An Introduction to the Theory of Smooth Dynamical Systems [1 ed.] 0471901172, 9780471901174

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An Introduction to the Theory of Smooth Dynamical Systems

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An Introduction to the Theory of Smooth Dynamical Systems W. Szlenk Institute of Mathematics University of Warsaw

A Wiley— Interscience Publication

JOHN WILEY % SONS Chichester

* New

PWN-POLISH

York e Brisbane e Toronto e Singapore

SCIENTIFIC Warszawa

PUBLISHERS

Translation from the Polish original Wstęp do teorii gładkich układów dynamicznych published in 1981 by Państwowe Wydawnictwo

Translated

Naukowe,

Warszawa

by Marcin E. Kuczma

Graphic design: Zygmunt Ziemka Copyright © by PWN-Polish All rights reserved.

Scientific Publishers, Warszawa

1984

No part of this publication may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the copyright owner. $

Library of Congress Cataloging in Publication Data: Szlenk. W. An introduction

to the theory of smooth

dynamical systems.

Translation of: Wstęp do teorii gładkich układów dynamicznych. 1. Differentiable dynamical systems. I. Title. QA614.8.59413 1984 516.3'6 82-23771 ISBN 0 471 90117 2 British Library Cataloguing in Publication

Data:

Szlenk, W. An introduction to the theory of smooth dynamical systems. 1. Differentiable dynamical systems I. Title 516.3'6

II. Wstęp do teorii gładkich układów dynamicznych QA614.8 z

ISBN 0 471 90117 2

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Printed in Poland by W.D.N.

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English

To the memory

of

Professor Stanislaw Mazur

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Preface

The theory of dynamical systems has arisen from an attempt at an adequate description of phenomena in the surrounding world. Therefore it is very intimately connected with various domains of knowledge stemming directly from the needs of reality; we mean thereby mechanics, statistical physics (classical and quantum), electrodynamics, hydromechanics, even biology (the problem of evolution of populations). Owing to its universal character, the theory uses methods that resort to nearly all branches of mathematical science. The interplay between the diverse methods in all their variety goes unexpectedly deep, providing a large number of very specific techniques in the investigation of dynamical systems. But also, conversely, problems posed in the theory of dynamical systems penetrate other mathematical theories, giving them a fresh impulse and opening new problems. The methodology is so widespread in mathematics that, for instance, dynamical systems are not given a separate heading in the Mathematical Reviews. The closest relation exists between the theory of dynamical systems and the theory of ordinary differential equations, functional analysis, diflerential geometry, differential topology, algebraic topology, measure theory, probability theory, and number theory (especially diophantine approximation). The present book is not a monograph, though certain topics are treated in it with considerable generality. Most of the material presented here was included in lectures held by the author at the Mathematical Institute of the Aarhus University in 1971/72. The book is addressed to readers who are familiar with standard undergraduate courses in mathematics. Thus the general prerequisites are: linear algebra, calculus in several real variables, abstract measure and integral, basic concepts of differential geometry (differentiable manifolds, tangent spaces, geodesic lines), basic concepts of functional analysis (Banach spaces, linear operators and functionals), elements of topology and, last but not least, ordinary differential equations within the usual elementary scope. At the end of the book the reader will find

VIII

Preface

a list of notes containing the definitions which are omitted in the text,

together with brief explanatory remarks. It is not to be expected that anyone would learn those concepts from the notes. They are meant as a reminder; the basic and operational knowledge of the underlying theories should be gathered from other sources. Of a similar type is the first section in Chapter III, containing a brief survey of facts and notions of functional analysis needed in the exposition of the material of that chapter. On the whole they exceed the customary scope of university

courses;

their

presentation

in Section

3.1,

however,

can

hardly serve as a unique source of information. The reader should turn to specialized literature. The book contains a number of exercises. If the reader really wishes to learn something, he ought to devote some energy to solving them; one of the reasons is that they often provide proofs omitted in the main text. Exercises marked by a capital A contain such complementary proofs (or parts of proofs) to be used in the sequel; they are referred to in subsequent considerations. Those marked by a capital B serve just as practice material giving the reader the necessary skill in handling the theory. Almost all the exercises are accompanied by hints. Those marked with an asterisk are relatively difficult, in spite of the hints. If the reader wants to get some idea as to the kind of mathematics which is cultivated in this book we recommend him to read Section 1.2 through. This may encourage him to further reading; and perhaps it will just save him from unnecessary loss of time and effort. It is the author’s pleasant duty to thank Dr. M. Krych and Dr. M. Misiurewicz for their valuable criticism and comments, which have helped a great deal in the preparation of this book. W.

SZLENK

Contents

Preface

VII

Introduction .

1. Basic Concepts and Facts of the Theory of Dynamical Systems bE Introductory Notions and Basic Examples. RZA Survey of the Topics Studied in this Book 1.2.1. Classification of dynamical systems . 1.2.2. Generic properties 1.2.3. Stability of dynamical ie 1.2.4. Theory of entropy .3. Survey of Basic Facts of the TRS, at Ordinary DUEkntial Equations . . Vector Fields and Thejt a Gunes on Nomóobe ; . The . The

Hadamard—Perron Stable Manifold

Theorem and the

. . Uinstables Mowiiożia of

a Hyperbolic Critical Point. The Case of a Cascade . .]. The

.

19. 1.10. rit.

Stable

Manifold

and

the

Unstable

Manifold

of

a Hyperbolic Critical Point. The Case of a Flow . Linearization of a Dynamical System in a ee of a Hyperbolic Critical Point . : Behaviour of a Flow in a R bolninad s a Cisse Orbit : Suspension of a BiómÓWK 3 Limit Sets

2. Dynamical Systems on Manifolds of Dimension 1 and 2 A Survey of Basic Facts of Differential Topology . 22, Limit Sets for Flows on Two-Dimensional Manifolds 23. Index of a Critical Point of a Flow on R* . : 24. Homeomorphisms of the Circle. 23; Differential Equations and Flows on the Doras! 2.6. Algebraic Automorphisms of the Torus. 3. Transversality Theory and Generic Properties. Sol Survey of Facts and Notions of Functional 2 Elements of the Transversality Theory .

RONEM

X

Contents

3.3. Differentiation of a Flow on a Manifold with Respect to 148 the Generating Vector Field. . . . Manifolds 154 3.4. Critical Elentents of Vector Fields on om 3.5. Generic Properties. Genericity of ee AKMMAC? Ge |. FAR JPY 472 36. Genericity-ol Property" 57% Sie Genericity of Property 9.4.38 172 3.8. Non-Wandering Points. Closing (maa iGeneretna s 40 42. Ce eee Properties "7; and "Gs", 4. Structural Stability. Homoclinic Points . . . SOR GAOL EG . 186 the! 4.1. The Idea of Stability. Structural Stability . e 4.2. Structural Stability of Diffeomorphisms of the Gircłe: rot hes) 4.3. Structural Stability. Necessary Conditions. . . . . . 196 SENS FA NZOZ PIĘ 4.4. Morse-Smale Systems . . . 4.5. Properties of Morse-Smale Siem: PER Zhi 4.6. Structural Stability of Flows on Two- pimensan| Mani

„58% fOIAS* > RYTY, SPB RPNE COR NERWICE NOO Poe 9% ak 6 2022 A) gO Semen, Re 4.7. Hyperbolic Sets i 229 4.8. Anosov Diffeomorphisms . . . 4.9. Smale’s Counter-Example. The BE bf SARE StablesDynamical*Systems: M eS ys PAU PER 9 FEIE 4.10. GeodesicaF lows "20. TEAS OPPs. SUM. e as MORZA 4.11. Expanding Mappings ZG . . . . . 264 4.12. Axiom A Diffeomorphisms. Remar on > Stab += 1266 4.13. Homoclinic Points Mr” 5. Ergodic Theory of Topological preai ski Entropy „8213 5.1. Measure Preserving Transformations. hers . . . . . . 277 5.2. Birkhoffs Theorem. Ergodicity. 5.3. Invariant Measures for Topological Dynamical Systems 286 p JEZU 5.4. Uniquely Ergodic Dynamical Systems . . . 5.5. Invariant Measures for Smooth Dynamical S ems pnp se 5.6. 8.1... 5.8. 5.9.

Flows on the Torus T? (II). Linearization oowriEknta Metgics BorOBy, Ys oe zwani t ość . . . Topological Entropy. Properties of Entropy. nauczi eo sratarnik and ‘MetriczEntnopy, iovent® bus:

NOTES a Bibliography. (00S List: of Symbols

Te ts o Gai fe ee,

co.

Subject Index Altthor IRdeEX

go

0

es «4

ata

. . AE

ge a

ge

ey

eae Se

ies,

ce

e

. . aie

.

Tanlogend dete a SO re ee 5

ne ee

316 ee 328

O ES

Introduction

The theory of dynamical systems has its origin in astronomy. To be more accurate: in celestial mechanics. The general laws of motion of planets about the Sun have been fairly well known since the time of Newton and Kepler, and the same can be said of the shapes of orbits in the simplest cases. According to Newton’s second law of dynamics, the force acting on a mobile body is directly proportional to acceleration: in symbols

ax

ATA where X = (x', x*, x3) denotes the position of the body in the space, F =(f', f, f°) is the force and m is a scalar coefficient called the mass of the body. The easiest to handle is the case of a single mobile body subjected, in accordance with the general law of gravitation, to the action of the force mMx

Ge

>el”

G denoting the gravitation constant, M being the mass of a large “immobile” body which attracts the body considered, and m standing for the mass of the latter (M > m); ||x|| is the length of the vector X. As a model one can view the case of the sun and the earth. From the mathematical viewpoint, the problem consists in solving a system of three differential equations of the second order, which can easily be reduced to a system of six equations of the first order. An explicit solution is found, and it turns out that the small body runs along a conical curve having the large body at its focus. This conclusion is confirmed

by astronomical

observation,

provided,

however,

that

the

time period is not too long. For periods extending to thousands of centuries one must take into consideration the influence of other factors: gravitational effects coming from other planets, the relativity phenomena (change of mass) and so on. In the case where there are several bodies considered and their

2

Introduction

mutual gravitation cannot be disregarded, the problem is more complicated. In the case of N bodies the motion of the system is described by N equations

a dx, F.=m—,

ate

ps2

KIN,

in which the forces F, depend on the positions of all these bodies: 5

PIE.

x

N

OBO

=

>

>

m,m; (X, — x;)

eee j*i

Thus we get a system of 3N differential equations of the second order, equivalent to a system of 6N equations of the first order. Already for N = 3 the system is unsolvable in quadratures. All that can be done is to find an approximate solution by numerical methods. Such numerical solutions are of great importance when applied to concrete problems arising in astronomy and technology: forecasting the configuration of celestial bodies, launching space vehicles etc. But they are of little use from the viewpoint of cosmogony of the solar system. They give no information which might permit one to foresee the evolution of the solar system in the distant future. We are facing here a very typical problem of the theory of qualitative properties of ordinary differential equations. The French mathematician Henri Poincarć may be regarded as the initiator of that theory, and certainly the first to apply it in celestial mechanics. In his famous book Les mćthodes nouvelles de la mécanique céleste (Poincaré, 1892-99) he comprised a large number of ingenious observations and intuitions concerning mechanics and the qualitative theory of differential equations. Even today the book is a source of creative inspiration to many mathematicians. A system whose motion is described in terms of ordinary differential equations is termed a classical dynamical system. A thorough study of such systems shows that they are closely connected with the behaviour of iterates of continuous mappings of metric spaces (and, in particular, diffeomorphisms of manifolds) as well as with the theory of measure preserving transformations (which are the object of research in the socalled ergodic theory). In general, the theory of dynamical systems deals with the action of groups of continuous transformations of topological spaces, a special role being played by groups of ‘diffeomorphisms of manifolds and

Introduction

3

groups of measure preserving transformations; the ergodic theory may be regarded as a domain within that general theory. Henri Poincaré, who can justly be called the creator of the theory of dynamical systems, had numerous successors, whose contribution to that theory cannot be overestimated: G. Birkhoff, E. Hopf, A. N. Kolmogorov, V.I. Arnold, D. V. Anosov, J. Moser, Y. G. Sinai, S.

Smale and Ch. Pugh, to list the most prominent. The readers who wish to gain a more profound knowledge of the theory of dynamical systems are invited to consult original papers and survey articles: Abraham and Marsden (1967), Abraham and Robbin (1967), Coddington and Levinson (1955), Hartman (a Só Katok (1967), Smale (1967).

Basic Concepts and Facts of the Theory of Dynamical Systems

1.1. Introductory

Notions and Basic Examples

Let X be a metric space and let g*: X > X be a continuous semigroup’ of continuous mappings of X into X, ae%. Definition 1.1.1 Any such pair (X, 9”) is called a topological dynamical system. The term: dynamical system is also used with regard to another class of objects. Namely: Definition 1.1.2 By a metric dynamical system we mean a quadruple (X, M, u, p”), where (X, Mi, w) is a normalized measure space (X denoting the space, M a o-algebra and u a measure with u(X) = 1) and g* denotes a continuous semi-group of measure preserving measurable mappings of X into X, ae. Of course, in either case, the semi-group can actually be a group (with all @* invertible). Throughout this book we will be concerned only with topological dynamical systems, and specifically with smooth systems, i.e., systems in

which X is a differentiable manifold and g* are differentiable mappings. Therefore the term “dynamical system” will in the sequel always stand for a “topological dynamical system”. We shall distinguish two principal types of dynamical systems, according as the index semi-group Y is discrete or continuous:

LL WS 2°

°S (0,1, 250. of WOSZENO

FL 2

..j3

io beng

Introductory Notions and Basic Examples

5

invertible. Then the semi-group |g*, ae] is generated by g' = g, o” = identity, and the system can be denoted shortly by (X, g). Such a

system is called a cascade. 2. W= R* (the non-negative reals) or 2 = R (the reals), g* being invertible. Such a system is called a flow and is written down simply in the form (X, @,). Examples of dynamical systems

1.1.1. Let S' denote the unit circle and let p: S' >S' be the rotation by an angle 2na:

OZ=zoz,

zze”ieŚ! « zże

(multiplicative notation).

Fic.

1.1

The circle is homeomorphic to a segment with endpoints identified. In other

words,

the circle can

be viewed

as the real line, numbers

differing by an integer identified; ie. S' = R/Z. We say that the real line R is a covering space” for the circle and we call the assignment

xt+ p(x) =e?"* projection of R onto S'. The mapping @ induces, in a natural way, a mapping @ of R into R (this is the so-called operation of lifting; the map @ is termed: the lift of p to the real line*). Then 6 is of the form: P(X) =x+a+m,

m denoting an arbitrary integer. Regarding the circle as the interval Asin2nx;

it also

1.1.2

1.1.3. Let T? be the two-dimensional torus. We can define T? in two equivalent ways:

oT

= SiS) = {eaw) cess west.

2° Let I denote the group of all translations of R* by vectors with integer coordinates:

P= WWR"

2 RA ave wind) = (cla? an), een

We define an equivalence relation x~ ysdyer:

y(x) =

Clearly, [ is isomorphic to Z x Z.

~

in R? as follows:

Introductory Notions and Basic Examples

W

Denoting by [x] the coset (mod +) containing a point x=(x*, x’), we see that

[x] = |(X +n, x*+m): n, meZ}. (In Fig. 1.1.3, marked

points belong to [x].)

Fic.

1.1.3

Dividing R” by ~ we obtain the torus

R2/~ = R2/T = (RxR)Y(Z xZ) = R/ZXR/Z = SxS! =T?. The division by ~ corresponds geometrically to the following procedure (see Fig. 1.1.4): consider the square with vertices (0, 0), (1, 0),

Fic.

1.1.4

(1, 1), (0, 1) and identify pairs of points whose coordinates differ by an integer (e.g. x and y in Fig. 1.1.4; no points in the interior of the square are identified, of course). This means that identification involves opposite sides of the square. This can be visualized sticking first the side I, and I, (results a cylinder) and then J, with J,, ie. the sides, or rather the circles, forming the edges of the cylinder. As a result we get a surface resembling that of a tyre-tube.

Basic Concepts and Facts

8

Just as the line in the case of the circle, the plane R* is a covering

space (universal, in fact 2) for the torus T?, with projection p: R* > T* given by Dion. x) As (SRR

SC).

Quite analogously one can define the m-dimensional torus T™ having R” as its covering space.

Let p: TT? p(z,

w) =

be a mapping defined as follows:

(ZoZ, wow)

for

(z; w)e KS

where zy

Ga.

wę

=

ALS

Such a mapping is called a translation (on the torus). It is easy to see that a lift of @ to the plane is given by

$(x, y) =(x+a, y+). Regarding

T* as R*/T we rewrite the mapping ¢ itself as

p(x, y) =(x+a, y+f)(mod

1).

2555

FiG.

1.1.5

1.1.4. Let S* be the two-dimensional sphere. Consider the stereographic

projection z onto R” with centre at the “north pole” N. Let A: R? > R2 be the linear transformation

defined by the matrix

Introductory Notions and Basic Examples Then the mapping

Oi)

9

@ given by:

tee OAOn (x), KOD xESZ.xZN,

p(NIEGN is a diffeomorphism

of S? onto itself.

FIG

le Ie6

1.1.5. Let X = T? and define

o (2, WE (C784z ie wi Then (X, g,) is a flow. Regarding

T* as R?/I we rewrite g, as:

~,(x', x?) =(x' +at, x* + Bt)(mod 1) and denoting by 6, the lift of o, to the plane, we have

@, (x, x?) =(x' +at, x*+ Bt). Now, let (X, p”) be a dynamical arbitrary point.

Definition 1.1.3

system

and

let xeX

be an

The set

(x) = iy: dae MW: y = g* (x); is called the trajectory or the orbit of the point x (or: the orbit passing through x).

et

2

Zor

WZ:

Definition 1.1.4 A point xeX is called a periodic point of a dynamical system (X, g) iff there exists an integer p> 1 such that

(x) = x.

Basic Concepts and Facts

10

Any such p is called a period of x and the least such p is called the prime period of x. The set of all points periodic with respect to the transformation p will be denoted by Per(g).

Now

let Ad=R"

or A=R.

Definition 1.1.5 (1) A point xeX is a fixed point (or: critical point) of the flow (X, g,) iff

000 = x

for all teR*

[teR],

Lee

(x) = 1x}. (2) The trajectory y(x) is called periodic iff there exists a number z > 0 such that

0:43)

OX)

for all teR*

[te R].

(eed

Any such z is called a period of the trajectory and the least such T is called the prime period (provided it exists i.e. x is not a fixed point). A point x for which (1.1.1) is satisfied with some t >.0 is called periodic.

Examples 1.1.6. Let (X, g) be as in Example 1.1.1. If x is rational (i.e., zo is a root

of unity), p,qeZ.

then

every

point

zeS'

is periodic.

Indeed,

let « = p/q,

Then

p'(z) = z$z = (Cz = z, and so g? is the identity. The number q is a common period for all z; it is a prime period if the ratio p/q is irreducible. If a is irrational, there are no periodic points. Moreover, every trajectory is a dense subset of S'. We shall prove this fact. Let I =(a, b) be any open arc of the circle. Write z, = o"(z). The sequence z, has a cluster point in S', and so there are two terms z, and z, such that |z,—z,| < |b—a| (the symbol |x—y| denoting the length of the arc (x, y) with endpoints x and y). Denote J = (z,, z,) and put w

= g*~'. Since (z) =o' '(p'(z)) = g*(z) = zy, the intervals J and W(J) are adjacent. So are w(J) and w*(J) etc. (see Fig. 1.1.7). All the w"(J) have the same length; thus their closures cover the whole circle. Since

Introductory Notions and Basic Examples

11

their common length is less than |b—a|, there is an endpoint of some w"(J) which belongs to (a, b). That means, the trajectory y(z) intersects I, and this ends the proof.

ZI

Fig.

IDZ

11.7. Let (X, g) be as in Example 1.1.3. Let zo and wy be roots of unity, thus there are integers p, q > O such that z =w$=1. Then

(z, w) = (20'Z, wg'w) = (z, w), which means that every point (z, w) is periodic. Suppose that zo and wo are algebraically independent; thereby that

we

mean

ZWo =l>n=m= 0.

If zy = e7"*, w, =e?"/, then zę and wo are algebraically independent iff a and f are independent

over the ring Z:

na+mbeZ>n=m=0. It can be proved that then every trajectory is a dense subset of T* (see Exercise 12.2 0n=p."22); 1.1.8. Let (X, g,) be as in Example 1.1.5. If B/a is rational, then every trajectory is periodic and all the prime periods are equal. If f/x is irrational, then every trajectory is dense in T* (see Exercise 1.2.5 on

p. 23).

1.1.9. Let K! be the unit disc:

KA = tee, x): (x1)? +(x2)?< 1}.

Basic Concepts and Facts

12

following

the

Consider

system

of differential

= a;

a> 0.

equations

(in polar

coordinates):

—

=r(1—nr),

=

The general solution of this system is:

r(1)

Re

ACZ FAW

ne

10)

ee

where 0 < ro M, I = = R, y(t) = M, is an integral curve of the field ®(x) iff

= 6(y(1)), )(1) = a8 t

(1.1.2)

$(t) denoting the tangent vector to y at y(t). If ©(x) satisfies some additional conditions, e.g. is of class C! (ie. ©(x) depends on x smoothly in local coordinates), then the equation (1.1.2) has, for any given x,¢M, a unique solution satisfying the initial condition (0) = X,. (This is a consequence of an analogous theorem for regions in R".) Denote this solution by (Xp, t). If, moreover, M is compact, then the solution y(t) is defined for all teR (see Theorem 1.4.2). The group of mappings

Pr (Xo) = 7(Xo, t)

Basic Concepts and Facts

14 is a flow on

M which is said to be generated

by the vector

field ®(x).

An equation of the form (1.1.2) is termed a differential equation on M. The curves ~,(Xo) are solutions of that equation. Now, suppose we are given a system of particles in a stationary field of forces. Assume that the particles are not free, i.e., that there are some

constraints imposed upon their positions. E.g.: a) the particles have to be in contact with a surface, or: b) the distance between any two particles has to be invariant (the particles are then said to constitute a rigid body). Any configuration of the particles in the space at a given moment can be specified in terms of a certain number of numerical quantities q = (41, -.., Im), Which we call the coordinates of the system. As a rule, the space of all possible configurations (the so called configuration space) forms a differentiable manifold. The parameters q,, ..., q„ may then be regarded as chart coordinates. The number m, equal to the dimension of the manifold, is called the number of degrees of freedom.

Examples

1.1.10. Consider the motion of a flat pendulum. Each state is determined by an angle we. The configuration space is a circle.

Fic.

1.1.10

1.1.11. For a spherical pendulum the configuration space is a sphere (of radius equal to the length of the pendulum) 1.1.12. Now consider the so-called double flat pendulum. Each state is described by two angles o, W, as shown in Fig. 1.1.11. Thus the

Introductory Notions and Basic Examples

15

configuration space can be identified with the product of two circles, i.e.. a two-dimensional

torus.

mite

JCJĘJNI

Returning to general considerations, let M with the configuration space and let q(t)eM system at a moment £ (then t + q(t) represents in time). The state q(t) can be written in the

be a manifold identified denote the state of the the “path” of the system coordinates involved:

q(t) = (4° (0), ..., q"'(0)). The quantities

jjrate 40=v(0).

—q'(t) = v'(t),

denote the component

ing a thePema

velocities (in coordinates

q'); the vector

dą (t) OND)j= 5area is a tangent vector to the path q(t). The motion of our system is fully determined by prescribing the initial values of position go = q(0) and velocity v9 = v(0). The pair (qo, Vo) is an element of the tangent bundle T(M). The equations of motion are

dq(t) _ do(t)_F(q, Wh aok Pr

v),

F(*,*) denoting the vector of force. For any x =(q, v)eT(M)

let us

write ©(x) = (v, F(q, v))e T.(T(M)). Then the motion is described by the single equation dx(t) -% dt

(x(t)),

which determines a flow on T(M); motion occurs on the integral curves

of the above equation. In the most commonly encountered dynamical

Basic Concepts and Facts

16

systems of mechanics motion is energy preserving (eg. in Hamiltonian systems, see formula (4.10.2) on p. 250).

Subsets of T(M) which are invariant with respect to the flow and such that the energy of the system is constant at each of their points are, as a rule, compact submanifolds of T(M). Therefore special attention will be paid to flows on compact manifolds. In the case of a single flat pendulum the tangent bundle T(M) can be identified (in fact is diffeomorphic) with a cylinder, one of its crosssections standing for the configuration space; see Fig. 1.1.12, where all possible types of trajectories are shown. There are two fixed points m =e (110)

xs = (0,0) Ries

(glu

(equilibrium points): x, = (0, 0) (stable equilibrium) and x, = (1, O) (unstable equilibrium — the pendulum “sticks up”). If the speed is great enough, the pendulum swings round without changing direction; its motion is rotation (clockwise or anti-clockwise). The trajectory is of type y, (Fig. 1.1.12). If the pendulum oscillates about the stable equilibrium point x,, the trajectory is of type y,. There is one more type of trajectory: running from the point x, (starting at the moment — oo), via x,, again to x, (attaining it at +00) — this is a curve of type Y3°

In mechanics it is often more convenient to consider the manifold (4, p), p denoting the impulse vector, instead of (q, v). The manifold (q, p) is called the phase space.

1.2. Survey of the Topics Studied in This Book We now list the main topics which will be dealt with in this study. They belong to the most important problems considered in the theory of dynamical systems.

Survey of the Topics Studied in This Book

17

1. Classification of dynamical systems from the viewpoint of the topological properties of their trajectories (the concept of topological conjugacy). 2. Generic properties. 3. Stability of systems. 4. Theory of entropy. 5. Specific properties of certain classes of dynamical systems. 6. Invariant measures for smooth dynamical systems. Let us now briefly comment on the particular topics.

1.2.1. Classification of dynamical systems Definition 1.2.1 Two dynamical systems (X, g*) and (Y, V*), «eM, are said to be topologically conjugate iff there exists a homeomorphism h: X >Y such that the following diagram commutes (for all ae 2): gp”

X

——

x

|

ie

—_—— a

Y

y

ee

eles

This means that 9% =h-'ow*oh or hog* = oh for all a. We then say that h conjugates the two systems, or that h conjugates g” with vy”. Obviously, if Wis Z” or Z, the above condition is equivalent to

o=h

oyoh

sor

hop=VWoh

(p and W denote the generators).

Examples 1.2.1. Let X and Y be segments on the real line: X = , W: > , D = Diff' (1) and let g be as in Fig. 1.2.4. Since p has no fixed points inside (a, b), p'(a) £ 1, o'(b) £ 1, every Y close to p in C'-topology has no fixed points in (a, b), either. It

follows from the considerations of Example 1.2.1 that p and w are then topologically conjugate, and thus

is structurally stable.

Survey of the Topics Studied in This Book

21

1.2.4. Theory of entropy Entropy is a certain numerical quantity which characterizes the intensiveness with which the points in X are “mixed” by a transformation @. In fact, there are two notions of entropy: 1) topological entropy,

2) metric

entropy.

Metric entropy is considered in the measure-theoretic case and is defined for measure preserving transformations. For more information on this subject see Section 5.7. We now restrict our attention to the concept of topological entropy. Let X be, as usual, a compact metric space. Given a family € = (U;} of open sets covering X, (i.e., a cover of X), let N(é) denote the least number

equal to the cardinality of a subcover

of ć. N(6) is finite, by

the compactness of X. If ć = {U;} and 4 = (V;} are two covers of X, there is automatically defined a third cover which we denote by Ć v n: Śvn=1U,NV;.

Let p: X >X be a continuous mapping; then o '(6), defined as the family ip *"(U;)), also is a cover of X. Every cover € = (U;} then induces a sequence of covers of X:

ćvo '(6)=(U,no

*(U;,)}

and by induction

SCREENS) W Ap? ZU

DO SU,

JazzD0v

(6) (BJ,

i,, ..., i, ranging over all possible n-tuples of indices. It can be proved that the following limit always exists:

log N(¢,) = h(g, ¢) = lim o n+

26

The topological entropy of @ is defined by:

h(9) = an h(g, Ś) (cf. Section 5.7).

Basic Concepts and Facts

22 Exercises

1.2.1.A. Show that the following conditions are equivalent:

(a) e?"* is not a root of unity. (b) For any Riemann-integrable function f: S!—C

and any point

zeS' we have: 1 nod lim gad

| ime)

2n 1 } = ie | rena.

