C, and I. This homeomorphism is near the identity, if Y is near X, and can, therefore, be extended to all of C, and Ij. For each singularity at, we define h(Qt) = Qt(Y) and, for each closed orbit o1, we define h(E1 n aj)
E; n Q,(Y). Finally we extend h to the whole of M using the conjugacy equation h = Y hX _t as in the first part of the proof. It follows, then, that h is one to one and onto. The continuity of h at the singularities and the stable manifolds of the saddles can be checked as in the first part. The continuity of h at the closed orbits follows from the invariance of the sections E, as we saw in the local stability of hyperbolic closed orbits (Section 1 of Chapter 3). (2.b) Finally let us suppose that X has an attracting closed orbit and a repelling closed orbit. This case becomes entirely analogous to the previous one after a further reparametrization of the fields X and Y. This reparametrization is necessary to enable us to extend the homeomorphism constructed in (2.a) to the repelling closed orbits. For this let us take transversal invariant sections t, associated to the repelling closed orbits it and let 1, be the corresponding fundamental domains. Each 11 decomposes as a union of closed
subintervals whose end-points interior to 1, are the ifttersections of the stable manifolds of the saddles with int I. Notice that all the points in each of these open subintervals have the same attractor as w-limit, as in Figure 25. Let p e W'(r) n 11 be one of the end-points of one of these subintervals. Let us consider a small interval [a, b] c I, around p such that the orbit through every point of [a, b] intersects the transversal section S. Using Lemma 1.3 of Chapter 3, we make a reparametrization of X in such a way that all the points of [a, b] reach Sty at the same time 1. Let X also denote this reparametrized field. Let [a', b'] c (a, b) be an interval containing p. By a new
reparametrization we can ensure that all points of [X,(a), X,(a')] a St reach E1 at time 1. Similarly we can do the same for [X,(b), X,(b')] reaching C1. Thus X2[a, a'] c E; and X2[b, b'] a C,. We repeat this construction for
Figure 25
132
4 Genericity and Stability of Morse-Smak Vector Fields
the various saddles whose stable manifolds intersect I,. Finally we reparametrize X so that all the points in the complement in 11 of the union of the intervals [a, b] above reach the various sections C, and E; at the same time
t=2.
We also make the same reparametrizations for fields Y near X. The conjugacy h between X, and Y, is now constructed exactly as in (2.4) with the
extra requirement h(E, n a,) = E, o v,(Y).
0
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces In this section we show that M-S is dense in 1'(M2) for an orientable surface M2. We use the Kupka-Smale Theorem, which permits some simplification
of Peixoto's original proof [81], although we must remark that Peixoto's work came earlier and served as motivation for that theorem. At the end of the section we analyse the case where M2 is nonorientable and discuss the corresponding results for diffeomorphisms. We begin the section by proving the density theorem for the sphere S2, which is much simpler and yet illustrates the general case.
Definition. Let y be an orbit of X e 3<'(M). We say that y is recurrent if w(y) - y or a(y) y. A critical element of X is always recurrent. In this case we say that the recurrent orbit is trivial. Any orbit of the irrational flow on the torus is recurrent and nontrivial. By the Poincare-Bendixson Theorem every recurrent orbit of a vector field X e 3E'(S2) is trivial. This fact simplifies considerably the proof that M-S is dense in X'(S2). 2.1 Theorem. If X E X'(S2) is a Kupka-Smale vector field then X is a MorseSmale field.
PROOF. As X is Kupka-Smale it has a finite number of singularities, all hyperbolic. By the Poincare-Bendixson theorem the co- and a-limit of any orbit is a singularity or a closed orbit. This is because if the w-limit of an orbit y contains more than one singularity then these singularities must be saddles and w(y) must also contain a regular orbit joining these saddles. As X is Kupka-Smale it has no saddle-connections, which proves the above statement. The closed orbits are hyperbolic attractors or repellors, so it remains to prove that there is only a finite number of them. To get a contradiction suppose X has infinitely many closed orbits. Let x,, x2, ... , x,,, .. . be a sequence of points in distinct closed orbits. By taking a subsequence we can suppose that x converges to some x e S2. Clearly, w(x) is a saddle since there cannot be a closed orbit in the stable manifold of an attracting singu-
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces
133
larity or closed orbit. Similarly, a(x) is a saddle. Thus x is itself a saddle for
otherwise its orbit would be a saddle-connection. On the other hand, the a.-limits of the unstable separatrices of x are sinks because, again, there are no saddle-connections. This gives a contradiction because an orbit through a point close to x but not in its stable separatrices has one of these sinks as co-limit. Therefore, x cannot be accumulated by closed orbits, which proves the theorem.
0
As the set of Kupka-Smale fields is dense in 1'(M) we have the next corollary. Corollary. M-S is dense in Xr(S2).
b
We shall follow the same line of argument in proving that M-S is dense in X'(M2). However, the proof has to be more delicate because of the presence
of nontrivial recurrent orbits. The irrational flow on the torus provides the simplest example of nontrivial recurrence. We suggest that the reader tries to show that the vector field generating an irrational flow can be approximated by a Morse-Smale field. Let us consider some examples of vector fields exhibiting nontrivial recurrence on other two-dimensional manifolds. EXAMPLE 11. We shall give a vector field on the pretzel (or sphere with two
handles) that has nontrivial recurrence. The following construction can be generalized to give examples of vector fields with nontrivial recurrence on the sphere with r handles for r > 2. Let A: ff -+ 083 be the reflection in the
plane x3 = 0, that is, AX 1, x2, x3) = (x1, x2, -x3). Let us consider the torus T2 embedded in R3 in such a way that A(T2) = T2 and T2 n (x; x3 = 0) is the union of two circles.
Let X be the gradient field of the height function measured above the plane x3 = 0. Clearly, X is a symmetric field, that is, A* X = -X. The singularities of X are the source r, the sink a and the saddles st and s2 as in Figure 26. Consider a circle Ct c T2 orthogonal to X and bounding a disc
Figure 26
134
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 27
D, that contains the sink. Let C2 = AC, and then D2 = A(D1) is the disc bounded by C2. By the symmetry of the field. the orbit through any point p e C2 (not in a separadrix of a saddle) intersects C1 in the point q = A(p). Let h: C, C2 be the diffeomorphism defined by It = R, ,, A. where R. is an irrational rotation of ('2. On T2 - (D1 U D2) consider the equivalence relation that identifies C, and C2 according to h. Let M be the quotient mani-
fold and P: T2 - (D, u D2)
M the canonical projection. Let Y = P,X.
Clearly, M is diffeomorphic to the pretzel. Moreover, all the orbits of Y are dense except for the two saddles s 1 and s2 and the unstable separatrices of s,. In fact all the other orbits intersect P(C2) and so it suffices to prove that the intersection of each orbit with P(C2) is dense in P(C2). Let p e P(C2) and take q e C 2 such that P(q) = p. The orbit of X through the point q intersects C, in the point A(q) but this point is identified with U(q) = R,(q). Thus, the
positive orbit through p intersects P(C2) for the first time at the point PRQ(q). By induction we see that the positive orbit through p intersects P(C2) for the nth time at the point PR=(q). As a is irrational. $,R;(q); n e N} is
dense in C2. This shows that the positive orbit of p is dense in P(C2) and, therefore, it is dense in the pretzel. EXAMPLE 12. We shall now describe a Cam' vector field X on the pretzel with
the following properties:
(a) X is a Kupka-Smale field; (b) X has only two singularities s, and s2 which are saddles; (c) every regular orbit is dense in the pretzel. We shall see in Lemma 2.5 of this section that X can be approximated by a vector field that has a saddle connection. (This fact can be verified directly.) Consequently, the set of Kupka-Smale fields is not open on the pretzel.
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces
C,
IN 2
135
I
\/11
71\
C
Figure 28
Similar examples can be obtained on the sphere with k handles for any k z 2. We construct X starting from a Morse-Smale field Y on the torus which has one source, one sink and two saddles. We cut out a disc around the sink and a disc around the source and then identify the boundaries of these discs by a
convenient diffeomorphism, which is equivalent to glueing a handle onto V. Let ir: 682 - T2 denote the canonical projection. First we describe a field £ on 682 and then consider the field Y = n# P. In order that £ projects to a field on T 2 we shall make ?(x) = £(y) if the coordinates of x, y e 682 differ by integers. Therefore, it suffices to describe 1 on the square in 682 with vertices (0, 0), (1, 0), (1, 1) and (0, 1). The field f has a source at the point (#, 1), a sink at the origin and saddles at the points (0, #) and (1, 0). Let C, and C2 be circles of radius S < I around the sources and sinks, respectively, as in Figure 28. If a E (0, Jx) c C1 then the positive orbit through a meets C2 at a point (p1(a). Thus, we have a diffeomorphism rp,: (0, }n) c C, -+ (- n, -1n). Similarly, we define rp2 : (ir, n) -+ (-In, 0), 93: (n, 3n/2)..27r, - 3n/2) and q : (3ir/2, 2ir) -. (- 3ir/2, - rr). By constructing the field 27r,
symmetrically we have V,(a) - -a - in if i =. 1, 3 and cp,(a) = -a + it
ifi=2,4.
Let tp: C2 -+ C, be the diffeomorphism sp(a) = -a + e where -/n is irrational. Let D, and D2 be the ripen discs whose boundaries are the circles C, and C2, respectively. We obtain the pretzel T2 by using the diffeomorphism
-p to identify the circles C, and C2 which form the boundary of T2 (D, v D2). Let X be the field induced on T2 by the field Y via this identifica-
tion. That is, X = P,, Y where P: T' - (D, u D2) -+ T2 = T2 - (D, u D2)/ is the projection. See Figure 29. We shall show that every regular orbit of X is dense in the pretzel. In the following argument C, denotes the circle C, in the torus as well as the circle
136
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 29
P(C1) in the pretzel. As every regular orbit meets the circle C, it is enough to
show that the intersection of a regular orbit with C, is dense in C,. Let D = {i(7t/2); i = 0 1, 2, 3}. The positive orbit of X through a point a e C, - D
meets C, again, for the first time, at the point qi(a), where t': C, -+ C, is defined by
ka) =
+e+if aE[0,2)vIn, 2) n e-2, if aEn IT
[3
a 27)
Let @+(a) = {rGm(a); m >_ 0} be the positive fi-orbit of a and 9_(a) _
{m 5 0} its negative fi-orbit. If 0+(a) n D # 0 then a belongs to the stable manifold of a saddle point of X and if (9_(a) - a) n D 0 0 to one of the unstable manifolds. If 8+(a) n D = 0 and 8+(a) is dense in C, then the positive X-orbit of a is dense in T2. Similarly for the negative b/r- and X-orbits. We will show that all positive (and negative) i'-orbits are dense and so the same is true for the X-orbits. In particular, X exhibits nontrivial recurrence.
Let us show that the positive ir-orbits are dense in C,. First, notice that 0 is 1-1, onto. discontinuous only at a finite set D c C, and it preserves lengths of segments (Lebesgue measure), i.e. the image of an interval of length 1. is a finite union of intervals whose lengths add up to L. We also claim that 0 has no periodic points and each orbit of 0 intersects Din at most one point. In fact, let x E C, (resp. x e D) and let m 0 0 be an integer such
that t(i"(x) = x (resp. t'm(x) ED). But t(i'"(t) = x + me + nm7t/2 for some
integer 05nm53.Thus x+me +nmlt/2=x+2kit,kc-Z(orx+me + nm7t/2 = jr; 2; j e Z) which is a contradiction because a/rt is irrational. Next we show, following the ideas in [45], that if i/i is a mapping of the circle
137
Density of Morse-Striate Vector Fields on Orientable Surface,
with the above properties then each positive (negative) v/i-orbit is dense. This is an immediate consequence of the statements (1) and (2) proved below.
(1) IfF c C, is a finite union of closed intervals such that cr(F) = F then F= C,. Suppose, if possible, that F # C1 and let x e C, be a boundary point of F. Since Vi(F) = F and the restriction of 0 to C1 - D is a homeomorphism it follows that +y-'(x) is either a boundary point of F or an element of D. Hence there is a positive integer k such that 0-`(x) E D because F has finitely many boundary points and 0 has no periodic points. Similarly either '4r(x) is a boundary point of F or x e D. Thus there is a nonnegative integer j such that O'(x) e D. As a consequence we get ,/,k+i(y) e D, where y = /-"(x). This contradicts one of the properties of i/i listed above. We conclude that F = C1. C1 is a closed interval then there exists a positive integer n such (2)l f I that U°=o '(1) = Ct.1n particular, 0+(x) is dense for every x e C1. The proof that. 0_(x) is dense is entirely similar. Let B = (d1) v D, where Of is the boundary of 1. For each x e B we set AX)
-
if ,"(x) # 1 otherwise. )inf{n > 0; *'(x) ,E I - al }, + 00,
a1
-
for all n > 0,
We can write I = U;=, Ij, where the Ijs are closed intervals with pairwise disjoint interiors and x e B, /3(x) < o). )13I = (a 1Tu {'4r(x); f
For each j, let n; = inf{n >,O; t0-"I; n I # 01. We claim that n; is finite. Suppose not. Then /-m1j n 0-"I,,; 0 for all 0 5 n < m because, if not, n I, 0 0 and since I; e I we would have n; finite. But 4k-"I,, n >- 0, cannot be all disjoint since the length of each of these sets (finite union of intervals) is the same as that of I; I. We conclude that nj must be finite. From the first part of this argument we also conclude that'Ij,
are pairwise disjoint. Finally, we claim that i-"JI; c 1. Otherwise, since
-",1i n 10 0, there is a point x E dl in the interior of 0J"4;. From the definition of n;, we have that '4"(x) #I, for 0 c i < n,, and '"J(x) is in the interior of I, c I. Since x e B, this implies that nj = /3(x) which is a contradiction because /i-(')(x) would be a boundary-point of some I", 1 5 k 5 n, and thus it could not belong to the interior of I. Now we set F =
U:=, Uk' o'0-kl;. Clearly +(.-'F c'F. Since fir-' preserves length of intervals it follows that c-1F = F. By (1) we have that F = C1 proving (2). We suggest that the reader show this vector field X can be approximated by Morse-Smale fields. EXAMPLE 13. .We are now going to describe very briefly an example due to Cherry of a C°° (or even analytic) vector field on the torus T2 that has non-
trivial recurrence and also a source. The construction of the vector field is quite complicated and. will be made. in the Appendix at the end of this
138
4 CAwricity and Stability of Morse-Smale Vector Fields
Figure 30
chapter. Let us represent thtetorus by a square in the plane with its opposite sides identified. The Cherry field X has one source f and one saddle s. The
unstable separatrices 1, and y2 of the saddle are c-recurrent. In fact the ar?imit of yt contains yl and y2. Moreover, X has no periodic orbit. Therefore X is a Kupka-Smale field. As we shall see at the end of this section, X can be approximated by a field that has a saddle-connection. Figure 30 shows the Cherry field on T2.
Let us consider a circle C transversal to X and bounding a disc D that contains the source f. Let M beg compact two-dimensional manifold and Y a vector field on M that has a hyperbolic attraetor. Let B be a disc containing a sink of Y with its boundary C transversal to Y. By glueing T2 - D into M - B by means of a, diffeomorphism h: C -i C we obtain a manifold M. The vector field 2 induced on M by X, Y and this identification has a nontrivial recurrent orbit. In this way we can construct vector fields with nontrivial recurrence on any two-dimensional manifold except the sphere, the projective plane and the Klein bottle where all recurrence is trivial. This is true in the sphere and projective plane by the Poincare-Bendixson Theorem and in the Klein bottle by [56]. Definition. Let X e 3:'(M). We say that K c M is a minimal set for X if K is closed, nonempty and invariant by X, and there does not exist a proper subset of K with these properties. If K is a critical element of X we say that K is a trivial minimal set.
We remark that if K is minimal and y is an orbit contained in K then y is recurrent. This is because co(y) is closed, nonempty and invariant by X, and co(y) c K. Thus a(y) = K y. Similarly, a(y) Y.
2.2 Lemma. Let F c M be closed, nonempty and invariant by X, where X e 1"(M). Then there exists a minimal set K c F. PROOF. Let .F be the set of closed subsets of F that are invariant by X, and let us consider in F the following partial order: if A, B e .F then A < B if
1A
§2 Density of Morse-Smn a Vector Fields on Orientable Surfaces
A c I Now let {A,} be a totally ordered family in F. By the BolzanoWeierstrass Theorem n, A, is nonempty. As n, A, is closed and invariant by X,, it belongs to .F and is thus a lower bound for {A,}. By Zorn's Lemma [46] there exists a minimal element in X. O We must mention the following important facts about minimal sets even though they will not be used in the text, except in the Appendix where a complete description of Cherry's flow is presented. In [16] Denjoy exhibited a C' vector field on the torus T2 with a nontrivial minimal set distinct from T2. On the other hand, Denjoy [16] and Schwartz [100] showed that a minimal set of a C2 field on M2 is either trivial or is the whole of M2 and, in
this case, M2 is the torus. Consequently the w-limit of an orbit will either contain singularities or be a closed orbit or be T2. We refer the reader to [31] for a converse of the Denjoy-Schwartz Theorem.
Definition. A graph for X E 3r'(M2) is a connected closed subset of M consisting of saddles and separatrices such that: (1) the w-limit and at-limit of each separatrix of the graph are saddles; (2) each saddle in the graph has at least one stable and one unstable separatrix in the graph. EXAMPLES. Figures 31 and 32 give four examples of graphs for vector fields on a chart of M2.