0

(c) Let I be an open arc on S’, let zeS” and let N,„(I, z) be the number

of all integers k, 0 4 k(b) for polynomials, then apply the fact that polynomials are dense in the family of continuous functions.

1.2.2.B. Let T? be the two-dimensional torus and let zg, Ww)eS'. Show that the following conditions are equivalent: (a) zw, =

(b) For

z,weS!

1l>r=s=0(r.sEZ).

any

Riemann-integrable

function

f: TY+>C

and

any

we have: 2n 2n

tim +" f(cbz, whw) = yt

Ray | ea

1

4n

||f(e*, &)dxdy. at

0;

(c) For any measurable l

le

0

set A c T* and any z, weS!

we have

1

lim —N,,(A, lm n(A, (2,(z, w)) =—|A|, =z—lAl

where N,(4, (z, w)) is the number of all integers k, 0 < k R*

be a linear

mapping

given by a matrix

that:

(a) p induces a mapping of T? = R?/I (see Example

1.1.3) into

itself iff a, b, c, deZ.

(b) p induces a bijection of T? iff detp = +1. In this case g is called an algebraic automorphism of T*. Hint. Let Q = «0, 1) x R" be a C'-mapping satisfying f(0) = 0. Prove that if all eigenvalues of the derivative df(0) are less than 1 in absolute value, then there is an ellipsoid B with centre at 0 such that for any x€B we have:

f"(o)eB for neZ*

and

lim f"(x) =0

Hint. For any sufficiently small e > O there is 6 > O such that f(x) = df (0)x +(x) where ||e(x)|| < e||x|| for ||x|| < 6. Apply Exercise 1.3.4.

1.3.6*.B. Let x denote the stereographic projection of the m-sphere S” onto R” having N as the north pole. Let A be a linear operator of R" into R”. Define a mapping g: S"—S”™ by:

p=n

'oAon.

(a) Prove that o is a diffeomorphism transformation of R”.

of S” iff A is a similarity

Vector Fields and Integral Curves

as

(b) Give examples of A for which: 1° @ is discontinuous, continuous but is not a diffeomorphism. Hint. Calculate the derivative dg at the point N.

2 og is

1.4. Vector Fields and Their Integral Curves on Manifolds Let M be a C”-manifold

(without boundary).

Definition 1.4.1 By a vector field of class C’ on M (r > 1) we mean any

C'-section ® © of the tangent bundle T(M):

®: M > T(M). The set of all C’-vector fields on M will be denoted by ¥’(M). To denote vector fields we shall use Greek capitals ©, Y. The symbol @(x)

will denote the value of the field at the point x. Roughly speaking, a vector field on M is the set of pairs (x, ®(x)), where xeM and each ©(x) is a vector tangent to M at x. The field is of class C' iff the assignment is of class C’ in every chart. Definition 1.4.2 Let ® be a vector field of class C’ on M, r>1. By an integral curve of © passing through a point xe M we mean a curve y(t) on M, te(a, b), such that y(to)= x for a certain tye(a, b) and

(1.4.1)

for all te(a, b);

(6) =P(y(t))

p(t) is the tangent vector to y(t) at t. We may always assume that the time interval is of type (—a, a) with ty = 0. The existence and uniqueness of integral curves is a direct

consequence of suitable theorems of differential equations: at any point xEM

we choose a chart (U, a), U3x,

and transfer equation (1.4.1) to

an open set of R”. Let c(t) =aoy(t) and f(y) = (da)(x)(9(x)) for y = a(x). Then (1.4.1) induces the equation:

elt) =f(c(0), in which c(t) eR”, f(y)eR”, and one can apply theorems formulated in

the preceding section. Definition 1.4.3

Let

Be ¥"(M) and let I(x) =(a,, b,) denote the maxi-

mal interval in which the integral curve passing through x is defined. The vector field © is called complete iff I(x) = R for any xeM, ie. a,

=—0,b,=

+00.

Basic Concepts and Facts

32

The above definition and considerations together with the theorems of the preceding section imply: Theorem 1.4.1 If ®e.F¥"(M) is complete, r > (1) for every point xeM there exists a unique such that y,(0) = x; (2) writing ~,(x) = p(x, t): =y,(t), we have p: MxR-M is of class C’; (3) for every teR the mapping Q,: M>M (4) the family of mappings o, is a group: Pr+s

=

POPs,

1, then: integral curve y(t) = y(t) py(x) = x; the mapping

is a diffeomorphism;

SIER.

If the completeness assumption is omitted, assertion (2) has to be reformulated as follows: (2) for every xyeM there are a neighbourhood U 3 x, and an interval I=(—a,a) such that (x, t)=@,(x): =y,(t) is a C'-mapping from U xI into M. Assertions (3) and (4) are not necessarily true in that case. Theorem 1.4.2 Let be.F'(M) Then © is complete.

be a vector field on a compact manifold.

Proof Suppose that for some x„eM the trajectory y(t) = g,(xę) with y(0) = x, is defined in the maximal existence interval te(«, 5) where, say, B < +o. Let t„e(a, 8), t, 7B as no. M being compact, the sequence (y(t,)) contains a convergent subsequence; we may assume for simplicity that y(t,) = x, itself converges to a limit yeM. Let U be a neighbourhood of y and I =(—a, a) an interval with properties stated in assertion (2). Then x„eU for n sufficiently large, and so the trajectory passing through x, exists for te(—a, a). Since x, =y(t,), the curve y(t) is defined, in particular, for all t with t, E be a linear operator. Definition 1.5.1 We say that 0OeE is a hyperbolic point (for A) iff all eigenvalues /,, ..., A, of A have moduli different from unity: |A,| £ 1 for c= lak: Suppose 0 is hyperbolic for A. It follows from classical theorems of linear algebra (e.g. the Jordan form for matrices) that E admits decomposition into the direct sum of two invariant subspaces E* and

EY: E=E

QE

such that all eigenvalues of A|E* are less than 1 in absolute value, while the eigenvalues of A|E~ are greater than 1 in absolute value. The subspace E* is said to be contracted and the subspace E~ expanded by A. Let gp:

M >M

be a diffeomorphism of a manifold

M, dimM =m,

and let x, be a fixed point: (Xo) = Xo.

Definition 1.5.2 The fixed point xg is called hyperbolic (for g) iff 0 is hyperbolic for the derivative dp (x) in the tangent space T.(M), i.e., iff every eigenvalue A of the operator d@(xo): T,(M) > T.„ (M) fulfils:

|A|

1.

The

diffeomorphism

@ is said

to be

contracting

at xo

iff all

The Hadamard—Perron

Theorem

39

eigenvalues of d@(xo) satisfy: |A| < 1; it is said to be expanding at x; iff all eigenvalues of d@(xq) satisfy: |A| > 1. We are going to show that, given any manifold M, any diffeomorphism yg: M>M and any hyperbolic fixed point x9, (Xo) = Xo, there exist two smooth submanifolds in a neighbourhood of xo, both containing xo, such that ¢ restricted to one of them is contracting and restricted to the other is expanding. This is undoubtedly one of the most important theorems of the theory of smooth dynamical systems. It was proved, in various versions, by several authors, but finally gained renown in mathematics as the Hadamard—Perron theorem. We shall retain this name although the version which we present is due to Sternberg (1955). Theorem 1.5.1 (Hadamard—Perron; abbreviated to H—P theorem) Let p: M>M be a diffeomorphism of a manifold M and let x, be a hyperbolic fixed point of o: Q(Xxg) = Xo, Xo€M. Then there exists an arbitrary small neighbourhood U of x, such that the set

ci={xeU:

loc

g"(x)eU for neZ*}

is a C!-submanifold of M with the property:

Tey(Wet) = E*, where E* denotes moreover, we have:

Weloc ={xeU: Hadamard and (1955) showed that and Robbin, 1967) We first prepare

the

subspace

contracted

by do(xg);

g"(x) >xg asn>+o0}. Perron proved that Wt is of class C°; W,* is of class C'; later it was proved that W), is of class C’,r > 1, provided a lemma, which we will need in further

Lemma 1.5.1 Let A: R">R" Ais ..., A, Suppose that |A;| < 1

of T,,(M)

be a linear operator

Sternberg (Abraham so is o. reasoning.

with eigenvalues

ii dwahyea5 8:

Then there exists an inner product €',*> in 'R" such that

IA] < 1, where |-| denotes the operator norm induced by .

Basic Concepts and Facts

40 Proof

To begin with, we show that the sequence of norms (\|A"Il)

tends to 0 exponentially, i.e. that there are numbers a >0,0(0,0)

as n> 00.

(1.5.13)

This gives inclusion = in our claim, provided that U is small enough. To obtain the opposite inclusion, assume that (u, v) is a point such that o"(u, v) =(u,, v„) EU for n=0, 1, ... Just as before, u, >0. Now we estimate ||v,,;—g(U,+1)|| from below, ne Z*. This time we apply to f, the Mean Value Theorem together with estimate (1.5.6). We get |

+ 1 —g(un+ dll

z

||Bo, +

(un, U,) — Bg(u,) -f (u, glu„)||

2 ||B(vn —g (un))|| NE 310, —g (u„)|| Ś

Since ||B"'|| = 1/b < 1 and ||B"'||||Bv|| > ||B""Bo|| = ||v||, it follows that ||Bo|| > b||v|| for any v. Taking 9 sufficiently small, we have b, =b— —9>

1, and so

lOn+1—G Uns Il 2 bllen—g (uli —9 lv —g (ul = b, |lv,—g(u,)|| > by”'|v—g(v]|.

(1.5.14)

By assumption, (u,, v„) EU for all n; the sequence (g(u,)) is bounded (in fact convergent to zero). Thus there is a constant C such that

lvn=glull< C

for

n=0,1,...

This yields, in view of (1.5.14),

C > b |v—g(u)||

for

n=0, 1,...

Since b; > 1, we obtain v = g(u). This settles the desired inclusion and ends the proof of our claim.

>

The Stable and Unstable Manifold. Cascade

|

51

This claim, together with relation (1.5.13), results in the following set equality:

Wa = (x: @"(x)EU, neZ*, lim o"(x) = 0}. Thus the proof of the Hadamard—Perron theorem is complete. On replacing o by g ' in the H-P theorem we obtain Corollary 1.5.1 Let xo be a hyperbolic fixed point of a diffeomorphism peDiff' (M). Then there exists an arbitrarily small neighbourhood U of Xq such that the set

Woe :Z1XxEU:

g"(xeU

for neZ|!

is a C'-submanifold of M with the property

T,(Wo)=E", where

E

moreover,

denotes we

the

subspace

of T,,(M)

expanded

by do(xv);

have

Woe = (xEU: g"(x) >Xo asn>—o0).

Exercise

1.5.1.4. Let E and F be linear subspaces of R” such that R"=EQ@F.

Let geC'(E, F). Show that the graph of g, i.e., the set {(u, g(u)): ue E} is a C'-submanifold of R” (r > 1).

1.6. The Stable Manifold and the Unstable Manifold of a Hyperbolic Critical Point. The Case of a Cascade Let M

and

N be two

manifolds.

Definition 1.6.1 A C'-mapping f: M > N is called an immersion iff the rank of the linear map df(x): T,(M) > T„„,(N) is equal to the dimen-

sion of M for every xeM:

rank df(x) =dimM For instance,

if M =R

for xeM. and

N is any

manifold,

dimN > 0, then

52

Basic Concepts and Facts

every curve y(t) in N, teR, such that y(t) ź 0 for all t is an immersion of the line into N.

Theorem 1.6.1 (Smale)

Let M be a manifold and let oeDiff' (M). For

every hyperbolic fixed point xyEM, g(Xxg) = Xo, there exists an injective

immersion W: R: > M such that:

(1) Y(0) = xo;

(2) y(R)=W*:=|lxeM:

lim g"(x) = Xo};

G) T,,(V(R5) = where

k

=dimE*

and E*

is the subspace of

T,,(M) contracted

by

de (Xp).

Proof

It is easy to see that

W*=

U 9" (Woe):

where W, is defined in the H-P theorem. Let V be a neighbourhood of

0 in R* and let wy: VW, be a diffeomorphism which maps V onto a neighbourhood of x, in Wt. We will not restrict generality by assuming that the image W,(V) is the entire We.Such a wy certainly exists (Exercise 1.5.1). We may assume wW,(0) = x9. Define

g=Wy

Opowy.

Since dg (0) = (dyy) * (x9)

odp (x) O dy, (0),

the operators dg(0) and do(xvg)|g+ have identical eigenvalues. By the definition of E*, the operator dp(Xg)|+ has eigenvalues with moduli less than 1; consequently, so are all the eigenvalues of dg(0). In virtue of Lemma 1.5.1 one can introduce an inner product in R* in which dg(0) is less than 1 in norm:

ldg(O|| < 1. Let Vy=V be a ball with centre at 0 such that Ildg(x)||

d@o(x) uniformly in xe M). Assume that x9eM is a hyperbolic fixed point for Qo. Then: (1) There exists a neighbourhood U of x, such that, for n large enough, each of the o, has exactly one hyperbolic fixed point x, in U

(see

Exercise

3.4.1

on

p.

160),

and

we

have

dimW*(x,, o,)

= dimW* (Xo, Po) ;the symbols W *(x,, @,) denote the stable and unstable manifolds of x, in the system (M, g„), n=0,1,2,... (2) There exists a sequence of injective immersions W,: R>M,

where

s=dimW* (x9, pg),

such

that

=(), 1, 2,...; the sequence (,) converges

w,(R')=W* (x,,@,),

n

to (Wo) in C'+topology,

uni-

formly on compact subsets of R*.

The proof of this theorem is difficult and therefore omitted here. We refer the reader to Hirsch, Pugh and Shub (1977).

Basic Concepts and Facts

56

1.7. The Stable Manifold and the Unstable Manifold of a Hyperbolic Critical Point. The Case of a Flow. In this section we prove certain theorems on flows, which can be regarded as counterparts to the theorems of the preceding section, concerning a cascade. Let be.7'(M) be a given vector field on a manifold M and let g, be the flow generated by ©. Suppose that xy EM is a critical point for ®, ie. P(x) =0. Then g;(xo) = Xo for all teR. Since GOD, (X). = Oe g (S)

for

xe M, t, seR,

we have at the point Xo dg, (Xo) O dg, (Xo) =

dQ,+(X0).

Thus the family of derivatives dg,(x9) constitutes a continuous group of linear operators on the space T,_ (M). It follows (see Exercise 1.8.4) that there exists a unique linear operator B: T,, (M) T,, (M) such that

do, (x) =e"

for teR.

|

(1.7.1)

Definition 1.7.1 The operator B in formula (1.7.1) is called the Hessian of the vector field © at x); the Hessian will be denoted by the symbol P(X). Our present objective is to represent the Hessian in a given coordinate system. Let (U, a) be a chart, x9 EU, «(X0) = yo ER”. The differential equation on M, do, (x) Aż go”

P (P, (x),

is transferred locally, via a, to «(U), a subset of R”. Write y = a(x) and let W,(y) be the flow generated by the vector field Y(y) on a(U) into which the field ®(x) is carried by a. We have

dy, (y) _ dt

F (4,9)

and ¥(yo)=0. Differentiating this equation with respect to y and changing the order of differentiation (we may do this on account of

The Stable and Unstable Manifold. Flow Theorem

1.3.2 on p. 25), we obtain

0 dy,(y)

PZP d oy,

dra

sy)

@

w (W,(y)), Ow,

W

d¥ (Wy,= yw i

We may view d¥ as a matrix; this is just the Jacobian matrix of the vector Y. For y = yo we get

Fe 0,00=APU)S00) O The matrix 9)

then satisfies the linear differential equation (just

y

w

obtained) with condition”:

constant

coefficients,

together

with

the

“initial

(OW o/Oy) (Yo) = identity. Thus By a © o) = ef? Oot.

If now

Y(y) is written in local coordinates as

FV) =(¥'*(),..., POD), then GP: |

d¥ (yo) =(

GW

hay

Hence the Jacobian matrix d¥(yo) is the required representation of the Hessian ®(x,) in local coordinates.

Definition 1.7.2, A critical point x»¢M of a vector field B€ F'(M) is called non-degenerate iff the Hessian (x9): T,,(M)— T,,(M) is a surjection. Theorem 1.7.1 If Xo is a non-degenerate critical point of a vector field $e.F'(M), then xg is an isolated point of the set of all critical points of @.

58

Basic Concepts and Facts

Proof

The theorem is of local character (it involves the behaviour of ©

in a neighbourhood of a given point). Therefore we may assume that M is simply a subset of R”, x, = 0, and that $ is a C'-vector field defined in a neighbourhood U of 0. Then © can be regarded as a mapping b: UR", 9(0)=0. Since the point O is _ non-degenerate, det ©(0) 40. Thus, by the Inverse Mapping Theorem. there is a neighbourhood

V of 0 such that ®(x) #0 for all xeV,

x 40.

Definition 1.7.3 A critical point xeM of a vector field Pe.7'(M) is called hyperbolic iff the Hessian ©(x9) does not have purely imaginary complex numbers as eigenvalues.

Remark 1.7.1 If Xo is a hyperbolic critical point of © (in the sense of Definition 1.7.3), then x, is a hyperbolic fixed point of each ¢,, for t # O (in the sense of Definition 1.5.2). Proof

Since

dope, every eigenvalue A of the operator dg,(Xxg) is of the form Aas oF.

u being an eigenvalue of (x,). Since Rep + 0,

parse

1.

Definition 1.7.4 The eigenvalues m;, ..., lm of the operator ©(x,) are called the characteristic exponents at the critical point xo; the numbers e“1,..., e"m are called the characteristic multipliers at xo. Theorem 1.7.2 Let ®e.F"(M) be a vector field and o, the flow generated by ®. Suppose that Xo is a hyperbolic critical point of ®. Then there exists an injective immersion %: R* > M such that:

(1) F(0)= Xo, (2) (RSS

Ws

aX

Shin 70, to

On

&

(3) T,)(%(R‘))=E*, where E* < T,,(M) is the eigenspace of the operator A

corresponding to the eigenvalues u such that Rep < 0.

Proof ln virtue of the H-P theorem, it suffices to show that W* = We i= (x2: limo, (x) =x}. Obviously, no

W*cW,.

The Stable and Unstable Manifold. Flow

59

To obtain the opposite inclusion, take an xe W,* and let (U, a) be a

chart for

Wc (at xo): Xo EU, a(xg) = 0, a(U) = R*. Let V be an open +

convex bounded subset of R* such that OeV=Vca(U) and let p„(x)ea '(V) for n> ny. The flow g,(x) is transferred by a to a(U); denote by w,(y) the resulting flow in a(U). Since the assignment (r. vy), (y) is of class C’ in Rxa(U), there is a constant Lsuch that

dy, (I
U,.

@(x) = Lx+f(x) is one-to-one:

to see this, suppose

@(x,) = G(x); then

Lx, — Lxą = f(x2)—f(%1) and by the Mean

Value Theorem

we get

LEG x231 = F%2) Fall < ella — l;

This can

Linearization in a Neighbourhood of a Hyperbolic

Point

63

on the other hand,

ILG — xa > 5 les — sl. and since 2e R"”,g

continuous,

lim IIxll>+ x

g(x)=0,g(0)=0;.

H becomes a complete metric space if the metric is defined by:

o(g, k) = sup (Ilg1 (0) —ky ll + Ilg2 ©) — ka Cll) xeR™ for

g =

(91; 92)»

k =

(ką, k,), where

gi (x), k, (EEX

; g2(x),

k,(x)eE”

If a homeomorphism h of the form (1.8.5) conjugates p with L= do(0), then

po(1+g) =(U+g)oL, (L+f)o(I+g) = (1+g)0L, L+Log+fo(l+g) = L+goL,

g=L 'o(goL—fo(1+g)).

(1.8.6)

64

Basic Concepts and Facts

The last ‘equality written “coordinatewise” g,=A*

o(9, oL-f; o(1+g))

92 =B™

0(9,0L-f, o(1+g)).

assumes

the form

and

The first of the resulting equalities can be rewritten as g,

= Aog,oL '+f,0(1+g)oL"'.

We now define

a map F: H—H

by putting

F(g) = (Fi(9), F2(9)), where

F,(g)=Aog,oL

'+fio(l+g)oL"",

F,(g) = B"'o(gaOL-f,0(1+9)). According to the preceding considerations, all that we need is to find a go€H satisfying: F (go) = go. We first show that F indeed maps H into H: it is readily seen that F (g)(0) = 0. To verify that

TEK IF (9) CJII = 9, suppose that ||x|| > +00; then

IL *xdl> +00 >g,(L 'x)>0>Aog,oL 'x>0; IIx > + o >|I+g)(L*x]|| =||L"x+g(L"'x)|| > + o and this, in view of (1.8.4), yields

Jio(I+g)oL"'(x) =0 for ||x|| large enough. Thus F,(g)(x) >0. Further, we have

||Lx|| > +00

g(x) > 0 >|

> g,0L(x) >0=>B''o0g,oL(x) >0;

+9) (ll > + oo,

whence (again by (1.8.4))

fo(1+g)(x) = 9 for ||x|| large enough, and so F,(g)(x) >0. It follows that F(g)eH.

Linearization in a Neighbourhood of a Hyperbolic Point Now gi

we

prove

CZE 92), k =

that

F

is a

contraction

on

H.

65 Let

g,keH,

(k,, k,). Then

o(F (9), F(k)) = sup (IF, (9) (x) — Fi(A) + IF (9) (9) —F2(k) (9!) On account

of (1.8.3) we have

sup IF; (9) (x) — Fy (k) (| = sup||A0g,0L '(x)-Aok;,oL '(x)+

+f, oI +g)oL *(x)—f, oI +kyoL *(x)|| < ||All sup |igs 0) —k, ll +sup|lefAll igo L*(x)—koL* (lI (g)(%) — Fz (k) (20)|| = sup||B''og,oL(x)—B

!ok;0L(x)+

+B"'o0f,0(1+9)(x)—B"' of,o(1+k)(0|| IB” *|| sup |lg2 ©) — ka COII+IIB" "sup I4G]|-llg ©) —k(0II < |IB""-ll92 — £zll+ellB""I-llg—Kll.
0 there is a basis in C” in which 0

B,

A=

B,

é

where B, =

4;

.

Dee

és i

uty

0

0

€

A;

\

6

Me

Having chosen a suitable e, introduce the inner product in which the basis corresponding to e is orthonormal, 1.8.3.B. Prove that any two contractions of the same space E are topologically conjugate. Assume that dimE < + o; the theorem is also true if E is any Banach space (Strelcyn, 1970b).

70

Basic Concepts and Facts

Him. Let S be the unit sphere in E = R” and let gy,g, be the contractions in question. Let V, denote the “annulus” limited by the surfaces S and y,(S), i= l, 2. Map V, homeomorphically onto V,; then extend the mapping to a homeomorphism of the entire R”; compare Example 1.2.1.

1.8.4.4. Let A(t) be a continuous group of linear transformations of R” (ie. A(t): R">R" for any teR, detA(t) 40, A(t+s) = A(HA(s), lim A(t,,) = A(t)). Show that there exists a linear transformation B: R" hl

— R” such that

A(t)=e

for all teR.

Hint. This is a special case of a general theorem on the representation of a continuous semi-group of operators in a Banach space; see (Dunford

and Schwartz,

1958; p. 614).

1.9. Behaviour of a Flow in a Neighbourhood of a Closed Orbit Let M be a manifold and let ©e.7F'(M). The vector field © generates the flow g,(x) = g(t, x). Suppose that y(t) is a closed orbit of db, with prime period t. Let $ denote the set of points lying on this curve (i.e. the set of values assumed by y; cf. Section 1.4, p. 35). Each point xQe$ is a fixed point of the diffeomorphism g,(x). Thus

dę,(x9): Tey(M) > Te,(M). Assume y(0) = Xo. Obviously, A = 1 is an eigenvalue of the operator d@,(Xo); the tangent vector »(0) is the corresponding eigenvector:

de, (Xo)? (0) = 7(0) = 7(9). Definition 1.9.1 (1) By a local transversal section (or a cross-section) of y at a point x9€) we mean any submanifold S = M of codimension 1 such that x»eS and ®(x)¢ T,(S) for all xeS. Then the tangent space T,(M), xeS, admits the decomposition

T.(M) = [®(x)] © T,(S), where [®(x)] denotes the one-dimensional space spanned by ©(x). (2) Let S be a cross-section of y at xo and let U, be a sufficiently

Behaviour in a Neighbourhood of a Closed Orbit small neighbourhood of xo in S (see the theorem define a mapping ©: U, >S by:

(x) = pli(x), x).

fil) that follows). We

for xeU;,

where t(x) denotes the least positive number t such that g(t, x)eS. The mapping © is called the Poincaré map at Xp. Theorem 1.9.1 (1) For every cross-section S of a closed orbit y at a point x9€), the Poincaré map ©: Uy +8 is correctly defined, provided that the neighbourhood Up» of Xo in S is small enough. (2) If S, and S, are two cross-sections of y at X,, x,€5 (respectively) and if @,, ©, are the corresponding Poincaré maps, then ©, and ©; are diffeomor phically conjugate, i.e., there is a diffeomorphism h: VW, > V, (V, denotes a suitable neighbourhood of x; in S;, i = 1,2) satisfying

©,oh=ho©,. Proof (1) The assertion is of local character. Thus we may assume that S is contained within a single chart. By an appropriate choice of the coordinate system, S can be written (in that chart) as follows:

S=(xER":

x" = 0};

ar (Onn, 0),

WO}

(Ola,

Osaka]!

Let

OE X10

(Eek) ce

eX):

Then o(t, x)ES iff p(t, x) =0. Now, we have g(t, 0) = 0 and

do”

“ae AOL. In virtue of the Implicit Function Theorem there exist a neighbourhood U, of 0, an interval (r—e, r+e) and a C'-function t(x) such that

g™(t(x),x)=0

and

t(x)e(t—e, t+8) for xeU5.

For any x, the number t(x) is indeed the least time it takes x to return to S because t is the prime period. Hence

O(x) = p(t(x), x)

is a well defined C'-map.

for xEUg

72

Basic Concepts and Facts

It is not difficult to see that © is in fact a diffeomorphism. (2) If x, £ x,, then shifting S, along trajectories of the flow g, we reduce the situation to the one where x, = x,. Again the setting is local, and so we may assume that S,, S, s R”. Locally, each integral curve intersects S, at exactly one point; the same is the case with S,. We define h as follows: we choose a neighbourhood V, of x, in S,; for any xeV, we draw a trajectory through x and define y = h(x) as the point where this trajectory meets S,; we put V, =h(V,). The neighbourhood V, has to be chosen small enough to ensure uniqueness of y and correctness of the definition.

NGESA

Clearly, h conjugates ©, and ©, (see Fig. 1.9.1). By construction, h is a diffeomorphism. Remark 1.9.1 (1) Let y, x,, Ś,, ©, be as above, i = 1, 2. Since ©, and ©, are diffeomorphically conjugate, we have sp(d0, (x,)) a sp(d@,(x,)).

(2) If S is a cross-section of a closed orbit y at xgef and if © denotes the corresponding Poincaré map, then sp (dO (xo) Ul; = sp(do,(xv)).

t denoting the prime period of y. Proof

(1) This is an immediate

the invariance formations.

of the

consequence

spectrum

under

of Theorem

non-singular

1.9.1 and of

linear

trans-

Behaviour in a Neighbourhood

of a Closed Orbit

(2) As pointed out in Definition admits the decomposition

13

1.9.1, the tangent

space

at xy

T„(M) = [50] © 7,,(8), where [}(0)] is the span of }(0). Thus every vector ye T.,(M) can be written as: J) =(V1 V2),

where

y,e[7(0)],

J+€ TS):

The vector (0) being invariant under the operator represent this operator in the matrix form

dgy,(xo), we may

> do,(X) = loA

where A is an (m— 1) x(m— 1)-matrix. Differentiating

the identity

O(x) = p(t(x), x) with respect to x and putting x = xg, we obtain (calculation carried out in “coordinates” yy, V>): 6 . dO (xo) = E(t , Xo) (i (xo)p||AA

=

se

Oy

70] „|+|

= (Oya

isa

for any

xe T,,(S)

0 B

|| 0

4 ||

+ bal

3 0 wnere.a(x) = i0)| |The resulting vector has to belong to T,,.(S). Hence a(x) = —ax mediate.

and

Ax =dO(x,)x.

Our

assertion

is now

im-

Examples

1.9.1. Let coordinates

M = R?, 9©(x) =[—x?, x']. The integral curves are (in polar r, w):

Q, (To, Mo) = (roCos(t

+ Wo), rosin(t +vv)).