2.3 Proposition. Let X e 3r'(M2) be a vector field whose singularities are all hyperbolic. If X only possesses trivial recurrent orbits then the w-limit of any orbit is a critical element or a graph. Similarly for the a-limit.
OF
110
Figure 31
qa4
3
140
4 Gencricity and Stability
Morse- Smale Vector Fields
l-igu e 32
PROOF. Pick any trajectory ; of X and suppo,k- that is not a critical element. It is clear that w(;) cannot contain an attracting singularity or a closed orbit for then it would reduce to one of these elernents. On the other hand, w(y) must contain a saddle. This is because a minimal set in w(y) must be a critical element and we have already dealt with the possibility of an attracting singularity or a closed orbit. Thus, co(y) contains saddles and, as it is not just one singularity, it contains separatrices of saddles. Clearly, the number of separatrices in co(,,,) is finite Let us first suppose that each separatrix in w(y) has a unique saddle as its w-limit. We shall show that w(y) is a graph. We claim that there exists a graph contained in e l y). In fact, let a, E w(y) be a saddle and y; an unstable separaLet a2 = w(y ,) and let 12 be an unstable separatrix trix of a, contained in of a2 contained in (:)(; ) and so on. As we only have a finite number of separatrices this process will give a sequence ;;, y;t t, ... , 77 = of separatrices in w(y) such that x(y;- ) and this defines a graph. Let G be a maximal graph in w(y), that is, there does not exist any graph in
w(y). containing G W e claim that w(;) = G. If not, there exists a saddle Q, e G and an unstable separatrix of a, not belonging to G. Consider the
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces
141
saddle a2 = w(y,) and a separatrix of a2i 2 c a(y). By continuing this such that 6,k = 5, for some argument we obtain a sequence j < k or a, a G. Either way we obtain a graph t; in w(y) properly containing G, namely, 6 is the union of G with the saddles a,_., d. and the separatrices Yt . ,yk- t This is a contradiction since G is maximal. Thus w(y) = G as claimed.
Now let us suppose that there exists a separatrix y, c w(y) whose w-limit
is not just one saddle. Then w(y,) c (o(y) and w(y,) does not contain y, because all recurrence is trivial. If w(y,) only contains separatrices that have a unique saddle as co-limit then, by the previous argument, w(y,) is a graph. This graph will also be the w-limit of y, which is absurd since w(y) y,. y,whose (o-limit is not a unique singularity. Thus, there must exist y2 c co()
Moreover, w02) c w(y,) c w(y). By continuing this argument we shall necessarily find a separatrix y c such that w(y) y since the number of separatrices is finite. This is absurd since there is no nontrivial recurrence.
Corollary. If X E X"(M2) is a Kupka-Smale field with all recurrent orbits trivial then X is a Morse-Smale field. PROOF. By the last proposition, the w-limit of any orbit is a critical element
(and so is its a-limit) because there cannot be any graphs as there are no saddle-connections. It remains to show that there only exist a finite number of closed orbits. This can be proved by the argument for the case M = S2 in Theorem 2.1. We now move on to the proof that any X e X'(M2) can be approximated by a Morse- Smale field, provided M is orientable. For this we shall exhibit a field Y near X with the following property: there exists a neighbourhood ?! c X'(M2) of Y such that any Z e l& only has trivial recurrence. Then we approximate Y by a Kupka-Smale field Z e V. By the corollary above Z is Morse- Smale. We shall use the next two lemmas. Their proofs will be given at the end of this section. 2.4 Lemma. If X E X'(MZ) is a vector _field without singularities then X can be
approximated hr afield Y that contains a closed orbit. 2.5 Lemma. Suppose M2 is orientahle. if X c- X'(M2) has .singularities all of which are hyperbolic and there is a nontrivial recurrent orbit then X can be approximated by afield Y that has one more saddle-conrection than X has.
2.6 Theoltem. The set of Morse--Sntale -fields is dense in X'(M2) for M2 orientable.
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4 Genericity and Stability of Morse-Smale Vector Fields
PROOF. Let X E X'(M2). By perturbing X, if necessary, we can suppose that it has only hyperbolic singularities.
Case 1. X hds no singularities. As M2 is orientable it must be the torus T2. By Lemma 2.4, X can be approximated by a field X, that has a closed orbit. By Lemma 2.5 of Chapter 3 we can approximate X, by Y that has a hyperbolic closed orbit y. As Y has no singularities, y does not bound a disc in T2 and, therefore, T2 - y is a cylinder. Thus Y has only trivial recurrence. Each field near Y enjoys the same property since it also has a closed orbit. We now take a Kupka-Smale field Z near Y. By the corollary of Proposition 2.3, Z is a Morse-Sntale field. Case 2. X has singularities, all of which are hyperbolic. Let y be an unstable separatrix of a saddle. We say that y is stabilized if ow(y) is a hyperbolic attractor (singularity or closed orbit). Similarly, a stable separatrix of a saddle is stabilized if its a-limit is a hyperbolic repellor.
(1) If X has all the separatrices of its saddles stabilized then X can be approximated by a Morse-Smale field. In fact, there exists a neighbourhood 9l of X such that any Y is-'L also has
all its separatrices stabilized. Thus, by Lemma 2.5, these fields have only trivial recurrence. Now approximate X by a Kupka-Smale field Y E V. By the corollary of Proposition 2.3, Y is Morse'-Smale. (2) If X has separatrices that are not stabilized then X can be approximated by a field Y that has one, more stabilized separatrix than X. Proving this claim will complete the theorem since there are only a finite number of separatrices and by stabilizing them one by one we arrive at (1). Let I be a neighbourhood of X such that each Y e & has at least as many
stabilized separatrices as X. By Lemma 2.5, we can approximate X by e cW which has only trivial recurrence because thereare only a finite number of iaddles that can be connected. There are four possibilities to consider: (a) Y has no saddle-connections. Let y be a separatrix of a saddle for Y tha. is not stabilized. (Y already satisfies (1) if there is no such y.) Then w(y) (or a(y)) is a nonhyperbolic closed orbit. We can approximate Y by Z E V to make this orbit hyperbolic and thus stabilize the separatrix of Z corresponding to y. (b) Y has a graph that is the co-limit (or a-limit) of some orbit. Consider a transversal section S through a regular point p of the graph. As there is a trajectory y whose co-limit (or a-limit) contains p, y must intersect S in a sequence a -+ p. For n large enough, the arc of the trajectory y between a and the segment (a,,, t) of S and the graph bound an open region
A e M2 that is homeomorphic to an annulus. Let F be asmall flow box containing p and let AY be a C'-small vector field on F that is transversal to Y at all points of the interior of F and vanishes outside F as in Figure 33. If
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces
143
Figure 33
Z = Y + AY then Z,(A) c A for t >_ 0 in the case p e w(y). The separatrix of a, (or a2 in the second case), which is initially contained in the graph, penetrates the annulus after the. perturbation. Thus the w-limit (or x-limit) of this separatrix becomes a certain closed orbit oT the field Z that forms in the annulus. This comes from the Poincare-Bendixsoh.Theorejn since Z has no singularities in A. Now by a further perturbation of the field Z we make this closed orbit hyperbolic and, in this way, stabilize another separatrix. (If p e x(y) we consider Z, for t < 0.) (c) Y has a graph that is accumulated by closed orbits, as in Figure 34. The annulus A bounded by the graph and a closed orbit close enough to it makes (c) entirely analogous to case (b). (d) The last possibility will now be analysed. Let y be a saddle-connection and S a transversal section through a point p e Y. as in Figure 35. Let us consider a small open interval (a, p) c S. All the points of (a. p) have the same w-limit which is an attracting singularity or a closed orbit. We remark that if this does not happen for sufficiently small (u, p) then we must have the situation} described in (b) or (c). In fact, if Y does not Lome
144
4 Gencricity and Stability of Morse-Smale Vector Fields
Figure 34
under case (b) then the @-limit of each point of (a, p) is a singularity or a closed orbit. Moreover, there is no stable separatrix of a saddle, except y, that accumulates onp since the a-limit of such a separatrix would contain y and we should be in case (b). Thus the a.--limit of each point of (a, p) is an attracting singularity or a closed orbit. As p is not accumulated by closed orbits, which corresponds to (c), it follows that all points in (a, p) have the same w-limit, which we shall denote by a. .
Proceeding as before,we perturb the field so that the o)-limit of the unstable separatrix of o, becomes o. If a is an Attracting singularity we have stabilized one more separatrix. If a is a closed orbit we makeit hyperbolic by a further
perturbation, if necessary, and again obtain one more stabilized separatrix. This concludes the proof of the theorem. G Before proving Lemmas 2.4 and 2.5 we shall make some comments on nontrivial recurrent orbits.. Let X E x'(141=) and let } be a nontrivial cu-recurrent orbit. We claim that there exists a circle transversal to X through any point p e y. In fact let us consider a flow box F, containing p. Let ab and cd denote the
Figure 35
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces
145
Figure 36
sides of Ft transversal to X: Asp a w(y), y intersects ab infinitely many times.
Let pt'be the first occasion that y returns to intersect ab. Let us take a flgw box F2 containing the arc of the trajectory qo pt. We shall suppose that pt is below po in ab, see Figure 36. The construction is similar in the other case. If M2 is orientable we can find in F2 an arc of a trajectory of X starting from a point qt of cd above qo and intersectin&ab at p2 above pt. In F2 we can take an arc g1P3, with p3 above p2 but below po, that is transversal to X and has positive slope at the ends as in Figure 37. Now we complete the required circle by joining p3 and qi by An arc in F1'that contains p and is transversal to X. This arc must have the same slope as the previous one at p3 and q1. We leave it to the reader to construct a transversal circle as above in the case where M2 is not orientable. In this case it is necessary to consider consecutive intersections of y with ab. We now denote by C a circle transversal to X through p e y. Let D c C be the subset of those points whose positive trajectories return to intersect C again. We define the Poincar6 map P: D - C to be the map that associates to each x e D the first point in which its positive trajectory intersects C. By the Tubular Flow Theorem, D is open. Thus, D = C or else D is a union of open intervals. Let us suppose that D 0 C apd let (at, a2) be a maximal interval in D. We show that w(al) is a saddle and so is (0(a2). If w(al) is a singularity it must be a saddle since a sink would also attract orbits near that of at whereas every orbit beginning in (at, a2) returns to intersect C. Therefore it will suffice to show that w(af) does not contain regular points. Let us suppose the contrary and let x be a regular point in w(al) and S a section transversal to X through x. The positive orbit of at intersects S infinitely many times since x e w(at). On the other hand, if q e (at, a2) the
Figure 37
146
4 Genericity and Stability of Morse-Smale Vector Fields
number of times N that the piece of trajectory qP(q) meets S is finite, since S is transversal to X and qP(q) is compact. By the Tubular Flow Theorem, this
number is constant on a neighbourhood of q and as (a a2) is connected it is constant on the whole of that interval. By applying the Tubular Flow' Theorem to an arc of the positive orbit of a, that cuts S some number of times
n > N we find points in (a,, a2) whose positive orbit intersects S at least n
times before returning to C. This is a contradiction, which proves that w(a,) is a saddle. Similarly w(a2) is a saddle. Thus D is a finite union of open intervals in C whose end-points belong to
the stable separatrices of saddles. If we consider the inverse P- t of P (the Poincare map for - X) its domain is a finite union of open intervals whose end-points belong to the unstable separatrices of saddles. We' should remark that if the Poincare-map is defined on the whole circle C then M2 is the torus T 2 or the Klein bottle K2. In fact, it is easy to see that the set saturated by C, U, E R X,(C), is open and closed and thus coincides with
M2. Therefore, the vector field X has no singularities, which proves our states tent, and we have M2 = T2 or M2 = K2 depending on whether P preserves or reverses the orientation of C. If P: C -+ C reverses orientation then P has a fixed point. This fixed point corresponds to a closed orbit y of X.
This closed orbit does not bound a disc in K2 and so K2 - y is a Mobius band, which shows that X has only trivial recurrent orbits. PR(x)F of LEMMA 2.4. If X has a closed orbit there is nothing to prove. If X has no closed orbit then it has a recurrent orbit y and M2 = T2. Take p e_y and let C be a transversal circle through p. Put C, = X -,,(C) and C2 = Xh(C), where S > 0 is small. Clearly, C, and C2 are transversal to X. Consider a flow box F containing p whose sides transversal to X lie in C, and C2. We define P: C2 C1 by associating to each point x of C2 the first point where its positive trajectory intersects C1. As we have seen P is well defined and preserves orientation. By this we mean that, given an orientation of C2, the maps P and X_ 2h: C2 --+ C, induce the same orientation on C,. Now let
Figure 38
§2 Density of Morse-Smale Vector Fields on Orientable Surfaces
147
pi = P'(qo) = P(qi-1) where qo = Xa(p). There exists a sequence n1 such that p,,, -i po and we can suppose that each p,,, lies below po on ab, see Figure 38. .
Let us consider the family of fields Z(u) = X + suY, where s > 0,
0 5 u < 1 and Y is a field that is transversal to X on the interior of F, points upwards and vanishes outside F. If a is small then Z(u) is near X for all
05u<1.
Now fix a closed interval I in ab with po in its interior. For each point x E I we consider the vertical distance in F between x e I and the point where the positive orbit of Z(1) through x intersects C2. As I is compact tuis distance
has a minimum p > 0. We claim that, for some 0 < u < 1, the orbit of Z(u) through q0 is closed. In fact, as Z(u) = X outside F for any u, we can define pk(u) = P(qk_ 1(u)) for k > 1, where qo(u) = qo and qk_ 1(u) is the point where the positive orbit of pk _ 1(u) E C1 meets C2 for the first time. Let us fix a number i such that pi = P'(qo) is below po and its distance from po is less then p. We should remark that pi(u) and qi(u) depend continuously on u. For small u, pi(u) is in I and below po and its height increases with u. In the same way q,{u) is below q0 and its height increases with u. Thus, either p.(uo) = po for some uo a (0, 1] or p,{u) e I for all u e (0, 1]. In the first case q.(uo) is above qo and -o there exists u1 < uo such that q,{u1) = q0. In the second case, q;(1) is above qo because the distance'from p((1) to po is less than p. Thus, there
exists u1 < I such that g1(u1) = qo. In either case Z(u1) has a closed orbit through qo. PROOF OF LEMMA 2.5. First we shall prove that, if y is a nontrivial ao-recurrent
orbit, there exists a stable separatrix that accumulates on y. That is, there exists a stable separatrix y2 such that a(y2) y. Consider a point p E y and a circle C transversal to X through p. Let P: D c C - C be the Poincare map defined on D. We have D # C since, otherwise, X has no singularities. Suppose, if possible, that y is not accumulated by stable separatrices; that is, suppose that there is an interval in C that contains p and is disjoint from the stable separatrices. Let I c C be the maximal interval with this property. As y is co-recurrent and p e y, we have P(p) e I for some integer k > 0. Now the interval J = Pa(l) is contained in I. This is because, on the one hand, I r J # 0 since Pk(p) E 1. On the other hand, if J c I, then J would contain
Figure 39
148
4 Genericity and Stability of Morse-Smale Vector Fields
w
an interval that has one of the erid-points of Iin its interior. M I is the maximal interval disjoint from the stable separatrices it would follow that J contains points of stable separatrices. As these separat> ices are invariant by the flow and, therefore, by Pk, it would follow that I also contains points of stable separatrices of saddles as J = P(1). This would contradict the definition of I and so, in fact, P(I) c I. From this we can construct a region A containing p, homeomorphic to an annulus and invariant by X t > 0. See Figure 40. This is a contradiction because y could not be nontrivially w-recurrent in A. Thus y is, in fact, accumulated by some stable separatrix y2. We also claim that y is accumulated by some unstable separatrix or is itself an unstable separatrix. In fact, let us suppose that this claim is false. If the second alternative does not occur we consider, as in the previous case, a point p e y and a maximal open arc I in C containing p and disjoint from unstable separatrices. As y is w-recurrent, there exists an integer k > 0 such that P}(I) n 10 0.-It follows from this that, for some q e 1, P-"(q) is well defined and P-k(q) a 1. There are two possibilities to consider. If P-k is not defined'on the whole arc I then there exists a point z e I whose negative orbit dies in a saddle. In particular,z e I n y for some unstable separatrix y which contradicts the definition of the arc I. The other possibility is that P-k is well' defined on the whole arc 1. In this case, as before, we shall have P-'(1) c 1. We leave the reader to.complete the proof of the claim from this statement.
We. should remark that the property just' proved also holds for nonorientable manifolds. The only difference is that the region A used above can be a Mobius band where, again, there cannot be nontrivial recurrence. Now let yt be an unstable separatrix that either is y or accumulates on y. Also let y2 be a stable separatrix that accumulates on y. As in the proof of Lemma 2.4 we consider the circles Ct = X -a(C) and C2 = X6(C) transversal to X. Let P: D .c CZ -+ C1 be the'Poincare map and F a flow box containing p e y. As the number of saddles is finite we can-take F disjoint from any saddle-
connections that X might have.
Let at and a2 6e'the saddles associated to the separatrices yt and y2 which-accumulate on y. Let us consider the family of fields Z(u) = X + euY,
§
Density of Morse-Smale Vector Fields on Orientable Surfaces
149
Figure 41
where e > 0, 0 < u 5 1 and Y is transversal to X in the interior of F, points upwards and vanishes outside F. If we take E small then Z(u) is near to X for all u e [0, 1]. We want to show that, for some 0 < u < 1, Z(u) has one saddleconnection more than X. We fix a small closed interval I in [a, b] containing po in its interior.,As before, let p > 0 be the minimum for x E I of the vertical distance in F between x and the point where the positive orbit of Z(1) through x first meets C2. Let xo and zo be the first times that yt intersects [a, b] and y2 intersects [c, d], respectively. Note that the arcs of separatrices otxa and Q2 z0 are not affected by the perturbations Eu Y above, see Figure 42. Now take a point x e yt n f near po and a point z E y2 n C2 such that the vertical distance between x and z is less than p. The point x corresponds to the
ith intersection of y1 with Ct for some integer i > 0. Similarly, the point z corresponds to the jth intersection.of y2 with C2 for some j > 0. We shall suppose that x is below z in F. If this is not possible then we must take the field Y pointing downwards.