For any closed orbit, the Poincaré map @ is the identity (Fig. 1.9.2).

Basic Concepts and Facts

74

1.9.2. Let M = R?. Consider the flow induced by the following system of differential equations (in polar coordinates): dr i — = sinr, dt

do — =const = c > 0. dt

Fic.

1.9.2

The integral curves are

log |tanżr| = t+log|tanżr|,

w=ct+@o,

where (ro, @o) is the initial position, rę 4 nn, neZ. For ry = 0 we have r(t) = 0; for rę =nn, neZ, n+ 0, the integral curves are circles y,. In

particular, y, is the circle r = n. Consider the point ry = 1, »y = 0 and the cross-section S$ defined as the segment perpendicular to y,. The corresponding Poincaré map (2) is, locally, a contraction. Indeed, after simple computation we get

d JW

1 s:

25

= |

n

which implies (see Exercise 1.3.5 on p. 30) that © is contractive (on S) at r =". This means that close to y,, the integral curves approach ), ; speaking intuitively, they “wind” towards y,. The situation is similar in the vicinity of any other y,; the trajectories approach y, as t > +00 (for n odd) or as t > — oo (for n even). We leave to the reader the examination of the behaviour of the flow in the vicinity of the origin.

Behaviour in a Neighbourhood of a Closed Orbit

75

Definition 1.9.2 (1) A closed orbit y of a vector field be.£F'(M), with prime period T, is called transversal (or: the period t is transversal) iff, for any x9€$, the unity is an eigenvalue of multiplicity 1 for the operator dgQ,(xg).

Fic.

1.9.3

(2) A closed orbit y is called hyperbolic (or elementary) iff y is transversal and, for any x9€%, the operator dg,(x9) has no eigenvalue A with AA1=|A. As an immediate consequence of Remark 1.9.1(2) we obtain: Corollary 1.9.1 _A closed orbit y is hyperbolic iff, for any xę€% and any cross-section S at Xo, the point Xo is a hyperbolic fixed point for the corresponding Poincarć map ©.

We now define the stable and the unstable manifoldsof a hyperbolic closed orbit y. Thus let y be such an orbit of a vector field ®c F"(M) and let o, be the flow generated by ©. Take any point xge$ and any cross-section S at xg. According to the last corollary, xg is a hyperbolic fixed point for© (the Poincaré map). In virtue of the H-P theorem, © defines locally two subsets of S: Wot (x9) and Wy: (X0). Define

W* (y) = U @: (Mice (Xo), =Wy) = U @ (Wee (%0)). teR

teR

It is not difficult to show that the sets W* (y), W (y) depend neither on the choice of xy nor on the choice of the cross-section S (because all the Poincaré maps are diffeomorphically conjugate).

76

Basic Concepts and Facts W* (+) and W” (y) are immersed

dimW

submanifolds of M and we have:

* (7) = dim W,loc (xo)+ 1,

dim W ~ (y) = dim W,. (Xo)+ 1, dimW * (;)+dimW

(y) = dimM +1.

Definition 1.9.3 The sets W” (y) and W~ (y) defined by formulas (1.9.1) are called (respectively); the stable manifold and the unstable manifold of the given hyperbolic closed orbit y. . Topologicallv. W” (7) and W (%) are immersion embeddings into M

either of the cylinder S! x R* or of the generalized Móbius band S' x R* (the so-called "twisted product”)'*.

1.10. Suspension of a Diffeomorphism In the preceding section we defined the Poincarć map for a crosssection of a closed orbit of a flow. That means, with a given flow and a given submanifold of codimension 1 (cross-section) we associated a certain diffeomorphism on that submanifold. Now consider a problem that is, in a sense, converse to the above: given a manifold M and a diffeomorphism p: M — M, can one find a super-manifold N and a flow g, on N such that M be a cross-section for p, and that p, = @ on M? The answer is positive. Consider the manifold M, = M xR and the map 4(x, s) =(@(x),

Put: [= Define N=

s—1)

for (x, s)EM,.

u”: neZ}.Then I is a group of diffeomorphisms on M,.

Ma.

and denote by n the natural projection from M; onto N. We have the quotient topology in N: if (x, s) is a point in M,, U is a neighbourhood of x in M and e >0, then the z-image of any of the sets g"(U) x (s+n=e,

s+n+e)

= My,

neZ,

is a neighbourhood of the z-image of the point (x, s). Thus, if M isa manifold of dimension m, N is a manifold of dimension m+1.

Suspension of a Diffeomor phism

We now

77

define a flow 6, on M, by putting:

P(x, s) =(x,

t+5).

This flow induces a flow g, on N. The resulting flow g, has the required properties: identifying M with z(M x {0}) in N, we see that M is a cross-section for g,. If z((x, s))EM, ie. s =0, then

p, z((x, 0) = 2(G, (x, 0)) = x(x, 1) = x(e(), 0). Definition 1.10.1 Given a cascade (M, g), the flow (N, g,) just constructed is called the suspension of the diffeomorphism og.

The similar vation with a The

flow g, obtained as a result of this procedure has properties to the properties of the original diffeomorphism g. This obserallows one to construct examples of flows sharing properties given diffeomorphism. operation of suspension is a particular case of a more general

procedure,

which

we

now

outline.

Let X be a compact metric space and let p: X >X be a continuous mapping. Further, let f(x) be a real-valued function on X,f(x)>0 for xeX. As before put:

eo

CAR,

a(x, s) =(p(x), s—f(x)) Leo?

for (x, s)e X,,

eZ.|.

Kel In other words, X” results from X, by identification of all pairs of the

type: (x, s) and (p(x), s-/(1)).

Let a be the natural projection of X, onto

X”. The formula

@, (x, 5) =(x, £+5)

defines a flow on X, which induces (via n) a flow g, on X7.

Definition 1.10.2

The resulting flow (X7, g,) is called the special flow

over the cascade (X, ~) with respect to f. In the case where f(x) = 1 for all xe X and o is a homeomorphism the special flow over (X, g) is precisely the suspension of 9.

78

Basic Concepts and Facts 1.11. Limit Sets

Definition 1.11.1 The sets

(1) Let M be a manifold and let ge Diff’(M),

x9eM.

(Xo) = (x: Fay ky +00, lim o':(X9) = x}, |

a(Xg) = (x: Fay ki > — 00, lime*n(xo) = x} are called the limit sets of xo (at +00 and —oo, respectively). (2) Let $e.F'(M), let g, be the flow generated by © and let xoEM. The sets: 0(Xg)::= 1x: Jay i > +0, lime, (xo) = x), (Xo) := {X: Jay t, > — 0, limo, (xo) = x} are called the limit sets of xo (at +o

Theorem 1.11.1

and

—o).

If M is compact and Ge 7'(M) (case (2) in Definition

1.11.1), then for every xyEM the sets w(Xo), «(Xo) are nonempty, closed and connected. In the case of a cascade (M, g), pe Diff'(M) (case (1) in Definition 1.11.1), the sets w(Xxg), X(Xx9) are nonempty and closed.

Proof Only connectedness requires a proof (case (2)); other statements are obvious. Thus assume that 0(X9) = X, UX?,

where X, and X, are nonempty, closed and disjoint. Then

dist(X,, X;)=0>0. Let U,, U, be the open sets given by:

U, = {xeM: dist(x, X,) 40. By the definition of w(xv), the trajectory g,(xg) enters and leaves each of the sets U,, U, infinitely many times as t > +00. Thus there

exists a pair of sequences

(x‘), (xi) such that x%eU,, xPeU,

(neZ*), xy) = 9. (Xo), X = Ps(%o)

and

@,(x9)¢U,UU,

for

te(t,, s). The piece of the trajectory ~,(x9)corresponding to te(t,, S,) certainly contains a point x, whose distance to either of the sets U,, U,

Limit Sets

79

exceeds 50. By compactness, the sequence (x,) has a cluster point x. Thus we arrive at a contradiction; on the one hand, xew(x5) by the definition of a limit set, while, on the other hand, x¢U, UU, > o(xg).

Example 1.11.1. Let y be a hyperbolic closed orbit for a vector field with prime period t. Suppose that for every x9€)

®¢ F’(M),

sp(dę,(x0))-{1}© (2: [4] < 1). Then there is a neighbourhood O(XI=?

U of $ such that

for all xeU.

We shall prove this. Choose any cross-section S at a point xg€$ and let © be the corresponding Poincarć map. According to Remark 1.9.1(2), the spectrum of © is contained in the unit circle of the complex plane. Hence, by Exercise 1.3.5 on p. 30, there is a neighbourhood U, of x9 in S such that lim O" (x) = Xo for all xeUy. Fix a 6>0 U=

and U

write:

—, (Uo).

—0.-

Basic Concepts and Facts

82

1.11.7.B. Give an example of a dynamical system such that for some x the set a(x) is not minimal. Hint. E.g. a flow on a two-dimensional manifold with trajectories running as shown in Fig. 3.8.2 on p. 180. We conclude

this section with one more

theorem

on minimal

sets.

Theorem 1.11.3. Let (X, g”) be a dynamical system. Suppose that X is connected and that each og”, «eM, is a homeomorphism. Then every minimal set S either is nowhere dense in X or is the entire X. Proof Assume that int S 4 ©. Since the o” are homeomorphisms, we have g” (intS) = intS. According to Exercise 1.11.6, the boundary of S, je. the set S—intS, is invariant under all the g*. Clearly, this set is closed and contained in S — hence it is empty, by the minimality of S. Consequently, $ is a closed-open set in X and thus, X being connected, we have S=X.

Dynamical Systems on Manifolds of

Dimension

1 and 2

2.1. Survey of Basic Facts of Differential Topology

In several points of this chapter we shall have to resort to some facts of differential topology. Therefore we devote the first section to a brief discussion of those concepts and theorems of this theory which will be needed in the sequel. Proofs are omitted; they can be found in Milnor (1965). Let M and N be oriented manifolds '? of equal dimension. Assume that M is compact, N is connected. Let f: M—N be a mapping of

class C”. Definition

—

2.1.1

A

point

xeM

is called

regular

iff df(x): T,(M)

15, (M) is a surjection. A value yef(M) is called regular iff every point x ef ""()) is regular. A point [a value] which is not regular is called critical.

It follows immediately that the set of regular points is open. Let x be a regular point for f: M>N. We define the symbol sgndf (x) as +1 iff df(x) preserves orientation, and as —1 iff df(x) alters orientation. For any regular value yeN we define:

deg(f; y):=

2

(2.1.1)

sgndf(x).

xef- 1(y)

In virtue of the Inverse Mapping Theorem, the points in f~‘(y) are isolated

and

so, by the compactness

of M, the set f~'(y)

formula (2.1.1) thus defines correctly a certain integer.

is finite;

84

Dynamical Systems on Manifolds of Dimension

1 and 2

Theorem 2.1.1 The number deg(f; y) depends only on f and does not depend on the choice of the regular value y (Milnor, 1965, Section 5, Theorem

A).

This theorem justifies the following definition: Definition 2.1.2 The number deg f= deg(f; y), y being an arbitrary regular value, is called the degree of f. Theorem 2.1.2 If f,g: M>N are smoothly homotopic? = degg (Milnor, 1965, Section 5, Theorem B).

then

degf

It follows that the concept of degree makes sense not only for a particular mapping f but also for the whole equivalence class of mappings pairwise smoothly homotopic. Since any C'-map is smoothly homotopic to one having regular values, the definition extends to all

C'-mappings.

| Examples

WISH M=N=S" /(2)=z". then dezf= m. 2.1.2. If M=N =S', f(x) = Asin2nx(mod 1), then degf=0. 2.1.3. If feDiff' (M), M any compact

ANNNFZN=N

"P>

b

manifold, then

E | a,b,c,deZ,

deg f= +1.

det f #0, then

deg j= det/.

Let 6c ¥'(U), U being an open subset of R”, and let xoeU be an isolated critical point of ®. For any sufficiently small sphere B with centre at xy we define a mapping F of B into S”~', the unit sphere in R”, by:

Fee

$ (x)

© = 16091

for xeB.

(21.2)

If B, and B, are two such spheres and neither of the corresponding balls contains critical points other than xg, then the corresponding mappings F, and F, are smoothly homotopic. Therefore, according to Theorem 2.1.2, degF, = degF,. This observation allows us to define the index of an isolated critical point:

Basic Facts of Differential Topology

85

Definition 2.1.3 Let x, be an isolated critical point of a vector field De F'(U). Let F be given by (2.1.2), B denoting any admissible sphere. The number lo (Xo) : 5 degF

is termed the index of © at xo (or: the index of x, with respect to ©). Examples

PO Lele? (x) =| —x, x |, X = , x-)eR. Lhe point (070) isvan isolated critical point of © (Fig. 2.1.1). Here F is the rotation of the unit

circle by the angle żnm, and so 310

desfe= 1.

FiG. 2.21

2.1.6. Let U =R*, $(x) = const 4 0. Then for any xeR? ly(X) = degF = 0. Now, let f: U; > U, be a diffeomorphism between two open sets in R" and let ©, e.F'(U;). Then f transports the vector field ©, to U,; let @, denote the resulting vector field

Pr (yy=df(f-*(y))o®(f-*())

for ye U2.

Theorem 2.1.3 If Xo is an isolated critical point of ©, then yo =f (Xo) is an isolated critical point of ®, and we have lp, (Xo) = le, (Yo). In view of this theorem, it is legitimate to introduce the following definition:

86

Dynamical Systems on Manifolds of Dimension

Definition 2.1.4 Let M be a manifold and let

1 and 2

6c ¥'(M). By the index

of an isolated critical point xy e M we mean the index of its image in any coordinate system (with respect to the vector field in R” induced by ® via the chart). To conclude the section, we formulate one more important theorem (Milnor, 1965).

Theorem 2.1.4 (Poincaré-Hopf) Let M be a compact manifold and let Ó©e F¥'(M) be a vector field with finitely many critical points: x,, ..., Xn. Then n

m

2 Io(x,)= x(M)= > (— 1) rank H, (M), where y(M) is the Euler characteristic of M and H,(M) denotes the i-th homology group of M (for the definitions see e.g. Spanier, 1965).

For M = S? (2-sphere) we have y(S*) = 2; for M = T? (2-torus) or M = K? (Klein bottle) we have y(M)=0; for other 2-dimensional manifolds

we have x(M) O such that 9,(x)eT, ie, p*(t(x), x) =0. By the Implicit Function Theorem, J is open in I’. Let us introduce the function

f(s) = o'(t(s, 0), (s, 0))

for (s, O)el.

This function maps I into I’. We shall show that f'(s) 4 O for all s. Each g, is a diffeomorphism and so we have

0p'

dg!

és

ćw

ap?

dg? 47

s 6u at each point. Obviously,

dg dt

has

dgidx

dg*dt | dg

haste eon

"a"

(2.2.1)

The equation 9” (t(x), x) = 0 is satisfied identically for x = (s, O)el. Differentation with respect to s yields

dg? 0x Les dy?dt |6g? iddp? dt DOC Ar.OWE: AE ques

222 >

Equations (2.2.1) and (2.2.2) form a system with a non-zero determinant. The vector [f’(s), 0] on the left side forces the non-vanishing solution [dt/ds, 1]. Consequently /'(s) 4 0 for any s, as claimed.

88

Dynamical Systems on Manifolds of Dimension

1 and 2

Since © is of class C*, so are t(x) and f(s). Write

K = ST.

The set K is closed and nowhere dense in I’: if K

were dense in some piece of I, S would be dense in an M, contrary to the assumption. We may also assume points of I do not belong to S. The properties of minimal sets imply that every point point (as t > +o) of the trajectory passing through x.

open subset of © that the endx€S is a limit Therefore, for

every xeK there is a t(x) > 0 such that (t(x), x)eT'. This means that K cI

and /(K)=K.

Let V be an open subset of I such that

Kee Ve Ver. Summing up, we see that the function f fulfils the following conditions: there are positive constants o, 3 and A such that

Po x, for all xeji OS (© denotes the Poincaré map) (see Fig. 2.2.1).

FIGs

2241

The theorem which we now present arose from a number of independent results, proved by several authors at different times and under different assumptions. Putting those results into a single statement, we obtain Theorem 2.2.3 (Poincarć-Bendixon-Schwartz) Let M be a compact oriented two-dimensional manifold. Let © e.F*(M) and denote by o(t, x) the flow generated by ©. Suppose that for a certain xęEeM the set w(Xo) contains no critical points and that w(xXo) # M. Then w(Xo) is a closed orbit and o(t, Xo) winds toward w(Xo).

Dynamical Systems on Manifolds of Dimension

92 Proof

By Theorem

According to Theorem

1.11.2 the set w(x9) contains

a minimal

1 and 2 subset.

2.2.1, this subset is a closed orbit. Denote it by

%9. Since M is oriented, a sufficiently small neighbourhood

of 7% is

diffeomorphic to a cylinder {(x!)?+(x?)? = 1, |x*| (8441)

—80,)).

n>ok=1

The index of © along J is the number

(J):= >:

ea?

this number is independent of parametrization and of the choice of a.

Dynamical Systems on Manifolds of Dimension

94

1 and 2

Evidently, 14(J) is an integer. It is not difficult to check that if the region encompassed by J contains exactly one critical point Xo, then ip(J) is equal to 1g(Xo) (cf. Definition 2.1.3); we leave the details to the reader. Thus, Theorem

2.1.2 implies

If J is a closed Jordan curve in R* and ®e F'(R?) is a

Theorem 2.3.1

field without critical points in the closure of the region D surrounded by J@thencts (l= 0: As a consequence we obtain further theorems: Theorem 2.3.2 If the region D surrounded by J contains a finite number of critical points x,,..., X, of a vector field ©, then n

Ip(J) =

) le (X,)k=1

Theorem 2.3.3 field ®, then

If J =$, where y(t) is a closed trajectory of a vector

AGATĘ Proof If J is a closed orbit, then the total change of angle between (x) and any fixed direction (as x runs over J) is 2n. Hence ig(J) = 1. Corollary 2.3.1 Every domain encompassed contains at least one critical point of ®.

by a closed

orbit

of ©

Corollary 2.3.2 If such a domain contains finitely many critical points, the sum of their indices is equal to 1.

Lemma 2.3.1

Let ®y, ©, e F'(R”) be two vector fields and let J be a

closed Jordan curve. If the vectors ®y(x), ®,(x) have opposite directions at no point xEJ, then

Proof

For any se consider the vector field

$, =(1—s)P,+sP,. The condition

P(x) #0

of the lemma

implies:

for all xeJ, se 0.

for all xeS'.

(2.4.2)

Define: F(x) = logę (W). The function

(2.4.3)

f(x) is of bounded

variation. To see this, take any

partition

Go, Gy, -..,6, Of S';ć,=ćo.

Theorem

to the function logu (for u > a) we get

y-

1

Applying

the

Mean

Value

wees

2 FE-PE = i=©M loge’ E+ 1) loge’ Ea < i=08 Y -lo'Gis1)—0'

i=0

v=2

I

I fee » = log TT o'((Xx-„) = log Il 0 (x) = log (x) k=0 j=-n dg" (Xo).

= log ay

(2.4.4)

Homeomorphisms of the Circle

99

The last two equalities together with (2.4.4) result in: d

n

log SE) dx

d

+ log

A

= (Xo)

loa

do” "2

dg"

3) )|

Hence

do" dp" gz (0) (xp:

(2.4.5)

Step III. Assume that A 4 S'. The set S'—A is open (see Exercise 1.11.1) and so

where

I,,n =0,1,...,

are

a sequence

of pairwise

disjoint

intervals

(arcs) on S*. The set S'—A is g-invariant. Thus ¢* (Ip) = I„, for some my ;for different k the numbers m, are also different ;otherwise o would

have a periodic point. Hence

2 ng! < +90.

(2.4.6)

The length of I„, can be estimated from below:

dep*(x) [my = | ao > po

„_dop*(x) = |I a. (6), Percent WIA

2.4.7 (2.4.7)

To

where ć,EI,,Denote by v; and u; the minimum and the maximum (respectively) of g' on I,,,. On account of (2.4.6) and (2.4.7) we have

x

Me/< +00.

(2.4.8)

de:

Note that, by (2.4.2),

= 21) D;

D;

MPE

-— D;

a

woja

100

Dynamical Systems on Manifolds of Dimension

1 and 2

Therefore

i

ET

1 Kat

1

SK

w

shows that for every

> x€EI5'

dx

Thus

do* nis ©)

Ę

a for every x€l5.

(x) =0

This holds in particular for x = x9, where xg denotes I. In view of Exercise 1.11.3, p. 81, there is a sequence that (2.4.1) holds with n=n,. According to what has Step II, inequality (2.4.5) is satisfied at this point for all arrive at a contradiction.

an endpoint of n, > + © such been shown in n = n,. Thus we

Remark 2.4.1 The assumption of o being of bounded variation is essential. There exist diffeomorphisms > 1(51) for which 4 4 S! (see Denjoy,

1932).

In our further discussion we shall need the concept of the so-called rotation number of a circle homeomorphism. Let us now work out the definition of this concept.

"Let

g: S'>S'

be an

orientation

preserving

homeomorphism.

Denote by f(x) the lifting of @ to the real line R:

JJ RZR:

oe

eS

orze

Clearly, there are infinitely many such continuous functions f(x), but any two of them differ by an integer. Since @ preserves orientation, / is monotonically increasing and fulfils

f(x+1) =f(x)+1.

(2.4.9)

Homeomorphisms

of the Circle

|

101

If we assumed that ¢ alters orientation, (2.4.9) ought to be replaced by

Pixar

i(Xt

|

f being monotonically Theorem

2.4.2

(2.4.10)

decreasing.

For every

x €R

the limit

lim (260) =0

(2.4.11)

[ni ec

exists and does not depend on the initial point x9. This limit is a rational number iff o has a periodic point.

Proof The proof is carried out in 4 steps. Step I. We first show that if the limit in (2.4.11) exists for some value of xy, then it exists for any other value and is the same. Fix Xo, Jo ER and write x, =f"(xg), Vx =/"(Vo). Let k and m be integers such that M £ Vo— (X0+k)

0, whence lim g"(x) |n| >©

= +w.

Since

lim

h(x)/x = +1,

we

obtain,

using

the identity f”

=hog’oh"',

Gol OQ,

OS n

|nir+a¥9

(x)

iis PO h

m->+o

n

= +1.

9709)

To see what this means in terms of o and w, write 0, =0,+k, 0, =Oytl, 0,, 0yE 40, 1), k, leZ. If w is orientation preserving (the plus sign in the last formula), we get o, +k = 0,+1, whence g, = g,. In the opposite case, 0, +k = —0,—l, whence g,+0y = 1.

Corollary 2.4.1 Two rotations of the circle are topologically conjugate iff they are either identical or mutually inverse.

Proof

Let o and W be two rotations of S', the first by an angle 2na,

the second by an angle 2x, a, B= S' is an orientation preserving homeomorphism, for which A = S' (see Theorem 2.4.1 or Exercise 1.11.1), then @ is topologically conjugate to the rotation by 2ng, where @ is the rotation number of o. Proof Let A and B be as in Lemma considered in the lemma:

ho: B>A, The condition:

2.4.1. Denote by ho the bijection

hy(f"(0) +m) = ne +m. 4 =S'

implies that every trajectory

(g"(z): neZ) is

dense in S! and consequently the set {f"(0) (mod 1): neZ! is dense in R h(x0) =

lim x>x9-

xeB

ho (Xo)-

(2.4.16) (2.4.17)

by

108

Dynamical Systems on Manifolds of Dimension

1 and 2

This is correct, since B is dense in R and hy is monotone on B. Clearly, h is monotone; it is also continuous, since otherwise it would have jumps, which is impossible because h(B)= A and A is dense in R. Consequently, equalities (2.4.16) and (2.4.17) remain valid, with h in place of ho, for all xe R. Thus h is a lift of a certain homeomorphism of the circle; moreover, h conjugates /, the lift of o, with r, the lift of the rotation by 2xg. Hence p and the rotation are topologically conjugate. Corollary 2.4.2 Every orientation preserving diffeomorphism o of the circle which has no periodic points and satisfies the condition Varg' < +00 is topologically conjugate to a rotation. Proof In virtue of Theorem 2.4.1, the limit set for @ is the entire circle. Hence, by Theorem 2.4.4, @ is topologically conjugate to rotation by the angle 2ng, where g is the rotation number of g.

Exercises

2.4.1.B. Let p: S' >S' be a homeomorphism for which

A = S', let w

be the rotation by 2ng, where g is the rotation number of g, and letf and r denote the respective lifts of @ and w to the real line. Show that if gy is not a rotation then there exists exactly one continuous function h: R>R such that hof=roh, h(x+1) =h(x)+1 and h(0) =0. Hint. Use the fact that ;f"(0)(mod 1): neZ! is a dense subset of S' have the same meaning as in the last theorem and let Q denote the rotation number of the homeomorphism @,.

If o is irrational, then there exists a homeomorphism h: T? > T*, h(x, t) =(u, v), such that the trajectories coordinates (u, v), are of the form

of (2.5.1), written additively in the

v(u, Vo) = (u, Vo +uQ), _ ie. they are rectilinear in the covering plane R?.

Proof In virtue of Theorem 2.5.1 and of Corollary 1.4.2, there is a homeomorphism of S* whose lift H: R=R fulfils the identity roH=Hog,;

(2.5.4)

here, as before, g,(*) = y(t, *), and r denotes the lift of the rotation by

112

Dynamical Systems on Manifolds of Dimension

2no. Now we introduce new coordinates: draw an integral curve of (2.5.1): x=

1 and 2

through any point (t, x) we

p(t, Xo);

where Xp) = p(—t, x) = g_,(x) and we replace the coordinates (t, x) by (t, Xo); clearly enough, this change of coordinates is defined by a homeomorphism. Let us write

hy (t, Xo) = (t, H (xo) +to). The desired homeomorphism follows:

h, or rather its lift to R”, is defined

as

h(t; 'x) = hy lt, e(—1, x)) = hy (t, 9-,(3)). es h(t, x) = (u, v), |

where u=t,

v=H(x9)+toę = Hog_,(x)+te. In order to check that h indeed defines torus, we

have to show

a homeomorphism

of the

that

h(t+1, x) = h(t, x +1) = h(t, x)(mod 1).

We have

A(t, x+1) =(t, H(x9+1)+to) =(t, H(xo)+te)+ (0, 1) = (t, H (x) + to) (mod 1).

i

As regards the other equality, it suffices to verify that

h(0, x) = h(1, x)(mod 1).

Writing xo = g_,(x) =g;'(x) we get, in view of (2.5.4), h(1, x) =(1, H(x9)+0) = (1, roH(x)) =(1, roH og; *(x))

=(1, H(9) =(0, H(9)+(1, 0) =(0, A(x) = h(0, x)(mod 1).

Finally, observe that the trajectory p(t, xo), written in coordinates

Differential Equations and Flows on the Torus

LES

(u, v), takes the form h(t, p(t, x0)) =

(u, H

(x0)

+ ug) =

(u, Vo

+ ug),

and this ends the proof. Remark 2.5.1 If the function f in equation (2.5.1) is of class C’, r > 2, and if H (in the proof above) is a C’-diffeomorphism, then also h (the rectifying homeomorphism in Theorem 2.5.2) is a C’-diffeomorphism. Remark 2.5.2 In general, little can be said about the regularity of the homeomorphism h. E.g., it is not known whether h must be of class C! when f is analytic. Theorem 2.5.3 (Siegel, 1945, Siegel and Moser, 1971) If ®€ F"(T?),r > 2, is a vector field without critical points or closed orbits, then there exists a closed curve s(t) on T* such that

ś(t)ttP(s(t))

for every t;

moreover, this curve intersects every trajectory disconnect the torus. The curve s(t) is called Siegels curve.

Proof

of ®

and

does

not

Let ®(x) =(f'(x), f*(x)). Consider the field ¥ orthogonal to ©

at each point, given by

i

F(x) =(—f7 (x), f7(x). w is also of class C’. We will examine the system of equations

dy(t, x) _

Uda

a x (W(t, x),

w(0,x)=x.

(2.5.5)

The integral curves of (2.5.5) are orthogonal to the trajectories of ©. Let W(t, Xo) be a fixed solution of (2.5.5). Since T* is compact, the set w(Xxg) is non-empty. Thus there are time instants t, and ty, ty < tą, such that the points Y(t;, Xo) and Y(t, Xo) lie arbitrarily close to each other. Since ¥(x) 4 0 for x¢ T*, we can choose a rectifying chart in which the field trajectories are as in Fig. 2.5.2. The instants t; and t, can be chosen so that, within the chart, there is no portion of trajectory w(t, Xo) corresponding to a time interval in (t,, ¢,) and situated between the two pieces marked in Fig. 2.5.2. It is not difficult to construct

114

Dynamical Systems on Manifolds of Dimension

1 and 2

y(t), Xo)

Fic.

2.5.2

a C'-curve J(t),0O0 2, be a vector field without critical

points or closed orbits and let p, be the flow generated by ©. Then there exists a coordinate system in which the trajectories of o, are rectilinear. Proof

The flow 9, is given by the system of equations (2.5.6). In view

of Lemma

2.5.1 we

may

assume

that f'(x', x?) > 0. Dividing

the

Algebraic Automorphisms of the Torus

117

second equation of (2.5.6) by the first we obtain

dae 460,3). Ce i

(2.5.7)

exer

the quotient on the right is of class C’, since so is the variable change in Lemma 2.5.1. Then by Theorem 2.5.2 there is a coordinate system on

T* in which the trajectories of (2.5.7), lifted to R?, are straight lines. This proves coincide.

our

theorem,

since the trajectories

of (2.5.6) and (2.5.7)

Corollary 2.5.1 The phase portrait of a field ®e ¥"(T’), r > 2, without critical elements is equivalent to the phase portrait of a certain constant vector field on the torus. Corollary 2.5.2 If the homeomorphism h in Theorem 2.5.2 is of class C’,r >2, then the system of equations (2.5.6) can be reduced by a suitable change of variables (of class C') to the system dy!

di

dy 2

—_=G

UA 2) GU)

ay

=

eG(y,

%

Ay y*)

(2.5.8) żyj

where Q is a constant. This is an immediate consequence of Remark 2.5.1 and the proof of Theorem 2.5.4. The rectilinearity of the trajectories of (2.5.7) means just

M0

5 x 4)/f «(x x) = const. Hence the form (2.5.8). Exercise

2.5.3.B. Let U c R* be a simply connected closed domain and let © =(f',f)e7'(U),r> 1, $(x) 40 for xeU. Consider the orthogonal vector field W =(—f?, f'). Show that there is a coordinate system in U in which the trajectories of © and Y are mutually perpendicular straight lines. Hint. The trajectories themselves can be taken as the coordinate axes (and their translates); the coordinate values are to be defined as the transition times along the trajectories.

2.6. Algebraic Automorphisms of the Torus In Chapter I (Exercise 1.2.4) we introduced the concept of an algebraic automorphism of the torus T*, In this section we study the properties of such an automorphism in more detail.

118

Dynamical Systems on Manifolds of Dimension

1 and 2

The automorphism p: T? > T* given by a matrix AP ad where a, b, c, deZ, detp = +1, is said to be ergodic iff neither of the eigenvalues of @ is a root of unity. Let A and u be the eigenvalues of g. These values are real and we have Au = +1 (see Exercise 1.2.5). E.g., let |A| > 1, |u| < 1, and let u and v be the corresponding eigenvectors:

pluj=lu,

=gp(v)=

The lines: a = |tu: te R}, b= {tv: te R} decompose the torus into two parallelograms, which will be denoted by U and V. Fig. 2.6.1 shows how U and Vare obtained; U is the shaded part of the unit square, V is the remaining part.

CY NE b

FiG.

2.6.1

Shifting U and V by integer vectors we cover R” by a network Z of parallelograms isometric to U or V. The sides of U and Vparallel to v (sides Sy, Sg, dy,dą in Fig. 2.6.1) are called contracted sides; those parallel to u are called expanded sides. Let us look at o(U) and @(V). We see that: 1° p(d;) T? such that

poA=AoV. Proof

The assumption

that p and yw are topologically conjugate on

T? implies that they are also topologically conjugate on R”, up to an integer shift; that is to say, there are a homeomorphism h: R? > R?

124

Dynamical Systems on Manifolds of Dimension

and an integer vector reZ* = Z xZ

poh(x)=hoy(x)+Hr

1 and 2

such that

for xeR”.

(See Fig. 2.6.4; Q denotes the unit square, which we identify with the torus.)

kę a,’

"RB —— b 4

h

h ZA

ZĘ

m.

FiG. 2.6.4

Let k be any integer vector. Then

h(x+k) = h(x)+ A(k, x), where A(k, x)€Z* and A(k, x) depends continuously A(k, x) = A(k) = const

on x. Hence

with respect to x.

Since h is a lift of a torus homeomorphism,

we have

h(x®Z*) =h(x)BZ*, and so A: Z* + Z? is surjective. We will show that A is linear in k. Let k, leZ?. Since h(x+k+lI =h(x+k)+A(l and

h(x+k+l =h(x)+

A(k+D),

= h(x)+ A(k) + A(I)

Algebraic Automorphisms of the Torus

j Os

it follows that

A(k+l) =A(k)+A(D.

Consequently A is given by a matrix with integer entries. As observed, A is a surjection of Z”; thus the determinant of that matrix is equal to

+1. This means that A is an algebraic automorphism It remains to show that A conjugates gy with w.

of T*.

Choose xeR* and keZ? arbitrarily. The following equalities are fulfilled:

p (h(x) +k) = h(W(x+k)+r =hl(y(x) +y'(k))+r, p (h(x) +A(k)) = poh(x)+poA(k) = p(h(x +h), how(x)+Aow(k)+r = h(W(x)+(k)+r. Hence

poA(k)=Aoy(k)

for all keZ?’.

The two mappings: poA and Aow are linear; the last equality shows that they coincide, in particular, on the elements of a basis for R*. Thus they are identical; the proof is complete.

Transversality Theory and Generic Properties

3.1. Survey of Facts and Notions of Functional Analysis The study of generic properties requires the use of several nontrivial facts of functional analysis. Customarily, those facts are dealt with in standard courses. To acquaint the reader with them is the aim of the first section in this chapter. We omit proofs; they can be found in Dieudonné (1960). Let X and Y be Banach spaces. This notation is kept throughout the section. The space of all continuous linear mappings from X to Yis denoted by £ (X, Y). This is also a Banach space with the usual operator norm. Definition 3.1.1 A mapping f: X >Y is said to be differentiable at a point x,€X iff there exists an operator A,,e£ (X, Y) such that

fom WL o+ WS C0) — Ang (Il 1m

h>0

al

= 0.

The operator A,, is called the derivative of f at x9 and is denoted by the symbol df(xo), and its evaluation by df(x9)h (for an increment he X). The definition makes sense also if f is defined on some neighbourhood of xy.

Survey of Facts and Notions of Functional Analysis

127

Examples

3.1.1. Let

X=, Y=R,f(x) = ) 63 for x=(ć,el?. Then we have

for xo =(é,), h=(h,)

, p 0)

J(%0+h)-f(%)

= x (6; +h,„) —ć )

-3Y 2h, + ¥ (3ć,h2+ 65) n=1

n=1

= df(x9)h + R(h). It is easy to see that ||R(h)||/|h|| 512

Let AeŚ(X,

313,

Let

> O as h>0.

Y). Then dA(x,) =A

for every xpEX.

1

X =€0,

1) =Y¥, 6) (0) = [t+5)x*(5)d5"

for xe C40, 1).

0

Then 1

(df (x) h)(t) = 2 |(t+s) x(s) h(s) ds

for tec 0

as

h—>0.

3.1.5. Let F be a continuous symmetric n-linear map of X x ...

x X to

n times

Y. Write f(x) = F(x, ..., x). (Any function of this shape is termed an n-homogeneous polynomial; cf. Definition 3.1.2). As in Example 3.1.4, it can be shown that df (x)h = nF(x, ..., x, h): n— 1 times

128

Transversality

Theory and Generic Properties

Exercises

that the function f(x) =||x||,f: C—R,

3.1.1.B. Show

differentiable at any point Hint. The

directional

3.1.2.B. Examine

USB;

0

is not

xeC ella, are norms obtained in the way just described, then these norms are equivalent, i.e.. there are constants a and b such that

a||P||y, < IIPl|y, < b||Plln, | for every

Pe F"(M).

We pass to the definition of the k-jet bundle. Let ©,, ®,¢.¥"(M), let xeM and let (U,a) be a chart, xeU, y=a(x). Denote Pia; Fa. the images of ©,, ©, in that chart. We will write 14 (y)

II] =

a d'¥,,(y)

b

=

by

(y)

d'¥, ,(y)

here k is an integer between

for

I =

0, +...

k;

O and r.

Lemma 3.1.1 Let (Uo, «9) be some fixed chart and let xeUy. vector fields ©,, b,e.F'(M) satisfy the relation k

Fa, (Vo) = zyj (Yo),

Jo = % (x),

If two

Survey of Facts and Notions of Functional Analysis

137

then they satisfy also:

k FW)

=

Paa (W).

y =

a(x),

for any other chart (U, a), xeU. We leave the proof to the reader. k

This lemma

shows

relation between

that the relation

=

may

be interpreted

vector fields on M (at a point xeM):

as a

by definition,

P, (x) = 9ą(X) k Pa

(W) =

Paa (0);

in some (equivalently: equivalence relation.

=

a (x),

any) chart

(U, a), xeU.

Clearly,

this is an

Definition 3.1.7 Given a point xeM, a vector field ©e.7'(M) and an integer k, O< k0,

N).

We will write g, in place of g(a); each g, is a C'-map from N: g,(x)eN for xeM. The evaluation map ev,: 4

and

M

to

XM +*N

is defined by the formula:

ev(a, x) = Qa (x). If the evaluation

map

ev, is of class

C’, we

say that

g is a C’-

representation.

Theorem 3.2.2 Let a, M, N be C®°-manifolds, let W-N be a closed subset (not necessarily a submanifold) and K = M a compact subset, and let

0: 4 +C°(M, N) be a C°-representation. Then A$fy :=lfaew: g,(K)0W=Ó) is an open subset of 4.

144

|

Transversality

Theory and Generic Properties

Proof Let x9EK,aqEAkw. Since Q,,(Xo)¢W, there exists a neighbourhood V, x U,, of the point (do, Xo) in ./ x M such that g,(x)gW for any aeV,, xeU,,. This is a consequence of the assumption that ev, is continuous and W is closed. Keep a, fixed and let xo vary in K. By compactness, finitely many U,,,..., U,, cover K. To each U,. there corresponds a neighbourhood V,,,. The set V=F,1 9 riMuREi

a neighbourhood of ay and Vc ARy. Theorem

3.2.3

Let 4, M, N be Banach

C'-manifolds,

dimM < +o,

let Wc N be a closed C'-submanifold and K c M a compact subset, and let

be a C'-representation.

Akw:=|laeW':

Then

og, W for xeK}

is an open subset of 4.

Proof We define a certain set Z c £(T(M), T(N)). By definition, a point (x, y, A,,) (=an element of £(T(M), T(N))) belongs to Z iff either

1 yéW, or 2 yeW and the image A, ,(T;(M)) contains a closed complement to T,(W) in T,(N). The set Z is open. Indeed, if yę W, the set Z contains the point (x, y, Ay.) together with any neighbourhood Up where

x BAL, 8),

V, o W=

@ and B(A, „» £) denotes the e-ball around

space Y(T.(M), T,(N)); if R

4,, in the

the image of T,(M) under a con-

tains a complement to 7,(W) in T,(N); thus every operator A sufficiently close to A, in £ (T,(M), T,(N)) shares this property (we leave the details to the reader), and again the set U,xV,xB(A is contained Consider

x,y? é)

in Z, provided the set

W' = £(T(M), T(N))—

that

U,, V, and ¢>0

are small enough.

Elements of the Transversality

This is a closed subset of representation, the mapping

Theory

145

¥(T(M), T(N)).

Since

g

is a

C--

g,: 4 >C'(M, £(T(M), T(N))) given by

@a (x) = dg, (x) is a C°-representation.

Oa hiaW

„ill.

By construction,

on(xé W's

the condition (in the definition of transversality) that T,(M) should be split (by a certain subspace) is here immaterial, because dimM < +00. Applying Theorem 3.2.2 to the representation g’ and the closed set ‘A

W' = £(T(M), T(N)), we conclude that Axy is an open subset of „4. Theorem 3.2.4 (Transversal Density Theorem) Let 4, M, N be Banach C'-manifolds and WEN a C’-submanifold, and let

0: 4 >C'(M, N) be a C'-representation.

1° dimM =n
max(0, n—q), 32 -€V, „RM

Then the set

Ap:= (aew: 0, hW) is residual in A.

Proof

Consider the set |

Be

ey, (W) ECM,

According to Corollary 3.2.1, B is a submanifold of codimension q in 4x M. Denote by n, the projection Ty:

WXM>M,

and write T = ny|B:

Tl, (a, x) =a,

146

Transversality

Theory and Generic Properties |

Then z is a C'-map. Denote by R, the set of all regular values of z. We shall show that: (a) x is a Fredholm mapping with index n—q, (b) Ay = R,. Hence,

by Theorem

3.1.12, follows our assertion.

In the proof of (a) we will need the following lemma, which we give without proof, inviting the reader to regard it as an exercise. Lemma 3.2.1 Let F and G be Banach spaces, dimG = n < +00, and let E c F xG be a subspace of codimension q. Denote by p, the projection of F xG onto the first factor and write p = pylg. Then p is a Fredholm operator with index n—q. The proof of the theorem will be carried out in 3 steps. Step I. Proof of (a). We have to show that

dz: Ta,x(B) > Ta(A) is a Fredholm operator. This follows directly from Lemma 3.2.1 applied to G = T,,(M), F = 1 (4), E = Tq, (B); the assumptions of the lemma are fulfilled, because codim T,,,,,(B) = codim W = q, and so (a) is proved. Step II. We now prove the inclusion R, = Ap. Let ae R,. We have to show that g, hh W. Since T,(M) is finite-dimensional, it suffices to show that

T,(W) + da,(T,(M)) = T,(N) for xe M, y =0,(x)eW. Thus let ye T,(N). All that we need is to find a pair of vectors: We T,(W) and xeT;,(M) such that

y =w+do,x;

(3.2.1)

as in the proof of Theorem 3.2.1, we write for brevity dg,x, instead of do,(x)x; this will cause no confusion, because x remains fixed. On account of assumption 3° there are: a vector w, € T,(W) and a vector (a, X,)€ 7,(A) x T,(M) satisfying the equality

y = w, +d(ev,)(a, x,).

(3.2.2)

(Here again d(ev,)(a, x) stands for d(ev,)(a, x)(d, X,); an analogous notational convention is used with respect to all derivative symbols occurring in this proof.) Since d(ev,) = 0,ev, + 0,ev,, equality (3.2.2) can be rewritten as y =

Wy i Og (ev,)a+dQ,X1

A

(3.2.3)

Elements of the Transversality

Theory

147

By assumption, aeR,; thus there exists a vector be T, „(B),b = (d, X»), such that da(d, x.) =a. Further, by the definition of B, we have ę d(ev,)(d,

X) € LA:

1..,

W2 : = 0,(€V,)d + dg, € T,(W).

From

0, (ev,)4, y =

(3.2.3) and (3.2.4) we

(Wy + W)

— do,X2

(3.2.4)

obtain, by eliminating

the summand

+ doaX+,

which is just (3.2.1) with

W=w,+WET(W),

x =x,—-X,€T,(M).

Step III. Finally, we prove the inclusion Ay = R,. Let ae Ay and let (a, x)eB. We have to show that dz(a, x) maps 1, „(B) onto T,(.V). Choose de 7,(./) arbitrarily. Let x, e T,(M) and w, e T,(W), where v= 0,(x).) Clearly, the vector y (=

w, +

0, (ev,)a + do,X,

(523)

belongs to 7,(N). On the other hand, since g, sh, W, this vector can be written in the form y a

(3.2.6)

wą +dQ,X2,

where w; and Xx, are some vectors in T,(W) and Equalities (3.2.5) and (3.2.6) result in

T,(M), respectively.

(,(ev,)d = do,X +W, where W =Ww,—W;, d(ev,)(4,

— x)

X = X,—X,. Consequently, =we T,(W),

whence

(a, —x)e(d(ev,)) *T,(W) = Ta,» (B). Thus a = dn(a, x)(d,

— x)

is in the image of T,,,,(B) under the operator dn(a, x). Since a was chosen arbitrarily, this operator is surjective, and this ends the proof.

Transversality

148

Theory and Generic Properties

Exercises

3.2.1.A. Let M be a finite-dimensional manifold. Every vector field de F¥'(M) can be regarded as a mapping of M into T(M). Let % denote the set of vector fields de.7"(M) whose critical points are all non-degenerate. Prove that

G = (M) := {@e F'(M): © h(TMy)), where

(TM),

denotes

the

O-section

of

T(M)

(ie,

(TM)o

=-(x,0): xEM.). Hint. If ©(xg) = O and x, is non-degenerate, than the matrix d$(xg) is non-singular.

3.2.2.B. Show that %5(M) is dense in -F'(M) (in C'-topology). Hint. Apply Theorem

3.2.4.

3.3. Differentiation of a Flow on a Manifold with Respect to the Generating Vector Field Let M be a finite-dimensional manifold of class may consider the map

C”, dimM =m. We

G: F'(M)xXMxR>M given by the formula G(©, x, t) = g,(x)

(3.3.1)

for be.F'(M), xeM, teR; as usual, g, denotes the flow generated by $. Theorem

Proof (®, x, proof: ential

3.3.1

The mapping

G defined by (3.3.1) is of class C’.

It suffices to prove that G is of class C! in each variable t) separately. That G is so with respect to x and t requires no it is a direct consequence of basic theorems on ordinary differequations. We thus focus attention on continuous differentiability

of G with respect to the “variable” ¢

¥'(M).

Let us fix a point xo e M and a time-value t = a, and let Ge.F'(M) be a given vector field. Differentiability is a property of a local character; therefore we are only interested .in the behaviour of © in a subregion of M which contains a neighbourhood of the trajectory

Differentiating of a Flow with Respect to the Vector Field (,(Xo) for

0 2

adi KĘ

t 0

brad 2: 0

y! 2

kol: ym

Critical Elements of Vector Fields on Compact

Manifolds

157%

where 1

y =

3

1

x. I

>

m

This equation can be rewritten as:

x

«|

: |+t x Me

(A-D)|

7 m

:

y?

Ją

zm

The above equation is solvable for any y iff 1 is not an eigenvalue of A. This establishes (2). Theorem 3.4.2 Let bye.F'(M), M compact, and let yo be a closed orbit of %, with prime period ty. Suppose that Ty is transversal. Then there exist: (1) a neighbourhood V of %, in £F'(M), (2) an arbitrarily small neighbourhood U of 5% in M, (3) an arbitrarily small neighbourhood I of t> in R, with the following property: for every BEV there is a unique closed orbit y of ® such that )$zU and tel,tT denoting the period of y. Proof Let x 9€¥o and let S be a transversal section of yo at Xp. Consider the manifolds .7”(M)xSxR and MxR* x M locally at the points (®o, Xo, To) and (Xo, To, Xo), respectively. Denote by F and E the model spaces for S and M; then clearly dimF =m—1, dimE =m. Passing to charts, we may assume that M and S are (locally) portions of E and F, and thus we may identify the manifolds 7'(M) x S x R and

M x R* xM with the spaces 7'(M)xFxR onal A is then a linear subspace of subspace

A= (0.0.0: yer} is a complement

to 4 in ExRxE.

and ExRxE.

The diag-

Ex RxE of dimension m+1. The

Transversality

158

Let x denote the projection of

Theory and Generic Properties

Ex RxE

onto 4* given by:

n(x, t, y) =(0, 0, y—x). We define a mapping

G: F'(M)xFxR>4* by the formula

|

G(®, x, t) = nofe(x, t) =(0, 0, 9,(x)— x),

(3.4.4)

where g, is the flow generated by ©. Of course, G(®o, Xo; To) = 0. In virtue of Theorem transversal,

we

3.3.1, G is a C'-map. Since the period To is

have, on account

of Theorem

3.4.1,

4+dfa, (Xo, To)(E XR) = EXRXE=A1@4.

(3.4.5)

It follows that the linear operator tO fo, (Xo, To)

=

maps ExR onto J. By assumption, S is a cross-section;

thus

E admits

the decom-

position

E = [6(x)] OF, where F is tangent to S at Xo, dimF = m—1. vector xeE is of the form

x=(,6), Furthermore,

where

This means that every

we[®(x,)], EF.

the coordinates

u, v can be chosen

so as to have

1

(xo) =[1,0,...,0],

dQ, (%0) = k a

where A is an operator from F into F, and a is an (m—1)-vector. Writing vectors columnwise, we have n odfgy (Xo, To) (X, t) = n(x, t, dQ, , (Xo) (X, t))

= n((%, t, )+(0, 0, do, (x9)(%, )— x) 0 =

e

OO)

RA

(00, Ot Pro (Xo)t a ax Pro (X0)X — z)

Critical Elements of Vector Fields on Compact

=

ores Z >

Vs

x

Manifolds

| pd ne

0.A || 0

EE

159

0 )

The number 1 is not an eigenvalue of A, and so A—I is a surjection of F onto F. Since t can be arbitrary, the operator TOdfg, (Xo, To) carries F xk onto 4*. Comparing the dimensions dimF xR =(m—1)+1=m=dim4,

we infer that the operator ©

TO fo, (Xo, To) ==——~ G (Po, Xo, To) (Xt)

is an isomorphism between the spaces F x R and A‘. Hence, by the Implicit Mapping Theorem, there are: a neighbourhood V of $, in ¥"(M), an arbitrarily small neighbourhood U; of x9 in S and an arbitrarily small interval I = (19—€, to +), for which there exist unique functions x(®) and t(®) (defined for ®E V) such that x($)eU;, t(®)el and

G(®, x(®), t()) =0

for every eV.

Thus, taking a sufficiently small neighbourhood U of 5, (U has to satisfy: UMS 0 and «> 0, (4) a neighbourhood V of %, in F'(M) with the following properties: (a) By has no closed orbits with periods t in the interval a P,, be defined as follows:

k(A) = characteristic polynomial of

Ae #(R”, R").

It is easy to see that k is an algebraic map from R” onto P,,. Let .4/ denote the set of all linear operators R” > R” which have at least one

purely imaginary eigenvalue. Then .% coincides with k~ *(v(B)). Now we employ the following two theorems of algebra; their proofs can be found in Abraham and Robbin (1967). Theorem 3.5.2 The image and the inverse-image of a semi-algebraic set under an algebraic map is also a semi-algebraic set.

Theorem 3.5.3 submanifolds.

Every

semi-algebraic

set

in R" is a finite union

In virtue of these theorems the set . = k~'(v(B))

of

c £(R”, R") is a

finite union of submanifolds. Denote them by S,, ..., S,. Every operator Ae./ can be made, by an arbitrarily small perturbation, into an operator that does not belong to ./. Therefore the codimension of each S, is at least 1. The set .w is closed. To see this, take a sequence A, >4, A,€-%. For every A, there is a vector z„,eC”, ||z,|| = 1, such that Az, = ia,z,, where a,€R. We may assume that z,>z9 and a, +a, (because lal < ||A,|| £ const). Letting no we obtain Azg =iagzo, and so AED, Step II. For

every

chart

(U, a) on

M,J'(M)

can

be locally re-

presented as the product a(U) x Ex £(E, E). In view of what has been shown in Step I, the set W is the finite union of submanifolds

W, =a(U)x+0)xS,

j=l,...,l.

Obviously, codimW; > m+ 1, since the second factor is of codimension m and the third is of codimension not W is closed because .w is closed.

less than

1.

Generic Properties. Genericity of Property G, Proof of Theorem

3.5.1

(continued)

The

following

;

167

conditions

are

equivalent: 1° de FY,(M),

2 Qo(M)NW=6, a Oe NW; for j= 1, 1:.„1. The implications 1° 2° and 2° = 3° are obvious. Now, assume that 3° holds and that for some xe M Qo(x) EW,

Le; 00(x) EW; for some j.

According to 3° we have

deo()(T;(M))+ T„(W) = T„(J* (M),

NE

where w = Q9(x). On the other hand, codim T,,(W;)

ż m+1

and dim T,(M) = m,

and this contradicts (3.5.1). Thus 3° >2. Since W is closed and M is compact, the set 4 (M) is open, in virtue of Theorem 3.2.2 and the equivalence between conditions 1° and 2°. Using the equivalence between 1° and 3°, we may write 4, (M) as the intersection

%(M) = ()Hy

|

(3.5.2)

where H, is the set of vector fields © such that gg is transversal to Wj. If r > 2, then each H, is residual in 7'(M) on account of Theorem 3.2.4 (00 is of class C''! and r—1 > 1). Hence, by (3.5.2), 4, (M) is residual in 7'(M), and thus is dense in ¥'(M). It remains to consider the case of r = 1. Every open set Vin 7'(M) contains a vector field ®, of class C? together with a set V, which is a neighbourhood of ©, in ¥*(M). By the conclusion of the preceding case (r > 2), there exists

BEV,

AG?(M)

CVA Gi(M).

This ends the proof. Corollary 3.5.1

Property 4, is C'-generic.

168

Transversality

Theory and Generic

Properties

Exercise

3.5.1.A. Let og(x) =j$(x), xeM,

Be F'(M), r > 2. Show that

d(ev,)(®, x): F'(M) x T(M) > TiA(J” (M)) is a surjection. Hint. This follows from the definition of a jet. 3.6. Genericity of Property *,

In this section we prove the following theorem: Theorem 3.6.1 If M is a compact manifold and if r > 1, then the set (a) is open and dense in F"'(M) for every a> 0. Proof We first present the general scheme of the proof. The entire procedure is divided into six steps and we now characterize briefly each of them. In Step I we show that the set %5(a) is open. In Step II we show that the set 45,(a) is open. In Step III we show that every © e 5 (M) has a neighbourhood V such that Vc Y3(ap) for some do. In Step IV we show that if Vc 4{(M) is an open set and if

Vn%5(a) is dense in V, then V1 %5,(3 a) is also dense in V. In Step V we show that if V O such that

G ha +94

for (©, x, the(G3(a)

\V)x M x(0, ża+2).

The set %;(a)nV is open in 7'(M) (as was shown in Step I), and so it is a Banach manifold. Thus

evfo(x, t) = G(®, x, 1) rhy,y4 for (x, theM x(0, ża+e) and Be w :=45(a)oV. In virtue of Theorem 3.2.4 the set A, consisting of all ©e%5(a)nV satisfying

fo hav4

for (x, t) EM x(0, Ża+e)

is residual in 45 (a) nV; a fortiori, it is dense in %5(a)oV. To conclude

Genericity of Property 4,

171

the proof of the assertion of this step, it remains to observe that AC

432 Ga).

Step V. Before formulating the claim of this step we state without proof an auxiliary fact; the proof will be given later on (Theorem 4.3.1 on p. 198): If de 43,,(a) and y is a closed orbit of © with period r < a and if U is a neighbourhood of , then there exists a vector field Y e.F'(M) such that 1° P|; =0, 2

Plm-vu=0,

3° for seR* sufficiently small, y is a hyperbolic closed orbit of the vector field ©+sY. Using this fact together with Theorem 3.4.7, we now show that if $ef3,,(a), then there exists a Ye.F'(M) such that the following conditions are fulfilled for seR* small enough: 4° critical elements with periods 0 and a neighbourhood Vof ® such that G',(ao) 0Vis dense in V. As we have shown in Step IV, 49312 (Fao) is dense in V; hence, in view of what was proved in Step V, also the set 4: (205) is dense in V. It follows by induction that for every keZ * the

set 45 ((3)* ao) is dense in V. In particular, taking ky such that (3)'oa, > a, we get the inclusion

4, (a) = %; ((3)'0a0), which shows that 45 (a) is dense in V. Since ® was chosen arbitrarily in r (M), it follows that (a) is dense in %;(M). By Theorem 3.5.1 4 (M) is a dense subset of .7”(M). Consequently the set 75 (a) is dense in 7'(M). This set is also open in 7'(M) (Step I). The theorem is thus proved. Corollary 3.6.1

The set Y,(M) is residual in ¥’(M).

This means that

Property G is C*-generic. This is an immediate consequence of the last theorem and the set equality

GM) = () G5(a) 3.7. Genericity of Property 9, The theorem presented below and asserting the genericity of Property 4, has been proved independently by two authors: Kupka (1963) and Smale (1963). This theorem seems to be the most significant of all the results concerning generic properties. Theorem 3.7.1 (Kupka-Smale) If M is a compact manifold and if r > 1, then the set 43 (M) is residual in ¥'(M), i.e. Property 43 is C'-generic.