Figure 42
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4 Geoericity and Stability of Morse-Smale Vector Fields
Consider the maps that associate to each u the ith intersection x(u) of the separatrix yt(u) of Z(u) with Ct and the jth intersection z(u) of the separatrix y2(u) of Z(u) with C2 . It is clear that x(u) and z(u) are well defined for small u. As M2 is orientable, x(u) is monotone increasing on [a, b] and z(u) is monotone decreasing on [c, d]. We have two situations to consider. Suppose first that x(u) and z(u) are
well defined for all u e [0, 1]. Then there exists uo E (0, 1) such that the vertical distance between x(uo) and z(uo) is zero. This is because x(u) and z(u) are continuous and the vertical distance between. = x(0) and jr = z(0) is less than p. Thus we have a saddle connection between at(uo) and a2(uo). Now suppose that one of the maps, x(h) for example, is not defined for all u e [0, 1]. This means that, for some uo a (0, 1), y, (it.) reaches one of the boundary points of the domain oft'; that is, it reaches a point whose positive orbit goes directly to a saddle a3. In this case it will be a saddle connection between at and a). The reasoning is similar for the case where it is z(u) that is not defined for all u e [0, 1]. This completes the proof of Lemma 2.5 and hence of Theorem 2.6., To close the section we remark that it follows from Theorem 2.6 that any structurally stable vector field X e £'(M2) is Morse-Stikale.
§3 Ge"neralizations Next we txtake' some comments on the density theorem for Morse-Smale
fields on orientable surfaces and its partial extension to nonorientable surfaces.
We shall also mention the theorems about the openness and stability of Morse-Smale fields on manifolds of any dimension. In particular, there exist structurally stable fields on any manifold. However, Morse-Smale fields are no longer dense in the space of vector fields on manifolds of dimension three
or more. This will be proved in the next section. We should, however, emphasize a useful special case in which Morse-Smale vector fields are .dense: namely in the space of gradient fields on any compact manifold. Our first remark is that in the proof of Lemma 2.5 we cannot guarantee
that after the perturbation there is a saddle connection between the first saddles considered. This leads us to formulate the following problem. Let
be an unstable separatrix and y2 a stable separatrix of saddles of X C :T,(M2). Suppose that w(yt) n y2 # 0 or w(yt) n 2(y2) # 0. It is a y,
difficult question to know whether it is possible to connect these two separatrices for a C' perturbation of X. This problem is open whether M2 is orientable or not and for any r >_ 1.
The difficulty in proving the density of Morse-Smale fields in X'(Af2) when M2 is not orientable lies in the proof of Lemma 2.5. All the other facts
§3 Generalizations
151
are true in this case. The question is open in this nonorientable case and it is an interesting problem whether the answer is positive or negative, although a negative answer would be a surprise. In this direction some partial results have been obtained as follows. (1) Morse-Smale fields are dense in X'(M2) whether M2 is orientable or not. Pugh obtained this result using the Closing Lemma as we shall show later. The restriction to the C' topology comes from the fact that the Closing Lemma has only been proved so far for this case. (2) It is easy to see that the theorem holds for £'(P2) and any r >- 1 where P2 is the projective plane. This is because the vector fields on P2, as in the case of the sphere S2, do not have nontrivial recurrence. Density is also true for the Klein bottle K2 as Markley showed in [56]. Gutierrez [30] simplified the proof for K2 and showed that in the nonorientable manifold
L2 of genus one more than K2 that is the torus with a cross cap, the nontrivial recurrences are "orientable". Thus the proof we presented for orientable manifolds also applies to this case. Therefore we have the
density of Morse-Smale fields in X'(M2) for any r >- 1 when M2 is orientable or M2 = P2, K2 or L2. We now describe the proof for X'(M2) using the Closing Lemma. manifold without Closing Lemma [88].Let M" be a compact boundary. Take X e X'(M) and let y be a nontrivial recurrent orbit of X. Given p e y and e > 0 there exists Y e '(M"), with I Y - X 1c, < s, having
the orbit through p closed.
The proof of the Closing Lemma is very delicate even in the case of surfaces. As regards the class of differentiability, the question is open for any
r _ 2andn2:2. In the case of surfaces M2 the closed orbit constructed from a nontrivial recurrent orbit cannot bound a disc. This is because of the existence of a circle transversal to the field and not bounding a disc as constructed in Section 2 of this chapter. 3.1 Theorem. The subset consisjing of Morse-Smale vector fields is dense in I'(M2) whether or not M2 is orientable.
PROOF. We shall show that any field X E £'(M2) can be approximated by a Kupka-Smale field with only trivial recurrence. Then the result will follow immediately from the Corollary to Proposition 2.3. Take a Kupka-Smale field X* near X. If X* has only trivial recurrence the proof is finished. Otherwise, consider a nontrivial recurrent orbit y, of X* and a point p e yi. By the Closing Lemma there exists X, near X* with' a closed orbit a, through p. Now approximate X, by a Kupka-Smale field
X* that has a hyperbolic closed orbit it near a,. If Xi has only trivial recurrence then X i is a Morse-Smale field. Otherwise, we repeat the above
4 Genericity and Stability of Morse-Smale Vector Fields
152
process starting from X i but without changing it on a neighbourhood of at. We claim that the number of steps in this process is finite, bounded by 29 where g is the genus (the number of handles) of the manifold M. In fact each stage of the process is like considering a Kupka-Smale field on a manifold N2 of lower genus (or, equivalently, of higher Euler-Poincare characteristic K(N2) since K(N2) = 2 - 2g(N2) for N2 orientable and K(N2) = 2 - g(N2) for N2 nonorientable, [59], [119]). As K(N2) < 2 for any N2 We shall obtain
a Kupka-Smale field with only trivial recurrence after a finite number of steps; this field will then be Morse-Smale as required. To see this reduction of genus we make a cut in M along the hyperbolic closed orbit o* that resulted from the nontrivial recurrence. There are two possibilities to consider: either we obtain,a manifold M10 with boundary or two manifolds M11 and M12 with boundary [59]. In the first case the boundary of M10 consists of one or two copies of o depending on whether at is orientable or not. In the second case the boundaries of M11 and M12 are each a copy of at. In the first case we glue in one or two discs D, and D2 so that M, 0 u D, or M 10 u D, u.D2 is a manifold without boundary. Thus we have K(M10 u DI) _ K(M) + K(DI) or K(M10 u DI u D2) _ K(M) + K(D1) + K(D2). As K(D1) = 1 for i = 1, 2, K(M10 u DI) or
K(M10 u D, u D2) is greater than K(M). In the second case K(M) = K(M11) + K(M12) and K(M11) < 1, K(M12) < l because otherwise rr* would bound a disc. Thus
.
K(M11 u DI) = K(M11) + 1 > K(M) and
K(M12 u D2) = K(M12) + 1 > K(M). Although it is not necessary for our purpose, we can complete the field X; defined on M10, M11 and M12 by putting in D, and D2 a sink or a source depending on whether a, is repelling or attracting. We continue the process with the manifolds Mi; (j = 0, 1, 2) obtained by glueing in these discs but without altering the field on- neighbourhoods of these discs. As the Euler-Poincare characteristic is bounded by 2 and grows with each cut made, the number of cuts is finite and bounded by 28 as claimed. This proves the theorem.
0
For Morse-Smale fields on M2 we should also mention that their equivalence classes were described by Peixoto [83] and Fleitas [19]. Also, the connected components of the Morse-Smale fields on M2 were classified in [32]. Now consider a manifold M of any dimension n endowed with a Riemannian metric. One basic question is whether there exists a structurally stable field on M. Results from [75], [79], [106] show that there exist many Morse-Smale fields and they are structurally stable; see also [57]. These results are as follows:
§4 General Comments on Structural Stability. Other Topics
153
(1) the set consisting of Morse-Smale fields is open and nonempfy in ?i'(M"), r Z 1; (2) if X e 3:'(M"), r >_ 1, is Morse-Smale then X is structurally stable; (3) the set of Morse-Smale gradient fields is open and dense in Grad'(M"),
rz1.
Here'Grad'(M") denotes the subset of 3f'(M") consisting of the gradient fields of C'+' maps from M to ]l8 with respect to a Riemannian metric on M. Consider the orbit structure of a Morse-Smale gradient field. As we saw in Section 1 of Chapter 1, a gradient field cannot have closed orbits. Moreover, every orbit has its a- and w-limit sets consisting of singularities. We leave it
to the reader to prove that even the nonwandering set consists only of singularities. Thus a Morse-Smale gradient, field is just a Kupka-Smale field with nonwandering set a finite number of hyperbolic singularities In contrast to what happens in 1'(M2), r = I or r > I and M2 orientable, or in Grad'(M"), the Morse-Smale fields are not dense in X'(M") for n >_ 3. This fact will be seen in the next section together with examples of structurally stable fields in 3`'(M') that are not Morse-Smale since they have infinitely many periodic orbits.
§4 General Comments on Structural Stability. Other Topics In this section we shall briefly describe Morse-Smale diffeomorphisms, Anosov diffeomorphisms and the diffeomorphisms that satisfy Axiom A and
the transversality condition. The first are analogous to the Morse-Smale fields described in previous sections. The last include the first two and constitute the most general class of structurally stable diffeomorphisms known. We shall describe in detail two famous examples that illustrate the last two classes well: one of them is due to Thom (an Anosov diffeomorphism of T2) and the other is Smale's horseshoe. Besides its intrinsic importance the study of diffeomorphisms has been of great relevance in understanding the orbit structures of vector fields. This was already emphasized by Poincare and Birkhoff in their pioneering work on the qualitative theory of dynamical systems. One example is the description of the orbit space of a vector field in a neighbourhood of a closed orbit. As we saw in Chapter 3 this is done by using the Poincare map (or local diffeomorphism) associated to a transversal section. At the end of Chapter 3 we generalized this idea by the process of suspending a diffeomorphism. Thus any diffeomorphism f of a manifold of dimension n represents the Poincare map of a field X f on a manifold of dimension n + 1. The field Xf is called the suspension off and its orbits are in a natural correspondence
154
4 Genericity and Stability of Moue-Srnale Vector Fields
with the orbits of f. In particular, X f is a Kupka-Smale field if and only if f is a Kupka-Smale diffeomorphism. Also, X f is structurally stable if and only if f is structurally stable. Let f e Diff'(M). A point p e M is nonwandering for f if, for any neighbourhood U of p and any integer no > 0, there exists an integer n such that in I > no and f "U n U 0.0. The set 0(f) of nonwandering points is closed and invariant, that is, it consists of complete orbits off. The limit sets w(q) and a(q), for any q e M, are contained in 2(k). In particular, every-fixed or periodic point off belongs to 0(f ). , We say that f e Diff'(M) is Morse-Sniale if
(a) 0(f) consists of a finite number of fixed and periodic points, all hyperbolic; (b) the stable and unstable manifolds of the fixed and periodic points are all. transversal to each other. Next'we list some important facts about Morse-Sinale diffeomorphisms. (1) The set of Morse-Smale diffeomorphisms is open (and nonempty) in Diff'(M) for any manifold M and any r z 1, [75]. (2) If f e Diff'(M) is Morse-Smale then f is structurally stable [75],179].
(3) The set of Morse-Smale diffeomorphisms is dense in Diff'(S'), r z 1. This fact, due to Peixoto, can be proved directly from the KupkaSmale Theorem for diffeomorphisms and an argument similar to that in the proof of Lemma 2.4 in this chapter. A more elegant proof is as follows. Let f e Diffr(S' ). Take a C°° diffeomorphism f C"-close to f. Consider the suspension X f off, which is a C`° field defined on T 2 or K2 depending on whether f preserves or reverses the orientation of S'. We can consider S' as a global transversal section of Xj on T2 or K2 with J`being the associated Poincarb map. If Y is a field on T. or K2 that is C'-close to X j then S' is also a transversal section for Y and the Poincare map g associated to Y is C'-close tot and so to f. By the density of Morse-Smale fields in X(T2) or 3:'(K 2) it is possible to choose Y Morse-Smale and C'-close to X f. This implies that g is a Morse-Smale difl'eomorphism C'-close to f. (4) The set of Motse-Smale diffeomorphisms is not dense in Diff'(M"),.
n a 2. We describe next a nonempty open set 'W c Diff'(S2) such that n M-S = 0..A similar example can be constructed on any manifold of
dimension n z 2. Consider on S2 a C°° field X with a saddle connection from one saddle to itself as in Figure 43; al and a2 are sinks, a4 is a source and a3 is a saddle, all hyperbolic. Let X, be the flow induced by X and f = Xl the time one diffeomorphism. Then a3 is a hyperbolic fixed point for f and one of the components of W'(a3) - a3 coincides with one of the components of W"(a3) - a3. We perturb f so as to obtain a diffeomorphism g that has a3
as a hyperbolic fixed point and has orbits of transversal intersection of W°(a3, g) with W"(a3, g) besides a3. To do this we take p e Ws(a3) n W"(63),
p # a3i and a small neighbourhood U of p with U n f U = 0. Let is S2 - S2 be a C' diffeomorphism supported on U (so that i is the identity on K =
§4 General Comments on Structural Stability. Other Topics
155
Figure 43
M - U) with i(p) = p and W = i(W"(a?)) transversal to W'(a3) at p. Define g = i o f. We claim that g and any diffeomorphism close enough to g are not Morse-Smale. As g = f outside U, a3 is a hyperbolic fixed point of g and the local stable and unstable manifolds of a3 for f and g coincide. But W c W"(a3, g). In fact if x e W then i-'(x) a Wu(a3) and so (i o f)-'(x) =
f -1i(x) e W(a3) n K. As i is the identity on K, (i o f)-"(x) = f -"i-'(x) e W"(a3) n K for n >_ 1. Thus, x e W°(a3, g) since g-"(x) _ (i o f)-"(x) converges to a3 as n -' oo and so W e W"(a3, g). On the other hand, W3(a3) n U C Ws(a3, g). In fact if y e W(a3) n U then f(y) e Ws(a3) n K and so g"(y) = (i G f)"(y) = f"(y) e W'(03) n K for n Z 1. Thus y e Ws(a3, g) since g"(y) converges to 63 as n -+oo and this shows that Ws(a3) n U c Ws(a3i g). Thus, W'(a3, g) is transversal to W"(a3, g) at p. Although it is not necessary we can,extend this argument a little to make Ws(a3, g) and W"(a3, g) transversal at all their points of intersection. This
corresponds to one of the assertions of the Kupka-Smale Theorem for diffeomorphisms. A point p, as above, of transversal intersection of W'(a3, y) and W"(a3, g) is called a transversal homoclinic point. The reader is challenged
Figure 44
156
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 45
to draw a diagram, with Figure 45 as a rough sketch, of the intersections ofthe stable and unstable manifolds along a transversal homoclinic orbit. Birkhoff showed that p is accumulated by hyperbolic periodic orbits of g; Smale generalized this result to higher dimensions [108]; see also [66]. Here we only need the fact that p is nonwanderi.ng. To see this consider the arc I of W"(a3, g) going from a3 to p. For any neighbourhood U of p choose a small arc 11 of W"(Q3, g) in U passing through p. Since ll is transversal to WS(r3, 9), g"(11) contains, by the A-lemma, an arc arbitrarily close to I for all n greater
than some no > 0. As 11 c U and l n U :A 0 we have g"U n U # 0 for n > no. Thus p e Q(g) and as p is not periodic g is not Morse-Smale. The same happens for all diffeomorphisms close enough to g since they also have transversal homoclinic points. This follows from the fact that compact parts of the stable and unstable manifolds of a saddle do not change much in the
C' topology when we perturb the diffeomorphism a little. We can thus guarantee that these manifolds still have an orbit of transversal intersectiondistinct from the perturbed saddle. From the nondensity of Morse-Smale diffeomorphisms on M2 we can deduce, using suspension, that Morse-Smale fields are not dense in Xr(M") forn _ 3. We shall now give another example, due to Thom, of a diffeomorphism with infinitely many periodic orbits. Then we shall show that this diffeomorphism is structurally stable.
This was one of the examples that motivated the definition given by AnOsov of a class of structurally stable dynamical systems with infinitely many periodic orbits [3]. In particular, there exist stable systems that are not Morse-Smale. Consider a linear isomorphism L of R2 that is represented with respect to the standard basis of 682 by a hyperbolic matrix with integer entries and determinant equal to 1. It is easy to see that the eigenvalues of L, A and 1/A with JAI < 1, are irrational and that their eigenspaces E' and E" have irrational slope. As det L = 1 it follows that L-' has the same properties. If
§4 General Comments on Structural Stability. Other Topics
157
c R2 denotes the set of points with integer coordinates then L(Z) Consider a manifold structure on T2 = R2/g2 for which the natural projection x: 682 -+ T2 is a local diffeomorphism. This manifold structure can be obtained by identifying R2/Z2 with a torus of revolution as in Example 2, Section 1 of Chapter 1. We recall that x(u, v) = x(u', v) if and only if u' u e Z and v' - v e Z. In this case x(L(u, v)) = x(L(u', v')), which enables us to define a map f : T' -+ T' by f (x(u, v)) = xL(u, v). As it is a<. C°° local diffeomorphism it follows that f is of class C. The same argument applies to L so f is, in fact, a C°° diffeomorphism. For each p e T2 and x e 682 with x(x) = p the curve W'(p) = x(x + E) is dense in T2. Thus {W'(p); p e T2} defines a foliation of T2, called the stable foliation, each leaf of which is dense in T2. Moreover, this foliation is invariant by f, that is, f W(p) = W'(f (p)). Similarly, we define the unstable foliation {W"(p); p c- T2} by W"(p) = x(x if E"). If we write Ep and E', Z2
for the tangent spaces to W'(p) and W"(p) at p then EP = dxx(E), Ep = dxX(E") and EI(,,) = df,(E,), E7(p) = dfo(E").