Before passing to the proof, let us introduce some notation and recall certain facts concerning genericity. I. Let ye Y,(a) for some a > 0. In virtue of the Hadamard—Perron theorem, of Definition 1.9.3 and of Theorem (a) a number e > 0;

3.4.7, there exist:

(b) a neighbourhood V(®9) of ©, in 45(a); (c) open sets U,,..., U, = M with disjoint closures;

Genericity of Property 43

173

(d) manifolds P{, ..., Pg, o = +, each P? being equal to either R* or R°xS!

or R*'*xS', where s is an integer not greater than

dim M;

(e) C'-immersion Vy”: P? + M, GeV(9,), such that:

(f) V(®o) = Fate); (g) every

vector

field

VP, ---.yę With periods BL

deV(©,)

©, in C'-topology, then 3? > Wg, uniformly on compact subsets of P/,o= +. II. Let w, and W, be bijective immersions of manifolds P, and P, into M. We say that w, and wW, are transversal iff the product map

F: P,

xP, > MxM

given by P(py, Pa) =(W1(P1), W2(p2)) is transver-

sal to the diagonal A in MxM. The transversality of Vy; and YW, is equivalent to the transversality of the submanifolds W, = w,(P,) and W, = W2(P2).

III.

Let

y, and

y, be two

critical

elements

of a vector

field

De GY,(M). Let W” (74) =" (P1), W (v2) = (P2), where y* and

W” are immersions, and P,, P, = R* or R*xS' or R°xS'. Let U; and

U, be neighbourhoods A

gp, =

of 7, and $,. Write

$+.%,

where

A ={VeF"(M):

P(x) =0 for xeU, VU}.

is a closed subset of ¥’(M). The construction of suitable immersions Section 1.6. We now briefly recall it.

Let’ 7, =x;, There exists

has

been

be a critical point. Then) P,-—*R*,.

described

s =dimW

in

(7).

a mapping f: U > D, where U is a small neighbourhood of

x, in W*(x,), f(x,) =0 and D is a neighbourhood of 0 in R'; the mapping f depends on the flow o; (generated by ©5) restricted to U. We define a map g: D>D by

g(p) =fogof *(p), where ¢ stands for pfo and peD; then we extend g to the whole of R’. We have ||dg|| < 1, and so we may require that the extended g be a

174

Transversality

Theory and Generic Properties

contraction of R* to 0; denote this extension by 4. Further, we define

Veo (p) = p-nof "' 0G"(),

(3.7.1)

where n is an integer large enough to have g”(p)eD. The mapping defined by (3.7.1) does not in fact depend on n; the value of the composition on the right of (3.7.1) is the same for any two values of n fulfilling the condition: g"(p)eD. Without restricting generality we may assume that U = U,. The immersion ~~ = Va, is constructed quite analogously. If y, is a closed orbit, the case can be reduced to the preceding one by considering suitable cross-sections. Since the vector fields e.g are identical with ©, on the set U, UU,, the immersions

Wa (p) = 92, 0f *og"(p) coincide with whenever Wo According (with respect

Va, (P) whenever Wg (p)eU,; similarly, Wo (p) = Va, (P) (p)eU>. to Theorem 3.3.1, the assignment ®+> 9%, is of class C! to de V(®,)). Hence, a fortiori, the assignment ¥+> 9%,

= so" is of class C! with respect to Yew. It follows that WZ, We are of class C', regarded

Now

as functions

of Few,

Y= @-—@p.

consider the product map

Fog: PyxP,>+MxM

given by Fa(P1. P,) = (Wo (P1), Vo (p2)).

We will prove Lemma

3.7.1

evr:

The evaluation map

A xPyxP,

>MxM

is transversal to the diagonal Proof

A c

MxM.

The transversality of W * (y;) and W” (y,) is invariant under the

action of the flow g?, and thus it suffices to examine it at some (freely chosen) point xeW* (y,) 7 W™(y2). Suppose that

Wż,(P), Va, (P2))e4 for some p,, pz. Denote the common value of Wo, (P1) and Wo, (p2) by x. If yy #y2, then (in view of U,; 1U,=@) we may assume that

Genericity of Property 43

+75

xgU,UU;. (Otherwise, say if xeU;,, by pushing x along the trajectory we find a moment at which the point has left U, without entering U,; and translation along a trajectory does not impair transversality.) If yy =y2, we take U, =U, small enough to be sure that (by hyperbolicity) there is no other orbit of ©, different from y, and entirely contained in U,. Thus also in this case we may assume that x¢U,

=U, uUU,. Let us write the derivative of ev;. at the point (¥ = 0, py, p):

d(evr)(0, P+, P2) (W, Pi, Pa) = Opevp (0, Pi, Pa) (H) + Oe, pCVF (0, P+, P2)(P1; Pa)The first summand

on the right is of the form:

ÓgeVp(0, P+, Pz)(P) = (Va, (PJ) (W), ów, (Pz) (©). Now, the field Y can be defined arbitrarily in small neighbourhoods of U, and U, (provided those neighbourhoods are disjoint). Thus for every pair of vectors x;,, X,¢T7,(M) there exists a vector field Pew such that Oy Wo, (P1)(¥) =>

wa, (P2)(¥) 02

(see Exercise 3.7.1 on p. 179; also Theorem

(3.7.2)

3.3.2). Hence

T(4)+ówevp (0, P+, P2)(-4 x 10; x 10;) = T,(M) x T,(M), ending the proof of the lemma. Proof of Theorem 3.7.1 Let a be any positive number. For every ó,e%5,(a) there exists a neighbourhood V(%,) satisfying conditions (a)(e) on p. 172. By Theorem 3.6.2 45(a) is an open subset of the separable Banach space ¥"(M); thus 45(a) is a separable paracompact space. Consequently the cover |V($%): de%75(a)) contains a countable subcover, which is locally finite. Let (V,),c=1,2,..., denote such subcover. Hence, to each V, there correspond: numbers e, and k,, manifolds P?.,i=1,...,k,,o = +, and immersions 97, satisfying conditions (f)-(h). Let b be any natural number. Each P?, is either R* or

R'xS'

or R'xS'. Let K(P;,, b) be the set in P?, defined as follows:

K(P?,, b) =(eeR': |lel|

G3 (a)

for a, < ap.

Therefore

G(M) = ()(a) Each of the sets 45 (a) is residual in ¥’(M); thus 43 (M) is also residual in .7'(M), and this is precisely the assertion of our theorem. The proof is complete. The Kupka-Smale theorem admits certain modifications. For instance, let M be a compact Riemannian manifold. We denote by grad’(M) the space of all gradient vector fields of class C’ on M. Clearly, grad'(M) is a closed linear subspace of #"(M); in fact, grad'(M) is isomorphic to the space C'(M) of all real-valued C’functions on M. The proof of Theorem 3.7.1 works without any essential changes in the case of the space grad’(M). And thus, we have: Theorem

3.7.2

If M

is compact,

then Property

4

is generic in the

space grad'(M),r>1. Let us now turn our attention to the space Diff’(M); this is a complete metric space. The concept of genericity is introduced in the space of diffeomorphisms exactly in the same way as in the space of vector fields. The Kupka-Smale theorem for diffeomorphisms reads as follows:

Theorem 3.7.3 If M is a compact manifold, then the following properties are generic in the space Diff”'(M),r>1: (1) every periodic point (of the diffeomorphism under consideration) is hyperbolic;

(2) for every pair of periodic points p, qeM, and W (p) intersect transversally.

Theorem 3.7.3. is easily derived from Theorem suspensions of diffeomorphisms qe Diff’(M).

the manifolds

W*(p)

3.7.1 by passing to

Non-Wandering

Points. Closing Lemma

179

Exercise

3.7.1.A. Prove the statement including formulas (3.7.2) in the proof of Lemma 3.7.1.

3.8. Non-Wandering

Points. Closing Lemma. G, and 7;

Genericity of Properties

We now introduce the important concept of non-wandering points. Let $e.7'(M) and let o, denote as usual the flow on M generated by ©. Definition 3.8.1 A point xeM is called wandering iff there exist a neighbourhood U of x and a number f, > O such that

U g,(UJnuU=0Q.

lt] >to

It immediately follows from this definition that the set of wandering points is open and 4g,-invariant. Definition 3.8.2 A point xeM which is not wandering is called nonwandering. The set of all non-wandering points with respect to a vector field © will be denoted by Q(@). In other words: a point xeM is non-wandering iff for every neighbourhood U of x there is a sequence (t,), |t,| > +00, such that each g, (U) intersects U (see Fig. 3.8.1).

Definition 3.8.3 A point xeM is called recurrent iff xeo(y,)Ua(yx), where y, is the trajectory through x and o(y,), «(y,) denote the limit sets defined in Section 1.11.

180

Transversality

Theory and Generic Properties

If xev(y,), we say that x is w-recurrent ;if xea(y,), we say that x is a-recurrent. i It is easy to note that every recurrent point is converse is not necessarily true. E.g. consider dimensional manifold M) behaving on a fragment Fig. 3.8.2 (x,, x2, X3, X4 are critical points). Then on the trajectories y;, 2, Y3, Y4 are non-wandering,

FiG.

non-wandering. The a flow (on a twoof M as shown in all points which lie yet not a-recurrent.

3.8.2

Exercises

3.8.1.A. Show that Definition 3.8.2 is equivalent to the following: a point xeM is non-wandering iff for every neighbourhood U of x there

is a sequence (t,), |t,| > + co, such that o, (U) VU # @, n=1, 2,... 3.8.2.B. Find an example of a point which is w-recurrent but not arecurrent. Hint. Take a constant vector field on the torus T? without closed orbits (then every orbit is dense in T*); multiply that field by a function vanishing at exactly one point.

Non-wandering points are defined quite similarly for the case of a single diffeomorphism ge Diff’ (M).

Definition 3.8.4 A point xeM is called wandering iff there exists a neighbourhood U of x such that

U o"(U)nu=6Q. |n| >O

Non-Wandering

Points. Closing Lemma

181

Definition 3.8.5 A point xeM which is not wandering is called nonwandering. The set of all non-wandering points with respect to a diffeomorphism o will be denoted by Q(g). As before, Q(@) is a closed set, invariant under og. We

now

state

a theorem

on

non-wandering

points,

the famous

“Closing Lemma” proved by Pugh (1967a, 1967b). The formulation is astonishingly simple, unlike the proof, which is long and extremely difficult; the two papers have jointly some 60 pages. Theorem 3.8.1 (Closing Lemma) Let ®)¢.¥'(M), M compact. Suppose that x EM is a non-wandering point for dg. Then every neighbourhood V of ©, in F'(M) contains a vector Ma ® such that x lies on a certain closed orbit of ©.

We do not present a proof. It should be emphasized that the theorem is concerned exclusively with vector fields of class C'. An analogous statement for fields of class C° (ie. continuous) is very easy to prove (Exercise 3.8.3); as regards fields of a higher class of regularity, the question is still open.

Problem Is the Closing More precisely: let xe M b,e7F'(M) r>2. Does necessarily contain a field

Lemma true for ®)¢4"(M), where r > 2? be a non-wandering point for a vector field every neighbourhood V of ®) in ¥"(M) © such that x lies on a closed orbit of ©? Exercise

3.8.3.B. Let ©, ¢ F¥°(M) be a vector field generating a flow g,. Suppose that xeM is a non-wandering point. Show that in every neigh-

bourhood Vof ©%, in ¥°(M) there is a field beV such that x lies on some closed orbit of ©. Hint. Perturbate ®, locally until the orbit through x closes.

|

We recall that the symbol Ig denotes the set of all points of M belonging to critical elements of ©.

PE

2 (B)=

3.8.6

We say that the field $e.7"”(M) has Property 4, iff

maaie going to es that Property 4, is generic. The proof requires some facts of general topology, which we now briefly discuss.

182

Transversality

Theory and Generic Properties

Let Y be a compact metric space. Denote by FY the class of all closed subsets of Y. Then FY is a metric space with metric

(3.8.1)

d(A, B) = max(supo(a, B), sup @(A, b)), beB

acA

Q denoting the metric in Y. We call d the Hausdorff metric in FY. Exercise 3.8.4.A. Show that formula (3.8.1) defines a metric in FY. For a sequence of sets A„eFY

define

liminfA, := {ye Y: y =limy,, where y,¢€A,}. This set is an element of FY. Let g: X >FY be a set-valued metric space.

function;

X is assumed

to be a

Definition 3.8.7 The function g is said to be lower semi-continuous (abbr.: l.s.c.) iff for every xe X and every sequence (x,) converging to x we have liminfg(x,) > g(x). n>

o

Lemma 3.8.1 For Y compact, every monotonically increasing sequence of L.s.c. functions converges to an l.s.c. function. This means that if g,: X > FY are l.s.c. and g,(x) € gn+1(%),n=l, 2,..., then the limit g given

by g(x) = lim g,(x) :=

Ugn(x)

n>©o

is LS.

Lemma 3.8.2 For Y compact, the set of points of continuity of any Ls.c. function is residual in X.

We omit the proofs of the two lemmas; they can be found in Kuratowski (1961), pp. 38—40. In the application that follows, .7' (M) will be taken for X and M for Y (by assumption, M is compact). Let T„(%) denote the set of all

Non-Wandering

Points. Closing Lemma

183

periodic points of ® with periods 1) by diffeomorphisms for which those values are different from those assumed by o. Also, if we considered Diff’(M) with C°-topology, no diffeomorphism would be structurally stable (in the modified meaning); diffeomorphisms which are close in C°-topology may have different . numbers (or sets) of periodic points. Now we can readily see that if the definition of structural stability for flows imposed topological conjugacy in place of phase portrait equivalence, we would arrive at a concept of little use, if any. The periods of closed orbits would be invariants. Thus e.g. every vector field @ having, say, finitely many closed orbits would be unstable (in that sense): the field (1+e)© is arbitrarily close to ©, but the periods of closed orbits of © and (1+¢)@ are different.

Exercise

4.3.1.B. Let © be a Pęz'(T7). Hint. Applying either a field whose dense, according as coordinate axes at

constant vector field on the torus T?. Show that | an arbitrarily small perturbation we can obtain orbits are all closed or a field whose orbits are all the trajectories of the perturbed field intersect the an angle « with tana rational or irrational.

After these preliminary remarks we discuss necessary for a system to be structurally stable. M denotes a compact manifold.

certain

conditions

198

Structural Stability. Homoclinic

Points

Definition 4.3.1 A closed orbit y of a vector field $e.7"(M) is said to be tubularly isolated iff there exists a tubular neighbourhood U of 7 such that no other closed orbit is entirely contained within U (we admit, however, a non-empty intersection with U).

Lemma 4.3.1 isolated.

If ®ed"(M),

then every closed orbit of ® is tubularly

Proof Each hyperbolic closed orbit is tubularly isolated. Thus the lemma is true for fields with Property 4,. Since this property is generic, G,(M) is dense in 7'(M). Consequently, for every ®e2’(M) one can find ©, €Y,(M) such that © and ©, have equivalent phase portraits. It is easy to see that tubular isolation is invariant under phase portrait equivalence. Hence the assertion. Theorem 4.3.1 (Markus, 1961) If ®)¢¥"(M) is structurally stable and satisfies condition (e), then ®) has Property G3.

Proof Step I. We first prove that all critical points bolic. Since 44 (M) is a dense subset of ¥’(M) and with Property 4, has finitely many critical points, the finitely many critical points. Denote these points by

of ©, are hypersince every field field ©, also has x,, ..., X,.

For each x,, i= 1,..., s, consider the Hessian ©,(x,) and examine its eigenvalues. Suppose that ©0(x,) has k, eigenvalues A with Red 0 and j; eigenvalues with Rea = 0. Fix an x; and carry our consideration locally in a chart. Thus assume that x;eR” and that $, is defined in a neighbourhood U of x; in R”. Let U, be a neighbourhood of x, containing no critical points of $, other than x,. Let g(x) be a C'-function which is equal to 0 outside U, and to 1 in some smaller neighbourhood of x,. For any neR write

Ó!(3) =n(x—x)

for xeU;,

Qi (x) = g(x)Q? (x); Q! is a C'-field defined for all xeU.

Define;

Pi (x) = Bo(x) +Q; (x). Clearly

*!(x,) = 0. Further, we have

AI —P(x) = AI — $9(x)-n1 =(1-n)l — bo(x).

Structural Stability. Necessary Conditions

199

Thus

sp? (x) = sp®o (x) +n. Choose

4 so as to have

Rei 40

for Aesp ¥"(x;)

(then x; is a hyperbolic critical point of ¥7). If 4 is close to 0 and negative, 7 (x;) has k,+j, eigenvalues A with Rea < 0; if 7 is close to 0 and positive, Y7(x,) has k; eigenvalues A with Red < 0. Now, returning to the manifold, we may regard Q! as a C’-vector field on M, equal to ©, outside a neighbourhood of x,. Put 6" = 0+

¥ Qi. i=1

For |n| small enough the field ©” is arbitrarily close to ); any two fields ©" have equivalent phase portraits, provided that |4| is sufficiently small. Further, for small |y|, the points x,, ..., x, are the only critical points of ©". Write

d(®") = > dimW* (4, 9). t

Obviously, this number is invariant under phase portrait equivalence of the ©". Supposing that |y| is small enough, we have: ), (kj +i)

if n > 0,

Yk;

if 7 M is a one-to-one immersion, P denoting some manifold, i(P) = M,, then a set Bc M, is a cell iff the set i '(B)c P has non-empty interior and smooth boundary and is simply connected ;

Morse-Smale Systems

in the case of P=R*

205

this means

that i'(B) is diffeomorphic

to

a ball in R". The theorem which we now present is known as the A-Lemma (see Palis, 1969); it is one of the crucial theorems of the theory of smooth dynamical systems and will be often resorted to in the sequel. Let ye Diff’(M) and let pe M be a hyperbolic fixed point of g. As usual, W * (p) and W” (p) denote the stable and the unstable manifolds of p.

Theorem 4.4.1 (A-Lemma) Suppose that dimW (p)=s,02(x)

| (4.4.4)

Condition (4.4.1) implies that F"*(x) = 0 for x =(0, x”). Thus, putting x =q,, we obtain

+

Bee a (4) 0

de (q,) = | F?4(q,)

Aa F2? MI

(4.4.5)

Let e > O be arbitrary. Using (4.4.1), we can find a neighbourhood of 0 of the form:

OBEB>xXODEU, where oJ denotes the ball of radius 6 in W*(0), 6 being sufficiently small for the following estimate to be valid: 2

kK, :=sup

ać

min (e, k).

(4.4.6) °

U,

Applying the representation of dg (x) in the form (4.4.4) together with inequalities (4.4.2) and (4.4.3), we obtain the following estimate, holding for any 0, € 7,(S), xeS such that o'(x)eSoU,, i =0; ..., m: (do (%n- 1) On— 1))*| (dp O

są

eA

1)©-1)) ||

Ee Oc” 1)) pea)

R WE? (m4) (07-1)

How all + kalln_11|

BE

24) (One | (x, GI

p A(Un—1) +k,

giby [lon41 ka Ilo_2ll i by —kA(v,-1)

(4.4.7)

Points

Structural Stability. Homoclinic

208

Setting x =q we get, in view of (4.4.5), WG

n

ees

A+ FT (4-0)

+ FP An 1)On—13]] -

(pares 1)) (07 i)|

celOn— BREST] all +s len all n, the set D,, contains a cell

D, such that B' is e(1+(b,—1)~*)-close to D,,. If dimS > s, we draw a hyperplane H through W (0) and the point Gn,; considering the submanifold S,;= HS and appealing to the conclusion of the preceding case we can find a cell D,, = D,, 0 H such

that B* is e(1+ (bz— 1) ")-close to D,,* The proof of the A-Lemma is complete. Corollary 4.4.1 Let py, pa, pa be three hyperbolic fixed points of 9. Suppose that W* (p,) and W” (p2) have a point of transversal intersection

and so do W* (py) and W~ (p3). Then also W* (p,) and W

(ps) have a

point of transversal intersection.

Proof Let aeM be a point of transversal intersection of W* (p,) and W (p). Let A,B=W (p,) be cell neighbourhoods of a and pp,

210

Structural Stability. Homoclinic

Points

respectively. Take ng¢Z~* such that o "°(A) c B; such an no certainly exists because O '|v- Wo») is a contraction (to p»). In virtue of the ALemma, for any 6 > O there exists a cell neighbourhood D=W (pz) such that B is 5-C!-close to D. Thus given e > 0, we may assert that A

is e-C!-close to y"°(D), just by choosing 6 appropriately to e. Since e may be arbitrary, we conclude that g”0(D) intersects W” (p,) transversally at some point a,¢@"°(D) lying close to a (see Fig. 4.4.4).

Fic. 4.4.4

Corollary 4.4.2 Let peM be a hyperbolic fixed point of a diffeomorphism ye Diff’(M), r>1. Suppose that for some qeM,q#p, we have W” (p) h „W (p). Then the set Q(@) is infinite. Proof Let B= W (p) be a cell neighbourhood of p and let U be an arbitrarily small neighbourhood of q such that the set B; = UAW (p)

FiG. 4.4.5

Properties of Morse-Smale

Systems

zu

is a cell. Since g '|y-(, is a contraction (to p), there is an nyeZ* such that p "°(B,) < B. According to the A-Lemma, for any e > 0 there exists a cell D= W (p) such that B is e-C' D (this follows from

the proof of the A-Lemma; see Fig. 4.4.5). Let D, c D be a cell such that D, is e-C'-close to og"0(B;). If e is small enough, we have g”o(D,)o U 4 © which in view-of the inclusion p "(D;)sU yields o"'"o(U)nU #4 @. Since U can be arbitrarily small, it follows that q is non-wandering. Hence also each g”(q), neZ*, is a non-wandering point. We have

lim o"(q)=p,

o"(q)źp (neZ*);

consequently

CardQ(9) > Card (g"(q): neZ*), which is infinite.

Remark 4.4.1 Corollaries 4.4.1 and 4.4.2 are true without changes for hyperbolic periodic points of a diffeomorphism oeDiff"(M); they also, retain validity in the case of flows generated by vector fields de F'(M),r>1. 4.5. Properties of Morse-Smale Let M be a compact manifold and denote all the periodic orbits of 9. Definition 4.5.1

Systems

let pes'(M).

Let y;,,..., yy

We say that an orbit y, is preceded by y, (in symbols,

y £ % iff W*GJOW

(y) # 0.

EIGS4DA

Structural Stability. Homoclinic

212 Theorem 4.5.1

Points

The relation < just introduced defines a partial order in

the Set 4445 2, Yup Proof

Since W*(yjoW

(y,)

>}; # ©, we have y, < y;. Transitivity

follows from Corollary 4.4.1. Assume that y, < y, and y, < ;. If y, £ Y%»

then

(see

Corollary

4.4.1)

Wt (y)AW

(y)—-3;# OM,

whence

by

Corollary 4.4.2 the set Q(g) is infinite, contrary to the definition of a Morse-Smale diffeomorphism.

Definition 4.5.2

The set

{y,,..., yy} together with the order


1. (2) If a vector field Y is sufficiently close to a vector field ®€S"(M), then W and © have isomorphic phase diagrams; a similar statement holds for diffeomorphisms.

Proof We prove the theorem for r = 1 only; our proof is based on the Closing Lemma.

214

Structural Stability. Homoclinic

Points

We consider only the vector field case. Let Be S'(M) and let y,, ..., yy be all the critical elements of ©. Let U be a neighbourhood of a hyperbolic critical point x9; U is called hyperbolic if there exists a larger neighbourhood U, > U such that the system (U,, g,) is topologically conjugate to the local system (UG) where Ue T.,(M) and L, is the flow generated by the vector field @(X,)x in the tangent space T,,(M). Now let U be a tubular neighbourhood of a hyperbolic closed orbit y; U is called hyperbolic iff, for every cross-section S$ of y the set SU is a hyperbolic neighbourhood of the point Sy relative to the Poincaré map Os. Suppose that U,,..., Uy are disjoint hyperbolic neighbourhoods of i> --:, Vw. Let V, be a neighbourhood of © in ¥'(M) such that every vector field WeV, has exactly one critical element in each U,, =i. 2Na(see" Lheoreny-34.2-0nm p17). In virtue of Theorem 3.3.1 (p. 148) the mapping

G: F'(M)xXMxR>M given by

G(®, x, t) = 9,(X) is of class C' at any point (®, x, t). The proof of Theorem 4.5.2 is carried out in four steps. Step I. We shall show that there exists a neighbourhood V, of % such that every YWeV, has just as many critical elements as ©. Assume the contrary. Then there exists a sequence of vector fields

Y,,n=1,2,...,

C'-convergent to © and such that each Y/, has N,

critical elements, where N,;4 N. We may assume that Y„eV, for all n, which means that each Y/„ has one critical element in every U,, and. * thus N, > N. Since Y, has as many critical points as ®, it follows that Y, has more closed orbits. For any n denote by £, such an extra orbit, i.e., one not contained within any U,. Pick a point x, in each 8, so as to obtain a sequence (x,) such that

(we may pass to a subsequence if necessary). Since P, > © uniformly, we have >a (Xp) —>@ (Xo).

Properties of Morse-Smale

Systems

215

Let l, denote the trajectory of © passing through x, (see Fig. 4.5.3).

FiG. 4.5.3

By the conclusion of Exercise 4.5.1 on p. 212 the sets x(/,) and o(l,) coincide with some critical elements of ®. Let y,, and Vi, be these elements (respectively). In view of the continuity of g,(x), both in x and in the ®-variable, each trajectory f, (for large n) intersects U i, and U;,,. Denote by y, the earliest point on f,, later than x,, at which 8, enters U,,. We may assume that Ve > Vo Chi @ EU:

Now, f, is a closed orbit not entirely contained in U;,. Thus there is a point at which f, leaves U;,; denote by z, the earliest such point, later than y,. Again we may assume that 2,

2g

Fr U;,:

Let l, denote the trajectory of ® passing through Zo, /,(0) = Zo. Then the “half-trajectory” l; defined as I, restricted to t < 0 is wholly contained

in U;,. Indeed:

assume

/(t,;)¢U;, for some

t, < 0; this, by

the continuity of G(®, x, t), implies 8,(t,)¢U;, for large n and for t, - near t,, showing that the piece of f, limited by y, and z, intersects the boundary of U;,, contrary to the construction of y, and z,. The trajectory l, is approached by the trajectories f, and its w-set also coincides with some critical element s; By repetition of the previous argument we obtain a sequence of critical elements EA

PERŁA

Since © has only finitely many critical elements, one of the elements y, must enter the sequence more than once, and this, in view of Theorem

Structural Stability. Homoclinic

216

Points

4.5.1, implies WA

ling

Consequently

ZAZIE

1, is contained

in

W*(y,)nW (y,).

Hence

by

Corollary 4.4.2 (in the formulation involving flows) all points of /, are non-wandering and some of them do not belong to the union of 7,, i=1,..., N. This contradicts the assumption that © is a Morse-Smale

field. Step II. Vector fields which are C'-close to © have Property 4,4. Suppose that ® can be C'-approached by fields ¥,, such that Py = Perv, e CCP.

Thus let x„eQ(Y,)—Ty,.

Without

loss of generality we may

assume

N

that x,¢ UJ U;; otherwise the whole trajectory y*"(x,) passing through i=1

x, would lie in some U;, whereas the only trajectory of ¥, (for large n) contained in U, is the closed orbit near to y; — a contradiction, because y*r(x,) cannot be periodic, by the choice of xy. By the Closing Lemma each Y/„ can be C'-approached by a field w, whose trajectory through x, is periodic. Choose ¥/, so as to have W,— Pin C!-topology. For n large enough each Y/, has exactly one critical element in every U; and one closed orbit passing through x,E€UU;, ie, has at least N+1 critical elements, contrary to the conclusion of Step I. Hence follows the claim. Step III. Now we show that if Y is C'-close to ©, then © and Y have isomorphic phase diagrams. That means, we show that then

W'YJOWYJDZÓ

— iff W* OF) OW" (1K) # ©;

here y? and y; (i = j, k) denote the unique critical elements of © and Y, respectively, contained in the neighbourhood U,.

Thus assume that W*'(yj)o0W (yę) 4 ©. Let 6,>@ in C'topology. Let iq”: P; > M be an immersion that carries P*, the underlying manifold, onto W*(yjn), n=0, 1, 2,..., where ©) stands for ®. Clearly, P/ is the same for all ®,, provided that n is sufficiently large. In fact, P; is equal either to R* or to R°xS' or to R°xS'.