On T2 we consider the metric induced from 632 by x; that is, if wt, w2 e T(T2)p_xtxl then we define <w1, w2>p to be . In this
metric we have Ildf,vll = Iz1IIvII,
Ildffwll = I2I-1IlwII,
if vcE'p; ifweEup.
It follows from this that if q e W(p) then d(f"(q), f"(p)) -i 0 as n - oo and that if q e W"(p) then d(f `(q), f -"(p)) -. 0 as n - oo. Hence every periodic point p off is hyperbolic and the stable and unstable manifolds of p are the W'(p) and W"(p) defined above. Moreover, for any p, q e T2 we have W(p) transversal to W"(q) and W'(p) n W"(q) is dense in T 2. In particular, p = x(0)
is a hyperbolic fixed point of f and its transversal homoclinic points are dense in T 2. As in the previous example this implies that f and any diffeomorphism near f are not Morse-Smale. Birkhoff's-result and the density of the transversal homoclinic points imply that the periodic points of f are dense in T2. We shall now give a direct proof of this.
4.1 Proposition. The periodic points of f : T2 - T2 are dense in T2.
PROOF. Let 2' be the set of points in P2 with rational coordinates. We shall show that x(2) coincides with the set Per(f) of periodic points of f. As 2' is dense in 182 it follows that Per(f) is dense in T2. If £" = {(m1/n, m2/n); m1, m2 a Z}'then 2' = U.> t Y". As L is a matrix with integer entries we have L(2'") = 2". Therefore f(x2") = x2". As x2'" = x{(m1/n, m2/n); m 1, m2 a 7L, 0 5 m 1 5 n, 0 5 m2 5 n} we deduce that x2'" is a finite invariant
subset of T2 and so its points are periodic. Hence x(2) c Per(f). On the other hand, let x e P2 satisfy f "(x(x)) = x(x) for some integer n. We claim that x has rational coordinates. In fact since x(L"x) = x(x) the point y = L"x - x has integer coordinates. As L is a hyperbolic matrix with integer
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4 Genericity and Stability of Morse-Smak Vector Fields
entries it follows that L" - I is an invertible matrix with integer entries and
so (L" - 1)-' is a matrix with rational entries. Thus x = (L" - I)-'y has rational coordinates and so Per(f) is contained in n(2'), which proves the claim.
At first sight it may seem very difficult to show that the diffeomorphism f,
which has infinitely many periodic orbits, is structurally stable. There is, however, one useful property: f has a global hyperbolic structure and is even induced by a linear isomorphism of R2. In particular, all the periodic orbits of f are saddles with stable manifolds of the same dimension. By contrast, a Morse-Smale diffeomorphism must have sources and sinks and, usually, saddles too. We shall now give a simple and elegant proof, due to Moser [65],
that f is structurally stable. This proof is very close to the analytic proof of the Grobman-Hartman Theorem in Chapter 2. We recall that f is induced by an isomorphism L of R2 where L is defined by a hyperbolic matrix with integer entries and determinant 1. 4.2 Theorem. The diffeomorphism f : T2 -' T2 is structurally stable. PROOF. Take g e Diff'(T2) near f. We claim that there exists a diffeomorphism
G: R2 -+ R2 near L that induces g on T2. In fact, for each x c- RZ we can consider fn(x) = nL(x) where it is the canonical projection from R2 to T2. As gn(x) is near fir(x) there exists a unique point y E RZ near L(x) with n(y) = gir(x). We define G(x) = y and get nG(x) = gn(x). It is easy to check that G and L are C'-close. We now write G = L + (D where 4) is a C' small map of R2. As L is hyperbolic we know from Lemma 4.3 of Chapter 2 that L and L + 0 are conjugate. That is, there exists a homeomorphism H of R2
such that HL = GH. It is, therefore, enough to check that H induces a homeomorphism h of T 2 with nH = hn because this will imply that hf = gh. In fact nHL = hnL = hfn; similarly nGH = gnH = ghnrand so hfn = ghir. Since x: 682 -+ T2 is surjective we obtain hf = gh. Now we check that the homeomorphism H of R2 induces a homeomorphism h of V. In solving the
equation HL = GH we write H = I + u and G = L + 4) and obtain uL = Lu + 4)(1 + u). We want a solution u e Cb(R2) and, for this theorem, we need I + u to project to a map of T 2. This last requirement is equivalent to the following: for each x e R2 and each point p with integer coordinates
there exists q with integer coordinates such that (I + u)(x + p) = q + (1 + uXx). That is, u(x + p) = q - p + u(x). But u will be constructed with small norm for g near f so we deduce that u(x + p) = u(x) for any x e R2 and any p with integer coordinates. Thus we are led to consider the subspace 9 of Cb(R2) consisting of periodic functions u e Cb(R2) satisfying u(x + p) = u(x) for any x and p in R2 where p has integer coordinates. It is immediate that 9 is closed in Cb(R2) and that 2(p) c Y where the operator Y: Cb(682) --+ Cb(R2) is
defined by 2(u) = uL - Lu. Moreover, . is
invertible because L is hyperbolic. On the other hand, as G = L + 4) does
§4 General Comments on Structural Stability. Other Topics
159
project to a map of T2 and 4) is C' small, we have 4) e Y for the same reason
as above. Therefore, the map p: Y - -,*, p(u) _'-'M(I + u)) is well defined and is a contraction. The unique fixed point u of p satisfies the equation uL - Lu = (D(I + u) or, equivalently, (I + u)L = (L + 4XI + u). The proof that H = I + u is a homeomorphism is as in Lemma 4.3 of Chapter 2. As u e Y the homeomorphism H`= I + u projects to a homeomorphism h of T 2 and hf = gh. This shows that f is structurally stable in 0 Diff'(T2), r z 1.
This diffeomorphism f : T2 -+ T2 is a good example of an Anosov diffeomorphism and we now give the general definition.
Definition. Let M be a compact manifold. We say that f c Diff'(M), r z 1, is an Anosov difeomorphism if:
(a) the tangent bundle of M decomposes as a continuous direct sum,
TM =E®E"; (b) the subbundles E' and E" are invariant by the derivative Df of f ; that is, DfXEl = E'f(X) and DfXEX = E"f(X) for all x e M;
(c) there exists a Riemannian metric on M and a constant 0 < A < 1 such that IIDf.vll < Apvll and IIDfx'ull < Allull for any xeM, veE; and
ueEu. Anosov was the first to prove that these diffeomorphisms are structurally stable in Diff'(M) for M of any dimension (see [3], [65] and the appendix by
Mather in [109]). He also defined an analogous class of vector fields and
proved their structural stability. The suspension of an Anosov diffeomorphism is an example of such a vector field. Another important case is that of the geodesic flow for a manifold of negative curvature [3].
We remark that the definition of Anosov diffeomorphism imposes strong restrictions on the manifold. For example, among the compact manifolds of dimension two only the torus admits Anosov diffeomorphisms. It is even believed that Anosov diffeomorphisms only exist on very special
manifolds such as the torus T" and nilmanifolds [109], [113]. By contrast there exist Morse-Smale diffeomorphisms on any manifold. It is also known that every Anosov diffeomorphism on T" is conjugate to one induced by a linear isomorphism of R" as in the example described above [55]. It is conjectured that any Anosov diffeomorphism has its periodic orbits dense in the manifold. On these questions various important results were obtained by Franks [20], Manning [55], Newhouse [67] and Farrell-Jones [18]. Recently, a very interesting class of Anosov flows was constructed by Franks and Williams [24] for which the nonwandering set is not all of the ambient manifold. The corresponding question for Anosov diffeomorphisms remains open. We thus have two classes of structurally stable diffeomorphisms, MorseSmale and Anosov, that have, as we have already emphasized, very different
160
4 Genericity and Stability of Morse-Smale Vector Fields
properties. Smale then introduced a new class of diffeomorphisms encompassing the previous two. These diffeomorphisms, which we shall soon describe, are structurally stable and it is conjectured that they include every structurally stable diffeomorphism.
Consider a compact manifold M. Let f e Diff'(M) and let A c M be a closed invariant set, f A = A. We say that A is hyperbolic for f if:
(a) the tangent bundle of M restricted to A decomposes as a continuous -direct sum, T,.M = E. $ E;,, which is invariant by Df; (b) there exists a Riemannian metric and a number 0 < A < 1 such that IIDfx vil s 2ll v;l and IIDf x lu 11 s 1Ilull for any x e A, v c- P. and u c -E'.
Now consider the nonwandering set Q = Q(f ), which is closed and invariant.
`
Definition. We say that f satisfies Axiom A if Q is hyperbolic for f and fl = Per(f ), that is, the periodic points off are dense in Q. If f satisfies Axiom A, Smale showed that Q = Q(f) decomposes as afinite disjoint. union of closed subsets which are invariant and transitive, Q = Q, v'Q2 v ... v Qk. For details see [109], [70], [72], [77], [101]. The sets Qi are called basic sets. Transitivity means that there exist dense orbits in each fi,. In each Qi all the periodic orbits have stable manifolds of the same dimension. In the case of Morse-Smale diffeomorphisms the basic sets are periodic orbits. In the case of the Anospv diffeomorphisms on T2 there is only one basic set, the whole.of T2. In general.these basin; sets can have a much more complicated structure as we shall see in examples. As in the
Morse-Smale case they can be of attractor, repellor or saddle type. In Examples 3 through 6 below, we exhibit interesting cases of nonperiodic basic sets, which are attractors, repellors and saddles. Examples 5 and 6 can be seen as special instances of a quite large class of Axiom A attractors whose structure was described by Williams [121].
With regard to the above definitibn of Axiom A it is known that, when
dim M ='2, Q(f) hyperbolic implies Per(f) = 1(f) and that this is not true in higher dimensions [71], [15]. Let us make one last remark about the transitivity requirement for basic sets. Since this is a difficult requirement to check directly, the following criterion may be more useful, as in Examples 4 and 5, below. If A is a hyper bolic set for f with periodic points dense and if A W'(p) n Wu(q) # 0 for every p, q E Per(f) n A, then A is transitive. In fact, for 'any
p e Per(f) n A, Ws(p) will accumulate' on Per(f) n A and so Ws(p) is dense in A. For each p E Per(f) n A choose a base of neighbourhoods U;, U5, ... , Ut,... of p in A. Then'f = U"E a f"U,, is open and dense in A. As A is closed it satisfies the Baire property and the periodic orbits of f in
A are countable since they are hyperbolic. Thus, D = np,k Vf, for pe Per(f) n A and k e N, is dense in A. It is easy to see that any x e D has a
§4 General Comments on Structural Stability. Other Topics
161
dense orbit in A. In fact, given a nonempty open subset A of A, there exists
some Uf c A and, as x E Vf = U. f"Uf, it follows that f -"(x) e U[ c A for some n c- Z.
With structural stability in mind we are going to generalize the concepts of stable and unstable manifold to nonperiodic orbits. Let f e Diff'(M) and choose x e M. We define. W,5(x) _ {y e M; d(f"x, f"y) -0 as n -" oo} and W'(x) = {y e M; d(f"x, f"y) -' 0 as n -+ - oo} where d denotes the metric induced by the Riemannian metric on M. When 0 = f2(f) is hyperbolic and x E n then W(x) and W"(x) are Cr injective immersions of Euclidean spaces l8' and 68" of complementary dimensions [39]. That is the justification
in this case for calling the sets W(x) and W(x) manifolds. In each basic set f1, of a diffeomorphism that satisfies Axiom A the stable manifolds of all the
orbits in Q, have the same dimension. Furthermore, their union coincides with the set of points whose oo-limit set is contained in f2,, see [9]. In particular,
if f2, is an attractor then the union of the stable manifolds of points in f2, is a neighbourhood of I. In.the case of an Anosov diffeomorphism of T2 induced by a linear isomorphism of R2 the stable manifolds are the projections to T2 of parallel lines in 182. The corresponding statements hold for unstable manifolds.
Definition. Let f e Diff'(M) satisfy Axiom A. We say that f satisfies the transversality condition if W'(x) and W(y) are transversal for any x, y E fI(f ).
Before discussing the structural stability of diffeomo.rphisms that satisfy Axiom A and the transversality condition we shall give some examples of them. EXAMPLE 1. Morse-Smale diffeomorphisns.
EXAMPLE 2. Anosov diffeomorphisms of the torus T2 induced by linear isomorphisms of U8'. It is also true, although we shall not prove it, that any ,Anosov diffeomorphism satisfies Axiom A and the transversality condition.
EXAMPLE 3. Let g: S' - S' be a Morse--Smale diffeomorphism with two fixed points, the north pole n is a repellor and the south pole s is an attractor.
Let f : T' - T2 be an Anosov diffeomorphism induced by a linear isomorphism of R2. Consider the product diffeomorphism g x f : S' x T2 S' x T2. It is easy to see that its nonwandering set has two parts, 01 = {n} x T' and f12 = {s} x T2 and that they are hyperbolic and transitive. Here f21 is a repellor and (12 is an attractor. Let us check the transversality
condition. Let (z, w) e S' x T' be a point of intersection of 43"(x) and W"(y) where x e f12 and y e.,. Now W'(x) is the product of W i in S' x {w} and W'2 in {z} x V. Similarly, let us denote by W, and W2 the f wtors of
W"(y) in S' x {w} and {z} x -T2. As W, = (S' - n) x {w} and W'= (S' - s) x {w}, W, and W, are transversal in S' x {w}. Moreover, WZ
162
4 Genencity and Stability of Morse- Smale Vector Fields
and W2 are transversal in (z) x T2 because they are the projections of lines
in R2 parallel to independent eigenvectors. This proves that W(x) and W"(y) are transversal. If x and y both belong to Sgt or both to f12 the transversality of Ws(x) and W"(y) is immediate. Thus the diffeomorphism g x f : T' T 3 satisfies Axiom A and the transversality condition. It is clear that g x f is neither Morse-Smale nor Anosov. EXAMPLE 4. In this example we describe a diffeomorphism,f of S2 satisfying Axiom A and the transversality condition. The nonwandering set !Q = S2(f) consists of three basic sets: S2, is a repelling fixed point, 03 is an attracting fixed point and 02 is a Cantor set in which the periodic saddles are dense. The important part of this example is the "Smale horseshoe". At the north pole of the sphere we put a hyperbolic source Q. and the whole northern
hemisphere H. including the equator, belongs to its unstable manifold W"(S21). Therefore.
if H_ denotes the southern hemisphere f(H_) e
Int H_ . We now describe f on If_. The map I is linear, compresses horizontal lines by a compression factor 0 < ) < ; and expands vertical lines by an expansion factor p > 4. The map g twists the rectangle 1(Q) at its middle quarter making a horseshoe F and translates it to the position indicated in Figure 46. Thus, Q n fQ has two rectangular components R, and R'2 that
f
Q
Figure 46
§4 General Comments on Structural Stability. Other Topics
163
-f . R,
I
Figure 47
are images of the rectangles R, and R2 in Q. In the rectangles Rt and R2 the diffeomorphism f is affine (a linear map composed with a translation), contracting horizontal lines by and expanding vertical lines by p. Finally, at the centre of the disc 6, = f D1 we put a hyperbolic attracting fixed point C13 and 6, c W`()3). Let us analyse the set f2 = f2(f ). If x e H+ and x is not the north pole then x is wandering, since it belongs to the unstable manifold of the source. If x E B, and x # f23 then x is wandering, since it belongs to the stable manifold of a sink. If x e D1 then f (x) a b, and x is wandering. If x e D2 then f (x) E D, and x is again wandering. We conclude that, when x e H - and x # 03, x can only, be nonwandering if its orbit is entirely contained in Q. Therefore, if x e f2 but x # f2, and x # f23 we have
xEn"Ezf"Q = A.
'
We now describe the set A. As Q n f Q has two rectangular components, Q n f Q n f 2Q has four rectangular components and so on. Schematically
we have: Q; Q n fQ = R, V R2; Q n fQ n f2Q = R11 V.R12 V R21 -
R22; etc. The suffixes are attached to the rectangles in the following
way: f R, n Q = R11 V R12 and R11 c R1. R12 n R1 = 0. Similarly,
f R2 n Q = R21 U R22 and R21 n R2 = 0, R22 C R2.