According to Theorem

1.6.2, we may assume that i? ig

topology on any set of the form K; (a) ea

(DEP;

: ||Pll < a}.

in C'-

Properties of Morse-Smale

Quite analogously,

we

Systems

217

have a sequence

of immersions

i, *: P,

> M.i, *(P,) = W (yen), such that i; '* >ig' in C'-topology on any set of the form

K, (a):= (peP,: ||pl| < a}, where P, denotes a manifold of one of the three types.

Let qe W* (y?°) 1 W (799). Then there exists a number a > 0 such that

.

q€ig"(K; (a)) Vig “(Ky (a). Since

W * (770) kW” (729) and since (i, "), (i, '*) converge C'-uniformly on Kj; (a), K, (a), we infer that for every n, sufficiently large, there exists a point q, such that

W* (yj5) ha, W” (w). To obtain the converse implication, assume that W* (y?) NW” (yg) = ©. Repeating the argument of Step I, we show that there is no sequence (@,) C'-convergent to © and such that W* (yn) OW” 0x9 ź © for all n. Step IV. It remains to prove that if ¥ is sufficiently close to ©, then Y satisfies the transversality condition (condition (3) of Definition 4.4.1

on p. 202). Again the proof is indirect. Suppose that there exists a sequence (©,) of vector fields C'-convergent to ® which do not satisfy the transversality condition. As shown in Step I, each ©, has just as many

critical elements as ©. By assumption, W* (y?n) intersects W (y?") nontransversally

at some

point g,. We

may

assume

that q,—-q,). Then

(yp?) OW (7%). Put W*e do

E, = T,,(W* (y?*))+ T,,(W~ (ve"))Since E, # 1,,(M), we 162,

have dimE, Tr(W* (i).

According

for i=j,k.

Hence

E, > Tyy(W* ($))+ GW” (78) = T,(M).

to Theorem

Structural Stability. Homoclinic

218

Points

This is the required contradiction: dim T,,(M) = dimM and dimE, < dimM. The theorem is thus proved. A significant result obtained by Palis and Smale in 1968 establishes structural stability of Morse-Smale systems. We only formulate the result; the proof is long and difficult and therefore omitted here. Theorem 4.5.3 Every diffeomorphism oes'(M) is C'-structurally stable. Every vector field d©eS'(M) is C'-structurally stable. This result is not the strongest possible with regard to structural _ stability. In Section 4.12 we formulate a theorem (Theorem 4.12.2 on p. 264) of which Theorem 4.5.3 is a particular case.

4.6. Structural Stability of Flows on Two-Dimensional

Manifolds

In the case of M being a two-dimensional manifold, the Morse— Smale conditions (1)-(3) (see Section 4.4, p. 202) admit an alternative restatement: Theorem 4.6.1 For ®e¥"(M), dimM =2, conditions (1)43) are equivalent to the following ones: (1’) for every trajectory y the sets «(y) and w(y) coincide with some critical elements,

(2) all critical elements are hyperbolic, (3) there is no trajectory joining two saddles. Proof (1)-3) imply (1’H(3’): (1’) is the conclusion of Exercise 4.5.1; (2) and (2’) are identical; (3') results from the following observation: if two saddles, call them x, and x2, were connected by a trajectory y, then 9 < W* (x1) 01W™ (x2) and consequently this intersection would not be transversal. (1'}H{3’) imply (1)3): (2) does not differ from (2’), (3) is an immediate consequence of (3’). It remains to prove (1). | To begin with, we show that © has only finitely many critical elements. Since all of them are hyperbolic, there cannot be infinitely many critical points. Thus assume that © has infinitely many closed orbits; denote them by f4, 8a, ... By compactness, the sets f, cluster about some point xg. According to condition (1’), the sets w(x) and

Structural Stability of Flows on Two-Dimensional

Manifolds

219

a(Xo) are critical elements. They must be saddles; to see this, notice that otherwise the dimension of the stable (unstable) manifold of w(x5) or a(x9) would be equal to 2, and so either the set w(x,) or the set a(X9) would attract its vicinity (either at +00 or at —oo), which is

Fic. 4.6.1

impossible, because every neighbourhood of w(xg) or of a(x9) is intersected by closed orbits B„. Thus the trajectory passing through x, connects the two saddles: a(x9) and w(xg), contrary to condition (3’). We have yet to show that every non-wandering point belongs to some critical element. Assume that xe (2(©) and that ®(x) ź 0. The sets a(x) and y(x) are critical elements. If a(x) # w(x), both sets are saddles (otherwise x has a “wandering neighbourhood”) and are connected by the trajectory passing through x; this contradicts (3’). Therefore a(x) equals a(x) and it is not a saddle. Since all critical elements are hyperbolic, w(x) is a closed orbit passing through x. This ends the proof. Exercise

4.6.1.B. Show that if M = S?, then condition (1’) is a consequence of (2) and (3). Hint. On account of Schwartz” theorem (Theorem 2.2.1) the set w(y) contains a critical element. If the stable manifold of that element is of dimension 2, then w(y) is equal to that element. If w(y) contains saddle points (one or more) but is itself not a saddle, then (S* being simply connected) there must exist a trajectory joining two saddles or a trajectory 7 such that w()) = a(¥) =a saddle.

Structural Stability. Homoclinic

220

Points

Example

4.6.1. Consider the vector field ®, defined in Exercise 1.4.4 on p. 37; we already came across that field in the discussion of Example 4.1.3 on p. 189. We recall that ©, results by projecting the gravitation field onto a tilted torus. In the conclusion of Example 4.1.3 we have stated that ©, is structurally stable. Now we show that it is a Morse Smale field. @. is a gradient field. Hence all of its trajectories begin and end at the critical points x,, x2, x3, X4. As we have already ascertained there is no trajectory joining the saddles x, and x,. It remains to verify that the critical points are hyperbolic. We will use the following parametrization of T*: x =(a+bcoss)cost, y =(a+bcoss)sint, z= Dsins, where

a, b, t, s are as in Fig. 4.6.2; a>b.

FiG. 4.6.2

The torus has to be tilted; thus we consider the gravitation field in

R* to be of the form P(x) = [cosa, 0, sina]

=const,

Oa

Hm.

Projecting Y onto T* we obtain the following vector field (written in the coordinates

$(t,sj=|

t, s; we omit

—

sintcos« a+bcoss

The critical points are: X4 =(0, &),

the calculations):

—sinscostcosa+coss sina

;

b

a

.

x3 =(0,a+n),

x, =(n, —a),

x, =(n, na).

Structural Stability of Flows on Two-Dimensional

Manifolds

503|

The Hessian is given by COS ft cos &

R P(t, S) =

sins sintcos«

a+bcoss ŻE sinssintcosa

b

hea Sipe —cosscostcosa—sinssina

b

b

At the critical points the Hessian takes on the form +cos a

P(x)

=

a+bcosa

0

1

b

b=] 23S 4,

+— wa:

It is readily seen that the eigenvalues of ©(x,) are real and non-zero. Thus the four critical points are hyperbolic. In virtue of the Palis-Smale theorem the field under consideration is structurally stable. Observe that the above conclusion is a result of the general theory and an elementary calculus. A straightforward proof would by no means be easier than general considerations, carried out at least with regard to gradient fields. The Palis-Smale theorem gives a sufficient condition for structural stability. As we shall see later on (see p. 264), this condition is not necessary

in general.

dimensional Theorem

However,

it is necessary

in the case

of two-

be a two-dimensional

oriented

orientable manifolds.

4.6.2 (Peixoto,

1962)

Let

M

manifold and let Bye. F"(M), r > 1. The field $, is C’-structurally stable if and only if it is a Morse-Smale field.

The “if” part is established by Theorem 4.5.3, which asserts the sufficiency of Morse-Smale conditions for stability in the genera! case. As regards the necessity (the "only if” part), we shall present a proof only for the case of r = 1; our proof will resort to the Closing Lemma and to Smale’s Theorem 4.3.4. A proof for the general case can be found in Palis and de Melo (1978). We will need some auxiliary facts. Let M

Lemma

be compact,

4.6.1

connected,

two-dimensional,

Suppose that a vector field dBe2"(M)

and oriented.

has infinitely many

222

Structural Stability. Homoclinic

Points

closed orbits: B,, Ba, ... Write F = UB,.—UB, and let xpeF) (xg) #0. Then: 1° the sets w(Xo) and a(xg) are reduced to saddlé points, 2° the trajectory y passing through xo is a submanifold of M (in the topology of M). Proof Assume that there is a point x, ev(x0) such that $(x,) 40.

Choose a rectifying chart (U, a), U3x,. The trajectories of ® in U are parallel straight line segments (see Fig. 4.6.3). Let I, denote the piece of

the trajectory through x, contained within U, x,e1,. Let i, =|,2,..., be a sequence of segments of y, converging to I,. Draw a cross-section $ through x,, perpendicular in U to Ig and denote by y, the cross-point of S and /,. It can be assumed that the sequence (),) converges to x, monotonically. Each 7, is the limit of a sequence of pieces of f,. Denote by z, the point of the (finite) set B.S which lies nearest to x,. Let (f,,) be a subsequence of (f,,) such that each z,, belongs to an interval (Vas FEU: the sequence (y,,) being monotone. It is well known (see e.g. Theorem 7.1 in Wallace, 1968) that M is homeomorphic to the sphere S* with a finite number of “handles”; denote this number (the genus of M) by p. There can exist at most p closed curves on M not disconnecting M (see Wallace, 1968). Consider the curves ,,, ..., Broa’ In view of what has just been said, one of these curves, say B,,. is a common boundary of two connected components of M. Since M is oriented, we may distinguish the two sides of f,.. The segments /, and I,,+1 lie on the opposite sides of B, (see Fig. 4.6.4) and thus they belong to different components. But I, and I, ,, are pieces of the same trajectory y. Thus it is possible to pass from I, to I, +, along y without crossing any of the curves f,. We have arrived at a contradiction; this shows that w(x) does not contain any non-critical point.

Structural Stability of Flows on Two-Dimensional

Fic.

Manifolds

228

4.6.4

The set w(Xxg) is connected (see Section 1.11). Thus it must be a singleton, because it consists entirely of critical points (as we have just shown) and all critical points are hyperbolic — here we use Theorem 4.3.1, namely the case considered in Step I of its proof (not involving condition (e)). The unique point in @(X9) is a saddle; otherwise it would attract some neighbourhood of xp, and this is impossible, since the closed orbits 8, cluster about xy. This proves assertion 1° for w(Xg); the reasoning for x(Xo) is the

same. Assertion 2° is a direct consequence of the proof of 1°: any segment of » contained in U is isolated from any other such segment, and hence y is a one-dimensional submanifold of M. Lemma 4.6.1 implies Corollary 4.6.1 If xq EM is a recurrent point of a field De F"(M), M being compact, two-dimensional, orientable, then x, is periodic. From

now

on we

restrict our

to r=lL1.

Lemma

4.6.2

Proof

Assume that © has infinitely many closed orbits B,, B,, .... By

Theorem

Let F P(X) points union either

4.3.1

If Bez'(M),

considerations

with Remark

then © has finitely many

4.3.1, all critical elements

closed orbits.

are hyperbolic.

be as in Lemma 4.6.1. Certainly there exists a point x„eF with ź 0; if there were no such points, F would consist only of critical which are hyperbolic and thus cannot be cluster points of the of closed orbits. In view of Lemma 4.6.1 the orbit y = y(Xo) is closed or connects two saddles. The first eventuality is not

224

Structural Stability. Homoclinic

Points

possible, since all closed orbits are hyperbolic (they are either attractive or repulsive, and so 7 cannot contain cluster points of £,). The second eventuality is contradicted by Theorem 4.3.4 (Smale’s): if y connects two saddle-points, then 7 is a subset of the stable manifold of one of these points and of the unstable manifold of the other.

Proof of Theorem 4.6.2 (for

r=1)

Assume that ®, is C'-structurally

stable. In virtue of Theorem 4.3.3 ©, has Property Y, (here is the Closing Lemma involved; the proof of Theorem 4.3.3 is based on it).

Thus Q($) = Per$,. Since ©, has and closed orbits (Lemma 4.6.2), we the Morse-Smale condition (1) of fulfilled in virtue of Theorem 4.3.1 follows from Theorem 4.3.4. As

we

have

seen,

structurally

only finitely many critical points have Per$, = Per®,. This proves Definition 4.4.1. Condition (2) is with Remark 4.3.1, condition (3) stable

vector

fields

on

two-

dimensional compact orientable manifolds have finitely many critical elements. This is not necessarily true in higher dimensions, e.g. the socalled Anosov flows (see Definition 4.8.2 on p. 242) on tori of dimension not less than 3 are structurally stable but have infinitely many closed orbits (see Anosov, 1967). The same remark applies to diffeomorphisms. Another interesting property of structurally stable vector fields on two-dimensional orientable manifolds is expressed by the following theorem: Theorem 4.6.3 (Peixoto, 1962) If M is a compact oriented twodimensional manifold, then 2” (M) is a dense open subset of -F'(M),r>1.

Proof Again we restrict ourselves to the case of r= l, referring to Palis and de Melo (1978) for the general case. In virtue of Theorem 4.6.2 structurally stable fields are precisely the. Morse-Smale fields, and the latter constitute an open subset of #'(M), by Theorem 4.5.2.

Now

we

prove

that

43(M)n4%4(M)eZ'(M).

Let

$e%1(M)

0%4(M). It follows from the definition of Property 4; and from Theorem 4.6.1 that then © satisfies the Morse-Smale conditions (2) and (3) (see p. 202). Hence by Corollary 4.6.2 © has only finitely many closed orbits, and thus finitely many critical elements in general. Consequently and by Property 44, we have Q(®) = Per®. This shows

that ® fulfils also condition (1), and so ©eZ'(M).

Hyperbolic Sets

225

Now we apply the Kupka-Smale theorem (Theorem 3.7.1) and Pugh’s theorem (Corollary 3.8.1). According to those theorems, 43 (M)

and %4(M) are residual subsets of #1(M). Since the set Z'(M) contains the common part of 4;(M) and %!(M), it is also residual, hence it is dense in 7'(M). This ends the proof. Remark 4.6.1 It is not known whether the Closing Lemma is true for C'-flows on compact oriented two-dimensional manifolds. If this is the case, the above proofs work without any .change in the general situation (concerning C'-flows and C’-stability, r > 2). 4.7. Hyperbolic Sets It has already been emphasized that Morse-Smale conditions are only sufficient for structural stability. In the next section we shall present a class of structurally stable diffeomorphisms which need not satisfy the Morse-Smale conditions — the so-called Anosov systems. To define and discuss those systems we shall need the concept of a hyperbolic set. Before introducing this concept, let us briefly recall some facts of differential geometry. Let M be a Riemannian manifold with scalar product (*, '),, xe M. Let f: T(M)—T(M). We call f a bundle map iff f preserves the fibres. A bundle map f is called: 1° contracting iff there exist constants c > 0 and A, 0 1 such that

| for ve T(M), neZ”.

Theorem 4.7.1 If M is compact and if f is contracting then f is also contracting in any other Riemannian metric; a similar statement holds for expanding maps. Proof On the fibre 7,(M) any two norms ||||, and |-|, are equivalent, since M is finite-dimensional. Thus, for every xeM there exist numbers a(x) and b(x) such that

a(x)I le SMI < FOOT I.

226

Structural Stability. Homoclinic

Points

The norms |-|, and ||-||,, depend continuously on x and M is compact. Also, we have dimT,(M) = dimM for all x. Fonseaugnuls there are constants a > O and b < +00 such that a < a(x)

and

b(x) M is an Anosov diffeomorphism, then there is a constant 9 > O such that for any x, yeM,x#y, we have

o(e"(x), o"'(9))29 Hint.

Locally,

for some neZ.

the decomposition

T,(M)=E; QE.

implies the

existence of subspaces W,* and W; such that T,(W*) =EZ,xeM. The manifolds W, and W, locally constitute a coordinate system;

230

Structural Stability. Homoclinic

Points

consider the metric o? := 0% +e2, where g+ denote the manifold metric restricted to W*. Then o, is equivalent to the original (see Bowen, 1975; cf. also Theorem 4.12.1).

metric

o

Theorem 4.8.1 Let M be a compact Riemannian manifold. Then the set of all Anosov diffeomorphisms is open in Diff'(M). Proof

Let oyeDiff' (M) be an Anosov

diffeomorphism

and suppose

that peDiff! (M) is C'-close to o. Define mappings F, Fy: F'(M) > ¥'(M) by:

F(®)(x) =dp(p"'(x)(P(p"'(X))), Fo(9)(x) = dpo(Po ' ())(®(@o * (29): The tangent bundle variant under dg:

T(M)=E*@E-,

T(M) admits

a hyperbolic decomposition,

T.(M)=E*@E,

for xeM.

in-

(48.1)

The mappings F and F, are linear; let us write them in the matrix form corresponding to the decomposition (4.8.1):

F-

PORE

Koask zl. This means

F. =

oo

Fo

2

Ota ee

that for

D (x)= (0(x)) x (x)= (RF) we ought to have

ED) (x)= (FAT PPE XE

ae PER);

and similarly for Fo. Since the decomposition (4.8.1) is dQo-invariant, and hence Fo-invariant, we have F$_ = F2 , =0. Applying the result of Exercise 4.7.1 (slightly modified), we can introduce a metric on M such that

IFS „o”|]-)'e ||

F°(M)

given by

p* (9) =(dp *og)(¥°9), p* (P(x)

Points

(4.8.6)

=do '(y(x))¥(p(x))

for xeM.

Finally, define ©: #°(M) > F”(M) by

F(P) = o* (¥)+Q(¥). Lemma

4.8.2

For every

W, We F(M)

s > 0 there exists

6> 0 such

that for any

with ||Fllo o)F(p(X)). Thus

dFO) =o".

F(0)=0.

to Y; we get

Anosov Diffeomor phisms Consequently to the estimate

ldQ (lo

|

235

Q (0) = 0 and dQ(0) = 0, which, by continuity, leads

Se

for ||PIlo ¥°(M)

(given ®) as

follows:

CP Lemma

=P

4.8.3

+ Pa (2): For

every

(4.8.8) e >0

there exists

6>0 such

that for any

@e F'(M), Y, Ve F°(M) with |||, < 6, ||Pllo .

n=0

These formulas define (I— y*)~' as a continuous linear operator on the

whole of F°(M). We use the opportunity to inform the reader that also conversely, if

the operator y* defined by (4.8.6) is such that ([—g*)"' exists on the entire F°(M), then o is an Anosov diffeomorphism; this result is due to Mather (1968). Now equation (4.8.10) obtains the form

=(1-g*) 'oR @(¥). Fix e > O and choose 6 > O in accordance

with Lemmas

4.8.3 and

4.8.4. We shall show that the operator (I—g*) ' oR, takes the ball ||¥llo File < 3117 — F'llo; the estimates follow from Lemmas 4.8.2 and 4.8.3 and from the choice

5 Thus (I-g*)

'oRg

is indeed a contraction

on

|Y: ||Fllo

0).

Consequently there exists a Y belonging to that ball and satisfying equation (4.8.10), which means that the mapping h = exp¥ fulfils the equation

woh=hog;

(4.8.11)

h can be arbitrarily close to identity because 6 can be arbitrarily small. It remains to prove that h is a homeomorphism. We first show that h is one-to-one. Assume that h(x) = h(y), x, YE M. From (4.8.11) we get by induction

W"oh=hog"

for all neZ.

Hence

ho g"(X) = V'oh(x) = p"oh(y) = hog"(j). If x # y, then by Exercise 4.8.1 we can find neZ

such that

o(9"(x), p"(y)) 2 9 > 0, 9 denoting a constant depending on @ only. Since h may be arbitrarily close to identity, we may assume

o(h(x),x) R* as follows: DA

20. 29)

{Orie (x

Further, writing go =(0, w, 0, ©) we define a new

B,x=go+Bx

for xeR*.

ex eR

mapping

B,:

240

Structural Stability. Homoclinic

Points

Let G be the least group of isometries of R* containing all translations by vectors vel and the mapping B,. It is easy to verify that

G=lu{Bi:

neZ}

(we identify a vector vel with the translation commutative: in general, we have

v+B,x 4 B,(v+x)

by v). G is non-

for ver, xeR*.

It is not difficult to see that every geG is of the form

g(x) =v+ngo+B'x

for xeR*,

where vel ,o=0,1 and n and o are simultaneously Hence g can be written as

g(x)=v+0g0+B'x

even

for xeR*,

or odd.

(4.8.12)

where vel and o = 0, 1 (since 2go€T). We will show that the following inequality holds for all xe R* and for any geG different from the identity:

lIx—g (oll > /2. Insert

(4.8.13)

= (a, B, ©. 0), a = a+b./3, b= c+d,/3, a, b, c, dEZ, into

(4.8.12) and calculate the square of the norm of x—g(x): for o = 0 we have

Ig (||? = lll"

= (a+b,/3)? +(c+d,/3)2 +(a—b./3)2+(c—d,/3)2 = 2(a? +c?)+6(b? +d”) > 2 (a, b, c, d cannot

all be zero, because g is not the identity); for o = 1

and x =(x!, x?, x3, x*)eR* we have

c= g(oll* = |x —v—go — BX||? = |\(2x'-a, —B—w, 28-46, —B-—@)||?

> (B+)?+ (B +0) = 2(c+ 4)? +6(d+ HS 2) This proves (4.8.13). Now we establish another property of the group G. Write

G(0) = {xeR*: x =g(0), geG}.

Anosov

Diffeomor phisms

241

We claim that

dist (x,G(0)) < 2+,/12 To show

for xe R*.

this take any point

(4.8.14)

xe R* and denote by x =(n, m, k, I)

the point with even integer coordinates which is nearest to x; clearly,

|x — X|| < 2. Consider the point x9 = (a, B, «, B), where a =a+b,/3,

ii

„pREELKĄ

We

B =c+d,/3,

b=

A

(>)

Rf

a

ares

4-5 (=)

have

0 a;(q)d'.

Then p acts as a functional on the space

T,(M):

(4, p) = 2T(4, 4), hence p is an element of the co-tangent bundle T*(M). The Lagrange equations (4.10.1) then transform into the system of Hamilton canonical equations: dq

¢H

UP,

deaOŚBR(zaldói

mate

naddz

(4.10.2)

where

H(p, q) = T+V= 3 X, a*(q)pip, +V, ij=l

(a'/(q)) denoting the inverse matrix to (a,;(q)). The function H is called the Hamiltonian function of the system (4.10.2); it expresses the total energy of the system. It is a first integral of the system under consideration, that is to say, the motion occurs on the submanifolds M,

= i(p, q): H(p, q) =h}. According to the Maupertuis variational principle (principle of least action, the motion trajectories are the curves minimalizing the

Geodesic

Flows

251

functional b

Jpdq,

(4.10.3)

where a, b are given points and

pdq = ), pidq'. i=1

More precisely, consider all smooth curves connecting a with b on M,. The space of those curves can be identified with a certain Banach space of vector fields on M,: namely, the space of those fields which possess an integral curve joining a with b. For any such curve y we have the quantity

F()) = |pdq. Vy

Then the motion are critical points For any curve =a, q(p) = b, we

is actually accomplished along those curves y which of the functional F. y in M, with parametrization q(t), te, q(x) have

A F(y) = |p(1)4(1)dt

(a(tat

I

q CRK B

| 2T(q(t), 4(t))dt B

© az(a(0)d (04 (0 de, ae |vie Via) |ij=1 a

which yields

ESA

+/h=V

| sea,;(q) dq'dq'. Lij=l

a

(4.10.4)

252

Structural Stability, Homoctinic We now

detine a new

by (a)

Riemannian

Points

metric on M by putting

(h -) (a) kay;(a),

X dby(akiq'dg’ = (h—Vi@) X, ay(a) dę'dq”. dj=1

(4.103)

i=

Consider only those points geM, for which k— Vig) * ‘ge My: h = V(g)) is closed, and every trajectory of (4.10.2) a nowhere dense set; for otherwise in some time-interval have dg/dt = CH Cp = Q. Le, g(t) = const, The length element dy assumes in the new metric the

ds? =(k-V(q)) ¥ ayiadg'de’.

0. The set meets it in we would

form

4100

Wet

It is readily seen from (4.104) and (4.106) that the curves of motion are the critical points of the functional 3

F(y) = | pdą > £

m

=v2 | Vh-F@ | © azladdądę = :

2 |ds.

wel

Clearly enough, this occurs if and only if the kngth of the curve is minimal (provided that a and b le close enough to each other) Thus the curves of motion are just the geodesic lines in the metric (4100 te. the motion in the set (ge M,: Fig) < 2} is accomplished along the geodesics, Since

i

= ele

adler

&

er)

dkór

, =

”

/Uh-V

=

i

aldZZ:

zę J

000 the SYŃCI USG OW to be 4 ZŁOŚCKE fiow. The grok is finished, Opviowsy, the vomvase TEM ts2460 true: every geodesic ow i mish by wome WAŃOGAŚŃC system; Ń Ace trophy to take V= ly, it, 60cones the 1408 in the tid A forces equal to ZErO (usiborm OLD SIBLE, tb Vy dong the EOÓGŃC). MOW we poms 2 (RONEM which 16 knows under several names,

Śozesóówę 0% bow X isSorted and werpróeć, M sighs rightly be eh

the Ano

Tiewem

4102

Catan HANIA

NIOUEYSKY thestem.

U M 66 4 comput Ricmsnion MUSA

wydmie , then the geodeóc fow is Anon.

f neyotive

We wil prove this only inthe case A dwoM =2 under the

WOO Cit M kas comtash curvature equal ŁO —1. ha usta A sith 2 math is the Lobachewsky plane Ly, In Poósczćs moth SL, 2s dette 25 tollows” Ly = 'zEC: imz > 0;

dt = ~ dry),

We shall need corti wroperóc prods: supply the musing

A SL, and we invite the reader to

rece

4161A

le zęc£3, SET (Źzj. Bo v ts at

parallel to the axis

z. Rez = 0), then the geodesic line passing through z, in the Gixection

254

Structural Stability. Homoclinic

of v is the semicircle with centre on the axis

Points

(z: Imz = 0} passing

through zy and tangent at zy to v. If v is parallel to (z: Rez = O}, then the geodesic is the half-line (z9+tv: teR} A L2. Hint. First show that any transformation F(z) = (az +b)/(cz +d) preserves the Riemannian metric ds*; then show that vertical lines are geodesics and use the fact that F is a conformal mapping taking straight lines and circles into straight lines and circles.

4.10.2.A. Every orientation form: F(z) =(az+b)/(cz+d),

preserving where

isometry

F of #2

is of the

ki1[ESL R) = ‘Ac L(R?, R?): detA = 1}. c

The group of orientation preserving isometries of /, can be identified with SL(2, R/Z,, where Z, is the cyclic group generated by -1 0

EMI

Hint. Use the same argument as in the preceding exercise (see the hint): transform the geodesics involved into vertical lines.

Now, let Wo = (qo, Vo)€ T(-£ >), Vo being non-vertical, and let ||vgl| = 1. Denote by y* the geodesic line through qo in the direction of vp and by y the geodesic through qo in the direction of —v . Then §* UP” is a semi-circle passing through g, and orthogonal to the real axis. Let q, and q_ denote the points at which y* and y meet the line

(z: Imz = 0}. Consider the family of all semicircles passing through q+, orthogonal to the real line, and the family of all semicircles passing through g_, orthogonal to the real line. Denote these families by S* and S~, respectively. S* and S~ are thus certain families of geodesics in BA

Definition 4.10.2 Let H* c T(4,) be any set such that 2(H*) = iq: (q,v)eH*) is a circle tangent to (z: Imz =0} at q, and, for every (q, v)€ H*, » is the unit normal vector to x(H*), sensed to the interior of x (H*). Any such set H* is called a positive (or contracting)

horocycle of the family S*. Putting everywhere the sign — in place of + and demanding that v be outward normal to x(H ) we define a negative (expanding) horocycle of the family S~.

Geodesic

Flows

253

Fic.

4.10.1

Fig. 4.10.1 shows a positive and a negative horocycle of

St and S .

Exercises

4.10.3.A. Let S* and H* be as above. Pushing each element

we H* by

a fixed distance t along the geodesic (determined by w), we obtain a set

H,’ c T(-£,), which is also a horocycle of S*. Hint. Apply an inversion with centre at q,.

4.10.4.A. Let w, w”eH* and denote by wi, w; the elements of H;* (see the preceding exercise) resulting from w”, w” by translation along their geodesics. Let |, denote the length of the arc of x(H/') limited by z(w;) and n(w,). Show that there exist constants a, £ > O such that

ee ps he Hint. Calculation becomes simple if we apply a suitable inversion, as in the preceding exercise. Proof of Theorem 4.10.2

We first prove the theorem in the case of M

= £,, the Lobachevsky

plane.