In general,
R,... epap.1 c fRa,_?p and ap+t = o'p if Rc,...apap.I c Rel... op and o'p+t # ap if they are disjoint. Here each 6; is I or 2. Note that, given any integer k > O, R,
Al 1 ... , where the suffix 1 is repeated k times. That is, R 1 t ... I e
fkRt n Q and, in particular, f 'R, n R, # 0. The same is true for RQ,Q2...p where each a, can be 1 or 2. Consider any horizontal line a such that a n Q # 0 and let [a, b] = a n Q. Then [a, b] rf Q is two closed segments and is obtained from [a, b] by removing three disjoint segments. From each of these two segments we remove three more segments to form [a, b] n fQ n f 2Q, and so on. The reader will recognize that [a, b] n (n. ,o f"Q) is a
164
4 Genericity and Stability of Morse-Smale Vector Fields
Cantor set. If we take a vertical line and follow the same reasoning we find
that the intersection of this line with n"
vertical segments are sent by f to vertical segments with an expansion greater than 1 and, dually, horizontal segments remain horizontal and are contracted. Now take x e A and let R be a rectangle in Q with two vertical sides of the same height as Q and with x e R. Note that however small the width of R it always contains one of the rectangles R,,, ,D. This is because A is contained in the intersection of all the rectangles,,, ,y. Let us now show that x e A is nonwandering. Let Qx c Q be a square containing x in its interior and let N be a positive integer. We shall show that f"QX n Q. 0
for some n > N. As f expands vertical segments there exists an integer m >_ Q such that f'Qx n Q contains a rectangle R X) of height equal to that of the original square Q. A is invariant by f so that f(x) e A. Thus c Rfw.x. As we have already noted, it is there exists a rectangle possible to choose an integer k > N + m such that
,, n R,,,2...,a #
0. This implies that fkRfTX n Rp.-X # 0 and so f"Qx n Qx # 0 where n = k m > N. This shows that any x e A is nonwandering. Let us next prove that the periodic points off are dense in A. In fact, by the previous argument, for any x e A and any square Qx containing x, we have f"Qx n Qx 96 0 for some large n. The map f" contracts the horizontal sides and expands the vertical sides of Qx linearly. From this expansion we deduce that there exists a horizontal segment 1k in Q. for which f"1', c 1k. From the contraction we deduce that there exists a vertical segment 1 in Qx such that f"1, 1,. Thus, !k n 1, is a fixed point of f", that is, a periodic point of f. This shows that the periodic points are dense in A. Before we can call A a basic set it remains to prove that there is a dense orbit in A. According to the criterion which we gave when we defined basic sets, in order to have transitivity in A it is sufficient that the stable and unstable manifolds of any
QX x
fn QXn Q Figure 48
0 General Comments on Structural Stability. Other Topics
165
periodic orbits should have nonempty intersection. But this is just what happens in this example because the stable manifolds contain horizontal segments and the unstable manifolds contain vertical segments right across the square Q. We deduce that the basic sets of f : Ss -+ S2 are 51,,112 = A, 113. The transversality condition is immediate since n is a repelling fixed point and 523 is an attracting fixed point. Thus f satisfies Axiom A and the transversality condition.
We remark that a construction like the horseshoe can also e made in dimensions greater than two [108], [66], [70], [72]. EXAMPLE 5. We describe here another significant example of a C' diffeo-
morphism on the torus T2 that satisfies Axiom A and the transversality condition. It is due to Smale (see [109], [122]) and known as the DA diffeo-
morphism, meaning "derived from Anosov". Its nonwandering set
is
hyperbolic and consists of two basic sets: a repelling fixed point and a onedimensional attractor which is locally homeomorphic to the product of an
interval and a Cantor set.- We start with in Anosov diffeomorphism g: T 2 -+ T 2 induced by a linear isomorphism L of [I 2, via the natural projection
x: @82 -+ T 2 as in Example 2. Let v' and v" be a contracting and a repelling eigenvector of L, respectively. Let e and e" be the vector fields on T2 defined
by e(n(x)) = d7t(x) v' and a"(rr(x)) = dn(x) 'v". On T2 we consider a Riemannian metric for which {e'(p), a"(p)} is an orthonormal basis of 7'(T2)p, for 'each p eT2. Hence, dg(p) e(p) = .ie'(g(p)) and dg(p) e(p) _ pe"(g(P)), where A and p = 1/A are the eigenvalues of L. Notice that e and e are C' vector fields and their orbits, which. are the leaves of the stable and the unstable foliations, respectively, are dense in T2. Now we consider a diffeomorphism f of T2 satisfying the following properties:
(1) f is equal to g in the complement of a small neighbourhood U of the fixed point po = n(O) of gi (2) f preserves the stable foliation of g inducing the same map on the space of leaves; i.e. f (W'(p)) = W'(g(p)) for each p e TI; (3) po is a repelling fixed point for f ;
(4) if we define a, /3: T 2 -. R by df(p) e(p) = a(p)e'(f (p)) and df (p) e(p) = p(p)e'(f (p)) + pe"(f (p)), then #2 <(,U1 - 1)(t - 1)2, where A = sup I /3(p) I for p E T2, and there exists a neighbourhood V of Po in the unstable manifold of po such that 0 < a(p) < a < 1 for some constant a and all p E. T2 - V.
Before proving the existence of a diffeomorphism f satisfying the four properties above, we shall describe its dynamics and show that it satisfies Axiom A. Let Lo denote the leaf of the stable foliation of g through the fixed point po. From properties (3) and (4), we have that the restriction of f to Lo is an expansion near po, a contraction on Lo - V and V c W"(po, f).
166
4 Genericity and Stability of Morse-Smale Vector Fields
It follows that f has one and only one fixed point in each connected component of Lo - {po}. These fixed points, p, and P2, are saddle points, and since
Lo - {po} = W'(p,, f) v W'(p2i f), we have that W'(p;,.f) is dense in the torus for i = 1, 2. If we set A = T2 - W"(po, f) and since V c W"(po, f), we conclude that S1(f) c (po) u A and that A contains the closure of the unstable manifolds of pi for i = 1, 2. We claim that the converse is also true and that, in fact, W"(po, f) is dense in T2 and thus A is the closure of the homoclinic points of pi for i = 1, 2. To prove the claim we start by observing that, since W'(pi) is dense in T 2, it intersects W"(pj) for i, j = 1, 2. Hence,
W"(pi) accumulates on W"(p) for i, j = 1, 2. Let now p e A and W be a neighbourhood of p. Let I c W be any small interval contained in' the stable manifold of some periodic point for g. Because of property (2), the leaf L, of the stable foliation which contains I is preserved under a power off. Since f - t expands L, in the complement of V and Lt is dense, there exists an integer n such that f -"(I) n V # 0. This proves that p is in the closure of W"(po, f). But p does not belong to W"(po, f) and so either W"(p,, f) or W"(p2, f) intersects W and, therefore, both of them intersect W. This proves that W"(po, f) is dense in T2 and that both W"(p,,.f) and W"(P2s f) are dense in A. From the density of W'(pi, f), we conclude that W contains transversal homoclinic points associated to pi as claimed. In fact, no component of W"(pi, f) n W can be contained in a stable leaf. Otherwise, by taking negative iterates, we get part of the local unstable manifold of W"(pi, f ) along the stable foliation, which is clearly impossible. Thus, the homoclinic
orbits associated to pi for i = 1, 2 are dense in A. In particular, Q(f) = {po} u A. Now let us prove that A has a hyperbolic structure. For a, p e R
with - I <- a < a + p < 1 and p e A, consider the cone Cp(a, p) = {xe'(p) + ye"(p); y 0 0 and a 5 x/y 5 a + p}. From property (4) it follows
that the image of Cp(a, p) by dfp is the cone Cf(p)(a', p'), where a' = (a(p)/µ)a + P(p) and p' = (a(p)/p)p. Since (a(p)/p) < d/u < 1, we have that n"a o df f - "(p)(C f - "(p)(-1, 1)) is a one-dimensional subspace EP e T(T 2) Clearly, dfp(ED) = Elf' (P). Thus, for each p e A, we have a decomposition E® Q ED of the tangent space of T2 at p, where EP is the subspace generated by e(p). Such a decomposition is invariant under the derivative of f and E' is contracted by a. It remains to show that dfp uniformly expands vectors in
E. To see this, let us estimate the slope of the subspace E. Let v = xes(p) + ye"(p) be a vector in Ep and let v" = dJ p "v = -(p)) + .
y"e"(f -"(p)). From the definition of EP, we get that On the other hand, xn- 1
= a(J -"(p)) X. + P
-"(P))
because v,_., = df (f -"(p)) v". Thus, IX"-1I <
alz"I+ - fnralin> 1_
5 1 for all n >- 0.
§4 General Comments on Structural Stability. Other Topics
167
By induction, for all n z 1 we conclude that X
X"
Y
Yn
+
µ)i
P i=
5 GY
+"-t
I
µ i=o l1/
Yn
1 - (1/p)n
Yn+lt 1 -(Ma) Hence, I x/y I < P/(p - 1). Now let us prove that df p uniformly expands vectors in E. Let v = xes(p) + ye"(p) be a vector in ED and let 6 = JCes(f (p))+ ye"(f (p)) be its image under df p. By property (4) and the expression above, we have that 116112
_ X2 + Y2
Ilvll2
x2 + y2
92}2
x2 + Y2
=
2
(x2/y2) + 1
(a/
µ2
132/(Et - 1)2) + 1
> 1.
This proves our assertion. We leave the reader to verify that the bundles Es and E" are continuous (see Exercise 44). Let us show that the periodic orbits are dense in A. This fact follows from the density of transversal homoclinic orbits proved above and Birkhof's Theorem stating that such homoclinic orbits are accumulated by periodic ones. However, we are going to present a very instructive proof following the so-called Anosov Closing Lemma. Let p e A and W be a neighbourhood of p. Let R c W be a closed rectangle, which contains p in its interior, whose bout}dary consists of four intervals: It and 12 contained in stable leaves, Jt and J2 transversal to the stable foliation. Each connected component of the intersection of a stable leaf with R is an interval which we call a stable fibre of R.
Since peA, f -"(R) intersects R for infinitely many values of n e N. For n sufficiently big, f -"(R) is.a very long (in the stable direction) and very thin rectangle fibred by intervals contained in the stable leaves. We can assume that f -"(R) n R is connected for, otherwise, we can shrink R in the stable direction, as indicated in Figure 49. Hence, we may find R and n e N such
Figure 49
168
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 50
that, for any stable fibre I c R, f -"(I) contains a stable fibre of R. If Ix intersects J, at a point a(x). Since a: J,.-+ J1 is continuous, it has a fixed point y; i.e. f -"(I,) contains I. The restriction of f" to the interval f -"(I,) is a continuous map denotes the stable fibre through x e J1, then f
onto I,, and, therefore, has a fixed point q. Thus, we have found a periodic point off in W. Using the same criterion as in Example 4 above, the reader can prove the existence of a dense orbit in A. Notice that A contains the onedimensional unstable manifolds of the points p, for i = 1,'2 (in fact, it contains the unstable manifold of any point p e A). Transversally, along the stable foliation, A locally contains a Cantor set. Indeed, let p e A and let I be a small interval along the stable leaf through p such that al c: W"(po, f). Finally, we prove the existence of a diffeomorphism f : T2 - T2, satisfying the four properties we mentioned at the beginning. Let U be a neighbourhood of po and :p: U -+ 682 be the local chart whose inverse is given by qp-'(x1, x2) 7G(xiv' + x2 v"), where Us and u" are the unit eigenvectors of L. Then, we have
(P ° g 0 lp-'(x1, x2) _ (As,, µx1). Let ': 68 -+ R be a C`° function such that O(68) c [0, 1], '(- t) = fi(t) f o r all t, i'(t) = I f o r I t j < }, 'G(t) = 0 for
j t j > 4 and tj'(t') 5 0(t) fort' z t >- 0. Now we set f = gin the complement of U and f = (p-' ° F ° (p, where F.(x1, x2) _ (F'(x1, x2), F2(xt, x2)) _ (axi + (2 - ).)iIi(x2)0(kx1)xi, µx2). In-this expression k is a positive real
number chosen so that j(aF'/ax2)(x)j < (µ -
for all x =
(x t, x2). Clearly, the origin is a repelling fixed point for F. It remains to show
the existence of a neighbourhood W e qp(U) of the origin such that W is contained in the unstable manifold W"(0, F) and supx, w (aF'/ax1Xx) = a < 1. Indeed, since ip is an isometry, after constructing W we just take V = (P- '(W). Let J, = (s; (tF'/axixs, t) Z 1) and .Ix, = {(x1, x2); x1 e Jx,}. We have that J, c J,, if t > t' 2: 0 and J, is either empty or a symmetric interval, say J, = [-a a,). Since (3F'/axixxi, x2) z I for all x, a J,,,,, and F'(-xl, x2) = -F'(xl, x2), it follows that F-'(Ix,) is a symmetric interval whose length is smaller than or equal to that of Ix,. Therefore, F"'(Ix,) is
§4 General Comments on Structural Stability. Other Topics
169
contained in I(1/µ)x2. Clearly 1 o c W"(0, F) and, thus, Ix2 c W"(0, F) for c x2 small. Hence, Ix2 c W"(0, F) for all x2 a [-1, 1] because I(1/µ)x2. Since the union of the intervals Ix, for x2 a [- 1, 1] is a compact set
and W"(0, F) is open, there exists 0 < a < 1 such that W = {(x1, x2); (aF1/ax1Xx1, x2) > a} is a neighbourhood of 0 in W"(0, F). It is now immediate to verify that f = q o F o (p satisfies properties (1) to (4) listed above. Notice that we set V = cp-1(W) as the neighbourhood mentioned in property (4). Thus, we have constructed a DA diffeomorphism in the torus T2. Remark 1. Let p, be another periodic point of the Anosov diffeomorphism g considered above. We can modify g simultaneously on neighbourhoods of po and pt in order to obtain a diffeomorphism f Axiom A with a nonwandering set that consists of three basic sets: a repelling fixed point
po, a repelling periodic orbit 0(p1) and a nonperiodic attractor A. Per. forming the same construction along different periodic orbits of g, we can get several examples of nonconjugate hyperbolic attractors (the number of' periodic orbits of a given period might be different ...). Remark 2. The same methods can be used to construct Axiom A attractors in
higher dimensions. We start with Anosov diffeornorphisms whose stable manifolds have dimension one and perform modifications similar to the ones above.
EXAMPLE 6. We now show that, even for the sphere S2, it is possible to construct an Axiom A diffeomorphism with a nonperiodic attractor. The result is due to Plykin (see [5] for references and a nice picture). Let 0: 612 -. 682 be the involution O(x) = -x. Since cb(Z2) = Z2, d induces an involution t of T2 = R2/Z2. Notice that tl has four fixed points: po = n(0, 0), pl = n(j;, 0), p2 = n(J, #) and P3 = n(0, 1). Let V, be a small cell neighbour-
hood of each of the points p,, 0 5 i 5 3, such that ,1(V) = V, and let N = T2 - U V. The restriction of t to N is an involution without fixed points. Let - be the equivalence relation on N that identifies two points if they belong to the same orbit of q. Let M be the quotient space N/- and p: N - M be the canonical projection. Considering M with the differentiable structure induced from N by p, we can see that M is diffeomorphic to the complement of four disjoint discs in the sphere S2. M corresponds to the shaded region, with the identification of the corresponding sides, indicated in Figure 51. We have that p is a two-sheet covering map. That is, p is a local diffeomorphism with each point having two pre-images. Therefore, if f : N -. N is a differentiable mapping such that A = q,f, then it induces a differentiable mapping f :
M - M such that pJ = fp. As in Example 5 (see Remark 1), we can construct a DA diffeomorphism I on the torus having po as a repelling fixed point and IN, P2, P31 as a repelling periodic orbit (of period three). Furthermore, it is easy to see that we may construct I so that Jil = tJ, since we can
170
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 51
start with an Anosov diffeomorphism g which is induced by a linear isomorphism and, hence, commutes with ri. For instance, start with the diffeomorphism of T2 induced by the linear isomorphism of 682 given, in canonical
coordinates, by L(x1, x2) _ (2x, + x2, x, + x2). Now, we take closed neighbourhoods V; of the points pi so that , (V1) = Vi, V; contained in the unstable manifold of p, for 0 < i < 3. We have that 7 is a diffeomorphism
of N = T2 - U, Vi onto f (N) c N, and A = n,,o ln(N) is the DA attractor. Thus, l induces a diffeomorphism f of M onto f (M) c M. We can extend f to the sphere S2, by putting a source in each of the four discs in the complement of M, one of them being periodic of period three. Then, it is not hard to see that S)(f) consists of these repelling fixed or periodic orbits and A = p(A), A being a one-dimensional attracting basic set. This last assertion follows from the fact that p is a two-sheet covering and PI = f p.
Now we give the results on structural stability of diffeomorphisms satisfying Axiom A and the transversality condition. Let d'(M) c Diff'(M)
§4 General Comments on Structural Stability. Other Topics
.