Let W be the sphere bundle over #,. For each w =(q, v)eW the tangent space T,,(W) admits the splitting

T„(W) = [a] ® [6] OTC]; in this formula, a is a vector (in T,,(W), i.e. 3-dimensional) tangent at w to H* (the positive horocycle passing through w viewed as a line in W), b is a vector tangent at w to H (the negative horocycle), and c is a vector tangent to the trajectory of the geodesic flow passing through

256

Structural Stability. Homoclinic

Points

w. The above decomposition is invariant under the action of the geodesic flow (Exercise 4.10.3). As follows from the result of Exercise 4.10.4 the norm ||dg,(w)a|| is exponentially decreasing and the norm |do, (w)b|| is exponentially increasing as t > +00. Thus the theorem is true-for Me—-2 5. Now let M be a compact two-dimensional manifold of constant curvature equal to —1. Then &, is a covering space for M and the covering projection is locally isometric (see Springer, 1957). Let W and W, denote the sphere bundles over M and &,, respectively. It follows that the decomposition of T(W,) into the contracting part (E*), the expanding part (E') and the neutral part (E°) can be projected onto T(W), yielding the corresponding decomposition for the geodesic flow over M. The proof in the general case (that of a compact Riemannian manifold with negative curvature at each point) is much more difficult; it can be found in Anosov (1967).

4.11. Expanding

Mappings

An interesting class of mappings of smooth by the so called expanding mappings. Let M

be a Riemannian

manifold;

M

need

Definition 4.11.1 A differentiable mapping p: ing iff there exist constants

ldp"(o9e|| > aA” |v

« > 0,

manifolds is furnished not

be compact.

M >M

is called expand-

A> 1 such that

for all xeM,

veT(M,n=l2.

The class of all expanding C'-mappings will be denoted by the symbol 6' (M).

Examples

4114. Lete “M.S ldp"(z)o|| = N*||v||.

QZ) = 2 aN

ZnSe

1

ns

Wexpandine:

4.11.2, Let M =S*, g(e”7*) = e*"VO, f: R>R, feC'(R), fx+)=N+ +/(x), NeZ”, N>l and let f’(x) >A>1 for all x; o is expanding:

for z =e*"*,

veT,(S')

Idp"(z)ol| > A” Jol].

we

have

||dp(z)v|| 2/'(xl|ol| > A||oj|, whence

Expanding Mappings

27

4.11.3. Let M=R", p(x) = Ax, AE L(R", R"), spAc JĄ: |A| > 1). By Lemma 1.5.1 there is a norm |-| in R” such that 9:=|A 3] 1. Since in R” all norms are equivalent, there are constants a, b such that a||:|| 7 bd > pe Pell.

SAS

Exercises

4.11.1.A. Let M be compact. Show that if oeć'(M) in some Riemannian metric then peć"(M) in every Riemannian metric. Hint. Cp. Example 4.11.3.

4.11.2.A. Let M be compact. Show that if o is expanding, then there’ exists a C’-smooth Riemannian metric ||: ||, and a constant 2, > 1 such that ido (9ol|+ > A, lull,

for all

Hint. See the proof of Lemma

xe M, ve T,(M). 1.5.1; cf. also Exercise 4.7.1.

4.11.3.A. Let M be a Riemannian manifold, compact or not. Show that if o is expanding, then @ is locally a diffeomorphism’. If M is connected and compact, then the sets p '(x), xe M, are of constant finite cardinality, equal to dego. Hint. That o is a local diffeomorphism follows from the Inverse Mapping Theorem. The function x cardog"" (x) assumes finite values (by the compactness of M) and is continuous. The conclusion of Exercise 4.11.2 yields: Corollary 4.11.1

For M compact, 6'(M) is an open subset of C'(M, M).

258

Structural Stability. Homoclinic

Points

Proof Let ||-||, and 4, be as in Exercise 4.11.2. If y is sufficiently close to o, then we have for some 4, with I |Il? (Olldr. 4

|

0

The distance between two points in M is equal to the infimum of the lengths of all connecting paths; hence

o(@ *(a), @*(b))
C, whence

by (4.11.2) we get

p"(U) > z(6"(U,)) > z($"(BG, r))) > z(C) = M. (4) Let B = B(x, r) be an arbitrary ball in M, xeM,r>0. Then there is an neZ* such that g”(B) > M, as we have just shown. Thus p "(B)\B #0. Choose a point xoeg "(B)OB. Then there exists a point x, e B such that g”(x4) = xo; similarly, there exists an x, eB such that p”(x,) = x,, and so on. Proceeding inductively, we find a sequence of points x,¢B such that Pp"(Xp41)

k=)"

= Xs

12"

Using condition (4.11.1) we obtain

1 O(%+1> %) < 190% RAJ:

1 M stands for the lift of « to a mapping between the universal covering spaces of N and M. Consider the following family of mappings:

V,=(B: N>M,B is a lift of a continuous map B: N > M, and ob? t=zem It is readily seen that V,; becomes a complete metric space if the distance between A, BeV, is defined by

d(A, B) = sup e@(A(x), B(x)) xeN

(see Exercise 4.11.4 below).

Lemma

let a:

4.114

Let f: M>M,g:

N>N

be expanding

N= M be a continuous map. If there exist lifts f.

mappings

and

M > M,g: N

Expanding

. >N,&:

Mappings

N>M

261

such that f*oa* =a* og*, thenVz contains a unique

B satisfying the condition fo B = Bog. Proof Write T(A)=f~'oAog for AeV;. To begin with, we prove that T(A) is a lift of some map N > M. By the definition of +, we have foj (jof for any jeDy (cf. statement 3° on p. 258). Hence j =f 'of*(j)of, because f is a homeomorphism (statement 2°) and so

fe

does exist. Thus; ifef*(jq) =f * G2; then j, = ję, showing that f*

is an invertible transformation definition is meaningful:

h(j:=(f*)

of Dy.

'oA*og*(j)

Therefore

(again we use statement

the

for je Dy.

In order to prove that T(A) is a lift of some enough to show that

TAVOI = h(fyoTtA)

the following

map

N-—M,

for all je Dy,

it is

(4.11.3)

3° on p. 258). Now, we have

Aogoj=(A*og*(j)joAog

for jeDy

and

Doi consequently,

T(A)oj

(Off. for any

for jeDx; jeDy we get

=f *oAogoj =f. OA Og

= fins

(loa og

007)

104" Ogt ())ofof „940g

=((f*) 'oA*og*(j))of 'oAog = h(j)o T(4), ie (4.1173). We claim that T(A)eV;. It is already settled that T(A) is a lift. It remains to show that T(A)* =a*, or, equivalently,

(T(A)* ())oa=a«oj

for jeDy.

The following identity is an easy consequence of the definition of the operation +:

(J7'oAog)* =(/7)*oA*og".

Structural Stability. Homoclinic

262

Points

je Dy, The expression on the left is nothing else than T(A)*. Thus, for

(T(A)* (joa =((F~')* 0A* 0g* (joa = (fF ')*oa* og*(J))oa. By hypothesis, a* og* =f*oa*.

Hence

(T(A)* (j))o@ =((F')* of *0a* (j))oa = a* (j)oa = aoj, proving the claim. The assignment A> T(A) thus defines a mapping T: V;—V;. As it is easy to guess, Tis a contraction. Indeed, let A, Be Vz. Since f ~' is contractive (with the contraction coefficient 1/4), we have

d(T(A), T(B)) = supe(f~'oAo0g(x), f-' oBog(x)) xeN

1 M are expanding mappings and f is homotopic to g, then the systems (M, f) and (M, g) are topologically conjugate.

Proof Select a point aeM. Let f,0 M. Choose yoen

‘(a) < M and write

y1 =f (yo). Denote by 7 the lift of y whose one endpoint is y, and by g the lift of g such that y, = @(yo) is the other endpoint of 7. For any

jeDy

we have foj =f,*(j)of,; the lifts f have to be chosen con-

tinuously with respect to t. It is evident that also f,*(j) depends continuously on t. On the other hand, the set Dy is discrete; hence

j* (j) = const = fot (1 = fi* (i) =f * (i) = 9* (U). Now apply Lemma 4.11.1 to the mappings f, g and « = identity: there is a unique Be K, such that fo B = Bog. Interchanging the roles of f and g and repeating the argument, we find an AeV, satisfying goA=Aof. Multiplying (in the sense of composition) the former equality by A and applying the latter, we get

foBoA=BogoA=BoAof.

Expanding

Since

Mappings

263

A* =id* =id, B* =id* =id,

we

have

(BoA)* =id

and

so

BoAeV,. By uniqueness (again in virtue of Lemma 4.11.1, now applied to the mappings f, f and « =id), BoA =id. Resorting once more to symmetry, we see that also 40 B = id, and thus A and B are homeomorphisms.

Corollary 4.11.2 Every expanding mapping Riemannian manifold is structurally stable.

of a connected

compact

Proof Let f: M > M be an expanding map. If g: M > M is close to f, then g is also an expanding map (Corollary 4.11.1), homotopic to f (see e.g. the proof of Lemma 4.8.1). Hence by Theorem 4.11.2 the systems (M, f) and (M, g) are topologically conjugate. Anosov diffeomorphisms and expanding mappings show certain similarities. Also, the two classes are subjected to a common extension to the class of so-called Anosov endomorphisms. We shall not formulate their precise definition; let us only mention that e.g. a mapping f: T” — T™ is an Anosov endomorphism whenever f is given by a matrix f =(a,), i,j =1,...,m, ajEeZ, |detf| > 1, spf {A: |A| =1} =. In particular, if |detf|]=1 then f is an Anosov diffeomorphism; if spf szla: |A| > 1}, then f is an expanding map. As it has been shown by F. Przytycki (1976), an Anosov endomorphism is structurally stable iff it is either a diffeomorphism or an expanding map. Exercises

4.11.4.A. Show that d(A, B) is finite for all A, BeV; (see the proof of Theorem 4.11.1) and defines a complete metric in F;. 4.11.5.B. Let fe &1(S'), degf = N. Show that there exist compact intervals A,,..., Ay with the following properties:

(a) intk A; ointA, = | © for i źj, (b) U4 =S', i=1

(c) f(A) = S! for each i, (d) f is one-to-one on each intA,, N

(e) da

itd > FrA; for each j.

Hint. Lift f to R; since f has a fixed point, we may assume that f(0)

= (). Examine the inverse images f~'(I;), where I, = 0 such that g*(x)eA. Suppose that the system is ergodic (see Definition 5.2.1). Prove that then |n(x)du=l1 (Kac A

theorem).

Hin “Write 4,=(x€4:n(x)=k|,k=l1,2,... pi (A,) 0 g'(4,) = O for i 1,

So(x) =0,

> 0;.

Then

| fdu>0 ANB

for every set AEM Proof

which is @-invariant.

Write

S, (x)= max S, 08 OSk 0 (since $,(x) = 0) and

I0+S; (939) 2/09+8,(9(X) = S,41(X) Thus for any xeB,:= x: S}(x)>0!

we have

/%0+S, (9(9)) > max S,(x) = Sy (x). 1Sk a}. |

Applying

nz1Ńhę=0

Lemma

5.2.1 to the function f(x)—a

we obtain

| fdu>au(A0B,) ANB,

for every set AeJt which is g-invariant. Further, set

F(x) =limsup- Y fog'(3), f(9 = liminf~ Yfogt(x), n— 1

n-1

no

Nę=0

"aj

n>wo

Exp = {x: f() a).

n k= 0

The set E, is measurable, p-invariant and contained in B,. Hence, by Lemma

5.2.1,

| fdu> op (E,,p). Ex.p

Repeating obtain

the

argument

| fdu < Bu(E,g). EB

with f,a,B

replaced

by

—f, —a, —B,

we

Ergodic

280

Hence p(E,,,) = 0 for B jeaid = PAC). 0

moreover,

JEZ

AU A

oli sdi = da"

If fe L?(X, W, u), the convergence

foo

holds in ¥?-norm.

In the class of metric dynamical systems we distinguish the so-called ergodic systems. Definition 5.2.1 A metric dynamical system (X, Wt, u, o”) is called ergodic iff every g”-invariant set is of measure either O or 1, i.e., iff the condition

p *(A)cA

for all aeA

implies

u(Aj=0

or

Thus ergodicity question does not Let (X, WŁ u, U: £P(X, M, u) >

Uf=jog

u(4)=l.

means just as much contain properly any p) be a metric F(X, M, w), p> 1,

JEŻOW,

as saying that the system in non-trivial subsystem. dynamical system and let be the operator defined by

p.

We leave the proof of the following statement to the reader as an easy exercise:

Ergodic

282

Theory. Entropy

Exercise

5.2.1.4. A system (X, Mt, u, p) is ergodic functions / satisfying the equation Uf =f Hint. If Uf =f and fis not equal a.e. to an a such that the set {x: f(x)

B,C

wAJ

neZm

re

>

eran:

neZm

The Fourier expansion is unique, and thus a,

=

dą:n

=

Lee

=

dą:

==

wee

Fix an n#0 and consider the sequence of indices (A**n). Two situations can occur: 1° The terms of this sequence take on finitely many values. Then A**n =n for some k, which means that spA contains a value 4 such

frat a= 1. 2° The terms of this sequence are different from each other. Then, in view of the equality

HH, = Z aż, neZm

we have ad, =0. Thus, if spA contains no root of unity, then a, = 0 for all n 4 O and

so f=const. If there is a AespA such that 4* = 1, then the matrix A* with integer entries has an eigenvector w (A“w = w) with rational components; multiplying by a common denominator we find an integer

vector neZ” such that A'n=n. The function KL

f(x) a

y

e27i(AYn,x)

>

satisfies equation (5.2.3) and is non-constant; the system is not ergodic.

The next theorem shows how the concept of ergodicity of metric dynamical systems is related to the ergodic theorem. Theorem 5.2.4 A system (X, DŁ, u, p) is ergodic if and only iffor every fe £'(X, M, u) the function f* given = (5.2.1) is constant a.e.: f* (x)

= |fdu.

Ergodic

284

Theory. Entropy

The proof is a direct consequence of Theorem 5.2.1 and Exercise Rab The condition expressed in Theorem 5.2.4 has a kefigite physical significance. It means that the time-average of fog* approaches a limit value equal to the mean value of f over the space X. In the kinetic theory of gas this property seems evident from the viewpoint of physics. On the basis of this observation, it was already Boltzman who posed the hypothesis that the motion of gas particles in a closed vessel is ergodic. This fact was first proved by Sinai (1963). Exercises

5.2.2.B. Show that the Bernoulli ergodic. Hint. Apply the relation

shift (see Example

5.1.2, p. 274) is

lim u(y "(A) 0B) = u(A)u(B), holding for any sets A, Be.

(First show that it holds for cylindric

sets; then use the fact that such sets generate

the entire o-field.)

5.2.3.B. Prove that each of the following two conditions is equivalent to the ergodicity of a given metric dynamical system (X, Wt, u, @):

(a) for any f, ge L7(X, MR, u)

lim —

n>c Nę=0

(UJ,9)=(f, DU, 9)

1 denoting the constant

function equal identically to 1;

(b) for any sets A, Bet

im > a u(@ *(A)O

B) = u(A)u(B).

Hint. Prove the condition in (a) for simple functions and then apply the fact that simple functions are dense in ¥?; appeal to Birkhoffs theorem. The concept of topological conjugacy has an analogue in the theory of metric dynamical systems. We only give the definition.

Birkhoff’s

Theorem. Ergodicity

285

Definition 5.2.2 Two metric dynamical systems (Y, R, v, w) are called isomorphic iff: (1) there exist sets XyeM, YyeMR such that WOJE

PIREZ GC

WPAŁOE

Yo,

(X, Wt, u, pg) and

U(X — Xo) = v(Y— Yo) = 0;

(2) there exists a measurable bijection h: X)— Yy sending Mmeasurable sets into Yt-measurable sets (and vice versa, with regard to

h~') and such that hog=woh;

(3) for every AcW

we have

(A) = v(h(A)). Exercises

5.2.4*.B. Prove that any two algebraic automorphisms of the torus T? with equal spectra are isomorphic (Adler and Weiss, 1967). Hint. Apply the results of Section 1.6. 5.2.5.B. Show

that the Baker’s

Transformation

(see Exercise

5.1.1) is

isomorphic to the Bernoulli shift (Example 5.1.2 with A = {0, 1), p(0) = p(1) = 4; see Halmos, 1956). Hint. Let (x, y)eQ. Combining the binary developments of x and y, produce a single +0, 1}-valued sequence € indexed by n ranging from —o to +o. The mapping h: (x, y)r>G carries rectangles into cylinders, preserving measure. 5.2.6.B. Let o be an algebraic automorphism of the torus T*. Show that the system (T*, £, |-|, g) is ergodic iff the spectrum of o has no points in common with the circle |A| = 1; if the system is ergodic, then spo = R. Hint. The characteristic equation for @ is quadratic with coefficients: 1, an integer, 1. Show that if Aespo and |A| = 1, then A is a root of unity; then use Exercise 5.2.1. 5.2.7.A. Prove that if the (X, Wt, v, p) are ergodic and

orthogonal *”.

dynamical systems (X, OR u, o) and m4 v, then the measures u and v are

286

Ergodic

Theory. Entropy

Hint. Assume the contrary. Then v = vg + v1, Vo ŚM, vy Lu. Take a set A such that v9(A) > O and v,(A) = 0 and show (applying Theorem 5.2.4) that x4 is not equal v-a.e. to a constant function. 5.3. Invariant Measures for Topological Dynamical Systems In Examples 5.1.1, 5.1.2, and 5.1.3, we encountered topological dynamical systems, each of them preserving a certain natural measure on the underlying space: the Lebesgue measure on S' and on T” (in Examples 5.1.1 and 5.1.3) and the product measure on X (in Example 5.1.2). The question arises whether every topological dynamical system preserves some measure, say, a Borel measure; in the case of a smooth system the problem concerns the existence of an invariant measure which is absolutely continuous with respect to the Riemannian measure on the manifold considered. We will examine this question in the present section. Let p: X + X be a continuous transformation of a compact metric space X; @ induces an operator U, on the space C(X):

(U,f)(x) =fop(x),

feC(X).

U, maps the space C(X) into itself; if p is a homeomorphism, U, is an isometry.

We shall be concerned with Borel measures on X. Thus the o-field of measurable sets will remain unaltered: the Borel field B(X). Speaking of metric dynamical systems we shall just write (X, u, 9)

instead of (X, B(X), n, 9). The symbol W will denote the class of normalized (= probability) Borel measures on X. Theorem 5.3.1 (Bogolyubov-Krylov, 1936)

let p:

X +X

be a continuous map.

Let X be compact metric and

Then there exists a @-invariant

probability measure on X. Proof Let U,: C(X)>C(X) be the operator defined above and let UZ: C(X)* > C(X)* be adjoint to U,. Then U* maps W into itself. Indeed, taking any ueW and writing v=U$(n), we have, by the definition of UŚ,

[U,fdn = [fdv,

Invariant Measures for Topological Dynamical Systems

287

and by the definition of U,,

JU, fdu = [fogdu; hence v(A) = u(p *(A)) for every A e B(X) and thus ve W. Further, the set Wis compact in the weak * topology*' and is convex: if ly, L,EW, then v = tu,+(1—t)y,¢W for te

wo

k=0

Uniquely Ergodic Dynamical Systems

291

is satisfied for every function feC(X). A point xgeX is generic for a flowa > Hs 9.) iff the limit condition ję

lim = o) dt = jra Moe

0

is satisfied for every function feC(X).

In the following we shall often write u(f) in place of [fdu. Theorem

5.4.1

(Oxtoby,

1952;

see

also

Furstenberg,

dynamical system (X, o), any of the following others: (1) the system (X, @) is uniquely ergodic; (2) for every function feC(X) the sequence

1961)

statements

Given

a

implies the

Go”) converges uniformly on X to a constant function; (3) there is an invariant measure u such that every point x€X generic for the system (X, u, Q).

is

Proof (1) implies (2). Let u be the unique .g-invariant probability measure on X. As before, consider the Spa U, Cay CX) given by: U,f=fog. Write

E, = {gEC(X):

g =U, f—f, feC(X)}.

E, is a linear subspace of C(X) and we have Ey 4 C(X), since the constant function f(x) = 1, for instance, does not belong to Eg; indeed,

the u-mean value of any geEp is zero. We

now

show

that

codimE,=1.

Assume

the

contrary:

codimE, > 1; then there are g,, g,¢C(X) such that

Eo © span {gi} # Eo © span {gp}. By the familiar theorems of functional analysis (see Dunford and Schwartz, 1958) there exist continuous linear functionals v,, vą such that VA PU

=

0

on

v2 (91) = V1 (92) = 9,

Eo,

(5.4.1) V1 (91) = ¥2(92) = 1.

292

Ergodic

Theory. Entropy

Hence

vi(Ugf) = vif)

ior ally ec(X),

REŻ

1.€., Vj, V2 are Q-invariant. By virtue of Theorem 5.3.2 the non-negative meastres v;*,i = 1, 2, occurring in the Hahn decomposition of v, are also g-invariant. By assumption, (X, @) is uniquely ergodic, and so every invariant finite measure is proportional to m. Thus v, = ap, v, = bu and v4, vą are linearly dependent, contrary to (5.4.1). This shows that C(X) = Eo ®span{1!

(I denotes the constant function f(x) =1; 1¢E) Now let g be any continuous function and let e > O be arbitrary. According to the above, g is the sum of a constant function and a member of Ey, i.e., there are feC(X) and a constant a such that

Jig —a— (Uf—f)II S! given by: F (z) =

e?riA((z9,2)) |

Since 4 vanishes on singletons, F is continuous. For any three points Z1, Z2, Z3€S' we have

A((z1, 22))+A((z2, z3)) = A((z1, Z3))(mod 1).

(5.4.2)

Uniquely Ergodic Dynamical Systems Thus, writing Wo = (Zo),

295

w = p(z), we have:

F ((2)) = F (w) = Eom

= @27ilA(zo,w)) +2(w9,w)] _ @2niA((z9,W)) @27iA((wo,W)) = ea @2ni M(zo,2)) — @2nia F(z),

where a = A((zqWo)). Suppose that « is rational. Then we get

F(9"(z)) = F (z)

for some n,

DE:

A((zo, w„) = A((zo, z))

for some n,

where w, =g"(z). Setting in (5.4.2) z; = zg, Z, = W,, Z3 = Z, WE obtain

A((z, w,)) = 0(mod 1).

Thus either A((z, w,)) = 0 or A(z, w,)) = 1. The function z >A((z, g”(z))) = /((z, w,)) is continuous, and so, if it is zero [one] for some z, then it is zero [one] permanently on S*. Hence every arc on S' is of measure zero [one]. Either possibility is absurd. Thus «@ is irrational. The system (S', g) is transformed by F into the system (S', r), r denoting the rotation by 2na: (MER

pe

ABK Z

AMOR

Let F(u), F(v), F(A) be the measures

Ss!

EA:

s!

induced by F:

F (u)(A) = u(F"'(4)), F()(A) =v(F"*(4)), F(4)(4) =A(F"'(A)). All these measures are invariant uniquely ergodic, and thus 1

under

r. Since a is irrational, r is

F (py) = F(v) = F(A) = z,(Lebesgue measure on S’).

296

Ergodic

Theory. Entropy

Let I be any arc on S!. Write J =F(I), I, =F"'(J). Then HU) =v)

=A).

It follows from the definition of F that A(I, —I) = 0. Since A = $(u+Vv), we get

HU, >) =v, —)). =A,

— 1) =0

Fic.

5.4.1

.

(see Fig. 5.4.1). Hence u(l) = v(I) for every arc I; consequently u(4) = (4) for every Borel set A =S', and thus u=v, concluding the proof. Given a cascade (X, 9), one can consider the special flow (X7, g,) over (X, g) relative to a given continuous positive function / (cf. Section 1.10). Recall that

X'=XxR/*,

= (x, s) ~ (P(X), s—f(>)).

If one of the two systems is uniquely ergodic, need the other be so? Here is the answer: sion | | Theorem 5.4.4 (Strelcyn, 1970a) The cascade (X, @) is uniquely ergodic if and only if the special flow (X‘, @,) is uniquely ergodic. Proof

The “only if” part. Assume that (X, @) is uniquely ergodic. Let

G(x, s) be any continuous

function

on

X”, let T>0

be any number

and (x9, s) any point of X”. Write x, = g'xg, k =1, 2, ..., and take n such that

TF

|

(Xo) —S HF (1) + ... +f) ST 1, H S'. A. Furstenberg (1961) has shown that if degf 4 0, the system is uniquely ergodic. Nevertheless, if deg f = 0, the system can admit more than one invariant measure (clearly, the system preserves the Lebesgue measure). We shall now show an example of such an f. Let v, be a sequence of integers increasing rapidly; say, let v,,,

=2™ vy) =1. Write

and define

fi) = Ż afer 1p",

(5.4.4)

where (a,) is any sequence with the properties

a=

+o,

Y»|jal=+©

(e.g. a, = 1/n). Then there exists a function H(t) satisfying the equation

H(t+a)—H(t) = f(t), namely + 2

1

MO); ==

Geant,

a2,

a,

oO:

Since the sequence (a,) is square-summable, the function H is square-

integrable: He The curves

?(S', &, |-|). .

Bo = {(x*, x”): x* = H(x')+C} are @-invariant:

(Oe H(x!)-+C)

(x! +a, H(x')+C+f(x'))

=(x' +a, H(x'+a)+C).

Uniquely Ergodic Dynamical Systems

301

The function H is equivalent to a Borel function; we may thus assume that it is Borel itself. The curves under discussion are then Borel subsets of the torus. For each C transport the Lebesgue measure from S' onto Be by the mapping

Stax't+(x', H(x')+C)eB; and denote the resulting measure

on B, by mc; then extend mę to a

measure on T? by putting jpic(A)= pe (A a Be) © for Ae B(T?). Clearly wc is 9-invariant. It follows that the system admits uncountably ergodic probability measures. Minimality is left to the reader as an exercise:

many

invariant

Exercise

5.4.4*.B. Prove that the transformation og of the torus defined by (5.4.3) with f given by (5.4.4) is minimal (i.e. the system (T*, g) is minimal). Hint. For any meN and k=0, 1,..., v„ consider the vector

Wee

2ni k (Cte k ieee

2ni

m

Tm

Take e > 0. Show that there exists an m such that for every we T” and every

n,

1o(g,(x)),

o(p(x)) = o(x) + grade(x)f(x)h+R(h),

we

may

(5.5.4)

where R(h)/h > 0 as h>0. Calculate 00,(x)/0x: since @,(x) is of class C? in h, we have

fe 2

Pr) =x+/(0h+> a e(XIM?,h? =

where

=

ć is a number

09„(X)

Ox

between

0 and h. Hence

= 1+df(x)h+ F(x, hh’,

F(x, h) denoting a function continuous OC, (x det oa ) Ox

in the region

Dx R. Thus

= det (I+ df(x)h+F (x, h)h’).

It follows from the definition of the determinant that the Jacobian under consideration is of the form

det OP (X) _ Óx

aa 09h+G(x, hh?

= L+ >

= 1 +divf(x)h+ G(x, h)h”,

(5.5.5)

304

Ergodic

G(x, h) being a continuous (5.5.3) we obtain

|

o(x)dx—

function.

Theory. Entropy

Inserting (5.5.5) and (5.5.4) into

| o(x)dx

91+ nA)

(A)

= | ((o(x)+grado(x)f(x)h+ R(h))(1+divf(xph+ (A)

+ G(x, h)h?)—@(x))dx

| ((erado(x)f(X)+o(x)div/(x))h+ L(x, h)h?)dx,

=

(A)

where L(x, h) is a continuous function. Dividing by h and passing with h to zero, we get in the limit

R |e09ax(A)

|(erade0009+ e(saivs

ax

9 (A)

=

|div (a (x)f(x)) dx. (A)

We immediately conclude, by the continuity of the last integrand, that the value of ) o(x) dx is independent of t (for any A c D) if and only

(A)

if

div(o(x)f(x))=0

for all xeD.

This is exactly the assertion of the theorem.

Corollary 5.5.1 If (5.5.1) is a Hamiltonian system, then the Lebesgue measure is invariant for the flow induced by the system. Proof

Indeed, if (5.5.1) is of the form

dq

0H

apo +.