171
be the subset of these diffeomorphisms with M compact. First, Robbin [92], [93] proved that any f e.sd2(M) is structurally stable. Then the same result was proved in [61] for f e sd'(MZ). This left the question of the stability of the diffeomorphisms f e .4'(M) for M of any dimension. This was settled positively by Robinson [97], and in [95], [96] he proved the corresponding result for vector fields. An important question, that still has no general solution, is the converse of the above results: if f is structurally stable, does it satisfy Axiom Aand the
transversality condition? In the particular case where f2(f) is finite, f e Diff'(M) is structurally stable if and only if f is Morse-Smale [79]. In the general case where f2(f) is not finite partial results were obtained by Pliss [86] and Mafle [53]. Using a very original idea, Maflb [54] has recently solved the question for two-manifolds. Another result in this direction is due
to Franks [21], Guckenheimer [26] and Mafle [52]. They introduced the concept below from which they obtained a characterization of stability similar to the one we have just formulated. We say that f e Diff'(M) is absolutely stable if there exists a neighbourhood V(f) e Diff'(M) and a number K > 0 such that, for every g e V(f), there exists a homeomorphism It of M with hf = gh and llh - 1110 s Kllf - gll0 In this expression I is the identity map of M and II 110 denotes Co distance. It was shown that if f e Diff'(M) is absolutely stable then f satisfies Axiom A and the transversality condition. Franks also observed from the proofs of structural stability of the diffeomorphisms f E .e(M) that these diffeomorphisms are absolutely stable. We can therefore say that f e Diff'(M) is absolutely stable if and only if f e .ar(M). In studying (structural) stability of diffeomorphisms and flows, special attention has been devoted to stability restricted to nonwandering sets. The motivation for such a study is based on the idea that the dynamics of a system is ultimately concentrated on the nonwandering sets. There lie all limit sets and in particular the periodic and recurrent orbits. We say that f e Diff'(M) is f2-stable if there exists a neighbourhood V(f) c Diff'(M) such that, for any g e V(f ), there exists a homeomorphism h: (2(f) -+ f2(g) with hf (x) = gh(x) for all x e i2(f ). An important concept here is that of cycles in the nonwandering set. Let f satisfy Axiom A and let 0 = f2t v
.U
S2k be the decomposition of 0 = f2(f) into basic sets. A cycle in f2 is a sequence of points p 16 f2k,, p2 a Elk, , . I Pk. C- f1k, = flk, such that W'(p,) n W"(p, + 1) #
0 for 1 5 i S.s - 1. In [110] Smale proved that if f satisfies Axiom A and has no cycles in f2(f) then f is fl-stable. The corresponding result for vector fields is due to Pugh-Shub [91]. Later Newhouse [69] showed that it is enough for f2-stability to require the hyperbolicity and nonexistence of
cycles on the limit set. More recently, Malta [50], [51] weakened this hypothesis to the Birkhoff centre, which is defined as the closure of those orbits that are simultaneously a- and w-recurrent. In the opposite direction, it is known that if f satisfies Axiom A and there exist cycles in n(f) then f is not f2-stable [76]. There remains the problem: if f is f2-stable does it
172
4 Genericity and Stability of Morse-Smale Vector Fields
satisfy Axiom A? There are results for 0-stability analogous to those described above characterizing structural stability.
Another natural question is whether the structurally stable diffeomorphisms and vector fields are dense (and, hence, open and dense) in Diff'(M) and P(M). The first counter-example to this was given by Smale [111]. f2-stability is also not generic as is shown by the examples of AbrahamSmale [2], Newhouse [68], Shub and Williams [104] and Simon [105]. Although they are not in general dense in Diff'(M) or F'(M) the stable systems are still plentiful as we shall indicate. In the first place, they exist on any differentiable manifold; the Morse-Smale diffeomorphisms are typical
examples. More than this, Smale [112] showed that there exist stable diffeomorphisms in every isotopy class in Diff'(M). That is, any diffeomorphism can be joined to a stable one by a continuous arc of diffeomob phisms. Another interesting fact is that any diffeomorphism can be C° approximated by a stable diffeomorphism (Shub [102]). Thus, in the space
of C diffeomorphisms, r > 1, with the Co topology, the stable diffeo, morphisms are dense. The isotopy classes in Diff'(M) that contain MorseSmale diffeomorphisms were analysed by Shub-Sullivan [103], FranksShub [23] and, in the case of surfaces, by Rocha [94]. In this last work a characterization is given in terms of the vanishing of the growth rate of the automorphism induced on the fundamental group. The types of periodic behaviour possible for stable diffeomorphisms in a given isotopy class are studied in [22]. For vector fields, Asimov [8] constructs Morse-Smale systems without singularities on every manifold with Euler characteristic zero and dimension bigger than three. Of course, this is possible for twodimensional surfaces (the torus and Klein bottle). Somewhat surprisingly, Morgan [60] showed that this cannot be done for certain three-dimensional manifolds.
Perhaps the most important generic property so far discovered in Dynamical Systems is due to Pugh. By combining the Kupka-Smale Theorem
with his Closing Lemma. Pugh [89] proved the following theorem: there
exists a generic set 9 c Diff'(M) such that, for any f e T, the periodic points of f are hyperbolic, they are dense in S)(f) and their stable and unstable manifolds are transversal. A very important question is to determine whether or not the same result is true in Diff'(M), r >_ 2. This result is known as the General Density Theorem.
Diffeomorphisms that satisfy Axiom A and, in particular, Anosov diffeomorphisms have also been studied using measures that are defined on
their nonwandering sets and are preserved by the diffeomorphisms. The branch of Mathematics, related to Dynamical Systems, that uses this technique to describe the orbit structure of a diffeomorphism is called Ergodic
Theory. It has its origin in Conservative Mechanics, where the diffeomorphisms that occur usually have the property that they preserve volume.
§4 General Comments on Structural Stability. Other Topics
173
The Ergodic Theory of diffeomorphisms that satisfy Axiom A begins with the work of Anosov, Sinai and Bowen. The reader will find accounts of this theory in [7], [9], [10], [120]. ' Another active area of research is Bifurcation Theory, which has been considered by several mathematicians as far back as Poincar6. Roughly speaking, the theory consists of describing how the phase portrait (space of orbits) can change when we consider perturbations of an initial dynamical system. In particular, how can we describe the equivalence classes of systems
near an initial one under topological eonjugacies or topological equivalences (unfolding). Questions of a similar flavour appear in other branches
of Mathematics, like Singularities of Mappings and Partial Differential Equations (see, for instance, [17], [28], [42], [99], [118]), but we will restrict ourselves to a few comments on bifurcations of vector fields and diffeomorphisms. A common point of view is to determine how the phase portrait of a system depending on several parameters evolves when the parameters vary. Bifurcation points are values of the parameters for which
the system goes through a topological change in its phase portrait. An increasing number of results is available in this direction, especially for one parameter families (arcs) of vector fields and diffeomorphisms. In order to give an idea of this topic, we will describe two of these results, the first being of a more local nature than the second. Let I = [0, 1] and let M be a compact manifold without boundary. We indicate by d the space of CS arcs of vector
fields : I -+ 1'(M) with the Cs topology, 1 < s < r and r _>_ 4. From the fact that generically (second category) a vector field is Kupka-Smale, we would expect for a generic arc e si that gy(p) E 3r'(M) is Kupka-Smale for most values of u e 1. If d(uo) is not Kupka-Smale for some po e 1, then a
periodic orbit of (po) is not hyperbolic or a pair of stable and unstable manifolds are not transversal. Moreover, if the arc is not "degenerate", either only one periodic orbit of (po) should be nonhyperbolic or all periodic orbits should be hyperbolic and only one pair of stable and unstable manifolds should be nontransversal along precisely one orbit of t(uo). The lack of hyperbolicity of a singularity p of s(uo) is due to one eigenvalue (or a pair of complex conjugate ones) of at p having real part zero. The lack of hyperbolicity of a closed orbit y is due to one eigenvalue (or a pair of complex conjugate ones) of DP at p e y having norm one, where
P is the Poincar6 map of a cross section through p. In both cases, let us denote this eigenvalue by k We now describe the phase portrait of µ for p near po. To do this we must assume certain nondegeneracy conditions on the higher order jets of gy(p), which are not discussed here. A crucial fact is the existence for all u near po of an invariant manifold for gy(p) associated to A.
It is called the centre manifold and it has dimension one if A is real and dimension two if not (see [40]). Essentially, the bifurcation occurs along the
centre manifold; normally to it we have hyperbolicity (see [80]). In the pictures below hyperbolicity is indicated by double arrows.
174
4 Genericity and Stability of Morse-Smale Vector Fields
Singularity.
(a) A = 0. In the central invariant line two saddles collapse and then disappear (or vice versa). This is called a saddle-node bifurcation.
Figure 52
(b) A = bi, b # 0. In the central invariant plane a hyperbolic attracting singularity becdmes nonhyperbolic but still attracting, then it changes into
a hyperbolic repellor and an attracting closed orbit appears. This is called a Hopf bifurcation.
Figure 53
Closed Orbit. We will restrict ourselves to the phase portrait of the associated Poincare map.
(a) A = 1. This is the analogue of the saddle-node singularity. The dots in the picture are just to stress the fact that the orbits of the Poincare map are discrete.
Figure 54
§4 General Comments on Structural Stability. Other Topics'
175
(b) A = e, 0 < ii < n. This is the analogue of the Hopf bifurcation for a singularity: after the bifurcation there appears a circle which is left. invariant by the Poincare map. This circle corresponds to a torus which is left invariant by the flow of the vector field.
'Figure 55
(c) A = -1. In the central invariant line, a hyperbolic attracting fixed point becomes nonhypxrbolic but still attracting, then it changes into a
hyperbolic repellor and an attracting periodic point of period two appears. This is called a flip bifurcation.
Figure 56
A nonhyperbolic singularity or closed orbit as above is called quasihyperbolic.
For a nontransversal orbit of intersection of a stable and an unstable manifold, we require the contact to be of second order (quadratic). Figure 57
Figure 57
4 Genericity and Stability of Morse -Smale Vector Fields
I
Figure 58
indicates a second-order contact (saddle connection) between a stable and an unstable manifold of two singularities of a three-dimensional vector fielA. Figure 58 indicates a saddle connection of two closed orbits, drawn in a two-dimensional cross-section. We can now state the following result, which is mostly due to Sotomayor
[115]; parts of it have been considered by several other authors, like Brunowsky [11]. There is a residual subset of arcs -d c .sat such that if E 5 8 ' then ( s) is Kupka-Smale for p E I except for a countable set p,, ... , µ,,, .... For each n c- N one of the following two possibilities hold. Either s(µ,) has all periodic orbits hyperbolic except one which is quasi-hyperbolic and their'stable and unstable manifolds meet transversally or all periodic orbits are hyperbolic and their stable and unstable manifolds meet transversalay except along one orbit of second-order contact. Let us now consider the concept of stability for arcs of vector fields and state one of the results that have recently been obtained in this direction. Two arcs It" c d are equivalent if there is a homeomorphism p: 1 - I such that for each p e I there is an equivalence h between y(p) and '(p(p)). Moreover, we demand that the homeomorphism h. should depend continuously on It E I. Let q t c/ denote the subset of arcs of gradient vector fields on M. The following recent theorem to appear is due to Palis and Takens: an open and dense subset of arcs in !r are stable. We refer the reader to [1], [5], [6], [27], [34], [58], [78], [84], [117] for accounts of this topic.
Etctsrs 1. Show that every Morse-Smale vector field on a compact manifold of dimension S has an attracting critical element and a repelling one. 2. Let X be a Kupka-Smale vector field on a compact manifold of dimension n. Show. that, if the nonwanddring set of X coincides with the union of its .critical eldnc ts' then X is a Morse-Smale field.
177
exercises
s3. Let J M" -.
be a C't' Morse 'function, that'ls, J has a finite ntrn ber of nondegenerate critical'points. Suppose that -,wo critical points always have distinct images. Show that there exists a°filtratlon for X.7 rd t.
4. Let X by a gradient-field on a compact manifold gUlimension n. Show that X can. be approximate4by a Morse= Smale field. I. We say that a differentiable furictlion f : M - 02 is a'first integral of the vector field X e X'(M) if df(p) - X(p) = 0 for all p e M. Show that if X is a Morse--Smale field then every first integral of X is bonsttnt. 6. Let M be a compact manifold of dimension n and let .X be a C' yector field on M two hyperbolic singularities p and q, p being an attractor'and q a repellor. Suppose that, for all.x e M - {p, q} the cry-limit of x is p and the a.hmit js q. Show that h1 is homeomorphic to S".
7. Let X e 3'(M2) be a Morse-Smale field. We say that X ita polar field if X hag only one source, only one sink and has no closed orbits. Let X e X'(M2) be a polar field, bounds a disc containing with sink p and let Cx be a circle that is transversal to p. Show that, in A orientable manifold of di>ztension Wo, polar fields'X and Y are topologically equivalent if hind only if there, exists a'homeomorphism.h: Ca Cr with the following property: x, y e Cy belong to the.un';lfble manifold of a saddle
pf Y if and only if h(x), h(y) t long to the unstable manifold of a saddle,of Y '(G, Pleat*s).
8. Show that two:potar fields on the torus ire topologically equivalent. ;9. Shdty that there-exiltt Iwo polar fields on the sphere with.two.handles (pretzel) that
are not topologically equivalent..
10. Describe all the e4uwalence classes of,pplir fields on the pretzel. Hint. See Example 10 in this.chapter, Section 1. . . y a 11. Consider & closeU disc D (= 022 and lµ C be its boun4at'y. A chordal system in q is a finite collection of disjoint arcs in D each of which joins two points of C and is tianSVusal to Cat these points. Show that, for any chordalsystem in D there exits a vector field X transversal tq C with the following propertiesi (i.) the-arcs are contained in the unstable manifolds of the saddles of X and each arc contains just one saddle; (ii) each connected component of the complement ofthe unionof the arots contains just one source; (iii) the a-limit of a point in C is either a single source or a-single saddle. ' .
'
12. Let X and Y be vector fields on the discs D 1 acid D2 associated to two chordal systems
as in Exercise It. Let h: 6D1 -+ 3D2 be a diffeomorphism such that if p e eD, is contained in' the unstable manifold of 'a saddle of X then h(p).is contained+in the. stable tpanifold of a sink of - Y. By giueing D1'and D2 together by means of h we obtain a field Z on S2 that coincides with X on D, and with - Y on D2. Show that (a) Z is a Morse-Smile field, (b) every Morse-Smale field without closed orbits on S2 is tbpolpgically equivalent to a field obtained by this construction.
178
4 Genericity and Stability.of Morse-Smale Vector Fields
Figute 59
13. Let X and Y be Morse-Smale fields on S2 with just one sink and such that the orbit structure on the complement of a disc contained in the stable manifold of the sink is as shown in Figure 59. Show that'X and Y have isomorphic phase diagrams but are not topologically equivalent 14. Describe all the equivalence classes of Morse-Smale fields without closed orbits on S2 with one sink, three saddles and four sources. 15. Describe all the equivalence classes of Morse-Smale fields without closed orbits on the torus T2 with one sink, two sources and three.saddles,
16. Consider a vector field X whose orbit structure is shown in Figure 60. The nonwandering set of X consists of the sources fl and f2, the saddles s, and s2 and the sinks p, and p2. Prove that X can be approximated by a field which has a closed orbit. This shows that X is not f2-stable.
17. Let X be a Kupka-Smale vector field with finitely many critical elements on a compact manifold of dithension n. Show that, if the limit set of X coincides with the set of critical elements then X is a Morse-Smale vector field. 18. Show that the set of Kupka-Smale fields is not open in X'(M2) if M2 is different from the sphere, the projective plane and the Klein bottle.
Dint. Use Cherry's example (Example 13 of this chapter).
I:
179
19. Let X e r(M2 ) be a field whose singularities are hyperbolic. If G is a graph of X and
p E M has to(p) = G then tae exist, a neighbourhood V of p such that a(q) = G for every q e V.
20. Show that, if M2 is a n.prientabk manifold and X e 3``(M2) is structurally stable, then X is Morse-Smale. Show also that if M is nonorientable and X e Xt(M2) is structurally stable"alien X is Morse-Smale: 21. Show that, if f e Diff'(M) is structurally stable and has a,finite numler of periodic poinp, then f is Morse-Smale.. , 22, L.et.X'(R2), r 2t1, be the set of vector fields on R2 with the Whitney topology (fee Exercise 17 in,Chapter 1). Show that there exist' structurally stable vector fields in
T(R2) Hint. Consider a triangulation of the plane and construct a field with singulirities at .the centres of the triangles, edges and vertices.
Remark. In [63], P. Mendes showed that there exist stable vector fields and diffeomorphisms on any open manifold. 23. Show that, if y is a nontrivial recurrent orbit of a field X on a nonorientable manifold of dimension two, then there exists a transversal circle through a point p e y.
24. Show that, if h: S' -. S' is a homeompephism that reverses orientation, then h has a fixed point.
25. Show that a vector field without siagularitie's on the Klein bottle has only trivial recurrence.
26. Show that the set. of Morse-Smak difeomorphisms is not dense on any manifold of dimension two. 27. Show that the set of Morse-Smale fields isnot dense on any manifold of dimension greater than or equal to three.
28. Show that if the limit set of a diffeomprphisnt or a vector fiekl.consists of finitely many orbits then the 13irkhoff centre consists of (faiitelymany) fixed and periodic orbits. (The Birkhoff centre is the closure of tJose orbits that are simultaneously w- and a-recurrent.)" t 29. Show that if a diffeomorphism or a vector field has finitely many fixed and periodic orbits, all of them hyperbolic, then there can be no cycles between these fined and periodic orbits along transversal intersections of their stable and unstable mamfoids.
30. Show that a structurally stable difleomorphism or vector field whose limit set consists of finitely many orbits must be Morse-Smale.
31. Show that the sets of Anosov d;ffeomorphisnts and of Morse-Smale diffeomorphisms are disjoint.
32. Show that the only compact manifold of dimension two that admits an AnosoN diffeomorphism is the torus. 33. Show that the Anosov ditTeomorphiim of Proposition 4.1 of this chapter has an orbit dense in T2. 34. Give an example of an Anosov diffeomorphism of the'torus T" = St x . . k S'.'
180
(3eneritgty and Stability ofMorse-Smale Vector Fields
35. Using the General Density Theorem'show that. an Attosov diffeomorphism satisfies Axiom A.
36. Show that on any manifold of dimension two there exists a diffeomorphism satisfying
Axiom A and the transyersality condition and .possessing an infinite number'of periodic points.
Hint. Use Smale's Horseshoe in S2 and a Morse-Smale diffeomorphism on M2.