JED

„CH

| edi a Sone

e

where H(y, p) is the energy function, then

di

OH Vv

Op

OH ,

—

0q

OH, =

oqóp

M is an expanding C?-map of a compact Riemannian manifold M, then: 1° @ preserves a certain measure u, which is equivalent to the Riemannian

measure,

2 the system (M, u, g) is ergodic.

Before passing to the proof we introduce some notation. Let ||-|| be the Riemannian metric such that

II\dp(x)(v)l|

c|jo| | for xe M, veT,(M),

where c > 1; let wy and d(-, -) denote the Riemannian measure and the distance (metric) induced by the Riemannian metric ||-||. Let u, be the Lebesgue measure in T,(M). We introduce the function (Dg) (x) =

Ho(x) (do (x) (B,)),

XE

M,

where B, is any set in T,(M) of Lebesgue measure function Do is called the scalar derivative of 9. We shall need the following lemma:

1: y,(B,) = 1. The

Lemma 5.5.1 Under the assumptions of Theorem 5.5.4 there exists a constant a > O such that for every Borel set A d(9'(x), o'().

(5.5.14)

On account of property (8), we have

d(g' (x), p' (9) 2 cd(g'*(x), g'*(y)), whence

IAA

PAW ZE "dlv'(950/(9).

— 1E0.MEL.

Estimate (5.5.14) together with the last inequality results in

(Dep")(x)

(Dey) 5

Li

dle" u: G.03,- u)

(2 * zy

ć

1 < exp (=a

mii

diam M )

This number is the desired constant b. Step III. We are now in a position to prove the lemma. On account of (4), (5) and (6) we may write

Lo(p"(4)=

¥

JE 1,0) dito(y),

(5.5.15)

1 Uo (A)

>,

(5.5.15)

and

313 (5.5.16)

we

get, for

"A (Do;ń...i)(X)

1 2, and that © has neither critical points nor closed orbits in T?. Further, we assume that there exists an integral inyariant g(x', x?) > 0

for system (5.6.1) and that g is of class C' 1. By Theorem 2.5.3 there exists a Siegel curve for the vector field ©.

Suppose that x’ = 0 is that curve (i.e., the lines x! = neZ are copies of

Flows on the Torus T? (II). Linearization that

curve

(x, 07)

in R”). In view

0m fore

of Lemma

(xi, x) ER

‘For,

317 2.5.1

we

may

assume

that

6x-=10; 2) f(xy x?) ra = 0}

denote by w(v) the abscissa of the earliest point at which the trajectory p(t, x) of (5.6.1) meets the line x’ = 1. Then the function (mod 1) is a C'-diffeomorphism of the circle S' (see Theorem 1.5.1); since © has no critical elements, has no periodic points. By Corollary 2.4.2 there is a homeomorphism H: S' >8' such that Hoy =roH, where r is the

FiG. 5.6.1

rotation of S' by 2rg, © denoting the rotation number We now prove:

of wp.

Lemma 5.6.1 Under the above assumptions on the system (5.6.1), H is a C’-diffeomor phism of the circle.

Proof

Instead of (5.6.1) we consider the system

abe

edpae. =

axe

ef Gx)

(5.6.2)

dt ft(x', x?)

(5.6.2) is equivalent to (5.6.1) in the sense that the trajectories of the two systems are identical and thus they induce the same diffeomorphism wy

of the circle. The first equation of (5.6.2) yields x'(t)=xg+t.

The

function

gi (x!, x?) =g(x!, x f't(x!, x3)>0,

(x!, x7)ER?,

is an integral invariant for the system (5.6.2); this can be easily verified, for instance with the use of Liouville’s theorem. Thus

(adores A

ide, dx~ =
dq'*

for

p,qeZ.

Consequently

! d je?" —1| > — where d, is some constant. This yields the following estimate for w,;: n” 2 |w,|

n° 2 |a,| n

S

Za

n* 2 |a,|n

Wal S jężwe_i] din?

=

1

n dy *|a,|.

SIEC

Observe that (2mi)* )) n*a,e**""* is the Fourier series of the continuous z

:

d

function

4

4

ast (s). S

Hence

the

series

:

of its

|

coefficients

is

square-

summable, + ©

DS ACE R=

=

SEE

oo

and thus also + ©

Dente WOS showing that K’”€1?(0, 1). It follows that KeC'(R), ending the proof of the lemma.

Proof of Theorem 5.6.2 According to Theorem 5.6.1, we may assume that the system under consideration is of the form (5.6.3). Denote by F (s) the time needed for getting from the point (0, s) to the point (1, s,) along the trajectory; of course, s, =s+@. The first equation of (5.6.3) gives 1

Rad s

dx! G(x!, ox" +5) 0

thus clearly FeC*(R). By Lemma

KEC'! (R).

5.6.2 equation (5.6.5) has a solution

322

Ergodic

Theory. Entropy

Let us consider the curve y = y(s) defined as follows:

y(5) = Pk (0, 5)); y is a closed C!-curve on the torus. We claim that y has the following property: for any point xe$, the earliest moment t > 0 at which the trajectory @,(x) again intersects y is

t =c = |F(s)ds. 0 We prove this. Choose xe. Considering the situation in the covering plane, we may assume that x belongs to the strip {0 < x!

< 1). The trajectory g,(x) intersects the lines {x' = 0} and {x' = 1} at the points which we denote by (0, s) and x,, respectively (see Fig. 5.6.2).

y

Fic.

5.6.2

Denote by x, the point at which g,(x), t>0, first meets 7. By the construction of y, the time of passing from (0, s) to x is K(s); the time of passing from x to x, is thus equal to F(s)—K(s). The x*-coordinate of the point x, is g@+s; hence the time of passing from x, to x, is equal to K(s+g). Summing up, we see that the time period t needed for

getting from x to x, is

t = F(s)—K(s)+K(s+0), and, since K satisfies equation (5.6.5), we obtain t Now we introduce the new coordinate system

u(x) = ~1(3),

=c, as claimed.

v(x) = x?— ox! + ou(x),

where t(x) denotes the time of getting from 7 to x =(x!, x?).

Flows on the Torus

T? (II). Linearization

323

In the original coordinates (x', x”) the torus is considered

as the

strip {0< x' (x*, x?) is differentiable: using equations (5.6.3) we obtain

Ou

Ou

O(u, v) NOx "Ox" — = det OSX") dv Ov

——

ex!

0x?

_1/

du dx! fs ou dx?

eG

Gxedts,.

soxeudt 1

= ag txt, x*()'= %

0,

and this proves that the mapping (x', x”) ->(u, v) is a diffeomorphism. In the coordinates (u, v) system (5.6.3) assumes the form of the system of equations

du agi

dt

|

c

and (since x? = gx' —const) do 4d

do

du

@

dada ydy id 6 Thus «a = l/c and f=ogyc proved.

are the desired constants;

the theorem

is

Theory. Entropy

Ergodic

324

Remark 5.6.1 The assumption that g satisfies condition (A3) is essential: M. Sklover (1967) proved that if g is very well approximated by rationals (i.e., if g does not fulfil any of the conditions (An)), then system (5.6.3) is not linearizable. Exercise

5.6.2.B. Consider

system

(5.6.1) supposing

that f',f?

are analytic

functions, that there exists an integral invariant which is also analytic, and that o satisfies condition (An) for some n. Prove that the system is linearizable. + oo

Hint. If a function

oo

)) a,e*"" is analytic, then n=

> n* |a| < + oo

Do)

for every k. Copy the proof of Theorem 5.6.2 (of foun s 6.2, in fact).

5.7. Metric Entropy

We present the briefly in the form Readers who want Billingsley (1965), Let (X, MW, w) subsets of X: € = A,;EM

A;

concept of metric entropy and list its properties very of isolated definitions and theorems, omitting proofs. to study this subject in more detail should resort to Sinai (1973, 1970) and Totoki (1970). be a given probability measure space. A family of {A;: i=1,..., n} is called a finite partition iff

(Orsio rs

A; =@

ION

ZJ.

Ah A, = X (modulo a p-nullset).

Let € = (A; and n = |Bj)y be two partitions. Define ĆVĄ:=|A;NB;:

A,EĆ, Ben}.

Clearly, € vy is also a partition. Given partitions ć = {A,\"_, and y =

(5.7.1)

B,\v +, we say that 7 is finer than ¢ iff each member of € is the union of some members of 4 (possibly, modulo nullsets):

= U B, (mod u = 0), jej;

SSN.

Metric Entropy

325

This is written symbolically as

on. Given i=

we

(5.7.2) a finite sequence (Ajitina

of partitions €,,..., &,

Pomel, aon: Kk,

write _k

Visss= 81 WARE J

dw nA,

0204ye‘ ie SeiG

For any finite partition

eae alta 1

€ = {A;}{ we introduce the quantity

H(é):= 2, u(Ajlogu(4,) i=1

called the entropy of € (if for some = 0). If €0, then clogc > aloga+ blogb. Further, let @ be a measure

p: X>X,

preserving transformation:

u(p '(4))=u(A)

for AEM.

Mć=(A,r is a partition, o *(6):= lo *(A)}T is a partition as well. It is readily seen that

(5.74)

H(9 = H(e7'(5). Theorem 5.7.1

For any partition

ae! 8 lim ~H(¢ v 9 MONS

2

ć = (A,jjy the following limit exists:

zaj Oe ae):

For the proof see e.g. Billingsley (1965), Ch. 2, Section 6. . Denote this limit by h,(o, ¢).

326

Ergodic Theory. Entropy

Definition 5.7.1 The metric entropy of a metric dynamical (X, M, u, p) is the quantity h,(0) defined by the formula

system

h,(9) = suph,(¢, 6), é

the supremum Now

we

being taken over all finite partitions. formulate

certain

theorems

concerning

the concept

3

introduced. Theorem 5.7.2 phic, then

If the systems (X, Mi, u, o) and (Y, It, v, W) are isomor-

h,(p) = hy). Theorem 5.7.3

For every metric dynamical system (X, Di, u, p)

h,(p”) =nh,(p)

for neZ”.

If @ is invertible, then also

h,(p") =lnlh,(p)

for neZ.

Theorem 5.7.4 Given a metric dynamical system (X, M, U, 0), @ invertible, and a sequence of partitions

TRZ.

eG, e,

Suppose that the o-field generated by all sets which are members of the partitions N

VAG

NG)

k=—N

EM

NS hee:

(i.e., the smallest o-field containing all such sets) is the entire IN. Then h,()

a

lim

hy (9, ÓW:

A partition $, is termed a generator iff the o-field generated by all the sets which enter the partitions Wan

(Eo),

NACZ,

coincides with 9. Theorem 5.7.4 implies

Metric Entropy Theorem 5.7.5 invertible, then

327 If & is a GENEL ALOT for a system (X, WN, u, g) with @

h, () c= h, (p, Ś0).

Krieger (1970) has proved that every ergodic dynamical system with an invertible p on a Lebesgue space (i.e., on a measure space isomorphic *° to ( p; log p;. i=1

Hint. Take A, = {(x,): Xo = aj], i = 1,..., m. The partition € =(A)f is generating and we have

H(\/ 970) = nH(©). n-1

5.7.3.A.

Show

that

if p, >0, » p, = 1, then

-2 p,logp,< logm, t=1

equality holding if and only iri5a= (=

jo, ==*]/M.

Hint. Apply Jensen's inequality * to the function points x, = 1/p, with the coefficients t, = pj,

—logx at the

i=l1,..., m

5.7.4.A. Show that

H($ v n)

£H(6)+H(n)

holds for any finite partitions ć, 4 | Hint. Let 6 =(A))t, 4 =(BJ)T, u(A,) 4 0. For each i consider the conditional measure yp; = u(*|4,) given by w,(B) = u(BO0A4/)/u(4,) and define

H(nlć) := — > (Aj) y u (Blog u, (B,).

Ergodic

328

Theory. Entropy

Show that H(ć v n) = H(6)+H(nlć) and that H(nić) < H(n); this follows from the convexity of the function xlogx; see Billingsley (1965). 5.7.5.A. Let A and v be probability measures on (X, W). Show that the measure p = td+(1—t)v, Ooh

k=0

is called the topological entropy of Q with respect to the cover a. Entropy can also be defined'relative to closed covers ;the character-

istic that results is of an entirely different nature. Theorem 5.8.2

The symbol h(Q, a) has the following properties:

(1) h(g, a) < Ho(a); (2) if « X; the „BATE is taken over all open covers a.

The entropy h(@) is a certain numerical parameter of a given dynamical system (X, o). It indicates how quickly the points of X are “mixed” by go. As we shall see in a moment, isometries have zero entropy (they preserve distance, hence do not mix points), whereas

Topological Entropy

algebraic automorphisms positive entropy. Let (a,),

331

of the torus, for instance,

n= 1, 2,..., be a sequence

of open

have in general

covers

such that:

le Oln =< dln+ 13

8° for every open cover 8 there is an a, with B Y be the conjugating homeomorphism: =wog, and let «a be any open cover of X. Then, writing

gog Y as

gogog ', we obtain

1 h(w, g(a))= lim „ Ho(a(0) vy”'(gla)) v ...

vy”"*'(g(a)))

1 = lim Hol(g — (a) v (gop ')(a)v ... v(gog "*")(a))

i | = lim —Ho(g(a vp '(a)v ... vop""*'(a))) n>»oh

n>aolt

= Mise noo

Voss

Y „wo

5 (4) = h(O, a).

Ergodic Theory. Entropy

332

Hence we immediately see that h(w) > h(g). By symmetry we have also

h(9) 2 h(y). Theorem 5.8.4

Let (X, g) be any dynamical system.

h(g*) =kh(o) If, moreover,

for every keZ”.

Q is a homeomorphism,

h(o*) =|klh(p) Proof

Then

then

for every keZ.

Let « be an open cover and let keZ*. Then

h(p) > h(g*,av ge‘(av ... vp *** (a) k

= lim —- Ho (a Vv iV n>w

o TATE(a

pu@)

sin

=

Views ada)

M

=kh(9, a). Hence

h(o*) > kh(Q). On

the other

hand, 1

h(g, a) = lim Pr n>

Me en

ome

(a)

oo

> lim

Ho (a v (o) '(a) v ... v (o) "** (a))

1 =

hot, a),

and so,

h(9) > "(o ). 1

k

The two inequalities result in the required equality. If p is a homeomorphism, then h(~) = h(p~'), as can easily be verified ;consequently

h(p*) = |k\h(@). Examples 5.8.1. Let p: X > X be an isometry. Then h(g) = 0. To see this, take a, to consist of all balls with radii S' be a homeomorphism. Now, for any covers a and f which consist of intervals we have N(a v B) < N(a)+N(f) (cf. Exercise 5.8.2). Consequently,

N(% ME (2)

VaV.

(Gy)

x k—(n-1)

V

VV

p (a)

j=-k=(n-1)

1 = lim — Ho( noah

> V

p'(a9))

j= —k-(n-1)

= lim Ho(o'(

V, „9 60)

= lim“Hol VPio, W yp? eo)

< lim —“(Hol V.o/(a))+ Hal V/p7i(ag)))=h(@, aw). n>

wo

Topological Entropy

333

It follows that h(9) = h(o, Xo).

@g /(A,) = © for Since p /(A,)A

s#r, we have

N(ao V @ 1 (a) V ... V 9 "*!(aq)) = m”, whence

Ho(% V gp'(a%) V ...

V 9 "** (a) = nlogm

and we arrive at the result h(~) = h(Q, 49) = logm.

If we took X to be the product of countably many copies of an infinite set ./, the entropy of the shift defined analogously would be infinite. ' 5.8.6. Let p be an algebraic automorphism of the torus T* and suppose that g is ergodic. Let A and ea ' denote the eigenvalues of g; e = signdet@. According to the conclusion of Exercise 5.2.6 A is a real number and |A| 4 1; we may assume that |2| > 1. We shall show that h(9) = log|A|. Let a, be the cover of T? consisting of open parallelograms P with sides of lengths hy(g, 6-1, (2) pla, 6) < 2. Proof!

zLetycózż4Gryraa, Geb.

Chooses

%compaci

wsdsuKZAIÓ,

I

=1,..., s, such that u(C,—K)) are small enough for (1) to hold for the partition a=

(Ki,

...y

BY

=

U

i=1

K;}.

Put U,,, =X-—,\) K, and U, =K,uU,;, i=1 A

ou Ue

Dias U

and define a:

248

Condition (2) is satisfied, as is easy to see.

Proof of Theorem 5.9.1

Let & be any finite partition and let = and a be

as in Lemma 5.9.2 with og” in place of o (n being fixed for the moment). In virtue of Lemmas 5.9.1 and 5.9.2 we get

h,(9", 6) +00)

h, (9) < h(9).

Properties of Entropy

339

Before further considerations we produce an equivalent definition of topological entropy. As usual, X is a compact metric space, o denoting the metric, and p: X >X is a continuous map. Definition 5.9.1

(1) A setE c X is said to be (n, e)-separated iff, for any

two distinct points x, yeE,

there is a j, 0 0

Theorem 5.9.2 (Bowen, 1971) following equality holds true:

h(@) =d(9).

(5.9.1)

e—>0

For every dynamical system (X, @) the

340

|

Proof

Ergodic

Theory. Entropy

Let a, be a finite cover consisting of balls with a

less than e

and let& be a minimal subcover in the cover« A, p *(a,). From

each member of 64 select one point; denote the bór easy to see that F is (n, e)-spanning. Therefore

set by F. It is

N(a;) = N(%) = cardF > r,(e). Taking obtain

logarithms, dividing by n and passing with n to infinity we

h(o) > hlo, a,) > Fe, whence, passing with e to 0,

h(9) 2 d(9).

(5.9.2)

Now let « = (4, be an arbitrary open cover of mesh less than e. Denote by 6 the Lebesgue number of a. Let F, be minimal (n, 4e)spanning. For xeF, put

= A,(yeX; o(p'(3), p*(Y)) < 36). Since 6 is the Lebesgue

number

for «a and since diamB, *ua 2 é The

measures

u, are normalized;

hence

the set (m:

n=1, 2,...} is

weak *-compact in C(X)*. Let u be a cluster point of the sequence (y,). We may require that p be the limit of a certain subsequence (,,) such that lim =, (2) =S(e). ivan

MN;

Clearly, u is an invariant measure,

which depends on e.

342

Ergodic Theory. Entropy We now construct a partition « with the following properties: diam (a) < e,

u(U FrA)=0

for all n

Aeal

(Fr denotes boundary). This is done as follows:

Let E = (x,,..., Xm} be a fe-net. The sets ty: o(x;;-y) =r}, de 2m>0,

and let

0 0 we get, on account of Theorem 5.9.2, h(9) = DIE < AP hye(p) < sup h, (¢). &>

a=

u

The opposite inequality is asserted by Theorem thus complete.

5.9.1. The proof is

Remark 5.9.1 B. Gurevich (1969) has constructed an example of a transitive dynamical system (X, g) in which the strict inequality

h,(p) < h(@) holds for every g-invariant probability measure u; in other words, the supremum suph,(@) is not attained. u

Properties of Entropy Invariant measures

345 u which enjoy the property

h,(p) = h(g) are called maximal. In Gurevich’s example (X, g) is not a smooth dynamical system (e, X is not a manifold). M. Misiurewicz (1973) proved that every manifold M of dimension > 4 admits a diffeomorphism pg: M>M without any maximal invariant measure. As before let Q = Q(@) denote the set of all nonwandering points of the system (X, @). The set Q is compact and g-invariant ; thus it makes sense to consider the dynamical system (Q, g). Theorem 5.9.4

The topological entropy of the system (X, g) is attained

on Q, i.e.,

h(~) =h(Qla). This is a direct consequence of Exercise 5.3.3 on p. 289 and of Theorem 5.9.3. Now we turn our attention to the entropy of smooth dynamical systems. Let M be a Riemannian

— M

manifold, dimM = m, and let p: M

satisfy the Lipschitz condition

o(p(X), p(y)) < Lo(x, v) where L is some constant the Riemannian metric. It ness of M, condition (5.9.5) the constant L itself may

for x, yeM,

(5.9.5)

and g denotes the metric on M induced by is not difficult to see that, by the compactdoes not depend on the choice of the metric; depend on it.

Theorem 5.9.5 (Krzyzewski, 1968) If o satisfies condition (5.9.5), then the topological entropy h(@) is finite. The

proof is based

on the following

lemma,

which

is sufficiently

obvious to justify the omission of a proof. Lemma 5.9.4

There exist a system of charts (U;, W;) on M,i=1,..., k,

an open set G = R”, and constants

(1) I"=G=y,(Uj), k

2 UW'IM=M,

UB

a, b > O such that

PEAT X be a continuous map. Then there exists a mappingf: Yo > Yo such that the following diagram commutes: ~

Yo a

Yo

xX —————_—> 7X

A

Such an f is called a lift of f to Yo (see Spanier,

1966).

350

Notes

+ Let X be a complete metric space with metric @(-, *). A mapping /: X > X is called a contraction

iff there is a number

e(f(-9, f(y) < xe (x, v)

a, 0 h.(b, x)ep '(b), where b, Uex

(see Spanier,

1966, Ch.

Hence, the bundle projection map p: E x F > U are locally equivalent. The fibre is demanded that for any beB the fibre F, the isomorphism being established by U denotes a suitable neighbourhood of

2, $7).

* By a section of a vector bundle (E. B. F, p) we mean any mapping s: B >E such that pos=idp. For any Uex (notation as in Note 7) we have s: U >p"'(U) and

hy! os(b) = (b, x(b)),

beU, x(b)eE.

If the mapping s is continuous, we say that the section is continuous; s is the zero-

section iff hy! os(h) = (b. 0) for all beB.

* Let M. N be topological spaces and let f.g: M +N be continuous mappings. We say that f and g are homotopic ill there exists a continuous function h: M x N

Notes

351

such that (GORZEJ:

h(x, 1)=g(x)

for all xeM.

If M,N have differentiable structure, we say that f,g are smoothly homotopic whenever there exists an h of class C' with properties as above. If, moreover, f, y and h(-, t) (for all t) are diffeomorphisms, then f and g are isotopic.

‘© Suppose that M is a compact differentiable manifold and denote by Diff’(M) the class of all C’-diffeomorphisms of M. We define the C'-iopology (C'-convergence) in Diff”' (M) as follows: PQ, >9o

in C'-topology,

@ns Po € Di” (M)

iff

d'p„(x) > dkpo(x) uniformly for

O M (J e R being an interval) is called a curve of class C’.s , which is a simple arc in M, define its length to be the quantity d

b

Sllp@ll de.

Notes

353

If M is connected, a metric can be introduced in M by means

of lengths of curves:

o(x, y) := inf{length of y: > M, y(a) = x, y(b) = y|.

If M is compact,

this infimum

is attained for some curve.

"© Let M be a Riemannian interval),

is a curve

such

manifold and suppose that y = y(t), teJ (J being an

point xoe$ has a neighbourhood U with the following property: if x, yeU 7, x = (a), y =7(b), are any two points such that the piece of 7 corresponding to the values of te is contained in U, then the length of that piece is equal to the distance g(x, y) (see Note 15). Any such curve $ is called a geodesic line in M. In local coordinates

that

every

x =(x', ..., x”),

m=dimM,

the geodesics are described

by the

differential equation: ad

dt?

ai

ud

>

dx dx/ Suenos

WOE, dt

where If ;(x) are called Christoffe?'s symbols; every solution of this equation is a geodesic line. Hence, for any xeM and any veT,(M) there is exactly one geodesic line passing through x and having v as its tangent vector at x. Every point in M has a neighbourhood such that any two points in it can be connected, in a unique way, by a geodesic line of length equal to g(x, y) (see Gromoll, Klingenberg and Meyer, 1968; Sternberg, 1964).

17 A very specific notion in Riemannian geometry is the so-called exponential map exp: T(M) > M (we assume that M is compact). It is defined as follows: given a pair (x, v) eT(M), we draw a geodesic line through x in the direction of v and we pick a point x’ on that line such that the distance from x to x’ is equal to t — a prescribed constant

value; by definition, x’ = exp,.v. Usually one takes t = 1; in general, t has to be so small that any point y whose distance from x is less than t could be uniquely joined with x by a geodesic path of length o(x, y). The mapping exp is of class C'. and we have dexp,(0) = identity. Moreover, for a fixed xeM,exp,(*) is a diffeomorphism of the ball B(0, z) = T,(M) onto a neighbourhood of x in: M (see Gromoll, Klingenberg and Meyer, 1968; Sternberg, 1964). 18 On

a Riemannian

manifold

M

we introduce

Borel set contained within a single chart (U, a) (i.e.

A(A):= | /det(g,,(y))dy,

the Riemannian

measure:

if A is a

A c U), we put

y=a(X);

a(A)

integration is performed with respect to the Lebesgue measure in R”; if A is not contained in a single chart, A(A) is defined “piecewise”. It can be proved that the value of A(A) does not depend on the choice of charts. 19 By the Gaussian curvature (or simply: the curvature) of a two-dimensional Riemannian manifold M at a point xe M we mean the number K (x) defined as follows: let U be a neighbourhood of x such that any two points in U can be joined uniquely by a

Notes

354

geodesic line realizing the distance ;for any pair of points x’, x” ¢U consider the geodesic triangle with vertices x, x’, x” (i.e. the curvilinear triangle whose sides are the corresponding geodesics) and denote by a, a’, a” the angles of that triangle, i.e., the angles between the respective tangent vectors in T,(M), T,(M), T,.(M) (they are well defined, since the tangent spaces are equipped with scalar products); then

5.

K(x) ="

GARG se

lim Bee Se

2

9

=

A(Axx'x')

where the expression in the denominator stands for the Riemannian measure of the geodesic triangle xxx". If dimM = m > 2, one can examine the curvatures of two-dimensional sections: given NeM. choose a two-dimensional subspace P of T,(M). project it locally (in a neighbourhood of 0) onto M via the mapping exp and compute the curvature Kp(x) of the resulting submanifold. If K p(x) = const for all Pe T,(M), dimP = 2, and for all xe M, we say that M has constant curvature; e.g. the m-sphere Sg with radius R has constant curvature equal to R'* (see Cartan, 1951).

+0 Let M be a Riemannian manifold. Denote by T*(M) the space of all linear functionals on the tangent space T,(M), xeM. The set of all pairs (x, p), where NEM. pe T*(M), is called the co-tangent bundle and is denoted by the symbol 7*(M). In the terminology of mechanics, the pair: position and velocity is an element of the tangent bundle, whereas the pair: position and momentum is an element of the co-tangent bundle. 2! Let M be a Riemannian manifold of class C* and let h(x) be a real-valued C'function on M. Fix xeM. With any tangent vector ve T,(M) we associate the directional derivative D = ©, (cf. Note 11); 0,h(x) depends linearly on v. Every linear functional f(v) on

a Euclidean

space

is of the form f(v) = (v, tg), where

vy is some

vector.

Hence.

in

particular. o.h(X) =(v, Uo); the vector vg is called the gradient of h at x. If M is (a piece of) R” and

x=(x',..., x”) are the natural coordinates,

vy takes on

the form ch 09 =

ch

FOR)

PODBU

cx!

ols

GH

*= A system of differential equations KAZ

on = jh(si).

is called =

f(x),

f:

xeR”,

autonomous R"

=

whenever

f: R"xR>R",

f(x,t) does

not

depend

explicitly

on

r. ie. f(x. t)

R".

** Whitney's Extension Theorem Let r be a given positive integer, D © R" an open set and VW 7 D a closed subset. Suppose that with any multi-index z =(2,. ..., x), x, being non-negative integers, |x| = Va; O such that al|x|| < [x] < bl|xl|

for all xe£.

26 Let E be a Banach space and let Ey be a closed subspace. We say that Ep is a complemented subspace of E iff there is another closed subspace E, cE such that E == [BG Ee Non-complemented subspaces exist; e.g., in C F

is called a linear operator iff

for x, yeE, a, BER (or O).

A is a bounded linear operator iff there exists a constant c such that ||Ax|| < ¢||x|] for all xeE. The least such constant is called the norm of A (denoted by |[A|l); |All = sup ||Ax||. A linear operator is continuous iff it is bounded. The space of all bounded {|x]] E* by the formula (A*y*) (x) = y* (Ax)

for xeE, yeF, y*eF*.

31 Let E be a separable Banach space. A sequence (x*) of functionals x* eE* is said to be weak* convergent to x*eE* iff for every xeE the numerical sequence (x*(x)) converges to x*(x). Accordingly, we define weak* closedness, weak* compactness, etc. Every set Bc E* which is bounded (ie. B = {x*:||x*||