37. Show that if f,: M, --. M, and f2: M2 - M. are two diffeomorphisrris that satisfy Axiom Aand the transversality condition then so does f : M, X M2 -, M, X M2 where f (p, q) = (f,(Cp),f2(q))
38. Let f M -. M be a diffeomothism which is C' stable and satisfies Axiom A. Prove that f must satisfy the Transversality Condition. 39. Show that any diffeomorphism satisfying Axiom A has an attracting basic set. 40. Show that, if a diffeortlerphia n f of a (compact) manifold M satisfies Axiom A, then the union of the stable mani(blds of the points in the attracting basic sets off is open and dense in M. ,
41. Show that in any manifold there is an open set of C' (r > I) diffeomorphisms isotopic to the identity that do not embed in a flow. Recall that a diffeomorphism f is isotopic to the- identity if there is a continuous arc of diffeomorphisms connecting jwith the identity. A flow of C' diffeomorphisms is a continuous, group homomorphism q+: (9 , +) (Diff'(M), ). We say that f embeds in a flow if f = (p(1) for sothe flow (p.
42. Show that two Cr (r 2 l) commuting vector fields on a surface of genus different from zero have a singularity in common (E. Lima). Ifint.'First show that a C' vector field on a surface can only have finitely many nontrivial minimal sets. For r >_ 2, this follows from the Denjoy-Schwartz Theorem since there can be only trivial minimal sets. Notice that, in Chapter 1, we posed the problem above for the 2-sphere. The corresponding question in higher dimensions Seems to be wide open. 43. Show that, in an orientable surface M2, the set of vector fields with trivial centralizer
contains an open and dense subset of 1 °(M2) (P. Sad). The vector field X has trivial centralizer if for any Y e I(M2) ' such that [X, Y) = 0 we have Y = cX for some c e R. Restricting to Morse-Smale vector fields (or; more generally, to Axiom A vector fields), the same result is true for an open and dense subset in higher dimensions. Notice that a pair of commuting vector fields generates an P82-action. The concept of'structural stability for W -actions is considered in [12].
44. Let X be a C' vector field on a compact manifold M. Suppose X has a first integral which is a Morse function. Show that X can be C' approximated by a Morse-Smale vector field without closed orbits. Recall that f :-M -- 9 is called a Morse function if all of its critical points' are nondegenerate or, equivalently, if allsingularities of grad f are hyperbolic. To be a first integral for X means that f is constant along the orbits of X.
45. Let f e Diff'(M) and A c M be a closed invariant'set for f. Suppose that there exist a Riemannian mStric on M, a number 0 < 2 < I and for each x e A a decomposition
181
Appendix: Rotltion Number and Cherry Flows
TM, = Es ® E, such that Dfr(E') = Ef(,j, Df:(Ex) =.Ef(x), IlDfxvll 5 A(Ivll for all v e Ex and llDf s ' wll S All wll for all w e Ef(X,. Show that A is hyperbolic, that is, the subspaces Ex and Ex of TM, vary contimiously with x. 46. For all n >_ 2 there exists f : S' - S", satisfying Axiom A and the Transversality Condition, whose nonwandering set contains a repelling fixed point and a nonperiodic attractor. Hint. Use Example 6 of Chapter 4, Section 4.
47. Show that the conclusion of Exercise 46 holds for all manifolds M of dimension
aZ2. 48. Show that the set of stable diffeomorphisms of S2 is not dense in Diff'(S2). Hint. Use Exercise 38 and a modification of Example 6 of Section 4: 49. Showthat the set of stable diffeomorphisms is not dense in Diff' (M) for any manifold
M of dimension n z 2.
Appendix : Rotation Number and Cherry Flows We shall now give the details of the construction of Cherry's example that we mentioned in Example 13 of this chapter. Firstly we shall show the existence of C°° fields on the torus that are transversal to a circle E and have
exactly two singularities: one sink and one saddle, both hyperbolic. The Poincare map of such a field is defined on the complement of a closed interval
in Y. and extends to a monotonic endomorphism of E of degree 1. The concept of rotation number, introduced by Poincar6 to study the dynamics of homeomorphisms of the circle, extends to such endomorphisms. We shall show that these fields have nontrivial recurrence if and only if the rotation numbers of the endomorphisms induced on E are irrational. As the rotation number varies continuously with the endomorphism, the existence of fields with nontrivial recurrence follows. Let n: 682 --* T2 be the covering map introduced in Example 2 of Section 1, Chapter 1. Thus it is a C°° local diffeomorphism, n(x, y) n(x', y') if and
only if x - x' e Z and y - y' e Z and n([O,1] x [0,1)) = V. If X is a C°° vector field on the torus we can define a C0° field Y
n*X on iY' by the
expression Y(z) = (dnr)- 'X(n(z)). Clearly the field Y defined like this satisfies the condition .
Y(x + n, y + m) = Y(x, y),
V(x, #E R2,
V(n, M) a V.
(*)
Conversely, if Y is a C" vector field on the plane satisfying condition (*) then there exists a unique C' field X on the torus such that Y = z''X. We can thus identify the vector fields on the torus with the vector fields on 68Z satisfying condition (*).
1 82
4 Genericity and Stability of Morse-Smale Vector Fields
Figure 61
Let `( be the set of vector fields X e II(RI) satisfying the following conditions:
(t) X(x + n,y + m) = X(x.y);d(x,y)aR2,(m,n)E1Z; (ii) X is trans,. ersal to the straight line (0) x 18 and has only two singularities p, s in the rectangle [0, 1] x [0, 1] where p is a sink and s a saddle, both hyperbolic; (iii) there exist a, b e l with a < b < a + I such that if y e (b, ,a + 1) then the positive orbit of X through the point (0, y) intersects the line {1} x R in the point (1, ff(y)) while if y e (a, b) the positive orbit through (0, y) goes directly to the sink without cutting {1} x K; (iv) limy -t, f (v) _ +oo andlimy-.+t f.(y) = +co.
In Figure 61 we sketch the orbits of a field X e W. We remark that it follows from condition (iii) that the (f)-limits of the points (0, a) and (0, b) are one and the same saddle. If (1, c) denotes the point where the unstable manifold of this saddle meets the line { 1) x K we can extend fx continuously
Figure 62
Appendix: Rotation Number and Cherry Flows
183
to the interval [a, a + 1] by defining fx(y) = c if y e [a; h]. Condition (i) implies that fx(g + 1) = fx(a) + I so we can extend fx continuously to R by defining fx(y : -, n).=. fx(y) + n if y e [a, a + 1 ] and n e Z.Let us denote by X the vector field on the torus induced by X E , that is
X = n'X.'Then X has exactly two singularities, both hyperbolic; n(p) is a sink and n(s) is a saddle. Moreover, X is transversal to the circle Y. _ n({0} x R) and the Poinvare map PX defined on n({0} x (b, a + 1)) can.be extended continuously to Z. The map fx is a lifting of P1. In fact, *: k E defined by i(y) = n(0, y) is a covering reap and i - fx = PX < n. Lemma 1. ' is nonempty.
PROOF..Consideq the vector field Y(x, v) _ (2x(x + 2), - y). The nonwandering set of Y consists of two singularities, a saddle (0, 0) and a sink (-3, 0). It is easy to check that Y is transversal to the unit circle all points of the arc C =' {(x, y); x2 + v2 = 1, x 5 f} See Figure 62. Let Z(-y, Y) _ Jc0(x, y)(2x2 + x) + (1 - (p(x y)Xx 2 + 1). - y i
where cp is a Cm-function such that (p(R2) c [0, 1]. rp(x, V) = 1 if x > i or if (x, y) E U, cp(x, y) = 0 if x < 4 and (x,.y) E 1R' - V. Here i and V are small neighbourhoods of C with U c V. If V. is sullicicntly ,rn,iil, the non.wandering set of Z is empiy. Take T > 0 such that 7,(C) c x.> l} for all t a T. Using the How of Z we can define a diffconior; hism H. (0.-1) x (0, 1) ;+ W c 1 2 by H(x, y) = Z, (h(i')) where h: [(l. j -, C is a di11'eo-
morphism. If ze(0, 1) x (0, 1) we define X(z) =,dH-'(1->i(:))- 1(II(0). As Z= Y in a- neighbourhood of C and in {(x, F);x > ',),, we have X(:) (1, 0) if z belongs to a small neighbourhood of the boundary of the rectangle
[0, 11 x [0,' 1]. We can now extend X to R2 by defining. X(z) = (1 0) if z beldngs to the boundary of [0, 1 ] x [0, 1 ] and V(x + ii, y + m) = A (x, y) if (x y) E°[0°1) x' [0, 41 and (n. to) E Z.2. One, checks im 'ediately that X satisfies conditions (i)'(iii).' Condition (iv) follows from the fact that the ,trace of 'dX(s) is kositive as the reader can verify. (Although thi is, not necess ry we can assume that X is linear in a small neighbourhood of s.)
Lemma 2. There exists, a vector field X e
such that, dfx(y) .- I for all
ye(b,a + 1). PRooR Take Y e "t " with Y(x, y) _ (I, 0) if 3 - x < 1. As f i. takes the interval
(b. a + 1) diffeomorphically onto an interval of length 1 and, by condition (iv) f;.(y) 54 near b and'a + 1, it follows that there exists a diffeomorphism.
f : (b, a + 1) -. (fy(b), fr(a + 1)) such that f'(y) > 1, bye (b, a + 1) and. fXy) = fy(y) if y is near b or a,+ 1. It remains to show the existence of a field
&19 withfx=f. Let rp,; (c, c +.1) - (c, c + 1) be given by (p,(y) = (1 - t)y + up(y) where cp = f e f Y ' and c = each t e [0, 1], cp, is a diffeomorphism and tp,(y) = y if y is near c ore + 1. Let a: [1,1] -+ [0, 1] be a (,'O function
1$4
4 Genericity and Stability of Morse-Smale Vector Fields
such that a = 0 in a neighbourhood of 4 and a = 1 in a neighbourhood of 1.
(4, 1) x (c, c + 1) given by
Consider the map H: (4, 1) x (c, c + 1)
H(x, y) _ (x, cp,(x)(y)). Then H is a diffeorntorphism. Let X be the vector field defined by
X(x, y) = dt(W '(x. y)) Y(H ' (x, y)) if (x, y) E (4, 1) x (c, c + 1), X(x + n, y + m) = X(x, y) if (x, ti-) e (, 1) x (c, c + 1) and (n, m) E 711,
X(x, Y).= Y(x, y) if ((x, y) + 1Z) n (j, 1) x (c, c + 1) = 0.
It is easy to check that X E6 and that fX = W f, _ 1. Lemma 3. Let f, g: R -+ R be monotonic continuous functions such that f (x + 1) = f (x) + 1 and g(x + 1) = g(x) + I for all x E R. Then (i) p(f) = lim".,X (f"(0)'n) exists and I(f"(O)/n) - p(f)I < 1/n; (ii) lim"_. (f "(x) - x)/n exists.for all x e lR and is equal to p(f ). (iii) p(f) = m/n with in, n e 71, it > 0 if and only if there exists x e R such that
f"(x)=x+m:
0 such that rf h f - glio = supxER 1f(x) (iv) given e > 0 g(x) I < 6 then I p(.f) - p(g) I < E; (v) p(f + n) = p(f) + n for any integer n. .
PROOF. Let Mk = max.. R (f k(x) - x) and m,t = minx E R (f k(x) - x). We claim that Mk. - mk < 1. In,fact, As f (x + 1) = f (x) + 1 we have f k(x + 1) _ f k`(?c) + 1. Therefore, cp = f k - id is periodic with period 1. Consequently there exist xk, Xk E R with 0 < xk - Xk < I such that cp(xk) = m{ and yi(Xk) = Mk. Since fk is also monotonic nondecreasing we have fk(Xk) < fk(xk). Hence Mk + Xk 5 Mk + Xk and so Mk - Mk < xk - Xk < 1 which ptaoves our claim.
We are nOw going to prove that
x
fk(y) - y - 1 _<.fk(x) In
fact,
fk(y)
)
Vx,yER.
(1)
fk(y) - y + 1. We next put 3' = 0 and x = fk''- "(0) in (1) and obtain fk(O) _ I s flo(p) _ flw'
11(0)
fk(o) + 1.
.
Thus
(fk(0) - 1) <
n(f'`(0) != i
j-1
(fki(O) -'f
n(fk(0) + 1):
From this we deduce that of k(0) - ms: f k"(0)
of k(0) + n.
We now divide by kn to obtain f'YO) k
I
k5
fk"(0) kn_ < .fk(0) k
1
+k
IV
Appendix: Rotation Number and Cherry Flows
f
{k"\0)
5 k.
k
(?)
Similarly 1
kn f_(O)
n fn(O)
n
The sequence f k(0)/k is a Cauchy sequence because If k(0)/k - f "(0)/n 11 9 Ilk + 1/n and to it converges to some limit p(f). By letting n tend to op in (2) we see that I p(f) - f k(0)/k I < 1/k which proves (i). t
By putting y = 0 iv (1) we obtain fk(0) - 1 < fk(x) - x < fk(0) +.1.
Thus f k(0)
I
f k(X) - X
k
k
k
f k(0) +
I
k
k
This shows-that (f k(x) - x)/k converges to p(f) and proves (ii).
To prove (iii) suppose that there exists x c f such that f "(x) = x + to with m, n e Z, n > 0. It follows easily by induction that f k"(x) = + kM Thus p(f) = limk...(f k"(x) - x)/kn = limk- m km- kn = m/n. Now let p(f) = m/n and suppose if possible that f "(x) A x + m, d X E A. Thus f "(x) - x > in,
V x e R (or f"(x) - x <em, V x e R). As f", - id is periodic there exists
a > 0 such that f"(x)-x _ m + a (orf"(x)-x<m-a)VveR.Thus fk"(x)-x - km+ka(or fk"(x)-x_ (m + a)/n (or p(f) < (m - a)/n) which is a contradiction. To prove (iv) we remark that
k0)
I P(f) - P(g) I S P(g)
+
- f (0)
I
If
P(f )
1 gk(0) - .f k(0) 1,+
Fix an iftteger k such that 2/k < e/2 and choose 8 > 0 such that Ig(0) f k(0)I < ke/2 if jig f Ila, < b., Then {p(g) - P(f) I < x if jig - f IIo < b. It remains to prove that p(f + n) = p(f) + n if n e Z. By induction we have (f + n)k(x) = f k(x) + kn. Thus p ( ,f
+ n ) = lien
(.f + n)k()
Jt-M
k
= li en k-.p
- ---- = P(f) +
f k(0) + kn k
n.
Lemma 3 allows us to introduce the rotation number for degree I mono-
tohic endomorphisms of the circle. In fact, let.n: 2 - S' be the covering map n(t) = exp(21tit). An endomorphism a: S' - S' is monotonic and has degree 1 Wand only If it has a lift x: R
R which is a continuous monotonic
function satisfying *x + )) = x(x) + k. We then define the rotation number of a as p(x) - np(a"). This does not.depend on the choice of the lift i because of Lemma 3(v).
4 Genericity and-Stability of Morse.Smale Vector Fields
186
.(y) > 1 for all y e (b, a + 1) and P(fx) is irrational.
PROOF. Choose X° e'B such that df,9(y) > 1 for' all y e (b, 'a + 1)' and X°(x, y) = (1, 0) if < x < 1. We shall construct a family of vector fields Xa le such that fxA = fxo + A. Let H: (1,1) x J - (4, 1) x R be given by ff(x, y) = (x, y + a(x)e,} where a: [4,1] -+ .[0, 1) is a C°° function such that a(x) = 0 if x is near I and a(x) _ I for x near 1 ` We define X't(x, y) = dH(H
y)) X °(H 1(x, y)) if (x, y) a (4, 1) x I8,
XA(x + n, y) = X''(x, y) if x e (4, 1),
XA(x,y)=X°(x,y)
if
(x+Z)n(2,1)=0.
As H(x, y + m) = H(x, y) + (0, m) for all m e Z. we have X x e 16. It follows
immediately that fxx = fxo 4- A. Consider the map h: R -+ 08 given by h(A),= P(fxx). By Lemma 3, h is continuous and h(l) = h(0) + 1. Thus there exists A e [0, 1] such that h(A) is irrational. It now suffices to take X = X. vector field Y'on. the torus with the following Theorem. There exists a orbit stracture. (1) Y has exactly two singularities, a sink P and a.saddle S, both hyperbolic; (2) W°(P) is dense in T2 and the compact set A = T2 - W5(P) is transitive, that is there exists q e A- - {S} with w(q) = a(q) = A; {3), if q e T 2 - (A u {P}) then.cQ(q), P and a(q) c A; (4) there exists a circle Y. transversal to Y such that E n A is 'a Cantor set.
PROOF. Take Ye X°°(T2) such that n*Y - X satisfies the con"icns'of Lemma 4. It'is clear that Y satisfies (1) with P = ri(p) and S = a(s), The circle -1 = ir({0} x l) is transversal to Y'and fx is the lift of the Poincarb map P'y: E E. If Q is the first.point_ufintersection of the unstable manifold of with E then Q = i0. c) and P.r--t(Q) = n({0} is [a, b]) IT w(Q) = S
or w(Q) = P there dxists a e N such that ,.,Q) = Q. Thus of X(c) = P",. n(c) = P"Y Q) = Q = ft(c) and so there exists me Z such that f X(c) = c + m. This contra'dicts the fact that p(fx) is .irrational. Hence `Y has no , saddle connection and A = w(Q) does not contain P. A similar argument shows that Y has no closed orbit. . We now show that W'(P) is dense in r. For this it is enough to prove
that the compact set K = E - W3(P). has empty interior. So we shall suppose, if possible, that the interior of .K is nonempty and take a maximal
interval J. c: K. Let J. = PX(J) = Px(J"_ ). As J n Ws(P) = 0, J, is a compact interval longer than J.-, Since J is a maximal interval and Y has intervals J" are pairwise, disjoint. This contradic ion no proves that W'(P) is dense. According to the Denjoy-Schwartz Theorem, Y has only trivial minimal Let us check this fact directly. Suppose L is a nontrivial minimal set for Then L n E-is a compact set disjoint from both the stable and the unstable
_
Appendix: Rotation Number and Cherry Flows
187
manifolds of the saddle S for otherwise L would contain the saddle. Since
W'(P) is dense and L n W(P) = Qs, W5(S) is dense in L. Thus if J is a maximal interval in E - L then PY(J) n J # 0 for some n. As the endpoints of J belong to L we have P"Y(J) = J which implies the existence of a closed orbit. This contradiction shows that P and S are the only minimal sets of Y.
The intervals 1" = PY (Q) are pairwise disjoint and U
i int(1")
W'(P) n Y. is dense in E. The end-points of the interval l" belong to different components of W'(S) - {S}. Hence W'+ (S) and W'_ (S) are both dense in
L - W'(P), where W+(S) and W'-(S) denote the components of W'(S) {S). It follows that W+(S) and WS (S) are both dense in A = T2 - W'(P) and so the, a-limit of any point in W'(S) - {S} is A. Let q e T2 - (A u P). As a(q) is a compact invariant set and the minimal sets of Y. are S and P it follows that a(q) contains S. Thus, either a(q) = St or a(q) contains some point of the stable manifold of S in which case *q) = 14r, It is easy to see that
A n E is a perfect set with empty interior. Thus A n 'T, is a Cantor set. Finally, since (W'(S) u W"(S)) n E is countable and A n E is uncountable, there exists q c- E - (W'(S) u W"(S)). Hence a(q) = co(q). = A. 0
Remark. Although the vector field Y in this theorem is not structurally stable we can describe completely all the topological equivalence classes of vector fields in a small neighbourhood . A( of Y. In fact, if Z e X and .,r is small enough then the circle E is still transversal to Z and the Poincare map Pz has derivative greater than 1 at all points of its domain. It follows that, if Z e ,,t ' is Morse-Smale, then Z has just one closed orbit and this is a repellor. (There certainly are Morse-Smale fields in .;V because they are dense in X°°(T?).) If Z is not Morse-Smale there are two possibilities: either Z has
a saddle-connection which is a repeHor and the nonwandering. set of Z reduces- to this saddle connection and the sink or else the rotation number of the endomorphism of E induced by Z is irrational and in this case Z satisfies conditions (1) to (4) of the theorem. It can be shown that if the endomorphisms of E induced by two fields Zt and Z2 near to Y have the same rotation number and this is irrational then Z, and Z2 are topologically equivalent. There is a partial converse: if there is a topological equivalence h
near, the identity betviteen Zt and Z2 and the rotation number of 'Z1 is irrational then the rotation number of Z2 is equal to that of Zt It is actually enough to require that h is homotopic to the identity. Thus, in a neighbourhood -of Y v'e have just one equivalence class of structurally stable fields, ' one equivalence class of fields that have a saddle-connection and infinitely many equivalence classes of fields that have nontrivial recurrence. These
last are characterized by a single real parameter, the rotation number of the endomorphism induced on E.
It is good to emphasize the impossibility of describing completely the topological equivalence classes in a neighbourhood of a field without
188
4 Gegericity and Stability of Morse-Smale Vector Fi"
singularities on the torus having a. nontrivial recurrent orbit. In fact, if Y is such a field there exists a circle E transversal to Y and the Poincare map Pr is defined on the whole circle L According to an important theorem of Herman [37], we can approximate Y by a field X that is C°° equivalent to an irrational flow. Then we can approximate X by a field with all orbits closed. It follows that the number of closed orbits of the Morse-Smale fields in any neighbourhood of Y is unbounded. Thus, in any neighbourhood of Y there are infinitely many equivalence classes of structurally stable fields.
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[871 V. PLISS, Sur la grossierete des equations differentielles definies sur le tore, Vestnik LGU, ser. mat. 13, (3), 1960. [88] C. PUGH, The closing lemma, Amer. J. Math., 89, 1967. [891 C. PUGH, An improved closing lemma and a general density theorem, Amer. J. Math., 89, 1967. [90) C. PUGH, On a tlieoremof P. Hartman, Amer. J. Math., 91, 1969. [911 C. PUGH and M. SHUB, The fl-stability theorem for flows, Invent. Math., 11, 1970.
[92) J. ROBBIN, A structural stability theorem, Ann. of Math., 94, 1971. [93] J. ROBBIN, Topological conjugacy and structural stability for discrete dynamical systems, Bull. Amer. Math. Soc., 78, 1972. [94] L. ROCHA, Characterization of Morse-Smale isotopy classes on two-manifolds, Anais Acad. Brasil. Ciencias, 50, 1978. [95] C. ROBINSON, Structural stability of vector fields, Ann. of Math., 99, 1974. [96] C. ROBINSON, Structural stability of C' flows, In: Dynamical Systems Warwick 1974, edited by A. Manning, Lecture Notes in Mathematics, vol. 468. SpringerVerlag, 1975. [97] C. ROBINSON, Structural stability of C' diffeomorphisms, J. Diff. Equations, 22, 1976.
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[99) D. RUELLE, Dynamical Systems with Turbulent Behavior. Lecture Notes in Physics, vol. 80. Springer-Verlag, 1978. [100) A. SCHWARTZ, A generalization of a Poincare-Bendixson theorem to closed two-dimensional manifolds, Amer. J. Math., 85, 1963. [101] M. SHUB, Stabilite globale des systemes dynamiques, Asterisque, 56, 1978. [102) M. SHUB, Structurally stable diffeomorphisms are dense, Bull. Amer. Math. Soc., 78, 1972.
[1031 M. SHUB and D. SULLIVAN, Homology theory and dynamical systems, Topology, 14, 1975.
[104] M. SHUB and R. WILLIAMS, Future stability is not generic, Proc. Amer. Math. Soc., 22, 1969. [105] C. SIMON, Instability in Diff'(T3) and the nongenericity of rational zeta functions, Trans. Amer. Math. Soc., 174, 1972.
Refcences
'
193
[106] S. SMALE, On gradierit dynamicai systems, :nn. o/ alath., "4. 196E {107] S. SMALE. Stable manifolds for differential cqu!i o:i :tn,i diffcomorphisms. Ann. Scuela Sup. Pi_a, 17. 1963.
[108] S. SMALE, Diffeomorphisms with many periodic Combinatorial Topology.. Princeton Univ. Press, 1965 [109] S. SMALE, Differentiable dynamical systems, Bull. Aou
IT,
Differential and
t-4ath Soc.. 73, 1967.
[110] S. SMALE, The f2-stability theorem. In: Global A:ialh : ,. Proc Symp. in Pure Math:, vol. XIV. American Math. Soc. 1970. [11.1] S. SMALE, Structurally stable systems are not dense, Am, , J. Math., 88, 1966. [112] S. SMALE. Stability and isotopy in discrete dvnamic,,! stems In - Dynamical Systems, edited by M. Peixoto. Academic Pres. i975 1113) S. SMALE, Essays on dynamical systems, ecdnomic processes aid related topics. The Mathematics of Time. Springer-Verlag, 1980
11141 J. SOTOMAYOR, Generic one parameter families of vecto
fields on twodimensional manifolds, Pub!. Math. Inst. Hautes Etudes Scici li ignes. 43, 1973. [1,15) J. SOTOMAYOR, Generic bifurcations of dynamical systems. In: Dynamical Systems, edited by M. Peixoto. Academic Press, 1973. [116] J. SOTOMAYOR, Lifdes de equafoes Diferenciais Ordindrias, Projeto Euclides, IMPA-CNPq, 1979. [117] F. TAKENS, Introduction to Global Analysis. Comm. no. 2, Math. lnst' Univ. Utrecht, 1973. [118] R. THOM, Structural Stability and Morphogenesis. Benjamin-Addisort Wesley, 1975.
[119] C. ZEEMAN, Uma introducao, informal a topologia das superficies, Monogr? de Mat., 20. IMPA. 1975. 11201 P. WALTERS, An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79,
1982.
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[122] R. WILLIAMS, The "DA" maps of Smale and structural stability. In: Global Analysis. Proc. Symp. in Pure Math., vol. XIV. American Math. Soc., 1970.
Index
absolute stability Anosov Closing Lemma diffeomorphi$m flow
Characteristic polynomial 44 chart 5, 8 Cherry flow 137, 178, 181 chordal system 177
171
167
159, 161, 179
159
approximating by a C' map
21
atlas, C'and C° 9 attracting singularity 30, 119 attractor basic set 160, 161, 165, 169 closed orbit 35, 119, 130 automorphism of fundamental group 172 Axiom A diffeomorphism 160, 170, 172,
composition map 22, 36 conjugacy between diffeomorphisms 33,63 conjugate vector fields 26
180
Baire space 20, 37 Baire's Theorem 3 basic set 160 Bifurcation Theory 173 Birkhoff centre 38, 171, 179 Bolzano-Weierstrass Theorem
bounded continuous maps 60 bump function
circle 30, 113, 122, 136, 161, 179 class C' 2, 6 dosed orbit 11,18 Closing Lemma 151, 167, 172 commuting vector fields 34, 180 complete 20 complex spectrum 44 complexification 42, 46
8.21,%
canonical form 43, 44, 48, 69 Cantor set 164, 165, 168, 186 centre Birkhoff 38 manifold 173 Chain Rule 2, 6
31,
constant vector field 68, 94 contraction 48, 61 coordinate neighbourhood 5 critical elements (singularities and closed orbits) 139
104, 106, 118
point 3 value 3 cycle 35, 171 cylinder 34, 35
DA diffeomorphism 165 decomposition 160, 166 Denjoy flow 139 Denjoy-Schwartz Theorem 139, 180, 186 195
196
Index
densPts
gradient
of s'arle diffeomorplasms 172 of (" n+aps 22 of !Morse Smile nelds 112 141 of periodi_ points 157, 160
of transversal homochnic points
157,
4, 26
direct sum decomposition
160
eigenvalues 44, 45, 46, 53 elementary fixed point 59 embed in a flow 89, 180 embedding 7, 9, 73 endomorphism of the circle
differential equation
Grobman-Mart man Theorem 60,69r74, 87. 100. 158 Gronwalt's inequality 64
' Hartman's Theorem 60 ' Hausdorff 8 height function 15, 116, 133 honloclinic point .155, 166
Hopf bifurcation horseshoe 162
130
185
4
fixed point 59, 60, 72, 75 linear isomorphism 48, 50, 63, 156 linear veetor'field 46, 49, 50. 52, 54, 68
nonwandering set
160
periodic points 110, 114 set Ib0 sfngulasity 30, St, 64, 100, 104, 106, 116, 119.
48
a
exponential of a linear map 42 family of discs 85 fibration 94 fibre 129 filtration 124, 177 first integral 180 fixed point of a diffeomorphism
immersed subiganif9ld 99 immersion 3, 7, 9, 73 Impl,,icit Funotion Theorem
3, 55, 78
Inclination Lemma 80, index 46; 51, 52, 68, 119, 195 59
11
inner product 13 integral curve 3, 10, 11 invariant section
box 93, 142, 146 foliation 157, 165 fundamental domain 81, 86, 130
set
96, 130
31
Inverse Function Theorem 3. 6 irrational flow an the torus
12, 28, 35. 110,
117,132, 133
group 172 neighbourhood 82
isomorphic phase diagrams isotopy class 172
General Density Theorem 172, 180 generic property 91, 107, 115, 172 genus 152 geodesic flow 159 geometric theory 26 global flow II transversal section
174, 175
hyperbolic closed orbit` 95, 98, 104, 106, 1J6, 119,
equator 18, 37 Ergodie Theory 172 Euler-Poincare characteristic 152 existence and uniqueness of solutions of a
flow
139,
142. 179
derivative 2. ', 19 derived from Anosov 165 difl'eomorphism 2, 6, 31, 48, 59, 68, 153 differentiable manifold 4, 9, 112, 169 map 2, 6
expansion
13, 116, 133, 150, 153, 176
graph of a map 76, 79 transform 80 (union of separatrices and saddles)
166, 167
differential equation dimension 52
12. 28
vector field
111
124, 126
Jordan
canonical form 44 Curve Theorem* 1.6
Klein bottle 138, 146, 151, 172, 179 Kupka-Smale diffeomorphism
110, 112, 114, 154
197
Indbx
Theorem vector field
107, 110" 106, 113, 132, 134, 141, 153,
154, 176, 178
d-lemma limit set a- or a j-
80, 82, 86, 156
nonhyperbolic periodic orbit 173 nonorientable surface 148 nontransversat intersection 116, 173 nontrivial minimal set 139 recurrence
132, 133, 136, 137, 141, 144,
151, 181
11, 12, 15, 18, 26, 31, 57, 73, 88,
113,116,117,139 L. or L. 117, 179
linear contraction
nonwandering set 118, 154, 160 north pole 18 south pole diffeomorphism 161 south pole field 12, 29, 119
31, 37, 48
vector field 27, 41, 46, 54, 88 Lipschitz 60, 64, 80
local chart 5, 8 classification 68 conjugacy 59, 63, 66, 72, 90 diffeomorphism 6 flow 4, 40 orbit structure 39, 54 stability 40, 59, 63, 64, 67, 88, 98, 131
stable manifold 73 topological behaviour 40, 68 unstable manifold 73, 166 locally finite cover 8 long tubular flow 93
maps, space of C' 19 matrix 43, 68 maximal integral curve
0-stability 171 open manifold 179 openness of diffeomorphisms
orbit of a diffeomorphism of a flow 10, 11 structure 26, 31, 115 orientable surface 141
orientation of orbit
II
meagre 3 Mean Value Theorem 78 measure preserved by a diffeomorphism 172
meridians 12 minimal set 138 Mobius band 34, 122, 146, 148 Morse function 89, 177, 180 Morse-Smale diffeomorphism 154 vector field 116, 118, 122, 127, 141, 152, 153,176,177,178 multiplicity of an eigenvalue 45, 53
23
of Morse-Smale "fields
153
31
39
parallels of latitude 12 parameter 4 parametrized neighbourhood partial order 123 partition of unity 8 Peixoto's Theorem 115 period 11, 31 periodic
behaviour possible points
172
31,154, 157
perturbation 52, 57, 79, 173 phase diagram 123, 126 portrait 173 Plykin attractor 169 Poincar6 compacrification 37 map 92, 100, 111, 113, 130, 145, 173, 174, 183, 188
Poincar6-Bendixson Theorem 143
nilrnanifold 159 no cycles 171 nondegenerate bilinear form 89 nondensity` of Morse-Smale diffeomorphisms
polar fields 123, 177 polynomial vector fields 34
pretzel, 123, 133,,134;177' 154,
projective plane
35
156
of structurally stable diffeomorphisms 172
8
qualitative theory 10, 26, 153 quasihyperbolic 175, 176
18, 132, 138,
Index
198
rational flow on the torus 12, Z8, 117 real canonical form 43, 45, 48, 69 part of eigenvalue 46 recurrent 38, 132
regular orbit of a flow 11, 18, 135 point of a flow 40,68,93 point of a map 3 value of a map 3, 7, 24 reparametrization 97, 130 repelling closed orbit 30, 119, 130 singularity 30,-119 residual 3, 20, 37, 104, 107, 110 rotation number 181, 185 saddle
Ti
12, 28, 110, 116, 137, 146, 157, 161, 165, 192, 179, 181, 186
of dimension n 35, 159, 179 with a cross-cap 151 trajectory 10 transitive 160. 164 transversal circle 144, 146, 151, 179, 188 homoclinic point 155, 166
119, 166
21
separatrix 128, 133, 142, 148 simple singularity 55 singularity I1, 18, 54 162, 180
129, 130 92, 125 Flow Theorem 40, 68. 93, 95, 101, 145
flow
neighbourhood
of bounded maps 60, 158 of C' maps 19 spectrum 44, 52 sphere 12, 18, 37, 74, 113, 119, 132, 154, 162, 169, 177, 178, 181 splitting 46, 48, 60, 160
stability of arcs of vector fields stabilized separatrix 142 stable foliation 157, 165 manifold 73, 98, 161 Manifold Theorem 75 separatrix 128
176
structurally stable
diffe4wrdMst,!i-, 3 e354t 1S& 170 1
132, 139, 141
recurrence
tubular family
s1 ece
Ii
section 92, 96, Ill, 130, 142 Transversality Condition 161, 170, 180 Theorem 25, 56 trivial centralizer 180 minimal set 138, 186
sink
30, 122 Smale horseshoe solution 4 source 30, 137
23, 36, 56, 82, 85, 106, 108, 110, 116, 154
intersection
saddle connection 120, 122, 141, 142 saddle-node bifurcation 174 Sard's Theorem 3, 8, 25 saturation III section 92 separable
tangent bundle 8 vector 9 Thom Traoversality Theorem 25 topological embedding 73 immersion 73 topologically equivalent 26, 39, 187 torus
r*cta UW4 tbi1Wld+
sub anifold
'
4-1 =3 V_
6
9
unstable foliation 157, 165 manifold 73, 98, 161 separatrix - 128 vector fields, examples 28
vector field
3, 10, 23, 41, 106, 116
wandering point 118 Whitney Embedding Theorem topology . 37, 179
sub erion 3,d su
sus
Ies
113r1A
nsion
I 11, 112, 114, 133
Zorn's Lemma
139
9, 2:
