Smooth Ergodic Theory of Random Dynamical Systems (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

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Pei-Dong Liu Min Qian

Smooth Ergodic Theory of Random Dynamical Systems

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Authors Pei-Dong Liu Min Qian Department of Mathematics and Institute of Mathematics Peking University Beijing 100871, R R. China E-mail: mathpu @bepc2.ihep.ac.cn

L i b r a r y of Congress C a t a l o g i n g - i n - P u b l i c a t i o n

Data

Ltu,

Pet-Dong, 1964Smooth e r g o d l c t h e o r y o f random d y n a m l c a ] s y s t e m s / P e l - D o n g L i u , Mln O i a n . p. cm. - - ( L e c t u r e n o t e s in m a t h e m a t i c s ; 1606) Includes bibliographical references and i n d e x , ISBN 3 - 5 4 0 - 6 0 0 0 4 - 3 (Berlin : acid-free). - - ISBN 0 - 3 8 7 - 6 0 0 0 4 - 3 (New Y o r k : a c i d - F r e e ) 1. E r g o d l c t h e o r y . 2. D 1 F F e r e n t l a b l e dynamical systems, 3. S t o c h a s t i c differential equations. I . Ch" l e n , M l n . II. Tltle. III, Series: Lecture n o t e s In m a t h e m a t i c s (Sprlnger-Verlag) : 1606. QAg.L28 n o . 1605 [0A641.5] 510 s - - d c 2 0 [514'.74] 95-21996 CIP

Mathematics Subject Classification (1991): 58Fll, 34F05, 34D08, 34C35, 60H10

ISBN 3-540-60004-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1995 Printcd in Great Britain SPIN: 10130336

46/3142-543210 - Printed on acid-free paper

Table of C o n t e n t s

Introduction ..............................................................

vii

C h a p t e r 0. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w Measure T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w Measurable Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w Conditional Entropies of Measurable Partitions . . . . . . . . . . . . . . . . . . . . . . w Conditional Entropies of Measure-Preserving Transformations: I . . . . . . w Conditional Entropies of Measure-Preserving Transformations: II . . . .

1 1 5 7 9 16

C h a p t e r I. E n t r o p y and L y a p u n o v Exponents of R a n d o m Diffeomorphisms . 22 w T h e Basic Measure Spaces and Invariant Measures . . . . . . . . . . . . . . . . . . 22 w Measure-Theoretic Entropies of R a n d o m Diffeomorphisrns . . . . . . . . . . . 31 w L y a p u n o v Exponents of R a n d o m Diffeomorphisms . . . . . . . . . . . . . . . . . . . 37 C h a p t e r II. Estimation of E n t r o p y from Above T h r o u g h L y a p u n o v Exponents ..................................................... w Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w P r o o f of T h e o r e m 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 51

C h a p t e r III. Stable Invariant Manifolds of R a n d o m Diffeomorphisms . . . . . . . 55 w Some Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 w Some Technical Facts A b o u t Contracting Maps . . . . . . . . . . . . . . . . . . . . . . 61 w Local and Global Stable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 w HSlder Continuity of Subbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 w Absolute Continuity of Families of Submanifolds . . . . . . . . . . . . . . . . . . . . 84 w Absolute Continuity of Conditional Measures . . . . . . . . . . . . . . . . . . . . . . . . 86 C h a p t e r IV. Estimation of E n t r o p y from Below T h r o u g h L y a p u n o v Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w I n t r o d u c t i o n and Formulation of the Main Result . . . . . . . . . . . . . . . . . . . . w Construction of A Measurable Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w E s t i m a t i o n of the E n t r o p y from Below . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 93 103

C h a p t e r V. Stochastic Flows of Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . w Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w L y a p u n o v Exponents and Stable Manifolds of Stochastic Flows of Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w E n t r o p y of Stochastic Flows of Diffeomorphisms . . . . . . . . . . . . . . . . . . . .

109 109

C h a p t e r VI. Characterization of Measures Satisfying E n t r o p y Formula . . . . w Basic Concepts and Formulation of the Main Result . . . . . . . . . . . . . . . . w S B R Sample Measures: Sufficiency for E n t r o p y Formula . . . . . . . . . . . . w Lyapunov Charts ..................................................

128 129 137 141

118 124

w w w w w

Local U n s t a b l e Manifolds and Center U n s t a b l e Sets . . . . . . . . . . . . . . . . Related Measurable Partitions ..................................... Some Consequences of Besicovitch's Covering T h e o r e m . . . . . . . . . . . . . The Main Proposition ............................................. S B R S a m p l e Measures: Necessity for E n t r o p y F o r m u l a . . . . . . . . . . . . .

146 154 166 170 175

C h a p t e r VII. R a n d o m P e r t u r b a t i o n s of H y p e r b o l i c A t t r a c t o r s . . . . . . . . . . . . w Definitions a n d S t a t e m e n t s of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w Technical P r e p a r a t i o n s for the P r o o f of the Main Result . . . . . . . . . . . . w P r o o f of the M a i n Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182 183 192 197

A p p e n d i x . A M a r g u l i s - R u e l l e I n e q u a l i t y for R a n d o m D y n a m i c a l S y s t e m s .. 207 w N o t i o n s and P r e l i m i n a r y Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 w T h e M a i n Result a n d I t ' s P r o o f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216

Subject Index ............................................................

219

vi

Introduction

This book aims to present a systematic treatment of a series of results concerning invariant measures, entropy and Lyapunov exponents of smooth random dynamical systems. We first try to give a short account about this subject and the brief history leading to it. Smooth ergodic theory of deterministic dynamical systems, i.e. the qualitative study of iterates of a single differentiable transformation on a smooth manifold is nowadays a well-developed theory. Among the major concepts of this theory are the notions of invariant measures, entropy and Lyapunov (characteristic) exponents which culminated in a theorem well known under the name of Oseledec, and there have been numerous relevant results interesting in theory itself as well as in applications. One of the most important classes of the results is Pesin's work on ergodic theory of differentiable dynamical systems possessing a smooth invariant measure. Another is related to the ergodic theory of Anosov diffeomorphisms or Axiom A attractors developed mainly by Sinai, Bowen and Ruelle. A brief review of these two classes of works is now given in the next two paragraphs. In his paper [Pes]l, Pesin proved some general theorems concerning the existence and absolute continuity of invariant families of stable and unstable manifolds of a smooth dynamical system, corresponding to the non-zero Lyapunov exponents. This set up the machinary for transferring the linear theory of Lyapunov exponents into non-linear results in neighbourhoods of typical trajectories. Using these tools Pesin then derived a series of deep results in ergodic theory of diffeomorphisms preserving a smooth measure ([Pes]~). Among these results is the remarkable Pesin's entropy formula which expresses the entropy of a smooth dynamical system in terms of its Lyapunov exponents. Part of the work above has been extended and applied to dynamical systems preserving only a Borel measure ([Kat], [Fat] and [Rue]2). We now turn to some results related to the ergodic theory of Axiom A attractors. Recall that for a given Axiom A attractor there exists a unique invariant measure, called Sinai-Bowen-Ruelle (or simply SBR) measure, that is characterized by each of the following properties ([Sin], [Bow]2 and [Rue]3): (1) Pesin's entropy formula holds true for the associated system. (2) Its conditional measures on unstable manifolds are absolutely continuous with respect to Lebesgue measures on these manifolds. (3) Lebesgue almost every point in an open neighbourhood of the attractor is generic with respect to this measure. Each one of these properties has been shown to be significant in its own right, but it is also remarkable that they are equivalent to one another. More crucially, Ledrappier and Young proved later in their well-known paper [Led]2 that the properties (1) and (2) above remain equivalent for all C 2 diffeomorphisms (That (2) implies (1) was proved by Ledrappier and Strelcyn in [Led]3). All results vii

mentioned above are fundamental and stand at the heart of smooth ergodic theory of deterministic dynamical systems. In recent years the counterpart in random dynamical systems has also been investigated. For an introduction to the scope of random dynamical systems, one can hardly find better description than that given by Walter [Wal]2 in reviewing the pioneering book Ergodic Theory of Random Transformations by Kifer ([Kif]D:

"Traditionally ergodic theory has been the qualitative study of iterates of an individual transformation, of a one-parameter flow of transformations (such as that obtained from the solutions of an autonomous ordinary differential equation), and more generally of a group of transformations of some state space. Usually ergodic theory denotes that part of the theory obtained by considering a measure on the state space which is invariant or quasi-invariant under the group of transformations. However in 1945 Ulam and yon Neumann pointed out the need to consider a more general situation when one applies in turn different transformations chosen at random from some space of transformations. Considerations along these lines have applications in the theory of products of random matrices, random Schrb'dinger operators, stochastic flows on manifolds, and differenliable dynamical systems". In his book [Kif]l, Kifer presented the first systematic treatment of ergodic theory of evolution processes generated by independent actions of transformations chosen at random from a certain class according to some probability distribution. Among the major contributions of this treatment are the introduction of the notions of invariant measures, entropy and Lyapunov exponents for such processes and a systematic exposition of some very useful properties of them. This pioneering book establishes a foundation for further study of this subject, especially for the purpose of the development of the present book. In this book we are mainly concerned with ergodic theory of random dynamical systems generated by (discrete or continuous) stochastic flows of diffeomorphisms on a smooth manifold, which we sometimes call smooth ergodic theory of random dynamical systems. Our main purpose here is to exhibit a systematic generalization to the case of such flows of a major part of the fundamental deterministic results described above. Most generalizations presented in this book turn out to be non-trivial and some are in sharp contradistinction with the deterministic case. This is described in a more detailed way in the following paragraphs. This book has the following structure. Chapter 0 consists of some necessary preliminaries. In this chapter we first present some basic concepts and theorems of measure theory. Proofs are only included when they cannot be found in standard references. Secondly, we give a quick review of the theory of measurable partitions of Lebesgue spaces and conditional entropies of such partitions.

Vlll

Contents of this part come from Rohlin's fundamental papers [Roh]l,~. The major part of this chapter is devoted to developing a general theory of conditional entropies of measure-preserving transformations on Lebesgue spaces. The concept of conditional entropies of measure-preserving transformations was first introduced by Kifer (see Chapter II of [Kif]l), but his treatment was only justified for finite measurable partitions of a probability space. Here we deal with the concept in the case of general measurable partitions (maybe uncountable) of Lebesgue spaces and prove some associated properties mainly along the line of [Roh]2, though the paper of Rohlin only deals with the usual entropies of measure-preserving transformations. Results presented in this chapter serve as a basis of the later chapters. The concepts in Chapter I are mainly adopted from Kifer's book [Kif]l. But for an adequate treatment of entropy formula (in Chapters II, IV, VI and VII) an extension of the notion of entropy to general measurable partitions is indispensible. So we have to formulate and prove the related theorems in this setting, which could be accomplished if the reader is familiar with the preliminaries in Chapter 0. In Section 1.1 we first introduce the random dynamical system X+(M, v) (see Section 1.1 for its precise meaning). Then we discuss some properties of invariant measures of X+(M, v). When associated with an invariant measure #, 2d+(M,v) will be referred to as z~+(M,v,#). Section 1.2 consists of the concept, of the (measure-theoretic) entropy hu( zY+ (M, v)) of zY+(M, v, p) and of some useful properties of it deduced from its relationship with conditional entropies of (deterministic) measure-preserving transformations. In Section 1.3 we introduce the notion of Lyapunov exponents of ?(.+(M,v,#) by adapting Oseledec multiplicative ergodic theorem to this random case. In Chapter II we carry out the estimation of the entropy of z~+(M,v,tt) from above through its Lyapunov exponents. We prove that for any given zle+(M, v, #) the following inequality holds true:

< /F_, i

where A(1)(x) < A(2)(x) < ... < A(~(~))(z) are the Lyapunov exponents of z~+(M, v, #) at point x E M and mi(x) is the multiplicity of A(0(ze). This is an extension to the present random case of the well-known Ruelle's (or MargulisRuelle) inequality in deterministic dynamical systems. As in the deterministic case the above inequality is sometimes also called Ruelle's (or Margutis-Ruette) inequality. In the random case this type of inequality was first considered by Kifer in Chapter V of [Kif]x (see Theorem 1.4 there), but the proof of his theorem contains a nontrivial mistake and this led the authors of the present book to an essentially different approach to this problem (see Chapter II for details). Our presentation here comes from [Liu]l. As compared with the deterministic case, it involves substantially new techniques (especially the introduction of relation numbers and the related estimates). After the first version of this book was completed, the authors received a preprint [Bah] by J. Banmiiller and T. Bogenschiitz which gives an alternative ix

treatment of Ruelle's inequality. Their argument shows that the mistake mentioned above is inessential and can be corrected with some careful modifications, and their argument is also carried out within a more general framework of "stationary" random dynamical systems. It turns out, then, that the correction of the mistake in the original Chapter II is at the expense of an extraneous hypothesis (see Remark 2.1 of Chapter II). However, the treatment in that chapter (for example, the argument about the C2-norms and relation numbers) is besides its own right very useful for the later chapters. For this consideration and in order not to change drastically the original (carefully organized) sketch of the book, we retain here the original Chapter II and also introduce Bahnmiiller and Bogenschiitz's argument (with some modifications) in the Appendix (it involves some results in Chapter VI). Chapter III deals with the theory of stable invariant manifolds of X + ( M , v, #). We present there an extension to the random case of Pesin's results concerning the existence and absolute continuity of invariant families of stable manifolds ([Pes]l). Although some new technical approaches are employed, our treatment goes mainly along Pesin's line with some ideas being adopted from [Kat], [Fat] and [Bri]. Besides their own rights, results in this chapter serve as powerful tools for the treatment of entropy formula given in later chapters. In Chapter IV we extend Pesin's entropy formula to the case of X + ( M , v, #), i.e. we prove that

h.(

v)) = j i

when # is absolutely continuous with respect to the Lebesgue measure on M. This formula takes the same form as in the deterministic case, but now the meaning of the invariant measure p is quite different since it is no longer necessarily invariant for individual sample diffeomorphisms; we also have to point out that the implication of this result exhibits a sharp contradistinction with that in the deterministic case (see the arguments in Section IV.1 and those at the end of Chapter V). This result was first proved by Ledrappier and Young ([Led]l) in the setting of the two-sided random dynamical system X(M,v,#) (see Chapter VI for its meaning), and a more readable treatment of it was later given in [Liu]2 within the present one-sided setting A~+(M, v, #). In this chapter we follow the latter paper. As compared with the deterministic case ([Pes]~ and ILeal]a), the proof of the result given here involves the new ideas of employing the theory of conditional entropies and of applying stable manifolds instead of unstable manifolds. Aside from these points, the proof follows essentially the same line as in the deterministic case, although the technical details are much more complicated. In Chapter V we apply our results obtained in the previous chapters to the case of stochastic flows of diffeomorphisms. Such flows arise essentially as solution flows of stochastic differential equations and all the assumptions made in the previous chapters can be automatically verified in this case. Thus we reach and finish with an important application to the theory of stochastic processes.

Chapter VI is devoted to an extension of the main result (Theorem A) of Ledrappier and Young's remarkable paper [Led]2 to the ease of random diffeomorphisms. Roughly speaking, in the deterministic case one has Theorem A in [Led]2 which asserts that Pesin's entropy formula holds true if and only if the associated invariant measure has SBR (Sinai-Bowen-Ruelle) property, i.e. it has absolutely continuous conditional measures on unstable manifolds; for the case of random diffeomorphisms we prove in this chapter that Pesin's entropy formula holds true if and only if the associated family of sample measures, i.e. the natural invariant family of measures associated with individual realizations of the random process has SBR property. This result looks to be a natural generalization of the deterministic result to the random case, but it has a nontrivial consequence (Corollary VI.1.2) which looks unnatural and which seems hopeless to obtain if one follows a similar way as in the deterministic case (i.e. by using the absolute continuity of unstable foliations). This generalization was actually known first to Ledrappier and Young themselves, though not clearly stated. Here we present the first detailed treatment of this result. Although the technical details are rather different, our treatment follows the line in the deterministic ease provided by [Led]2,3. The sources of this chapter are [Led] 1,2,3 and [Liu]a. In Chapter VII we study the case when a hyperbolic attractor is subjected to certain random perturbations. Based on our elaboration given in the previous chapters, a random version of the deterministic results mentioned above for Axiom A attractors is derived here. The idea of this chapter comes from [You] and [Liu]5. Random dynamical systems, though only at an early stage of development by now, have been widely used and taken care of, especially in applications. In this book, our intention is to touch upon only a part of this subject which we can treat with mathematical rigor. For this reason, we naturally restrict ourselves to the finite dimensional case. Infinite dimensional dynamical systems with random effect should be more interesting from a physical point of view. Scientists from both probability theory and partial differential equations have already paid jointly sufficient attention to this new and important field (a conference was organized by P. L. Chow and Skorohod in 1994). We hope their efforts will lead to a substantially new mathematical theory which, we believe, could be considered as the core of the so-called "Nonlinear Science". We would like to express our sincere thanks to Prof. Ludwig Arnold since conversations with him were very useful for the preparation of Chapter VII. Our gratitude also goes to Profs. Qian Min-Ping and Gong Guang-Lu for helpful discussions. During the elaboration of this book the first author is supported by the National Natural Science Foundation of China and also by the Peking University Science Foundation for Young Scientists. Finally, it is acknowledged that part of the work on this book was done while the first author was in the Institute of Mathematics, Academia Sinica as a postdoctor and he expresses here his gratitude for its hospitality.

xi

Chapter

0

Preliminaries

In this chapter we first present some basic concepts and facts from measure theory. Then we give a quick review of the theory of measurable partitions of Lebesgue spaces and conditional entropies of such partitions. A detailed t r e a t m e n t of this theory is presented in Rohlin's fundamental papers [Roh]l,2. The rest of this chapter is devoted to developing, following the scheme of [Roh]2, a general theory of conditional entropies of measure-preserving transformations on Lebesgue spaces.

w Measure Theory Let (X,13,/~) be a measure space. Two sets B1,B2 E 13 are said to be equivalent modulo zero, written B1 = B2 (mod 0), if the symmetric difference B1/kB2 has measure zero. When we write A1 = A2 (mod 0) for two subsets -41,A2 of 13 we mean that for each A1 E A1 there exists A2 C A2 such that A1 = A2 (mod 0) and vice versa. Let A be a subset of 13, we say that A generates 13 (mod 0) if 13 = 130 (rood 0), where 130 is the a-algebra generated by -4. The following is the well-known approximation theorem (see, e.g. [Rud]): 1.1. If(X, 13,#) is a probability space, a subalgebra .A C 13 generates 13 (mod O) if and only if, for every B E 13 and ~ > O, there exists A E A such that p ( A A B ) <_e. Theorem

Before going further, let us first review some definitions and simple facts about function spaces on a measure space. Let (X, 13, p) be a measure space and let 1 < p < +oo. We denote by LP(X, 13, #) the quotient of the set of functions f : X --+ C such that Ifl p is integrable, under the equivalence relation that identifies functions which coincide a.e. We endow LP(X, 13, t*) with the norm

II. lip defined by 1

Ilfllp =

IflPd~

which makes LP(X,13,#) a Banach space. For p = 2, the norm I1" 112 comes from an inner product / , < f ' g > = .Ix fgd# with respect to which L2(X, 13, #) is a Hilbert space. Given a probability space (X,13,#), if there exists a countable subset of 13 which generates 13 (mod 0), then we say it is separable, several equivalent descriptions of this kind of separability are given by the following

1.2. For a probability space (X,/3,#), the following properties are equivalent: 1) (X, 13, #) is separable; 2) L I ( X , 13, #) is separable; 3) L P ( X , B , # ) is separable for every 1 <_ p < +oe; d) There exists a countable subalgebra .4 C 13 which generates/3 (mod 0).

Theorem

A complete separable metric space is known as a Polish space. This kind of spaces provide an important class of separable measure spaces by the following theorem ( see, for instance, [Man]t): 1.3. If X is a Polish space, 13 is the Borel ~r-algebra of X and # is a probability measure on/3, then (X,/3, #) is separable. Theorem

We formulate below a theorem concerning the regularity of finite Borel measures on Polish spaces (see [Coh]). 1.4. Let (X,13,#) be a probability space, where X is a Polish space and/3 is the Borel ~-algebra of X . Then for every Borel set B E/3 and e > 0 there exists a compact set K C B such that # ( B \ K ) < ~.

Theorem

Let (X, 13) be a measurable space and let # : /3 --+ [0,+oc] and u : 13 [0, +oc] be measures. We say that # is absolutely continuous with respect to u, and we w r i t e # < < u, i f B E B and u(B) = 0 i m p l y # ( B ) = 0. The following Radon-Nikodym theorem characterizes this kind of absolute continuity. T h e o r e m 1.5. If (X, 13, u) is ~-finite, then # << u if and only if there exists f : X --* R +, integrable with respect to u on allsets B E 13 such that u(B) < +oc,

satisfying the following condition for every B E 13: #(B) = / ,

fdu.

The function f is unique a,e. (with respect to u), and is denoted by d#/du. A function g : X ---* C is in L I ( X , B , # ) ~f and only if g f is in LI(X,/3, u), and in this ease we have /xgdp=/xgfdu" If # ( X ) < +oo then f E Ll(X,/3, t~). The function f is called the Radon-Nikodym derivative of # with respect to u. Let (X, B, #) be a measure space and let A be a sub-a-algebra of B. For every f E L I ( X , B , la), the Radon-Nikodym theorem allows us to prove easily that there exists a unique function, written E(fl~4), in L I ( X , . A , # ) such that fA E(f[.A)d# = fA f d # for every A E .4. This function E(fI.A ) is called the

conditional expectation of f with respect to .4. We now define the conditional expectation operator E(.IAM): LI(X,B, tO --, L~(X,AM,tL), f ~ E(flAM). T h e o r e m 1.6. Let (X, B, #) be a probability space and let AMbe a sub
Proof. For f ~ LI(X,B,#) we know that E(flAM) is the only AM-measurable function such that fA E(IIAM)d~ = fA fd~ for every A E ,4. Let P denote the orthogonal projection of L2(X,B,#) onto the closed subspace L2(X, AM,p). If f C L2(X,B,p), then P f is AM-measurable and if A E .4 AfdP = < f, XA > = < f, PXA > = < Pf, XA > = /A Pfd#. Therefore P f = E(flAM).

[]

The Radon-Nikodym theorem also allows us to introduce the general definition of Jacobian of absolutely continuous maps between measure spaces. Let (X,/3, #) and (Y, .4, u) be two a-finite measure spaces, and let T : X Y be a map. We say that T is absolutely continuous if the following three conditions hold: (i) T is injective; (it) if B C B then TB E .4; (iii) B e B and # ( B ) = 0 imply ~(TB) = O. Assume that T is absolutely continuous. We now define on B a new measure # r by the formula #T(B) = u(TB). The measure #T is absolutely continuous with respect to the measure #. Thus by the Radon-Nikodym theorem one can introduce the measurable function J(T) = d#T/d# defined on X, it is called the Jacobian of the map T. It is easy to see that, when the absolutely continuous m a p T is bijective and T -1 is also absolutely continuous, we have 1

J(T) - j ( r _ l ) o T for # almost all points of X (we admit here 1/0 = +oo and 1 / + oo = 0). When X and Y are two Riemannian manifolds without boundary and of the same finite dimension, f : X -* Y is a C 1 diffeomorphism, and Ax and Ay are the respective Lebesgue measures on X and Y induced by the Riemannian metrics, then in this particular case it is easy to see that for any x C X one has

J(f)(x) - d(Av o f ) ( x ) = I detT~f], dAx where T~f is the derivative of f at x, and for any h E LI(Y, Ay) one has

I x ( h o /)(x)l det T~fldAx(x ) = / r h(y)dky(y).

Next, we have the following Lebesgue-Vitali theorem on differentiation (see, e.g., [Shi]): T h e o r e m 1.7. Let A C R ~ be a Borel set, and h : A --+ C an integrable function with respect to the Lebesgue measure A of R '~ Then the following holds true for A-almost every x E A : lim 1 /B hd.~ = h(x), ~-o A(B~(x) n A) r(.)nA where r e ( x ) = {y e a '~ :

d(~, y) < r}.

A simple application of the Lebesgue-Vitali theorem yields the following: T h e o r e m 1.8. Let T : (X, B, #) --* (Y,.4, u) be an absolutely continuous map, where X is a Borel subset of R '~ with A(X) < +oc, I3 is the (r-algebra of Borel subsets of X and # is absolutely continuous with respect to ,klx. Then there holds the following formula for p-almost every x E X : lira pT(B~(x) n X) ~-~o # ( B r ( x ) n X ) = J ( T ) ( x )

Proof Let h = dp/d)h then lim pT( B~(x) n X ) ~-0 p ( m ( ~ ) n x ) A(Br(x) O X ) -1 fBr(~)nX J(T)hdA = lira ~--o

A(B~(x) N X ) -1 fB~(z)nX hd),

=J(T)(x), This completes the proof.

p-

a.e.x r X .

[]

An easy application of Theorem 1.7 also gives T h e o r e m 1.9. I r A C R n is a Borel set and p is a Borel measure on R n which is absolutely continuous with respect to .~, then the limit lim p(Br(x) N A)

r-0

p(m(x))

exists #-almost everywhere in R '~, and is equal to 0 if x ~ A and to 1 if x E A. When the above limit exists for x E A and is equal to 1, we call x a density point of A with respect to p.

We conclude this section with the notion of Lebesgue spaces. A map between two measure spaces is called an invertible measure-preserving transformation if it is bijective and measure-preserving and so is its inversion. Two measure spaces ( X i , B i , P i ) , i = 1,2 are said to be isomorphic mod 0 if there exist Y~ E B~,Y~ E B2 with p~(XI\Y~) = 0 = p2(X2\Y2) and there exists an invertible measure-preserving transformation r : (Y1, Blly,,#lly~) --+ (Y2,B21Y~,~2IY:). Given a probability space ( X , B , p ) , let X0 = X \ { x : x E X with {x} e B and #({~}) > 0} and s = #(X0). We call ( X , B , # ) aLebesgue space if (X0, BlXo, #lXo) is isomorphic mod 0 to the space ([0, s],/:;([0,s]), l), where ~;([0, s]) is the cr-algebra of Lebesgue measurable subsets of [0, s] and l is the usual Lebesgue measure. There is now the following important theorem (see [Roy]): T h e o r e m 1.10. Let X be a Polish space, p a Borel probability measure on X ,

and 13u(X ) the completion of the Borel c~-algebra of X with respect to #. Then (X, 13u(X), #) is a Lebesgue space. Throughout the remaining sections of this chapter it is always assumed that (X, B,/z) is a Lebesgue space.

w Measurable

Partitions

Let (X, B, #) be a Lebesgue space. Any collection of non-empty disjoint sets that covers X is said to be a partition of X. Subsets of X that are unions of elements of a partition ( are called (-sets. A countable system {B~ : a E .A} of measurable 4-sets is said to be a basis of 4 if, for any two elements C and C ~ of 4, there exists an a E .4 such that either C C B ~ , C ' ~ B~ or C ~ B ~ , C t C B~. A p a r t i t i o n with a basis is said to be measurable. Obviously, every element of a measurable partition is a measurable set. For x E X we will denote by 4(x) the element of a partition 4 which contains x. If ~, ~' are measurable partitions of X, we write 4 _< 4' if 4'(x) C 4(x) for #-almost every x E X, 4 = 4 ~ is also considered up to rood 0. For any system of measurable partitions {4~} of X there exists a product ~/~ (~, defined as the measurable partition ( satisfying the following two conditions : 1) 4~ _< 4 for all a; 2) if44 <_ 4' for all a, then 4 ~ 4 I. For any system of measurable partitions {4~} of X there exists an intersection Aa (a, defined as the measurable partition ~ satisfying the conditions : 1) 4~ >_4 for a l l a ; 2 ) if4~_>4' for a l l a , then~>__41 . For measurable partitions (~, n C N and 4 of X, the symbol 4n / z 4 +oo indicates that ~1 ~ 42 <_ " " and Vn=l 4~ = 4, the symbol 4~ "x~ ( indicates that +oo 41 >_ s >_ "'" and A~__14~ = 4. If { B 1 , B 2 , . . . } is a basis for the partition 4 and fl,~ is the partition of X n into the sets B~ and X \ B n , then the partitions 4n = Vi=l ~i form an increasing

..~oo

sequence and V,~=I ~ = (. Thus, for any measurable partition ( there exists a sequence of finite partitions ( , such that ~,~/z f. One can define an equivalence relation on 13 by saying that A and B are equivalent (A ~ B) if and only if la(A/XB) = 0. We denote the set of equivalence classes by 13. The operations of countable union, countable intersection and subtraction of sets in 13 can be carried over to the same operations on classes, making /~ a a-algebra. Any part of/~ that is closed with respect to these operations is said to be a sub-a-algebra of B. It is clear that the intersection n~/~, of any system of sub-a-algebras {/~} is a sub-a-algebra of/}. The sum V~ B~ of a system of sub-a-algebras {B~} is defined to be the intersection of all sub-a-alg2bras each of which contains all -I-oo N the / ~ . If B1 C B2 C " . and V,~=113,, = 13', then we write t},~ /z ~,. If +oo N = / ~ , then we write/~,~ x,~ ~t. B1 D /~2 D ..- and M,,=IB~ For any measurable partition ~, we denote by/~(~) the sub-a-algebra of/~ consisting of classes of measurable ~-sets. If/~(~) = B(~'), then ~ = ~', and for any sub-a-algebra of B there exists a measurable partition f such that/~(() is this sub-a-algebra. Thus, the sub-a-algebras of/~ are in one-to-one correspondence with the classes of mod 0-equal measurable partitions. Here B(~) C B(~') if and only if ~ _< ~'. We also have

In particular, /~((n) ,7/~(() if and only if ~,, /z ~ and/~(~n) "N/~(() if and only iff~ "Nh. The factor-space of X with respect to a partition ( is the measure space whose points are the elements of ( and whose measurable structure and measure #r are defined as follows: Let p be the map taking each point x E X to ~(x), a set Z is considered to be measurable if p-l(Z) E 13, and we define #~(Z) = # ( p - l ( Z ) ) . This factor-space will be denoted by X/~. When ~ is a measurable partition X/# is a Legesgue space. A very important property of measurable partitions of a Lebesgue space (X, 13, p) is that associated with every such partition { there exists a unique system of measures {Pc}ce~ satisfying the following two conditions: i) (C, BIo, pc) is a Lebesgue space for #(-a.e. C E X/~; ii) for every A E 13,#c(A MC) is measurable on X/~ and

I~(A) = f #c(A MC)dp~. Jx /r Such a system of measures {#c}ce~ is called a canonical syslem of conditional measures of # associated with ~. Its uniqueness implies that any two systems {#c}ce~ and {#~c}ce~ satisfying i) and ii) above are rood O-equal (that is, Pc = Pb for #r C E X/~). Let ~ and r/ be measurable partitions such that ~ < r/ and let A be an element of 77 and C an element of ~ containing A. As an element of the partition

of X, (A,/~IA, #A) is a Lebesque space. On the other hand, as an element of the partition rllc of C with the a-algebra Blc and the measure #c, (A, NA, (;~C)A) is also a Lebesgue space. The uniqueness of canonical systems of conditional measures implies that (#C)A = #A for t~,-a.e. A E X/71. This property is called the transitivity of canonical systems of conditional measures. Let {tLc}ce~ be a canonical system of conditional measures of # associated with a measurable partition ~. By the standard method of measure theory we know that for any g 6 L 1(X, B, #) , the section gc defined as gc(

) =

if

e c

is integrable on (C, BIc,t*c) for tLe-a.e. C ~ X/5, fcgcd~c is measurable on X / ( and

L g(,)dt,=fx/ (L gcd,c)

(2.1)

w Conditional Entropies of Measurable Partitions In this section we give a brief review of several notions about entropy for measurable partitions of Lebesgue spaces. Here we assume again that (X, B, #) is a Lebesgue space. Let ~ be a measurable partition of X and let C1, C 2 , . . . be the elements of of positive/~ measure. We put G(~)=

{ -E~(G)log~(G) +oo

if if

~(X\U~G)=0 /z(X\Uk Ck) > 0

(3.1)

The sum in the first part of (3.1) can be finite or infinite. H,(~) is called the entropy of (. If { and ~ are two measurable partitions of X, then almost every partition {B, defined as the restriction {IB of~ to B, B 6 X/~, has a well-defined entropy H..(~B). This is a non-negative measurable function on the factor-space X/rl, called the conditional entropy of { with respect to rj. Put

H,(Si~) = f IG~(5.)dt~,. Jx /,

(3.2)

This integral can be finite or infinite. We call it the mean conditional entropy of ( with respect to rI. Obviously, when q is the trivial partition whose single element is X, H.((}r/) coincides with the entropy H , ( ( ) . We put log 0 = - 0 % 0 log 0 = 0 and define for each x E X re(x,

=

m(x,,q,)

=

n

then they are measurable functions on X and (3.1), tively in the forms: f H,(4) = - / ~

(3.2)

can be written respec-

log re(x, ~)d#,

(3.3)

H.(41.) = -/• log re(z, 4[rl)d#.

(3.4)

The mean conditional entropy has, in particular, the following properties: 1) If 4,~ 7 4, then for any measurable partition r/ Hu(G~]r/) /

H~(~]r/);

(3.5)

2) If 4,~ "N 4, then for any measurable partition 77 with H,(41 [rl) < +0% Hu(4~lr]) \

Hu(~lr/);

(3.6)

3) For any measurable partitions 4, 77 and (, Hu(4 v ~[C) = H.(41C) + H.(r/14 V/); 4) If rt~ /

(3.7)

71 and 4 is a measurable partition such that H.(4lr/1) < + ~ ,

then

H,(41~) \ H.(41~);

(3.8)

5) If r/.~ "N r/, then for any measurable partition H.(4I~) /

H.(4Io);

(3.9)

6) If (Xi,Bi, pi),i = 1,2 are two Lebesgue spaces and T is a measurepreserving transformation from (X1,/~a,pl) to (X2,B2,#2), then for any measurable partitions 4 and r/of X2, Hm (T-14]T-l~)

=

H,~(41").

(3.10)

The proofs of 1)-5) can be found in [Roh]2. Now we give a proof of 6). When 4 and r~ are finite partitions, it is easy to see that

m(x,T-14[T-l~7)

= m(Zx,4[rl),

#1 - a.e.

x E Xl.

Since T/~I = #2, then (3.10) follows from (3.4). In addition, it is obvious that H.~(~I~) < +oo In general, let 4 and 7/be any two measurable partitions of X2. Choose two +co1 and {r/,~},~=1 +co such that ~,~ .7 ( sequences of finite measurable partitions {4n }n= and rl,,/z rj. Since for any k, n E N H m ( T - l~k IT- 1rh~) = Hu~ (4k It/n), letting n -+ +cx), by (3.8) we have for every k E N H.~ (T- 14k IT- it/) = H.~ (4k [r]), then (3.10) is deduced from (3.5) when k --* +co.

w Conditional

Entropies of Measure-Preserving Transformations: I

Now we begin to deal with the concept of conditional entropies of measurepreserving transformations. Let T be a measure-preserving transformation on a Lebesgue space (X, B, #) and A a sub-a-algebra of B satisfying (4.1)

T - 1 A C A.

Then there exists a unique (mod 0) measurable partition f0 of X such that/3((0) is the ~-algebra .A consisting of classes of sets in ,4, and from (4.1) it follows that T - i f 0 _< f0. (4.2) Let f0 be as obtained above. We denote by Z(f0) the set of measurable partitions ( of X satisfying Hu((lf0 ) < +ec. For ~,~ 9 Z(f0), put p((, 7?) : Hu(([~ V (0) +

Hu(,;I( v f0).

(4.3)

Then it can be easily shown that for any (, 7, f 9 Z(r

IHu(~lf V fo) - H.(01f V G)I <- P(~, n),

(4.4)

[Hu(e[0 V G) - H , ( ~ l f V f0)l _< ;(~, f),

(4.5)

and if ~n,~ 9 Z(fo) with ~n 7 ~ or ~n "N ~ then

p(~,~, r --~ 0

n ---++oo.

as

(4.6)

In fact, (4.4) and (4.5) follows respectively from

H~,(~lf V G) - H~,(0lf v fo) _< H~,(s v 0If V fro) - H~,(rllf v Co) =

H,(~I~

v f v

G) <_ H,(~IT/v f0)

and

H.(~I~ v r - Hu(~[( V ~o) ~ H,(~ v r v ~o) - H~(~I~v r v
n-1

: V

= V

k=0

= {x},

k=0

and by e we denote the partition of X into single points. Sometimes we use the simpler notations ~n,~- and ~0 to denote ~ , (T and (~ respectively. T h e o r e m 4.1.

Let T,.A and (o be as given above. Then for any ~ E Z((o)

h~(T,~) =

lim n--~ + c o

1Hu(5=~lf0) n

(4.7)

1 H U(4TI(O)n exists and equals i n f n k l ~This limit is called the .A-conditional entropy o f t with respect to ~.

Proof Let an = H , ( [ } l ( 0 ) , then by (3.7), (3.9) and (3.10) /n +m - 1

_ an+m <

. ( e T ] ( 0 ) q- H #

V k----rt

"~

)

T - k ~1(o

<_ an + H l , ( T - n ~ l T - n @ ) ~- an -k a m

for any m, n E N . F i x p C N arbitrarily. Each n C N can be w r i t t e n as n = kp+i with 0 < i < p. T h e n an

ai+kp

ai

akp

ai

]Cap

kp =

ai

ap

-.p

L e t t i n g n ~ + e c , we have an

lira s u p - n~+oo

ap

< --

n

p

a n d therefore an

lim s u p - n~+oo

But inf

ap

p_>l p

n

ap

< inf - - ,

--p_>l p a~

< lim inf '~ , -- n ~ + o o n

so l i m n ~ + o o 2ann exists and equals infp_>l La p p. Definition

[]

4.1. Let T and.A be as given above. Then

he(T) =

sup h.A

(4.8)

(eZ((o)

is called the A-conditional entropy o f T . R e m a r k 4.1. T h e r i g h t - h a n d side of (4.8) does not change if the s u p r e m u m is t a k e n only over the set of finite m e a s u r a b l e p a r t i t i o n s . In fact, this r e m a r k can be d e d u c e d from the following l e m m a and (4.6). L e m m a 4.1. Let h~(T,~) be as defined by (4.7). Then for any ~,~ e Z((o) ] h e ( T , ( ) - h~u(T, , ) l < P((, 7/)-

10

(4.9)

Proof. From (3.7) it follows that H.((~IC0) + H.(~'+I~~ v G) = H.(~ ~ v ~lCo) = H.(,7~IG)+ H.((~lrl ~ v (o)

(4.10)

which yields I H . ( C I G ) - H.(,PIG)I <_ H . ( ~ I ~ ~ v Co) + H . ( ~ I C

vr

(4.11)

But n-1

H.(('~irf + V G) <_ ~ H.(T-k(lrf + V G) k=O rz-1

<_ ~ H.(T-k(IT-kT1V T-kCo) k=O

= nHt,(glr~ v Co),

(4.12)

and similarly Ht,(r/~ IZ~~ v G) _< nHu(rll(

v Co),

then (4.9) follows from (4.7) and (4.11)-(4.13).

(4.13)

[]

Now we give some further properties of the A-conditional entropy. T h e o r e m 4.2. Let T , A and Co be as given before. Then for any ~, rI E Z(Co)

the following hold true: 1) hr <_H.((]C0). 2) hr ~ V q) <_he(T, ~) + hA. (T, rl). 3) ~ <_rI implies hr <_hr q). ~) he(T, ~) <_hr rl) + H.(~lr I V Co). 5) hr162 < hr162 5) hA. (T, r = h..4 (T,~T) k for any k E N. 7) h. (T , ~ T+) = khr ~) for any k e N. 8) If~ 9 Z(T-1Co), then h~A(T,~) = h. (,~). 9) If T is invertible and T - 1 A = A, then h. (T -'

= hf(T,

Proof. 1) From r~--I

I H.((~Ko) <- -nl E H"(T-k(l(~ k:O

<- - E H" (T-k(IT-k(~ : H~'((l(~ n

k:o

11

and (4.7) we obtain 1). 2) Since H.((( V ~)~I(o) = H.(( ~ V ,~I
Hu(~ '~ V r/~](0) =

H,~(r/'~ Ko) +

Hu(CD"v

G)

and n-1

Hu(,g'~lrf~ Vr

G

~_,H,(T-khlrf ~ Vr k=O rL--1

<_ ~_, H~,(T-<~lT-k(rl V ~o)) k=O

= nH.(~b

v G),

this together with (4.7) yields 4). 5)

hA.(T,T-I~)=

lim n

<

lim

--

n--~+oo

"

T-k(K~

1H.(T-I~IT-1Co)= hr n

lim 1H.(r 6) h.A ( T , ~ )k = ~-~+oo n =

7)

lim 1 ~--*+~n+k-1

H O:'~+k-1 "'" Ko)=

.4 k 1H .-i k-i h~(T ' ~ k ) = n--.+oolimn ~ ( V T - k J ( i y o T - i ~

hr 1

().

~o)

\ j =0

=

8)

lim klH~(~k~[~o) =

n--*+oo

IV

kh~(T,~).

hT-1A(T,,~)= lim 1Ht,(,~r*lT-l(,o ) n ---* + o o

=

lim

n~+oo

r$

ln g~, V T-k,~IT-I(.o +H u ,~ \k----1

/

= lira !H.(~-IK0): n~+cc

hr

f/

12

T-l(~o k----1

9) I f T is invertible and T-I.A = ,4, then hr

1H.((~lC0)= lim 1Hu(T"-~(~[T'~-Ir

lim =

lim 1H,(~_,lff0) ; hr

This completes the proof. Theorem

[]

4.3. Let T and .A be as given before. Then for any k E N

hr

k) = khr

(4.14)

I f T is invertible and T - 1 A = A, then hr

(4.15)

Proof. From 7) of T h e o r e m 4.2 it follows obviously that hr

k) > khr

(4.16)

On the other hand, for any ~ E Z(~0)

k = kh~(T,~). <_ h,04(T k ,~r)

hr So

hr which together with

[]

k) <_ khr

(4.16) proves (4.14). (4.t5) follows from 9) of T h e o r e m 4.2.

T h e o r e m 4.4. Let T and .d be as given before. If T is invertible and we let ..4+ = Vk=o +oo TkA, then

he(T) = h.04+ (T)

Proof. From 8) of T h e o r e m 4.2 we know that for any finite measurable partition and any k C N

hr

= h,T~O4(T,~).

(4.17)

Let ~+ = Vk=0 +oo Tk(0 " Since for every n C N

1H

1H~,(~'~ I~+ )

(4.18)

as k --* + o c , by T h e o r e m 4.1 we can easily show that for any ~ > 0 there exists k E N such that

G~+(T,~)>h.

(,~) 13

~

Hence

h.`4+ (~,~) >_ hr On the other hand, by (3.8)

h.`4+ (V,r <_ he(T,{). Thus, for any finite measurable partition

h..4 + (T,~) = h r This together with R e m a r k 4.1 yields

h.`4+ (T) = h e ( T ) We complete the proof.

[]

T h e o r e m 4.5. Let T and .4 be as given before, ff {l <_ ~2 <_ "'" is an increasing +oo sequence of partitions in Z(~o) such that (V,~=a ~,~)v ~0 = c, then

hr

/

he(T)

as

~ ~

+oo

Proof For any r/ E Z(~0), by 4) of T h e o r e m 4.2

he(T,,) _
".(,1~. v;0)

Since, according to (3.8), H.(,l~

v/o) % 0

as

n ---, +oo,

and by 3) of T h e o r e m 4.2 the sequence h e (T, ~1), h e (T, ~2), 9' " is increasing, we h ave h e ( T , r/) < lim h~(T,{n). --

n ~ - o o

Since r1 is arbitrary in Z(~0) we see that

hr completing the proof. Theorem then

/

hO(T)

as ,~ -~ +oo,

[]

4.6. Let T and.4 be as given before, i f { C Z(~o) satisfies ~T V(:o = e,

h.~(v,~) = he(T).

Proof This t h e o r e m follows from T h e o r e m 4.5 and 6) of T h e o r e m 4.2.

[]

T h e o r e m 4.7. Assume that X is a Borel subset of a Polish space Y , # is a Borel probability measure on X and B is the completion of the Borel c~-algebra of

14

X with respect to #. Let T be a measure-preserving transformation on (X, B, #), A a sub-~-algebraofB satisfying T - i A C A and r the measurable partition of X with 13((o) = .4. I f ~ l , ~ ; , . . . is a sequence of countable partitions in Z(r such that lira diam(~,~) = 0 where diam(~,~)= supce~ diam(C), then he(T)=

lim

hr

To prove this t h e o r e m we need the following 4.2. Let r >_ 1 be a fixed integer. Then for each e > 0 there exists (5 > 0 such thai if { = { A t , . . . ,Ar}, 71 = { C i , ,C~} are two measurable partitions of X with E [ = l IX(AiACi) < (5 then H,(el~) + H,(rl[~) < e.

Lemma

Proof. Let E > 0 be given. Choose (5 > 0 so t h a t (5 < 88 a n d - r ( r 1)(5 l o g 6 (1 - (5)log(1 - (5) < ~. Let r be the partition into the sets Ai N Cj(i r j) and U~=i(A~ n C~). T h e n ~ V 7/= r V r/ and since Ai ;'1Cj C U~=I(A,~AC,~)(i r j) we have p(AiACj)<(5(i~j)

p(,U=l(A,~nC~)) > 1 - 5 .

and

Hence H,,(r < - r ( r - 1)(51og(5 - (1 - ( 5 ) log(1 - (5) < ~. Therefore H~,(T/) § H~(~lr/) = H~(~VT/) = H , ( 7 ] V r _< H,(r/) + H~(r < a~(r/) + 2' and so < Similarly we have H,(~]~) < }. T h e n the proof is completed. [] P r o o f o f T h e o r e m 4.7. Note t h a t # can be regarded as a Borel probability measure on Y and then (X, B, #) is a Lebesgue space. Fix now c > 0 arbitrarily. We can choose a finite partition ~ = { A i , - " , A ~ } of X such t h a t its elements are Borel sets and hA,(T,~) > he(T) - c if he(T) < + o c , o r hr > c - i if hr = +oo. Take (5 > 0 as corresponding to c and r in the sense explained in L e m m a 4.2. Since every finite Borel measure on a Polish space is regular (see T h e o r e m 1.4) and p can be regarded as a Borel measure on Y s u p p o r t e d by X, we can choose c o m p a c t sets Ki C Ai with # ( A i \ K i ) < 6(2r) -1, 1 < i < r. Let (5' = infi#j d(Ki, Kj) and choose n such t h a t d i a m ( ( , ) < 2-1(5 '. For 1 <_ i < r let E (i) be the union of all the elements of 4~ t h a t intersect Ki and let E~(r) be the union of the remaining elements of ~,~. Since d i a m ( ~ , ) < 2-1(5 ' each C E (,~ can intersect at most one Ki. Then ~ = {E(~0, -.. , E~(r)} is a measurable partition of X such t h a t ~" < ~ and Er_l #(E(~i)/kAi) = (i) ~ ~ Ei=m #(E,~ \A~) + E i = l #(Ai\E~ O) <- Ix(X\ Ui= 1 Ki) + Ei=~ Ix(Ai\Ki) < (5.

15

From Lemma 4.2 we have H u ( ~ W ) + Hu({~ 14) < g. Therefore, if n is such that diam({,~) < T then

_< he(r, e:) + H (gle: v t0) _< hO(r,

+

~t

Thus diam(~,~) < T implies hA(T,{,~ ) > h e ( T ) 2e if he(T) < +oc or hA(T,~,, ) > 6"- 1 - - g if hAu(T) = +oo. So limn-++oo hAu(T,~,~) exists and equals

hA(T ).

[]

w Conditional Entropies of Measure-Preserving Transformations: II Let T and .4 be as in Section 4. In this section we generalize the definition of the entropy function hA(T, {), which was justified in Section 4 only for partitions in Z((0), so that it is defined on the set of all measurable partitions of X. In order to do this we have to confine us to the case when T - 1 A = .,4. D e f i n i t i o n 5.1. Let T,.,4 and 4o be as in Section 4 and assume that T-1.,4 = .'4.

For every measurable partition ~ of X, put h~(T,{) = Hu({IT-14T V 40),

(5.1)

we call this the A-conditional entropy o f t with respect to ~. R e m a r k 5.1. If ( E Z(Co), then h A ( T , ( ) defined by (5A) coincides with that defined by Theorem 4.I. Indeed, if~ E Z(40), from (3.7), (3,10) and T - 1 A = .,4 it follows that for every n ~ N

k k=l

k=l

= H,(T-'C~-'IT-'(o)+

H u(elT-le n-1 v Co)

= H , ( ( " - * I C 0 ) + Hu({[T-14 "-1 V C0), then an easy induction shows n--1

H,(C~I(0) = H,(dC0) + ~'~ H , ( g [ T - ' g ' V C0).

(5.2)

/=1

Since

Hu(~IT-I( l VCo) "N H u ( ~ I T - ~ - V (o) as l --* +0% by (5.2) we have

1

-H~(~'~[(0) "N Hu(~lT-l~ - V(0)

as

n --~ +oo.

n

This shows that the conclusion of Remark 5.1 is true.

16

[]

(5.3)

Theorem

5.1. Let T and .4 be as given before and assume that T - 1 ` 4 = `4.

Then

he(T) = suphr where the s u p r e m u m is taken over the set of all measurable partitions of X . Proof It is enough to show that for any measurable partition ~ and any number a < hr there exists a finite measurable partition r/such that h e ( T , 71) > a.

Let ( 1 , ~ 2 , 9' be a sequence of finite measurable partitions such that ~ / / ~ . Since (,~ <_ ~ we have he(T,&)

= H , ( & I T - I ( & ) - V~0) > H u ( ~ I T - ~

- Vr

From H u ( ~ I T - I ~ - V ~o)

it follows that hr

, Hu(gIT-lg

- VG)

as

n ~

+oo

> a for sufficiently large n and we can take r1 = ~,~.

[]

Before discussing further properties of h r given by Definition 5.1, we first give some preliminary results (Lemmas 5.1-5.3). L e m m a 5.1. Let T , A

and ~o be as given before and assume that T - 1 A If ~, 77 are two measurable partitions with 77 < (, then

= A.

rt--1

H,(('IT-~,j - VG)=

~ " Hu(~IT-I(u - V f f ) V G )

(5.4)

k=0

In particular, H.(~"IT-~

- v ~0) = ~ h r

Proof Since ~k = ~ V T - I ~ k-1 and rI <_ ~, we have

H~(~ k I T - k . - v ~o)

= H , ( T - I ~ k - l l r - k r / - V (o) q- Hu(~lT-kr] - V ~o V T - I ~ k - l ) = H u ( ( k - l l T - ( k - 1 ) r l - V ~0) + H u ( ( l T - l ( r l -

V

~k-1) V ~0),

(5.4) is then derived from this equation by an obvious induction.

[]

L e m m a 5.2. Let T , M and ~o be as given before and assume that T - 1 ` 4 = .4. I f ~,r I are two measurable partitions with rl <_ ~ and Hu(~IT-177 - V ~o) < +0% then 1

--H,(~IT-"rl - VG) n

\

h~(T,~)

17

as

n ~ +oo.

(5.5)

Proof. Since ('~ 7 ( - and r / < (, we have r/- V (~ ,7 ( - and by (3.8) H.((IT-~(~

-

v ( " ) v G) \ H.(dT-~(

Then (5.5) follows from (5.4).

- v G).

[]

L e m m a 5.3. Let T,.A and (o be as given before and assume that T - 1 M = .,4. If (,r] are two measurable partitions with ~ < r] and H , ( ~ I T - I ( - V io) < +oc, then lim 1 H , ( ( ' ~ I T - ' b - V G) = hi(T,(). n~+oo

rt

Proof. Let 6 be a positive number. Since by Lemma 5.1

~-H.(("IT-~-v G)<

~H,((~IT-"CVG)=hr

and by Lemma 5.2

!H.(~IT-~(- V
as n-~ +oo,

n

it suffices to show that the inequality

1H.(('~lT-"rl- V G ) n

>

hi(T,()-

6

holds true for all n for which

I-H.(v"IT-~(- V r n

hJ(T,~) + 6.

This is clear from the following chain of relations: 1

-H.(('~IT-"rj - V (o) n

:88

- v

>hr

rl) -

v

1H,,(~" I(" V T - ' ( -

v

V
k-n1 H .(~,, IT-"(-vG)-6-- n1 H .(~,~lOvT-n(-V(o) 1

= - H . ( ( '~I T - " ( - V G) n

:hi(T,() - ~ The proof is completed.

[]

We are now prepared to deduce some properties of h e ( T , ( ) given by Definition 5.1.

18

T h e o r e m 5.2. Let T,.A and (o be as given before and assume that T - I J I = .4.

Then for any measurable partitions ~ and 1) h ~ ( T , [ ) < H , ( [ l ( 0 ). 2) h~(T, ~ V ,) ~ h f ( T , ~) + h~(T, ,). _ h, (T,~T) for all n e N. 5) h.04(T " , ~ ) = .h~. (T, ~). 5) If ~ ~ ~? and H , ( ~ I T - I [ T V(0) < +oo, then hA~(T,[) ~ h~(T,~?).

Proof. 1) h~(T,~) = H , ( [ I T - I [ - V (o) <_ H~(~lCo). 2)

h f f ( T , [ V q ) = H ~ ( [ V ~ I T - I ( ~ V ~ ) - V(0)

~ H~(~IT-I[ - V (o) + H , ( ~ I T - I ~ - V (o) = h~(T,[) + h~(T, q). 3) Since (r 04

= ( - we have

n

h. (T,r ) = H.(g~IT-I~ - Vr 4)

> H.(r

h~(T,T-I() : H,(T-I(tT-I(T-I()

- Vr

= h.04 (T,r

- V(0)

= H , ( ( I T - I ( - V (o) = h~(T,(). 5) Since ((~.)~o = ~ ,

we have

h.A (T n ,~.) = H~(~IT-'(~.)~o v (o) : H ~ ( ~ . I T - ~

v (o),

then we get the desired result by applying Lemma 5.1. 6) Since ( < ~? we have

vr

<

vr

By Lemma 5.1 the right-hand side is equal to h~(T, z~) and by Lemma 5.3 the limit of the left-hand side as n ---* +oo is h~(T, ~), then 6 ) i s proved. [] A measurable partition ~ of X is said to be fixed under T if every element C of y satisfies T - 1 C = C. Let r] be such a partition of X, we denote by Tc the measure-preserving transformation on (C, BIc, #c) induced by T for # ~ - a . e . C . T h e o r e m 5.3. Let T , A and (o be as given before and assume that T-1.A = .A.

If ~ E Z((o) and q is a measurable partition fixed under T, then

h~(T,~ v ~) = h~(T,~)

19

Proof First we assume that r/is a finite measurable partition. Since ! [ H . ( C v ~ ] T - ~ ( ( - V ~-) v G ) - H.((~IT-~(( - V ~-) v 4o)] n

1 =-H.(~=IO

vT-=( - VT-=~

/%

1 =-H.(v'~I,~

-

- V(o)

V T - ' b / - V(o),

(5.6)

n

and since T - l r ] = r], the last term of (5.6) is zero. T h e first t e r m of (5.6), by L e m m a 5.1 and L e m m a 5.2, converges as n --+ + o c to h J ( T , ( V r~) - h J ( T , ( ) . Thus hAg (T,~ V q) = hr If r/is an arbitrary measurable partition fixed under T, then there exists a sequence of finite measurable partitions r h , r h , - - , such that r/~ ,7 q and every rl~ is fixed under T. For every n 6 N we first have

hr

(5.7)

V r/N) = h r

Then, from

hr

V q~) = H,(gD~ V T - l g - V G ) ,

he(T, ~ V ~) = H , ( g b V

T-lg

-

v r

and (3.8) and (5.7) we obtain

hA.(T,~ V r/) = h.A(T,~) by letting n --+ +oo.

[]

5.4. Let T,`4 and (o be as given before and assume that T-1M = .4. If ~ E Z((o) and q is a measurable partition fixed under T, then Theorem

hr

= Ix

/7

h.B .4B (TB,~B)d#v

(5.8)

where .4B = .4lB. Proof. Since hA'(TB,~B)u,

= H,,(~BITgI((B)TB V (~O)B) = H , B ( ~ B I T ~ I ( ~ ) B V ((O)B) = - - / B log rn(x, (B ITt~ 1(~T)B V ((0)B)d#B,

and since the function under the integral sign is the restriction to B of log re(x, ~lq V T - I ~ - V(0), we have

20

i x l , h~2 (TB, ~B)d#7

=-Ix /7 d'" /. l~ -

log

v r176 v T-I

- v r

=Hu(~lrl V T-I~ - V(o) = h~(T,~ V q). Then by Theorem 5.3 we get (5.8).

[]

T h e o r e m 5.5. Let T,`4 be as given before and assume that T-1`4 = .4. If y is

a measurable partition fixed under T, then h~(T) = Jxf/7 h~:(Ts)dPT"

(5.9)

Proof Let ( 1 , ~ 2 , " " be a sequence of finite measurable partitions such that ( ~ / z c. According to Theorem 5.4, for every n E N we have hr

=f hr Jx /7

7.

(5.10)

By Theorem 4.5, for #7-a.e. B E X/77 bus (

,(~)B)/z

hu,.

)

(5.11)

as n --* +oo. Then (5.9) follows from (5.10), (5.11) and the monotone convergence theorem. [] We end this section with the notion of ergodic decompositions of measure -preserving transformations. Let T be a measure-preserving transformation on the Lebesgue space ( X , B , # ) . T is called ergodic if every measurable set A satisfying T-1A = A has either measure 1 or measure 0, If T is not ergodic, then it can be decomposed into ergodic components in the following sense (see [Roh]3): There exists a unique (rood 0) measurable partition 71of X fixed under T such that Tc : (C, BIG, Pc) ~-' is ergodic for #7-a.e. C E X/~.

21

Chapter I

Entropy and Lyapunov Exponents of Random Diffeomorphisms

The purpose of this chapter is to introduce some basic notions, especially entropy and Lyapunov exponents, of ergodic theory of random diffeomorphisms and to present some basic properties of them. This chapter serves as the foundation of this book. It is assumed throughout that the reader is familiar with the contents of Chapter I of Kifer's book [Kill1.

w T h e Basic M e a s u r e Spaces and Invariant M e a s u r e s T h r o u g h o u t the remaining part of this book M is always (except mentioned otherwise) a C ~ compact and connected Riemannian manifold without boundary, Diff2(M) denotes the space of C S diffeomorphisms on M equipped with the C2-topology (see, e.g., [Hit]i). To simplify notations we shall denote Diff2(M) by f2. Let v be a Borel probability measure on (~2,B(f2)), where we denote by B ( X ) the Borel , - a l g e b r a of a topological space X. We are mainly concerned with in this book the ergodic theory of the evolution process generated by the successive applications of randomly chosen maps in f2. These maps will be assumed to be independent and identically distributed with law v. There are now two cases to be distinguished when the process evolutes noninvertibly or invertibly in time. The former case seems to be more natural than the latter since many important physical processes such as diffusions are in general noninvertible in time. We are first mainly concerned with the former one and in Chapter VI we shall touch on the latter and give an important result about entropy formula and Sinai-Bowen-Ruelle measures for this case. Now let us first describe the former case in a more precise way. Let +co

(fiN,/3(f~)N, v TM) = 1-I (fl, B(f2), v) 1

be the infinite product of copies of the measure space (f~,B(fl), v). For each

w = (fo(w), f l ( w ) , . . . ) C f i N and n > 0, define fo = id,

f ~ = f,~-l(~z) o fn-~(w) o . . . o fo(w).

Our purpose is to study the dynamical behaviour of these composed maps as n ~ + o c for v N - a . e , w, and the random system generated by {f~ : n > 0,w E ( f ~ N , B ( ~ ) N , v N ) } will be referred to as ?~+(M,v). Throughout this book, associated with any system ?d+(M,v) it is always assumed that v satisfies

f

log

Iflc=dv(f) < q - ~ 22

(1.1)

where log + x = max{log x, 0} and ]flc ~ is the usual C2-norm of f E Q whose definition is given in the following paragraph for the convenience of the reader. We shall always denote dimM by m0. Let h : M --+ R 2m~ be an C ~ embedding. Choose a system of charts {(Ui , ~ i ) } i =ll which covers M and let t l be an open cover of M such that V i C Ui, 1 < i < I. Then for every { V~ i}i= f E f2 define

Iflc~

= max

max

sup

2 (0 Of! i) I Of. 2~uk (u) ,

:l
1 _< j _< 2m0 + 1} whereEr kJl

r

:J2

J ' ' ' ~ J 2rm o - l - l ] ~ T = h o f o # ~ - i

andu=(ut,

"" 9 ~ Urno)T " I t i s c l e a r

that I'lc~ depends on the choice of {(Ui, ~i), Vi}~:_z and h , but if {(U[, ~;), V/' }~'=1 and h' is another choice and let l" Ib~ is the norm corresponding to it, then [. Ic~ and I" I ~ are equivalent, i.e., there exists a number C > 0 such that for all f E a

C-1[flc ~ _< ]fibs < cLflc~. Since we do not pay any attention to the difference between ]. ]c2 and 1. Ibm, such a norm [. [c~ will be called the C2-norm on fL Another equivalent definition will be given in Chapter II without using charts. Condition (1.1) will play a fundamental role in the discussions about entropy, Lyapunov exponents and stable manifolds of z~+ (M, v) carried out in the later chapters. And it can actually be varified in the case when ~ ' + ( M , v ) arises from a diffusion process (see Chapter V). We remark that, when v is supported by one point of f~, z~+(M, v) reduces to the deterministic case. Now we introduce two important spaces f i n x M and flZ x M and let them equipped respectively with the product g-algebras B(f~) N x B(M) and B(f~) Z x B(M) and with the product topologies. The standard knowledge of differential topology (see, for instance, [Hir]l ) tells us that f2 is an open subset of C2(M, M), which is the set of all C 2 maps on M endowed with the C 2topology, and C2(M,M) can be metrized to a Polish space, i.e., a complete separable metric space. Therefore, a can be metrized to a separable metric space. Hence we have B(fl) N x B(M) = B ( a N x M), B(f2) z x B(M) = B ( a z x M). Let r denote both the left shift operators on f2N and ft Z, namely,

fn(rW) = f,~+l(w) for all ~ = (.fo(~o), k ( ~ o ) , - . . )

~ a N and r~ _> O, and

23

for all w = ( . . . , f _ ~ ( w ) , f o ( w ) , f ~ ( w ) , . . . )

E ~2Z a n d n E Z. Define then

F

f~N • M ~ ~ N • M , ( ~ , x )

~ (~,fo(~)x);

G

a z x M --~ a z x M , ( w , x ) ~ ( r w , f o ( w ) x ) .

The two systems (f2 N x M, F) and (f2 Z x M, G) will act as important bridges between ergodic theory of random dynamical systems and that of deterministic dynamical systems. We now begin to discuss the invariant measures of X+(M, v).

A Borel probability measure # on M is called an invariant measure of vY+(M,v) if

D e f i n i t i o n 1.1.

L f#dv(f) = P

(1.2)

where ( f p ) ( E ) -- # ( f - l E ) for all E E 13(M) and f E ~. We denote by A4( X + ( M , v)) the set of all invariant measures of X + ( M , v). Since M is compact and v is a probability measure on F2, by Lemma 1.2.2 of [Kif]l we know that A/f(zl!+(M,v)) ~ r Furthermore, A 4 ( ? d + ( M , v ) ) can be proved to be a compact convex set with respect to the weak topology. In fact, for any #1,#2 E .A4( A'+(M, v)) and 0 _~ t < 1, it is clear that t/~l + (1 - t)#2 E A/t( A'+(M, v)), so 3//( A'+(M, v)) is a convex set. What remains is to show the compactness of it. Let A t ( M ) be the space of Borel probability measures on M equipped with the weak topology (see, e.g., Chapter VI of [Wall1), and assume that {P,~},~=I +oo is a sequence in A t ( A ' + ( M , v ) ) such that #,~ --~ # in A t ( M ) as n --* +oo. Since for every g E C~

J

;

/J

Vn E N,

and as n --, +oc

J

J

VfE~,

by the dominated convergence theorem we have

f g(x)dp(x) = f j g(x)d(f.)(x)dv(f). By the standard methods of measure theory (1.3) implies that

:jf.d~(f), 24

(1.3)

that is,

~ M( X+(M, v)). Hence .44( ?~+(M, v)) is a closed subset of J~4(M). This together with the compactness of A,~(M) shows that M/I( X+(M, v)) is compact. P r o p o s i t i o n I.I.

Let p E A 4 ( M ) . Then p E 2d( Ai+(M, v)) if and only if v N x # is F-invariant.

See Lemma 1.2.3 of [Kif]z for a proof. Define now projection operators P I : f ~ Z xM---*f~Z,

(w,x)~w,

p~:~Z xM~M, p:~Z

• M~N

(w,x)~-+x, • M, (w, x) ~ (w +,x)

where w + = ( f o ( w ) , f l ( w ) , . . . ) for w E f2Z. Then we have P r o p o s i t i o n 1.2.

For every p E AJ( • + ( M , v ) ) , there exists a unique Borel probability measure #* on ~ Z x M such that G#* = #* and P#* = v N • #. Moreover, PI#* = v Z , P 2 P * = P , and G~(v z x #) converges weakly to #* as

Proof. For every n > 0 let Bn denote the a-algebra {I-IZ~- i f~ x W : W E/3

(1-I+~ f2x M ) } and p* the probability measure G'~(vZ x #)it3,. Clearly B~ C B,~+I for all n > 0. Now we show that for every n > 0 tt*+l Iz= : #~.

(i.4)

Since B~ is generated by the semi-algebra Am = { (l-IZ~-1 ~ • I-IJ~ F i • I-[j+l ~) • Fi E B ( ~ ) , - n < i < j , j > - n , B E B(M)}, b y t h e extention theorems of measure theory (see, for example, Theorems 0.2, 0.3 and 0.4 of [Wal]l), in order to prove (1.4) it is enough to show that

,~+I(E) : , * . ( E ) ,

VE c X~.

(1.5)

Now for every E = (1-I-~-1 f~ x 1-IJ_. ri x I-[~++~fO xB e A,~, since, e M( ~+

25

(M, v)) we have

#~+1(E) =

(~Oz X

IA)(G-n-IE)

_-/

z

o...o

xIIF, xII -r,

/-I-[1 ~ -oo

X

~

F/ X +~176 H ~ #(f-n-l(w)-I

-n

/n Fi

o... o f_l(w)-lB)dv Z

j+l

1

j

.i+1

#(f-n-1 o.-- o

fZ~B)dv(f-,~-l)

dv(fi) -n

J #(f:~ o... o f2) B) H dv(Si).

J-EF,

(1.6) In the same way as above we easily see t h a t the last term of (1.6) equals # * ( E ) , (1.5) is then proved. Let Boo = Un+__~/3n. Obviously, /3oo is a sub algebra o f / 3 ( ~ Z x M ) and it generates /3(~Z x M). (1.4) allows one to define a probability measure # ~ on Boo by letting Now we show t h a t it is indeed a measure on/300. In order to do this it is sufficient to confirm t h a t for every decreasing sequence E0 D E1 D - - - of m e m b e r s of Boo +oo with A~=0 E , = r we have # ~ o ( E . ) --* 0 as n --* +oo. Suppose t h a t there exists a decreasing sequence E0 D E1 D ... of m e m b e r s +co of/3oo such t h a t N,,=oE,~ = r and p~(E,,) > Eo for some E0 > 0,Vn _> 0. T h e n +oo C Z + such t h a t E~ C Bzn for all one can choose an increasing sequence {/n}n=0 n >__ 0. Since fl is an open subset of C2(M, M), 1-I-l~ +oo f~ x M is a Borel subset +co

of 1-I-zn C2(M, M) x M, which is equipped with the p r o d u c t topology and is then also a Polish space. T h e n by the regularity of Borel probability measures on Polish spaces (see T h e o r e m 0.1.4), for every n > 0 we can find a c o m p a c t subset W~ of H + ~ f~ x M satisfying -I~-1

H f~xWncE,~ --00

26

and p;"

E,\

~ x W,

< 3-(~+:):0.

Put F,, =

ft x Wi i=0

)

for every n _> 0 , by a simple calculation we have

>__ 3 ' hence F~ ~ r Then by using the diagonalization process we can choose a sequence {(w. ,Xn)}n=l +oo such that ( w , , , x ~ ) E F,,,Vn >__0 and ( w n , x , , ) --* (wo,xo) +oo +oo as n ~ +oo. Clearly (w0, x0) E [~,,=0 iv, C Nn=0 E,,. This contradicts the fact -t-oo , that N,~=0 E,~ = r . Therefore, too is indeed a measure on Boo. By the Hahn-Kolmogorov extension theorem p * can be uniquely extended to a Borel probability measure on flZ x M and we denote it by p*. It is obvious that P#* = v N x # , P : # * = v Z,P2#* = # . Moreover, since for all E E Boo (G~,*)(E) = # * ( E ) , and Boo generates/3(ft Z x M), we have G#* = #*. From the discussion above it also follows clearly that G'~(v Z x p) weakly converges to #* as n ---++oo. On the other hand, from the above construction of #* we know that it is the uniclue probability measure on B(ft Z x M ) such that G#* : #* and P#* = v N x #. The proof of the proposition is then completed.

[]

From now on, X + ( M , v) associated with # E A~I( X + ( M , v)) will be referred to as X + ( M , v, #). Next we shall discuss the ergodicity of the invariant measures of X + (M, v). Let us first give a definition of this in a more intuitive way. A Borel set A E B ( M ) is said to be X + ( M , v ) - i n v a r i a n t if for #-a.e. x E M , x E A ::::::~f(~:) E A

for

x E AC :::::=~f ( x ) E A C

v - a . e . f E ft, for

v - a . e . f E ft.

More generally, a bounded measurable function g on ( M , B ( M ) ) A~+(M, v) invariant if g(f(x))dv(f)

= g(x), 27

#-

a.e.x.

is said to be

D e f i n i t i o n 1.2. An invariant measure It of ~d+(M, v) is said to be ergodic if every X + ( M , v)-invariant set has either #-measure 1 or #-measure 0 . We also call AZ+(M, v, #) ergodic when # is ergodic. P r o p o s i t i o n 1.3. Let X + ( M , v, #) be given. Then the following conditions are equivalent: 1) # is ergodic; 2) If g is an X + ( M , v, #)-invariant function, then g = consl.

#-

a.e,;

3) F : (f~N x M, v N x #) ~ is ergodic," 4) G : (f~Z x M,#*) ~-" is ergodic; 5) p is an extreme point of Ad( X + ( M , v ) ) . Proof. See [Kif]l for a detailed proof of the equivalence of 1), 2) and 3). We complete the proof of this proposition by showing that 3) < > 4), 3) ~ 5) and 5) ==~ 1). 3) ==~ 4). Let E E B(Q Z x M) be a set satisfying G - 1 E = E. By the approximation theorem of measure theory ( see Theorem 0.7 of [Wall]), for any given c > 0 there exists W C B(f~ N x M) such that

(1.7)

#*(E A P-zW) <6

and v N x # ( W A F - ' ~ W ) < e,

Yn > 0.

(1.8)

Since F is ergodic, by Birkhoff Ergodic Theorem in-1

lim

n--*+oo

nZxw~

vN x p-a.e.

k=v N xp(W)

(1.9)

k=0

Then from (1.8) and (1.9) it follows that /

)~w - v N x p ( W ) dv N x # < E,

thus v N x p(W) > ~(1 + x/1 - 2s) or

1

vm

x g(W) < 2 ( 1 - ,/1

-

2@

The arbitrariness ofe > 0 together with (1.7) yields that #*(E) = 1 or 0. Hence G is ergodic.

28

4) ==~ 3) . This follows easily from P#* = v N x # and P o G = F o P. 3) =:=ez 5) . Assume that # = tpl + (1 - t)#2 with #1,#2 E A 4 ( i u and 0 < t < 1. We have #1 < < P since t 5s 0. Let g = dpl/d# be the Radon-Nikodym derivative. Obviously v N x # l < < v N xtt and i = d(v N x #l)/d(v N x p)where i : f2N x M --* R + is defined by if(w,x) = g(x). Since v N x #1 and v N x p are both F-invariant, we have

vN

ioF=F

x

# - a.e.

and then ~" = const.

v N x # - a.e.,

thus

I t - a.e.

g = const.

So # = #1 = #2. The proof of 3) :=~ 5) is completed. 5) > 1). Suppose that # is not ergodic, then there exists an i~+(M,v,#) invariant set A0 with 0 < #(A0) < 1. Define measures #1 and #2 by

#I(A) = # ( 1 0 ) # ( A N A0), #2(A)

=

1 #(AC)P(A

MA C )

for all A E B(M). Since for every A e B(M)

/ f#l(A)dv(f) : ~

/ #(f-lANAO)d~)(f) 1

/ / XA(f(z))XA0 ( x ) d # ( z ) d v ( f )

-

#(do)

-

1 f f XA(I(x))XAo(I(x))d#(x)dv(I ) lz(Ao)

- ~Ao) / f#(AnAo)dv(f) = . 1 ~#(AnAo)= #I(A) and similarly

j fp2(A)dv(f) : p2(A), we know that #1,#2 E Ad( 2r

v)). Obviously =

+

this shows that # is not extremal. Thus 5) implies 1). [] Finally we touch briefly on the ergodic decomposition of an invariant measure of X + ( M , v). Let Ad( ~ + ( M , v)) have the smallest measurable structure 29

F

among those with respect to which every function of the form Og(it) = [ gdit J

on A d ( ? d + ( M , v ) ) is measurable, where g is a bounded function on M measurable with respect to any B,(M) which denotes the completion of B(M) for # E M ( ~'+(M, v)). The following theorem is adopted from Appendix A.1 of [Kif]l.

The set A4~ of all ergodic measures is a measurable subset of A4( A'+(M,v)) and each measure # from M ( X + ( M , v ) ) can be uniquely represented as an integral

T h e o r e m 1.1.

i

.

it =/.. pd-~,

(1.9)

e

where %, is a probability measure on f14( X+(M, v)) concentrated on .s (1.9) means that

it(A) = / ~

p(A)dT,(p) e

for all A E B(M), or equivalently (1.10)

/M g(X)dit(x) = ] t ~ /M g(x)dp(x)dT~*(P) for any bounded measurable function g on (M, 13(M)).

C o r o l l a r y 1.1. Let p C All( 2d+(M, v)). Then for any Borel function h on ~ N x M with h + E Ll(t'l N • M,v N x tt) and

hoF=h

vN x # - a . e .

(1.11)

we have h(w,x)= f h(w,x)dvN(w)

Proof.

vN •

Let Wh = { ( w , x ) : h(0a, x) r f h ( w ' x ) d v N ( w ) }

(1.12)

and it = f .

pdT,,

J

be the ergodic decomposition of #. By the standard method of measure theory it is easy to show that for any W E B(~ N x M ) , v N • p(W) is a measurable function on A4~ and

v IN x #(W) = f .

v N • p(W)dT,(p ). e

(1.11) and (1.13) yield that for 7 , - a.e. p E A/I~

hoF=h

vNxp-a.e. 30

(1.13)

and then by the ergodicness of F : (fiN x M, v N x p)

vN • Hence

vN xp(Wh)=

f . v N xp(Wh)d7~(p ) = 0 J2r e

which implies (1.12). []

w Measure-Theoretic Entropies of R a n d o m Diffeomorphisms In this section we discuss entropies (measure-theoretic entropies) of random diffeomorphisms. As in ergodic theory of deterministic dynamical systems, the entropy of a system generated by random diffeomorphisms should be a number which describes to some extent the "complexity" of the associated system. In the random case there may be different choices for the definition of the entropy, but in this book we are concerned with the entropy defined by Y. Kifer in [Kif]l for random transformations. We shall first give a review of this concept and then prove some useful properties of it. Let ~ + ( M , v , # ) be a system as defined in Section 1. T h e o r e m 2.1. For any finite measurable partition ~ of M the limit

o_1

)

= - aim

(2.1)

exists. This limit is called the entropy of ~ ' + ( M , v , p ) with respect to ~. Proof. Define g(x) = ( 0

( -xlogx

if if

x = 0 Let x>0 "

then

an+m ~.. / Hu /n+rn-1

31

"~

~

)

+SH.

(f2)

-1

V

(fk~w)-l~

dvN(w)

k=0

k----O = an + / i

Z g(#(('f~)-'C))dvN(w)dvN(w') C6~.(r

where ~m(w') = V~= m-1 k -1 ~. Since g is concave on [0, +oo), one has o (f.~,)

CE~(~')

= an + S

E

g(#(C))dvN(w')

~rt--i =

)

a n -~ a m .

Then from the proof of Theorem 0.4.1 we know that l i m n ~ + ~ li~a equals i n f , >-- l la [] n gt" D e f i n i t i o n 2.1. h,( zY+(M, v)) -=- sup h u ( z Y + ( M , v ) , ~ ) X + ( M , v,#), where the supremum is taken over the set

I'l

exists and

is called the entropy of of all finite measurable

partitions of M. From this definition we see that h , ( ~ + ( M , v)) describes to a certain extent the mean "complexity" of the dynamical behaviour of the systems {fo, f ~ , . . . } , w 6 f2N and it is a generalization to the random case of the entropy of a measure-preserving transformation in the deterministic ergodic theory. Now we proceed to prove some properties of the entropy h,(X+(M,v)). Henceforth in this book, for a given system X + ( M , v , / ~ ) the associated ~ algebra on f2N • M will be the completion of B(f2 N x M) with respect to v N • # and that on f2Z • M will be the completion of B(f2 Z • M ) with respect to/~*, except mentioned otherwise. We introduce now the space C2(M, M) N • M with the product topology. The general topology tells that it can be metrized to a Polish space. Since f2N x M is a Borel subset of C2(M, M) N • M and v N x # can be regarded as a Borel probability measure on C2(M, M ) N • M which is supported by f~N • M, then we know that (f2 N • M , v N • /z) is a Lebesgue space (see T h e o r e m 0.1.10). Similarly, (f2 Z • M, p*) is a Lebesgue space as well. Let F : f~N • M ~ and G : f2z • M ~ be as defined in Section 1 and let c~0,(r+ and ~ denote respectively the or-algebras {F • M : F 6 B(f2N)}, 32

{ r I _- 1 Q x V x M : r e B(1-I +~ a ) } and { U x M : r' e

aaz)}. Obviously, in

the sense as explained in Section 0.2 ~0, ~+ and a correspond respectively to the measurable partitions {{co} x M : co e a N) of a N x M and {1-I - 1 a x {co} xMco

e 1-I+~176 Q} and {{w} x M : w 9 a z} o f a z x M. We shall denote these

partitions still by the corresponding a0, ~+ and c~. T h e o r e m 2.2. The following hold true for zY+(M,v,#) : 1) If ( = { A I , . . . , A , } is a finite measurable partition of M and ~ = {ri,V2, ...} is a countable (or finite ) measurable partition of ~ N , then = h soN x , (F,r •

h,(X+(M,v),()

(2.2)

where ~ x ( = {Fi x A j : Fi E I , A j E (}. h,( ~'+(M, v)) = h:~q xu(F).

(2.3)

Proof. Since F preserves v N x #, F-lcro C ao and H v N x v ( ( x ~lCro) --- H~,(~) < +oo, the right-hand sides of (2.2) and (2.3) are well-defined (see Chapter 0). 1). Since for each w E f~N (v N x ~){wlxM = # and

n--1 k=O

where we regard {co) x M as M, then

n--1

)

This together with (2.1) proves (2.2). 2) From (2.2) it follows obviously that

h~,( 9[+(M,v)) <_ h vsoN • (F). Then what remains is to show that

hu( X+(M, v)) >__h : ~ x u ( F ) , and, by (2.2) and Remark 0.4.1, it is sufficient to prove that for every finite measurable partition a of fiN x M and every e > 0 there exists a measurable partition ~ of the type ~ x ( as explained above of f~N x M such that

hvNxu

- h Nxu 33

Since M is a compact metric space (with respect to the metric d( , ) induced by the Riemannian metric on M), one can easily find an increasing sequence of finite measurable partitions { ~ } , +=o1o of M such that Vn=l(, +oo is the partition of M into single points. Define ~ = { ~ N } • ( , , n > 1. By (3.8) of Chapter 0 one has as n ---~+oc H N• V #0) --~ 0. This together with 4) of Theorem 0.4.2 yields that when n is large enough

h:~qxu(F,o~) <_h:~qxu(F,~,~)+

HNxu(c~lfl,~ Vcr0)

•

The proof is completed. T h e o r e m 2.3.

[]

For 7Y+(M,v,#) it holds that

0"-t-(a) = h..(a) O" h.. and

(2.4)

o+

hoN•176(F) = h..(a)

(2.5) -I-oo

Proof.

Since G is invertible, G - I ~ + C (y+ and Vn=0 G'~cr+ = or, by Theorem 0.4.4 we know that (2.4) holds true. Let ~ be a finite measurable partition of fiN • M, from the relation P o G = F o P it follows that for every n E N

o_1

)

I

rt--1

)

Hu" p-a V F-k~lP-%r~ k=0

and then a+ ha oNxu(F,~)=hu.(a,P

Therefore h . aN• 0

a+

--1

(F) < hu. (G).

~). (2.6)

+oo Let {~n }n=l be an increasing sequence of finite measurable partitions of M such +oo is the partition of M into single points. Define ~,~ = {~Z} • ~,~, n > that V~=a~, 1 and let c > 0 be fixed arbitrarily. Since for every finite measurable partition of ~ Z • M, by 4) of Theorem 0.4.2,

34

and by (3.8) of C h a p t e r 0 H..(~IZ.

V ~) -~ 0

as

. -~ +~,

then one has for sufficiently large n

h~.(a) < h..(C,Z.)+~ Let ft, = { ~ N } x ~n,n k 1. Since p=lfl, = fl~, one has for every k E N

H N• u

\i:0

)

F-i/3.1~o =

Hr"

p-1

)

V Y-ifl'~lP-l~~

i=0

= H..

V a-~/~) ~+

\i=0 k-1

>- Hu" ( V G-i~'~la

) '

\i=O

hence for sufficiently large n _ >

oo hvN•

fln) _> h o~.(G, fln)

>_h~. ( a )

- c.

Since r is arbitrary, then

h~xu(F)>-h~*(G) which together with (2.4) and (2.6) yields (2.5). " []

If ~1,~2,'" is a sequence of countable measurable partitions of ~ N x M such that HvNxu(~nlc% ) < + ~ for all n E N and

Theorem

2.4.

diam (~n) = O,

lim

lhen ~o

hvNxu

(y)

=

9 hm h ~o~,T ( F , ~ ) . n-~+~ v~xu

Proof. This t h e o r e m follows from T h e o r e m 0.4.7. D Theorem

2.5.

If ~l,~2, "" is a sequence of finite measurable partitions of M

with lim

n---*-~- o o

diam(~n) = 0, 35

then hu(X+(M,v))=

lim h u ( X + ( M , v ) , ( n ) .

(2.7)

Proof.

Since f~N is a separable metric space, for each ~n we can choose a countable measurable partition (n of f~N such that diam(~n) _< diam((n). (2.7) follows then from Theorems 2.2 and 2.4 since lim diam((,~ x (,~) = 0 and HN• The proof is completed.

x (n]~0) = Hu(~,~) < +0%

VnEN.

[]

For X+(M, v, p), let p =/3a PdTu(P) be the ergodic decomposition of p, then h~..(G) = / ~ h~.(G)d'r,(p). (2.8)

T h e o r e m 2.6.

e

e

Proof. According to Appendix A.1 of [Kif]l and [Roh]3 and (1.13), we know that, if # = / . . pd%,(p) is the ergodic decomposition of #, then e

v N x p = f . v N xpdTu(p )

(2.9)

e

is the ergodic decomposition of (F, v N x #) and there exists a measurable partition r/ of gt N • M fixed under F such that (v N x p : p e A d e } = ( ( v

N X#)c:C 9

N x

M/O} mod 0,

where "rood 0" means that there exist a measurable set .M'~ C 2Me with %(3/[') = 1 and a measurable set ~" C ~ N x M/~ with (v N x #), (2-) = 1 such that {v N x p : p 9

N X#)c

:C 9

It is easy to see that P - l r / i s a measurable partition of f~Z x M fixed under G and {p* : p 9 :D 9 xM/p-I~7} mod0. (2.8) is then derived from Theorem 0.5.5. []

36

w Lyapunov

Exponents

of Random

Diffeomorphisms

The linear theory of Lyapunov exponents and the associated nonlinear theory of stable manifolds play a fundamental role in smooth ergodic theory of deterministic dynamical systems. In this section we introduce the concept of Lyapunov exponents of random diffeomorphisms and give some useful properties of these exponents. The theory of stable manifolds of random diffeomorphisms associated with these exponents will be discussed in Chapter III. Lyapunov exponents of random diffeomorphisms describe the exponential growth rates of the norms of vectors under successive actions of the derivatives of the random diffeomorphisms. The definition of them is based on the following proposition which can be easily derived from Theorem 3.2. P r o p o s i t i o n 3.1. exist numbers

Let • + ( M , v,l~) be given. Then for #-a.e. x E M there <

<..

<

(~O)(x) may be {-cx)}) such that for v N - a . e . w E a N there exists a sequence of linear subspaces of % M

(o}

=

V~(~

v2!. > c ..- c V.('(~)) - T ~ M

c

-

satisfying

lira

n~+oo

1 log IT~f2~I = ~(g)(x)

n

(i) \ v ( i - U) , 1 < i < r(x) . In addition, dim~),~) for a l l ~ E V,6o,~)~,6o,~ m i ( x ) depends only on x, 1 < i < r(x).

, )

D e f i n i t i o n 3.1. The numbers $(i)(x) : 1 < i < r(x) given m Proposition 3.1 are called the Lyapunov exponents of ? ( + ( M , v , p ) at x. The number m i ( x ) is called the multiplicity of A(O(x) . Before proceeding to discuss the properties of such exponents, we first review the well-known subadditive ergodic theorem which was first given by Kingman in [Kin]. It will play a very important role in this and the following chapters. T h e o r e m 3.1. (Subadditive ergodic theorem ) Let (X, 13, p) be a probability +c~ space and T a measure-preserving transformation on (X, 13, p). Let {gn}n=] be a sequence of measurable functions gn : X ---+R U {-cx~} satisfying the conditions: i ) Integrability: g + E L I ( X , 13, tt); 2) Subadditivity: gm+~ <_ g,~ + gn o T m #-a.e. for all m, n > 1. Then there exists a measurable function g : X ~ R U {-cr such that g+ E L I ( X , B , # ) , g o

1

T = g t z - a . e . , n~+oo lim -n gn = g # - a.e.

37

and

.JJ~ool /g'dp=infl /gnd#=

Now we formulate a kind of multiplicative ergodic theorem for random diffeomorphisms (Theorem 3.2) which is a reformulation in the case of zt'+ (M, v, #) of the Oseledec theorem stated in Appendix 2 of [Kat]. Let us first explain how the Oseledec theorem applies to the case of Pd+(M, v, #). Take a system of charts {(Ui, ~i)}[=1 which covers M such that for each x E Ui there exists an orthonorreal basis {e~i)(x)}~=~ 1 of T,:M which depends continuously on x E Ui, 1 < i < I. Put C i j ~-- {(W,Z) C ~ N x M : x C Ui,fo(w)x E Uj} for 1 _< i,j < l, and by C~,C~,...,C[2 we denote C l l , " - , C l l , " ' , C t l , ' " , C t z respectively. For every C~, assuming that C~ = C~j, if (w,x) E C~ then with respect to bases {e!i)(x)},'~~1 and {e!J)(fo(w)x)}~=~1 the map T, fo(w) can be expressed as an mo x mo matrix, written Ak(w, x). Then define A : ~ N x M ~ M(mo, R) by

A(w,z)=

{

A~(w,x)

if(w,z) eCi,

Ak(w,z)

if(w,z) eC'k\UC

k-1

r

r=l

where M ( m 0 , R ) denotes the set of all rn0 x m0 matrices with real entries. Obviously it is a measurable map and by condition (1.1) we have

log

IIA@, x)llodv N x t* < +oo

(3.1)

where 11. I10 denotes the usual Eucledean norm. Then we can apply the Oseledec theorem in Appendix 2 of [Kat] together with Proposition 1.1 and Corollary 1.1 to get the following multiplicative ergodic theorem for ~ + ( M , v,p). Some explainations about it will be given in Remark 3.1. T h e o r e m 3.2. For the given system zY+(M,v,p) there exists a Borel set A0 C f~N x M with v N x p(A0) = 1,FA0 C Ao such that: 1) For every (w,x) E Ao there exist a sequence of linear subspaces of T~M = V,(~

9

V,(r(~)) = T~M

(3.2)

and numbers AO)(z) < A(2)(z) <

(:r

may be { - ~ } ) ,

9 < A("(~))(z)

(3.3)

which ,tepend onty on x, such that lim n--.* + ~

-1 log[T~f,~l'~ = A(0(x) n

38

(3.4)

\W(i-1) , 1 < i < for all ~ E w(i) .(~,~)\.(~,~)

r(x),

and in addition,

aim 1 l o g [T~f~[ = ~('(~))(x),

(3.5)

n---* + o o ~2

~(')(x)mi(x)

lim 1 log l d e t ( T j ~ ) [ = ~

n--* + o o n

(3.6)

i

where mi(x) = dim V(~)'k ,x) - dim ~-xl) ) , , which depends only on x as well. Moreover, r(x),A(i)(x) and V((i),z) depend measurably on (w,x) 9 A0 and ~' Iw~V, (i) = V F(~,~) (i) r(fo(w)x) = r(x),A(i)(fo(w)x ) = ~(i)(x), T ~Jo~ ) (~,x)

(3.7)

for each (w,x) 9 h0, 1 < i < r(x). 2) For each

to denote

(~, ~) 9 Ao, we introduce p(1)(~) <_p(2)(~) < . . .

<

p(~o)(~)

~O)(~),..., ~(~)(~),..., ~(,)(~),..., ~(,)(~),..., ~(~(~))(~),...,

A(~(~))(x) with A(i)(x) being repealed mi(x) times. {~1,"" ,~mo} is a basis of T~M which satisfies

Now, for (w,x) 9 A0, /f

lim -1 log IT~f~il = p(i)(x)

n-*+oo

n

for every 1 < i < mo, then for every two non-empty disjoint subsets P,Q c {1,.-.,m0} we have lim l l o g T ( T j ~ E p , T ~ f ~ E Q ) = 0

n---* + c~ n

where Ep and EQ denote the subspaces of T~:M spanned by the vectors {~i}iEP and {~j}jeQ respectively and 7(', ") denotes the angle between the two associated subspaces. R e m a r k 3.1. 1) Given X + ( M , v , # ) , according to the Oseledec theorem in Appendix 2 of [Kat] we can first find a Borcl set A{) C ~ N x M with v N x #(A~) = 1, FA~ C A~ and with the following properties: For every (w, x) e A~ there exist measurable (in (w, x)) linear subspaces of T , M v(~ c V,(~,~) (1) c . . - c { o } = ,(~,~)

V~(~,~) (r('~

= T~M

and measurable (in (r x)) numbers

~(1)(~,~) < ~(2)(~,~) < . . < ~(~(~,~))(~,~) such that (3.4)-(3.6) and 2) of Theorem 3.2 hold true with ~(0(x), 1 < i < r(x) and mi(x) being replaced by ~(i)(w, x), 1 < i < r(w, x) and rni(oa, x) respectively, and that for each 1 < i < r(w, x)

r(F(w,x)) = r ( w , x ) , ~ ( i ) ( F ( w , x ) ) =

~(')(w,x),T.fo(w)V(~),.)= 39

V('~,.).

According to the subadditive ergodic theorem, from (3.1) and (3.5) it follows that A@(~,'))(w,x)+ E Ll(f~ N x M,v N x #), then by Corollary 1.1 we can easily find Ao C A~ satisfying the requirements of Theorem 3.2. 2) For each 1 < i < m0, the measurability of A(i)(z) and l~(i!,) on (w,z) E A0 means that the maps

~j:

{ ( w , x ) E Ao: r(x) > /,dim ~ ! ~ ) =

j}---* G r ( M , j ) , ( w , x ) ~

V((~I~),

i _< j < m0 are measurable, where Gr(M,j) denotes the Grassman bundle of M with the fibre at x E M being the Grassman manifold of j-dimensional subspaces of T~M. 3) Let x E M and E , E ~ be two subspaces of T~M. The angle between E and E ~ is defined by 7 ( E , E ' ) : inf{cos-~ I < ~,~' > I: ~ E E,~' E U,I~I = I~'l = 1} where < .,. > is the Riemannian metric on M. For x E M and 1 < p < m0, let (T~M)^, be the p-th ezteriorpower space of T~M, namely, (T~M) ^~ is the linear sp~ce of all linear combinations of elements in {~1 A ... A ~v : ~i E T~M, 1 < i < p} in which the following relations hold: (1)

~lA-..A(o~i+/3~)A-..A~v=

C~lA...A~iA..-A~p

+3&

A

. . . ^ ~

A

. . . A 4p

for a l l c ~ , / 3 E R a n d l < i < p ;

(2) & ^ . . . A ~ A . . . A ~ ^ . . . A G = - & A . . . A ( ~ ^ . . . ^ & ^ .

^~v

for all l _ < i , j _ < p . Obviously, if{~i : 1 < i < m0} is a basis of T,:M then {~x A . . A ( i , : 1 _< il < .-- < ip _< m0} is a basis of (T,~M)^,. Now, if {el : 1 < i < m0} is an orthonormal basis of T,:M, then by letting { l < eia A 9 9 9 A eip, ejl

A . 9 A ejv

>=

0

if ( i l , . . . , i p ) = ( J l , " ' , j v ) otherwise

we can define an inner product (., .) on (T~M) ^p, and it is clearly independent of the choice of the orthonormal basis {el " 1 < i < m0}. We shall denote also by 1" ] the norm on (T~M) ^~ induced by this inner product. Iff:M~MisaC 1 map, we define for x E M a n d 1
(T~f) ^, : (T~M) ^, --+ (TI(~)M) ^,,

~1 A - - - A ~ p ~ (Txf~l) A... A (Txf~p) 40

and define

~'~0

I(T.f)^l = 1 + E I(T*f)^'l p=l

Then a reformulation in the case of X + ( M , v , It) of an important conclusion from [RUe]l gives P r o p o s i t i o n 3.2.

Let ~ + ( M , v , I t ) be given. Then we have

lim -1 logl(T.f2)" ^ I= ~-~ ~(')(x)+m,(x)

v N xIt-a.e.

n---* + o o r t

i

and

lim 1 S l~

= i E J(i)(z)+mi(x)dIt"

n --+ + o o

i

Before going further, we first give a lemma which comes from Part III of [Kat] with a slight modification. L e m m a 3.1. Suppose that ( X , B , It) is a probability space and T is a measurepreserving transformation on (X, 13, It). Let g be a It-a.e. positive finite measurable function defined on X such that log + g o T 9 LI(X,B,It)" g

(3.8)

Then lim -1 logg(T"z) = 0

It - a.e.

(3.9)

n---* + oo n

and

f

log g o T dtl = 0. g

(3.10)

If (3.8) is replaced by log- ~ e Ll(X,13, It) where log- a = min{loga, 0}, then (3.9) and (3.10) still hold true. Proof. Since log + g - ~ E L I ( X , B, It), by the subadditive ergodic theorem the following limit exists It-a.e. lim

n--*-~-O0

1 ,,-_~1 g o T k+: 1_ g o T ~ a~f. log go~--s -,~-~+oolimn log g - h k=O

and moreover

goT.

41

where both sides may be - o o . Since lim

-I l o g g = 0

#

-- a.e.~

n ~ - { - cx) n

we have lim

'~)=h

llog(goT

n--+ + o o

#-a.e..

/l

On the other hand, from 0 < g < + o c # -

a.e. it follows that as n ---++oo

1 n converges to 0 in measure because T is measure-preserving. sequence {hi : i E N} C N such that - log(g o T ~)

lira --1 l o g ( g o T ' ~ ' ) = 0 i~+oo

Thus there is a

#-a.e..

n i

a.e. and proves (3.9) and (3.10). When (3.8) is replaced gOT LI(X, B,#), the discussions above remain true with log g

This implies h = 0 # by log- ~

E

and prove that (3.9) and (3.10) still hold true.

being replaced by - log ~

[]

The following result for ergodic case was given before in [Bax]l and Chapter V of [Kif]l. Now we prove it for the general case. We denote by Leb. the normalized Lebesgue measure on M induced by the Riemannian metric. 3.3. Let X+(M,v,#) J) E i ~(~)(x)mi(~) <_ 0 # - a . e

Proposition

be given. If# < < Leb. , then

2) EiA(i)(x)rni(x) = 0 # - a.e. if and only if ftz = # for v-a.e, f E ~. Proof. 1) In view of (3.7), in order to prove 1) it suffices to show that for any Borel set A C A0 with FA C A and v N x #(A) > 0 we have faE,~(i)(x)rni(x)dv N x # < 0.

(3.11)

z

Since for each (w, x) E f~N x M Idet(T=s

<

1s

mo

where Iflc, = max{lT, fl : x E M} for f E ft, from condition (1.1) it follows that log + I d e t ( T , s E Ll(f~ N x M,v N x #). Since F [ a : A --+ h preserves (v N x #)la, by the subadditive ergodic theorem

fa l~

x # = ~--.+~olim-nl fAlOgldet(T~f~)ldvN x # (3.12)

= LEi

~(O(x)mi(x)dvNx#. 42

If

f/A l~

x # = - c o , then (3.11) holds true obviously. Now we

log- I det(T.fZ)l e LI(A, (v N • Denote Leb. by ,~ and let q = d#/d,~, then

assume that

_

1

_

1

dv N x # -

v N x #(A)

v N x #(A)

q(x)dv N x )~

1

/aN~wq(x)d~dvN

1

faN ~(F-~A)~ q(f:x)ldet(T~fl)ld$dvN

1

/a

v N • #(A)

-

vN

x ,u(A)

-

vN

• #(A)

N

~F

q(f~x) ldet(r~f~)]d#dvN

-~h)~ q(x)

where A~ = {x : (w, x) e A} and (F-1A)~o has the similar meaning. From the concavity of the function log x it follows that = log v i x # ( i )

i

-'A)~

1 faN > v N x #(A) 1 -

vN

log

/ A [ log

q(x) [det(T~f~)]dpdvi q(x)

dpdv N

+ log ldet(T.yZ)]

q(flx) + log ldet(T~f~)[ ] dv N x t~. q(x)

• #(h)

(3.13) Since log +

q(f~x) < log + (q(f~x)]det(T~f~)])\ + (- log- I det(T~f~)]), q(x) \ q(x) /

we have

log+ -

-

(5 L I ( A , ( v

N

x #)IA).

Then by Lemma 3.1

A log ~q(f~x) dv N • p = 0 which together with (3.12) and (3.13) yields (3.11). 2) Since 1) holds true, ~ i

)~(i)(x)rni(x) = 0 It-a.e. if and only i f / E J

mi(x) d~ = 0. From the proof of 1) we know that this is equivalent to 43

,~(i)(x) i

1x

q(f~ )

detrT el~

= 1,

w h i c h m e a n s f # = # for v - a . e , f C ~ . []

44

v N x # - a.e.

Chapter II

Estimation of Entropy from Above Through Lyapunov Exponents

The relationship between entropy and Lyapunov exponents has been well studied in smooth ergodic theory of deterministic dynamical systems. A wellknown theorem in [Rue]l asserts that, if f : M --~ M is a C 1 map and p is an f-invariant Borel probability measure on M, then

h.(f) _
(0.1) i

where ~(1)(x) < ~(2)(x) < ... < A(r(~))(x) denote the Lyapunov exponents of f at x,m~(x), 1 < i < r(x) their multiplicities respectively and h , ( f ) the usual entropy of the system f : (M, #) ~ . (0.1) is sometimes called Ruelle's inequality. Our purpose in this chapter is to prove that Ruelle's inequality remains true for A/+(M, v, #) generated by random diffeomorphisms as defined in Chapter I, that is, the following result holds true: T h e o r e m 0.1. For any system ~ + ( M , v , # ) it holds that

h,( i~+(M, v)) <_ f f E )~(i)(x)+mi(x)d#" i

The Lyapunov exponents A(1)(z) < ... < A(r(~))(x) of X + ( M , v , p ) at x, their multiplicities mi(x), 1 < i < r(x) and the entropy h , ( r ~ + ( M , v ) ) are as introduced in Chapter I. Now we begin to prove Theorem 0.1 and we shall divide the proof into two sections.

w Preliminaries By the compactness of M there exists P0 > 0 such that the exponential map exp~ : {~ E T~:M : I~] < P0} --~ B(x, po) is a C ~176 diffeomorphism for every x E M, where B(x, po) = {Y E M : d(y,x) < p0}. The following l e m m a i s a basic fact of differential geometry.

For every 0 < ro < Po there exists b = b(ro) > 0 such that for every x E M and any y,z E B(x, r0)

L e m m a 1.1.

b-ld(y,z) <

l e x P ; 1 y --

45

exp~-1 z I < bd(y,z).

In o r d e r to prove the m a i n l e m m a ( L e m m a 1.3) of this section we need to i n t r o d u c e a new and equivalent definition of the C 2 - n o r m for C 2 d i f f e o m o r p h i s m s on

M.

Let T T M be the t a n g e n t space of TM. On M and T M we can i n t r o d u c e r e s p e c t i v e l y the R i e m a n n i a n m e t r i c s (I) and qJ to be defined in t h e following two paragraphs. Take a s y s t e m of c h a r t s {(Ui,~i)}i=l l on M with ~ i ( U i ) = B a ~ 1 < i< l such t h a t {V/ = ~P'i ~-lBm~ 1 ~, ] J i : l cover M , where B~0(O) = {{ 6 R m~ 9 I1(11o < r}. Let {Pi}~=I be a Coo p a r t i t i o n of unity on M s u b o r d i n a t e to {Ui, Vi, Y~i}[=l, i.e. all pi are Coo functions defined on M and satisfying Pi >_ 0 on M, pi > 0 on ~ , Pi = 0 o u t s i d e !o~-lB~n~ and P,iPi(X) = 1 for every x E M. For each 1 < i < l t h e bilinear form

Oi:(~,~)~---~<>o

for

~,rlET,:M

and

xEUi

defines a R i e m a n n i a n m e t r i c on Ui, and clearly pidPi c a n be e x t e n d e d to a Coo s y m m e t r i c bilinear f o r m on all of M which vanishes o u t s i d e ~ - l B ~ ~ b u t is positive definite at every p o i n t of V/. T h e n it is easy to check t h a t (I) = ~i=lPi~i~ l defined precisely by l

if

~,rl E T~M, x E M,

i=1

is a C ~ R i e m a n n i a n m e t r i c on M . Note t h a t {('Fu,M,T~i)}~=I is a s y s t e m of charts on T M , a n d for each 1 < i < I the bilinear form

qdi:(C,p) H<>o

for

(,pET(TM

and

(ETu, M

defines a R i e m a n n i a n m e t r i c on Tu, M. Now we define Pi : T M --* R , ( ~ pi(x) if ( E T~;M. T h e n Pi@i can be e x t e n d e d to a Coo s y m m e t r i c bilinear f o r m on all of T M which vanishes o u t s i d e T~o-(,B,~O(o)M a n d is positive definite at every p o i n t of Tv, M. Define q/ = ~i=lPiIuuwi 1 by 1

9((,P) = ~-~Pi(()~i(C,P)

if

(,P E

T~TM,~ETM,

i=i

then qJ can be easily verified to be a C ~ R i e m a n n i a n m e t r i c on TM. Note t h a t our discussions a b o u t L y a p u n o v e x p o n e n t s a n d e n t r o p y in this b o o k will not d e p e n d on the choice of a R i e m a n n i a n m e t r i c on M since any two such m e t r i c s are equivalent. Hence we shall assume t h a t the a s s o c i a t e d R i e m a n n i a n m e t r i c on M is always (I). For s i m p l i c i t y of n o t a t i o n we d e n o t e by I I b o t h the n o r m s on T M and T T M i n d u c e d by (I) and ~ respectively. Let dT( , ) d e n o t e t h e m e t r i c on

46

T M induced by q~. T h e n it is easy to see t h a t for every x C M and any E T~M we have dT(~,O) = I([ where 0 e T~M is the zero vector. For f E f2, define

Ilfllc= = sup{[T~Tfl: ~ E TM, I~l _< 1} where T~Tf is the tangent m a p of T f at ~ C TM. Let defined in Section 1.1. Proposition

Iflc=

(1.1) be the C 2 - n o r m

There exists a number Co > 0 such that:

1.1.

1) For every f E

Co~lflc ~ < ]tfllc~ < Colflc=;

(1.2)

sup{lT~Tf[: ~ E TM, I~l <_ r} < Corllfllc~,Vr ~ [1, +co);

(1.3)

2) For every f G

3) For any f,g C f~ IIf o

where Iglc,

gllc~ <_ Co m a x { I g l c , , 1}llfllc~llgllc~

(1.4)

supxEM IT~gl,

=

Proof. 1) Let {Ui, ~ , ~i}i=l t be as taken above in the definitions of r and q/. For each f E 12 define Ifl)~

=

sup m a x

xEM

i,j

max

:~(x))

l<_s,k,r<_mo

,

~(x

: x G Vi Cl f - l V j } where ( f ~ ( u ) , . . . , f~j~ = :j o f o w;l(u), u = (u 1 , U~o) T e R ~~ . T h e n it is easy to verify t h a t I" I~2 is equivalent to I" Ic 2. Since for every f C ~, i f x E k~ M f - l V j and ~ E T~M, the local expression of T~Tf with respect to the charts (To, M, T:~) and (TujM, T~j) is given by the value at the point T~i~ = (u,v) E ~i(Ui) x R "~~ of the following 2m0 x 2m0 m a t r i x function:

0f:~ ~

Of~'~~

OukOul vk k=l

mo --

a ~3

OukOul

Vk k=l

0U~0~mo V~ 2

mo

0 fii

Ou~O~mo ~ 47

0

""

0

0

-.

0

~Ofi"?~

~

0f:'~ o

and since on every TT~i(TTv, MTM) = T~i(Tv.M) x R 2m~ the norm [. [ induced by q is equivalent to the usual Euclidean norm 11" I10, i.e., there exists ag > 0 such that a}-l]~l _< Ilr]]10_< ail~l,v~ e TT~i(TTvMTM), we know that II" [Ic~ is equivalent to 1. ]bz and hence to 1-]c~. So there exists C1 > 0 satisfying

C f l l f l c ~ < Hfllc~ <- Cl[flc 2,

Vf E fi.

2) It can also be easily seen from the above discussion that one can find a number C2 > 0 such that for every f E fi

sup{[T(Tf[: ~ E TM,

KI < r} < C2rllflIc:, Vr 9

[1, +oo).

(1.5)

3) For any f , g 9 fi we have

Tg{~ 9 T M : ]([ _< 1} C {7/ 9 T M : [~/[ < Iglc'} and

T~T(I o g) = (TTe(Tf) o (TeTg),V( 9 TM, this together with (1.5) yields that

IIf o gllc~ < c2 max{Iglc,, 1}llfllc~llgllc~. Finally, letting Co = max{C1, C2}, we complete the proof. C] L e m m a 1.2.

For any given zu flog

we have

[[f,~[]c~dvN(w) < +oo, Vn

N.

Proof. Let n E N be given arbitrarily. By 3) of Proposition 1.1, for each w E fi N n--2

n-1

Ilf2[[c~ _< C~'-~ YI max{Ih(~)lc~, 1} I I IIA(~)llc~ k=O

k=O

and therefore n--2

log + IIf2llc= _< (-

-

n-1

1)log + Co + E l o g +

Ih(~)lc, + Y~log + IIh(~)llc=. k=O

k=0

Then by condition (1.1)in Chapter I and 1) of Proposition 1.1 we have log + E Ll(fi N, vN). The proof is completed, rn

IIf211c~

Let a > 0 be a number given arbitrarily . For each f E fi it can be easily verified that there exists 0 < r < m i n { ~ , 1} such that for every x E M the map 48

def H(],,) =

exp)-(lx) o f o exp, " {~ 9 T,;M: I~1 ~ r}

p0}

--+ ( r / 9 TS(~)M : I~1 is well-defined and

ITCH(:,. ) - ToH(:,.)I < a.

sup

(eT~M,l~l<_r

We denote by r a ( f ) the supremum of all possible r as explained above and call this the relation number of f with respect to a > 0. It is easy to show that ra(f) as a function of f E f* is lower-semicontinuous and so it is measurable with respect to B(f~). Let b0 > 0 be a number given by Lemma 1.1 associated with r0 = ,]P0 and let a0 = min{bo 1, 1}. One can easily check that for any f E ~, if x, y E M satisfy d(z,y) < rao(f), then

d(f(y), exp](~) OTxf o e x p ; 1 y) <_ d(x, y).

(1.6)

T h e following lemma will play an i m p o r t a n t role in the proof of T h e o r e m A.

Lemma

1.3.

For any given aV+(M,v) we have

-flogra(f2)dvN(~)

< +oo

(1.7)

for all a > 0 and all n E N. Proof. Let (U, ~) be a chart on M. In (U, ~) the system of geodesic equations is equivalent to the following

{ d~k ~ Vk dt mo

dt

+

i,j=l

1 < k < m0,

(1.8)

vj = 0,

where (u,v) E T ~ T u M = ~o(U) x R m~ and I'ikj(u),l _< i , j , k < m o are the Christoffel symbols (see, for instance, [Boo]). By S(u, v, t) = (a(u, v, t),/~(u, v, t)) we denote the solution of (1.8) with initial value (u, v) at t = 0 , then for every x E U and any 0 < r < P0 with B(x, r) C U the local expression of exp~ ITeM(r) in (U, ~) is e x p . : v ~-* ~(~o(x),v, 1), Vv E T ~ T ~ M ( r ) where T ~ M ( r ) = {~ E T~M : ]~[ < r}. By this property of exponential maps and by the definitions of the Riemannian metrics 9 and 9 one can find numbers 0 < r < 89 and A > 0 such that:

49

(i)

For every x 9 M, if~l,~p. 9 T . M ( r ) and ~ E T . M with I~l ~ 1, then

I(Te exp=) l < AI [ and

dT((T~I exp~)~, (Tr exp~)~) _< A[~I - ~2[[(], where Tr exp. : T ~ M ~ Texp= c M is the derivative of exp. at ~ E T ~ M ( r ) . (ii) For every x 9 M, if y, z E B(x, r), then for any 7/E T y M and ( 9 T . M

I(T v exp~-l)r/- (Tz

exp;~)(I_

(1 + I~])dT(~, ().

(iii) For any t > 0, if 77 9 T v M and ~ E T ~ M with ]q] _< t,](] < t and d ( y , z ) < 2r , then any piecewise C 1 curve in T M which is from 7/ to ( and whose length is less than 2dT(T/,() lies in {p E T M : ]p] < A t ) . Now for every f E ~ the following hold true for any x 9 M : (a)

H(I,~ ) = exp}-(1.) o f o expx :

{ rI 9 T . M

:

171

T

m a x { [ f l c , , 1}

}

--+ {( 9 TS(=)M : I(I _< r} is well-defined. and ~ E T,:M with 1 < (b) If ( t , ( 2 e rI 9 T,~M : 171 < m a x { l i l t 1 , 1 } l~I -< 2, letting y = exp~ (1 and z = exp~ ~2, we have

ITs, H(],,:)( - T(~ H(j,,~)(I =

[(Tl(y ) exp;(1 )) o ( T y f ) o (T{I expz)~ - (TI(z) exp}-(l~)) o ( T z f ) o (T{= exp=)g[

<

All + I(Tyf) o (Tr e x p = ) { l ] d r ( ( T J ) o (Tr exp.){, ( T z f ) o (Tg2 exp~)~)

<

All + If]c~Al~l]sup{[TcTfl:

( E T M , ](I -< 2A2}

92dT((T{~ exp.){, (T~= exp~)~) <-

8CoAS[ 1 +

Iflc,]llflIc=l~l

- <~t.

From this it follows that ITe, H ( j , ~ ) - Te=H(I,~)I _< 8COA5[1 + Iflcl]]]f]]c=14l -~21. For any given a > 0 we have now the estimate

r~(f) >_ min

8COA5( 1 + Iflc')llfllc2 ' m a x { I f l c l , 50

1}

(1.9)

which implies

-logro(y)

_< B + log + I f l c , + tog + tt/llc

where B is a number independent of f 9 ~2. Lemma 1.3 follows then immediately from Lemma 1.2. []

w Proof

of Theorem

0.1

The standard knowledge of linear algebra tells us that every m0 • m0 invertible matrix A with real entries can be decomposed as A = Q1 A Q 2 where A is a diagonal matrix with positive diagonal elements and Q1, Q2 are unitary matrics. We denote by 0 < 61(A) < /i2(A) < ... < 6m0(A) the diagonal elements of A. I f f E f~ and x 9 M, 6i(T,f),l < i <mo can be well defined by expressing T~f as a m0 x m0 matrix with respect to an orthonormal basis of T , M and TI(,)M respectively. Proof of Theorem enough, define

0.1. Fix n E N arbitrarily. For any given r > 0 small

rl =

9

: ,o0(f2) > ~ ~ i"} , k = 2 , 3 , . . N :~: < r ~o(f~)--< E-E-"

Fk={w 9

"

For every k 9 N, take a maximal ~:-seperated set E~ of M, i.e., a subset E~ of M such that if x , y 9 E~ with x ~ y then d(x,y) > ~ and for any z 9 M there exists an element z 9 E~ satisfying d(x, z) < ~. We then define a measurable partition a,k = {a~(x):x 9 Z~} of M such that c~(x) C int(a~(x)) and int(a~(x)) = {y 9 M : d(y,x) < d(y, xi) if x r xi 9 E~} for every x 9 E~. Furthermore, we take a countable measurable partition {Fkl, Fk2,.' "} of Fk satisfying diam(Fki) < ~ for all i 9 N. This is possible since f~N is a separable metric space. Put ~, = {Fki • a ~ ( x ) : x 9 Z ~ , k , i 9 N}. Obviously it is a countable measurable partition of f~N x M and diam(a~) < ~. Now we show that H v N x , ( a ~ ] a 0 ) < +co. In fact, HvN•

[cb)

=

H~'(a~)dvN(w) ~- E vN(Fk)log [a~l k

k=l

51

where I~1 denotes the number of elements in %k. It is easy to show that there exist C > 0 and to > 0 such that for any 0 < t _< to M contains at most [C(1) m~ disjoint balls with diameters t where [a] is the integral part of a > 0. Hence when is small enough we have la~[< c ( k ) m~ Therefore H Nxu(a~l~0)

+c~

+co

<

)m~ = l~ cc-rn~ + E

vN(rk) l ~

k=l Since

vN(rk)l~

k=l

.Jffoo

}--~vN(rk)logk

lograo(f~)dv N,

< log2 - [ k=l a{ . . . . . (]2)<~} by Lemma 1.3 we have H Nxu(a~lG0) < +oo.

(2.1)

Then by Theorem 0.4.3, Theorem 1.2.4 and 2) of Theorem 1.2.2 one has

nhu(

W+(M, v)) = ~__.0 N x uhl~176 i m ( r " , a,).

(2.2)

From the properties of conditional entropy (see Section 0.3 ) it follows that

V

H~Nxu

\k=O

)

F-kn~

= H Nxu(a~lO'o) + H N x

(

(F-'~c~la~Vo-o)+...

+H~Nx u Y-(I-1)'~~ V _< H N• De~ne g(~) = ,f 0 [ -xlogx

F-k~a~

)

Vo'o

k=0

])HNx,(F-n~l~.

)+(1-

(2.3)

Voo).

if x -- 0 by (2.3) we have if x > 0 '

~o (F . ~ ) hvNxu ,

<_ H N•

V~o)

fF k,i

<- E E k

where

(1.6)

1,j

n -1OJ k i n r - ~ r l J xEE~

Efr l rEEk

N~,k,'(w)

N

)

yEE~

#("~(x))l~ kVIT-nFI

is the number of elements of a~* which intersect

f~'~ak(x , e s, ,~ C f2 expz B(0, ~) k C expy2,:B(T=f~B(O,-s163 52

~ k f~a,(x). e

By

where

B(Q, 5) is the

TI2,;M

5-neighbourhood of Q c

in

T/2,~M.

If

1 n k ,~(v)n f2~(~) # r

we have

6

B(y, ~ll) FIexp.t2 , B(T~fnB(O, Then B(exp)-J,

8

6

g"

~), b0~ + b0 7) # r

E

6

~

y, bo'-~) ClB(T,:I2B(O , ~),bo~ + boT) # r

Since unitary operators preserve distances, it is easy to verify that the number of disjoint balls which intersect B(T,:J:"~B(O,~), bo~ + bo~) and whose diameters b-1~0 l does not exceed 61 1--[~~ = max{Si(T,;f2), 1 } m a x { / , 1} where 61 is constant depending only on b0 and m0. Therefore are

<

<

[ m0 Oll-Imax{ i(T,f:),l}

E E E jr, o._.., k

l

3

xEE~

i=1

logC1+ ~ ~ vN(rk n ,-~rz) logmax{~, 1} k

l

mo

+ E / H~i)(w'y)dvN x # i=1

de].

=

log C1 + A1 + ZS2

k where H(i)~w~ v , y) = log + 5i(T,:f2)if (w,y) E rk x %(x). From the definition of F~,k E N one can deduce that for any k,l E N

vN(i,k n r-"i,,) = vN(I'k)vN(Fz), this together with (2.1) yields A1

< }--"~ vN(i,k)vN(i,z)(log k + log/) k

l

= 2~vN(i,~)logk k

<_

2 log 2 - 2 f{

log to0 (/2)dv N. . . . . o(::)_<~}

By Lemma 1.3 one has limsup/~1

~

71---~ ~- OO

53

21og2.

(2.4)

As for /k2, since for every 1 < i < m0

]H!i)(w, y)[ < sup log + 61(T,:f~) = log + [f~[c~ xEM

and lim

H!i)(w, y) = log + 5i(Tu]2)

for any (w, y) G ~ N • M, by the dominated convergence theorem rn o

lim A 2 ----E

6i(TYf~)dvN • It.

f l~

~ 0

(2.5)

i=1

Since for every (w, y) E O N x M 1Tt o

I(

=)

I=

II

6i(Tyf2),

1 <_p < too,

i=rno-p+ l

one has rn o

I(Tyf2)^l > H max{Si(Tu]•), 1}.

(2.6)

i=1

Letting e -~ 0, by (2.2)-(2.6) we get

nh~,( X+(M, v)) <_log 61 + 2 log 2 + / log t(Tyf~) ^ IdvN x # , then by Proposition 1.3.2

h.(

v)) _
[-q

R e m a r k 2.1. Ruelle's inequality for random diffeomorphisms was first considered by Y.Kifer for ergodic case (Theorem V.1.4 of [Kif]l), but his proof of that theorem seems to be unacceptable. For example, the key estimate (1.23) on Page 164 of [Kif]l seems to be a nontrivial mistake. This led the authors of the present book to an essentially different treatment of this problem in the paper [Liu]l and this chapter comes from that paper. In our treatment here we make the assumption f l o g + [f[c2dv(f) < + ~ rather than f l o g + lfIcldv(f) < +c~, which was set down in Kifer's theorem. A recent preprint [Bah] shows that our this assumption is extraneous and Theorem 0.1 also holds true under Kifer's assumption (see the Appendix for the argument). However, the treatment in this chapter (especially the introduction of the relation number r~(f) and the related estimates) will be, besides its own right, very useful in the later chapters.

54

Chapter III

Stable Invariant Manifolds of Random Diffeomorphisms

In the development of smooth ergodic theory of deterministic dynamical systems, one of the remarkable landmarks is Yd. Pesin's work [Pes]l which translated the linear theory of Lyapunov exponents into the non-linear theory of stable and unstable invariant manifolds. Pesin developed there a general theory concerning the existence and the so-called "absolute continuity" of invariant families of stable and unstable manifolds of a smooth dynamical system, corresponding to its non-zero Lyapunov exponents, and thus paved the way to deep results in ergodic theory of arbitrary diffeomorphisms preserving a smooth measure ([Pes]2). A theorem concerning the existence of such families was proved later by D. Ruelle ([Rue]2) for dynamical systems preserving only a Borel measure, through a rather different and in some sense more profitable approach. Pesin's above work has also been generalized to a broad class of dynamical systems with singularities ([Kat]) along his original scheme but in a technically much more detailed way. The purpose of this chapter is to present an extension of Pesin's work [Pes]l to the random case, i.e. to carry out along Pesin's scheme some results concerning the existence and absolute continuity of invariant families of stable manifolds for random diffeomorphisms. Let us emphasize that Ruelle's theorem mentioned above can also be adopted to our present situation by a trivialization argument to obtain the existence of the stable manifolds. However, in this chapter we will prove some more subtle results, which besides their own rights are necessary for the treatments of Chapters IV, VI and VII, and deal with the absolute continuity problem. Although we will not provide here the detailed proof of our absolute continuity theorem ( Theorem 5.1 ) because it involves too much work and is completely parallel to the treatment in Part II of [Kat] of the deterministic case, we will present in Section 4 a detailed discussion of the H61der continuity of tangent spaces of the random stable manifolds and this will naturally lead the reader to the absolute continuity theorem. Results of this chapter will play a fundamental role in Chapters IV, VI and VII.

w Some Preliminary Lemmas Let ?d+(M, v,#) be given as in Chapter I, and let [a, b], a < b _< 0, be a closed interval of R. Denote by A~,b the subset of A0 (see Theorem 1.3.2) which consists of points (w, x) such that ,~(i)(x) ~ [a, b], i = 1 , . . . , r(x) . It is clear that FA~,b C A~,b. For (w, x) E A~,b and n, l E Z + we sometimes use the following

55

notations:

E.(~,~) = T~I2E0(~,~),

f~

H.(~,~) = T.f2Ho(,O,x),

f~(w)=f~+t_l(w)o...ofn(w),

n > 0;

l>0;

u~(~, ~) = T~(~, ~)1..(~,~) We now fix arbitrarily k E { 1 , . . . , m 0 } and 0 < c _< min{1, (b - a)/200} and assume that the set A~,b,k dr {(w, x) E A~,b : dim Eo(w, x) = k} r r 1.1. There exists a measurable function l : Aa,b,k x Z + ~ (0, +oc) such that for each (w,x) E Aa,b,k and n,l E Z + we have

Lemma

2) Iu~(~,~),71 _> l(~o, ~,,~)-~e(b-r

e

H~(w,x),"

3) 7(E~+~(~, ~), H~+~(~, ~)) > l(~, ~, n)-~e-~; ~) I(w, x, n + t) < l(w, x, n)e ~. Proof.

Let (w, x) be a point in A~,b,k and let n, l E Z +. We choose a basis mo {(i}~__~1 of T, M such that {~i}~=1 C E0(w, x), {(j }j=k+l C Ho(w, x) and for each lira

m~+oo

--1 log lT, f~m~il = P(O(x)

(1.1)

m

where p(i)(x), 1 < i < m0 are as introduced in T h e o r e m 1.3.2. According to Theorem 1.3.2, one has for every two n o n m m p t y disjoint subsets P, Q c { 1 , . . . , m0} lim

m--*+co

--1 log T ( T , f ~ Ep, T~f~ EQ) = 0 m

where Ep and EQ denote respectively the subspaces of T , M spanned by {~i}ieP and {~j}jeq. From this it follows that A ( w , x , n ) da = inf inf y~ - ' T ~j~ ~:,~+rEp, T,~f~+rEQ)e2'~o P,QrEZ+

> 0,

(1.2)

and A(~ , ~, ~ + l) > A(~, ~, n ) e - 2,~o "---'.

56

(1.3)

Particularly, we define

l~(w,x,n)= inf 7(E,~+~(w,x),H,~+~Oz,x))e~o , --K..-~ r

feZ+

then it is an everywhere positive measurable function on A~,b,k x Z +. Let m E Z +, For each ( = ~ i ~iT~f2~(i E Ts2~M, from (1.2) one has

i

<

[ 4 A ( w , x , m ) - l ] "~~ ~ .

(1.4)

oqr~z2e~

B(~,a:, m)KI where B(w,x,m) = [4A(w, x, m ) - l ] '~~ Indeed, let E be a vector an inner product ( , ), and let I1 II be the norm deduced from rL ( E E satisfy 7(r/,-t-() >__q-l, an simple geometrical consideration I1~11+ I1~11-< 4qll~ + ~11. Then an easy induction yields (1.4). It follows from (1.1) that there exists C(w, x, n) > 0 such that and r E Z +

c(~, ~,-)-*e(/')(*)-~ > _
< C(~o,~,,~)e(/~

space with { , ). If shows that for each ~

(1.5)

By these inequalities a simple computation yields that for any r, s E Z + we have

(1.6) for each

{i E Eo(w, x) and 9 t%f2n § 5Jl < C(w,x,n)2l%f2+'+~Sal e-(b-z)~+}~

(1.7)

~j E Ho(w,x). From (1.4) and (1.6) it follows that for each ~ = ~ i ctiT~:f~fi E E,(w, x) and any r, s E Z +

for each

I:K+~(~, x)~l = i

i

< < Thus the function

{ IT~+~(~'*)4[~-(~

~-~ 57

z+

E.(~ .)}

is finite at each point of A~,b,k. By the same way one can show t h a t the function

is also finite at each point of Aa,b,k. Finally, we define

l(w, x, n) = max{ll(w, x, n) -1 , 12(w, x, n), 13(w,x, n)}. T h e n it is easy to verify t h a t it is a measurable function on A~,b,k x Z + satisfying 1) 4). [] lI

Let l ~ _> 1 be a n u m b e r such t h a t the set A~,b,k, ~

~(~, x, 0) _< r} r r

~~ {(~,r

~ Ao,~,k :

l! Eo(w, x) and Ho(w, x) depend continuously on (w, x) E Aa,b,k, ~.

Lemma

1.2.

Proof.

If {(w,~, Xn)}n=l + ~ is a sequence of points in A~,b,k, " ~ such t h a t (w~, xn)

I' converges to (w, x) E A,,b,k, ~ and Eo(w,,x,~) converges to a subspace of T~M as n --~ + o a , by 1) and 2) of L e m m a 1.1 one can easily verify t h a t this subspace coincides with Eo(w, x). From this the continuity of Eo(w, x), and also of 1j Ho(w, x), with respect to (w, x) E A~,b,k, ~ follows obviously. [] 1I

Let (w, x) E A~,b,k, ~ and n E Z +. L e m m a 1.1 allows us to introduce an inner p r o d u c t ( , )(~,~),~ on Tf2~M such that ~-oo

<~,~'>(~,~),- : E e-~(~

~, ~)(, si(~, ~)~'>, ~ , ( ' E E,~(w,x),

(1.8)

l----0

I [w ,X)]-I~, [UI-I(02,x)]-lTl/), (71,17 )(w,x),n = ~/ ~ e2(b-2E)l/ru \L n-lk l--0

~, 71' C H.(w, x),

(1.9)

and E,~(w, x) and H,~(w, x) are orthogonal with respect to ( we define a n o r m II" II(~,.),n on TSo,:M such t h a t

,

)(~,x),,~ 9 T h e n

II~licw,x),. = [(~,~)(w,~),.]~,.' c E~(~,x),

(1.1o)

UT/l[(~,~),. = [(~, ~)(~,.),,~] 3 , 7 / c H,~(w, x),

(1.11)

IlCll(~,~),.~ = max{l]r

IIr/ll(~,~),.},

r = [ + ~ C E.(w,x) @ H.(w,x).

58

(1.12)

The sequence of norms {H "ll(~0,.),~}n=0 +oo is usually called a Lyapunov metric at the point (.~, x). It follows from Lemma 1.2 and (1.8), (1.9) that for each fixed n 9 Z + the inner product ( , )(~,z),~ depends continuously on (w,x) 9 Aa,b,k,E. 1~

L,,mm~ La.

Let (~o,,:) 9 Ao,b,k,,. rh~n th~ seq,,~nce of .orm~ satisfies for each n 9 Z +

-boo {ll'll(<:> L=0

2) IlVa(~,~).ll(~,.),~+l ___eb-:~ll'lll(,.o,.),,,., '7 9 U,~(w,x)," 3) 11~1 _< I1~11(~,=> _< ae~'"14l,~ 9 Tfg,,M, where A = 4(/')2(1 - e-2') - 89 Proof.

For each ~ 9 En(w,x) we have :t

IIS&(~o,~)gll(~,.),~+l =

~-2(~+2~)z,~,~+,,-,cz t ~)Sa(~o,~)gl =

= e~

[~--'(~ ' ~

'+~)'~ ,oo ,~,x)~:l~]89

and hence 1) holds. Similarly for each r / 9 H,~(w,x)

IIUa(~o, ~),711(~,=),,~+x [,,+1 =

|~'~ e2(b-2e)lJ[Ul {w x V l - l u l / o j /~.~ IL n + l - - l k , )J nk

] 89 ,Z)T]I 2

L/=O !

=

IUa(~,~)~l 2+~e

2(b-2s)l

/--1 l[u'n_(,_l)(~,~)]

--i

vl 2

1----1 I

-- -

Ig~(,.,.,,~,)~12 + d (b-2~) ~,~2(~'-2~)('-") I trun-1 ,,~ ' x)1-1,712 n-(l-1)\ /=1

_> eb-2~llnll(~,=> which proves 2). Now let ~ E Tfa,:M and write ~ = {+~7 with { E E,~(w, x) and 779 Hn(w,z). From (1.8)- (1.12) it can be easily seen that

KI < Ig[ + 1,71< Ilell(~,.),n + 11,711(~,,~),.< 211r which implies the first inequality in 3). We now prove the second one. By 1) and 4) of Lemma 1.1 and (1.10)

LI:0

59

and similarly, by 2) and 4) of L e m m a 1.1 and (1.11),

< [l'(1 -

e-2~)-&]e"~l,71

/=0

Since

-r(/~,,(~, ~), H~(~, ~)) >_ (l'~'") -~ we have

<

+ ol = Ze2~lCI.

[l'(1 - e-2~)-89

[]

The proof is completed.

Finally we prove the following important lemma. We use Lip (.) as usual to denote the Lipschitz constant of a Lipschitz map, and the norm we use is I " I except mentioned otherwise. L e m m a 1.4. There ezist a Borel set Fo C f2N (independent of ~) and a measurable function r : Fo --* (0, +oc) such that vN(Fo) = 1,rFo C Fo and the following hold true: 1) For each w E Fo and x G M, the map

F(w,x),O de_..fexp~o}co)z ofo(w) o exp:: " T~M(r(w) -1) ~ T/o(~o)zM is well defined and Lip(T.F(o~,~),o) <_ r(w), where T~M(r(w) -1) = {~ E T r M : [4[ < r(w) -1} and T.F(~,~),o T[r@,~),o,[ E T ~ M ( r ( ~ ) - I ) ; 2) r(Tn~) < r(~)e~', n e Z+,~ ~ to.

:4t

,

Proof. From the proof of L e m m a II. 1.3 we see that there exist numbers r > 0 and C > 0 such that for each f E ~ and x E M the map g(/,~) = exp/(1 ) o f o e x p , _

{

4 e T~M "141 -<

max{ff-]c~,r l} }

--~ T/(,:)M

makes sense and Lip(T.H(],~)) _< Cmax{[f[c1,1} max{[[f[[c~, 1}. Define now r I : f t N ---, (0, +oo) by the formula

r'(w) = max {r - t max{lfo(w)[c, , 1}, C m a x { l f o ( w ) l c l , 1} max{llfo(~o)llc=, 1}}. 60

Then it follows that for each w E f2N and x G M the map F~,x),0 clef expy_o}~)~ ofo(w) o e x p , Tr

-1) ~ T]o(~)xM

is well defined and Lip(T.F~,~),0) <_ r'(w), and moreover, by condition (1.1) in Section 1.1, logr' E L I ( f l N , B ( f l N ) , vN). According to Birkhoff ergodic theorem, lim

llogr,(rnw)=0

n--* + o o

vN - a . e .

r~

Then there exists a Borel set F0 C f~N such that vN(F0) = 1, vFo C F0 and for each w E F0 lim

-I logr'(T~w) = O.

(1.13)

n---, + oo n

From (1.13) it follows that

def=

:

>_ 0}

is finite at each point w E F0. Then it can be easily verified that the Borel set P0 and the function r : I'0 ~ (0, +oc) satisfy the requirements of this lemma. []

w Some

Technical

Facts

About

Contracting

Maps

In this section we introduce some additional preliminaries which consist of some technical facts about contracting maps. We first have the following simple result. L e m m a 2.1. Let X and Y be complete metric spaces and let X • Y have the product metric, i.e. d((x, y), (x', y')) = max{d(x, x'), d(y, y')), Let O : X x Y ~ Y be a continuous map. contraction (A < 1) on the second factor, i.e. d(O(x, y), O(x, y')) <__Ad(y, y'),

(x, y), (x', y') E X • Y.

Suppose that O is a uniform AVx E X, Vy, y' E Y.

For e a c h x E X , denote byO~ the map: Y ~ Y , y , ,O(x,y) and l e t ~ ( z ) be the unique fixed point of O~. Then the following hold true: 1) The map ~ : X ~ Y, z t , ~(x) is continuous; 2) When O is Lipschitz, ~ is also Lipschitz with Lip (~) < T~-~xLip(O). Moreover, if Lip(O) <_ )% then Lip(~) <_ A. 61

Proof.

Let x and x' be two points of X. Then

d(O(., ~(x)), 0(~', ~(.')))

d(~(.), ~(~')) =

<_ d(O(x, ~(x)), O(x', ~(x))) + d(O(x', ~(x)), O(x', ~o(x')))

d(O(x, ~(~)), 0(x', ~(.))) + Ad(~(.), ~(x')).

<

(2.1) Thus

1

d(~(x), ~(z')) <_ ~ _ ~ d(O(x, ~(z)), 0(z', ~(z)))

(2.2)

from which 1) follows. When 0 is Lipschitz, from (2.2) we obtain d(~o(x), ,^t

1

~tx,))< 1 - ALip(O)d(x'x')"~

Hence ~v is Lipschitz with Lip (~v) < ~-~Lip(O). Moreover, if Lip(O) < A, we have

d(~(~), ~(~')) = <

d(O(~, ~(~)), 0(x', ~(~'))) A max{d(x, x'),d(~(x),~,(x'))}.

Since A < 1, we obtain

d(~(.), ~(x')) ___~d(x, .'). Thus 2) is proved.

[]

L e m m a 2.2. Under the circumstances of Lemma 2.1, we suppose moreover that X and Y are closed subsets of two respective Banach spaces ( E, II . II) and (F,I[. I1) and that 0 is Lipschitz. We restrict below 0 and ~ respectively to int(x • Y) and int ( X ) n ~-xint(X), where int() denotes the inte,~or of a set. If O is of class C 1, then ~ is of class C 1. In addition, ifT.O is Lipschitz, then T.7~ is also Lipschitz with Lip(T.q~) <_ CLip(T.O), where C = (1 - A)-3[1 + Lip(O)] 2.

Proof. Let (x, y) C int(X x Y). Denote by T10(z, y) the partial derivative of 0 with respect to the first factor at the point (x, y).T20(z, y) is defined analogously. We first remark that [[T20(x, y)[] _< A since 0 is a uniform A-contraction on the second factor. Consequently id-T~0(x, y) is invertible and its inversion is given by the formula +oo

[id - T20(x, y)]-i = ~-~[T20(x, y)]n.

(2.3)

r~-~-O

We now prove that ~o is of class C a with

Tx~a = [id - T20(x,

~(x))]-lTlO(X, ~O(x)). 62

(2.4)

In fact, (x, ~o(x)) 9 int(X x Y) since x 9 int(X) r) ~o-lint(V). Let A x e E such that x + Ax E int(X), then we have

II~o(x + A~) - ~o(~) _<

II[id -

[id

- T~0(x, ~o(~))]-~T~0(x, ~o(~))A~ll

T~O(x, ~o(x))]-~llll~o(~

[~o(x + A~)

-

~o(~)] -

TlO(~,

+ A~) - ~(r

- T~0(~, ~o(~))

~o(x))A~l I

_< II[id- T~0(x,~(x))]-~tlll0(:~ + A~,~o(x + Ax)) -- 0(x,~(~))

This last expression is o(ll(Ax, ~(x + ZX~) - ~(~))11) and is hence o(llA~ll) since ~o is Lipschitz. Therefore, the derivative of ~o at x exists and is given by formula (2.4). From (2.3) and (2.4) it follows that T,~o depends continuously on z E int(X) M ~o-lint(Y). Thus ~o is of class C t. Now suppose that T.0 is Lipschitz. Let x, x' E int(X) M ~o-lint(Y), then by (2.3) and (2.4) we have

IIT~m - T~,mlI +oo

-<

+oo

~_-o[T~~

-,,=o~--~'[T~0(~"~'(x'))]n IlT~0(~,m(x))ll

+ ]~=o[T20(x',~(z'))] ~ _<

(n + 1)A"

IITIO(Z,~(z)) -

T~O(x',~(x'))[l

Lip(T.O)Lip(O)d((x,~o(x)),(x', ~(x')))

+ [+~=oA'~]Lip(T.O)d((x,~o(x)),(x',~o(x'))) <

(1 - A)-3[1 +

Lip(O)]2Lip(TO)d(x,x').

Thus T.~o is Lipschitz with Lip(T.~o) < (1 - A)-3[1 + Lip(0)]2Lip(T.0).

w Local and Global Stable M a n i f o l d s Let C, r ~ >__max{ 1,2p~ 1}, be a number such that the Borel set It r t

def

Aa;b,k,~ = {(w,x) E

Aa,b,k,e :W E FO i~

63

and

r(o~) < r'}

E]

~l rJ

is not empty. In this section we fix such a set A~,b,k, ~ and confine ourselves to it. We shall write A s = At', ~' a,b,k,E

for simplicity of notations. Before formulating the main results of this section, we first recall the definition of a continuous family of C 1 e m b e d d e d k-dimensional discs. 3.1. Let X be a metric space and let {D~}~ex be a collection of subsets of M. We call { D x } z e x a continuous family of C 1 embedded kdimensional discs in M if there is a finite open cover {Ui}[=l of X such that for each Ui there exists a conlinuous map Oi : Ui ~ E m b l ( D k , M ) such that Oi(x)D k = D~,x E Ui, where D ~ = {~ C R k : 1[~[[0 < I}. Definition

A b o u t local stable manifolds of r a n d o m diffeomorphisrns we have the following 3.1 . For each n E Z +, there exists a conlinuous family of C 1 embedded k-dimensional discs {Wn(w, x)}(~,~)eh, in M and there exist numbers c~,~,fl,~ and 7,~ which depend only on a,b,k,g,F and r s such that the following hold true for every (w, x) C A~: 1) There exists a C 1'1 map Theorem

where O,~(r x) is an open subset of E,~(w, x) which contains {~ e En(w,x) : < i)

such that = o;

it) Lip(h(~o,.),,~) < fl,~, Lip(T.h(~o,x),n) <_ fl,~; iii) Wn(w,x) = e x p . Graph (h(~o,.:),.~) and W n ( w , x ) is tangent to E n ( w , x ) at the point f ~ x ; O f,,(w)W,,(w, ~:) C W,,+i(w, x)," 3) d'(fln(w)y, fln(w)z ) < 7~c(a+4E)'d'(y,z),y,z G W,~(w,x),l E Z +, where d'( , ) is the distance along Wm(w,x) for rn E Z + ; 4) 4,+1 = ~.e-5~,r = fl.eT~,7.~+l = 7~e 2~. Proof. We complete the p r o o f by several steps. Step 1. Let CO ~-- e a+4r -- e a + 2 r

e0 = 4 A / e 2E,

ro = Co1r .

T h e n for every (w, x) E A ~ and l E Z +, by 3) of L e m m a 1.3 and L a m i n a 1.4 we have t h a t def

F(~,x),z =

- 1

e x p y ~ + , ofz(w) o exP.t.~ :

64

is well defined and LiPll.ll(T.F(~,~),l ) < c0e 3~z, where LiPll.ll(. ) is defined with respect to I1' II<~,~),z and I1 11(~,~>,,+~- Then from this it follows that LiPll.ll(F(~,~:)3 - ToF6o,~:)3 ) <_ Co. Step 2. Let (a),x) E A~ and n C Z + be fixed once and for all from this step to Step 5. We sometimes denote I1" I1(~,~),~ simply by I1 IIzLet t = e ~+6~ and define

r(z) =

{~ = {~(z)}+~ .~(t) 9 z.+,(w,~),~ >_ 1, [i

ii

def

and u~H = sup Ilt-'~(l)ll=+z < +oo}, />1

r(~) =

{~ = {~(t))+__7 : ,(l) ~ ~/~+,@, ~), t _ 0, Ii

ii

def

and [[rll = sup Ilt-%(/)[[,~+~ < +oo}. 1>0

It can be easily verified that (F(E), I1"II) and (F(H), I1"II) are both Banach spaces with respect to the natural operations of addition and scalar multiplication. Define X -- {a e r ( Z ) : II~ll_ r0e-ar

z = {~ e E~(~, x): I1s

<- r0~-3'~} -

They are obviously closed subsets of F(E), F(H) and En(W , x) respectively. Step 3. For the point (w,z), we denote F@,~),m simply by F,~ and denote Utm(w,z) by UZm for m , l C Z +. We now define

o:z•215215

(~,~,~),

,(~',~')

where

{

# ( 1 ) = wlF=(~, r(0)), ~rt(/) = 7 r l F n + / _ l ( O ' ( / -

1), r ( l - 1)),1 > 2,

r'(O) = [U1]-l[T(X) + U ~r ( O ) - 7r2F~(~, r(O))], Tt(l) [Ul+l]--l[T(l "4- 1) + U~+t'r(l ) -- 7r2F,~+z(a(l), r(/))], l >_ 1, =

where 7rl and r2 are respectively the projections of Era(w, z) @ Hm(w, z) to Em(w,z) and H m ( w , z ) , m E Z +. It is easy to verify that 0 is well defined and is a Lipschitz map satisfying Lip(0) <_ e -2~ 65

with respect to the product metrics on Z x X x Y and X x Y. We now show that 0 is of class C 1 in int(Z x X x Y) and T.O is Lipschitz. Let (~0, a0, to) be a point in int(Z x X x Y). Define a map A((0,~r0,vo) : E~(w,x) x F(E) x F(H) ---, P(E) x C(H),((,cr, r) ~ , (a", r") by letting

{ ii(1) =

cr (l)= 7rtTr,~+~_,(r

1),To(l- 1 ) ) ( ~ ( l - 1), 7-(1-- 1)),1 > 2,

where T,F,~ is denoted by TF,,(r]), r/ E T I 3 , M , rn > O, and { ~-"(0) = [U~]-~[r(1) + U~ ~-(0) - 7r~TF,(~o, 7-0(0))(~, r(0))J, 7-"(0 = [V~+,]-l[r( / + 1) + g~+,r(l) - 7r2TF,+,(r 7-0(1))(z(l), r(/))], l > 1. From Step 1 it follows that A(~o, c'o, 7-0) is a well-defined bounded linear operator and is the derivative of 0 at the point (~0, tr0, to) and obviously A(0,0,0) = 0. By a simple computation we obtain that T.O, which is defined on int(Z x X x Y), is a Lipschitz map such that Lip(T.0) < coe-~e 3~. Step 4. Since Lip(0) < e -2r by Lemma 2.1 there exists a Lipschitz map

~:Z---*XxY with L i p ( ~ ) s e -2~

such that for each ~ E Z, ~(~) is the unique fixed point of the map 0~ : X x Y ---* X x Y,(cr, 7") ~ 0(~,~,7-). Since ~v(0) = (0,0) and Lip(w) < e -2r we have ~(int(Z)) C int(X x Y). Thus by Lemma 2.2 the map T.~ is well defined on Jut(Z) and is a Lipschitz map such that Lip(T.~) < (1 - e-2~)-3(1 + e-2r

< De 3~n

where D = (1 - e-2r + e-2~)2co e-~. Step 5. Let ~o(~) = (c~(~), 7-(~)) for ~ G int(Z). Define

From Step 4 it follows that h(~,~),~ is a C 1'1 map satisfying h(~,~),,~(0) :

O, Toh(~,x), ~ : 0 and LipiI.H(h(~,~),~) < e -2~,

(3.1)

Lipll.ll(T.h(~o,~:),n) < De 3''~.

(3.2)

For l E Z +, let

66

defined wherever they make sense. Since for every ~ E int(Z) there is an only point i/E H~(w, x) such that

llf~@, ~)(L,~)II(~,~),,,+~ <

ro e-3eneca§

l E Z +,

we have Graph(h(~,z),,~) =

{(e, ~) e E . ( ~ , ~ ) 9 H.(~,~) :

IIF'(~, ~)(~, .)ii(~,-),-+, < ~0~-~*"~~

t ~ Z+}.

(3.3)

Step 6. First let us notice that what we have done in Steps 1-5 holds for every (co,x) E A' and n E Z +. Now Iet (w,x) E h' and n E Z +. Since for each (~, r/) E Graph(h(~,~),,~)

<_ roe-3Ene(a+6E)(t+l)

< roe-a~(n+l)e( a+o~)t, l E

Z +,

then by (3.3) we have

F~(~, ~)Graph(h(~,~),.) C Gr~ph(h(~,~),~+~).

(3.4)

We now prove that for every ( E Graph(h(~,~),,~)

I[TcF~(w, X)[T,a~phCh(~,~),,)[[ <_e=+4~,

(3.5)

where ii" II is defined with respect to I1 I1(~,~),. and I1 I1<~,~),-+1 and T~ Graph(h(~o,~),n) is the tangent space of Graph(h(~,~),,~) at the point (. In fact, let ((,~,r].) E TcGraph(h(~,~),.) and let ((,~+,,,,~+1) = TcF~(w,x)(e.,rl.). It follows from Step 1 and (3.1) and (3.4) that

Ii(,5~+~, r/,~+t)ll(,~,x),,~+1 :

t[~n+l[[(w,x),n+l

3~n

which implies (3.5). 67

Step 7. In this step we present the counterpart of the above results in terms of the norm [ . I. Let (w,x) E A ~ and n 9 Z +. Define

o~(~,~) = {~ 9 E=(~o,~): I1~11(~,~),~ < ~oe-~=},

Wn(w, x) = expI2~ Graph(h@,.),n) and let C~n _-- A - i t 0

e-5en.

Then by 3) of Lemma 1.3 we have

{~ e E.(~,~): I(I < a.} c o.(~, ~). And since Toh@,~),,~ = O, Wn(w,x) is tangent to Er~(W,x) at the point f~x. Moreover, from (3.4) we obtain immediately

A(~)w~(~, ~) c w.+l(~, ~). Let

flr~ = 2DA2 e 7~'~. Then from (3.1), (3.2) and 3) of Lemma 1.3 it follows that Lip(h(~,~),~) _
d~(f,(w,x)y,f~(w,x)z) < 7r~e(~'+4~)'d'(u,z),

U,Z E W,~(w,x),

I E Z +,

where

7,~ = 2[b(po/2)] 2Ae2~n and b(po/2) is as introduced in Lemma II.l.1. Hence, for every (w,x) E A' and n E Z +, Wry(W, X) and the numbers an, fl~ and 7,~ satisfy 1) - 4) of the theorem. Step 8. In this step we complete the proof by showing that {Wn(w, x)}(~,~)eA, is a continuous family of C 1 embedded k-dimensionM discs in M for each n E Z +Let n E Z +. By Lemma 1.2 we know that E,~(w,x) and H,~(w,x) depend r continuously on (w,x) E A'. Then there exists a finite open cover {A / t}t=j of A' such that for each A~z we can find a basis of E,~(w,x) and a basis of Hn(w, x), (w, x) E A't which are continuous with respect to (w, x) E A't. Let A'p be a set in {A't}~'=l. Since ( , )(,o,~),n depends continuously on (w,x) C A', then for each (w, x) E A'p there exist, with respect to ( , )(~,~),,~,an orthonormal basis {~i (w, x) }/k=1 of En (w, x) and an orthonormal basis {~j (w , x)}j=k+ ,no 1 of 68

H,~(w, x) such t h a t they are continuous with respect to (co, x) E A' v. For each (w, x) E A'v, let

A(co,x) : R k | R rn~

~ Z,~(w,x) @ H,~(co,x)

be a linear m a p satisfying A(w,x)e~ = {,(co,x), 1 < s < rn0, where {e,}~__~ is the n a t u r a l basis of R k q) R rn~ Define a m a p

0v : A' v --~ Emb 1(D ~, M) by the formula 0v(W , x) = exPj2~ o(id, h(~,~),,~) o A(w, x)lok. T h e n it is clear t h a t for each (w,x) E A'v,Ov(w,x) is a C 1,1 e m b e d d i n g with

x)D

= w.(co,

x).

Now we show t h a t 0v is continuous. For (co, z) E A' v, let

h'(~o,,:),,~ = A(co,x) -1 o h(~o,~).,~ o A(co,x)lD~. Suppose t h a t {(COrn , Xrn)}rn=l +oo is a sequence of points in A'v such t h a t (con, xrn) (coo, x0) E A'v as m --~ +c~. According to Arzela-Ascoli T h e o r e m , by (3.1)-(3.3) we know t h a t h'( . . . . . ),~ and T.h'( . . . . . ),,~ converge uniformly to h'(~o,~o),,~ and T.h'(~oo,~:o),,~ respectively. This together with the continuity of A(co, x) with respect to (co, x) E A' v implies t h a t 0v is continuous. [] 111rt

Let us r e m a r k t h a t the set Aa,b& e defined above depends only on a, b, k, e, l' and r'. Denote Ao=Aon(ro

• M),

S_o,b,k = A o , b , ~ n A o

(3.6)

and let {ltrn }m=l +oo and Ir l+c~ be sequences of positive n u m b e r s such t h a t t IrnJm=l I'm / +oe and r'rn / + o c as m ---, +oo. T h e n we have t

i

A I m,r ~ ~a,b,k,e

i

C

i

A 1 ~+a,r ~a,b,k,e

and

m+l

~

m

EN

-~O(3

Xo, ,k

U ~A".,r'm a,b,k,e

"

rn=l

If we write

{(an,bn)}n=l = {(a,b) : a < b < 0, a and b are rational} and let

= 1 min{1,

1

69

b -

(3.7)

then

X0 =

/ / A '''-,~'-" an,bn,k,en

kn=l

u {(w, x) E A0: A(i)(x) > O, 1 < i < r(x)) --

"

k=l m=l

(3.8) Needless to say, Theorem 3.1 holds for each A2=;~"~ . nj ny ~n The following is a theorem about global stable manifolds of random diffeomorphisms. T h e o r e m 3.2. let )~(1)(X) < . . .

Define

ws'l(0),x)

<

Let (w, x) E Ao\{(w, x) E Ao : )~(i)(x) >_ 0, 1 < i < r(x)} and ,~(P)(z) be the strictly negative Lyapunov exponents at (w, x). C "'" C

Ws'P(o2,x) by

W " i ( w ' x ) = { y E M : limsup -nl l~

<- A(')(x)

(3.9)

for 1 < i < p. Then w ' , i ( w , z ) is the image of V((~!=) under an injective immersion of class C 1,1 and is tangent to V,(~,~) (i) at x. In addition, if y E w*'i(w, x), then lim sup 1 log d" (]~x, ],~y) <_ ~(i)(x) (3.10) n~+oo

/Z

where d'( , ) is the distance along the submanifold f n w ' , i ( w , x). Proof. We carry out the proof by several steps. Now let i E { 1 , . . . , p} be fixed arbitrarily and let k = dim V((~!,). Step 1. Let a, b, c, l' and r' be numbers with the following properties: )~(i)(x) < a < b < min{0,~(i+a)(x)} (we admit here ~(r(=)+l)(x) = +oc), 0 < E < rain{l, (b - a)/200} and (w, z) E Aa'b,k,~. Corresponding to these numbers, we have the sequence of norms {It 9 I](~o,=),,~},~=0 +o0 and numbers A and r0 as defined in the previous sections. Let {Wn(w, x)}n= +~o0 be the sequence of embedded it rl k-dimensional discs obtained by applying Theorem 3.1 to the set Aa,'b,k,~, and finally we put +oo

w= U rt-----0

Since for each n E Z + c

we know W is the union of an increasing sequence of C 1'1 embedded discs 1/-i) under an injective imand hence is the image of "((w.,) {[f2]-lwn( , mersion of class C 1,1 (see Chapter VIII of [Hir]l). From 3) of Theorem 3.1 it follows clearly that for every y E W limsup 1 l o g d ' ( f ~ x , f ~ y ) n----++ oo n

70

< a + 4~.

(3.11)

We now show that

W~'i(~, z) c W.

(3.12)

In fact, if y 9 W~'i(w, x), it is clear that there exists no 9 N such that

d(fg~, f ~ v ) < e (~

W > ~o

and e(a+~) n~ < A - l r o e-Sen~ Then we have for each l 9 Z +

d(f~o+Ix, f~O+Zy) < e(~+~).Oe(~+~)t _< A-lroe-5~nOe(~+6~)l which together with (3.3) and 3) of Lemma 1.3 proves that f~Oy 9 W.o(W, x) and hence y 9 W. Therefore, (3.12) holds true. Step 2. Let a',e~,l" and r" be numbers such that )~(i)(z) < a ~ <_ a,e ~ < l" r" l " r tt e,l" >_ l',r" >_ r', and (w,x) 9 Aa,lb,k,~,. Corresponding to Aa,lb,k,e, , we obtain analogously the sequence of norms {]0 . , ' (~,~),.s.=0, i+oo the numbers A' and r~0 and the sequence of embedded discs { W'n (w, x)}.= +~0. Define -[-oo

w' = U [fg]-' w ' . (~, ~). n---~0

We now claim that Wt=W.

(3.13)

Indeed, it is clear that for every n E Z + ]](11(~,~),.~ -< I] I[(~,~),. <- 2A e 2~ " ( (w,x),n,

( E Ty2~M.

For each n E Z +, write

W.(w, x) = exp]2~ Graph(h(~,=),n), W'n(w, x) = expf2~ Graph(h'(w,~),n) and define U . ( ~ , z ) = {~ 9 E . ( w , ~ ) : II~ II(~,~),n ' <-- min{roe-a~n,r'oe-ae'"}} 9 Since for each ( 9 Graph(h'(~,.),..Iu.(~,.) )

IIF~(~, ~);11(~,.),-+, <-- IIF.' (~, ~)r <

li(in(w,=),n'e(a'+4e')l <- ro e-3e"e(a+6~)l

for all I C Z +, we have Graph(h'(~,.),n ]u.(~,=)) C Graph(h(~,.),n) 71

and hence Graph(h'(~,~),,~ Iu~(,~,~)) = Graph(h(~,~),,~ ]g~(~o,~)).

(3.14)

For each n E Z +, when I is sufficiently large we have for all ( E Graph(h'(~,~),~)

liE4 (~, ~)r

<_ rio e-3t'n e(a'4-4E')l <

min{roe -3e(n+l),r'oe-aE'(n+l)}

and hence, by (3.14),

f~(~)w~(~,.)

c

w~+~(~, ~).

Therefore, we obtain

(3.15)

W'cW. Similarly, for each n E Z +, when l is large enough we have

IIf~(~,x)4l[}~o,~),.~+~ < 2A'e2~'~ll#.(~,z)4ll(~,~),.~ +, <

2A~e2E'nroe-3ene(a+4e) t

<_ min{roe -3e(n+l),r'Oe-3#(n+O} for all

~ E Graph(h(,o,.),,~), and hence, by (3.14), fln(w)W,~(w , x) C W'n+l(w, x).

Thus we obtain

WCW

I

which together with (3.15) yields (3.13). Step 3. Considering that (3.11) holds for every y E W ~ with a -F 4E being replaced by a ~ + 4e ~, by (3.13) we have for every y E W

1

lim sup - logdS(fgx, f2Y) < a' + 4e'. n~+oo rl Since a ~ and e ~ can be chosen such that a ~ + 4e ~ is arbitrarily close to ,~(i)(x), then y E W implies limsup -1 logd" (f[~~x , f2Y) n.-.* + o o

< ~(i)(x).

(3.16)

/2

Thus we obtain

W C ws'i(w, x).

(3.17)

Since W is tangent to --.V((~!~)at x, the theorem follows from (3.12), (3.17) and [] (3.16). 72

R e m a r k 3.1. Let A' = A~,'b,k, l' r' ~ be a set as considered in Theorem 3.1. For (w,x) C A' let A0)(x) < ... < A(0(x) be the Lyapunov exponents which are smaller than a. From the proof of Theorem 3.2 it follows obviously that

w s ' i ( w , x ) = {y E M : limsup l l o g d ( f ~ x , f ~ y ) < a}. n~+oo

(3.18)

n

R e m a r k 3.2. Given X + ( M , v , p ) and a point (w,x) E 12N x M, the global stable manifold WS(w, x) is defined by

WS(o.), x) de_____{y f E M : limsup -1 log d(f~x, f~y) < 0}. n.--.* + oo

(3.19)

/2

From (3.18) we know that, if (w,x) E A0\{(w,x) C A0 : A(i)(x) _> 0,1 < i < r(x)) and A0)(x) < ... < A(V)(x) are the strictly negative exponents at (w, x), then WS(w,x) = W',P(w, x) and hence W S ( w , x ) i s the image of t~(~,)) under an injective immersion of class v(v) C 1,1 and is tangent to ~(~o,r) at the point x.

w H~ilder

Continuity

of Subbundles

Here we consider the H61der continuity of subbundles of T M constituted of tangent spaces of the submanifolds W',i(w, x). This kind of continuity is necessary for dealing with the absolute continuity of foliations formed by invariant manifolds as introduced in Section 3. Some ideas of this section are adopted from [Bri]. We first introduce the notion of tt61der continuity for subbundles of a trivial vector bundle. D e f i n i t i o n 4.1. Let A be a metric space, H a Hilbert space and {E~}~e~ a family of subspaces of H. The family {Ex}xea is called Hhlder continuous in x with exponent c~, a > O, and constant L, L > O, if for any x, y E A

d(E~, Ey) d~j max{F(E~, Ey), F(Eu, E~)} _< L[d(x, y)]~, where we define for two subspaces E and F of H

r(E, F) = sup inf I1~ - ~]l eeE fIEF II~ll=a

and call it the aperture between E and F.

73

Let X be a metric space with d i a m ( X ) _< 1, H a Hilbert space and {T{(x)}~ex, i = 1, 2 , . . . a sequence of families of bounded linear operators T/(x) : H --~ H. For each x E X and n E N, put

T~

= ~d,

T~(~:) = T~(x) o . . . o T~(~).

P r o p o s i t i o n 4.1. For numbers C >_ i and "d < "b let A~,-g~ C X be the (maybe empty) set of points x for which there exist splittings H = E~ | E ~

such that for any positive integer n ^

^

IIT"(~),~II _< c~O'~ll,~ll,

~ c E~,

Suppose that there is a number ~ > "~ and ~3 > 0 such that IIT"(x) - T~(y)II _< e ~ [ d ( x , y)] ~

for any positive integer n and any x , y E A~-2,?. Then the family { E ~ } ~ e A ^ ~ C~a,b

is Hb'lder continuous in x on A~-g,? with exponent [ ( a - g ) / ( a - ~ ) ] f l

and constant

3~2e%'-~'.

Proof.

For z E A~,a, ~ and n E N, set I4~n = ( ( E H : IITn(z)r

_ 2@fi~1[r

Let ( E Kzn and let ( = ~' 4- r/' where (I 6 Ez and r / 6 E ~ . We have

IIT~(z)r

= IIT'(z)(~' +,')11 ~

IIT"(z),'ll - IIT~(z)~'ll

Hence

117'11_< Ce-'~'~(ltW'~(z)r + Ce~'~llr -< 3C~e(~-'~)'~llr which implies d ( ( , E z ) de__f inf 1[( - ~ l l ~ ,tEE,

3c2eC~-W)nllCll.

Let x, y E AC.-g"~. Define = ( a - ~)/fl.

74

(4.1)

Since d(x, y) < 1 and 3' < 0, there is a unique non-negative integer m = re(x, y) such that e~(~+~) < d(~, y) _< e T M . (4.2) Then for any ~ E Ey

IITm(x)~'ll _< IIT"'(y),fll + IITm(x) - Tm(y)llll,fll

_< C~J-'ll,fll + J-'[d(~, y)]~ll,fll _<

(ce o'' + e~"'e~="~')ll,,'ll

_< 2C~~

.

Thus ~ E K ~ and Ey C K ~ . By symmetry we also have E~ C K ~ . It follows then from (4.1) and (4.2) that

d(E,, Eu) <_ 3@2e (~-b~" < 3C2eb-~[d(x, y)][(~-b)/(~-r []

The proof is completed.

Now we turn to the case of subbundles of T M . As before, let P0 > 0 be as introduced in Section II.1. If x, y E M with d(x, y) < P0, we denote by P(x, y) the isometry from T~M to TyM defined by the parallel displacement along the unique shortest geodesic connecting x and y. Then for any x, y E M, if E~ is a subspace of T~M and Ey is a subspace of TyM, define 1

d(E~, Ey) =

d(E,, P(y,x)Ey)

if d(x, y) > po/4, if d(x,y) < po/4.

(4.3)

Analogously to Definition 4.1 we introduce D e f i n i t i o n 4.2. Let A C M be a set. A family {E,}~eA of subspaces E~ C T~M is called Hhlder continuous in x on A with exponent ~ > 0 and constant L > O, if for any x, y E A

d(E~, Ey) <_ L[d(x,y)] ~.

Let x,x~,y,y ~ E M and let A : T , M ---* T , , M , B : TyM ---* Ty,M be linear maps. We introduce the following distance:

d(A, B)

if max{d(x, y), d(x', y')} _> p0/4,

f [A I + [B I

IA - P(y', x') o B o P(x, Y)I otherwise.

(4.4) And then for a differentiable map f : M --+ M and for a number a, 0 < a < 1, we define I T f l g ~ = I f l c ' + sup d(T~f,T~f) (4.5)

~,y~M [d(x, y)V

75

where If[el is defined to be s u p { I r ~ f I 9 x E M} as usual and we admit d(T~f, Tyf)/[d(z,y)] ~ = l if z = y. As a consequence of Proposition 4.1 we have the following C o r o l l a r y 4.1. Let {fl}i=l +o0 be a sequence of differentiable maps fi : M --+ M such that the derivatives Tfi satisfy ^ r , IT flirt ~ <_ Coe

(4.6)

Vn E N

i=1

where Co > 1 , F > 0,0 < tr < 1. For each positive integer n, put fo = id, f " = fn o . . . o fl. Fix C >__ 1 and "d < "b and let A~,.~(g be the (maybe empty) set of paints x for which there exist splittings T.M = E.,Z such that for any n E N

I%f= l <_

e' nl l,

I%f'~ol > C-leb'~lr/I ,

7/E E ~ .

Then the family {E~} is Hb'lder continuous in z on A~,~,~ with constant 3 C ~ f i -~" and exponent a = [(a - b ) / ( ~ - d')]a, where d ' = ln(2Co2) + 2 F + icr ln(po/4)t + lal-

In order to deduce this corollary from Proposition 4.1 we need the following lemma. Lemma

4.1.

For any x, y E M and any n C N

d(%f",Tyf Proof. Then

Let x , y E M.

<_J

[d(x, y)]a.

(4.7)

Assume t h a t d(fmx, f'~y) > po/4 for some m > 0.

po/4 <__d(fmx, f m y ) <_ ]fmlcld(x,y)

<

{

d(i~, y) rn

ITfilH~d(x,y)

i=1

and hence, by (4.6), we have

d(x, y) > ~ C ~ l e - ' ; m . 76

if m = 0 if m > 0 ,

Therefore, by (4.4)-(4.6) and by the choice of the number d, we have for any n > max{m, 1}

d(T~:Y'~,TvY '~) <

[Z~f'~[ +

Ir~,f'~l <_2fi ITf, IH~ i=1

<

2Coe r < edn[d(x,y)] ~'.

Hence it suffices to prove (4.7) for all positive integers n which satisfy d(fix, fiy) < po/4 for every 0 < i < n. It is easy to see that for such an integer n,(4.7) follows from (4.6) and the following inequality

d(T~:Y'~,Tvf '~) < ~ [TIilH-

[d(x,Y)] ~

(4.8)

i=1

which we now prove by induction. For n = 1 (4.8) follows from (4.5). Suppose now that (4.8) holds true for n = k. Then, by (4.4), (4.5) and the inductive assumption, we have for n = k + 1

d(T.:f k+l , Try k+l) =

iT~fk+l _ p(fk+ly, fk+lx) o T y f k+l o P ( x , y ) ]

=

[Tyk.f~+l o T.:f k - p(fk+ly, fk+xx) o Ty~vfk+ 1 o P ( f k z , fky) oP(fky, fkx) o Tyy k o P(x, y)[

<_ [Tyk~fk+l o T~:f k - p(fk+ly, f k + l x ) o Tykvfk+l o P(fkx, fky) o T.:fkl +lP(fk+ly, fk+lx) o Ty~yfk+l o P(fkx, fky) o T.:f k _ p ( f k + l y , fk+lx) o Tfkvh+l o P(fkx, fky) o p ( f k y , fkx) o Tvf k o P(x,y)l <_ d(T!~.A+I,TI~vA+I)IT~:fk] + [Tykyfk+l[d(T~ff:,Tvf t:) _< [[Tfk+x [H- -- Ifk+l[C~][d(f ~x, f~:Y)]~ [T~:fkl

+lfk+xlcl [i=~ lTfiln~]2[d(x,V)] ~

77

1-t-a

[ITA+,IH"-- IA+~Ic,]E=~If~Ic']

[d(x,

y)]"

+lfk+llc'[~=~lTf~lH']2[d(x,Y)]~ 2

ki----1

[]

The proof is completed.

Proof of Corollary 4.1. Let x and y be two points in A~,~.~-. If d(x, y) > po/4 or d(x, y) > 1, it holds obviously that d(E~, Ey) < 3C2eb-~[d(x, y)]%

(4.9)

We now assume that d(x,y) < min{p0/4, 1}. By means of parallel displacement we can find for every n > 0 an isometry P ( f n x , x ) : T I . ~ M --* T~M and an isometry P ( f " y , f " x ) : T / . y M --* T y , ~ M such that P ( z , x ) = id and, if d ( f ~ y , f " x ) < po/4, P ( p y , f ~ x ) is the isometry defined by the parallel displacement along the unique shortest geodesic connecting f n y and f n x . Set X - {x, y}, H = T~M and for every i E N define Ti(x) = e ( f i x , x) o T l , - l ~ f i o P ( f i - a x , x) -1, Ti(y) = P ( f i x , x) o p ( f i y , f i x ) o Ty,_luf i o p ( f i - l y , f i - l x ) - i

o P ( f i - l x , x) -1.

For each n E N, by (4.4) and Lemma 4.1, we have IT"(x) - Tn(y)l

:

IP(f=x, x) o T~f" - P ( f " x , x) o P ( f " y , f " x ) o T y f " o P(x, Y)I

=

iT~fn _ p ( f n y , f n x ) o T y f n o P(x, Y)I

<

d(Txf'~,Tyf n) <_ e~n[d(x,y)] ~

Then, by (4.3) and Proposition 4.1, we obtain (4.9) in this case. Thus, the [] corollary is proved. L e m m a 4.2. For system ~ + ( M , v) the following hold true: 1) f l o g ITfo(w)lH,dv N deal5o < +~," (4.10) 2) There exists a Borel set P'0 C f2N with v N ( r ~ ) = 1, rr~ c r~ and there exists a Borel function C : F~o ~ (0, +co) such that for every w e F~o and n e N n--1

A

l ' I ITf'(w)l"' -< C(w)e2C~ i=0

78

(4.11)

Proof.

1) Choose a finite n u m b e r of points {xi}i=l t such t h a t {B(z~,p0/4)}~=l cover M . For each 1 <_ i <__ I, put Ui = B(xi,p~),Vi = B(xi,7po/12) and W~ = B(x~,po/2) and let (U~, ~i) be a normal coordinate neighborhood. Given 1 < i < l, for z 6 V~ let p(t),O < t < 1 be a geodesic satisfying p(0) = z and (dp/dt)t=o = ~ with I~1 < p0/3. Set p(1) = z'. We write ~i(z) = q,T~oi~ = u and ~i(z') = q' and, with respect to the natural basis {ej }j=lm~ of R m~ we express the linear m a p T,,~oi o P(z, z') o ( T ~ i ) -~ as an m0 x m0 m a t r i x Ai(q, u), where P(z, z') is the isometry from T , M to Tz,M defined by the parallel displacement along p(t), 0 < t < 1. From the characterizations in local charts of geodesics and parallel displacement (see C h a p t e r VII of [Boo]) it follows t h a t A~(q, u) is a C ~ m a t r i x function of (q, u) 6 T~i{rl 9 Tv, M " Ir/I < p073}. Therefore, using the n o t a t i o n Ai(q, q') to denote Ai(q, u), we have

IIAi(q, q') - idllo < Ci]lq

-

q']lo

(4.12)

for any (q,u) C T~i{rl 9 Tw, M :[rl I < p0/4}, where Ci is a constant depending only on the chart (Ui, ~i). Now let x, y 9 M and f 9 ft. If m a x { d ( x , y), d(fx, fy)} >_ po/4, then

d(T~f, Tyf) d(x,v)

[T~f[ + [Tyf[ <~ 8(max{l, ]flc,})2po 1. d(x, y)

(4.13)

We assume now t h a t m a x { d ( x , y), d(fx, fy)} < po/4 and x -r y. Suppose t h a t x 6 B(xi,po/4) and f x E B(xj,Po/4) and let Fq denote the restriction of ~ o f o ~ - 1 to ~ ( w ~ n f-*Wj). P u t p = ~i(x) and q = Fi(Y). T h e n

]lTpFij - Aj(Fij(q), Fij(p)) o TqFij o Ai(p, q)]]o lithely - Tqf~yll0 + I[Tqf~y - r~f~j o Ai(p, q)]]o

+llTqFij o Ai(p, q) - Aj(F~j(q ), Fij(p)) o TqFij o Ai(p, q)]]o <_ ]]TpFij - TqFij]lo + ]lTqFij[Iollid- Ai(p,q)llo +]]TqFij o Ai(p,q)[lollid - Aj(Fij(q), Fij(p))l]o which together with (4.12) yields

IT~f - p ( f y , fx) o T J o P(x,y)] <_ Cij(max{1,]flc~})2d(x,y)

(4.14)

where Cij is a constant depending only on the charts (Ui, ~o~), (Uj, ~j) and the definition of If]c~ (see Section I. 1). Therefore, by (4.5), (4.13) and (4.14), there exists a n u m b e r C~ > 0 such t h a t for every f E f t

]Tf]H, < C~(max{1, Iflc~}) 2

79

which together with the condition f log + ]f[c~dv(f) < +oo yields (4.10). 2) Since v: ( ~ N v N ) ~ is ergodic, by Birkhoff ergodic theorem and (4.10) we have as n ~ +oo n--1

i

f

log H [fo(TkW)IH' ~ ] log [fo(w)[uldv N n J k=0

(4.15)

v N -- a.e.

Then it follows that there exists n Borel set F~0 C ~2N such that vN(Fr0) = 1, rrl0 C FI0 and (4.15) holds for each w E F%. For each w E PI0 define c ( ~ ) = sup

IA(~)I~,

e-2~~

"

,~ = 1 , 2 , . . . .

kk=0

It is obvious that U0 and the function C(w) satisfy our requirements.

[]

In order to prove the main result of this section (Theorem 4.1) we also need the following two auxiliary lemmas. L e m m a 4.3. Let (Ei, [[ 9 ]]) and (Fi, ][ 9 ]l),i -- 1,2 be Banach spaces, and let E = E1 G E2 and F = F1 ~3 F2 with respectively the norms [](~,q)[] = max{[[~[I , [Ir/l[}, (~, r/) C K1 (9 N2 and II(~', r]')l[-- max{iWl[, Ilr/'[I}, (~', r/t) C F1 9 F2. Assume that

JAil

A = [A~l

A12]

A2~ J : E1 9 E2 ~ F~ 9 F2,

(~, ~),

, (A~

+ A~2V, A 2 ~ + A 2 ~ )

is a linear map satisfying

[[A[~lll _< bo 1,

[[A~2[[_< ao, [[AI2][ ~ (50,[[A21[[ _~ ~o

where 0 < ao < bo,O < 6o < bo and do %f ( a o + 6 o ) / ( b o - 6 0 ) <_ 1. Then, if P : E1 ~ E2 is a linear map with HPH < 1, there exists a linear map Q : F1 ~ F2 such that HQII- 1 and

AGraph(P) = Graph(Q). Moreover, for each (~, 77) E Graph(P) we have

tlA(L v)ll _> (bo - ~o)11(~,'7)11. Proof.

Since [[A12P[[ < [[AI~[lttPH < 60 < [[A]-ll[[-1,

we know that An + AI2P : E~ ~

F~ is invertible and II(AH +

(bo - 60) -1. Define Q = (A21 + A 2 2 P ) ( A l l + A12P) -1 : F1 ~ F2. 80

AI~P)-lll

<_

Then it is clear that IlOll _~ do < 1 and AGraph(P) = Graph(Q). In addition, since [IPI] < 1 and [[Qll < 1, for each ({, r/) e Graph(P) we have IIA(g,,7)ll =

][Axl~r + Al~r/ll > boil5[] - 6o11'711 bo[l(~, ,J)ll- ~oll,~ll > (bo- ~o)ll(~,,7)ll. []

The proof is completed.

L e m m a 4.4. Let (El, ( , )),(E2, ( , )) be inner product spaces of finite dimension and let F1 C E l , F 2 C E2 be subspaces. Write E1 = F1 | F1~ and E2 = F2 | F ~ . Given 0 < a < 1, suppose that A : E1 --+ E2 is an invertible linear map such that max{llA]l, HA-I[[} < (1 + ~ ) ~

(4.16)

and AF1 = Graph(P) where P : F2 -+ F2~ is a linear map with IIPll <_ ~/2. Then there exists a linear map Q : F ~ -+ [12 such that ]IQ]] <- a and A F ~ = Graph(Q).

Let ~ E A F ~ N [ ' 2 and assume that 77 # 0. Since A-lr/ e F~ and Proof. A - l ( r / + Pr/) C F1, we have ( A - l q , A-l(r] + P~/)) = (A-lr], A - I ~ ) + (A-lrl, A-1pT]) = 0

which together with (4.16) and [IP[I- a / 2 yields that 1<-. -6

4

This is a contradiction. Hence A F ~ n F2 = {0}.

(4.17)

(4.17) together with dimAF~ = dim F r yields immediately that there exists a linear map Q : F~ ---* F2 such that AF1l = Graph(Q).

81

We now show that IIQII < ~ , Let 7/ 9 F~ with and A-I(Qr] + PQr]) E F1, we have

tl,711 =

1, Since A-I(Q~+r]) 9 F~

IIA-~(Q~ + ~) + A-I(Q~ + PQu)II 2 = IIA-'(Q~ + ~)11~ + IIA-~(Q~ + PQ~)II 2. Using (4.16) and IIPII < ~ / 2 , by a simple calculation we obtain from the above equation

IIQ~II _< which implies

IIQII _< ~, []

completing the proof.

From now on we keep the notations of Sections 1 and 3. Fix P'0, 0 < P'0 < P0, such t h a t for every x 9 M, if ~ 9 T~:M and I~1 -< P~0, then

1 1 2x max{JTe exp~ I, I(T~ e x p ~ ) - l l ) -< [1 + ~(~--~) ]~.

{(1

And define

. . . ~. . .1 + e ~ r'o . min

1}

r0,~P0

9

(4.18)

(4.19)

Let Aa,b,k, ~ be a set as considered in T h e o r e m 3.1. For point (w,x) in this set we put W(w, x) = exp~ Graph(h6o,~),0]KcEo(~,~):ll~ll(~,~),0 1 such that the Borel set IJ,r j

1~ r ~ C j d e f

Aa,'b,s ~ = {(w, x) 9 Aa,b,k, ~ " w 9 r'0 and C(w) < C'}

(4.21) 11 r '

C'

is not empty. From now up to the end of this chapter we fix such a set Aa'b,~, ~ and write 1~ r ~ C ~ A = Aa,'b,~, ~ (4,22) for simplicity of notations. Given w E 9t N , we put

and assume t h a t A~o # r

We then introduce the Borel set

ff(,~) A

= U ~EA~ A

~(~, ~).

(4.23)

Let y E W ( A ~ ) and assume that y C W(w, x). Denote by E~(y) the subspace of TyM tangent to W(w, x). Notice that if y also lies in W(w, x') then the tangent 82

A

A

spaces of W(w, x') and W(w, x) at point y coincide, hence E~(y) is defined independent of the choice of W(w, x) which contains y. T h e o r e m 4.1.

Let {E~(y)}ye~(A ) be as defined above. Then this family is

Hb'lder continuous in y on W(A~) with constant 12A2[b(po/2)] 2 and exponent a = (a - b+ 20e)/(a+ 10c - d), where d = ln(2C') 2 + 2~'o + I ln(po/4)[ + [a + 10el. A

Proo]_

Let y E W(A~) and assume that y E W(w, x). Write

T~M = E~(x) (~ E~(x) •

TyM = E~(y) @ E~(y) •

By 3) of Lemma 1.3, (3.2) and (4.19) we have d(x, y) < P'0 and (Ty e x p ; t ) E ~ ( y ) = Graph(P) where P : E~(x) ~ E~(x) • is a linear map satisfying IF[ < (4A) -1. According to Lemma 4.4, from (4.18) it follows then that there exists a linear map Q : E~(x) • ---* E~(x) such that ]QI -< (2A) -1 and (Ty exp;1)E~(y) • = Graph(Q). By 3) of Lemma 1.3 we have I[QIl(~,~),0 <: 1. Denote ~ = e x p ; 1 y and write for IcZ + r~(0 ,t(0] TF~(~'~)(F(~'~)'I = [ A~'~I'~11 "'2~4(0~12 ] : H,(w, x)@E,(w,x)--+ Hl+t(w,x)@E,+l(w,x). From Step 1 of the proof of Theorem 3.1 and (3.5) and Lemma 1.3 it follows that

II(A~7)-III_ (e b-2~ - ~ o ) -~,

IIA~'~)I,< e ~

+~o,

max{llA~ll,[IA~Tll} _< eo,

where II "11is defined with respect to ][ .ll(~,~),t and I1" I1(~,~),,+~- Applying Lemma 4.3, we obtain for every 7/E Graph(Q) and n E Z +

IlT(F~( w, x)TlII(~,~),,~ > ( eb-2~ - r which together with 3) of Lemma 1.3 yields

n , C-le(b-10e)nlr / h ITyf/~zll>-

Vq' 9 E~(y) •

where C = 2A[b(po/2)] 2. (3.5) together with 3) of Lemma 1.3 results in

IT y f 2~~ l' _ < Ce (~+1~

I~'1, v~' 9

The theorem follows then from Corollary 4.1,

83

[]

w Absolute

Continuity

of Families

of Submanifolds lI

t. I

C t

We keep here the previous notations. Let A = Aa'b,~, ~ be the Borel set introduced in (4.22). Theorem 0.1.4 allows us to choose a sequence of compact + ~ such that A1 C A, A t C Az+a and v N x # ( A \ A ' ) _< i-1 for every sets {A 1}/=1 l _> 1. We now fix arbitrarily such a set A I. For (w,x) E A and sufficiently small r > 0 we put Ua,~(x, r) = exp~{r C % M :

IICII(~,~),0 <

r},

(5.1)

and if (w,x) E A t we put

v,,,((,o, ~), r) = {(,,.,', x') c/',': d(~',,~) < r,x'~ UA,~(x,,')}.

(5.2)

From the formulation and proof of Theorem 3.1 together with the compactness of A ~ it follows immediately that there exists a number 5A~ > 0 such that for each (w,x) E A t , if (w',x') E VA,((w,x),q/2),O < q _< hA', there is a C 1 map F: {~ G Eo(w, x ) : ][~l[(~,~),0 < q} ~ H0(w, x) satisfying

and

e x p ; l [ W ( w ', x') VI Ua,~o(x, q)] = Graph(F)

(5.3)

1 sup{llTcFll(~,~),0 : ~ E E0(w, z), II~[[(~,~),0 < q} -< g.

(5.4)

Let (w, x) E A ~ and 0 < q < 6~,. We denote by :PAL(z, q) the collection of submanifolds W(w, y) passing through y G A~ M U~,~(x, q/2). Set

7,~(~, q) =

U

w(~, v) n u~,~(~, q)

(5.5)

ycaLnu~.,~(z,,//2)

D e f i n i t i o n 5.1. A submanifold W of M is called transversal to the family UAL(z,q ) if the following hold true: i) W C UA~(x,q) and exp;1W is the graph of a C 1 map r : {~ E Ho(w,x) : 117/]](~,~),0 < q} --~ Eo(w,x); ii) W intersects any W(w, y),y E A~ fq Ua,~(x,q/2), at exactly one point and this intersection is transversal, i.e. T z W O % W ( w , y ) = T z M where z = W M W ( w , y ) . For a submanifold W transversal to "~aL (x, q) we define

IlWll = sup IIr

+ sup IIT,r

(5.6)

where the supremurns are taken over the set {7? E H0(w, x ) : I1~11(~,~),0< q} and ~b is defined as above. And we shall denote by Aw the Lebesgue measure on W induced by the Riemannian metric on W inherited from M.

84

Consider now two submanifolds W 1 and W 2 transversal to r

(X, q). Since,

A

by T h e o r e m 3.1 and (4.20), {W(w, y ) } u e A is a continuous family of C 1 embedded discs, there exist two open submanifolds ~ i and ~ 2 respectively of W i and W 2 such that we can well define a so-called Poincar~ map

P~+, ~+2 " ~ l n A~(x,q) --~ W~ r"l A~(x,q) by letting

P+,,+~z, for z = ~ l n

,W~nW(~,y)

~ ( ~ , y ) , y e ,~5 n UA,~(~:,q/2), and moreover, P+,,+: is a

homeomorphism.

The family .TA L (x, q) is said to be absolutely continuous if there exists a number EA~ (x, q) > 0 such that, for any two submanifolds W 1 and W 2 transversal tO.~A~(x,q) and satisfying IIWill <_ CAL(x,q),i = 1,2, every Poincard map P~.~ ~2 constructed as above is absolutely continuous with respect to Aw, and Aw 2.

D e f i n i t i o n 5.2.

Besides Theorem 3.1, which describes the existence of invariant families of local stable manifolds corresponding to Lyapunov exponents smaller than a fixed number a < 0, we have now the following other main result of this chapter which deals with absolute continuity of such families. As before, let A denote the Lebesgue measure on M. 5.1. Let A t be given as above. There exist constants 0 < qA~ <_ 5A~,~AZ > 0 and JAz > 0 such that the following hold true for each (w,x) E At: 1) The family :P/,L (x, q/,, ) is absolutely continuous. 2) If A(A~) > 0 and x is a density point of A~ with respect to A, then for every two submanifolds W 1 and W 2 transversal to .TAL (X,qAZ ) and satisfying [[W ll _ i = 1,2 any Poincar map P+, is absolutely continuous and the Jacobian J ( P ~ , , ~ 2 ) satisfies the inequality Theorem

J;) <_J(P , +2)(v) <_

(5.7)

for Aw,-almost all points 9 E ~ i A 2xS(x,qa, ). Because a detailed proof of the above absolute continuity theorem would involve too much work and it can be carried out by a completely parallel argument with that of Part II of [Kat] (for the deterministic case), we omit it here and we refer the interested reader to Part II of [Kat] for an analogous argument. Roughly speaking, in smooth ergodic theory of deterministic dynamical systems, presence in a smooth system of an absolutely continuous family of local stable manifolds ensures many important ergodic properties of the system, for example, the positiveness of entropy and the Bernoulli property etc. (see [Pes]2). 85

It would be seen that, in ergodic theory of systems generated by random diffeomorphisms as considered in this book, the presence in such a system of an absolutely continuous family of local stable manifolds also plays a very important role. For example, it will enable us to estimate the entropy of the system through its Lyapunov exponents (see Chapter IV).

w A b s o l u t e Continuity of Conditional Measures The purpose of the present section is to prove an important theorem (Theorem 6.1) which is a consequence of the absolute continuity theorem (Theorem 5.1) and of Fubini Theorem. Roughly speaking, this theorem asserts that the conditional measures induced on local stable manifolds of random diffeomorphisms by an absolutely continuous measure on M are absolutely continuous on these submanifolds. The proof of this theorem presented here is adopted from that of a similar result for the deterministic case (Theorem II.11.1) in [Kat], whose idea goes back to Ya. G. Sinai ([Ano]). We shall need the following basic proposition which is a straightforward corollary of the definition of conditional measures (see Section 0.2).

Proposition 6.1. Let (X, B, u) be a Lebesgue space and let c~ be a measurable partition of X. If ~ is another probability measure on B which is absolutely continuous with respect to u, then for ~-almost all x C X the conditional measure ~(~) is absolutely continuous with respect to u~(~) and (6.1)

du~(~) -- ~(~) gduc~@) where g = d~/du. Let A I be a set as introduced in Section 5. Beginning from now, we suppose that the numbers qaz and Ca~ in Theorem 5.1 satisfy qa, = caz. Note that this last assumption does not present any restriction of generality. Henceforth, we confine ourselves to an arbitrarily fixed point (w, x) E A t which is such that ~(A~) > 0 and x is a density point of A~ with respect to A. We introduce now the following notations. uA :

c

: IICIl(

,=),o < q A , } ;

~2 : {rI C Ho(w, x ) :

< qa, } ;

13: the measurable partition {exp~({~} x ~2)}r

of U;

I : fl(x) M ANtw (x, q,~,.); a : the partition {W(w, y) M 5}ueaLnv~,~(,,qa~/2 ) of A h ( x , q a , ) ; [N]: U~eNa(Z) for N C I; 86

/3I : the restriction of/3 to [I]; Ax: the normalized Lebesgue measure A / A ( X ) on a Borel subset X of M with A(X) > 0; A~: the normalized Lebesgue measure on/3(y), y 9 g induced by the inherited Riemannian metric; A~' : A~/A~(/31(z)) for z 9 [1]; A~: the normalized Lebesgue measure on a(z), z 9 [I] induced by the inherited Riemannian metric. R e m a r k 6.1. 1) It is easy to see that [I] = 2X~(x, qA,). 2) Since x is a density point of A~ with respect to A, one has A(A~ gl UA,~(x, qA~/2)) > 0 and hence A([I]) > 0. In addition, one easily sees that a is a measurable partition of [I] since { W ( w , y)}u~AL is a continuous family of C 1 embedded discs. 3) From Proposition 6.1, Fubini Theorem, Theorem 5.1 and from the fact A([I]) > 0 it follows clearly that Az~(/3x(z)) > 0 for every z 9 [I]. Now we formulate our main result of this section as follows: T h e o r e m 6.1.

~x[I] Let t"a(~)}~e[I] be a canonical system of conditional measures

of A[I] associated with lhe measurable partition a. Then for X-almost every z C [I] the measure A[I] ~(~) is equivalent to A~, moreover, the following estimate dA[I]

R;d _< dA---T-holds A~-almost everywhere on a(z), where R~z > 0 is a number depending only on the set A z but not on individual (w,x) G A z.

Proof. In the present proof let us admit that a number marked with the subscript A 1 such as R ~ means that it is a number which depends only on A 1 but not on individual (w, x) E Al. We complete the proof by two steps. Step 1. Let us first notice that for every z e [I] there exist Y E a ( x ) and E I such that z = /3(7) N a(~), and ~ and ~ are uniquely determined by z. Thus we may use (~,y) as coordinates of z E [I] and we shall sometimes write (~, ~) instead of z. Let {A~ Z(y)}ye~ be a canonical system of conditional measures of A~ associated with the partition /3. From Fubini Theorem, Proposition 6.1 and 3) of Lemma 1.3 it follows clearly that for X-almost all y C U the measure A~U(y) is equivalent to Au~ and there exists a number R ~ > 0 such that

<- R2!

< 87

(6.3)

holds Aye-almost everywhere on ~(y). Let 1cA[t]flz(z)JzE[I]l be a canonical system of conditional measures of A[I] associated with the partition /31. Then by Proposition 6.1 and (6.3) we obtain immediately that for ,~-almost all z C [I] the measure A[~](z)is equivalent to , ~ and there exists a number R~! > 0 such that dA[d

(~(2!)--1 < flZ(Z) def h!')< -

dA~

R~)

(6.4)

=

holds A~'-almost everywhere on ~i(z). Now let ~ E a(x). Noticing that I =/~(x) M [I], we define a Poincar6 map P : ~ : I --*/3(5) r-i [I],

2 ~/~(5)

rh a(2).

Since we admit q~z = ea~, then by the absolute continuity theorem (Theorem 5.1) we know that ,~-~ o P~% is equivalent to , ~ and there exists a number x R~! > 0 such that (R2!)-'

_<

d),~,

=

-

(6.5)

holds A~'-almost everywhere on fl,(x). For every ~ E I we also define a Poincar~ map

@:Y o~(~) --, o4~),

5,

, ~'(5) n o4~).

From the definition of the partition/3 it follows clearly that A~ o P;~ is equivalent to A~ and there exists a number R~) > 0 such that

(•

2!)

-' -< d ( ~

dA~

def h(3) < R2 )

o P;~)

(6.6)

Y -

holds A~-almost everywhere on a(x). We introduce below two other measures on a(x) and I. Firstly, for a Borel set K C a(x) we put K(13) = U~eg/3(5) and define

~,~(K) : :;'](K(~') n [I]). Secondly, for a Borel set N C I we define

.~(N) = ~m([N]). Then ux and ur are clearly Borel probability measures on a(x) and I respectively. The absolute continuity theorem easily implies that u, is equivalent to , ~ and

88

PI t o )~x~I . Moreover, (5.7) together with Fubini Theorem yields that there exists a number R~! > 0 such that

d.= =of h(4) < ~(2~ (R(2))-' <<_ dA; = -

(6.7)

holds A~-almost everywhere on o~(x) and

(R~!)-I--

d~, d%-;- dof = h(~)<- R~!

(6.s)

holds ul-almost everywhere on I. Step 2. Let Q c [I] be an arbitrary Borel subset. From the uniqueness of the canonical system of conditional measures we know that to prove Theorem 6.1 it is sufficient to show that

,~[I](Q)=-ffl{~(g)[XQna(g)('~)Gg(~)]dAy(~)}dui(y)

(6.9)

where {Gg : a(~) ~ [0, + e c ) } y e l is a family of functions which are such that the right-hand side of (6.9) is well defined and for ui-almost every y E I the following estimate

RT,~ <_@('2) <_R,,,

holds for ),~-almost all ~ G a(~), where R a, > 0 is a number as described in the formulation of Theorem 6.1. We now begin to prove (6.9). From the properties of conditional measures, from Fubini Theorem and from (6.4)-(6.8) it follows that

a[I](Q)

89

(where H(~, ~) =

hO)(P~(~))h(2)(~)h(4)(2)h(~)(~))

The fact that the last integral is equal to A[X](Q) implies (6.9) with the family {G~-: a(~) ---* [0, +(x~)}~-el being defined by

GF" "~' ' H((P~g) -I"~,='h(a)HP~y) g tt

r

~ e a(~)

s R(i) for each ~ E I and with RA, = I-Ii=1 The proof is completed. ~ A !

"

90

[]

Chapter IV Estimation of Entropy from Below Through Lyapunov Exponents

In this chapter we carry out the estimation of the entropy h~( •+(M, v)) from below through the Lyapunov exponents for a system ~ + ( M , v , p ) . This together with the estimation from above (see Theorem II.0.1) leads to a generalization of the well-known Pesin's entropy formula to the random case. A significant gain of our elaboration is when applied to the case of diffusion processes no conditions whatever like hyperbolicity are needed anymore (see Corollary 1.1 below and see also Chapter V).

w I n t r o d u c t i o n and Formulation of t h e M a i n Result In smooth ergodic theory of deterministic dynamical systems, Pesin's entropy formula asserts that, if f is a C 2 (or H51der C 1) diffeomorphism on a compact Riemannian manifold N and if it preserves an absolutely continuous (with respect to the Lebesgue measure on N induced by the Riemannian metric) Borel probability measure #, then

h~(f) = IN E ~(')(x)+mi(x)dp i

where A(1)(x) < A(2)(x) < .-. < A('(~))(x) denote the Lyapunov exponents of f .... ~(~) at point x, lmi[x)L=l their multiplicities respectively, and ht,(f ) denotes the usual measure-theoretic entropy of the system (N, f, #). The purpose of this chapter is to prove the above entropy formula in the random case of zl:'+(M, v, p). The formula takes the same form as in the deterministic case, but the meaning of tt is quite different since it is no longer invariant for individual random diffeomorphisms. The main result is the following T h e o r e m 1.1. Let zY+(M,v,#) be given and assume that v also satisfies the condition logldet T . f l LI( x M , v x If p is absolutely continuous with

respect to the Lebesgue measure on M, then hi, ( z~+(M, v)) = /M E )~(i)(x)+m'(x)dP" i

We also call (1.1) Pesin's entropy formula.

91

(1.1)

R e m a r k 1.1. Given 7t!+(M,v,#), if v satisfies log + I f - l l c ~ E Ll(fl, v), then we have log I det r~fl ~ L 1(• x M, v x ~), since log + [ det T~fl <_log + [ f l c ~ and log-IdetT~f[ > - l o g + ~e - 1 m0 for every (f,x) E f~ x M. _ C ~ The above theorem was actually first given by F.Ledrappier and L.-S. Young in article [Led]l. But their result was formulated and proved in the setting of a system generated by two-sided (time set T = Z or R) compositions of r a n d o m diffeomorphisms, and the proof is somewhat sketchy. Considering that working on time set T = Z or R is unusual from the point of view of diffusion processes, we present here a detailed treatment of this result and carry out it mainly within the more general one-sided setting of X + ( M , v, p). We hope it will be accessible to a larger audience. We first consider a case when the condition "/~ < < Leb." in T h e o r e m 1.1 is met. Suppose that W+(M, v) is a system as introduced in Section 1.1. The transition probabilities P(x,-), x E M of AZ+(M, v) are defined by

P(x,A) = v({f E a : f x E A})

(1.2)

for x E M and Borel subset A of M. For each x E M the formula (1.2) defines a probability measure on M, and for each A E B(M) the function P(x,A) is measurable in x. We say that the transition probabilities of X + ( M , v) have a density if there exists a Borel function p : M x M ~ R + such that for every x E M one has P

p(x, A) = ]A p(x, y)dA(y) for each A E B ( M ) , where A denotes as before the Lebesgue measure on M. In this case, any W+(M, v)-invariant measure is actually absolutely continuous with respect to ~. In fact, let # be such an invariant measure. Then for any

A E B(M) #(A)=/a

fp(A)dv(f)= /a/M

xA(fx)d~(x)dv(f)

= /M /aXA(f~)dv(f)d~(~)= /MP(~,A)d~(x)

which implies # < < A. Thus a consequence of Theorem 1.1 is 1.1. Let X + ( M , v , # ) be given with v satisfying l o g l d e t T ~ f l G Ll(fl x M,v x #). If the transition probabilities of X+(M,v) have a density, Pesin's entropy formula (1.1) holds true.

Corollary

Now let us show how the random case is different from the deterministic one. The assumption that P ( x , . ) , x E M have a density is natural from the probabilistic point of view. For example, it is satisfied when X+(M, v) is derived

92

from a diffusion process with elliptic generator. In contrast, for a deterministic map P(x, .) is always a &measure. The remainning part of this chapter is devoted to the proof of Theorem 1.1. Since we have proved in Chapter II that Ruelle's inequality holds true for any system A'+(M, v, it), it remains to prove hu( X+(M, v)) >_ f ~

)~(O(x)+mi(x)dit

(1.3)

i

under the conditions stated in the formulation of Theorem 1.t.

w C o n s t r u c t i o n of A M e a s u r a b l e

Partition

In this section we construct a special measurable partition of ~'~N • M , by means of which the estimate (1.3) will be achieved. Roughly speaking, this partition is invariant under the action of F : ~N x M ~ and almost every element of it is an mod 0 open piece of a stable manifold of the random diffeomorphisms. The construction is accomplished by using local stable manifolds as introduced in Section III.3. Now we go into the precise treatment. Let ~ + ( M , v , i t ) be given and let A0 be as defined by (3.6) in Chapter III. To repeat, from the definition of/k0 it is easy to see that v N x it(/\0) = 1 and F/k0 C Ao. Put hi = {(w,x) 9 A0: ~(1)(x) < 0}. (2.1) If (w, x) E /~1, Remark III.3.2 asserts that the (global) stable manifold We(w, x) is the image of R ~ ( ~ <~ m,(~) under an injective immersion of class C 11. We put WS(w,x) = {x}, if (w,x) r D e f i n i t i o n 2.1. A measurable partition r1 of ~N x M is said to be subordinate

to W~-manifolds o f ~ + ( M , v , i t ) , f f f o r v ~ • c

a n d it

(w, x), ~(x)",dof.~= lY : (w,Y) C contains an open neighbourhood of in

this neighbourhood being taken in the submanifold topology of W~(w,x). We say thai the Borel probability measure # has absolutely continuous conditional measures on W s-manifolds of X+(M, v,#), if for any measurable partition ~7 subordinate to W s-manifolds of ?~+(M, v, it) one has for

D e f i n i t i o n 2.2.

v N - a . e , o) E ~ N <<

it -

a e.x 9 M

(2.2)

where { # ~ }~zEM is a (essentially unique) canonical system of conditional measures of # associated with the partition {r/~(x)}~eM of M, and A~w,x) is the Lebesgue measure on WS(w, x) induced by its inherited Riemannian metric as a submanifold of M ((~,,) = 6~ ~f(~,x) r A1). R e m a r k 2.1. It is a fairly straightforward fact that, if there is a measurable partition ~ subordinate to WS-manifolds of ~ + ( M , v , i t ) such that it satisfies

93

(2.2) for v N -a.e. w E ~-~N,then # has generally absolutely continuous conditional measures on WS-manifolds of X+(M, V,l_t). In fact, let rf be another partition subordinate to WS-manifolds of X + ( M , v , # ) . Since for v N x # -a.e. (w, x) the partitions (rI V rl')~lnw(~) and (77 V r/')~J~,(~) are countable, from (2.1) and from the transitivity of conditional measures it follows immediately that

~0')~ << ~,~), and then

t

$

p ~ < < A(~,~),

v N x ~ - a.e.

v N x ~ - a.e.

(~, x)

(~, ~).

This confirms the above assertion. The main purpose of this section is to prove the following P r o p o s i t i o n 2.1. Let ~ + ( M , v , # ) be given. Then there exists a measurable partition 77 of ~ N x M which has the following properties: 1) F - l q < ~, {{w} x M : w E fiN} < 7/; 2) 0 is subordinate to WS-manifolds of X + ( M , v, #); 3) For every B E B(~ N x M ) the function

PB(~, ~) = ~L,.~(.~(~) n B~) is measurable and v N x # almost everywhere finite, where B~ is the section {Y: (w,Y) e B } ; 4) / f # < < Lab., then f o r v N x # -a.e. (w,x)

Before going to the proof, let us first remark that property 4) above together with Remark 2.1 implies that, if # < < Lab., then/z has absolutely continuous conditional measures on WS-manifolds of zu v, p). This is actually a consequence of the absolute continuity theorem (Theorem III. 5.1) (see the proof of Proposition 2.1). In order to prove Proposition 2.1 we need some preliminaries. We first formulate a general lemma from measure theory (see also [Led]3). A proof is included here for the sake of completeness. L e m m a 2.1. Let ro > 0 be given and let v be a finite Borel measure on R such that u(R\[0, r0]) = 0. Then for any a, 0 < a < 1, the Lebesgue measure of the set

kmO

is equal to to.

94

Proof.

Let a 9 (0, 1) be given. Define for k 9 N N.,k = { r : r 9 [0, ro], v([r - c ~ , r + otk]) > k~v([0, roD}.

It is easy to see that Na,~ can be covered by a finite number of intervals [ri a k, rl + c~k], 1 < i < s(k) such that each ri lies in N~,k and any point of R meets at most two of these intervals. It follows then that 8(]r k2

~([0, r0]) <- ~

~([r, - ~ , r, +

~]) < 2~([0, r0])

i=1

which implies

~(k) ~ 2k 2.

Thus [N~,~I < 2s(k)~ k < 4k2~ k where IKI denotes the Lebesgue measure for Borel subset K of R. obtain

We then

+00

IN=,kl < - + ~ . k=l

According to Borel-Cantelli lemma, it follows that Lebesgue almost every r E [0, r0] belongs only to a finite number of N~,k and thus satisfies

~([r - ~ , r + ~ ] ) < + + . k=l

The proof is completed.

[]

Let zE+(M, v, p) be given. Notice that the partition of ft N x M into global stable manifolds {w} x WS(w, z), (w, x) E fin x M is in general not measurable. But we may consider the g-algebra consisting of measurable subsets of f2N x M which are unions of some global stable manifolds, i.e. the g-algebra

Bs = { B E B'mxu(~N x M) : B = where BvN• Put now

N x M) is the completion of B(f~N x M) with respect to v N x ft.

BI

= {A 9 BvNxtt(~'~N X M) : F-1A

we have then the following useful fact. Lemma2.2.

U(,,,,=)~B{w}x Ws(w'x) }

BI C B

s,v N x # -

rood0.

95

= A},

Proof. Put a N x /3"(M) = {f2N x B : B E B . ( M ) } where B.(M) is the completion of B(M) with respect to #. Since M is a compact metric space, there exists a countable set 5 ~- { g i :gi : [2N • M ---+ R is a continuous function and

gi(w,x)

depends only on x for each

(w,x) E a N x M,i

E N}

which is dense in L2(f2 N x M, f2r~ x B.(M), v N x/z). For each gi E J:, according to Birkhoff ergodic theorem and the general properties of conditional expectations, one has n--I

lim

-1 ~-~gio

n ~ + o o /l

Fk(w,x ) = Z(gilBI)(w,x)

(2.3)

k=0

for each point (w, x) of a set Ag, E B I with v N x #(Ag,) = 1. Denote Aj: = niAg~. If two points (w, y), (w, z) E Aj: belong to a same stable manifold, i.e. there exists (w,x) such that (w,y),(w,z) E {w} x WS(w,x), by (2.3) we have for each gi =

z(g,

z)

since lim,~--+oo d(fgy, fgz) = 0. Therefore, each E(g, lt31)l^,: (the restriction of E(giIB I) to A~) is measurable with respect to B ' [^~ and hence

{E(gill3I)l^7

:

gi 6 .T'} C

L2(A,,~sI^T,vN

X #).

(2.4)

On the other hand, according to Corollary I.l.1, we have

L2(aNxM, BI,vNxp) cL2(f~NxM, a N x ] 3 , ( M ) , v N x , ) .

(2.5)

Since 5c is a dense subset of the right-hand space in (2.5), by T h e o r e m 0.1.6, {E(gil BI) : gi E ~-} is dense in L2(V~N x M, BI,v N x #). Then from (2.4) it follows that

L~(A>,,BII^~,vN

x p) C L2(Aj,,B'I^,~,v N x tt)

which implies B I C B s,v N x # - m o d 0 since v N X /z(AT) = 1.

FI

To conclude the preliminaries, we review some facts about local stable manifolds introduced in Chapter III. Let ~2+(M,v,#) be given. From (3.8) in Chapter III and the arguments at the beginning of Section III.5 we know that there exist a countable number of compact sets {A i : A i C A l , i E N} such that VN X P(]~I\ U/'F_-~ Ai) --'-- 0 and each A/ is a set of the type A z as considered in Section III.5 but with E0(~, z) = U ~(0(x)<0 V.(~,x) (i) for each (w, x) E Ai. Suppose that A i E {A i : i E N} and write ki = dim E 0 ( w , x ) , ( w , x ) E Ai. Let {W~r x)}(~,~)e ^, be a continuous family of C 1 embedded ki-dimensional discs given by T h e o r e m III. 3.1, corresponding to n = 0. Recall that this family of embedded discs have the following properties: 96

(I) There exist Ai > 0 and 7i > 0 such that for each (~,x) E Ai, if V,Z C

Wl*oc(w, x), then for all l >__0 dS(f~y, ftz) < Vie-X'Zd'(y,z). (II) Assume that (w, x) e h i. L e t / ~ (x, q^,) and its partition ~(i) be defined analogously to/k~ (x, qzx~) and the partition c~ considered in Section III.6. Denote i (~,~)]l / ~i ~ (X, q ^ , ) by [I(,~,x)] i A for simplicity, and let rx[ 1"~(,)(~)sze[x~.~)] be a canonical / system of conditional measures of A[i (~.J associated with the partition ~(i).

~)] Then, by Theorems III.5.1 and III.6.1, for A-a.e. z E [ii(~,~)] the measure a. [Z~ (,)i,)

is equivalent to Az~('). (III) For (w, x) 9 Ai and r > 0 we introduce

B^,((~,~),~)

=

{(~', ~') 9 ^': d(~,J)

<

~, d(~, ~')

<

~}

and, to repeat, define

B(x,r) = {y C M d(x,y) < r}, UA,,~(x, r) = exp,{( E T~M: IIr

< ~}.

Then, owing to the compactness of A i, there exist numbers ri, 0 < ri < po/4 (see Section [[.1 for the definition of Po), and r 0 < r < 1, such that the following hold true: (i) Let (0a, x ) 9 A i. If(w',x') 9 Bn,((w,x),ri), then

B(~, ~) c U^,,~,(~', q^,/2); (ii) For any r 9 [ri/2, ri] and each (0a, x) G Ai, if (w', x') 9 B^, ((w, x), sir), s , , x') Cl B(x, r) is connected and the map then Wlo~(w

( J , ~') ~ W~or ', ~') n B(~,~) is a continuous map from B^,((w,x),eir) to the space of subsets of B ( z , r ) (endowed with the Hausdorff topology); (iii) Let r E [ri/2, ri] and (w, x) 9 Ai. If (w', x'), (w', x") 9 B^, ((w, x), gir), then either W,~r x') n B(x, r) = W~oc(~, ' ' x") n B(~, ~) or the two terms in the above equation are disjoint. In the latter case, if it is assumed moreover that x" G WS(w ', x'), then

d ~(y, z) > 2ri for any y 9 Wloc(W ~ i , x') N B(x, r) and any z 9 Wloc(W, , t x") N B(x, r); (iv) There exists Ri > 0 such that for each (w,z) 9 Ai, if (w',m') E B^,((w,z),ri) and y E Wl*o~(W',x')Cl B(x,ri), then Wl*or contains the closed ball of centre y and d' radius Ri in WS(w ', x'). We are now prepared to prove Proposition 2.1.

97

Proof of Proposition 2.1. Let A i G {A i : i G N} be fixed arbitrarily. Since A i is compact, the open cover {B^,((w,x),eiri/2)}(~o,~)en, has a finite s u b c o v e r / / ^ , of A i. Fix arbitrarily BA,((Wo,Xo),eiri/2) C ll^,. For each r E [ri/2, ri] put =

n B ( x 0 , r)] :

•

9

Let (r denote the partition of s N x M into all the sets {w} x [Wl~,c (w,x) and the set f~N X M\Sr. From properties (III) (ii) and (III) (iii) it follows clearly that (r is a measurable partition of f~N x M. Define

NB(xo, r)], (w, x) 9 BA,((wo, xo), eir)

fir= ( ~

F-'~r) V {{w} x M : w 9 ftN}.

We claim t h a t there exists r 9 [ri/2, ri] such that r/r has the following properties: (1) F - l ~ r < r/r, {{w} x M : w 9 l~N} < rb; (2) Put Sr = ~,~=o-IP+~~ ~'-'~r T h e n for v N x g -a.e.(w, y) 9 St, (rk)~o(Y)d~= { z : (w,z) 9 ~Tr(w,y)} C W~(w,y) and it contains an open neighbourhood of y in (3) For any B 9 B ( ~ N x M), the function

PB(w, y)

= ~ , y ) ( ( r b ) ~ ( y ) n B~)

is measurable and finite v N x # almost everywhere on St; (4) Let ~r = r/rl~ and for w 9 f~N let {P(~)~(Y)}Ue(S-)~ be a canonical system of conditional measures of #1($,)~ associated with the partition (Or)~. If # < < )~, then for v N- a.e. w 9 ~N it holds t h a t

In fact, rb satisfies (1)-(4) for Lebesgue almost every r 9 [vii2, ri]. We now prove this fact in four steps. Step I. From the definition of T/r it follows clearly that for each r 9 [ri/2, ri], fir has p r o p e r t y (1). Step 2. Let r 9 [ri/2, ri]. It is clear that for v N x # -a.e. (w, y) 9 .~,, (rk)~(y) C W*(w, y) since (r/r)~(U) C (f2)-lWl*or (2.6) for some n > 0 and some (r"w,x) 9 B^,((wo,zo),eir). On the other hand, we first claim that there exists a function/3~ : Sr --+ R + such t h a t for each (w,y) 9 Sr,z 9 W*(w,y) and d'(y,z) < t3~(w,y) imply t h a t z 9 (rl~)w(y). Indeed, define for (w,y) G S~ /3r(w,y) = inf n>0

{Ri ~--~id(f2y, OB(xo,r))e "~' ~ii }

98

Suppose that (w,y) e S~ and z 9 W~(w,y) with 0 < d~(y,z) <_ fl~(w,y). We will check that for every n >_ 0 (2.7) Since dS(y, z) < 174, by property (III) (iv), there exists (w, z) e B^,((w0, x0), cir) such that y, z 6- WlSoc(w,z) and hence for every n > 0

dS(f2y, f2z) < 7ie-'~'dS(y, z) < -~d(f[~y, OB(xo, r))

(2.8)

and n ds (f~oy, L~z) <_7ie - ~ ' -

T

71

< r.

(2.9)

We have now three cases to consider: (a) If F~(w,y) and F"(w,z) both belong to S,, we have (2.7) by property (III) (iii) and (2.9). (b) If neither Fn(w,y) nor F~(w,z) belong to St, we have (2.7) by the definition of (~. (c) If one of Fn(w, y) and F"(w, z) belongs to S~ but the other does not, we should have d S (f~n y, f~~ z) ->- d (f~n y, 0B(x0, r)) which would contradict (2.8). Thus case (c) is impossible. Our first claim is confirmed. We next claim that fir > 0 v N x # almost everywhere on Sr for Lebesgue almost every r E [ri/2, ri]. In fact, let u be the finite non-negative Borel measure on [ri/2, ri] defined by

v(A) = #({x 6_ M : d(x,xo) 9 A}) for each Borel subset A of [ri/2, ri]. According to Lemma 2.1, Lebesgue almost every r 9 [ri/2, ri] satisfies +co E tt({z 9 M : Id(z, Xo) - r I < e-"~'}) < +co.

(2.10)

rt=0

Let Ko = {r : r 9 [ri/2, ri], r satisfies (2.10) and #(OB(xo, r)) = 0}. Clearly IK01 = ri/2. Let r 9 K0. Since 0 < ri < po/4, the standard knowledge about Riemannian metrics tells that there exists a constant D > 0 such that

d(x, OB(xo, p)) < T implies Id(x,

- pl <

for p and r satisfying 0 < r < p < ri. Thus from (2.10) we obtain +co

E#({x

: d(x, OB(~o,r)) < D-le-n~'}) < +(x~.

99

(2.11)

Since for every n > 0

P~ : (a"

M, v"

x

x

#) ~ (M, #), (~, v) ~ J2v

is a measure-preserving map, from (2.11) it follows that +c~

E v N x #({(w,y) : d(f~y, OB(xo,r)) < D - l e - " ; ~ ' } ) < + o c . n=0

Then, by Borel-Cantelli lemma, we know t h a t v N x # -a.e. (w,y) E f~N x M satisfies

d(f2y, aB(xo, r)) _> D-'e -'~' when n is sufficiently large. Therefore, fir (W, y) > 0 for v N X # -a.e. (w, y) C St. T h e second claim is proved. Let r E K0. T h e two above claims together imply t h a t for v N x # -a.e. (w, y) G St, (rlr)w(y) contains an open neighbourhood o f y in WS(w, y). Furthermore, for every n > 0 we have

=

,,~=o

F-k~)

v (F-"r/~)] F-

(i~i F-%) F--~, v ( F - n (,tls,)).

(2.12)

Since #(OB(xo,r)) = 0 implies t h a t v N x # ( { ( w , y ) G a N x M : f~y G OB(xo, r) for some l > 0}) = 0, from (2.12) it is easy to see t h a t for v N x # -a.e. (w, y) C F-"S~,(r]~)~(y) contains an open neighbourhood of y in WS(w,y). T h u s rk satisfies the requirements in (2). Step 3. Let r E K0. From (2.6) it is easy to see t h a t for each B E /~(f~S x M ) the function PB(W, y) is finite for v N x # -a.e. (w, y) G St. Let n G Z +, and put (~ = Vk=oF-k~t ,-1 , p - x op --k r ~t. From the and S~ = v~= definition of St it is clear that, if U is an open ball in f~N x M , the function

Pu,.(w, y) = A ~ , y ) ( ( ( ~ ) ~ ( y ) N G ) is m e a s u r a b l e on S~. T h e n the s t a n d a r d a r g u m e n t s from m e a s u r e theory ensure t h a t so is PB,.(w,y) for any B E B(f~ N x M ) . Noticing t h a t for any B C B(fl s x M )

PB,.(~, ~) > PB,.+I(~, ~) ^n

for each (w, y) E S t , and lim

PB,. (w, y) =

PB (w, y)

for v N x # - a . e . (w,y) C St, we know for each B C B(f~ N x M ) the function PB(w, y) is measurable and finite v N x # almost everywhere on St.

100

Step 4. Fix arbitrarily r E K0. Now assume that # < < A. Let (w,x) E B^,((wo, xo), 6ir). From property (III) (i) and the definition of Sr we know that (S~)~ is a measurable subset of Ai(x,q^,/2) and Then, denoting by {A(~lsr)~(z)}ze(s~) ~ the conditional measures of A[(s~)~ associated --

with the partition (~r [s~)~, we have A(~js~)~(z ) < < A~.~),

A-a.e.

z E (Sr)oa

which implies, by Proposition III. 6.1, #-a.e. z E (Sr)w.

P(~ls~)~(z) < < A~,;),

Since for p-a.e.z E (S~)~, (T/~)~[(~)~(,) is a countable partition, we have

Noticing that A is quasi-invariant under the action of any C 1 diffeomorphism on M, i.e. fA is equivalent to A for any f E Diffl(M), we see easily that for every n > 0 and w E ~N p(,~lF_.s~)~(~ ) < < A~oa,~),p-a.e. z E (F-nSr)w. From this it follows clearly that 7/~ satisfies (4). To finish the proof of Proposition 2.1, notice that the treatment above holds true for every element of U+__~UA~ = {U1, U ~ , ' " }. For each Un we denote by rb~ the associated partition ~ satisfying (1)-(4) constructed above and denote by 5;~ the associated set S~. For each n E N put I~ = M+=~F-1S,~. It is obvious that F-1I~ = I,~,n > 1 and v N x /~(hl\ U~=l+ooI~) = 0. By LemmaS.2 we may assume that I . E B ' for every n > 1 since otherwise we may find I~ E B ' such that F - 1 I . ' = I~' and v N x p(I~AI~) = 0 and we may restrict the procedure of constructing r/~ to/In. Set ~)~ = r/~]1, for each n >_ 1 and define partition r / o f fin x M by

~(~,.) =

[

~(w,x),

if

(w,x) e h

~.(~,.),

if

(~, x) ~ I~\ U~-~ k = l I,

{(~o, x)},

if

+oo (w, x) E 12N x M \ Un= 1 I,.

Then one can easily check that r/satisfies the requirements of Proposition 2.1, completing the proof. [] The conclusion 3) of Proposition 2.1 allows us to define a Borel measure ~* on ~N x M by

;(K) = f

n

101

•

x)

(2.13)

for each Borel subset K of ~'~N X M. It is easy to see that A* is q-finite. Also , recall that by the definition of conditional measures we have

v s x # ( K ) = i # ~ ( r l o ~ ( x ) O K~)dv s x # ( w , x )

(2.14)

for each Borel subset K off2 N • M. Since, by Proposition 2.1 4), p ~ < < As for v N • p -a.e.(w, x), we have VN X p < <

A*.

Define

dv N x # gdA* The next proposition follows from a measure-theoretic observation. P r o p o s i t i o n 2.2. For

VN

X

(2.15)

#-a.e.(w, x), we have d#7 ~ g-

dA~,,~)

As(u,x) almost everywhere on ~?u(x). Proof. Let us first notice that (2.13) can be written equivalently as

for K E ~(~'~N • M). Then using standard methods of measure theory we easily obtain

for each h E L l ( f l s x M , B ( O N x M),A*). Let A E B(TI),B E B(fl N x M ) be two arbitrary sets. From (2.13)-(2.16) it follows that

= i JA nB

dv N x p : i

# ~ ' ( ~ ' ( ' ) M B ~ ' ) dvN x p(w,x). (2.17)

JA

Since ~N X M is a Borel subset of a Polish space (see Section 1.1), by Theorem 0.1.3, the measure space (l] N x M, B(~ N x M), v N x p) is separable. By Theorem 0.1.2, B(• N x M) can be generated (v N x/~- mod 0 ) by a countable subalgebra {Bj}+__~ of B ( n N x i ) . Fixing 1 < j < +oo, we apply (2.17) to an arbitrary set A E B(U) and to B : Bj. As A is arbitrary, (2.17) implies that there exists

102

a measurable subset Z/ of ~2N x M such that v N • /~(Zj) = 1 and for each (~, x) E Zj one has

g(w, z)d~o~,~)(z) ,,, (:t)n (B j),,,

Then, according to Theorem 0.1.1, we know that for v N x #-a.e.

f~

.

,7,..

(w,x),

z B~)

holds for any B E B(~2N x M), and therefore d#~ ~ g - d $ (~, )=. ~ , ~ ) almost everywhere on T/~(x).

[]

w E s t i m a t i o n of the Entropy from B e l o w In this section we complete the proof of Theorem 1.1.

Proof of Theorem 1.1. Let i,Y+(M,v,#) be given with v also satisfying log I detT~fl E Lt(f~ x M,v x #), and assume that # < < A. In view of Ruelle's inequality (Theorem II.0.1) it remains to prove h~( X+(M,

v)) > f E )~(i)(x)+mi(x)d#"

(3.1)

i

In what follows we keep the notations in Chapter I. Let T/ be a partition of f~N x M of the type discussed in Proposition 2.1. Denote by r/+ the partition P - i T / o f f~z x M. For every integer n > 1, we now assume that H~N x.(T/[F-%/V ~0) < +cr

(3.2)

then from the general properties of conditional entropies (see Section 0.3) we have

1

-H,,N •

v ,,4

n

1 =-Hg.(P-lrl]p-1F-%I

V P-la0)

n

=IH,.(n+IG-",+ V or+) n 1

= - H , - (C%+1~+ V G"- +) n

103

n

~., H..(Girt+lGi_lrt+ V G"~r +)

1

i=1 r~--I

-_-1n

~ t&. (,7+ ic_,,7+ v c~,,+)

(3.3)

i=0

Since H ~ . ( ~ / + [ G - l q + V ~+) = H ~ N x . ( ~ ] F - ' q V ~0) < + o e and Gio "+ //~ c~ as i ~ +oo, from (3.3) we obtain lim n~+oo

v ~0)

1H~• n

=H,,. (,7+ IC- ~,7+ v o9 = h~.(C, ,7+) <_h;.(C) = h~,( X+(M,v)). Therefore, in order to prove (3.1) it suffices to prove (3.2) and

/~

v ~o) >

nHv,•

A(i)(x)+mi(x)d#

(3.4)

i

for every n >_ 1. Fix arbitrarily n _> 1. We now begin to prove (3.2) and (3.4). By the definition of conditional entropies we have

v O'o)

H~N• = -

=-

F_~ l o g ( v N x #)(,.,,~)

Va ~

(,7(~,x))d,~ N x #

f.,,

dv"

(3.5)

Let {h}+__~ be the sets introduced in the construction of T/, i.e. in the p r o o f +oo of Proposition 2.1. P u t I = Ul= 1 I~ and I0 = ~N x M \ I . Clearly, F - 1 I = I , F - 1 I o = Io. Since q and F - ' ~ q V ~0 both refine the partition {I, I0} and their restrictions to I0 are partitions into single points, we know t h a t for each (w, x) E I0 l o g / ~0%)

,r~

( ~ ( x ) ) = 0.

On the other hand, by Proposition I. 3.3,

],o~(~

NX~

=

0.

i

Hence we m a y assume v N x # ( I ) = 1 without any loss of generality. Let ~ = dl~/dA be the R a d o n - N i k o d y m derivative. P u t A = { x : ~ ( z ) = 0}. Since

f 3 # ( A ) d v N ( w ) -- it(A) -- 0,

104

we o b t a i n f2p(A) = 0 for vN-a.e, w 6 f~N. Let B C M \ A be an a r b i t r a r y Borel set. If p ( B ) = 0, then for any w 6 ~-~N

~(B) = o,

f 2 a ( B ) = o,

f 2 , ( B ) = O.

It follows then t h a t

f2# < < #,

# < < (f2)-l#

for each point ~o of a Borel subset r ' of f~N with ~)N(F') ---- 1. Let w E r ' . It is easy to verify t h a t

d ( ~ , _ l # ( z ) _ ~(f2z) ldetTzf[~]- =(I)~(w,z). Then, by Proposition III.6.1, we have

~(~, ,)I((::)-,.ro~)(~)

d . (J : ) - 1 " : ~ --1

n

d((f,~)_1#)(:~),7. ~'

---- f

](( (I).(w,z'd""'~'-1) ((f~) #)~.(f2)-',.-~ f2)-',:~)(,:)

for #-a.e. x E M . For v N x #-a.e. (w,y) 6 f~N x M we can define

w~(~, y) : x ~ ( ~ , ~) =

,/:)_1,..~(,~(y)), :(y) g(F~(.,,y)) :(L"y) g(~, ~) '

gn(~o, y) = I det(Tuf21E~ I det(Zyf~)l

'

'(:2)-~'7~ Z,~(~o,y) = f(( ::)-',r~)(y) ~.(w, z'd ) t` t' :' ~ ~ )' - 1 ~)y It is easy to see t h a t Wn, Xn, Yn and Zn are all measurable and v N x p-a.e, finite functions on f2N x M. We now present several claims, whose proofs will he given a little later. C l a i m 3.1. W , = XnYn z, ,

VN x p almost everywhere on ~2N x M.

C l a i m 3.2. (a) - l o g ] : , 6 L1(~2 N • M,v N x #); (b) - f ~ logYndv N x tt = f ~ i A(i)(x)+mi(x)d#"

C l a i m 3.3.(a) l o g Z . E LI(~ N x M , v N x p); (b) f l o g Z , dv N x # >_ O. C l a i m 3.4. (a) l o g X , E Ll(f2 s x M,v N x #); (b) f log Xndv N x I.t = O. T h e n we immediately obtain (3.2) and (3.4) from (3.5) and Claims 3.1-3.4. This proves T h e o r e m 1.1.

105

Finally we give proofs of Claims 3.1-3.4.

Proof of Claim 3.1. It suffices to prove that for w~(~, ~) = In fact, for v N

X

X~(w, y)Yn(w, y) z~(~, y)

'

#-a.e.(w, x) we have for any

VN

X

#-a.e.

#~-a.e.y E

(w, x) one has

rl~(x).

B C B(M)

,~-(~) 1

L

d-(f=)-%'~

1

[

d

n

= w . , ( ~ , . ) z ~ ( ~ , ~) Jn~(~)~. r

1

.-1

#]2~ (y)

=

1

f

(I,.(w,

,~ -1 y)dup~ ,7.-~ (y) (f:)

=

1

fI

q~,~(w,(f,~)-ly)g(TnW, y)dMg.(~,~)(y )

=

1

f

n O~(w'y)g(F~(w'y))ldet(Tyf[ilE~

dA~(~'~)(Y)

and, on the other hand, J, Since B is arbitrary in

B(M), one has

1

Wn(w, x)Z~(w, x)~(w, y)g(F~(w, Y))I det(Tyf2 IEo(~,u))l = g(w, y) for A~,,)-a.e. y E r]~(x). Since W~(w,y) for any y E q~(x), it follows then that

= W~(w,x) and Z~(w,y) = Z~(w,x)

W,,@, y) = X,, (,,,, y)V,,(w, Y) Z . ( ~ , y) , Claim 3.1 is proved.

.';-a.e.u ~ ~(~).

[]

Proof of Claim 3.2. Noting that log + ]f~]c1 C L](f~N,v N) and for v N x #-a.e.

(w,y) 6

~n • M

ITyf2l~o(~,~)l _< If2lc,,

106

we know log + Izv]~lEo(~,v)l 6 LI(Q N x M, VN X #). By Oseledec multiplicative ergodic theorem we have /,

1 f

II~

A(i)(x)mi(x)d#

dvN • P = ]

n j

j

(3.6)

i

and

1 flogldet(T~y~lEo(~,~))ld~N• .=/~(')(~)-m~(~ld.

n

(3.7)

(both sides of (3.6 / and (3.7) may be - o c ) . From log ] det Tuf I E 51(~2 x M, v • #) it is easy to see that logldetTyfn~l E Li(12 N • M,v N • #). Hence, by (3.6),

E ~(i)(x)rni(x) e LI(M, #1 i

which together with (3.7) implies log Idet(TuEl~o(~,u)) I e L ~ ( f ? ~ • M, v N • ~). Claim 3.2 follows then from (3.6) and (3.7/. []

Proof of Claim 3.3. We first prove that log(I)n 6 Ll(f2 N x M,v N x p). Since /log-

~n(w, y)dpdv N

//[log- ~(~, y)]~.(~, y)d(f~ )-l #dv N

ff [~(~, y)log~(~, y)]-d(f~)-lpdv N > --(X)~ we obtain log- (I)~ E L 1(f~S X M, v N x #). Noticing that for v N x #-a.e. (w, y) E ~N•

log I det T~f[~ln

log (I)~(w, y) = log - ~(y) -

(3.8)

~(f2y)

which implies log-

~(y)

> log- ~n(w,y) + l o g - [ d e t T y f ~ [ ,

~(f2~) -

we know that log- ~ooF" ~o E Ll(f~ N x M, v N x p). Then, by Lemma 1.3.1, log ~o~is integrable with respect to v N x # and f log ~o-~r, dv N x # = 0. Thus, from (3.8) ~p the integrability of log (I),, follows. We now remark that, given a probability space (X, 13, u) and a sub-e-algebra .4 of B, we sometimes use E~ (.IA) to denote the conditional expectation operator. Define O(x) = ~" 0 [ x l o g x xx => 00 ' and define (I)(u) = min{~,~, N} for each integer N )_ 1. Considering that for v N • p -a.e. (w, y) E ~ N X M ,

Z,~(w, y) = E(y2)-,u(r

.)lB((f~)-t~?r-~))(y), 107

by the convexity of 0 we have

f

log ~ ( w , y)dv N x #

=//O(~n(W,

:

/

y))d(fn~)-t#dv s ]3

E(s-~-~.(O(r

n--1 ((f~) ,~ n ~ ) ) ( v )d( f ~n)- - 1

dv N

ff o(zo(, v))d(fn)-l#dv N - N--~+oo lim f /

Z,~(w,y)logE(:3)-~u(rI,(N)(w,

'~ -~ o ~ . ~ o ) ) ( y ) d ( f ~ )' ~1 -/ - t d v N ')16((fS)

=

r

, ~ - 1 Or , ~o))(y)(fS) d ~ -1 # dv N ")lB ((fS)

lira

N---++cof f

= ff

N

= f f log Z~dv N • p which proves log Z,~ E Ll(f2 N x M , v N • #). On the other hand, also by the convexity of 0, we have

/ = ff

logZ,~dv s • # O(Zo(w, y ) ) d ( f : ) - t # d v

N

>f o(f

'

~0. The proof is completed.

[]

Proof of Claim 3.4. By Claim 3.1, one has

log W,~ = log X~ + log Y,~ - log Zn < 0,

v N • #-a.e.

which yields log + X n < - l o g - Y , ~ + l o g +Zn,

v N x p-a.e.

Hence, by Claims 3.2 and 3.3, log + X~ C Ll(f2 N x M, v N x #). Then, according to Lemma 1.3.1, we know that logXn is integrable a n d f l o g X n d v N x tz = O. []

108

Chapter V

Stochastic Flows of Diffeomorphisms

In the previous chapters we introduced the notions of entropy and Lyapunov exponents and presented some related ergodic-theoretic results for a random dynamical system generated by i.i.d. (independent and identically distributed) random diffeomorphisms. The main purpose of this chapter is to develop a generalization of the theory to the case of a (continuous time) stochastic flow of diffeomorphisms. Roughly speaking, all stochastic flows of diffeomorphisms are essentially solution flows of stochastic differential equations ([Kun]l). The theory of stochastic differential equations was initiated by K. It6 in 1942. Since then the theory has been developed in various directions, of which an important one is the application to the study of diffusion processes associated with certain second order partial differential operators. A stochastic differential equation can also be used to describe a dynamical system disturbed by noise. In this chapter we adopt the latter point of view and we are mainly concerned with the random dynamical systems generated by solution flows of stochastic differential equations. Here we assume that the reader has a reasonable background of random processes and stochastic differential equations. The notion of a stochastic flow in Section 1 needs a basic knowledge of random processes; Remarks 1.1, 1.2 and the proofs of Propositions 1.1 and 1.2 in Section 1 demand the reader being familiar with the standard machinery of Markov processes; when we deal with stochastic flows arising from stochastic differential equations, it is assumed that the reader has a reasonable acquaintance with the theory of stochastic differential equations (see the references therein).

w Preliminaries Throughout this chapter, M is still a C ~~ compact connected Riemannian manifold without boundary. As usual, let Diffr(M)(r >__ 1) denote the space of C ~ diffeomorphisms of M, equipped with the C ~ topology (see [Hit]l). We first mention the fact that DitV(M) can be metrized in such a way that it becomes a Polish space. Indeed, let p be a metric on C ~(M, M) such that C r (M, M) is separable and complete with respect to p. Define a metric /~ on DitV(M) by ~(f,g) = p(f,g) + p ( f - l g - 1 ) . Then the topology of f~ is still the C ~ topology, but now DitF (M) is complete with respect to t~. This fact will be useful when we deal with random processes taking values in DifU(M). To begin with, we give the definition of a stochastic flow of diffeomorphisms on M.

109

D e f i n i t i o n 1.1. Let ( W , ~ , P ) be a probability space. A random process {9t : (W, .T, P) ~ Diff~(M)}t>0 is called a stochastic flow o f C r diffeomorphisms if it has the following properties (i)-(iv): (i) f o r any 0 _< to _< t~ _< ... _< tn, 9,, o ~:-~l, 1 < i < n are independent random variables; (ii) For any 0 < s < t, the distribution of gt o 9~ 1 depends only on t - s; (iii) With probability one {gt}t>o has continuous sample paths, i.e. the map R + --~ DittO(M) given by t , , 9 t ( w ) is continuous (with respect to the C ~ topology on DitV(M)) for P almost all w 9 W ; (iv) 9o = id, P - a . e . Obviously, a stochastic flow of C ~ diffeomorphisms is also a stochastic flow of C ~' diffeomorphisms if 1 < C < r. In this chapter we shall discuss stochastic flows of diffeomorphisms mainly from the point of view of ergodic theory of dynamical systems, that is, we shall mainly consider ergodic properties of dynamical systems generated by actions on M of diffeomorphisms of such flows. R e m a r k 1.1. A stochastic flow of diffeomorphisms can be characterized (up to equivalence of random processes with the same finite dimensional distributions) by a one-parameter convolution semigroup of Borel probability measures on the space of diffeomorphisms. First, let {~t " (W,.T, P) ~ Diff.(M)}~>_0 be a random process with Po = id, a.e. such that properties (i) and (ii) above hold true. Let vt be the distribution of ~t, i.e. vt(F) = P { w : ~t(w) E F} for all Borel subsets F of Diff.(M). One can easily prove that {gt}t_>0 is a temporally homogeneous Markov process with transition probability functions P ( t , g , F ) = v t ( F g - 1 ) , t >_ 0, F e B ( D i f V ( M ) ) , g 9 Diffr(M). The property (i) and (ii) imply clearly the Chapman-Kolmogorov equation

P ( s + t, g, r )

= f P(t, f, r)dP(s, g,

(1. 1)

Then, by a standard argument from measure theory, from (1.1) it follows that {Vt}t> 0 is a convolution semigroup, i.e. for all s , t >_ 0 vt * v, = vs+~

(1.2)

which means

f l(• o g)dv,(f)dvs(g) = f for all bounded measurable functions I : Diff.(M) ---+R. Conversely, assume that {vt}t>_o is a convolution semigroup of Borel probability measures on Diff.(M) with vo({id}) = 1 . Define P ( t , g , r ) -- v~(rg-1),t >_ 0 , r c B(Difff(M)),g E Diff.(M). This is a family of transition probability functions since the convolution property (1.2) also imply the Chapman-Kolmogorov equation (1.1) for {P(t, g, F)}. We may use this family of transition probability functions to construct a temporally homogeneous Markov process {pt}t>0 with values in 110

D i t V ( M ) and with 90 = id, a.e. This process has then the properties (i) and (ii). Moreover, the process {~t}t>0 has a modification with continuous sample p a t h s almost surely if and only if for all neighbourhood U of the identity of DitV(M),

~ vt(Ditt~ ( M ) \ U ) ~ 0 as t --+ 0. We refer the reader to [Bax]2 for a detailed t r e a t m e n t of this topic. R e m a r k 1.2. Properties (i) and (ii) above for a r a n d o m process {~t}t>_0 valued in D i t F ( M ) can also be characterized by Markov properties of all the npoint motions of {~t}t_>0 in the following way. Suppose t h a t {~t ' (W, iP, P) D i f g ( M ) } t > 0 is a r a n d o m process with ~0 = id, a.e. Let n _> 1 and let X(") : ( X l ~ ' ' " , x n ) be a point in M n. Set ~tx ('~) = ( ~ t x l , ' " , ~ t x , ~ ) . T h e n {~,tx('0 : ( W , C - , P ) -~ M~}t>_o is a r a n d o m process starting at x('0 at time 0. It is called an n-point motion of the process {~r If {~t}t>_0 satisfies (i) and (ii) above, then it follows easily that for each n >_ 1 and x (n) = ( x l , . . . , x,~) E M '~, {~tx('~)}t>_o is a temporally homogeneous Markov process with transition probability functions Pn(t, y('O, A) = P { w : pt(w)y (~) E A } , t >_ O, A E B(M'~), y(~) E M '~. Conversely, if for each n >_ 1 and x('0 = (Xl, ' ' -, x~) E M '~ the n-point motion {~otx('0}t>0 is a temporally homogeneous Markov process with transition probability functions P,~(t,y('O,A) = P { w : ~(w)y('~) E A } , t >_ O,A E B ( M n ) , y (~) E M '~, then one can show t h a t all the C h a p m a n K o l m o g o r o v equations for {P,~(t,y('O,A) : t >_ O,A E B(M'~),y ('0 E M'~},n >_ 1 put together lead to the convolution p r o p e r t y (1.2) and consequently {~t}t_>0 satisfies properties (i) and (ii). Let {Wt : (l/V, r P ) ---, DitV(M)}t>_0 be a stochastic flow. We now introduce the notion of invariant measures for the flow. D e f i n i t i o n 1.2. A Borel probability measure p on M is called an invariant measure of {~t}t_>o /f

for all t >0. Let P(t,x, .),t >_ 0, x E M be the transition probabilities of the one-point motions of {~t}t>_0. Denote by B ( M ) the space of all b o u n d e d m e a s u r a b l e functions on M. We now introduce the linear operators Tt : B ( M ) --+ B ( M ) , t >_ 0 by the formula

(Ttg)(x) = J g(y)dP(t, x, .)(y)

(1.3)

for g E B ( M ) and x E M. T h e family of linear operators {Tt}t>_o satisfies the semigroup p r o p e r t y Tt o T~ = T~+t, s, t >_ 0 (1.4)

111

because of the Markov property of the one-point motions, it is called the semigroup of linear operators for the one-point motions of {~ot}t_>0. Because of property (iii) in Definition 1.1 and the compactness of M one has TiC(M) C C(M) for all t > 0 and lim sup I(Ttg)(x) - g(x)l = 0 (1.5) t--+O xE M

for any g E C(M), where C(M) denotes the space of all continuous functions on M. Corresponding to Tt defined above, there is an adjoint operator Tt* : Jt4(M) ---+ .M(M) defined by

(T;p)(A) = j P(t, x, A)dp(x) for p E M ( M ) and A E B(M). The family of operators {Tt*}t_>0 also has the semigroup property T; o7-; = Ti*+t, s,t > O. (1.6) Clearly, a measure p E Ad(M) is invariant for {Tt}t>0 if and only if T;# = I~ for all t > 0. i.i. Let {~ot : (W, f , P) --+ DifJr(M)}t>o be a stochastic flow. Then there is at least one invariant measure of the flow.

Proposition

Proof. Analogously as in the case of X+(M,v) considered in Section 1.1, for every positive integer n there exists a T;/ -invariant measure p , E M ( M ) . Now take a subsequenee {#,~,}i>1 of {#,,},~>1 such that #,~, weakly converges to some probability measure # as i --* +oo. For each fixed t > 0, if g is a continuous function on M, then using the semigroup properties (1.4) and (1.6) we have

f gdTt*tt = / T,gdp = ~r~oo/ T~gdp~, =

i--.+oolimfT,_[m,]~gdT[*t,~,]~Iz,~ ,

=

i-+oolim/rt_[tnil_~Tgdl~ni

gdp by the Tl*/,~jinvariance of #.~ and (1.5). This implies that Tt*# = # for all t _> 0. []

In the sequel a stochastic flow {~ot}t>0 will be denoted by ({~ot}t>0, #) when associated with an invariant measure #. In order to carry the notions and results presented in Chapters I - I V for a system X + ( M , v, #) over to the case of a stochastic flow of C r diffeomorphisms, we shall prove a proposition (Proposition 1.2) which asserts that for a stochastic flow of C r (r = 1,2) diffeomorphisms the C%norms of the diffeomorphisms in 112

the flow satisfy automatically an integrability condition. For this purpose we first present some preliminary facts. Let {~t : ( W , $ ' , P ) --* Diff~(M)}~_>~ be a stochastic flow. As we have said in R e m a r k 1.1, it is a temporally homogeneous Markov process with transition probability functions P ( t , g , r ) = vt(rg-~),t _> 0,r E B ( D i f f r ( M ) ) , g 6 Diffr(M), where vt is as introduced in Remark 1.1. Let {Tt}t___0 be the semigroup of linear operators for {Pt}t_>0 defined analogously as in (1.3), i.e.

(Ttl)(g) = f l ( f ) d P ( t , g, .)(f) for each bounded measurable function l : Ditt~(M) ~ R and g 6 Diffr(M). It is easy to see that, if l is a bounded continuous function, so is Ttl. Hence {Tt}t>_o is a Feller semigroup. This together with property (iii) in Definition 1.1 implies that {~t}t>_0 is a strong Markov process ([Dyn]). Proposition

1.2.

Let {~t : (W, ~', P) ----, DiffC(M)}t>o(r = 1, 2) be a stochas-

tic flow. Then

f[

sup log + II~,(w)llc= + sup log + Ir~,(w)-allcr]dr(w) < + ~ 0
(1.7)

0
for any T ~ O, where ]]f]lcr = sup~cM IT~f[ = [flcl if r = 1, and Ilfllc~ is as defined in Section IL 1 if r = 2.

Proof.

For any f , g E Diffr(M), by Proposition II. 1.1 ( i f r = 2), IIfo gllc r <_ CllflIcrllgllcr max{llgllcx , 1}

(1.8)

where C > 1 is a constant. Define U = {f E Diff'(M): max{llfllc~,llf-lllcr,llflicl,llf-lllc

1} < K }

where I< = Ilidllc ~ + 2. By properties (iii) and (iv) in Definition 1.1, for any given ~ > 0 there exists t(~) > 0 such that

P{~l _< t(6)} _< 6

(1.9)

where rl is defined to be the stopping (Markov) time inf{t : ~t ~ U}. positive integer n > 1, put U '~ = { f 6 DitV(M) : f = f,~ of,~_l o . . . O f l , f /

For

E U, 1 < i < n}.

Then, by (1.8), max l < k < r

{ll/ILc~ IIf-lllc ~} < C~K2~ ~

(1.10)

--

for any f E U '~. Define r~ = inf{t : ~ ~ U~}, then, using the strong Markov property of the process {~t}~_>0 at the stopping time v~, we have for any given 113

to>0

P{T. < to} =

P { ~ t ~- U n for some t _< to}

--1 _< P{T._I < to and ~at o ~ar._, r U for some t satisfying r . - 1 < t < to}

<_ P{'r,~-i <_ to}P{71 <_ to}. Thus, by induction, P{w. < to} _< (P{T~ _< to}) ~.

(1.11)

Taking to = t(6) and using (1.9) we obtain

P{~. < t(6)} < 5".

(1.12)

Now, by (1.10) and (1.12), max ~ sup log+ll~,,llc,, sup log + l_
J

<_

11~7~11o,} dP (1.13)

rl=] oo

< ~-, log(CK ~) ~ ~ "

<

+oo

n=l

if5 < 1, where r0 = 0. This clearly imply (1.7) for T <_ t(5). In order to prove (1.7) for all T > 0, notice that for any n _> 1, by (1.8), sup

0_
sup

0
sup

log + II~,llc~

log + II~,llc, +

0
sup

t(~)_
sup

~-',(~)o ~,(~)ll~-

+11~,(~)11~'

log +ll~,llc,+logc+log

t(~)
log + I1~, o

(1.14)

~,(~)ttc--

log + [[~ot o -I

-1 then {r is also a stochastic Notice that if we define Cs = Tt(~)+sTt(t) flow. It is a temporally homogeneous Markov process with transition probability ! functions P(s, g, F) = v,' ( F g - 1 ) , s _> 0, F E B ( D i ~ ( M ) ) , g E DiW(M), where v, is the distribution of r Because of property (ii) in Definition 1.1 we have v s = v~ for all s > 0. Hence {gt}t_>0 and {r have the same finite dimensional distributions. This together with property (iii) in Definition 1.1 implies that /

sup log+ H~t t( 6)< t<(,+ l )t( 6)

o~(~)llc.dP= -1 ff 114

sup log+ll~t[[c.dP. o< t<_,~t(6)

Then by (1.13), (1.14) and an inductive argument we get log + II~otl}crdP < +oo sup 0
f

for all n > 1, proving the integrability of the first expression in (1.7). We now prove the integrability of the second one. For any n > 1, by (1.8), sup

max log +

O
<

sup

max log +

O
sup

+

11~7111c~

I1~7111c~

max log + ..~ll~t]~) o

t(6)
sup

2

(~ o ~(~))

lick

max log +11~o;111C, + l o g C

O
+2

sup

max log + II(Tt o ~,(~)) -1 -1 I1~

t(,5)
Then, for the same reason as above, we obtain

f

sup

max

0
l~ + ll~,7~llc~dP < + ~

for all n _> 1. The proof of the proposition is completed.

[]

R e m a r k 1.3. It can be shown that Proposition 1.2 holds true for all r ~ 1. In fact, let Iflc~ be the usual C~-norm for f E Diff~(M)(r >_ 1) (see [Fra] for the definition). By Lemma 3.2 of [Fra] one has

If ogler <_ C~l/Icr(Igl~c~ + 1) (1.15)

<_ 2C,.Iflcr max{Igl~r,1} for any f,g EDifff(M), where C~ is a constant depending only on r. Using (1.15) in place of (1.8) one can prove /[[.0 0 by almost the same argument as the proof of Proposition 1.2. R e m a r k 1.4. Y. Kifer presented in [Kif]3 an even stronger result which says that, if {~ot : ( W , ~ , P ) -~ Dif~(M)}t>_o(r > 1) is a stochastic flow, then

] [ sup I ,l r + sup LO
O
< +oo

for all T _> 0 and 1 E N. But we are not sure if this result in [Kif]3 and the p r o o f provided there are correct. Actually, the proof there is based on the following two statements: >_ 1) are independent; (1) and suPt(6)
I~r

_

_

// max [ O
sup

I~g-~l~ } dP

< +oo.

O
On the one hand, we are dubious if statement (1) is correct. On the other hand, Y. Kifer did not give a proper justification of statement (2) since he only proved t h a t for each fixed 1 E N there exists 51 > 0 such that

/m x/( O<_t
sup

O
We now pass to stochastic flows of diffeomorphisms arising from solutions of stochastic differential equations. This subject has been discussed in m a n y recent papers and books from various points of view. Here we shall neither a t t e m p t to touch on all the topics of the subject nor a t t e m p t to give the full bibliography on them. We shall only give the basic idea of how stochastic flows of diffeomorphisms are related to solutions of stochastic differential equations. Now we assume t h a t the reader has a standard knowledge of stochastic differential equations. For the sake of presentation we first review briefly some basic knowledge about stochastic differential equations defined on M , which can be found in [Ike], [Elw]2 or [Kun]2. Suppose now we are given vector fields Xo, X1,..., Xd on M . We assume t h a t X I , .. ",Xd are of class C a, i.e. with a local coordinate x = ( x l , . . . ,xmo) (where rn0 = d i m M as before), these vector fields are expressed as rno

xk(x) =

0

x (x)o i i=l

where X ki ( x ) , i = 1 , - . . , m 0 , k = 1,--. ,d are C a functions of x. As to vector field X0, we assume it is of class C 2. Let {Bt = (Btl, ... , Bd)}t_>0 be a d-dimensional s t a n d a r d Brownian motion defined on a probability space (W,.T, P) and let {-Pt}t>0 be the proper reference family of {Bt}t>0, i.e. ~ct = N~>0o-{B~ " u _< t + e} where c~{B~ : u _< t + s} is the smallest sub-a-algebra of r with respect to which By is measurable for each u < t + c. We now consider a Stratonovich SDE (stochastic differential equation) on the manifold M d

d~t = Xo(~t)dt + E Xk(~t) o dBkt. k=l

116

(1.16)

By a solution to SDE (1.16) starting at x at time s we mean a sample continuous, {Srt}t>_0-adapted r a n d o m process {~os,~(x) ( W , ~ , P ) -+ M}t>_~ with ~o~,~(z) = x, P-a.e. such t h a t for any C a m a p F : M ---+ R there holds the following It5 formula =

/.'(

F(x) + d

t

+}2 [ k=l

Js

for a l l t > s. For any given a: C M and s E R +, SDE (1.16) always has a unique solution starting at x at time s. T h e uniqueness holds in the sense that, if {~o,,,(x)},>_, ! and {~,,,t(x)}t_>, are b o t h solutions to SDE (1.16) starting at x at time s, then P { ~ , , , ( x ) = ~o',,,(z) for all t _> s} = 1. It is remarkable that, in this special case and with specific modifications, there exists a s y s t e m of r a n d o m processes {~o,,,(x) : (W, ~-, P ) --+ M } t > , , s >__O,x e M such t h a t the following (1) and (2) hold true: (1) For each x 9 M and s >_ 0 , { ~ , , t ( x ) } t > , is a solution to SDE (1.16) starting at x at time s; (2) We write ~o,,t(x) : (W, J:, P) ~ M as ~o,,t(x, .) for the sake of presentation. T h e n there exists a measurable set N 9 ~ with P ( N ) = 1 such t h a t for e a c h fixed w 9 N , ~ , , t ( . , w ) : M --+ M , x , , ~ , , t ( x , w ) defines a C 1 diffeomorphism on M for a l l 0 < s < t . More crucially, the Markov p r o p e r t y of the s y s t e m of solutions {~o,,t(x)},>_,, s > 0, x ~ M manifests itself as the independence p r o p e r t y (the p r o p e r t y (d) t h a t follows) of the family of C ~ diffeomorphisms {~',,t(', w) : 0 <_ s <_ t, w 9 N } . T h e detailed properties of the family of diffeomorphisms are s u m m a r i z e d in the following p a r a g r a p h . We write ~o,,t(., w) simply as ~o,,t(w). T h e n the set N of full P measure can be chosen such that the family of C ~ diffeomorphisms {~,,t(w)} has the following properties: (a) ~o~,t(w) = ~Ou,,(w)o~O~,u(W),S < u < t for each w 9 N; (b) ~o,,~(w) = id for each w e N; (c) p,,t(w) depends continuously on (s,t) (with respect to the C ~ topology of D i f f l ( M ) ) for each w 9 N; (d) ~t,q+~(') : N --* D i f f l ( M ) , i = O , - . . , n - 1 are independent for any 0__ 0. T h e p r o o f of the above facts is far from being easy as c o m p a r e d to the case of an ordinary differential equation. It requires a lot of careful a r g u m e n t s a b o u t null sets. We refer the reader to the i m p o r t a n t treatise [Kun]l of K u n i t a for a detailed proof. Let {~o,,t(w) : 0 < s < t , w 9 N } be as o b t a i n e d above. Define ~ot = ~o0,t for t _> 0. T h e n {~ot}t>0 is clearly a stochastic flow of C 1 diffeomorphisms and 117

(~s,t(W) ---- ~ t ( W ) 0 ~ s ( W ) - 1 for all 0 _~ s _~ t and w E N. We call {~t)t_>0 a stochastic flow generated by solutions of SDE (1.16). When the coefficients X1, ..., Xd are vector fields of class Cr+2(r _> 1) and X0 is of class C r+l, the stochastic flow {~t}t_>0 can be chosen such that it is a stochastic flow of C ~ diffeomorphisms (see [Kun]2). Conversely, for a given stochastic flow of Cr(r >__1) diffeomorphisms with a suitable regularity condition, one can write down an SDE such that its solutions generate the given flow. In the general case, however, such an SDE has to be based on an infinite number of Brownian motions, or equivalently a Brownian motion with values in vector fields. In this sense, there exists essentially a one-toone correspondence between stochastic flows of diffeomorphisms and stochastic differential equations. For a detailed treatment of the precise relationship we refer the reader to [Kun]1. Let {~}t>_0 be a stochastic flow generated by solutions of SDE (1.16). The transition probability functions of the solutions of SDE (1.16) are clearly given by P ( t , x , A ) = P { w : ~t(w)x E A},t > O,A E B ( i ) , x E M. They are uniquely determined by the generator of SDE (1.16) 1

d

L = ~E

X2 § X0.

(1.17)

k----1

Hence L determines all the invariant measures of {~t}t>0. If L is elliptic, or equivalently SDE (1.16) is non-degenerate, i.e. { X l ( x ) , . . . , Xd(X)} spans T , M for each x E M, then {~at}t>0 has a unique invariant measure #. The measure # has a smooth density with respect to the Lebesgue measure )~ on M induced by the Riemannian metric, i.e. d#/d$ = p for some smooth function p : M --+ R +. This is because p is a solution to the adjoint elliptic partial differential equation and Schwartz-Weyl l e m m a holds true (see, e.g. Chapter V of [Ike]). R e m a r k 1.5. Since we regard in this chapter stochastic differential equations as dynamical systems disturbed by noise, we consider here a Stratonovich SDE rather than an It5 SDE. If we want to consider an It5 SDE on M, then we should not regard the coefficients of the equation as vector fields (see Section II. 8 of [Kun]~).

w Lyapunov Exponents and Stable Manifolds of Stochastic Flows of Diffeomorphisms In this section we present a version of Oseledec multiplicative ergodic theorem for a stochastic flow of diffeomorphisms. This theorem establishes the existence of Lyapunov exponents for the flow, which describe the almost-sure limiting exponential growth rates Of tangent vectors under the flow. Then we establish the existence of (global and local) stable manifolds for the flow associated with the Lyapunov exponents. 118

Now suppose that we are given a probability space (IV, U, P), a stochastic flow { ~ t : (W, Jz, P) --* Diffc(i)}t>0(r _> 1) and an invariant measure # of the flow. Let (W x M , U x B(M), P x #) be the product space of (W,.~, P) and (M, B ( M ) , #). T h e o r e m 2.1. Let ({~ot}t>0, p) be as given above with r >_ 1. Then we have a measurable set A0 C W • M of full P • p measure such that for each (w, x) E A0, there exist numbers depending only on x -~

< ~(1)(x) < a(2)(x) < ... < a(~(~))(x) < + ~

(2.1)

and an associated filtration by linear subspaces of T~M {0} :

V,(~,~) (0) c

K(r(~)) (w,~) =

V,(~,~) (1) C ... C

T~M

(2.2)

such that lim _1log IT~o,(w)r

t~+oo t

for each ~ E

= A(~)(x)

(2.3)

v'(i)(w,x)\"(w,x)\1), lz(i-l < i < r(x) and ]im 1 log I T ~ t ( w ) l = A(r(~))(x),

t--*+~

(2.4)

~(~) lim

t-*+oo

llogldet T~,,(w)l = ~ a(~)(x)m~(x)

(2.5)

i=1

where mi(x) = dim E(i)(~,~) - dim l~.~) ),,

which also depends only on x. In ad-

dition, the numbers r(x),~(O(x) and the subspaces v((i),,:) all depend measurably

on (w, x). R e m a r k 2.1. According to Fubini theorem, Theorem 2.1 implies that for # almost every x E M there exist the numbers in (2.1) such that for P almost all w E W there exists a filtration (2.2) satisfying (2.3)-(2.5). We shall call the numbers ~O)(x) < ... < ~(r(~))(x) the Lyapunov exponents of ({~}t_>0,P) at point x E M and call mi(x) the multiplicity of ~(O(x). Proof We prove the theorem first for discrete time steps of length 1. Denote Diff1(M) by X for simplicity of notation. Let vl be the distribution of ~1. Then by Proposition 1.2 it holds that x[log+ [flc1 + log + [f-l[c~]dvl(f) < +oo.

(2.6)

(X N , B ( X ) N , vN) = H ( x , B ( x ) , vl)

(2.7)

Put 1

119

and define a map Y]I

" (W:~t~,P)

~

(xN,I~(x)N,vN), (2.8)

el

) (~t:~l(W) 0 ~ 0 ( w ) - l , ( / 9 2 ( W )

O ~I(W)-I,

..-).

By Properties (i) and (ii) in Definition 1.1 we know that E 1 is a measurepreserving map. Now note that Theorem 3.2 in Chapter I also holds true if [2 = DiffU(M) is replaced by X = Diffl(M) and / . i log + I f l c , d v ~ ( f ) < +o0. J A

This together with (2.6) shows immediately that Theorem 2.1 holds true if we consider only t E Z +. In order to obtain the full continuous time) result, note that if ( w , x ) C Wx M and~CT~MthenforallnE Z + and t E [n, n + l] we have

- log + IT~,(~o)~(~+l(W) o ~t(w)-l)l

log I Z ~ + l ( w ) r log IT~,(w)~l

(2.9)

_< log IT~p,~(w)~C[ + log + IT~o.(~)~:(~t(w) o ~ ( w ) - l ) [ , log IT,~=+l(w)[ - log + IT~o,(~),(~',~+~(w) o ~t(w)-l)l

log IT~,(w)l

(2.10)

log I T ~ ( w ) l + log + IT~.C~)~(~,(w) o ~ ( w ) - l ) l , and log I d e t T ~ . + l ( W ) l - log + IT~,(w)~(~o~+a(w) o ~ot(w)-~)[ ~o _< log IdetT~t(w)l

(2.11)

log I d e t T ~ ( w ) l + log + [T~.(w)~(~t(w) o

~fln(W)-l)l

rn~ .

Thus, if we put r

=

sup

log + I~t(w) o~o,~(w)-llcl

(2.12)

n
and 9 ,(w) =

sup

log +

I~+l(w)o~(w)-llc1,

(2.13)

n
then the desired full result follows from (2.9)-(2.11) provided that for P almost all w E W lim 1 0 ~ ( w ) = 0 (2.14) n~+oo

n

and lira l~/n(w) -- O.

n--* +oo //

120

(2.15)

So it remains to prove (2.14)_and (2.15). Put 2 = { f : f - - {ft : ft 9 X , t > 0} such that fo = id and t l ' ft is a continuous map from R + to X (with respect to the C 1 topology o n X ) _ } . Let B(X) be the smallest c~-algebra containing all the cylinder sets { f : ft, 9 Fi,1 < i < . ) , r l , : . . , r , 9 t (x),0 < < 9.. < t~,n > 1 and let P be the measure on ( X , B ( X ) ) which on a cylinder set { f : j~, 9 Fi, 1 < i < n} is given by

/r

P(t~ - tn-l'gn-l'dg'O'

P(tl'id'dgl) ~r P(t2 - tl'gl'dg2) " "ffv 1

2

n

where P(t,g,V) = v,(rg-1),t >_0 , r 9 B(X),g 9 x and vt is the distribution o f ~ t . For each s >__0 we define a map 0, : X ~ )f by (Osf)t = L+t o f-s-l, t >_ 0, f 9 )(. _By properties (i)-(iv) in Definition 1.1 it is easy to verify that 0, preserves P for each s > 0 and the map

E~ : (W, .T, P) --, (f(, B(X), P),

w,

is measure-preserving. We now define (I),kO : X ,

, { ~ ( w ) : t > 0} , R + by

( ~ ( f ) = sup log+lJ~lc,, 0
9 (]) = sup log + IL o f? lc . O
From Proposition 1.2 it follows that (I),~ E L I ( ) ( , B(X), P). Then, by Birkhoff ergodic theorem, one has lim •

=0

n---* + oo n

and lim l ~ ( o ~ f ) = 0 n~+oo

n

for /5 almost all f E X. This together with the measure-preserving property of E i proves (2.14) and (2.15). [] In what follows we show that, to the negative elements of the Lyapunov exponents (which indicate the exponential shrinking rates of the tangent vectors under the flow), there correspond stable manifolds in M on which the trajectories of the flow cluster together at exponential rates. We first have the following theorem concerning the global stable manifolds of the flow.

Let ({~t}t>0, #) be as given above with r > 2. For (w, x) E A0 (see Theorem 2.1), if )~(1)(x) < 0 and ,~(1)(X) < < )t(P-)(X) are the strictly negative Lyapunov exponents at x, we define W " l ( w , x ) C ... C W"P(w,x) by

T h e o r e m 2.2.

'

'

'

W S ' i ( w ' x ) = ( Y E M : limsup ~ l~ d(pt(w)x'~t(w)Y) < A(i)(x) } 121

1 < i < p. Then for P • # almost every (w, x), w " i ( w , x) is the image oCV,(~,~) (0 under an injective immersion of class C 1,1 and is tangent to V,(~ (i),~) at point x. In addition, if y E W~'i(w,x), then limsup _1log dS(~ot(w)x, ~ot(w)y) < A(i)(x) t---* +oo

t

where dS( , ) is the distance along the submanifold pt(w)W~'i(w,x). Proof. We also prove the result first for discrete time steps of length 1. Denote f2 = Diff2(M) and let vl be the distribution of ~1. By Proposition 1.2 one has log+ If[c2dva(f) < +oc. Put (f~N, B(f~)N, v N) = X-~(~, B(f2), vl) 1

and, similarly as in (2.8), define a map E2:

(W, 5r , P ) ~ (a N.B(fl) N , v N ) ,

(2.16) W l

) (~I(W)

O ~90(W) - 1 , ~ 2 ( w )

O (pl(w)--l,

.. .).

Then E2 is also a measure-preserving map. Thus, by applying Theorem III. 3.2, the theorem is proved if we consider only t E Z +. In order to prove the full (continuous time) result, note that, if y , z G WS,i(w, x), then for all n k 0 and t E [n, n + 1]

=

<

dS(r176176176176

sup

n
z)

(2.17)

I~ot(w) o ~ n ( w ) - l l c l d ' ( ~ ( w ) y , ~ ( w ) z ) .

Put Cn(w) =

sup

n<_t
]~ot(w)o ~ , ( w ) - l I c ,.

(2.18)

From (2.14) it follows that lim 1_ log + C,~(w) = O,

n--*+oo

This together with (2.17) proves the theorem.

P - a.e.w.

(2.19) []

As for the local properties of the stable manifolds WS'i(w, x) we have the following local stable manifold theorem. 122

T h e o r e m 2.3. Let ({~ot}t>0,/~) be as given above with r > 2. Given A < O, we put A0~ = { ( w , x ) 9 A0 : AO)(x) < ~ and,~(O(z) r ,~,1 _< i < r(x)} (A0 is given in Theorem 2.1). Then we have a measurable set A ~ C A~o with P x #(Ao~\A A) = 0, and measurable functions c~, fl, 7 : Ax --~ (0, +Go) such that s~)t for each (w, x) 9 A A there exists a C 1 embedded disc WIG c (w, x) which contains

x and has the following properties: 1) WiG ,,xc ( w , x ) is tangent to V((:) ) at point x, where i is such that A(i)(x) < < ~('+~)(~), and W;o2(W,X) = V'(i)(w,x) : I~[ < a(w, x)} ---* tiv,(i)(,~,,)~ ~{Lip(h(,~,,)), Lip(T.h(~,~))} < fl(w, 2) I l y , z 9 Wio~ (~,~), then for

exp~ Graph (h(~,~)) where h(~,~) : {~ 9 is a C 1,1 map with h(,~,~)(0) = 0 and max x); ant > o

d~(~ot(w)y, ~ot(w)z) <_ 7(w, x)d'(y, z)e xt where d~( , ) is the distance along the submanifold ~,(w)W~oXc (w,x). Proof. P u t A0~'1 = {(w,x) 9 A0X: ,~(i)(x) 9 ( - c o , , \ - 1)} and A0~'' = { ( w , x ) 9 A0~ : ,~(i)(x) 9 [ A - ( l - 1)-1,,~ - l - l ) } , / > 2 where / i s such t h a t )gO(x) < A < ,~('+l)(x) ()gi+U(x) = + c o if i = r(x)). In a way analogous to t h a t in the proof of T h e o r e m 2.2, it follows from T h e o r e m III. 3.1 that for each A0~'t we have a measurable set A x,l C A0~'z with P x #(A0~'Z\A x,z) = 0, and measurable functions at,flz,Tl : A x'l ~ ( 0 , + c o ) such t h a t for each (w,x) 9 A x't there exists a C 1 embedded disc WiGc (w, x) which has the following properties: z l 1)' WiG~,), ~ (~,x) _- exp.~ Graph (h(,~,=)) where h(,.,=) : {~ 9 Tx(i) "(,o,=) : I~I < i --, (IS.(i) ~x is a C 1'1 map such that hl~,=)(0 ) = 0, To h (~,~) = 0 and max {Lip(hl,o,,)), Lip(T.hl~,~)) } < Z~(w, z);

~'~" w, x) and n 9 2) ~For a n y y , z 9 W,loot d'(~.(~)~, ~.(~)z)

+,

< 7t(w, ~)d'(y, z)e ~'"

where ~z = ,~ - (2l) -1. Now note t h a t by (2.19) we may define a measurable and P almost everywhere finite function K : W --* (0, +co] by

K(,,,) = sup C~(,,,)e _ l__k} ~, , k>O where Ck(w) is defined in (2.18). Choose a measurable set r c w with P(F) = 1 such t h a t K ( w ) < + c o for each w E r . T h e n for each l >_ 1, if ( w , x ) E A x,i A (F x M) and y, z e Wlo c (w, x), we have

dS(~.ot(w)y, ~ot(w)z) < C.(w)dS(~.on(w)y, ~n(w)z) < K(,o)e~"7,(w, x)d'(y, z)e ~'" < e-~K(w)~z(w, x)d'(y, z)e ~ 123

for a l l n E Z + a n d t E [ n , n + l ] . Setting A ~ = U1>1[A~'I N (F x M)] and defining a(w, z) = e~,(w, z), 13(w, z) = /3t(w, z), 7(w, x) = e-~K(w)Tl(w, x) and h(w,~) = h'(~o,~) if (w, x) E A~'Z 71 (F x M), we get the proof completed. [] The results above of this section are actually an extension of the work of Ruelle [Rue]2 to the case of stochastic flows of diffeomorphisms, though our techniques used in dealing with the local stable manifolds are rather different from those of Ruelle. The programme of extending Ruelle's ergodic theory of (deterministic) differentiable dynamical systems to the stochastic case was suggested by L. Arnold at Les Houehes, June 1980 and was fulfilled by A. Carverhill [Car] for stochastic flows of diffeomorphisms generated by stochastic differential equations. By the present time, there have been many papers and books concerning various applications as well as estimation and calculation of Lyapunov exponents of random dynamical systems. For further information we refer the reader to the recent volume [Arn] and [Elw]l and the references therein.

w Entropy

of Stochastic

Flows

of Diffeomorphisms

We now turn to the notion of (measure-theoretic) entropy of a stochastic flow of diffeomorphisms. This is an extension to the continuous time case of what was introduced in Section 1.2 for a random dynamical system generated by i.i.d. random diffeomorphisms. Now let {~t : (W,.T,P) ---* Difff(M)}t>_o(r >_ 1) be a stochastic flow and # an invariant measure of the flow. D e f i n i t i o n 3.1.

Let to > 0 be given, ff ( is a finite measurable partition of

M, then the limit htu~

n--.+colim- - n l / H u ( V ~ O k t o ( w ) - l { )

(3.1)

is called the to-time-step entropy of ({~ot}t>o,p) with respect to ~. The limit (3.1) does exist and it holds that

ht~176

~> ,ninf -_, 1 / H ~

(~kt~ ',k=o

(3.2)

In fact, we put

o_1

) (3.3)

124

Since for any s > 0 the map E~ :

(W, .T, P ) --~ (X N , B(X) N , vN )

(3.4) w,

, (~.(w)o~o(w)-1,~2~(~)

o~.(w) -1, ...)

(where X = Diffl(M) and v~ is the distribution of ~,) is a measure-preserving transformation, by almost the same argument as in the proof of Theorem I. 2.1 one has

an+m(to,~) < a,(to,~) + am(to,~) for all n, m > 1. Thus, according to the proof of Theorem 0.4.1, lim

1

- a . ( t o , ~ ) = inf l a n ( t o , ~ ) .

n----~-4-oo n

n>l

n

This proves (3.2).

Let ({~t}t>_0,P) be as given above and let to > 0 be given.

D e f i n i t i o n 3.2.

Then

~),

h~O({~,},>_o) do__rsup h~,~

where the supremum is laken over the set of all finite measurable partitions of M, is called the to-time-step entropy of ({~Ot}t>_0, I-t). As in the case of a deterministic flow of diffeomorphisms, we have the following result, which is adopted from [Kif]l. P r o p o s i t i o n 3.1.

For any to > 0, ht~

= toh~({~vt}t>_o).

Proof. Still write X = Diff1(M) for simplicity of notation. s > 0 let v, be the distribution of ~,. Define Fs:

(3.~) For any given

(X N x M , B ( X ) TM x B(M),v TM x #)

(~, x) ~-~ (r~, f0(~)x) where we write w = (fo(w), fl(w),...) and ~- is the left shift operator on X N. Analogously to Proposition I.l.1, v TM x p is F,-invariant. Since E~ defined by (3.4) is a measure-preserving map, in the same way as the proof of Theorem 1.2.2 we have s h.({~,},>0) =

h ~v 1N7 6•

125

(F~) ,

(3.6)

where (r0 is the o'-algebra {P x M : F 6 B(X) N } on X N x M. Also, by T h e o r e m 0.4.3, for any k _> 1 hks({~t}t>o)

(7 0

=

O0

N

hv~" xu

=

(Fks) = hvN • (F~)

kh "~

v ~'~l X l~

"

(3.7)

(F,) = kh;({~vt}t>o).

Form this it follows that (3.5) holds true for all rational numbers to > 0. In order to prove (3.5) for all real number to > 0, choose an increasing sequence of finite measurable partitions ~1 ~ ~2 _~ '" " of M such that the union of the boundaries OA~ = A ik \ m . t ( A ki ) of elements A ik of the partitions (k has # measure zero and lim diam((k) = 0. k~+ov

Note that Theorem 1.2.5 also holds true if ft is replaced by X = Diffl(M). Then for any s > 0 we have

h~({~t}t>_o)=

lim h~({~ot}t>>_o,(k)

k~+oo

=

lim

k~+oo

=

lim

la,~(s,(k)

lim

n~+oo

rt

(3.8)

inf la,~(s,~k)

k--++oo n > 1 /2

by (3.2), where a~(s,{k) is defined by (3.3). Since the union of the boundaries of all elements of the partitions {k has # measure zero, we know that for any given s > 0 the union of the boundaries of all elements of the partitions ~o, (w)-l{k has # measure zero for P almost all w E W. / ,

Indeed, if A E B(M) and a(A) = 0, then #(A)

=

/#(~t(w)-lA)dP(w)

=

0

t /

implies #(~ot(w)-ld) = 0 for P almost all w. Thus the above assertion holds true. This together with property (iii) in Definition 1.1 yields that a~(t,{k) is continuous in t. Then, by (3.8), " lim hu({~ot}t>0 ) =

8-'*80

< --

=

lim lira i nn f l na n ( S , ~ k ) ,--.,ok--*+oo 1

lira inf lim - a n ( s , ~ k ) k~+(x:,

n

s'~'Jo n

(3.9)

lim inf =-a,~(So,~k)

k--,+oo

n

n

=

for any So > 0. = Consider the function r = 7lhs~ ({~t}t>0). By (3.7) we know t h a t r r for any rational number r > 0. On the other hand, from (3.9) it follows 126

that r is upper semi-continuous. Since the rational numbers are dense in R, these two conditions together imply that r = r for all s > 0. The proof is completed. [] Ruelle's inequality and Pesin's entropy formula respectively worked out in Chapter II and Chapter IV for i.i.d, random diffeomorphisms can be easily carried over to the case of a stochastic flow of C 2 diffeomorphisms. In fact, assume that ({~'t}t>0, It) is a stochastic flow of C 2 diffeomorphisms. Let D = Diff2(M) and let vl be the distribution of ~1- Then, as we have seen in the previous discussions in this and the last sections, from the measure-preserving property of the map ~2 defined by (2.16) it follows that for #-a.e. x the Lyapunov exponents /~(1)(X) < - . . < /~(r(x))(X) together with their respective multiplicities m i ( x ) , l < i < r(x) of ({~t}t>_o,it) coincide with those of X + ( i , vl,#) (see Section I. 1) at point x and h~({~t}t>_o) = h~,( zY+(M, vl)). In addition, Propo/ t

sition 1.2 asserts t h a t / l o g + Iflc~dvl(f) < +cx~ and log IdetT~fl is integrable , /

in (f, x) with respect to Vl x It (see Remark IV. 1.1). Thus, by Theorem lI. 0.1 and Theorem IV. 1.1, we obtain T h e o r e m 3.1. Assume that {~Pt}t_>0 is a stochastic flow of C 2 diffeomorphisms and It is an invariant measure of the flow. Then


(3.10) i

Moreover, if p << Lab., then

=

f

(3.11) i

We shall call respectively (3.10) and (3.11) Ruelle's inequality and Pesin's formula for the stochastic flow ({~t}t>0,p)If {~t}t>0 is a stochastic flow of C ~ diffeomorphisms arising from a nondegenerate SDE of the form (1.16) and/~ is the unique invariant measure of the flow (see Section 1), then Pesin's formula (3.11) holds true for ({~vt}t>0, #). It is remarkable that (3.11) is valid for arbitrary SDE's satisfying only the smooth and non-degenerate conditions, no other conditions like hyperbolicity or its like are required. This is in sharp contradistinction with the deterministic case.

127

Chapter VI Characterization of Measures Satisfying Entropy Formula

We consider in this chapter systems generated by two-sided compositions of random diffeomorphisms. Our main purpose here is to prove that Pesin's entropy formula holds true in this random case if and only if the sample measures, i.e. the natural invariant family of measures associated with individual realizations of the random process have Sinai-Bowen-Ruelle (SBR) property. Roughly speaking, we say that the sample measures have SBR property if their conditional measures on unstable manifolds are absolutely continuous with respect to Lebesgue measures on these manifolds. The idea of the above result goes back to the ergodic theory of Axiom-A attractors. Recall that, if f is a twice differentiable diffeomorphism on a compact manifold N and A is an Axiom-A attractor of f with besin of attraction U, then there is a unique f-invariant measure p with support in A that is characterized by each of the following properties: (a) p has absolutely continuous conditional measures on unstable manifolds; (b) Pesin's entropy formula holds true for the system (N, f,p); (c) There exists a set S C U 1 ~--~n-- 1 such that U\S has Lebesgue measure zero and lim~--.+oo ~ z_~k=o/51k~ -- p whenever z E S. The measures with the above properties were first shown to exist by Sinai ([Sin]) for Anosov diffeomorphisms and this result was later extended to Axiom-A attractors by Bowen and Ruelle ([Bowl2 and [Rue]3). These measures are then called SBR measures. Let us emphasize here that each one of properties (a)-(c) has been shown to be significant in its own right, but it is also striking that they are equivalent to one another. In addition, we remark that Y.Kifer gives another equivalent characterization of such measures via their stochastic perturbations (see [Kif]4). Some of these results for deterministic uniformly hyperbolic systems have been shown to remain valid in more general frameworks. A well-known theorem of Ledrappier and Young (Theorem A of [Led]2) asserts that properties (a) and (b) remain equivalent for all C 2 diffeomorphisms on compact manifolds. What this means is that, if N is a compact Riemannian manifold without boundary and f is a C 2 diffeomorphism on N preserving a Borel probability measure m, then a sufficient and necessary condition for the validity of entropy formula hm(f) -- f Ei A(0(z)+mi(z) dm is that m has absolutely continuous conditional measures on unstable manifolds (That (a) implies (b) for this case is proved by Ledrappier and Strelcyn in [Led]3). The main result of this chapter thus turns out to be a generalization of the above theorem to the random case. This generalization was actually first mentioned by Ledrappier and Young themselves, though not clearly stated (see [Led]l for the idea). We present here a (first) detailed treatment. Although the technical details are quite different, our proof here follows the main ideas in the deterministic ease given by [Led]3 and [Led]2. This chapter is organized as follows. In the first part (Section 1) we introduce the relevant concepts of ergodic theory of systems generated by two-sided 128

compositions of random diffeomorphisms, then we formulate the main result (Theorem 1.1) of this chapter and give an important consequence (Corollary 1.2) of this result. The second part (Section 2) is devoted to the proof of the " if " part of Theorem 1.1. The third part (Sections 3-8) consists of a detailed proof of the " only if " part of Theorem 1.1. The first part is basic. The second part is fairly easy for readers who are familiar with Chapter IV, and the conclusion and arguments of this part will be very useful when we deal with in Chapter VII hyperbolic attractors subjected to random perturbations. The third part is not essential for those readers who are not interested in technical details of the proof of the " only i f " part of Theorem 1.1. Such readers can omit this part on the first reading.

w Basic C o n c e p t s and F o r m u l a t i o n of t h e M a i n R e s u l t For the sake of clearity of presentation, we divide this section into several subsections.

A. The General Setting As in the previous chapters, let M be a C ~176 connected compact Riemannian manifold without boundary, and write m0 = dim M and f2 = Diff2(M). Suppose that v is a Borel probability measure on ~ satisfying

{ f~log + Iflc~dv(f) < f~log + I f - ' l c ~ d v ( f )

+oo < +oo.

(1.1)

In this chapter we consider the evolution process generated by forward and backward successive applications of randomly chosen maps from f2, these maps being independent and identically distributed with law v. More precisely, let v z) =

+oo

v)

be the bi-infinite product of copies of (ft,B(ft), v). For each w = ( . . - , f - l ( w ) , fo(w), fl(w),...) 6 gtz and n > 0, define fo = id,

f~

----f n - l ( w ) o f n - 2 ( w ) o . . . o fo(w), f w n = f _ n ( W ) - 1 o f _ n + l ( W ) -1 o . . . o f _ l ( W ) -1. We are here concerned with the random system generated by actions on M of {f~ : n 6 Z,w 6 (~2z,B(f~)Z,vZ)} and we denote this set-up by A'(M,v). Let us notice that, when dealing with a system ~ ( M , v), one can consider simultaneously the associated forward system ~ + ( M , v) as discussed in previous chapters. Relationship between these two systems will play an important role in this chapter.

129

Throughout this chapter, the spaces a z and a z x M are always endowed with the product topology. Recall that

~(~)z = ~(az) and B(f~) z x B(M) = B ( a z

x

M)

(see Section 1.1). Also, to repeat, put for each w = ( . . . , f - l ( w ) , fo(w),fl(w), ...)ca z w + ----(f0(w), f l ( w ) , . . . ) ,

W- = (.'. ,f__2(w),f-l(W))

and define maps P1 :f~z x M - - , a

z,

(w,x)~-,w,

P2 : a z x M - + M , e

:a z xM~a

(w,x)~-+x, N x M,

(w,x)~-+(w +,x)

and

G:a z xM~f~z xM,

(w,x)~-~(vw, fo(w)x),

where ~- is the shift operator on a z.

B. Invariant Measures, Sample Measures

A Borel probability measure It on M is called an invariant measure of 2d(M, v) if

D e f i n i t i o n 1.1.

~ f#dv(f) = # where (fit)(E) = # ( f - l E ) for all E e B(M) and f 9 n. We denote by A d ( • ( M , v)) the set of all invariant measures of 7Y(M, v). Obviously, M ( ~ ( M , v)) = Nt( X + ( M , v)). When associated with an invariant measure p, &Z(M, v) will be referred to as A'(M, v, #). Given X ( M , v, p), by Proposition 1.1.2 we know that there exists a unique Borel probability measure It* on a z x M which satisfies Git* = It* and Pit* = v N x #. Henceforth, unless indicated otherwise, the g-algebra associated with ( a z x M, #*) is always understood to be the completion B~. ( a z x M) o f B ( a z x M) with respect to It*. Since a z x M is a Borel subset of the Polish space H+~C2(M, M) x M, by Theorem 0.1.10 we know that ( a z x M, It*) is a Lebesgue space. Let {#~w}• : w E a z} be a (essentially unique) canonical system of conditional measures of #* associated with the measurable partition ~ = {{w} x M : w E a z } . Identifying any {w} x M with M and denoting It~w}xM simply by #w, we obtain a family of Borel probability measures {it~}weaz on M.

130

Definition

1.2.

{#w}weaZ

is called the family of sample measures of

X(M,v,I~). T h e following proposition says t h a t {#~}wef~z is intuitively a n a t u r a l invariant family of measures associated with individual realizations of the r a n d o m process ~ ' ( M , v, #). P r o p o s i t i o n 1.1. Let X ( M , v, p) be given. Then the family of sample measures {/z,0},oenz of X ( M , v , # ) is the vZ-mod 0 unique family of Borel probability measures on M such that the following 1)-4) hold true: 1) ~o ~

~(^~)

is a measurable function on

(nz,t~.~(nz))

f o r any ^ 9

B ( ~ z x M), where Bvz(f2 z) is the completion of B(~ z) with respect to vZ; 2) fo(w)#w -: #rt~, v z -a.e. w; 3) I~,~ depends only on w - for v z -a.e. w;

4) f ~ d v Z ( ~ ) =

~-

Moreover, for v z -a.e. w we have f ~ _ , ~ # ---+#~ as n ---++oo. Proof. Let { # ~ } w e a z be the family of sample measures of X ( M , v , p ) . By the definition of {#~},~enz and Proposition 1.1.2, 1),2) and 4) follow i m m e d i a t e l y f r o m the general properties of conditional measures. We now prove the last conclusion which implies 3) clearly. Recall t h a t we use c~+ in C h a p t e r I to denote the partition P - l { { w } x M : 0~ 9 ~ N } of D z x M . Now for each n > 0 we put cr+ = Gnu+ and denote also by a + the e - a l g e b r a consisting of all measurable e+-sets. Let {g{ : i 9 N} be a dense subset of C ( M ) (the space of all continuous functions g : M --+ R ) . For each i 9 N we define 9{ : ~2z x M ---* R , (w, z) ~ gi(x). Since or+ 7 ~, one has for e a c h i E N lim E(Oila +) = E(Oila), p* - a.e. which implies

,,mj

n~Jro0

gidf~-~# =

/

gidpw, v z - a.e.w.

Since {gi : i 9 N } is dense in C ( M ) , then for v z - a.e.w 9 Dz we have

for all g E C ( M ) . This means t h a t

as n ~ -t-c<) for v z - a.e.w. On the other hand, let {#~ }~enz be a family of Borel probability m e a s u r e s on M with properties 1)-4). Define a Borel probability measure/~* on Dz x M by /9,*(A)

/ [

^ e B(r z x M).

-

J J 131

From Properties 2)-4) it is easy to see that/~* is G-invariant and Pt~* = vN x ft. By Proposition 1.1.2 we then obtain ft* = ~*. Thus, by the essential uniqueness of conditional measures, {p~}~eaz is the family of sample measures of Pt'(M, v, #). [] D e f i n i t i o n 1.3. Let p 9 .M( ?d(M,v)). We say that p is ergodic if p is an ex-

treme point of.M( X ( i , v)). In this case we also say that zY(M, v, p) is ergodic. From Proposition 1.1.3 we see that the ergodicity of p E .M( X ( M , v)) coincides with that of p as an invariant measure of z~+(M,v). Recall that if p E .M( Pd(M, v)) then the following three conditions are equivalent: (t) p is ergodic; (2) F : (QN • M, v N x p) ,--, is ergodic; (3) G : (Qz x M,p*) ~ is ergodic. We now denote by A4~(P~(M,v)) the set of all ergodic invariant measures of Pt'(M, v). In what follows we discuss briefly ergodic decompositions of invariant measures of ~ ( M , v). Consider now ~'+ (M, v) simultaneously. Note that the transition probabilities P(x, .), x E M of ~ + ( M , v) as defined in Section IV.1 can be viewed as transition probabilities of a Markov chain {X~}~>0, i.e. Xk+l has the distribution P(x,.) provided Xk = x, k _> 0. Employing ergodic decompositions of Markov processes (see Appendix A.1 of [Kill1 and see also Chapter 13 of [Yos]), we know that there exists a measurable partition r/0 of M into a collection of disjoint sets {C~}~eA and M \ U~e.a C~ such that the following hold true: i) U~e.aCo has full measure with respect to any p E M ( 9 : ' ( M , v)); ii) For each a E A and x E C~,P(x, Co) = 1; iii) For each a E ,4 there exists a unique invariant measure p~ of AZ(M, v) concentrated on C~ and this measure is ergodic; iv) For any # E M ( 7Y(M, v)), {P~}~eA is a canonical system of conditional measures of # associated with the partition r/0. In view of Proposition 1.1.2 and Rohlin's ergodic decomposition theorem about measure -preserving transformations (see the arguments at the end of Section 0.5 or [Roh]3), one can show that, corresponding to the partition r/0, there exists a measurable partition 00 of Dz x M into a collection of disjoint sets {C~}~e.a and f~z x M \ U,~EA C~ such that: i)' has full p* measure for any tt 9 .M( Ae(M, v)); ii)' For each a 9 . A , G - 1 C a = Ca,P*a is concentrated on d~ and G " (O~, p ; ) ~ is ergodic. iii)' For any # 9 Ad( zY(M, v)), {P~}~eA is a canonical system of conditional measures of tt* associated with 7)o; iv)' For any # E .M( Y ( M , v)), the g-algebra consisting of all measurable 00-sets equals #*-rood 0 the C-algebra {A E Bt,- (ft z x M) : G-~A = A}. As an easy consequence of properties i)'-iv) t we have

U,~eAd~

C o r o l l a r y 1.1. Let # E Ad( Pd(M,v)). H " ~ z x M ---+R U {co} satisfying

Then for each measurable function

H o G = H,l~* - a.e. 132

there exists a measurable function h " M --, R U {cr

such that

H ( w , ~) = h(~), ~* - a e.(~, ~).

C. Entropy, Lyapunov Exponents Let z~(M, v, #) be given. From Theorem 1.2.1 it follows clearly that for each finite measurable partition ~ of M the following limit exists n-1

hu(X(M,v),~)d-~

lim _I / H .

n~+oo n

)

( y o ( f ~ ) -1r

dvZ(w).

Definition 1.4. hu( ?d(M, v)) def' sup{hu( z~(M, v),~) : ~ is a finite measurable partition of M} is called the (measure-theoretic) entropy of z~(M, v, #). By this definition and Theorems 1.2.2 and 1.2.3 one has hu ( X(M, v)) = h u ( zY+ (M, v)) = h i. (G).

(1.2)

We now turn to Lyapunov exponents of z't'(M, v, #). Condition (1.1) implies that f[log

Ifie1 + log + If-llcl]dv(f) < +cx~.

Then applying Oseledec multiplicative ergodic theorem about invertible systems (see Appendix 2 of [Kat]) in a way completely analogous to Theorem 1.3.2, we obtain P r o p o s i t i o n 1.2. Let z~(M, V,l_t) be given. Then there exists a Borel set Ao C f~z • M satisfying p*(A0) = 1,GA0 = A0 and having the following properties: 1) For each (w, x) E Ao there exist a decomposition TxM = El(W, x) • E2(w, x) @... 9 Er(~o,x)(w, x)

(1.3)

< AO)(w,x) < ,\(2)(w,x) < . . . < ,~(r(t~

(1.4)

and numbers -~

z) < +cxD

such that

lira 1 log IT~fn~l = A(i)(w, x)

n ---*4-~ n

for any r e E,(w, ~) with ~ # 0, 1 < i < r(w, x). Moreover, the decomposition (1.3) and the numbers in (1.4) all depend measurably on (w,x) E Ao. They also satisfy for each (w, x) C Ao r(C(w, x)) = r(w, x)

133

and A(i)(G(w, x)) = A(i)(w, x), %fo(w)E,(w, x) = Ei(G(w, x)), 1 < i < r(w, x).

2) Let (w, x) E Ao and let p(1)(w, x) <_... <_ p(m~ x) denote A(1)(w, x) _< 9.. <_ AO)(w,x) <_ ... <_ A(~(~,~))(w,x) <_ ... < A(~(""~))(w,x) with A(')(w,x) being repeated m~(w, z)% f dim Ei(w, x) times. If{{~,. 99 , {too} is a basis of T,:M satisfying lim n ....-*4- o o

ln l o g l % f ~ i l =

p(i)(w,~),

1 < i < too,

then for any two non-empty disjoint subsets P,Q c { 1 , . - . , t o o } we have lim l logT(%f~Ep,T~f~EQ) = 0 n ---, -t- c ~

n

where Ep and EQ respectively denote the subspaces of % M spanned by {~i}ieP and

D e f i n i t i o n 1.5. The numbers A(i)(w, x), 1 < i < r(w, x) introduced above are

called the iyapunov exponents of 7Y(i, v,#) at point (w, x), rni(w, x) is called the multiplicity of A(i)(w, x). R e m a r k 1.1. Given X(M,v,tt), let AO)(x) < A(2)(x) < ... < A(r(~))(x) be the Lyapunov exponents of 7t'+(M, v, g) at point x E M, as introduced by Theorem 1.3.2, and let rni(x) be the multiplicity of A(i)(x). From 1) of Proposition 1.2 and Proposition 1.1.2 it is easy to see that for #*-a.e. (w,x) E ftz • M

T(w, x) = r(x) and

A(')(w,x) = A(i)(x),mi(w,x) = mi(x),

1 < i < r(x).

Moreover, the following equation holds clearly true: (1.5) i

'D. Unstable Manifolds Let X(M, v, it) be given. By an argument completely analogous to Lemma III.1.4, one can find a Borel set F0 C 12z with vZ(F0) = 1 and rF0 = F0 such that for any given e > 0 there exists a Borel function r : F0 --~ (0, +oo) with the following properties: For each w E F0 and x E M, the map 1 def. ~,~) = exp~l~ of~ 1 o exp~ : T~U(r(w) -1) --, TIja::M

F(-

134

is well defined, Lip(T.FCwl)) < r(w) and r(v-nw) < r(w)e ~n for all n e Z +. Let ~0 be as introduced in Proposition 1.2. Then we put s

• M)

and

s

Clearly, ~*(s

----{(W,X) 9 s

= 1, Gs = s

)t(i)(W, x) and Gs

> 0 for some

i}.

(1.6)

= s

Let [b, c], 0 ~ b < c, be a closed interval of R. Denote by Ab,~ the subset of s consisting of points (w, x) such that )~(i)(w, x) ~_ [b, c] for all 1 < i < r(w, x). If (w, x) 9 Ab,~, we put

E(w,~) =

r,

+E,(w, =),~+(~,~) :

~(,)(w,~)
+E,(~,~)

~ ~(~)(~,~)>~

Let k 9 {1,... ,rn0} and suppose that

/~b,~,kd~=f'{(W,X) 9 /~b,~ : d i m S ( w , x ) = k} ~ r Substituting G -1 : (~z x M,p*) ,-~ for F : (l) N • M,v N x #) ,-~ and the decompositions T~:M = U(w, x) @ E(w, x), (w, x) 9 /kb,r for T~M = Eo(w, x) H0(w, x), (w, x) 9 A~,b, one can easily adapt the arguments in Sections III.l-III.3 to the case of X(M, v,#) to obtain the following two results. P r o p o s i t i o n 1.3. Let Ab,c,k be given as above and let 0 < ~ < m i n { 1 , ( c b)/200}. Then Ab,c, k can be divided into a countable number of Borel subsets

{ A~b,c,k)i> l with each A~,c,k having the following properties: 1) E(~, ~) and Y(~, ~) depend continuously on (~, ~) 9 Ai,c, ~, 2) There exists a continuous family of C 1 embedded k-dimensional discs {W(w,x) )(w,x)ea[,o,~ in M together with numbers hi,& and 7i such that for each (w, x) 9 Aib,c,k, the following hold true: i) W(w, x) = expz Graph(h(w,~ ) : U(w, x) ---+E(w, x)) where U(w, x) is an open subset of g ( w , x ) that contains {~ 9 H(w,x) : I~1 < ~,} and h(,,,~) is of class C 1,1 and satisfies h(w,x)(0) = 0,T0h(~,~) -- 0, Lip(h(w,x)) <_ fll and Lip(T.h(~,~)) < ~; ii) d"(f~ty, f~Iz) < 7ie-(r for all y, z 9 W(w,x) and l 9 Z +, where d~'(., .) denotes the distance along fg,'W(w, z) for each I 9 X +. P r o p o s i t i o n 1.4. Let (w,x) G ~kl and let A(q)(w,x) < ..- < A(r(w,x))(w,x)

be the strictly positive Lyapunov exponents of X ( M , v , # ) at (w,x). w~,r(~,~)(w, ~) c ... c w~,q(w, ~) by WU'i(w'x)= { y e M : l i m s u p ln l~

135

<-

Define

for q < i <_ r(w,x). Then W~,i(w,x) is the image of ~j>i | x) under an injective immersion of class C 1,1. In addition, if y G W tt~i (w,x), then limsup I log d~'(ff~'~x, f f n y ) < _ ~(O(w, x), n---* + o o

n

where d~(., .) is the distance along the submanifold fg,'~w~,i(w, x ) = W~,i(G -'~ f o r each n C Z + .

R e m a r k 1.2. Given X ( M , v , # ) and (w,x) C f2z x M, the global unstable manifold WU(w, x) is defined by W"(w,z)--

{

yeM:limsup n~+co

llogd(f,; x,f; n

y)
}

.

In a way similar to Remark III.3.2 we know that, if (w, x) C A1 and k(q)(w, x) < --- < )~(~(~'~))(w, x) are the strictly positive exponents at (w, x), then =

x)

and hence W~(w, x) is the image of ~(0(w,~)>0 | x) under an injective immersion of class C 1,1, and in addition, W~'(w, x) is tangent to this subspace of T~M at point x.

E. S B R Sample Measures, F o r m u l a t i o n of t h e M a i n R e s u l t Let ?((M,v,p) be given and let s be as introduced in (1.6). For each (w, x) E f~z x M, if (w, x) r s we define W~'(w, x) = {x}. D e f i n i t i o n 1.6. A measurable partition ~ of ~2z x M with r] > cr is said to be subordinate to W~-manifolds of h~(M,v,#) if for p*-a.e. (w,x),r],o(z)d-~{y : (w,y) E r/(w,x)} C WU(w,x) and contains an open neighbourhood of x in W~'(w, x), this neighbourhood being taken in the topology of W~(w, x) as a submanifold of M. Now Suppose that r] is a partition of f2z x M subordinate to W=-manifolds of zY(M, v,#). For each w E f2z we denote by rh0 the partition {rho(x ) : x E M} of M. Let l(i.t*~o t~ J(~0,~)}(w,~)eaz • be a canonical system of conditional measures of #* associated with r] and let {(#~)~}zeM,W E f2z be defined analogously. Identifying {w} • rho (x) with r]~ (x), by the transitivity of conditional measures we have 9 o = for , * -a.e. D e f i n i t i o n 1.7. We say that the sample measures pw,w E f~z of X(M,v,/~) have absolutely continuous conditional measures on WU-manifolds, or equivalently that the family of sample measures {#w}weaZ has SBR property, if for every measurable partition r] subordinate to W~-manifolds of ?d(M, v, #) we have

136

for v z -a.e. w E ~2z, It ( # ~ ) ~ < < ~(~,~),

#,~-a.e.x E M

where )tu(w,z) denotes the Lebesgue measure on Wit(w, x) induced by its inherited hit Riemannian structure as a submanifold of M ((~,~) = 5~ if (w,x) ~ iX1). The main purpose of this chapter is to prove the following 1.1. For any given ~ ( M , v , # ) , the following two conditions are equivalent: 1) The family of sample measures {#w}wenz has SBR property; 2) Pesin's entropy formula holds true, i.e.

Theorem

h,( X(M,v)) = f E A(')(w,x)+rni(w,x)d# *. i

R e m a r k 1.3. We show in fact that, if the entropy formula in Theorem 1.1 is satisfied, then for #*-a.e. (w, x) the density d ( # ~ / d ) ~ ~. . is a strictly positive function that is locally Lipschitz along WU(w, x) (see Corollary 8.2). Theorem 1.1 together with Theorem IV.I.1, (1.2) and (1.5) implies the following C o r o l l a r y 1.2. Let z~(M,v,#) be given and suppose that tt < < Leb. Then the family of sample measures {ttw}we•z has SBR property. The remainning part of this chapter is devoted to the proof of Theorem 1.1.

w SBR Sample Measures: Sufficiency for Entropy Formula Our purpose in this section is to prove that Theorem 1.1 1) implies Theorem 1.1 2). Let r~(M, v, #) be given. First let us notice that Theorem II.0.1 together with (1.2) and (1.5)implies

hu(zE(M,v)) < / E )~(i)(w'x)+mi(w'x)d#*. i

Thus, the point is to prove that, if the family of sample measures {#~}~enz has SBR property, we have

h~,( 2d(M, v)) >_/ E ~(i)(w' x)+mi(w' x)d#*.

(2.1)

i

Analogously to the proof of Theorem IV.I.1, we need to construct a measurable partition of gt z x M subordinate to Wit-manifolds of X ( M , v,#), by means of 137

which (2.1) will be achieved. The construction is accomplished by means of local unstable manifolds. We present below the necessary arguments. First notice that the partition of ft z x M into global unstable manifolds {w} x W = ( w , z ) , ( w , x ) 9 f~z x M is in general not measurable, but we may consider the e-algebra consisting of measurable subsets of flz x M which are unions of some global unstable manifolds, i.e. the a-algebra

B"( Pd(M, v,#)) = ( B C Bu.(~Z x M ) : B = ( ~ , ~ B {w} x W~'(w,x) } . In addition, put

13l(?r

= {A 9 Bu.(f~ z x M ) : G - 1 A = A}.

We then have the following useful fact: P r o p o s i t i o n 2.1. BI( ?d(M, v, #)) C B"( zY(M, v, p)), p * - m o d O. Keeping Corollary 1.1 in mind, one can easily adapt the proof of Lemma IV.2.2 to the present case to prove Proposition 2.1. Since the arguments are completely analogous, they are omitted here. P r o p o s i t i o n 2.2. There exists a measurable partition TI of ~ z • M with the following properties: 1) ~7 < G - l q , a < 7; 2) is subordinate to W"-manifolds of 3) For every B 9 B(~ z • M ) the function

P B ( w , x ) = ~(~,.)(O~(x) M B~) is measurable and #* almost everywhere finite, where B~ is the section {y : (w, Y) 9 B}. Considering G -1 : ( ~ Z X M, #*) ~ instead of F : (f~N x M, v N x #) *--, and applying Proposition 1.3 and Proposition 2.1 instead of Theorem III.3.1 and Lemma IV.2.2 one can easily adapt those arguments in Section IV.2 concerning the existence of a partition satisfying 1)-3) of Proposition IV.2.1 to the present case to accomplish the proof of Proposition 2.2. Details are left to the reader. P r o p o s i t i o n 2.3. Suppose that the family of sample measures {#~}wenz has

SBR property. Let TI be a partition of the type as introduced in Proposition 2.2. Then there exists a Borel function 9 : fl z x M ~ R + such thai for #* -a.e. (w, x) E f~z x M, dtp*'~" g(w,z) -

9 ..(x)

(~,~)

where ( # )* ( w " , z ) is regarded as a measure on qw(x). 138

Proof of this proposition is completely analogous to that of Proposition IV.2.2. Now we turn to the main part of this section.

Proof of Theorem 1.1 1)=V 2). It is sufficient to prove (2.1). Let r/be as introduced in Proposition 2.2. By (1.3) and Theorem 0.4.3, we have hu( 7Y(M, v)) = h~.(G)

= h~.(G -1)

a - 1 ,~) = Hu*(qlGq) = Hu*(G-I~Iq) >_hu.(G I

= - /log(#*)~,~)((a-%)(w,

(2.2)

x))d#*.

Let ~1 be as introduced above. Put A2 -- f~z • M \ A 1 . Clearly G - l ~ i = Ai, i = 1,2. According to Proposition 2.1, q and G - l q refine (#*-mod 0) the partition {/kl,/k2}. Since their restrictions to ~ are the partition into single points, one has for g* -a.e. (w, x) E ~2 log(p*)~,~)((G-lr/)(w, x)) = O. On the other hand,

fx ~ ~(')(~, ~)+m~(~, ~)du* = o. 2 i

Therefore, we may assume #*(~1) = 1 without loss of generality. For #*-a.e. (w, x) E flz x M we may define

x(w, x) = (#*)~o,.)((a-~,7)(w,.)), y(~, ~) _

g(w,x) g(a(w, x))'

z(~, ~) = Idet(T~f0(w)l.~. ))h where E(~,,) =

E

|

A(0(w,z)>0 It is easy to see that X, Y and Z are all measurable and #*-a.e. finite functions on flz x M. We first claim the following results, whose proofs will be given a little later. C l a i m 2.1. X = YZ-l,p*-a.e. C l a i m 2.2. (a) logZ E Ll(fl z x M,#*);

(b) f log Zd#* = f E, ~(~)(w, ~)+m,(w, x)d#*. C l a i m 2.3. (a) logY E Ll(fl z x M,p*); (b) f log Yd#* = O.

139

From these and (2.2) one immediately obtains (2.1) and completes the proof of Theorem 1.1 1)::~ 2). []

Proof of Claim 2.1. First notice that for p*-a.e. (w',x'), (G-lo)l,(w,,~,) is ((#*)~,~,)-mod 0) a countable partition. Then for #*-a.e.(w,z), identifying {w} x Th~(z) with r/~(x) and {w} x (G-1T/)~(x) with (G-l~)~(z), we have for any B E B(M) (#.~a-', 1

-X(w, x)(/~*)~,*)((G-I~)~ (x) _

1

X(w, Z)

f((

M B)

g(w, z)d)~(~,~)(z);

G-xO)~(x)NB

on the other hand, , G-I~/ (~)(~,.)(B)

=(I-t*)~(w,~)(fo(w)B) =[ g(vw, z)d~(~,~)(z) J, ,~(fo(w)x)N(fo(w)B) =/_ J(a

g(Tw, fo(w)z)l det(Tzfo(w)[E~..))id~(~,,)(z).

Since B is arbitrarily chosen from

B(M),

we obtain for #*-a.e. (w, x)

1

X(w, z) g(w' z) = g(rw, fo(w)z)l det(T~fo(w)lE~,,))l, This implies that for tt*-a.e.

(w, x)

X(w, z)

for each z E

Proof of Claim 2.2. Since

(G-b?)~(x).

one has

X(w,z) : Y(w,z)X(w,z)-l,k since X(w, x) = clearly. []

A(~,~)-a.e.z E

](w,x) -a.e.z

(G-1T1)~(x).

E (G-177)w(X)

From this Claim 2.1 follows

for p*-a.e. (w, x) E flz x M

IT=fo(w)[E(~,.)l < If0(w)[c~ and log + If0(w)lc~ ~ La(~ z, vZ), by Oseledec multiplicative ergodic theorem we know that logZ E Ll(f~ z x M,#*) and

S lo.z... = / z

9,

i

completing the proof.

[]

140

Proof of Claim 2.3.

By Claim 2.1, one has logX -- logY - l o g Z < O,#*-a.e.

which implies log + Y < log + Z,

#*-a.e.

Hence, by Claim 2.2, log+Y E L I ( ~ z x from Lemma 1.3.1. []

w Lyapunov

M,#*).

This claim follows then

Charts

In this and the subsequent sections we address ourselves to proving Theorem 1.1 2)=~ 1). The idea and outline of the proof are as follows. Qualitatively, negative Lyapunov exponents indicate stability in the sence that certain points converge towards one another asymptotically under repeated applications of random diffeomorphisms. On the other hand, positive Lyapunov exponents are associated with chaotic instability due to "sensitive dependence on initial conditions". Since entropy measures the degree of chaotic instability, it is mainly determined by positive Lyapunov exponents. In fact, we shall first prove in Section 7 that for a certain class of measurable partitions ~'s subordinate to W~-manifolds of zV(M, v, #) it holds true that hu( z~(M, v)) = Hu.(~IG(). Then we shall prove in Section 8 that this together with entropy formula implies ( # ~ ) ~ < < A~(~,~) for p*-a.e. (w,x). To carry out the first part of the proof is not easy. It is necessary to consider explicitly the role played by the zero exponent as well as by the positive exponents. So we shall introduce in Section 4 some nonlinear constructions, i.e. unstable manifolds and center unstable sets related to these exponents. These constructions are worked out by means of Lyapunov charts which are treated in this section. Two classes of needed measurable partitions connected with these constructions are introduced in Section 5. Some averaging results in Euclidean spaces are given in Section 6. Finally, in view of Proposition 2.1, we shall complete the proof by reducing the problem to ergodic case. While not at all essential, this line of approach simplifies the presentation, especially where notation is concerned. Thus we now declare the following H y p o t h e s i s for S e c t i o n s 3-7: 9 (M, v, #) is g i v e n e r g o d i c . Under this hypothesis we know that there exists a Borel set /k~ C A0 (see Subsection 1. C) with #*(/k~) = 1 and GA~ = /k~ such that for each (w, x) E /Vo, r(w, x), ~(i)(w, x) and m,(w, x) equal respectively constants r0, ~(') and mi, l
141

For (w, x) 9 A~, let

E(L,x) : ~

eE,(~,~),

u

----

d"l m E~t o , x )",

A(1)>0

E(C~) = Eio(W,X)

where

A(io) = 0,c = dim E(to,.), 9

c

.

$

s = dim E(to,~), A(1)<0

A+ = min{A(i): A(i) > 0},

A- = max{A(/): A(i) < 0}.

It is easy to see t h a t T h e o r e m 1.1 2) =~ 1) is completely trivial if u = 0. So we assume t h a t u > 0. We now begin the proof by constructing in this section L y a p u n o v charts, which will be particularly useful in the subsequent sections. The construction needs the following two lemmas.

3.1.There exists a Borel set r'o C f~z with vZ(F~) = 1 and 7"F'o = F'o h such that for any given 6 > 0 one can define a Borel function B : F~ ---+ [1, + o o ) with the following properties:

Lemma

A

1) For each w 9 F'o and x 9 M. the maps F(

- 1 w,x) d=e f . exp]L x of~ o exp. " T ~ M ( B ( w ) -1) ---+T I L . M ,

F(-1

def.

--1

to,x) -- expt=~ ~ Ofw 1 o exp.: : T ~ M ( B ( w ) -1) --~ T ] 7 ~ M

are well defined and Lip(T.F(to,~)) <_ B(w),

Lip(T.U(to~ )) < B(~); 2) B(~-•

<_ B(~)~ 6 fo~ all w 9

to.

Proof. By Condition (1.1) and by arguments completely analogous to the p r o o f of L e m m a III.1.4 we know that there exists a Borel function /~ : f~z ~ [1, + o e ) such t h a t for each w 6 f2 z and x 6 M the maps /~(

def.

w,~:) = exo:~ 1 o f 1 o exp.: : T r M ( B ( w ) -1) --~ T I L z M , ~]wx

~(-~ -~ z ~ -~ o exp~ : T~M(~(~) w,x) ~~ = exp]~l

-~)

--* T~,:~.M

are well defined, max{Lip(T.!~(to,x)), Lip(T.F(-tol,,))} _
vZ). According to Birkhoff ergodic theorem, lim n~4-oo

llogB(r,w) n

142

= 0,

vZ-a.e.w.

Then there exists a Borel set F~ C a z such that vZ(?~) = 1, rF~ = F~ and for each w E r~ lira n---* 4- ~

-1 logB(T'~w) = O. n

Given 6 > 0, define B :F~ ---* [1, +oo) by the formula

B(~) : sup{~(,'~w)e -N~ :~ 9 z}. Then one can easily verify that F~ and the function B satisfy the requirements of the lemma. [] L e m m a 3.2. For any given 6 > 0 there exists a Borel function C : /Vo ---, [1, +~x~) such that : 1) For each ( w , x ) G / V o we have

IT~Y~I <_C(w,x)e(-)'~%~)'l,11 for a l l n >_ 0 and all~ e E i ( w , x ) , l < i < to; 2)7(Ei(w,x),~j#i| >_ C ( w , x ) - l , 1 < i < ro for a l l ( w , x ) G A~o; 3) C ( G + l ( w , x ) ) 5 C ( w , x ) e ~ for all (w,x) 9 / Vo. Proof. By Proposition 1.2 and by arguments analogous to the proof of Lemma III.l.1 we know that the functions C + , C [ C[ : /X~ ~ [ 0 , + o c ) , l _< i _< r0 defined by

C+(w,x) =

k+n

sup

CF(w,x) = s u p

Ir'f~+'~5le- ( ~ ( ' ) + e ) " - 6 l ~ l ,., > o,k e z,o r

e Eg(w,x)},

k"t-n

Irxf~ ~[e-(~(')-6)n-~lk , n _< o,k ~ z,o r ,~ c E,(~,~)},

are all measurable, everywhere finite, and positive functions. Define C : ~ [1, +cx~) by

---*

C(w, ~) = mulC+(w, x), c,-(w, ~), c~(~, ~)-' }. Then one can easily check that the function C satisfies the requirements of this lemma. [] For (~,r/,() E R u x R e x R e we define H((,~,~)]] = max{l](llu, lit/lie, II~l],}

143

where II-[l~, I[-[[c and I['[[, are the Euclidean norms on R u , R c and R' respectively. And for r > 0 we put l (r) =

• R (r) • R"(r)

where l ~ ( r ) , R ~ ( r ) and l ~ ( r ) denote respectively the closed discs in R~',R c and R ~ of radius r centered at 0. Let 0 < e < min{1, A+/100mo,-A-/100mo} be fixed arbitrarily, and let po be a number as introduced at the beginning of Section II.1. Put A;' = A oCI(F~ x M) which satisfies clearly p*(Ag) = 1 and GA~ = Ag. In what follows we define for each (w, x) 6 A~' a change of coordinates in some neighbourhood of x in M. The size of the neighbourhood, the local chart and the related estimates will vary with (w, x) 6 A~. This is the following P r o p o s i t i o n 3.1. There exists a measurable function 1 : A~' ---* [1, +co) satisfying l(G=l:Z(w, x)) <_ l(w, x)e ~ for all (w, x) G Ag, and for each (w, x) E A'o' there is a C ~ embedding r ) : I~(l(w, x) -1) ---+M with the following properties: ~ E (~r c 1) ~(,,,,~)(0) = x, To~(w,~) maps R ~', R ~ and R s onto E (,~r and E~w,x ) respectively. 2) Let -1 H(~r = (I)c(~,~) o f~1 o (I,(~r H (,o,~,) -1 = (I,a-~(,~,~) -1 o f,r I o (I)(~o,~,),

defined wherever they make sense. Then /) e~+-'ll~[ [ _< [IToH(~,=f[[ for ~ e R u, e-'[[([] < ]lToH(w,=)([[ _< e ll [i for e R [[ToH(,,,x)~]] _< eX-+e[[~[[ for ~ e RS; ii) Lip(H(,~r - ToH(~r <_ r

Lip(M(-&)-

Lip(T.Hoo,~:)) <_ el(w, x), Lip(T.H(~,~)) < el(w, x); iii) max{[[T{g(wr [[T(g(-1,x)l[} < e ;~~ for all ( 6 R ( e - ~ ~ x ) - l ) , where Ao > 0 is a number depending only on e and the exponents. 3) For any (,r; 6 I:t(l(w,x) -1) one has

Kold(r

O(w,=)r/) ~ ]l~- r/I[ <

l(w, x)d(O(w,=)~,r

for some universal constant Ko > O. A

Proof. Let B : F~ ---* [1, +co) and C : A~ --~ [1, +oo) be functions of the types as discussed in Lemmas 3.1 and 3.2 respectively, corresponding to 6 = r -t- 1).

144

We now define a new inner product ( Let (w, x) ~ Ag. First we define

,

t )(~,~) on T~:M for each (w, x) ~ Ag.

-~-oo

--cx5

n----0

n-----1

for ~,~ E E ~ ( w , x ) , l < i < ro where ( , ) is the Riemannian metric on M. ! Then we extend ( , )(w,x) to all of T ~ M b y demanding that the subspaces E i ( w , x), 1 < i < ro are orthogonal to one another with respect to ( , )(~,~).J Let ]-I~w,~) he the norm on T ~ M defined by

= max{((~o,~o)(~,~))~ :~ = ~,c,~} for each ~ = ~u + ~ + ~s E E(~,~) | E(~,~) | E~(w,~) Then it can be verified that

~(')-~151(~,.) _< Ir~f;(Ic(~,.)_< for all ~ E Ei(w, x), 1 < i < r0 and

3x/-/----~l~l_ I I(~,~) < +cx)

for all ~ e T~M, where ~(~, ~) = 4~~ ~)~o+1 and Co = ( E ~ _ - ~ c ~ < ) ~ the last estimate coming from Lemma 3.2 and a consideration analogous to (1.4) in Chapter III. Let b(po/~) be a number introduced by Lemma II.l.1. Define for each (w, x) 9 t(~, ~) = m ~ x { 6 v ~ P o ~, ~ - ' ( 3 ~ b ( p 0 / 2 ) e ~ / ~ ( ~ ,

~)B(~)}

and let

:~o =

max{l~X(OI : 1 < i <

ro} §

2r

Ko = 3,/;;b(po/2). Next, for each (w,x) E /k~~ take a linear map L ( w # ) : T ~ M ~ R u • R r • R s _(~,~) onto R ~ x {o} x {o), {o} x w • (o} such that it takes E (~,~), u ,zc and E (~,~) ~ and {0} x {0} x R s respectively and satisfies ((L(~,=)~, L(=,=)~)) = ((, ff)~=,=) for all ~,T] E T ~ M , where (( we set for each (w, x) E Ag

,

)) is the usual Euclidean inner product. Then

-1 (I)(~, ~) = expx oL(w,~)]R(l(w#)-i ).

With the entries defined above, one can easily check that 1)-3) of the proposition are satisfied. []

145

From here on, we shall refer to any system of local charts {O(w,~)}(~,~)ezx~' satisfying 1)-3) of Proposition 3.1 as a system of (e, /)-charts, and A0 and K0 will be defined as above.

w Local

Unstable

Manifolds

and

Center

Unstable

Sets

As we have said in Section 3, in order to prove Theorem 1.1 2)=>1) it is necessary to consider explicitly the role played by the zero exponent as well as by the positive Lyapunov exponents. In this section we use Lyapunov charts described in Section 3 to introduce some nonlinear constructions related to these exponents. These constructions will be used in the next section to deal with some measurable partitions of f2z x M which play central roles in the whole proof. For the sake of clearity of presentation, we divide this section into three subsections. A. Local U n s t a b l e M a n i f o l d s a n d C e n t e r U n s t a b l e Sets Let {r be a system of (~,/)-charts. Sometimes it is necessary to reduce the size of the charts. Let 0 < 6 < 1 be a reduction factor. For (w, z) 9 ZX~ we define

S ~ ( w , x) -- {~ 9 t3~(l(w, X) -1 ) :11 G-~

o f: o r <_6 l ( a - " ( ~ , ~ ) ) - ~ , n 9

z+},

that is, r consists of those points in M whose backward orbit under actions f g n , n > 0 stays inside the domains of the charts at G-'~(w, x) for all n 3, 0. It is called the center unstable set of X(M, v,#) at (w, x) associated with ({O(~0,~)}(~,,)ezx;,, 6). On S~'(w, z) and for all n _> 1, one clearly has H(~,~)d~'r

1 o . . . o H - 1v-~(~,,~) o H(-~,~). ) o f:'~ o (I,(,o,:0 = H -G--+'(~,~)

We next introduce the local unstable manifold of Y ( M , v, U) at (w, z) E Ag associated with ({ff(~,~)}(~o,~)ezx~',5) . It is defined to be the component of W"(w, z) N r z) -1) that contains x. The if-1 (~,~) -image of this set is denoted by W ~ . , ) . , (x). L e m m a 4.1. Let {~(w,x)}(w,~)ezx~' be a system of @, O-charts. 1 ) / f 0 < 5 < e -(~~ and (w,x) C Ag, then u X ) is the graph of a i) Wiw,~),6(

C 1 function

g(~,~),~ : ft~(bt(w, x)_l) --+ Rc+,(bt(w, ~)-1) with g(w,~),~(O) -= 0 and Lip(g(w,~),~:) < 1;

ii) w~,~),~(~) c s~(w,~); 2) IfO < 6 <_ e-2(~o+~) and

(w,x) ~ /%', then

H (~,.)wi~,~),~(.) u n r t O l ( a ( ~ , .))-~) =

146

u 1 w~,.),~(f~x).

Proof. Let 0 < 5 _< e -(~0+~) and (w, x) E Ag. By Proposition 3.1 we have [[ToH~-~,~)~[[ _~ e-~++~[[~[[ []ToH(--~,~)~/[[_ e-~[[7/[[

for

~ 9 Rr

for

~/ 9 R ~+~

and L i p ( H ~ l ) - ToH~I,~)) ~_ el(w, x) . 51(w, x) -1

~

=

< rnin{e -~++2~ _ e-~++~, e-~ _ e-~++2~}, where H -~ is restricted to f~(Sl(w,x)-l). Then by arguments analogous to those in Steps 1-6 in the proof of Theorem III.3.1 one easily see that there is a C 1 function g(~,~),~ : R~(~l(~, x)-~) -~ R~+'(~l(~, x)-~) with g(~,~),~(0) = 0 and Lip(g(~,~),r < 1 such that O(~,~)Graph(g(,~,~),~) C W"(w, x) and moreover

(~,~)r

Graph(g(w,.),.) = {r E R(Sl(w,x) -1) :11H--n

<_ 51(w, x ) - l e -(~+-2~)", n 9 Z+}. Then 1) follows immediately. If 0 < 5 < e -2(Ao+~) and (w, x) 9 Ag, then u

where/f ~ = e -(~~

1

H

u

This together with 1) yields 2).

L e m m a 4.2. IfO < 5 ~_ e -2(~~

u

1

[]

then for #*-a.e.(w,x) E Ag one has

s~(w, x) n ~ 1 ~)w~(w, x) = wi~,~),,(~) Proof. In view of it) in Lemma 4.1, it suffices to prove that for p*-a.e.(w, x) E Ag one has s ~ " ( ~ , . ) n a-1 (~,~).,w~r~ , , ~) c wi~,~),~(~). Let r E S~'~(w, x)n(K(~,~)W"(w, x) and let d~'( , ) denote the Riemannian distance along W~-manifolds of z~(M, v,#). Since r162 E W~(w, x), by Proposition 1.4 and Remark 1.2 one has d~(f~"(I)(~,,)(, f ~ " x ) --* 0 as n --- +oc. But, according to Poincar~ Recurrence Theorem, l(G-'~(w, x)) -1 does not tend to 0 as n ~ +oo for #*-a.e. (w, x) E A~. This implies that for #*-a.e. (w, x) E Ag -k there is some k >_ 0 such that H(~,~)( E W~_k(~,~),6(f~x). Let k = k(w,x) be the smallest one among such nonnegative integers. If k > 0, then by Lemma 4.1 2), H (~,~) - k + l f~ qL R(~l(G-k+l(w,x))-l), which contradicts ~ E S$~(w,x). So

147

k = 0, or equivalently, ~ C Wi~,,:),~(x ). The conclusion in Lemma 4.2 is thus obtained. [] Let (w,x) E A~. Consider now y E (~(~,~)S~(w,x) with (w,y) e A~, where 0 < 5 _< 1/4. Let W(U~o,~),26(y)be the @~-~,~)-image of the component of W~(w, y) N @(~,~)[15t~(251(w, x) -1) x Rr x)-a)] that contains y. Then ~5(~,~)Wi~,~),26(y ) contains an open neighbourhood of y in W~(w, y) and is also referred to as a local unstable manifold of W(M, v,#) at (w,y) (although in ~ general (~(w,y)W~w,y),25(y) 7s (~(w,x) W(w,x),26 (Y))" A reduction factor 0 < 5 < 1/4 is taken because when working in G-'~(w, x)-charts we cannot control the unstable manifolds of points whose backward orbits under the actions f ~ n come too close to the boundaries of r -~ (w, x))-~). Another technical nuisance is that ]](I)--i c--(w,~)f~--rt YI] r 0. Aside from these, we have the following analogue of Lemmas 4.1 and 4.2. L e m m a 4.3. Let ((I)(w,~)}(~,~)eA o, be a system of (~, O-charts.

~ t , x) with and (w,x) E A~. If y E O(~,x)o6

1) Let 0 < 5 ~ 88 -(x~

(w, v) e Ag, then i) W~' ,x),28(y ) is the graph of a C 1 function g(~,~),v : R"(25/(w, x) -1) ~ RC+~(461(w, x) -1)

with Lip(9(w,~),y) < 1;

iO w~,~),~(v) c s~(w, ~),. 2) Let 0 < 6 <_ le-2(x~

and let (w,x) e Ag. If y E ~(~,~)S~(w,x) with

(w,v) ~ /vg and f~v ~ ~c(~,~)s~(C(w, ~)), then H(~,~)W~,~),2~(V) M [R~(251(G(w, x)) -1) • RC+~(451(G(w, 2:))-1)] CW~(w,~),26(f~Y); r

3) Let 0 < 6 _< 88 -2(~~ Then for p*-a.e. with (w,y) E A~, one has

(w,x) E Ag, g y E

i Mfu (,W ,y) c wi~,~),~(y) u cu s ~c u( ~ , ~ ) n (I)-(~,~),, c s~(w,~)n~r~(,)w~(w,u).

Proof. Let 0 < 8 < 88 -(~~ (w,x) E Nor and y C ~(~,~)S~(w,x) with -1 Write (w, y) E A~. Denote (u = (I)(~,~)y. [Hli

T~ n~-~,~)-- LH~,

H12] : R ~

H~J

148

•

R~+~

-~

R~

• R~+~'

r(~,~),y

(~'~)-

[%.o] H22

:{{ C

: I1(- (yH <-361(w, x) -1 }

_._,RU+C+s. By Proposition 3.1 one can easily verify the following estimates: [)H11~ii _< ( e-x++E + r

for

IIH22,11I _ (e -~ - ~)ll,II

for

(~ E R u, r/ C R c+s,

Lip(r(~o,~),y) < min{e -x++3~ - (e -x++, + e~f), (e -~ - r

- e-A++3~}.

Then by arguments analogous to those in Steps 1-6 in the proof of Theorem III.3.1 one can prove that there is a C 1 map h(,~,~),y : K~'(351(w,x) -1) with

~ Kc+~(351(w, z) -1)

(4.1)

h(w?:),y(O) = 0 and Lip(h(~,~),y) < 1 such that r162

+ Graph(h(~,~),u))

C W'(w, y)

and moreover ~u + Graph(h(~,~),u)

={r e P.(I(~,~)-') : IIH -( .~, ~ ) r

H '(-.;,~fl.ll ~ _< 3~I(~,~)-1~-(~+-~)~, ~ 9 z+} .

Define g(~,~),y :l~U(26l(w, x) -1) --~ RC+S(461(w, x)-l), ~ h(~,~),~(~ - ~y) + %

where (y = (~y,qu) 9 R~ • R~+~. Then 1) follows immediately. Let now 0 < 5 < ~ 1~-2(~o+~) We have

H(,o,~)W~o,~),2~(y) C H(~o,~)(ffy+ Graph(h0o,~),y)) C ~flwy -~ Graph(]~a(~,.),f~y)

(4.2)

where hG(,~,~),/~y is the function defined analogously to (4.1), corresponding to = 88 -(x~ But, if f~y 1 9 ~a(.~,..)S~U(G(w, x)), then [~f=y + Graph(ha(w,,.),f=y)] 0

[RU(2M(G(w, x)) -1) x RC+'(451(G(w, x))-l)]

=w~(~,.),.~(fiy). This proves 2). The proof of 3) is almost identical to that of Lemma 4.2.

[]

We remark that in general S~"(w, x) is a rather messy set. Among other things we think of it as containing pieces of local unstable manifolds (in view of Lemma 4.3 1) ii)). In the case when there is not zero exponent, S$U(w,x) is equal to W~,:~),,~(x).

149

B. Some Estimates We present here some estimates which will be used in later sections. Let {O(w,,)}(~o,,)eA~' be a system of (E, /)-charts. When working in charts, we use ( , to denote the u-coordinate of the point ( 9 R ( l ( w , x ) - l ) . Other notations such as ~s and ~cu are understood to have analogous meanings.

4.4. Let 0 < 6 < e -()'~ and let (w, x) 9 A~o~. i) If (,~' 9 R(Tl(w,z) -1) and I1r = IlC~ -C'll, then

Lemma

Iln(,,,.=)r -

H

(,,.:~)C t II-- II(H(,,,,~,)4),, > d'+-~llr

t (H(~,~,)ff),,ll

-

- r

2) If u in i) is replaced by cu, then the conclusion holds with A+ being replaced by O;

3) sfr162 9 s$-(~, ~), then

IIn(-~,~,)r

-

H(L~),':'II < e~llr

r

-

Proof. By Proposition 3.1 2) ii) one has Lip((H(~,~) - ToH(~,~))lR.(Tl(~,=)-,)) _< c6 which together with Proposition 3.1 2) i) yields 1) and 2) by a simple calculation. We now prove 3). First we claim that [ [ ( - ~']i = ll(~- -~'~][- Indeed, if it is not this case, by applying 1) to H-1 (~,~) we have

IIH(~I,=)~'_H (,o -1 =)',#',,

=

gH -1 f'~ ~ (,~,=)...11

II(H(-=~.:~)r

_> e-~'--~llr

- r

which implies by induction -n " /'tll,, = I1( H -(,~,~,)r " II H (,~,~,)r - H -(..,~).. >_ ~-(~'-+~)"11r

-n 'LII - ( H (,,,,~)r

- r

for all n > 0. This contradicts the fact that ~ , ( ' 9 S~U(w, x). Now this argument -1 t also applies to H-1 (w,x)~ and H(w,r)ff, since they belong to S2U(G-I(w, x)). Then it follows from 2) that ]]( - ('II -> e-2e

which is the desired conclusion. Lemma

r

4.5.

H(-I~)(

- H-l(w x)',r'"

[]

Assume that 0 < 6 <_ 88 -2(x~

and (w,x) E A~o'. Let y E

(w, x) with (w, y) 9 a~, ~ = ((0} • rto+,) n w(L,~),2~(v) and ,' =

150

({0} x R r

n W~a(~,.),2~ (f~y), where 5 = 88 -(~~

Then

Proof. Since, by (4.2), H(w,z)W~,~,.),26 (y) C (l~u + Graph(ha(~,~),]'~y)

(4.3)

and

=[(]~y + Graph(]~a(~,~)jh~)] N [Ft=(251(G(w, x)) -1) x RC+~(451(G(w, x))-l)],

(4.4) one has By Proposition 3.1 2), II(H(~,~)~)~II < (e ~ + ~)11~11 and II(H(w,~)~)~ll -< ~ll~ll. Therefore, I1~'11-< ( e~ + 2E)II~II < e3~ll~ll. [] C. Lipschitz P r o p e r t y of U n s t a b l e S u b s p a c e s W i t h i n C e n t e r U n s t a b l e Sets Let {eP(~,~)}(~,~)eA~, be a system of (r (w,x) E A~' and 5 = 88 -(x~ Denote by L ( R =, R c+s) the space of all linear maps from R ~ to R c+'. By Lemma 4.3 1) we know t h a t , if y E ~(~,~)S~=(w, x) with (w, y) E Ag, then there exists a unique Py E L(R =, R c+') with ][Py[[ < 1 such that -1

=

= Graph(Py).

Define s

:{y: y E '~(~,,~)S~'(w, x)

with

(w, y) E A~} --~ L(R ~', R~+~),

y ~-+ Py.

In the sequel we show that the map L:(~,~) is Lipschitz. For this purpose we make use of the following lemma: L e m m a 4.6. Let ( E,I . ]) be a Banach space, E = E 1 @ E 2 a decomposition of E into the direct sum of two subspaces E 1 and E 2 and suppose that the norm ]. I satisfies ]~] = max{]~t],]~2]} for each ~ = ~1 + ~2 E E 1 @ E 2. Let L ( E , E ) ( L ( E 1, E2)) denote the space of all bounded linear operators from E ( E 1) to itself (E2). Given positive numbers A , K and 5 such that e ~ < K and 0 <

151

c ~ f ( e 6 + 6)/(e x-~ - 6) < 1, we define X:{A:{An}~eN:A~EL(E,E)

isinvertiblewith

AN : LA~I )

A~)j

: E1 ~

IA~l<_K, and

-~

*

satisfies

I(A~I))-11 _< e -~+~, IA~I _< e ~, IA~)I _< ~ a~d [m~r~)] < 8, n E N}, Y={P:{P~},~eN:P,~EL(E1,E

e)

and

IP~l
Let X and Y have respectively the following metrics: +oo

d(A,A') = y ~ ~'~[m,~ - d~],

A,.4' E X,

n:l

1 where ~o = ~(1 + ~).

p, -i

,def.

.),~eS(

=

~(A)) E

Y

Then for each ~ ~ X , ~here e~ists a unique ~ -

such that A~ 1 Graph(Pn) = Graph(P,~+l)

for all n E N. This unique P --- ~(A) also satisfies for each ( E Graph(P1) and nEN

I~-'(I < 7"1(I, where A-~ ---- A n I o . . . OAl I and 7 = ( ex-~ - 5) -1. Moreover, the map ~ : X -+ y , A e-+ ~(A) is Lipschitz and Lip(~) <_ C with respect to the metrics defined above, where C is a number depending only on A, K and 6. Proof Clearly, both (X, d) and (Y, d) are complete metric spaces. Define o: x • y ---, Y , ( X , ~ ) ~ 0

where O = { Q . } . e N is given by ~(")D

Qn = (A~I) ~-.,'122

"

WA~)

A(~)~

n + 1 ) l , zl-11 21--"~12"

,,-1

n+l)

,

n

E N.

Let us first check that this definition makes sense. Suppose that (A, P) E X x Y. Since A ('~)~ < 6 < e ~-~ ~ 1 2 " n'-'bl __

I(A~'~))-l1-1,

A(n)O A~ ) + ~ 1 2 . ~+1 is invertible and

I(A~'~)+ ~(")D -"lu" n+l)~-a I~___ (ex-6_ 6)-1

152

Thus Qn E L( E 1, E ~) and ]Qn] < (e 6 + 5)/(e ~-~ - 5 ) = a. So 0 is well defined. Secondly, w e verify that 0 is a uniform contraction on the factor/3. Note that for each A E X and n E N the map FA~ : { S E L ( E 1 , E 2 ) : IS I < 1} ~ { S C L ( E I , E 2 ) : ISI < 1} ~(~)r is a C 1 map with (~) ) S ( A l(~) ~(~)c~o,~-1 , ( T s o r A , ) S = (A~2) - rA,(S0)A~2 l + ~12

S E L ( E 1 E 2)

for each So E L ( E 1 , E 2) with IS0] < 1. By a simple calculation one has [TsoFA, I < ~ for each So E {S E L ( E 1 , E 2) : IS[ < 1}. FA~ is therefore a Lipschitz map with Lip(PA~) < a. From this it follows that, if A E X, then d(e(A, P), ~(A, P')) < ~ d ( P , P ) for any P , P ' E Y, i.e. ~ is an a/s0-contraction on the second factor. Let .4 C X. Then there is a unique point ~5 = { p , ~ } , e N d ~ ( ~ ) E Y such that =

i.e.

A ~ G r a p h ( P , 0 = Graph(P~+~) for all n E N. Also, if n E N and (~, P,~+I~) C Graph(P,~+~), then (~) ]A,~(~, Pn+l~)] = ](A~ ) + A~2 P-+~)~I

_> (e

- ,S)l, l = (e;'-"

- 5)I(,L

P,-,+l,0

I.

This proves the first two conclusions of the lemma. We now prove the last conclusion. Endow X x Y with the product metric d( , ) (see Lemma III.2.1). Then by a simple calculation one can check that : (X • Y, d) ~ (Y, d) is a Lipschitz map and there is a number I > 0 depending only on ~, K and 5 such that Lip(P) < 1. This, by Lemma III.2.1, asserts that ~ : X ---* Y, A ~-* ~o(A) is Lipschitz and Lip(~) ~ (1 - a / a o ) - t l d ~ c .

[]

Lel {(I)(w,~)}(~,~)eAo, be a system of (e,l)-charts with e being small enough so that e -~++1~ + e 5~ < 2. Then for each (w, x) E /Vo', f-.(w,x) is a Lipschitz map and Lip(L(~,~)) < Dol(w, x) 2

Lemma

4.6.

where Do > 0 is a number depending only on the exponents and e.

153

Proof Here we keep the notations introduced in L e m m a 4.5. P u t A = A+, K = e ~~ and 5 = 2c. Let now (w,x) E ~ . For y E ~(,~,~)S~"(w,x) with (w,y) E/Vo', set (v = (I)-l(w,x)y and define A(~,x)(Y) = {TH(~)r vHG-~(w,x)}neN. It is easy to see t h a t A(~,x)(Y) E X. L e m m a 4.5 we also have

Write then ~(A(~,~)(y)) = {P(w,x),n(Y)}nEN.

By

c(~,~)(y) = P(~,~),,(y). H e n c e , f o r any z , z ' 6

{ y : y E - e (9~~ , . ) o ~C'CU[~w , X ~)

with

( w , y ) 6/k~'},

II&~,.)(z) - &~,~)(z')ll

--IIP(~,.),z(z) - P(~,,),I (z')ll +co

<_C ~--:~~3IITH;:~)~ Ha-~(~,. ) -- TH(,:~)r

I

(by L e m m a 4.5 /

r~=J.

+co _< C E

a~eI(G-~(w,x))[IH(~)G - y (~,~)s~ -n z , 1[

(by Proposition 3.1 2) )

rt---~l

+oo

<_c ~_. ,~'&l(G-'~(w, ~))(e=~PIlr - r

(by L e m m a 4.4 3) )

n=l

+co

_
+co


x)2d(z' z')

rt~- i

=DoZ(m,

~)~d(z, z') +oo

where Do = C ~ , ~ = 1 r []

'~ < + o o since c~0ea~ < 1. T h e proof is completed.

w Related Measurable Partitions A. P a r t i t i o n s A d a p t e d t o L y a p u n o v C h a r t s In order to make use of the g e o m e t r y of L y a p u n o v charts in the calculation of entropy, it is convenient to have partitions whose elements lie in charts. If 7) is a m e a s u r a b l e partition of ~ z • M, write 7)+ = Y+=~GnT) and denote by 71~ the p a r t i t i o n 711{~}• for w E ~ z . In w h a t follows we shall identify {w} • M with M for any w E ~ z . Let {O(w,~)}(~#)6zx~' be a s y s t e m of (G /)-charts and 0<6<1. D e f i n i t i o n 5.1. A measurable partition 7) off~ z • M is said to be adapted to ({O(~,x)}(w#)eZ~o, , 6) ifT)+(x) C (w,~)SC~(w, x) for #*-a.e. (w, z).

154

Our purpose in this subsection is to show t h a t there always exists such a partition 79 with H~. (791cr) < + c o . We shall use the following two preliminary lemmas whose ideas are given in [Man]2 in the deterministic case. Lemma

5 . 1 . / f zn E [0, 1] for n > O, then +co

+co

(5.1)

-E olog o _<Enxo+c0 r~----O

n----1

where we admit O l o g O = 0 and where co

=

2[e(1 -

~

-1-~ ) ] -1

.

+co

q-oo

Proof. If }--~n=l nxn = +c~, (5.1) is trivial. We now assume t h a t ~,~=1 nx,~ < +c~. Let S be the set of integers n such t h a t x , > 0 and - log xn < n. T h e n +co

-- E

X,, Iog xn = -- E

X,~ Iog xn -- E

nES

n=0

_< E

xo- E ologxo

n6S

Note t h a t n r S implies x~ _< e - ' .

xn log x,~

nr

nr

On the other hand,

-vqlogt _< 2 C

for all t E [0, 1]. Hence, we obtain +co

-E

lotto_<E

ntis

This completes the proof.

riffs

n=0

[]

L e m m a 5.2.Let p : f2 z x M ~ (0, 1] be a measurable function with f l o g p d # * > -cx~. Then there ezists a measurable partition P of ft z x M such that: 1) d i a m P ~ ( z ) < p ( w , z ) for each ( w , x ) G a z x M ; 2) g , . ( P l ~ r ) < + o c .

Proof. Take numbers C > 0 and r0 > 0 s u c h t h a t i f 0 < r_< r0 there exists a measurable partition ~ of M which satisfies dian~(x)

<_ r

for all x E M and

I~rl _< c(~) mo

155

where I(rl denotes the number of elements in (~. For each n >__ 0, put Un = {(w, x) E I2z x M " e -(n+l) < p(w, x) < e-'~}. The integrability of log p implies that for every N > 1, N

N

En#*(Un)<E- s n=l

n=l

logpd#* _< - ~

log pd#*,

~

ZxM

so that -t-co

rL:l

Define a partition P of Qz x M by demanding that 7) >_ {U,~ : n > 0} and ?)Iv. = {ft z x A - A E g r . } l u . , where n >_ 0 and r,~ = e -('~+1). Then 7~ is clearly a measurable partition satisfying 1). We now verify that Hu.(PI(r ) < +ec. In fact, we first have H,.(PI~) = f

Hm.(P,~)dvZ(w)

(5.2)

and for each w E ~ z +co

Hu~(7)w) = E ( n=O

E

p~(P) logp~(P)).

PE'P~

Pc(g~)~

For each w E ~ z and n _> 0,

-

E

#w(P)log#w(P)

PET~ PC(U~)~

(U )(log I&. I - log #w _<#~ (U~)(log C + rn0 log(1/r,) - log #~ (U~)) _<#,, (U,~)log C + mo(n + 1)#,~ (U,~) - #~ (Un)log #~ (U,~). Summing over {n : n > 0}, we obtain +co

H~,,. (P~) < log C + m0 E ( n

+co

+ 1)#w (U,~) - E

n----O

#~,(U,~)log ju~,(U,~)

n=O

which, by Lemma 5.1, implies +co

Hu,,('Pw) <_logC

+ m0 + co + (m0 + 1) E n----1

156

n#w(Un).

(5.3)

Since for every N _> 1 N

N

f F_,

=

n=l

n=l

..*(v.)

<

<

n=l

by the monotone convergence theorem, we have -t-oo

f

~.,~(V.)d,Z(w

<

rim1

<

(54)

n=l

T h a t H , . ( P l c , ) < +o~ follows then from (5.2)-(5.4).

[]

R e m a r k 5.1. As can be seen from the proof above, Lamina 5.2 also holds true if r v, #) is not ergodic. P r o p o s i t i o n 5.1. Let {O(W,=)}(~,=)eA~' be a system of (~, l)-charts and 0 < 5 <

1. Then there exists a measurable partition ~P of Qz x M such that: 1) "P is adapted to ({(I)(w,=)}(~:)ez~;,, 5); e) H~,-(~'I~)< +co. Proof. F i x l 0 > 1 such that A = {(w,z) e A~' : l(w,x) <_ 10} has positive #* measure. The ergodieness of G : (ft z x M, #*) ~ implies that there is a Borel set A C ft z x M with #*(A) = 1 such that each (w, z) E A satisfies Gk(w, x) E A for some k > 0. If (w,x) e A, put r(w,x) = min{k : k > O, Gk(w,x) 9 A}. Let AI = A N A . Define ~ : f ~ z x M ~ ( 0 , 1 ] b y 5

~,(w,x) =

(w,x)~A', if (w, x) 9 ^'. if

~lo2e_(~0+~)r(w: )

log~o is #* -integrable since f^,r(w,x)d#* = 1. By Lemma 5.2, there is a measurable partition P of f~z x M such that H..(PI~) < +oo and ;%(x) c B(x, ~o(w, x)) for each (w, x) e f t z x M. We now prove that this P is adapted to ({~(w,,)}(~,,)ez~',5). In fact, it suffices to show 7)+(x) C ~(~:)R(61(w,x) -1) +oo n for each (w, x) 9 [U,,:0C AI ] \[U+__~oG'(A\A')]. First consider (w, x) 9 A'. By the choice of 7~, we have P + (x) C :P,, (x) C B(x, ~o(w, x)) which is contained in (~(~:)R(61(w, x) -~) because ~(w, x) <_ 51(w, +oo n +c~ n I x) -2. Suppose now that (w, x) ~ A' but (w, x) 9 [U,~=0G AI ]\[U,,=0G (A\A)]. Let n > 0 be the smallest positive integer such that G-~(w,x) 9 A~. Then P+ (f-"x~ C B(fg~x,~o(C-'~(w,x))). Now

157

I: B(I;

--n

--ti

,~ _,~ (w, x ) ) - ~e-(Xo+~)~(a-~(,~,~))) C (~,~)HG_.(~,~)R(SI(G

c~(~,~)R(61(w, x ) - l e ~ e-(~~176 since n < r(G-n(w, x)). Note that the computation above makes sense because for every 1 _< k < n,H~_.(~,.)ft(~l(G-n(w,x))-le-(~o+~)~(G-~(w,':))) C

R(l(G-n+k(w, x)) -1 e-(~~

The proof is completed.

[]

B. Increasing P a r t i t i o n s S u b o r d i n a t e to W~-manifolds A measurable partition ~ of Qz x M is said to be increasing if ~ > G~. In order to prove Theorem 1.1 2)==~ 1), we require a family of increasing partitions of f~z x M that are not only subordinate to W~-manifolds of zl:'(M, v, p) but also of some additional properties. This family of partitions are described in the proof of the following proposition. P r o p o s i t i o n 5.2. There exist measurable partitions of f~z x M each of which,

written ~, has the following properties: 1) ~ is increasing and subordinate to W"-manifolds of Pd(M, v, #); 2) For every B E B(f~ z x M), the function PB(w,x) = A(~,.,.)(~w(x) Cl B~) (well defined for u*-a.e. (w, x)) is measurable and P* almost everywhere finite; 3) -n=o \]+oo 'J f:,-nrs is equal to the partir into single poinls, d) +oo ~ +oo n B(An:oG ~) = B~(zg(M,v,l~)), ,*-mod 0 where B(An=oG ~) is the c~algebra consisting of all measurable A+__O~ Proof. Let {q~(w,,)}(w,,)ezx~, be a system of (r Fix l0 > 0 such that the set {(w,x) E /k~ : l(w,x) <_ 10} has positive p* measure. Proposition 1.3 asserts that there exist a compact set A C {(w,x) E /kg : l(w,x) < 10} with p*(A) > 0 and a continuous family of C 1 embedded u-dimensional disks {ValiSe(w, x)}(~,~)e^ such that the following (i)-(v) hold true: (i) W1oc(w, x) C kh(w,~)W(~ ,~),$(x) for all (w, x) e A, where 8 = Xe-(~~ (ii) There exist i > 0 and ~ > 0 such that for each (w,x) E A, if y,z E x), then for all 1 > 0

Wl~or

A

dU(f~ly, fwlZ) ~ ;ye-AldU(y, z); (iii) There exist numbers § g and d with 0 < § < p0/4, 0 < g < 1 and d > 2~ such that for any r E (0,§ and (w,x) E A, if (w',x') E B^((w,x),gr)d-~{(w',x ')

158

E A: max{d(w, w'), d(x, x')} < i t } , then d~-diameter is less than d and the map

Wl"oc(w',x') M B(x, r) is connected, its

(w', z') ~ W~r

x') n B(x, r)

is a continuous map from B^((w,x),gr) to the space of subsets of B(x,r) (endowed with the Hausdorff topology); (iv) Let r E (0, § and (w, x) E A. If (w', x'), (w', z") E B^ ((w, z), gr), then either

W~oc(~ ,*') n B(~, r) = W~oc(~, ~") n B(~, r) or otherwise the two terms in the above equation are disjoint. In the latter ease, if it is assumed moreover that x" E W"(w', x'), then

d~(v,z) > d > 2 ~ for any y E Wioc(W, " t ~') n B(~, ~) and z e W~oc(W, u t x,p) n S(x, ~); (v) There exists/~ > 0 such that for each (w, x) E A, if (w', x') E BA((W, x), g§ and y E Wlor x') n ?), then Wlo~(W,x ) contains the closed ball of center y and d" radius/~ in W"(w', xl). We now choose (wo,xo) E A such that B^((wo,xo),g§ has positive #* measure. For each r E [~/2, § put S r = U{{w} X

[WI~c(w,x) n B(x0, ~)]: (w, x) 9 B^((w0, ~0), gr)}

and let ~r denote the partition of f2z x M into all the sets {w} x [Wl~c(W, z) f'l B(xo,r)],(w, z) 9 B^((wo,xo),gr) and the set f2z x M\Sr. We now define a measurable function f r : Sr --+ R + by

By arguments completely analogous to those in the proof of Proposition IV.2.1 concerning the existence of a partition satisfying 1) and 2) there, we know that there exists r' 9 [~/2, § such that fir, > 0 /~* almost everywhere on St,, this implying that ~"def''h" = ~ , = \/q'cx~c:_"c -,,=0 U ~ , satisfies 1) of this proposition since -1-OO ?1 # * (U,=0G St,) = 1. 2) is verified similarly to Proposition IV.2.1 3). We now verify that ~ also satisfies 3) and 4). Put ~- = \~+oo ,n=o ~U- , ~~.. Since G : (12z x M,p*) ~ is ergodic, for #*-a.e. (w, y) 9 f~z x M there exist infinitely many positive integers {n~ : i = 1 , 2 , - . - } such that G'~'(w,y) 9 S~, for all i _> 1. Then from property (ii) just above it follows that the d u diameter o f ~ g ( y ) is less than -~de-~n~ for all i >_ 1 and hence is equal to 0. This proves that ~- is equal to the partition into single points. In order to prove that B(A+=~G"~) C B~(X(M,v,/~)), #* -mod 0, it suffices to ensure that for/~*-a.e. (w,y) 9 ~, if z 9 W"(w,y), then there exists k > 0 such that G-k(w,z) C ~(G-~(w,y)). In fact, if (w,y) 9 and z 9 W"(w,y), we first have limsup -1 logd~(f~,y,f~nz) < _A + < 0. n....-* +c:x~

r/,

159

Define now/~', : Qz x M ~ R + by j3,r,(w, Y) = { /3r,(w,0 y)

otherwise.if (w, y) E St,,

According to Birkhoff ergodic theorem, one has for p*-a.e. (w, y) E /kg

ln-1

/

lim n ~-~/3"'(G-k(w'Y)) = gt --~ -~- O 0

~r'd#* > 0

k=O

and hence, i f z E W~(w,y), there will be some k > 0 such that

/3~r,(G-k(w, y)) > d ~ ( f ~ y , fWnz) which implies that G-k(w, y) E St, and G-k(w, z) E ~(G-k(w, y)). On the other hand, it is clear that B~(zli(M,v,#)) C B(G'~()(#*-mod 0) for all n > 0 and +~ " ~) = B ( A + ~ G ~ ) ( p * - m o d 0). The proof is hence Bu( •(M, v, #)) C N~=0B(G completed. I-1 R e m a r k 5.2. Each partition ~ we just constructed has the following additional characterization: Let S denote the set St, introduced in the construction of ~. If r = V~=o G+co ~ {S,^ f2z • M \ S ) , then for every (w, x) E f2z x M, (w, y) E ~(w, x) if and only if (w,y) E ~(w,x) and d ~ ( f ~ x , f ~ y ) <_ d whenever G - ~ ( w , x ) E

(we define d ~ ( f ~ ,

fg~y) = + ~ if f ; " y r W~(C-~(~, ~)))

L e m m a 5.3. Let ~1 and ~2 be partitions constructed in the proof of Proposition 5.2. Then o - 1,~1) = h #-I, o tG-1 ,q2}" c hu.(G

Proof It suffices to prove h u.~(G-1 , (1 V (2) = h u.q(G-1,~1). Using the fact that h~.(G -1) -- h~.(G) <_ ~-~x(,)>oA(i)mi < +oc, by Theorem 0.5.2 we have for every n _> 1, h~.(G -1,~l V ~2) : ha~*(G -1, Gn(~l V ~2))

= hC.r . ( a

--1

,& v a ~ 2 )

= H,*(~I[G~I V G'~+1~2) + Hi,. (~]G~2 V G - ' ~ I ) . By Proposition 5.2 4), one has as n --~ +c~ ( Al=0 + ~ G z~ ~ 2~) : G~I.

G~VG'~+~2\G~V~ Hence

H~. (~ IG~ v G"+~,'0 ~ H~. (,'~IG,'x)

160

as n ---, +co. Also, by Proposition 5.2 3), G~2 V G - ' ( 1 tends increasingly to the partition of f~z x M into single points. Thus

as n ~ +co. This completes the proof.

[]

C. T w o U s e f u l P a r t i t i o n s Let {@(~,~)}(w,x)ezx~' be a system of (E, /)-charts with E being small enough so that e -~++1~ + e s~ < 2. Let ~ be a partition of the type constructed in the proof of Proposition 5.2, with 10, S and d having the same meaning as in Subsection 5.B. Fix 0 < 5 < rrfin{~e-2(~~ and let :P be a partition adapted to ({eP(,0,x)}(w,~)ezx;,,5) with g~,*('Plcr) < +co. We require that P refines { S , a z x M \ S } V a and {/~,f~z x M \ / ) } V a , where /) is a Borel set of positive #* measure which will be specified in Subsection 5.E. Define

rh = ( V P +, q2 = P+These two partitions will play central roles in Section 7. Some of their properties are described in the following lemma. L e m m a 5.4. 1) G~h < rh, Grl2 < rlz; 2) ~h _> rl2;

3) ~2w(X) C r

:) glzd 771w(X) C (~(w,r

(~,~);

for #*-g.e.

4) h~. (a-', ~2) = h~. (a- 1, ~), h~. (G- 1,7,) = h;. (a -1, ~). Proof Properties 1) and 2) and the first half of 3) follow from the definitions of r/1 and ~h. The second half of 3) is a consequence of Lemma 4.2. The first half of 4) is straightforward. We now prove the last assertion. In view of the fact that h~. (G -1) < +oo and H~. (:plcr) < +co, by Theorem 0.5.2 we have for every n > 1 h;.(G-l,~l) = h;.(G-',G~

V C n P +)

V G"*'p+)

= h~.(G-l,(

= H~.(~ V GnP+IG ~ V Gn+I'P + V Or) < H , - ( ( I G ( ) + H,.(I:'+IG-"~ v GP+). H , . (P+ I G - n ( v G1p+) ~ 0 as n --* +co since G - " ( increasingly tends to the partition into single points and Hu-(:P+I( V G:P +) _< H,~ < +co. Hence

h~..(C-~,,TD <_H . . ( ~ I G ( )

=

h~

On the other hand, also by Theorem 0.5.2, we have h•.(G-l,rh)

= h~176

161

> h~

since Hu.(/],I V.=l +oo G " ( { V P ) V ~) < +oo and Hu.(~ V PlG~ V G) < +oo. This completes the proof. [] D. Q u o t i e n t S t r u c t u r e of/]2/rh Since /]1 >__ /]2, for each w 9 f2z and x 9 M we can view /]lw restricted to /]2~(x), written /]1~10~(~), as a subpartition of /]2~(x). Let (w,x) E AN such that /]2~o(x) C (b(~o,~:)S~"(w,x). Recall that for every y E ~(,o,~:)S~'(w, x) with (w, y) E AN, Wiu,x),2~(y ) is the graph of a function from RU(261(w, x) -1) to R~+~(461(w,x)-l).

The restriction of these graphs to Oi-~,~)/]2~(x) gives,

roughly speaking, a natural partition of Oi-~,~)/]2w(x). The next lemma says that this corresponds to/]1~ I,~(~)-

L e m m a 5.5. For p'-a.e. (w,x) 9 AN, if y 9 /]2~(x) with (w,y) 9 /Vo' and ~ (~) C W ~ (w, U), th~

(~(w,x)W~w,x),26(y)N/]2w(X)

~-- /liT(Y).

Proof. First consider z 9 O(~,~)W~,~),2t(y ) M/]2w(x). We shall prove that z E ~ ( y ) . Since P refines { ~ , ~ z x M \ S } and z E P+(y), in view of Remark 5.2, it suffices to show that d~(f~"y, fj~z) < d whenever G-'~(w, y) 9 S. This is in fact true for all n > 0, since by Lemma 4.3 3) and Lemma 4.4 1) one has for all n > 0

-< II~C~,.)y- o~J,~)zll < 251(w, x) -1 which implies d"(fC~'~y, fj~'~z) < Ko261(w, x) -~ < d. Thus '~(,~,~:)W~,,~),:~(y) M /]2~(x) C rh~0(y). The reverse containment follows from ~o(y) C W"(w,y) and Lemma 4.3 3). Noting that the above argument holds true for #*-a.e. (w, x), we complete the proof. [] This lemma allows us to regard the factor-space /]2w(x)/(rhw],2~(~)) , or written simply /]2~(x)//]aw, as a subset of R c+s via the correspondence /]l~(y) ~-~ W~,~,~:),26(y)M ({0} x Re+S). If we identify/]l(W, x) and/]2(w, x) with /]l~(X) and /]2~(x) respectively, the next lemma tells then that GIG-~(n=(~o,.)) : G-l(/]2(w,x)) ---* ~?~(w,x) acts like a skew product with respect to the above quotient structure.

L e m m a 5.6. For#*-a.e. (w,x) E AN, if(w,y) E /]2(w,x) with (w,y) E AN, ~w(Y) C WU(w,y) and~r-:,w(f~ly) C WU(G-l(w,y)), then we have

G-l(/]l(W, y)) :/]I(G-1 (w, y)) n G-l(/]2(w, x)). 162

Proof. From the definitions of q~ and 7/2 it follows clearly that G-~(q~(w, y)) C ~l(G-l(w, y)) [3 G-l(q2(w,x)) for every (w,x) E f~z x M and any (w,y) E q2(w, x). On the other hand, if both (w, x) and G-l(w, x) meet the requirement of Lemma 5.5 and (w,y) is a point in q2(w,x) such that (w,y) E A~, f ~ ( y ) C W~(w,y) and f~_,~(f~ly) C WU(G-I(w,y)), then the reverse containment follows from Lemma 5.5 and Lemma 4.3 2).

[]

E. T r a n s v e r s e M e t r i c s

As we have said at the beginning of Section 3, the first main point for the proof of Theorem 1.1 2)==> 1) is to prove that the entropy h , ( ~ ( M , v)) is determined by actions of f~, w E flz, n E Z on the W~-manifolds of X ( M , v, #), or more precisely, to prove that hg.(G) = H,.(~IG~ ) where ~ is a certain increasing partition subordinate to W=-manifolds of zF(M, v,~u). In order to use the fact that all the expansion of X(M, v,p) occurs along the W~-manifolds to prove this assertion, we need to show that the action induced by G on (G-l(q2(w,x)))/ql ---+ ~12(w,x)/7h does not expand distances. For this purpose we define in this subsection a metric on the factor-space q2(w, x)/ql for #* -a.e.(w, x). This will be referred to as a transverse metric. We shall actually deal with ~1 and q2 restricted to a certain measurable set of full #* measure. Now we choose a measurable set A~' C /k~ with p*(A~') = 1 and G/k~' = A~' such that for each (w,x) E /Vo',~(x) C W~(w,x),~2~(x) C 9 (w,~)S$U(w,x) and the requirements of Lemmas 5.5 and 5.6 are satisfied. We then put ~1' = ql Ia~", I

~2 : T/2lag". In what follows we define a transverse metric on r/2' ( w , )/ql' for #*-a.e.

E

A~'. First we give a point-dependent definition. Let (w, x) E /k~'. From Lemma ! 4.3 we know that for every y E q2~(x), W(~,~),26(y) intersects {0} x R r at exactly one point. We denote this point by (u. For (w,y), (w,y') E q~(w, x), define

y), (w, y')) = I1r - r By Lemma 5.5, d(~ ~)( , ) induces a metric on ~(w,x)/~l~, but in general, d(~,~,)(, ) r d(~,~)(' , ) for (w, x') E q~(w, x) with (w, x') r (w, x). Now we need to rectify this situation to give a point-independent definition. To this end we shall first specify ~; (see Subsection 5.C) and then choose a reference plane T and standardize all measurements with respect to T. By Proposition 1.3, there exists a compact set A C {(w,x) E /k~ : l(w,x) <_ /0} with/~*(A) > 0 and meeting the following two requirements: u ~-~c+s def.Ec s depend continuously on (w, x) E A; (i) E(~,~) and ~(~,~) (~,~) @ E (~,~)

163

po/8

(ii) There exists a number 0 < ao < exists a C 1 map

and for each ( w , x ) E A there

u

~,c+s

such t h a t h(.0,.)(0) = 0, Lip(h(,~,.)) _< 1/3, D(w,~)defexp.Graph(h(w,~)) C W~(w,x) and {D(,o,~)}(,.,.)eA is a continuous family of C 1 embedded disks of dimension u. For ( w , x ) 6 A and p > 0, we put E(w,.)(p) = E~,.)(p).x E~+,~)(p) where

E~,.)(p) : {( E E (.o,~) ~ 9 [([ < P] and E (,.,.)W~) (~,~): r t-~ = {• 6 E~+ ~ 1771< p}. From the compactness of A it follows that there are positive numbers to and so with to < ao/2 such that for each (w,x) 6 A and any ( w ' , x ' ) 6 Ba((w,x),so) the following (a) and (b) bold true: (a) exp c (b) T h e map I(.o,,~,),(w,::)dL--t exp,,: oexp~ : E(w,~)(~0) --* T~,M is well defined with Lip(I(~,,~,),(~,~)) _< 2. exp,-} x 6 E(w,~,)(to) and e x p , } D(w,~) intersects {0] x E C(~',x') +S at exactly one point. Moreover, exp,-} D(~,~) n

E(~,,~,)(to)

:

Graph(h(~, .,),(w,~))

where h(,~,,~,),(.,,~)

is a C 1 map with

-E(w,,~,)(o) t

Lip(h(~,.,),(w,~)) ~_ 1/2,

---.

, ) (to)

(

and

I(~, ~,),(~,~)({0] • Z(C+,~x)(2to)) M E(~,,z,)(to) =

Graph(g(w,~,),(w,~))

cTs where g(~,,~,),(~,~) : E(w,,~,)(to) ---* E~,,~,)(to) is a C 1 map with Lip(g(~, ~,),(~,~)) < 1/100. Choose now (w0, Xo) E A such that BA((wo, xo), So/2) has positive #* measure. T h e n we define

~: : BA((wo, x0), so/2), T : eXp

o({0} •

W i t h / ~ and T thus specified, noting t h a t / ~ is required to refine {/~, ~ z x M \ / ~ } (see Subsection 5.C), we define now a metric on ~ !2 ( w , x ) / q l! for every (w, x) E +oo n ^t U,~=oG E where

E' = E N Ag'. First take an isomorphism I(~ .... ) : 7r : ~ = 0 ~

~

E (~o,~o) c+~

~

R c+s and define

as follows: For (w, x) 6 / ~ ' , let ~r(w, x) = (I( . . . . o) o exp~-ol){T n

D(~,~)}

and in general, let ~r(w, x) -- r(G-'~(w':O(w, x))

164

a

function

where n(w, x) is the smallest nonnegative integer such that G -'~(~',~) (w, x) E / ) ' . Then define for each (w, x) 9 ~ = o ' - "

d(~,~)((w, y), (w, y')) = II~(w, y) - ~(w, Y')II if (w, y), (w, y') E r/~(w, z), where I1" II denotes the usual Euclidean distance. We now explain why d~(~o,~)( , ) induces a metric on rl~(W,X)/rl[. ~ ~+oo~,~/,, Let (w,x) 9 ~.=0"-" - 9 Since 79 -> {/),f~z x M \ / ) } V cr and r/~ = P+Jzx ~'" , for every n > 0 either G-~(rl~(w, x)) C [~' or G-~(rl~(W, x ) ) n / ) ' = 0. Moreover, when G - " ( w , z ) 9 F,', for each (w,y) 9 r /t 2 W( , x ) one can inductively prove by using L e m m a 5.6

a - ~ ( o ~ ( ~ , y ) ) = ~ ( ,c

_~ ( ~ , y ) ) n a

_~ ( ~,( ~ , ~ ) )

and, using L e m m a 5.5, one can easily obtain

~ i ( a - ~ ( ~ , y)) = ( { ~ - ~ }

• D~-o(~,~)) n ~ ; ( a - " ( ~ , ~)),

hence I G-n(l'~l(w,y)) = ({T--nW} X OG-n(w,y)) n a

--I~

(~2(W,X)). I

This guarantees that <w,~)( , ) induces a genuine metric on y~(w,x)/~?[ and t h a t for any (w,x') 9 ~ ( w , x ) , d~(w.,)( , ) = d~(~.~,)( , ). 5.7. Let F/ and T be as introduced above. Then there is a number N = N(lo) such that for all (w, x) 9 Lemma

1

~d(~.~)(

,

T ) _< d(,o,,) (

,

_< Nd(~.,)(

,

).

Proof. Let (w, x) E / ) ' . We define the Poincar6 map 0: (({0} x Z~+;)) f-I Graph(h(w.~).(,~,~,)) : x' 9 r/~o(x)} ~ exp~- 1 T by sliding along Graph(h(w,,),(,o,,,)). L e m m a 4.6 tells us that there is a number D' = D'(Ao, K0, A+, r such that max{Lip(O), Lip(O-i)} _< D'l(w, x) 2 where O- 1 is understood to be defined on the image of O. Thus, if (w, y), (w, y') 9 r/~(w,x) and ((~0,y) and ((,o,y,) are respectively the points of intersection of Graph(h(~o.,:),(,~,y)) and Graph(h(~,,~:),(,o,y,)) with exp~-1 T, then

d~•

r

<_ Kob(po/2)D'l~d(,o,~)((w, y), (w, y'))

165

where dexpT~T( we have

,

) is the distance along the submanifold exp;1 T. Therefore

v), (w, v')) : II (w, v) -

v')ll

< 2Kob(po/2)D'l~d(~,,)((w, y), (w, y')). The other inequality is proved similarly.

[]

As is evident from the proof, the number N depends only on the charts and on 10. It is independent of rh and r/2 , or the choice o f / ~ and T (provided of course that everything is as described before). Finally, what we have done in the last two subsections is, roughly speaking, to present (flz • M ) / ~ i as a subset of (f~z x M)/rl2 x R c+s, and to define transverse metrics on rl2(w , x ) / ~ l that correspond to the Euclidean distance on R c+s. This Euclidean space geometry plays a role in some of the averaging arguments in the next section.

w Some

Consequences

of Besicovitch's

Covering

Theorem

For x E R ", let B ( x , r) denote the ball of radius r centered at x and let /~(x, r) denote the associated closed ball. All distances are the usual Euclidean ones in this section. The covering theorem of Besicovitch (see [Guz]) t h a t follows is a valuable tool in the theory of differentiation and in m a n y other fields of analysis. T h e o r e m . ( B e s i c o v i t c h ' s C o v e r i n g T h e o r e m ( B C T ) ) Let A be a bounded subset o f R '~. For each x E A a closed ball B ( x , r ( x ) ) with center x and radius r(x) is given. Put .4 = {/}(x,r(x))}xEA. Then there exists a subset .,4p of.,4 such that .4 ~ covers A and no point in R n lies in more than c(n) elements of .A', c(n) depending only on n. Remark

6.1. If in B C T A is not bounded but sup{r(x) : x E A} = R < +ec,

the above covering theorem is still valid with the constant c(n) changing conveniently. To show this , it is sufficient to partition R n into disjoint sets A i d ~ { x E R " : 3 i R <_ Ilxll < 3 ( i + 1)R}, i E Z + and apply B C T to the intersection of A with each one of these sets Ai. In what follows we derive some useful lemmas from BCT. Now let m be a Borel probability measure on R n. The next two lemmas are standard when m is Lebesgue. When working with arbitrary finite Borel measures, we use B C T

166

instead of Vitali's covering theorem. 6>0

Let g E L I ( R n, m) and define for each

___1 /B (~,6) gdm, g~(~) = m(B(~,5)~ x E R ~ (we admit here that 0/0 = 1). If g >_ 0 m almost everywhere, we further define g* = sup g~ 5>0

and g, = inf g~. 6>0

g* and g, are Borel measurable functions since for each t E R the sets {x : g*(x) > t} and { x : g,(x) < t} are open. L e m m a 6.1. 1) For any t > O,

c(n)/

m({g* > t}) < - - ~ -

gdm;

2) Let v be defined by d~ = gdm. Then for any t >_O,

~,({g, < t}) _< c(~)t. Proof. Let A = {g* > t} M C where C is an arbitrarily fixed bounded For each x E A we can choose 5(x) such that f ~ ( , , ~ ( , ) ) g d m >

Borel set.

tm(/~(z, 5(x))). Letting .4 = {B(x, 5(x))},eA and choosing .4' as in BCT, we h ave

re(A) < ~

re(B)

BE.4 j

< ~

-[1~ gdm < c(n)

gdm

BEA~

which together with the arbitrariness of C proves 1). Part 2) is proved similarly. [] L e m m a 6.2. Let g E L l ( R n , m ) .

Then g~ ~ g m almost everywhere as 5 ---*O.

Proof. It is sufficient to verify that for each t > 0, the set At =

x E R n : limsup~__.0 m(B(x, 6))

(x,6)

is of m measure zero. We prove m(At) = 0 in the following way. Given e > 0, we take a continuous function f such that hd~'g -- f satisfies [[hHa _< e where [[h[tl

167

is the L 1 -norm of h in L I ( R n, m). For f we have obviously at rn-a.e, x C R '~ lim

1

~ o m(B(~,6))

/B

fdm =/(x),

(~,~)

and so re(At) = re(B,) where

Bt~ {x E

R '~ :

1 ~ hdm-h(x) m(B(x,a)) (~,6)

lim sup

6~0

>t}.

Notice that

Bt C { x E R n : lim6__.0sup

>

(~,6)

u

e a " :lh(x)l>

Since for each x E R '~

lim sup640 m ( B ( 1x , 6)) /B (~,6) hdm -< [h[*(x) where lh]* is defined analogously to g*, by Lemma 6.1 1) we have

re(B2) _<m(llhl*

> 2}) _< 2~t(n)llhll 1 _<2c(n)tC.

Also

re(B,=) < ~ll hll~ < -2~ --

--

t

Since e > 0 is arbitrary, we obtain re(At) = O.

[]

The next lemma (see also [Led]2) is usually stated in a slightly different way in the literature. For geometric reasons we average over balls instead of taking conditional expectations with respect to fixed partitions. L e m m a 6.3. Suppose that (X, B, m) is a Lebesgue space and 7r : X --+ R '~ is a measurable map. Let {rn~}~el:t~ be a canonical system of conditional measures of m associated with the partition {~r-a{~}}~eR.. Let a be a measurable partition

of X with Hm(a) < +oo. For ~ E R '~ and A C a, define gA(~) = rn~(A).

168

Let gA and gA be functions on R n defined as above. Define g, g~ and g, : X ~ R by g(x) -~ E XA(x)ga(Trx)' AGa g,(x) =

g,(x) -~ E XA(x)gA(Trx)" AEa

Then g6 ---*g m almost everywhere on X and -logg, d m < Hm(o~) + log c(n) + 1 where c(n) is as in BCT. Proof. First by L e m m a 6.2 we have ga ~ gA 7rm-a.e. on P~" as 5 ~ 0 for each A E cr and hence g~ --* g m-a.e, on X since H m ( a ) < +cr Note also t h a t the function h : R + --* R + , s ~ m ( { - l o g g , > s}) is continuous almost everywhere (in the sense of Lebesgue) and hence is R i e m a n n integrable on any interval [0, b], b > 0. From this it follows t h a t

f

- logg, dm = fo +~~m ( { - l o g g ,

> s))ds

_~ fO +c~ AGa

Now for each A E

re(An {gA o 7r < e - ' } ) _< re(A) and f

m(A M {g,A o r < e - e } ) = J XAX{g2o,~<e-.}dm = / Em(XAX{g,A~

]B({Tr-l{~}}~ert~))dm

= ff(gA o 7r)X{gAoTr<e-.)dm <

the last e s t i m a t e follows from L e m m a 6.1 2). T h u s

f -logg, dm < E

~ +~176

AE o~ 0

< g m (c~) + log c(n) + 1

169

by a simple calculation.

[]

A n o t h e r consequence of B C T is the following result. Lemma

6.4. Let m be a finite Borel measure on R '~. Then

inf

m(B(x,c)) > 0

0
Cn

for m-a.e.x 6 R n. In particular, limsup l o g m ( B ( x , e ) ) < n ~-o loge -

for rn-a.e.x E R n. Proof Let N be a positive integer. P u t AN =

x e R n : Ilxll < N -

and

-

inf

O<e
En

- 0

.

It is sufficient to prove t h a t re(AN) = 0. Let now 6 > 0 be given arbitrarily. For each x E AN there exists a number 0 < e(x) < 1 such t h a t rn(B(z, e(z))) _< & ( x ) " . Letting .4 = {/3(X,e(Z))}~eAN and choosing .4' C A as in B C T , we h ave

re(AN) < E

rn(B)

B6A'

_<

<

N + 1))

B6A'

where i is the Lebesgue measure on R u. Since 6 is arbitrary, we obtain re(AN) = 0. []

w The Main Proposition Using the m a c h i n a r y developed in Sections 3-6 we can now complete the first step of the proof of T h e o r e m 1.1 2) ::~ 1), i.e. we can now prove t h a t h~u.(G -1) is equal to the a-conditional entropies of G -1 with respect to certain partitions subordinate to W~-manifolds of X ( M , v, #). P r o p o s i t i o n 7.1.Let z ~ ( M , v , # ) be given with # being ergodic. Then for any /3 > O, there exists a measurable partition ~ of ft z x M and of the type as constructed in the proof of Proposition 5.2 such that / 3 ( c + s) >_ (1 - fl)[h~.(G -1) - h ~ u . ( G - l , ~ ) -/3].

170

Proof. The strategy is to construct ~Z as in Subsection 5.B and to use it to construct Yl and ~2 as in Subsection 5.C with h~.(G,l,y2) >_ h~.(G -~) /3/3. Let {#* . . . . } and {#* . . . . } be respectively (p*-mod 0 unique ) canonical systems of conditional measures of #* associated with ~1 and ~2 and denote them respectively by { ~1 (~,~)} and { #2 (~,~)} for simplicity of notations. We shall prove that,

ifBT((w,x),p)

= {(w,y) E ~ 2 ( w , x ) :

d~(,~,~:)((w,x),(w,y))

/3. lim sup log #~,~,~)BT((w, x), p) > (1 --/3)[h~. (G -1 , 7/2) p-.0 log p -

< p}, then

hl. (G -~ , ~71)-

2/3/3]

for #* -a.e. (w,x). The desired conclusion then follows immediately from this and Lemmas 5.4 and 6.4. We divide the proof into five parts. (A) We start by enumerating the specifications on ~ , rh and r/2. First fix c > 0 such that ~ < min(fl/3,A+/lOOmo,-A-/lOOmo} and e -x++l~ + e ~ < 2(see L e m m a 4.6). Let {(I)(w,x)}(to,x)ezx;, be a system of (r l) -charts as described in Section 3. Using these charts, we construct an increasing measurable partition ~p as in the proof of Proposition 5.2 with 10, S and d having the same meaning as in that proof. Let 50 = min{(Xe-(~~ Choose /~ and T as in Subsection 5.E. We assume that e-P~N 4tF(~:) < 1 where N = N(lo) is the number introduced in L e m m a 5.7. Now we take a measurable partition P of f~z x M adapted to ({(I)(w,~)}(~,~)ez~,,50) such that Hu.(T'[~ ) < +oc, P > { ~ , ~ z x M \ S } V{/~,fl z x M\/~} V a and hu.(G-l,P ) >_ hu.(G -1) - r Then we set 7h = ~ V P + and r]2 = P + . Let /k~ ~ be a s e t as chosen in Subsection 5.E. Recalling that p*(A~') = 1 and GA~' = A~', for the sake of presentation we may assume that A~ ~ = f~z x M since otherwise the discussions below also apply to the system G : (A~",#*) ~ and lead to the same conclusion. With rh and r]2 so constructed, O2(w, x)/rh has then a nice quotient structure endowed with a transverse metric 4 ~ , , ) ( , ) for #*-a.e. (w,x). (B) Before proceeding with the main argument, we record some estimates derived from the results of Section 6. For 6 > 0, define g, g~ and g. : ft z x M --* R by g(w,y) z #~w,y)(G-11]2)(w,y), 1 /_ 1 g6(w,y) = #~,,)BT((w,y),5) ~,T((~,,),~)#(~'~)(G

-1

2

~?2)(w,y)d#(~,u)(w,z)

g.(w, y) = inf g6(w, y) 5eQ where Q = {e-ZZN 2j : l,j E Z+}. By L e m m a 5.6 we know that g(w,y) is also equal to p~,u)(G-17h)(w,y) for p*-a.e. (w,y). For each 5 > 0, one can check that the functions

(w,y) ~-~ tt(w,u)B ~ T ((w, y), 5) and #~,u)BT((w, y), 5) >

(w, y), 5) are measurable and

171

* BT( (w, y) ~-* #(G-~,~)(w,~) 0 for tt*-a.e. (w, y). Since

H~,.(G-'r/~lr/2 ) < + o c , for F*-a.e. hence

92

(w,y)

one has

#~w,y)(G-l~12)(w,y)

> 0 and

tBTtt w y), 6)

_

-

9

T

6

y), 6)

g~ is therefore m e a s u r a b l e for each fixed 6 > 0. T h e measurability of g, is obvious. We claim t h a t g6 -* g P* -a.e. on f~z x M when 6 E Q and 6 --* 0 and t h a t f - l o g g , d#* < +oc. To see this, first consider one element of r]2 at a time. Fix ( w , x ) . Substitute (r/2(w , x ) , p ( ~ , , ) ) for (X, m) in L e m m a 6.3, let ~r: rl2(w, x) -* R c+s be the ~r defined in Subsection 5.E and let a = (G-lr/2)]n2(w,x). T h e n we can conclude t h a t g~ --* g #2(w,x) -a.e. as 6 E Q and 6 -* 0 and t h a t f-log

g, dp(w,,)2 _< f-log(infge(w,y))dp~,~:)(y)6>o -<

Hiq~.*)(G-lr}2)+l~

1. I n t e g r a t i n g over flz x M , this gives f - log g, dr* _< H , , ( G - ~r]2 ]7/2)+log c(n)+ 1<+~. (C) T h e purpose of this step is to s t u d y the induced action of G on G-I(TI2(w, x ) ) / ~ l ~ ~2(w,x)/th with respect to the metrics ~ a - ~ ( ~ , , ) ( , ) and ~ ~)(, ). Consider (w, x) 9 a z x M . T h e point (w, x) will be subjected to a finite n u m b e r of a.e. assumptions. Let r0 < rl < r2 < 99 9 be the successive times when Gr x) 9 8, with r0 _< 0 < rl. Note t h a t r0 is constant on r]2(w, x). For large n and 0 _< k < n, define a((w, x), k) as follows: If rj _< k < ri+~, then

a((w, x), k) ----BT(ek(w, x), e-Z(n-rS)N2J). We now claim t h a t

a((w,x),k)N(G-l~2)(Gk(w,x)) C V-la((w,x),k + 1). In fact, if k r rj - 1 for any j, then we have a((w, x), k + 1) a u t o m a t i c a l l y since d~a,(~,~)(

(7.1)

Ga((w, x), k) rl r]2(Gk+l(w, ,

) and a~ak+,(,~,,) (

,

x)) = ) are

defined by pulling back t o / ~ . T h e case when k = rj - 1 for some j reduces to the following consideration : Let (w, y) 9 E and let r > 0 be the smallest integer such t h a t G~(w,y) 9 E. Let ( w , z ) 9 (G-rr]2)(w,y). It suffices to show t h a t

z)) <

z))

First we have d(w,u)((w, y), (w, z)) < Nd((~,y)((w, y), (w, z)) ( for the definition of d(~ u)( , ) see Subsection 5.E). T h e n for i = 1 , 2 , . . - ,r, L e m m a 4.5 tells us t h a t da,(~,y)(ai(w, y), ai(w, z)) <_e~id(~,u)((w, y), (w, z)). We pick up a n o t h e r factor of N when converting back to the dT-metric at G~(w, y). W h a t we claimed above is thus proved.

172

(D) It is easy to see t h a t there exists a Borel set A C ft z x M with tt*(A) = 1 , 2 BTttw and G A = A such that, if (w,x) E A, then ~(~,~) ~ , x), 6) > 0 for all 5 E Q. We now e s t i m a t e tt~,~)BT((w, z ) , e -~( . . . . (~'~))) = tt~w,~)a((w, z), O) for (w, z) E A which will be subjected to a finite n u m b e r of a.e. assumptions. Write

~(~,~)~((~, ~), 0) ~-~ ~,(~,~)a((~, ~), k) = k:oH i.t2+~(w,~)a((w,

x), k +

1) " P2aP(w'~)a((w' x),p)

where p = [n(1 - e)]. First note t h a t the last t e r m _< 1. For each 0 _< k < p,

I-t2k(w,x)a((w,x), k) tt2k+a(w,r)a((w, x), k +

It2k(W'x)a((w' x), k). P2k('~ x))) p2a~(w,~)G-l(a((w , x), k + 1))

1) =

by the G-invariance of it* and by uniqueness of conditional measures. This is

I't2k(w,r) a((w'x)'k) 2 < ~ a-~ ,2)( a k( w , ~)) n a((~,~),k)) "~a~(~,~)(C ~.~)(ak(~,~)) - ~a.(~..)(( -

(7.2) by (7.1). If g~ is defined as in (B), the first quotient in (7.2) is equal to

[g6((w,r),n,k)(Ck(w,

X))] - 1

where

5((w, x),n, k) = e-Z('~-rs(w'~))N2J and

< i < k: Gi(w,x) E E). I(w,x) = -log#~,~,,)(G-lr12)(w,x). T h e n the second to e-t(ak(w,~)). T h u s j = #{0

Write equal

t e r m in ( 7 . 2 ) i s

2 B T H((w , x), e -fl( . . . . ( w , r ) ) ) log#(w,~) p--i

p--i

_ - ~ log g,((~,~),.,~)(a~(~, ~)) - ~ ~(a~(w, ~)). k=O

Multiplying by fl. lim sup p--. 0 >/3. lim inf ,~--.+oo

k=O

-1/n and taking liminf ,2 BTttw log ~(,o,~) ~ , x), p)

on b o t h sides of this inequality, we have

log p

Iog.~ ~)B~((w, x) e-~(--,o(~,~))) '

' log e-Z,~

[nO-~)] _>liminfn_..+oo1_n

E

l~

]~+m n

n=O

n=O

173

(G-1) --f).

The last limit= ( 1 - r (G-lr12[r12) > (1-r 7.1 is proved if we show that limsup - - 1

E

n--*+~

n=0

72

Thus Proposition

log g6((~o,,:),n,k)(G k (w,x)) < ( 1 - e ) ( h ~ . ( a -1, rh)+2e). (7.3)

(E) We now prove this last assertion (7.3). It follows from (B) that there is a measurable function 5 : f2z x M ~ R + such that for #*-a.e.(w,x), if 6 e Q and 5 < 5(w,x), then -logge(w,x) <_ - l o g g ( w , x ) + e. Also, since f-logg.a ," < there is a number 61 such that i f A = { ( w , z ) 6 ( w , x ) > 51} then f a Z x M \ A - logg.dtt* < e. We claim that for #*-a.e. (w, x), if n is sufficiently large, then 5((w, x), n, k) < 61 for all k < n(1 - e). First, by Birkhoff ergodic theorem, there is a positive integer N(w,x) such that for n > N(w,x), # { i " 0 <_ i < n,Gi(w,x) E E} <_ 2n#*(/)). l f n > g ( w , x ) , then for each k < n(1 - e ) 6((w,

n, k) = e

< c-~nN22nu*(E). Since e - Z ' N 4"'(~) < 1, 5((w, x), n, k) is less than 51 for sufficiently large n. Thus ["0-~)1

E

- log g,((~,~),~,~)(a k (w, x))

n=O

-<

Z

( - l~

z)) + ~) +

k=0

~

-l~

k=O

ak(w,z)~A

ak(w,z)r

and the limsup we wish to estimate in (7.3) is bounded above by (1

e)

-loggd#* +e+

ZxM\A

Recalling that g(w, x) = #~w,x)(G-l~l)(W, X) for #*-a.e.(w, x), we have f - l o g g d#* = h ~ . ( G - l , r h ) which is equal to h~.(G-I,~z) by Lemma 5.4. This completes the proof. [] C o r o l l a r y 7.1. Let A'(M,v,#) be given with # being ergodic. Then for any partition ~ of the type as constructed in the proof of Proposition 5.2, we have = h;.(a-1).

Proof. For any fl > 0, by Lemma 5.3 we have h ; . ( G - t , ~ ) = h~.(G-I,~Z) where ~ is as in Proposition 7.1. Letting fl ~ 0, we obtain the desired conclusion.

174

w SBR Sample Measures: Necessity For Entropy Formula In this section we complete the proof of Theorem 1.1 2) ::~ 1). A. P r o o f o f T h e o r e m 1.1 2) V 1): T h e E r g o d i c C a s e We have indicated that Theorem 1.1 2) ::~ 1) is completely trivial if u = 0. We now assume that u > 0. Let ~ be a partition of ft z x M subordinate to W~-manifolds of X ( M , v , # ) , as constructed in the proof of Proposition 5.2 with the associated c satisfying e -~++1~ + e 5~ < 2. By Corollary 7.1, o" -- 1 ,~) = H..(~IG() = h o" h~,.(a . . ( a -~) = h~.(G). Let {p *~(w,.)} be a (essentially unique) canonical system of conditional measures of #* associated with ~ and let 1 ~(~,.) be the Lebesgue measure on W~(w,x). We shall prove that

Ht'*(~IG~) = E

A(i)rni ~ p*((~,~) < < A(w,, " ) for #* - a.e.(w,x)

A(i)>O

~(w, x) is identified with f~(x). The idea of the proof is as follows. Put J"(w,z) = I det(Z, fo(w)l~;,~))l for p*-a.e.(w, z) 9 a z x M. T h e n by Oseledec multiplicative ergodic theorem, f log J~d#* = ~A(.)>0 "~(i)mi" Suppose we know that #~(w,z) < < Au(t0,.) for #*-a.e.(w, x). Then dP~(w,,)* = PdA}'w(,z) , p*-a.e.(w, x) where

for some function p : ft z x M ---* R + . This function must satisfy for p*-a.e.(w, x)

~(~)

p(w, y)dA(~,~)(y) = 1

and on ~ (x) 1

fl(G-l(w,Y))

u

by the formula for change of variables (see the proof of Claim 2.1). From this one can guess that for #*-a.e.(w, x), if (w, y) E ~(w, x),

,,def. p(w, ~) _ ~-7 J" ( a - k ( ~ ,

x))

A candidate for p is then

zx((w, x), (w, ~)) p(w, y) =

L(w, x)

if (w, y) E ((w, x), where

L(w,x) = ~

~(~)

/k((w,x),(w,y))d),(L,x)(y ).

In the sequel we prove rigorously that all this makes sense.

175

L e m m a 8.1. For #*-a.e. Lipschitz function on r

(w,x), y H l o g A ( ( w , x ) , ( w , y ) ) is a well-defined where 5 = 88 -()'~ It follows from this that for #*-a.e. (w,x), y ~-~ A ( ( w , x ) , ( w , y ) ) is a well-defined function on

~ ( x ) and is uniformly bounded away from 0 and +oo on ~ ( x ) .

Proof. First, by Condition (1.1) there is a Borel set F' C f~z with vZ(F ') = 1 and vF ~ = F' and there is a Borel function l t : F' ---+ [1, +oo) such that for each w E F ' and e a c h n _ > 0 , max{lfo(w)-a[c1, [fo(w)[c=} < l'(w) and

1;(,--'~) <_ t'(w)e 'n.

Secondly, by Lemma 4.6, there is a number Co > 0 such that for each (w, x) G A g (see Section 3), if y, z e (I)(~,~)W(~ ,~),i(x), then

IJ"(w, y) - J"(w, z)l < c0l(w, x)~lf0(w)lc~d(y, z) and also

max{J~(w,y) -~, J"(w, z) -~ ) <_ [f0(w)-i [c ~ Let now (w,z) E Z~ with w E U. For any y,z E ~(~o,~)WI~,~),~(x ), by Lemma 4.1 and Lemma 4.4 1) we have

+o~ log J~(a-~(w'Y)) k=l +oo

Ilog ,r'(a-~(w, ~,)) - log Jo(a-~(w, z))[

=~ k=l +co

<_~ Ifo(r-kw)l"~?lJ"(G-k(w, y)) - J"(G-k(w, z))l k----1 +oo

< ~ If0(~-~)l~~

x))~l/0(<~w)l~d(f:~y, f?o~z)

k=l +oo

< ~ CoKor(w)~o+~l(w,~)%C-~++(mo+~)~l~d(~,z) k----1

From this the first part of the lemma follows clearly. Since for #*-a.e. (w, x) there is n > 0 such that G - ' ( ( ( w , x ) ) C {w'} x ~(~,~,)W~,,,),~(x') for some (w', x') 9

gl (F' x M), the second part follows immediately.

[]

L e m m a 8.2. There exists a measurable function p : f2 z x M --+ R + such that for #*-a. e. (w, x), p(w, y) = A((w, x), (w, y))/L(w, x) for each (w, y) 9 ~(w, x).

176

Proof We define a sequence of functions p,~ " ft z x M ~ R +, n _> 1 in the following way: Let (w, x) E f~z x M. If G-t(((w, x)) C {w'} x q)(~o, ~,)W~t t , ,, ; ( x I ) for some l >_ 0 and some and (w, y) e ~(w, x)

(w',x') e ag ffl (V'

M),

x

then define for each n >__ 1

~] j ~ ( a - k ( w , y ) ) p,~(w,y) =

n

-1

k=l

J~

x))

J"(a-~(~,~))

f~(,) k = l - j - ~ ~ d ) ~ ( ~ , ~ ) ( y ) Otherwise, we define

p.(w,

y)

= 1

for each n >__ 1 and (w, y) E ((w, x). From the construction of ~ it is easy to see t h a t Pn is measurable on f~z x M for all n >_ 1 and, by L e m m a 8.1, for each (w, x) the limit lim

p,~(w,y)agp(w,y)

(8.1)

~ ----4nL O o

exists for all (w, y) C ~(w, x). Let p : ~ z x M ---, R + be defined by (8.1). T h e n it satisfies clearly the requirement of this lemma, cl Suppose t h a t p : ft z x M -* R + is as defined above. m e a s u r e u on f~z x M by

We now define a

By Proposition 5.2 2), using s t a n d a r d arguments from m e a s u r e theory one can easily verify t h a t u is indeed a well-defined Borel probability measure. Also, from the definition of u it follows clearly that, if {u~(~,~)} is a canonical s y s t e m of conditional measures of u associated with (, then du~(,.,,~) = pd)~(~,,=) for p*a.e. (w, x) and t h a t u coincides with #* on B(() (the G-algebra consisting of all m e a s u r a b l e (-sets). Lemma

Proof

8.3. f - log u~(w,~)((G-l{)(w, Define

q(w, x)

x))dp* = f

=/~(w,x)((G-l~)(w,X)).

q(~, ~) = f(~_.r L(G(w,x)) L(w,x)

log

J~'dp*.

T h e n for #*-a.e. (w, x)

A((w, ~), (,~, y))d~' ,~)(y) L(~, ~) 1 JU(w,x) 177

(8.3)

Since, by L e m m a 8.1,

L(w,z)

/I ju(G_k(w ' y))

= n--*+~ lim

~(z) k=l

dA(~,~)(y)

for p*-a.e.(w, x), it follows that L is a positive finite-valued measurable function on f~z x M with

LoG . / l o g + - - - ~ d # _ 0 and hence they are equal on +oo -n +c~ -n Vn=oB(G () = = Bu.(gt z M).

B(V.=oG

L e m m a 8.4.

Proof.

For

flogJ~d# * =

(w, y)

~)

x

Hu.((IG()

implies u = p* on B(G-I~).

E ft z x M, define

P(w, y) = "~(~'~)((a-~)(~' y)) 9 a.~(~,~)((

P is well defined #* almost everywhere since f o r # * - a.e . ( w , z ) (, G - 1~)1~(,) ~ is ! tt* ~ ( ~ , )~ m ~ convexity of the function log x we nave

~ ) ( ~ , y))

Hu.(G-I~I~)

< +cx). Noting that

acountable partition, by the

f logPdp* <_log f Pdtt* <_O (we admit here log 0 = - c ~ ) with f log Pd#* = 0 if and only if P = 1 #* almost everywhere. But we know that f log Pd#* = 0, since L e m m a 8.3 says that

- flog.e(~,.)((G-l~)(w,~))d.

* =

flogJOa. *

= Huo (C-1~t~)

-- - f log,~(~,,)((G-lr Thus u = p* on B ( G - I ( ) .

x))du*.

[]

Now let ~/be an arbitrary measurable partition of ~ z • M subordinate to WU-manifolds of ~ ( M , v , p ) and let {P~(w z)} be a canonical system of conditional measures o f # * associated with ~ 9 In order to prove that #*~ ( ~ ,, ) < < ~ (~,~) for #* -a.e. (w,x), we take a partition ~ as dealt with above. Suppose that {tt~vv)(w,,)} is a canonical system of conditional measures of #* associated with

178

V 7/. Noting that (~ V r/)l~(w,, ) and (~ V r/)lo(,~,, ) are countable partitions for #*-a.e.(w, x), we have #~(,~,,)((~ V 7/)(w, z)) > 0, P~(,o,,)((~ V r/)(w, z)) > 0 and

~(~,.)() "(~")(~,')() = ~,~(~,.)((~ v ~)(w,

~;(~,~)()

.

~)) = .;(~,.)((~

v ~)(w,

~))

for /z*-a.e. (w,x). From this it follows clearly that #*~(w,,) < < A(w,,),Pu ,_ a.e.(w, x). The proof of Theorem 1.1 2) =~ 1) of the ergodic case is completed. [] C o r o l l a r y 8.1. Let 2d(M,v,tt) be given ergodic such that Pesin's entropy formula holds true. Let rl be a partition of ft z • M subordinate to W ~manifolds of 2d(M,v,#) and let p be the density of #*~(~o,~:) with respect to A"(~,~). Then for ls there exist a countable number of disjoint

open subsets V . ( w , . ) ,

~,~)(~(.)\ U . ~

n e N of W~(w,.) such that U.~NU.(w,*) C ~(~), V.(w,~)) = 0 and on each U.(w,~) p is a strictly positive

function satisfying

p(~,~) _ ~ J~(c-~(~'z)) p(~,z)

..=

j~(c_k(~,y)),

~,z~U.(~,~),

in particular, logp restricted to Vn(w, x) is Lipschitz along W"(w,x). B. P r o o f o f T h e o r e m 1.1 2)=:r 1): T h e G e n e r a l Case We reduce the general case to its ergodic one. Let ~(M,v,t~) be given. Let 7)0 be the measurable partition of f~z x M into disjoint sets {C~}~eA and f2z x M \ UoeA C~, as introduced in Subsection 1.B for ~ ( M , v , # ) . Suppose that

h~.(G)

A(i)(w, x)+mi(w, x)d#*.

[ ~ J

(8.4)

i

By Theorem 1.2.6 and (8.4) we have f

h~..(V ) = j h;; (C) ~o (4) = ] f ~w ()'(x,+)m~(ix,a)p*~w (x,~).o~(.) where p~o is the measure induced by p* on the factor-space f2z • M/C/0. But, by Theorem II.0.1 together with Remark 1.1 and (1.2),

h~.(C) < f ~ ~(')(~,x)+..,(w,~)~p; i

179

for each a E ,4. Hence, there exists a measurable set .A1 C .A with #}0(.A1) = 1 such t h a t for each a E ,41

h;.(G) Using a variant of T h e o r e m III.3.1 for X ( M , v, #) and a procedure analogous to p a r t of the proof of Proposition IV.2.1, one can construct a m e a s u r a b l e partition ~ satisfying the requirements of Proposition 2.1. Moreover, the partition can be constructed to have the additional p r o p e r t y t h a t there is a m e a s u r a b l e function p : flz x M --* R + such t h a t for #* -a.e. (w, x),

p(w,y) =

(8.5)

L(w,x)

if ,~(O(w,x) > 0 for some i and

p(w, y) = 1, V(w, y) E 4(w, x)

(8.6)

if A(/)(w, x) < 0 for all 1 < i < r(w,x). T h e n we can define a m e a s u r e ~ z x M in a way completely analogous to (8.2), i.e.

~ on

for each A E B(f~ z x M). (8.7) can be also written as

for each A E B(a z x M). Clearly, if {~(~,~)} is a canonical s y s t e m of conditional measures of 1) associated with 4, then di~(w ~) -- pd~ E ( , ) ~ for p*-a .e. (w, x) Let ~ be as given above. T h e n there is a measurable set .A2 C .41 with #~o(.A2) = 1 such t h a t for every a E .A2, ~w(X) C WU(w, x) and contains an open n e i g h b o u r h o o d of x in W"(w, x) for p*-a.e.(w, x),i.e. ~ is indeed a m e a s u r a b l e partition subordinate to W~-manifolds of ~ ( M , v , p ~ ) . Let now A C ~ z • M be a m e a s u r a b l e set of full p* measure such t h a t (8.5) and (8.6) hold true for each (w, x) E A. P u t .43 = {a E .42 : p~(A) = 1}. T h e n for each a E .43, we can appeal to the proof of the ergodic case and conclude t h a t ,

p~(A) =

.

~(~)

This together with (8.8) yields #*_-~. Therefore, if {#~(w,x)} is a canonical s y s t e m of conditional measures of it* associated with ~, we have it*~(~,z)<< )~(w,~) and dit*~(w,z)/d~yw(,~) = p for it*-a.e.

180

Let now 77 be an arbitrary measurable partition of ~ z x M subordinate to W~-manifolds of X(M, v, p). By arguments analogous to those in the paragraph before Corollary 8.1 one easily proves that #*,(~o,x) < < Au(~,x) for p*-a.e. (w, x). This completes the proof of Theorem 1.1 2) ==~ 1) in the general situation. [] As can be seen from the discussion just above, we have the following C o r o l l a r y 8.2. Corollary 8.1 holds true in the nonergodic case.

181

Chapter VII

R a n d o m Perturbations of Hyperbolic Attractors

While in the preceding chapters we intended to make the presentation selfcontained as much as we could, in this chapter, it is not the case. It is actually devoted to the derivation of a set of new results, based on what we have already obtained. We shall study invariant measures for random perturbations of hyperbolic attractors. Leaving precise statements for later, we first give a description of the main result of this chapter. Let f be a twice differentiable diffeomorphism on a Riemannian manifold N and let A be a hyperbolic attractor of f with basin of attraction U. As we have indicated at the beginning of Chapter VI, there is a unique f-invariant measure p with support in A that is characterized by each of the following properties: (a) p has absolutely continuous conditional measures on unstable manifolds; (b) Pesin's entropy formula holds true for the system (X, f,p); (c) For Leb. -a.e. x E U one has lim,~__.+~ L,~ z_~k=0 ~-,,~-1 ~fk~ = p. The measure p is called the SBR measure on the attractor A. Our main purpose here is to show that the same kind of result holds true as well if f : U ~ U is subjected to certain random perturbations. This is described in the following paragraph. Now we consider the case when the system f : U ~ U is subjected to certain random perturbation which can be viewed as random compositions of maps from U into itself nearby f . Let v be a Borel probability measure on Q = C2(U, U) concentrated on those maps which are sufficiently nearly f. Let G : flZ x U *--' be as defined in Chapter VI and let P1 : QZ x U ---* flZ be the projection on the first factor. Then our main result of this chapter says that there is a unique G-invariant Borel probability measure /5 with P1/5 = v z and with support in = n,~>0G'~(fl Z x U) such that it is characterized by each of the following properties: (1) /5 has absolutely continuous conditional measures on unstable manifolds; (2) An entropy formula of Pesin's type holds true for the system (QZ x U, G,/5), i.e. h~(G) = f EiA(i)(w, x)+mi(w, z)dfi; (3) For v Z x Leb. -a.e. 1 ~-~n-- 1

(w, x) E ~2z x U it holds that lim,__.+oo ~ z-,k=0 6Gk(w,~) = P" In addition, if # is an absolutely continuous (with respect to the Lebesgue measure on U) invariant measure of the system •(U, v) (see Section 1 of this chapter) then/5 is just the measure #* introduced in a way analogous to Proposition I. 1.2. The existence and the characterizations (1) and (3) of such a measure ~ are due to [You] and the characterization (2) of the measure is due to [Liu]5. One of the purposes of this chapter is to apply our general argument to the measure t5 given by [You]. Finally we remark that in this chapter we study random perturbations of the hyperbolic attractor A as random compositions of maps nearby f : U ~ and the main result described above is then actually a consequence of the persistence of 182

the hyperbolic structure on A for the r a n d o m l y composed maps. Y. Kifer also studied r a n d o m p e r t u r b a t i o n s of hyperbolic attractors from the point of view of Markov processes. For further information we refer the reader to Kifer's book [Kif]2.

w Definitions and S t a t e m e n t s of R e s u l t s In this chapter we assume t h a t M is a C ~ (maybe not c o m p a c t ) connected R i e m a n n i a n manifold w i t h o u t boundary. Let U0 be an open subset of M with c o m p a c t closure U0, and let f : Uo ~ fUo be a C 2 diffeomorphism. A set A C U0 is said to be f - i n v a r i a n t if f A = A. 1.1. A compact f-invariant set A C Uo is said to be uniformly hyperbolic or simply hyperbolic if there is a continuous splitting of the tangent bundle T A M over A into a direct sum of two subbundles E " @ E ~ and there are also numbers 0 < )to < 1 and C1, C2 > 0 such that for all x 9 A and n > 0

Definition

T z f E ~ = E~(~:), T~fE~ = E~](z) and

IT~f%I ~ C2)t31~1, 0 9 Eg. R e m a r k 1.1. Via a change of Riemannian metric we m a y - a n d will-assume t h a t C~ = C2 = 1. This means t h a t there always exists an equivalent (to ( , /) R i e m a n n i a n metric < < , > > on M , called an adapted Riemannian metric, such t h a t for all z 9 A and n > 0

IITj%II ~ ~"11~11, ~ 9 E~, constant and II. II is the n o r m

where 0 < ~ < 1 is a on T M induced by < < , > > . This can be shown in the following way. Take a C ~ function 19 : M --+ R + such t h a t 0(x) = 1 for all x 9 A, O(x) = 0 for all z 9 M \ U o and 0 < 0(x) < 1 for all x 9 M . T h e n we define for x 9 M and ~, r] 9 T~M q--1

<< ~, 7/>>=

(~, 7/) + E ( O ( x ) T ~ f ~ , O(x)T~f~?)

if z 9 U0

/=1

(~, r/)

if x r U0

where q is a positive integer such t h a t C1)to q > 1 and C2)t0q < 1. Let [I. II be the n o r m on T M induced by << , >>. T h e n one can easily check t h a t for some number 0
_> ~-111r

for r 9 E ~, 183

IITf011

_< ~11011 for

7] 9 E ' .

(1.1)

Note t h a t the metric << , >> thus defined may not be smooth. However, one can a p p r o x i m a t e << , >> by a s m o o t h Pdemannian metric, denoted by the same notation << , >>, such that (1.1) will remain true but with possibly a little bigger a < 1. 1.2. An f-invariant set A C Uo is called a hyperbolic attractor with basin of attraction U if the following hold true: 1) U is an open set such that U C Uo,fU C U and

Definition

N f'~U = A; n>0

2)

A is a uniformly hyperbolic set; 3) f [ h has a dense orbit.

In the sequel we shall always assume t h a t A is a hyperbolic a t t r a c t o r of f with basin of attraction U. We now review briefly some relevant results concerning the S B R measure on h. To this end we first present some related concepts. For x E A, the unstable manifold W ~ ( f , x ) o f f at x is defined as

W ~ ( f , x ) = {y C Uo : d ( f - ~ x , f - ~ y )

~ 0 as n ~ + ~ } .

T h e n for each x E A, W ~ ( f , x) is an immersed C 1 (actually C 2) submanifold of Uo and W ~ ( f , x ) C A ([Hir]2). Let # be a Borel probability measure concentrated on h.

A measurable partition 77 of h is said to be subordinate to W umanifolds with respect to # if for p-a.e, x E A,~(x) C W ~ ( f , x ) and it contains an open neighbourhood of x in W ~ ( f , x ) , this neighbourhood being taken in the submanifold topology of W ~ ( f , x).

D e f i n i t i o n 1.3.

We say that # has absolutely continuous conditional measures on WU-manifolds if for any measurable partition q of A subordinate to W umanifolds with respect to # one has

D e f i n i t i o n 1.4.

~

<< ~(~,~)

f o r ~ - a.e.x ~ A,

where {#~}~eA is a canonical system of conditional measures of # associated with q and ~],~) is the Lebesgue measure on W ~ ( f , x ) induced by its inherited Riemannian metric as a submanifold of M . 1.1. Let A be a hyperbolic attractor o f f with basin of attraction U. Then there exists a unique f-invariant Borel probability measure p with support in A such that it is characterized by each of the following properties: 1) p has absolutely continuous conditional measures on WU-manifolds; 2) Pesin's entropy formula holds true for the system f : (U,p) *--', i.e.

Proposition

hp(f) = / h ~

A(O(x)+mi(x)dp

184

where ,~(~)(x) < --. < .~(~(~))(x) denote the Lyapunov ezponents of f at x, , ,~(~) mil, x) ~i= 1 their multiplicities respectively, and ho(f) denotes the usual measuretheoretic entropy of the system f : (U,p) ~-0," 3) For Leb.-a.e. x C U, 1, ~ k = 0 8fk~ weakly converges to p as n -~ +co. This above result is due to Sinai [Sin], Bowen and Ruelle [Rue]3, [Bowl2. T h e measure p is thus called the SBR measure on the a t t r a c t o r A. As a particular case, a p r o o f of the proposition will also be included in Section 3, where we shall prove a more general r a n d o m version of the result. Now we begin to consider the case when f : U ~ is subjected to certain r a n d o m p e r t u r b a t i o n which can be viewed as r a n d o m compositions of m a p s from U into itself nearby f . Denote by C~(U, U)(r > 1) the space of all C r m a p s from U into itself, equipped with the C r c o m p a c t - o p e n topology (see [Hir]~). If g 9 C~(U, U) and g : U ---* gU is a C 2 diffeomorphism, in a way analogous to (1.1) in C h a p t e r II we define

Ilgllc~,v

= sup{T~Tg : ~ 9 TaM, I~1 ~ 1}

and

Igl c~ g = sup{TCTg -1 : ( 9 TguM, Is _< 1}. An equivalent definition of the C2-norms I1

IIc~,v

and II 9IIc~,v can be given by

using local charts as in the definition of I' Ic ~ in Section I. 1. Choose a neighbourhood L/t(f) of f in CI(U, U) such t h a t if g 9 g : U --~ gU is a C 1 diffeomorphism. Write now

then

Bf,u = max{lKllc~ v, Ilfllc=,v}. Given a n e i g h b o u r h o o d U ( f ) of f in CI(U, U) with U ( f ) C U l ( f ) and a n u m b e r B >_ Bf,u, we put

au(f),B : {g e U(f) n C2(U, U) " Ilgllo2,u _< B, I g 32 a -< B] and let ~'~Id(.f),B h a v e the inherited C 1 topology as a subset of C I(U, U). By Arzela-Ascoli theorem, it is easy to see t h a t f~u(f),B is a c o m p a c t subset of CI(U,U). Also, if g C f~u(/),B then g is a uniformly continuous m a p from U into itself and so it can be uniquely extended to a continuous m a p defined on U. In the sequel, for a given neighbourhood /d(f) of f in CI(U, U) with /,/(f) C U l ( f ) and a given n u m b e r B >_ Bf,u we shall write

12 = l)u(f),B for simplicity of notation. This will not cause any confusion. Given 12 = f2u(f),B, let f~Z be the bi-infinite product of copies of 12 and let f~Z have the p r o d u c t a - a l g e b r a B(fl) z and the p r o d u c t topology. For w =

185

( ' - - , g - l ( w ) , go(w),gl(w),...) E ~ Z and n > 0 we write

gO = id, g~ = g , - l ( w ) o " ' o g 0 ( w ) , gw n --~ g _ n ( W ) - 1 o . . . O g _ l ( ~ )

-1

defined wherever they make sense. If v is a Borel probability measure on ~, we shall denote by 2d(U, v) the random dynamical system generated by actions on U of g ~ , n E Z with w being chosen according to law v z , where v Z is the bi-infinite product of copies of v. Let 7Y(U, v) be given. We define

(~,~),

a : a z x u - ~ ~ z • u,

, (~,g0(~)~),

where r is the shift operator on flZ (see Section I. 1), and projections PI:~Zxu--~

z,

P2:~ZxU---~U,

(w,x) , (w,x),

, w, )x.

Now suppose that/5 is a Borel probability measure on f~Z • U such that

Gf~ = ~,

Rift = vZ.

(1.2)

By the definition of ~ there holds clearly the following integrability condition:

/~Z

l~

IT~g~ IdP < +co.

xV

From this and Oseledec multiplicative ergodic theorem we obtain the following

Let G : ( ~ Z x U,/5) ~ be as given above. Then for ~-a.e. (w, x) E ~ Z • U there exist measurable (in (w, x)) numbers r(w, x) and P r o p o s i t i o n 1.2.

~(1)(~, ~) < . . . < ~(r(~,~))(~, ~), and also an associated measurable (in (w,x)) filtration by linear subspaces of T~M such that lim

1 l o g IT~g~l = ~(O(w, x)

n--*+oo n

\ I / U - I ) , 1 < i < r(w,x). for each ~ E l/(i) "(w,z)\'(w,x)

186

As usual, A(i)(w,x), 1 < i < r(w,x) are called the Lyapunov exponents of G : (f2 z x U,/5) +--' at point (w,x), and mi(w,x) &f=dim V,(i)(~,.) - d i m ~ . ~ ) is called the multiplicity of A(i)(w, x). We next turn to an (measure-theoretic) entropy characteristic of the system G : (f2 z x U, fi) ~-'. Let {#~o}~oeaZ be a (vZ-mod 0 unique) canonical system of conditional measures of/5 associated with the partition {{w} x U :w ~ f~Z} of ~ Z x U. Identifying {w} x U with U, we regard p~o as a Borel probability measure on U and call {#w }weaZ the famdy of sample measures of/5. From the G-invariance of/5 it follows clearly that for each k C Z +

gw#w = #rkw ,

v z - a.e.w.

(1.3)

P r o p o s i t i o n 1.3. Let G : (~2z x U,/5) ~-" be as given above. 1) Let ~ be a finite measurable partition of U. Then the limit

o_1

)

exists and is constant for vZ-a.e, w E f2Z, 2) Define hs( X ( U , v ) ) - - s u p { h ~ ( X(U,v),~) : ~ is a finite measurable partition of U}. (1.4) Then hs( hi(U, v)) = h~(G) (1.5) where o- = { r • u : r ~ t ~ ( a ) z )

R e m a r k 1.2. Bogenschiitz introduced in [Bog] the notion of measure-theoretic entropy for a general random dynamical system. The definition of h~( ~(U, v)) in (1.4) fits into that context.

Proo~

1) Put

an(W,~) ----H,,~

,~-1g ,~).

187

Then

['n+m-1

"~

an+m(W,~) = <- gu~

g

+ Hu~

-=

=

g~o)-1

m-1

)

g -,o) -1

k=O

a.(w,~)+a~(T~w,~)

for v Z - a . e . w . By the subadditive ergodic theorem (Theorem I. 3.1), the limit function a(w,~) = lim,~--++oo•,~ ,~(w,() exists and is r-invariant and thus constant v Z almost everywhere on f~Z. 2) We first prove that, if ~ is a finite measurable partition of U and ~ is a countable measurable partition of f~Z then h~( A'(U, v),~) = where~x~={rxA:I'E~,AE~).

h;(G,~ x ~),

(1.6)

In fact,

h~ (a, ~ • ~) =

1.Iv

lim n -.l- CX:~

n ~

u

)

G-k(~x()

~r

\k=O

=

,;-~+oolimIu/ n H

~(

n--1

n--.*+oon [

lim - 1 j H u , ~

=

~( ~) ) ' , kG-k(~ =0 x

{~}xv) dvZ(w)

)

(w

\k~o(g~)-i ~ dvZ(w)

hr,(~:(U,v),~).

This proves (1.6). Now from (1.6) it follows clearly that

h~( ~(u, v)) < h~(a). Then what remains is to prove that

h~( x(U, ,4) >_ h'~(a). By (1.6) it suffices to show that for every finite measurable partition c~ of F~Z x U and every ~ > 0 there exists a measurable partition fl of F~Z x U of the type x ~ as explained above such that

h,~(a,,~) < h~(V,/~)+~. 188

Since U is compact, one can easily find an increasing sequence of finite measurable partitions {(,~},~=1 +o~ of U such that V,~=I +o~ (,~ is the partition of U into single points. Define j3,~ = {~Z} • ~n,n >__ 1. By (3.8) of Chapter 0 one has H~(c~ I ~,~ V a) --+ 0 as n + +oo. This together with 4) of Theorem 0.4.2 yields that h~(G,c~) _< h~(G,~3,~)+H;,(c~ 19~ V~)

<

h~(C,Z~)+~

for sufficiently large n. The proof is completed.

[]

Given ~ = ~U(I),B, we put

X~(l),. = ~ a~(a z • u) n>0

and also write

= ~'u(I),B for simplicity of notation. For (w, x) G/~, the unstable manifold W " ( w , x) of G at (w, x) is defined as w"(~,x)

= {y 9 u : d ( g ; ' x , 9 ; ~ v )

--+ 0 as ~ ~ + o o } .

I f g / ( f ) is given sufficiently small, then for any given B > B I , u , A = AU(I),B is a compact subset of flZ x U, G/~ =/k, and W ~ (w, x) is a C ~ immersed submanifold of U and WU(w, x) C A for each (w, x) 9 h. Proofs of these results will be given in the next section. Now let b/(f) be such a sufficiently small neighbourhood of f in C l ( U , U ) and let B >_ BI,u. Let/5 be a G-invariant Borel probability measure on ~ Z x U (where 9t = f~u(]).s). Clearly, /5 is concentrated on /~ = -~u(/),B. By as we denote the measurable partition {{w} x U : w 9 f~Z} Is

A measurable partition rl of A is said to be subordinate to W"-manifolds wi~h respect to/5 if , >_ ~;~ and for/5-a.e. (w,x) 9 ;~,~0:) ~f {V : (w,V) C O ( ~ , x ) } C W"(~,x) and it contains an open neighbourhood of x in W~'(w,x), this neighbourhood being taken in the submanifold topology of w~(~,x). D e f i n i t i o n 1.5.

We say that/5 has absolutely continuous conditional measures on WU-manifolds if for any measurable partition rI of A subordinate to I/Vumanifolds with respect to/5 one has

D e f i n i t i o n 1.6.

/sr/

u

6g[

where { ~7(~,~)}(w,~)eA is a canonical system of conditional measures of ft associated with ~7,It(w,x)-' is treated as a measure on 71~(x) by identifying {w} • r/w(x) with Tl~(x), and )~(~,~) is the Lebesgue measure on WU(w,x) induced by its inherited Riemannian metric as a submanifold of M . Clearly, this p r o p e r t y of/5 can also be characterized by the corresponding property of its sample measures. We now state the main result of this chapter in the following theorem: Theorem

1.1.

Let A be a hyperbolic attractor o f f with basin of attraction U.

Given B >_ B~,~, flU(f) is a su~ciently small neighbourhood o f f in C~(U, U) and v is a Borel probability measure on CI(U, U) which is concentrated on ~ ---~U(]),B, then there exists a unique G-invariant Borel probability measure ~ with support in A -- Au(I),B and with Plfi = vZ such that it is characterized by each of the following properties: 1) fi has absolutely continuous conditional measures on W~-manifolds;

3) For vZ•

Leb.-a.e. ( w , x ) C ~ Z x U one has as n --~ +oo n-1

1 ~ ~G,(~,.) -~ ~ n

(:.7)

k=0

In addition, G : (A,~) ~-~ is ergodie. T h e p r o o f of this t h e o r e m will be given in the next two sections. Given ~ = ~U(I),B and a Boret probability measure v on ~, the transition probabilities P(x, .),x E U of X(U, v) are defined by

P ( x , A ) = v({g E ~ : gx e A}) for z C U and A E B(U). We say t h a t the transition probabilities of 7Y(U, v) have a density if there is a Borel function p : U • U --* R + such that for every x E U one has / .

P(x, A) = ]A p(x, y)dA(y) for all A E B(U), where ~ denotes the Lebesgue measure on U. A Borel probability m e a s u r e It on U is said to be Az(U, v)-invariant if

f gitdv(g) -= It or equivalently

/p(~,

A)ait(~) = It(A) 190

for all A E B(U). It is easy to see that if H ( f ) is given sufficiently small such that gU C W for all g E H ( f ) and for some compact neighbourhood W of A in U then there exists at least one X(U, v)-invariant measure. As we have shown in Section IV. 1, if the transition probabilities of X(U, v) have a density then any zV(U, v)-invariant measure is absolutely continuous with respect to the Lebesgue measure on U. For a reason analogous to Proposition I. 1.2, if # is an zV(U,v)-invariant measure then G~(v z x p ) weakly converges as n --* +cx~ to a G-invariant measure #* on f2 Z x U which satisfies PiP* = v Z and P~p* = #. As a consequence of T h e o r e m 1.1 we have

In the circumstances of Theorem 1.1, if we assume moreover that the transition probabilities of ~(U, v) have a density and # is an zV(U, v)invariant measure, then C o r o l l a r y 1.1.

~=~*

where ~ is the measure defined in Theorem 1.1 and #* is as introduced just above. Proof. Since the transition probabilities of rV(U, v) have a density, we know that # << Leb.. By 3) of Theorem 1.1, this implies that v z x ~u-a.e. (w,~) E f~Z x U satisfies (1.7). Let ~, : f~Z x U ~ R be a bounded continuous function. Then n--1

f~d~=

/,~._.+mlim-n 1 E~~ k=0

=

lim / ~ d

Gk(v z x #) k=O

n - - * + oo

=

)

f ~dp*

and thus ~ = #*.

[]

R e m a r k 1.3. Given B >__Bf,u, let l i ( f ) be a sufficiently small neighbourhood of f in CI(U, U), and let v~,e > 0 be a family of Borel probability measures on CI(U, U) with support in f~ = f~U(I),B and with v~ --~ 6] as c ---* 0. Let t~ and /5/ be the measures on ft Z x U given by T h e o r e m 1.1 corresponding to v = v~ and v = 61 respectively. Then it can be shown that

as e + 0 ([You]). Thus, if for each e > 0 the transition probabilities of 7Y(U,v~) have a density and Pc is an ~ ( U , ve)-invariant measure, then, by Corollary 1.1, p* - , p * as c ---* 0, where p is the SBR measure on A. From this it follows that

2J1~ ~ P2P* 191

as ~ ---* 0, i.e. #~ --~p as c ~ 0. This result can be interpreted as a statement of stochastic stability of SBR measures on hyperbolic attractors ([You]).

w

Technical

Preparations for the of the Main Rusult

Proof

Here we present some technical preparations for the proof of T h e o r e m 1.1. We shall only outline main arguments or principle ideas of the proofs of the results presented in this section, leaving details to the reader. T h r o u g h o u t this section we shall always assume that f : U0 --* M is as given in Section 1, A is a hyperbolic attractor of f with basin of attraction U, and 0 < ,~0 < 1 is the number introduced in Definition 1.1 corresponding to A. By using standard machinery of fibre bundle theory, one can extend T A M = E u | E s to a continuous splitting Tu, M = E 1 | E 2 of the tangent bundle T u , M over an open neighbourhood U' of A (see [Hir]3). We now fix an open neighbourhood V of A such that V C U Cl U ~. On Tv M = E 1 | E 2 we introduce a new n o r m II II by

The norms [. and II" II on T v M are clearly equivalent. Given f~ = f2u(f),B , let B_ = ~-u(f),B be as defined in Section 1. It is easy to see that there exists a neighbourhood/,/2(f) of f in CI(U, U) with U~(f) C / A l ( f ) (see Section 1) such that i f U ( f ) C U s ( f ) and B >_ B j , g then C ~ Z x V.

(2.1)

When (2.1) holds true, we denote by Es the pull-back of T v M by means of the projection P2 : A ---* V, (w, x ) , , x. P r o p o s i t i o n 2.1. There exists a neighbourhood U3(f) of f in CI(u, U) with H3(f) C H2(f) such that, if H ( f ) C H3(f) and B > B f , v , then there is a continuous splitting of Eh into E h 9 E A and there exists a number 0 < ~ < 1 (depending only o n / / 3 ( f ) ) such that the following hold true: 1) T~go(w)E~w,x ) = E G(w,~) ~ and T~go(w) E (~,~) s s for each (w, x) E = Ec(w,~) A;

2) For each (w, x) e

192

Proof.

Let E 1 and E 2 be respectively the pull-backs of E ~ and E 2 by means

of the map P2 : X --, V, (w, z) ,

, x. As usual, by L(E}t , E]) we denote the

fibre bundle over A whose fibre at point (w, z) is the space of all linear maps from E~w,~ ) to E 200,~), and by a continuous section c~ of L ( E ~ , E ] ) w e mean a continuous map ~r :/~ L(Es1 Es2 satisfying ~ro o" = id where ~r: L(E[, E]) --+ /~ is the natural projection. Let now S ~ denote the space of all continuous sections of L(Es1 Es2 equipped with the norm H" II defined by

I1~(~0, ~)11 < +o~.

I1~11 de~ sup

It is easy to see that (S ~, I1. II) is a Banach space with respect to the natural operations of addition and scalar multipIication. For each (w, x) E A, write

r.gL = [c,,(,.,,, ~) G12(w,x)~] E1 (~E~wx) E1 2 , ---' a(w,x) (~ EG(w,x) ' d;~l (~, z) G22(w,x) : (w,~:)

:r,~zl

[~,,(~, ~')

= La~(w, x)

G22(w,:Q]:

(~,,~)|

(~,~)

a-,(~,~)|

a-x(~,~)"

Choose a number 0 < e < 1 satisfying A0+2e< and write

1,

(Ao+e)-i -c>l

A~ + 2~

<1,

# =

(AO J- e) -1 -- C

9~ =

[('~0 nt- E) - 1 - - C] - 1

~

1.

Let now Ha(f) be a neighbourhood of f in C~(U, U) w i t h H 3 ( f ) C H2(f) such that if H(f) C H3(f) and B >_ Bf,u then for each (w, x) E A

max{lla~1(w,z)-111,lla22(w,x)ll,

IId,(~,~)ll,

max{lla~(w,~)ll, llC=l(W,~)ll, IId,2(~0,~)ll,

Ild~Xw,~)-~ll} < Ao+E, IIa2,(w,~)ll}<~.

Put S~ = {cr e So: I1~11 --< 1) and define a map F Sf --+ Sf by

(r~)(~, x) =

[a~(a-,(~, ~)) + a~(a-'(~, ~)) o ~ ( a - , ( ~ , x))] o[a,,(a-'(w, x)) + a12(G-'(w, x)) o ~(a-'(m, x))]-'

for ~ E S t and (w, x) E A. In a way analogous to the proof of Lemma VI. 4.6 one can check that F is a #-contraction and the unique fixed point, written a u, has the following properties: (i) IIo'~'ll <_ ~; 193

(ii) The vector bundle E h defined by

(w, x) 9 i satisfies

(w, .) 9 i;

T~go(w)E(%,~) : E~(~,~), Off) If (w, x) 9

and ~ 9 E (~,~), ~ then

IIT~g0(w)~ll _> ~-~11~11. By a completely analogous argument one can also prove that there exists a continuous section cr~ of L(E~, E l ) such that: (i)' I]~'ll _< p; (ii) ~ The vector bundle E A defined by

E~,~)

:

(a~(w, x), id)E~..,~),

satisfies

T~go(w)E~,.) = ES(~,.),

(w, ~) 9 i;

(iii)' If (w, x) 9 X and 77 9 E~w,~), then

IIT~g0(wbll ~ ~lbll. Thus we obtain a continuous splitting Es = Es 9 Es~ which obviously satisfies the requirements of the proposition. [] P r o p o s i t i o n 2.2. There exists a neighbourhoodLQ(f) o f f in CI(U,U) with //4(f) C U3(f) such that ff U(f) C //4(f) and B >_ Bj,u then there holds the following conclusion: There are numbers (depending only on U4(f) and B) C~u,j3=,Tu > 0 and 0 < O~ < 1, and for each 0 < 6 ~ c~, there exists a continuous family of C 1 embedded k~-dimensional (k= = d i m E ~( ,' )~ , (w, x) 9 A) discs

{W~'(w,x)}(~,~)e X in U such that the following hold true for each (w,x) 9 A." 1) W~(w, x) = exp. Graph (h(w,~)l{(eE~,~):ll(ll<e}) where

iS a C 1,1 m a p

satisfying

(i) h(~,.)(0) = O,Toh(~,~) = 0; (ii) Lip(h(~,~)) < fl~, Lip(T.h(~,.)) < fl~,, where Lip(.) is taken with respect to the norm I" I;

2) g;1w~(w,x) c w ~ ( a - l ( w , x ) ) ; 3) d~(g~ny, g~"z) <_ %0~d"(y,z),y,z E W~(w,x),n 6 Z +, where d~'( , ) denotes the distances along W~(V-"(w, ~)), ~ 9 Z+.

194

The proof of this proposition is actually a simplification of that of Theorem III. 3.1. The details are omitted here. We shall call W~'(w, x) a local unstable manifold of G : ~ Z x U ~ at (w,x). Let/A(f) C/A4(f) and B > B],u. From 2) of Proposition 2.2 it follows clearly that for each (w, x) 9 A and 0 < 6 < c~,

{~} x w~(w, ~) c X.

(2.2)

Suppose that ( w , x ) 9 [~ and W~'(w,x) is denned as in Section 1. Then, in a way analogous to the proof of Theorem III. 3.2, one has

W"(w,x) = U gL~ n>O

for each 0 < 6 < a~. Thus W " ( w , x ) is the image of E~'~,.) under an injective immersion of class C 1,1 and is tangent to E ~ , . ) at point x. Moreover, by (2.2) we have {,,,} x w " ( ~ , ~) c ~. The next result is an analogue of Proposition 2.2 for local stable manifolds of G : 12Z x U +-~ and the proof is also completely analogous. P r o p o s i t i o n 2.3. There exists a neighbourhood Ll5(f) of f in C I ( u , U) with /As(f) C Ud(f) such that if U ( f ) C /As(f) and B >_ B L u then there holds the following conclusion: There are numbers (depending only on ~As(f) and B ) c~s,fl~,% > 0 and 0 < Os < 1, and for each 0 < 6 <_ c~s there exists a continuous family of C 1 embedded k s - d i m e n s i o n a l ( k s ~- dim E(~,~), (w, x) 9 [k) discs

{W~(w,x)}(w,,)es in U such that the following hold true for each ( w , x ) 9 fk: 1) W~(w, x) = exp~ Graph(l(~,~)l{neE~,~):lloll<6} ) where 1(~,,:) : {7 e E ~

"

---,

~

is a C 1,1 map satisfying (i) l(~0,~)(0) = 0, Tol(~,~) = 0; (it) Lip(l(~0,~)) BLU. For (w,x) e A and y 9 W ~ , ( w , x ) , we denote by E (~,y) s the subspace Of TyM tangent to WS~,(w,x) at point y. P r o p o s i t i o n 2.4. There is a neighbourhood /A6(f) of f in C I ( U , U ) with U6(f) C Us(f), for which there holds the following conclusion: f l U ( f ) C/A6(f) and B >_ BI,u , then there exist numbers (depending only on/A6(f) and t3) 195

0 < 60 < min{a~, a s } , a 0 > 0 and Lo > 0 such that for each (w,x) 9 A and 0 < ~ < 6o the family of spaces { ~ s(~,y) : y 9 Uzew{(w,=)W~s (w,z)} is Hhlder continuous in y on Uzew;(~,~)W~(w, z) with exponent ao and constant Lo (for the definition see Definition IlL ~.2).

Proof. Let V be as given at the beginning of this section. By an argument analogous to the proof of Lemma III. 4.2 we know that for any given ~ = ~u(]),B there is a constant C > 0 (depending only on B) such that

IT(g~(~)lv)lm

<

c"

(2.3)

i----0

for a l l w 9 Z a n d n > 1. Now fix another open neighbourhood W of h which satisfies W C V. Clearly, there exists a neighbourhood t/6(f) of f in CI(u, U) w i t h / / 6 ( f ) C / I s ( f ) such that i f t / ( f ) = / / 6 ( f ) and B _> B],~r then for some 0 < 5 < min{c~,a~} one has W~(w,x) C W for all (w,x) 9 A. In this case, note that, i f ( w , x ) 9 /~ and ( , /(~o,=) is an inner product on E0o,= ) = E (~,~) @ E (~,~) which coincides with ~' ~ the Riemannian metric ( , ) on E~,=) and E ~(~,~) and which makes E ~(~,=) and E(~,=) "orthogonal, then the norm [. [(~,~) induced by ( , )(~,=) is equivalent to I" I uniformly for (w,x) 9 A. Note also that if E~ in Proposition III. 4.1 is replaced by a subspace F= of H which satisfies "y(E=, F=) > 7o > 0 then the conclusion of that proposition holds true as well with the corresponding exponent and constant changing suitably with 7o. Clearly, the same kind of argument also holds true for Corollary III. 4.1. Then, by (2.3) and Propositions 2.2 and 2.3, one can obtain the desired conclusion from an argument analogous to the proof of Theorem III. 4.1. [] Proposition 2.4 implies that the family of embedded discs {W;(w, z)}zew;(~,=) is absolutely continuous for each 0 < 5 _< 50 and (w, x) C A if U ( f ) C U6(f) and B >_ BI, u. To be precise, we first explain the idea of absolutely continuous family of C 1 embedded k-dimensional discs in M. Let A C M be a set and let {D=}=ea be a continuous family of C 1 embedded k-dimensional (1 _< k < m0, where m0 = d i m M ) discs in M such that Dy n Dz = 0 if y,z E A and y ~ z. Let x0 C A and p, q C Dxo, and let Wv, Wq be two smoothly embedded (m0 - k)dimensional discs transverse to D~o at p and q respectively. Then there exist two open submanifolds l?dv and Wq of W v and Wq respectively such that the so-called Poinca% map

~ND~I

) ~VqnDx

is a homeomorphism between % n and n The family of C 1 embedded discs {Dz}=cA is said to be absolutely continuous if each of its 196

Poincard maps Pw~wq is absolutely continuous (with respect to the Lebesgue measures on Wp and Wq). P r o p o s i t i o n 2.5. Let U ( f ) C U6(f) and t3 > 13I,v. Then for each (w, x) e and 0 < 6 < 60 (see Proposition 2.~), the family of C ~ embedded discs {W~(w, z)}~ewg(~,~ ) is absolutely continuous. Like in the case of Theorem III. 5.1, the proof of this proposition follows the line of the arguments of Part II of [Kat]. We also omit the details here. P r o p o s i t i o n 2.6. There is a neighbourhood UT(f) of f in Ca(U, U) with NT(f) C bl6(f), for which there holds the following conclusion: I f U ( f ) = liT(f) and B >__BI,u , then there exist an open neighbourhood N of A in M, a number 0 < &u < 60 (see Proposition 2.~), a continuous family of C 1 embedded k~,dimensional discs {W~(w,z)}(~,z)eaZxN for each 0 < 6 < &~, such that the following holds true: 1) W ~ . ( w , x ) C N for each (w,x) e iX; 2) go(w)lfV~(w,z) D t]V~(G(w,z)) if z e g and go(w)z E N; 3) If (w,z) e A, then l}V~'(w,z) = W•(w,z); 4) For each (w,~) e ;X, Uz~w;(~,~)W~'(w,z) contains an open neighbourhood of x in M; "~ ~) C U~ew;(~,y)W~~( ~,z) if 5) For each (w,x) E A,O~ewi/,(~,,)W~/4(w, y ~ wh,(~,

~) with (~, y) e ;t.

The idea of the proof of the proposition is the same as that of the proof of [Che] Proposition 2.1 or [Hir]3 Theorem 4.2, which deal with the result in the case of deterministic dynamical systems (for endomorphisms and diffeomorphisms respectively). We refer the reader to those references for an analogous proof.

w Proof

of the Main

Result

In this section we complete the proof of Theorem 1.1. Let bli(f), 1 < i < 7 be as introduced in the last section. Put 7

Uo(f)=Au, iy)=w(y). i=1

Lemma 3.1 Let U ( f ) C Ho(f) and B >__ BLu, and let (Cu,~) C A be a point such that rn~v # (v for all n > 0 and W-~((v,~) a local unstable manifold introduced in Proposition 2.2 for some 0 < 6 < oLu. If we write L -: W-~(,b, ~)

197

and let ~L be the measure on {tb} x W~U(~b,~:) defined by identifying ~L with the normalized Lebesgue measure )~L on L, then any limit measure/5 of 1 n-1

g ~G~,

nZN

k=O

has absolutely continuous conditional measures on W~'-manifolds. Proof We first give some useful estimates. From the definition of f~ = f~u(y),B and 1) (ii) of Proposition 2.2 it follows that there exists A > 0 such that for every 0 < 6 < ~ , , if (w,x) E A and y,z C W~'(w,x), then

IJ"(w,y) - J"(w,z)l <_Ad~(y,z) where JU(w,y) = Idet(Tugo(w)lE~,~))l and J~(w,z)is defined analogously. By this fact and 2), 3) of Proposition 2.2, in a way similar to the proof of Lemma VI. 8.1 one can see that there is C > 0 such that for each (w,x) E/~ and 0 < 6 _< C~u there holds the estimate 1

-c <

r I JU(G-k(w'Y)) < C J u ( c - ~ ( w , z)) -

(3.1)

k:l

for all y,z 6 W~(w,x) and n E N, and moreover,

+~ j " ( c - ~ ( w , x))

A((~,x),(~,~)) ~~ 1-[ z.(c-~(~,u)),

u e w~'(w,x)

k=l

is a well-defined function of y C W~(w, x). Suppose that/5 is a limit measure of L ~ - ~

Gk~L,n G N and

ni--i

1 ~

Gk~L__,#

Tli k=O

as i --~ +cx~ for some subsequence {ni}i>_O. Let V be a Borel subset of A of positive/5 measure such that it is the disjoint union of {w} • W~(w, z), (w, x) E E for some 0 < 6 < au and some subset E of/~. For (w,y) E r~, let V(~0,u) denote the element of the partition { {w} x W~ (w, x)}(t0,~)e~--]~ of Q that contains (w, y), and let (/519)(~o,u) denote the conditional probability measure of/519 on V(t0,u)- We may regard (/hlg)(~,u) as a measure on W"(w,y) by identifying {w} • (V(~,u))~ with (V(~,u))~o C WU(w,y). It is easy to see that, if we can find a finite cover of/k by Borel sets V/, i = 1, .. 9 l of the type of Q such that for each V/ one has for/5-a.e. (w, y) E ~ , then/5 has absolutely continuous conditional measures on W~-manifolds. 198

We now prove the existence of such a finite cover by constructing certain canonical neighbourhoods of A. Let (w0, x0) E A and 0 < 5 ___min{a~, a~}, and let W be an open neighbourhood of w0 in ~Z. Writting T = W[(wo,xo), we put S = ( W x T) n..~. and

=

U

{w} x W~(w,~).

(w,~)cz By Propositions 2.2 and 2.3 it is easy to see that we can choose 6 and W sufficiently small so that V is the disjoint union of {w} x W~'(w, x), (w, x) E E. Let V be a Borel subset of/~ thus obtained. It clearly contains an open neighbourhood of (w0, x0) in 3.. We may assume that /5(V) > 0, and we may also assume t h a t / 5 ( 0 i 7) = 0 (8(.) is taken in the topology of/~) by shrinking 6 or W if necessary. For each n E Z +, let L,~ = {z C L : Gn(@,z) E {w} x W~(w,x) for some (w,x) C ~ but G'~({dJ} x L) ~ {w} x W~(w,x)}. From 3) of Proposition 2.2 it follows that AL(Ln) --~ 0 as n --* +cx~. So we have nl --i

lira --1 E i---.+oo ni

k=0

Gk(iLI~}•

= ~.

(3.2)

Also, since/5(0~') = 0, it holds that lim i~+oo

1 E ~/

Gk(]XL I{~}x(L\Lk))

(l)) = ]5(I9).

(3.3)

k=0

Suppose that C"({~} x (L\L.)) ~ {w} x W~'(~, ~) for some (~,~) ~ E . Denote by ~,~,(~,~) the conditional probability measure of [G"(~L ]{al• on {w) X W~'(w,x) and let drh~,(~ ,~) Pn -tt dA(~,~) where we regard A(u,~) as a measure on {w} x with {w} x W~(w,x). Then

W~(w, x) by identifying W~(w, x) 1

pn(w,y) =

~ J~(C-k(~,~))

k=l,~ /w ;(~,~) ~= J~(C,-k(w, 1 y)) dA~,,~)(y) (3.4)

lei J~(a-'( w, x)) k=l J"(a-k(~, y))

fw(w,~)fl J~(a-~(w'x)) 199

9

•

W (w,x)

For each n 9 Z +, let p~ : l) ~ R + be defined by (3.4). In view of (3.1), it is not difficult to prove that p,~, n 9 Z + are uniformly continuous functions and p,~ uniformly converges as n ---, + o o to a continuous function p : l) ~ R + which is defined by

y) =

(w, y)) A((w, x), (w, y))dA(L,~)(y )

w

if ( w , y ) E {w) x W~'(w.x) and (w, x) E E on l) by letting

We now define a Borel measure 5

for A E B(I)). Then, taking (3.2) and (3.3) into consideration, one can easily show t h a t hi--1 i~+oo

n i

k=0

and hence S i n c e / k is compact, we can find a finite number of Borel subsets Q 1 , " ' , of A of the type of V constructed above such t h a t they cover A. T h e proof of this proposition is then completed. [] P r o p o s i t i o n 3.1. Let 14(f) C Llo(f) and t3 > 132,u, and let v be a Borel probability measure on ~ = ~HU),B. Then there exists a G-invariant Borel

probability measure/5 on ~ Z • U satisfying:

1) Pl/5 = vZ; 2) /5 has absolutely continuous conditional measures on WU-manifolds.

Proof.

Since v : ( ~ Z , v Z) ~

is ergodic, for vZ-a.e, w E ~ Z one has n-1

1 ~ 6~w n

--~ v z

(3.5)

k=O

as n ---* + ~ . Let L = W~(t~,~) for some 0 < 5 < a= and some (tb,~) E ,~ with ~b satisfying (3.5), and let /5 be a measure constructed as in L e m m a 3.1. T h e n /5 clearly satisfies the requirements of the proposition. [] P r o p o s i t i o n 3.2. Let h i ( f ) C N o ( f ) and B > B],u, and let v be a Borel probability measure on Q = QU(S),B. I f / 5 is a G-invariant Borel probability 200

measure on f~Z x U satisfying Plfi = v Z, then (3.6)

Proof.

By (2.1) we know t h a t Ac~Zxv.

Write

do = d(V, OU) d~f i n f { d ( x , y ) : x E V , y C 0U} > 0. Fix two numbers 0 < p0 < do and b0 > 0 such t h a t for each x E U, the exponential m a p exp~ : {~ C T~M 1~] < P0} ---+ B(x, po) is a diffeomorphism and bold(y,z) < [exp~-~ y - e x p ~ -~ z[ _< bod(y,z) for any y, z E B(x, Po). For n r N we define

f~'~ = { g : g = g , ~ o . . . o g l , g i C f ~ , l


By a r g u m e n t s similar to those in Section II. 1 and by the definition of ~ = f~u(]),B, for each n E N there exists 0 < r,~ < po/2 with the following properties: If x E U with d(x, V) < r~ and g G ~'~, then the map def

is well defined and sup

ITeH(g,~ ) - ToH(g,~)l < ao

where a0 = min{bo I , 1}. From this it follows t h a t if x, y E U satisfy d(x, V) < r,~ and d(z, y) < v,~ then for any g E f2'~ it holds t h a t

d(g(y), expg(x ) oTxg o exp~-1 y) < d(x, y).

(3.7)

Now fix n E N arbitrarily. For any given sufficiently small r > 0, take a m a x i m a l c-separated set E~ of U and then define a measurable partition a t = {o~(z) : z E E~} of U such t h a t a t ( x ) C int(a~(z)) and int (c~(x)) = {y E U : d(y, x) < d(y, zi) if x # xi E E~ }. By Proposition 1.3 we have

nh

(x(u,v)) = =

= h (C lim h,~ G '~,&~ )

201

(3.s)

where 5~ = {~Z x c~e(x) : x C E~}. Let 0 < g < r,~/4(1 + b0). In a way analogous to (2.3) of Chapter II one has for all l > 1

\k=O

< Define

H~(S~ I~) + (l - 1)H~(G-~&~ I ~ V ~r). if x = 0

g(x) = { 0

- x log x

if x > 0

'

by

foe/\

V~

we

have

h~(c",<) Hz(C-"< I ~ v ~)

<

H~((g~)

=

=/

o~ l e~)dvZ(w)

(3.10)

~(~(~)) ~ g \ xEE~

E

~(-~(~))

yEE~

#~(oLs(x))log Nn(w)dvZ(w),

xEE~

where N~(w) is the number of elements of c~ which intersect suppose that a~(x) n V r 0. Then, by (3.r),

g~o~r where

g,~o~(x). Now

C gn exp~ B(0, c) C expg:~ B(%~g,~B(O, e), bog)

B(Q, 5) is the 5-neighbourhood of Q c Tg:,:M in Tg~,~M. If ~(y) n 9 ~ ( x ) # 0,

we have B(y, bo 2 2) N e x p g ~

B(T,:g w B(O, c), 2bog) r 0

and then B(exp~=

g y, b0-a ~)nB(Y.g~B(0,~),Ub0e) r 0.

Since unitary operators preserve distances, it is easy to see that the number of disjoint balls which intersect B(T,:g,~ B(O, g), 2b0g) and whose diameters are bole mo does not exceed CI-I,=o max{6i(T,:gp~), 1} where C is a constant depending only on b0 and too(too = d i m M ) and 5i(T,:g~o), 1 < i < mo are as introduced at the beginning of Section II. 2. Note that /5(A) = 1 and A C ~ Z x V. Thus, by

202

(3.10),

.eE,E

<-f

Pw(~176

i=1

~r (=)nv~:$ ITI o

=

logC + E

/ H}i)(w'y)d/5

i----1

where H(i)(w,y) = log + 6i(Txgn~) if y C c~(x). In a way similar to the proof of (2.5) in Chapter II, we have

H!i)(w, y)d/5 =

lim ~

log + 5i(Tygn)d/5

~..., 0 ~ . . ~ r

i=1

z:l

<

f log I(Tug~) n ^ Id•~

= .f

yl+m,(w, y)d/5. i

Hence, by (3.8), for all n E N

x(v, v)) _< 1logo

+f

which implies (3.6) by letting n -~ +co.

[]

Let H(f) C Ho(f) and B > BI,u, and let v be a Borel probability measure on f~ = ~U(I),B. If/5 is a G-invariant Borel probability measure on f2 Z x U with P1/5 = vZ, then the following two conditions are equivalent: 1)/5 has absolutely continuous conditional measures on WU-manifolds; 2) h~( z~(V,v)) = f E i ~(i)(w,x)+mi(w,x)d/5 9 P r o p o s i t i o n 3.3.

As for the proof of this proposition, note that the proof of Theorem VI. 1.1 actually yields the following more general result: Let Yd(M, v) and G : f2Z x M ~ be as defined in Section VI. 1 and let/5 be a G-invariant Borel probability measure on f~Z x M satisfying P1/5 = v Z, then/5 has absolutely continuous conditional measures on W~-manifolds if and only if h~(G) = f ~ i A(i)( w, x)+m~( w, x)d/5, where A(/)(w,x), 1 < i < r(w,x) are the Lyapunov exponents of G : (~2z x M,/5) ~ at (w, x) and mi(w, x) is the multiplicity of A(/)(w, x). It is then easy to see that Proposition 3.4 can be proved by a simplification of the arguments of Sections VI. 2-8, the simplification being due to the non-existence of zero 203

Lyapunov exponent for the present system G : (f2 z x U,/5) *--~. We leave the details of the proof to the reader. Analogous to Corollary VI. 8.2, there also follows the following result: C o r o l l a r y 3.1. In the circumstances of Proposition 3.3, let/5 be a G-invariant Borel probability measure on f~Z x U with P1/5 = v z such that 2) of Proposition 3.3 holds true. If q is a partition of A subordinate to W~-manifolds with respect to /5 and let p be the density of/sn with respect to ,~(u,~), then for/5-a.e. (w, x), there exist a countable number of disjoint open subsets Un(w, x), n E N of W ~ ( w , x ) such that U e N U n ( w , x ) C rl~(x), A~o,~)(rho(x)\ U,~eN Un(w,x)) = 0 and on each Un(w,x) p is a strictly positive function satisfying _

71=

w, y ) ) '

y' z c

u=(w,x),

in particular, log p restricted to each U,(w, x) is Lipschitz along WU(w, x). P r o p o s i t i o n 3.4. Let B >_ By,or be given. If hi(f) C Llo(f) is sufficiently small, v is a Borel probability measure on Q = f2U(f),S and/5 is a G-invariant Borel probability measure on f2z x U satisfying 1) and 2) of Proposition 3.1, then l r~--i

- ~ n

6a~(~,y) -+/5

k=0

as n ---* +co for v Z x Leb.-a.e. (w,y) E f2 Z x U. Proof. First consider flA. By the spectral decomposition theorem (see Section 3.B of [BoW]l), every hyperbolic attractor can be decomposed into finitely many components A1,...,Ato such that fAi = Ai+l, 1 < i < l0 (where Ato+l = A1) and for each 1 < i < lo,ft~ is topologically mixing, i.e. for any two subsets V, W of Ai, open in the topology of Ai, there is a positive integer K such that f-t~ 7~ 0 for all k > K. The basin of attraction U of A can also be written as the union of open sets U1,..., Ulo which satisfy U/ D Ai and f U i C Ui+l for each 1 < i < l0 (where Uzo+l = UI). Correspondingly, if U ( f ) C U0(f) is sufficiently small and B > By,u, then G(f~ z x Ui) C ~ Z x Ui+i,1 < i < l0 and there exist disjoint compact sets /~l,'",/~Zo such that /~ = O ilo= 0 A~i , ~i C f2z • Ui and G[ki = Ai+l for all 1 < i < l0 (where/~zo+l = A1). In this case, if 15 is a G-invariant Borel probability measure on f2Z x U with P1/5 = v Z, then/5 has the expression 1

to

/5 V0.= 204

with ~i being a Borel probability measure on f2z x Ui which has support in P.i and satisfies G~i = ~i+1 and P1/hi = v Z, 1 _< i < l0 (where tht0+l =/~l)To prove the conclusion of the proposition, we first consider the case l0 = 1. Now let B > BI,u be given and let U ( f ) C H0(f). Fix 0 < 6 < &~/2 (see Proposition 2.6) and let {lTd,(w, z)}00,~)eaZ• be a continuous family of C 1 embedded k~-dimensional discs in U given by Proposition 2.6. For each (w, x) E /~, put N~(w,x) = Uzew;(~,~)i,;d~(w,z ) and N6(w,x) = N~(w,x)M ~.~, where / ~ = { x : (w, x) C A}. According to Proposition 2.6, N6(w, x) contains an open neighbourhood of x in U. Denote by w I the point (-.-, f, f, f , . . . ) in f~Z. Since A is compact, there exist x l , - . 9 xt E A such that U~=INUs(WI, xi) constitutes a neighbourhood of A in U. By the topological mixing property of flA and 3) of Proposition 2.2, there exists K > 0 such that for all 1 < i,j < 1, fKN6/s(Wf,xi) intersects N~/s(wy, xj) with fK N~/s(ws, xi) D W~(w I , z) for some z E W;/s(w I , Xj)N A. From this and Proposition 2.6 it follows that for the given B > BI,~] and 6 > 0, i f / / ( [ ) C / / 0 ( f ) is sufficiently small then for each w E f2Z there exist Yl,'" ", Yt E A~ and z l , . . -, zt E / ~ K w such that U~=lN614(w, yi) is a neighbourhood ofAw, - Ui=INh/4(T z g w, zi) is a neighbourhood o f / ~ K ~ and for all 1 ~ i,j < l, g~K N~/4(w, for yi) intersects /Y6(vKw, zj) with gwg Nh/4(w, Yi) D w T ( ~ K ~ , z ) some z E W~/4('rKw, zj) M A~K,~. Now let t7 be an ergodic component of/~. For a reason analogous to Proposition VI.2.1, /5' also has absolutely continuous conditional measures on W"manifolds. Moreover, by (3.5) we may assume that Plft' = v Z. Consider w E flZ. The point w satisfies (3.5) and will be subjected to a finite number of vZ-a.e, assumptions. Let # " be the conditional probability measure of ~' on /~,, (identified with {w} x ~_~) and let i be such that ' ^ ~ (w, z)] n Mo) > 0. From Corollary 3.1 one can see that ~,o([U~w;/,(,~,~,)w~ there exists a local unstable manifold W~(w,p),p E W~14(w, y~) with the property that A~w,v)-a.e. y E W~(w,p) satisfies 1

6ak(~,y ) --~

as n --+ +cr

(3.11)

n k=0

The absolute continuity property of the embedded discs {W~(w, Z)}zeW2(w,p) (see Proposition 2.5) together with 3) of Proposition 2.3 and 5) of Proposition 2.6 implies that A~'~z e y E l?V~/ 4(w, z) satisfies (3.11) for each z E W~/4(w, Yi). ( , ) -a "" From this and our choice of the yi's and zj's it follows that for each zj there exists s K a local unstable manifold W~(rKw,pj),pj E W~I4(r w, zj) with the property that for A~rK~,vD-a.e. y E W~(rKw,pj)

6ck(r~,.w,y) ---.

1

n

k=o

205

as n ---* +oo. For each 1 < j < l, the absolute continuity property of the family of embedded discs {W~(7"gw, Z)}zeW{(rKw,pj ) together with 3) of Proposition 2.3 implies that Leb.-a.e. y 6 tJzeWf(TKw,pDW~(TKw, z) satisfies (3.12), and hence, by 5) of Proposition 2.6, Leb. a.e. y E N6/4(TKw, Zj) satisfies (3.12). It is easy to see that for the given B > B/,u and 5 > 0, if N(f) C No(f) is sufficiently small then there exists an open subset W of U such that W C tJ}=~Ne/4(w, zj) for vZ-a.e, w E f~Z and ft Z x W D 3,. In this case it holds clearly that vZ• Leb.-a.e. (w,y) E f2z • W satisfies (3.11). The same is true for v Z x Leb.-a.e. (w,y) E f2 z x U since Gn(f2 z x U) C f2Z x W for all n greater than some no. This also clearly implies that/5~ =/5. The conclusion of the proposition is then proved for the case l0 = 1. We now prove the proposition for the case l0 > 1. Let B > BLu be given. Consider flo : U~ ~-~ and G t~ : f2z x U~ *--', 1 < i < l0 for U ( f ) C U 0 ( f ) . From the arguments above we see that, i f U ( f ) is sufficiently small, v is a Borel probability measure on f~ = f2u(l),B and t5 is a G-invariant Borel probability measure on flZ • U satisfying 1) and 2) of Proposition 3.1, then for each 1 < i < 10,vZx Leb.-a.e. (w, y) 6 f2Z x Ui satisfies n--1

n -lim +~

-1 n

6c~o~(~,,u ) = / 5 i k=0

and thus satisfies nlo-1

lim

lo

1

1 k=O

i=1

which implies n-1

lim -1 ~ n--,+oo n

6ak(~,y) =/5.

k=0

This completes the proof of the proposition. Theorem 1.1 follows clearly from Propositions 3.1-3.4.

206

[]

Appendix A Margulis-Ruelle Random

Inequality

Dynamical

for

Systems

As is described in the Introduction and Remark II.2.1, in this appendix we adopt with some modifications J. Bahnmiiller and T. Bogenschiitz's argument [Bah] about Ruelle's inequality. Their argument is carried out within a more general (than the "i. i. d." case) framework, due to the point of view of L. Arnold, of "stationary" random dynamical systems (see [Arn] for an introduction to this subject).

w

Notions and Preliminary Results

Let (f~, .T, P ) be a probability space, O: (fl, ~', P) ~ an invertible and ergodic measure-preserving transformation. We call (f~,.T, P, O) a Polish system if fl is a Polish space and .T is it's Borel ~-algebra. Let (X, B) be a probability space. We call a map r

(n, x)

,X,

x)

a (measurable) random dynamical system (RDS) on (X, B) over ( a , 9v, P, O) if the following hold true: 1) r is measurable; 2) Define r : Z ~ Z,x H r for n G Z + and w E ~. Then r n E Z + is a cocycle over 0, i.e. r w) = id and

r

+

=r

o

for all n, rn E Z + and all w E ~. Let r be an RDS defined above. It is said to be continuous if X is a topological space (with it's Borel G-algebra) and if r : X -+ X is continuous for all n C Z + and all w E ~. It is said to be C" (r >_ 1) if X is a C" manifold and r w) is C" differentiable for all n E Z + and all w E ~. In the sequel we often omit mentioning (~, $', P, 0) when speaking of an RDS r Let r be an RDS on X. Define O:~xX

(w, x) ~ (Ow, r

,~xX,

and call it the skew product transformation induced by r A probability measure # on (12 x X, ~ x B) is said to be r if it is invariant under O and if 207

it has marginal P on ft. Invariant measures always exist for continuous RDS on a compact metric space X (which is in complete analogy with deterministic dynamical systems). Denote by P r ( X ) the space of probability measures on (X, B), endowed with the smallest o--algebra making the maps P r ( X ) ~ R , v ~-* f x hdv measurable with h varying over the bounded measurable functions on X. Given a probability measure # on f~ x X with marginal P on f~, we will call a measurable map #.: ~2 ---, P r ( X ) , w ,--, #~ a disintegration of p (with respect to P ) if

.(A) = s 1 6 3

x)

dP( )

for all A C r x B. Disintegrations exist and are unique ( P - a . e . ) in almost all interesting cases, e. g., if X is a Polish space. In such a situation the r of # is equivalent to the validity of the equation r

w)#~o = p0~0,

n C Z +, P - a.e.w.

(1.1)

Henceforth in this appendix we will always assume existenc~ and uniqueness of a disintegration. Let # be a r measure. For every finite measurable partition ~ of X, one can show that the limit rt--1

h,(r

ae=f lira 1 - H ~ ( ~ / rl --+ O~ n

r

k=0

exists and is constant for P-a.e. co (see [Bog] Theorem 2.2). The number h,(r

d=~fsup{h,(r

: ~ is a finite partition of X}

is called the (measure-theoretic) entropy of r with respect to #. It is shown that the entropy h , ( r coincides with the conditional entropy of (9 with respect to ~r-19r , i.e. h~(r = h ; - ~ ( O ) (1).) where ~r: f t x X --+ fl is the natural projection on the first factor (see [Kit]l Theorem II.1.4 or [Bog] Theorem 3.1). L e I n I n a 1.1.

Let r be an R D S on ( X , J c) over (fI, J:,P,O), invariant measure.

1) De~ne Ck by Ck(n,~) = r

and # a r

E Z + , ~ e a for k ~ N. Then for all

k E N one has

h . ( r k) = kh.(r 2) A s s u m e that X is a compact metric space and (Ft,~',P, 0) is a Polish system. I f {~k}k+__~ is a sequence of finite partitions of X with limk--.+oo diam ~k = O, then

h.(.,., :: I,~,, h.(r k--++oo

208

Proof: The proof of 1) is analogous to that of Theorem 0.4.3. The proof of 2) is analogous to that of Theorem 1.2.5 []

Now let r be measure. If

a C 1

RDS on a C ~ Riemannian manifold X and # a r a

dp < +o~

log + IT~r

(1.3)

xX

holds true, by Oseledec multiplicative ergodic theorem we know that for #-a.e. (a~, x) there exist measurable (in (w, x)) numbers

~(1)(~,.) < ~(~)(~, ~) < ... < a(~(~,.))(~,.) (A(1)(w,x) may be - o o ) and an associated measurable ( i n (w, x)) filtration by linear subspaces of T~M {0} ~-~ V,(~,~) (0) c V~(~,~) (1) c . . - c

V.(~,.) (r(w'x)) = T . M

such that lim 1 log n ---+ O 0

IT~r162

= A(~)(~o, x)

n

if ~ 6 !~(;!,,) \ V(~:~')), 1 < i < r(w, x). The numbers A(i)(w,x), 1 < i < r(w, x) are called the Lyapunov exponents of r at point (w, x) and mi(w, x) = dim V((~!~) -

dim ~ ; 1 ) ) i s called the multiplicity of )~(i)(w,x).

w

T h e M a i n R e s u l t and It's P r o o f

The main result of this appendix is the following random version of the Margulis-Ruelle inequality. T h e o r e m 2.1 (Margulis-Ruelle inequality for RDS). Let X be a d-dimensional compact C ~ Riemannian manifold without boundary. Let r be a C 1 RDS on X over (~, jc p, O) and # a r measure. Assume that

alog+1r where [r

dP(w) < +(x~

= supxex ]T~r

Then we have

h,(r <s

~(')(~,x)+r~(~,~)d,. •

i

209

(2.1)

(2.2)

Let Pg+(M,v,#) be as defined in Section 1.1 with Diff2(M) being replaced by Diffl(M). Assume that v satisfies

C o r o l l a r y 2.1.

lift(M)

l~ + Iflc~ dr(f) < + ~ .

Then

h.(X+(M,v)) < fM E/~(i)(x)+mi(x)d#" i

Proof:

Put W = Diffl(M). Define an RDS r on M over (W z , / 3 ( w ) z , v z , r )

by r

Z+ x W Z x M

( n , w , x ) ~-* f~x,

, M,

where r and f~ is as introduced in Section VI.1.A. By Proposition 1.1.2, #* is a r measure. Applying Theorem 2.1 to r with invariant measure #*, we prove the corollary by (VI.1.2) and (VI.1.5). [] R e m a r k 2.1. Clearly, Corollary 2.1 confirms in part what we have said in the Introduction and Remark II.2.1 about Ruelle's inequality. R e m a r k 2.2. Noting that the random maps r n C Z+,w C ~ in Theorem 2.1 are assumed to be only C 1 maps, one can show that the conclusion of Corollary 2.1 also holds true if Diffl(M) is replaced by CI(M, M). Proof of this fact is analogous to that of Corollary 2.1. Now we proceed to prove Theorem 2.1. We first reduce the problem to the case when (~,hv, P, 0) is a Polish system. Let r and # be as given in the formulation of Theorem 2.1. We now introduce a new, Polish, invertible and ergodic measure-preserving system (fl, .T', P, 0) by defining

=

C I ( X , X ) z endowed with the product topology,

5~

=

the Borel ~-algebra o f ~ ,

/5

=

~ p where # : (~,Y') ~ ( ~ , ~ ) is the measurable map defined by #w = ( - . . , r

t~ =

O-lw), r

w), r

Ow),...),

the left shift operator on ~.

Then we may define a C 1 RDS r on X over the Polish system (~, ~',/5, tg) by

•215 where f~={

id f~-l(~)) o . . . o f0(~b)

210

ifn=0 if n > 0

for r = (... ,f_l(do),fo(da),fl(&),...) E (2. It is clear that EoO=@oE where @ is the skew product transformation induced by r and E : ~2x X --~ (2 x X is a map defined by E : (w, x) ~-~ (#w, z). Put /~ = E~. It is easy to see that/~ is a r

measure and that, by (1.2)

=

>

= h.(r

Since 7r: (~, J-, P) --+ ((2, ~ , / 5 ) is measure-preserving, from (2.1) it follows that

alog+ Ir

dP(~) < +c~.

This ensures the existence of the Lyapunov exponents ~(1)(~b, x) < --. < A(r(o,~)) (&, x) of r at/2-a.e. (&, x). Clearly,

/ z ><,,(.,.,+..,(.,.,..: i z ><,,(.,.,+..,(.,.,... i

i

Hence, in order to prove Theorem 2.1 it is sufficient to consider the case when (f2, ~', P, 0) is Polish. Next, we will prove the Margulis-Ruelle inequality for a C 1 RDS r on a compact subset of R d and then deduce our main result by extending r to a tubular neighbourhood of X. Considerations of this type are standard in differential geometry. In this appendix l[" II will denote the usual Euclidean norm and B ( X , E ) the closed c-neighbourhood of X in R d if X C R d. T h e o r e m 2.2. Let X be a compact subset of R d, U an open neighbourhood of X , and r a C 1 R D S on U over a Polish system (f~, J:, P, O) with r C X for allw C ~. Let # be a O-invariant measure with support in ~ x X and assume that there is an Co > 0 with B ( X , ~ o ) C U such that r C B(X,~o) for all w E f2 and ri log+

liT~r

sup

) < +oo.

(2.3)

xEB(X,~o)

Then it holds that hu(r

<- i n

E'~(i)(w'x)+mi(w'x)d"" •

i

211

(2.4)

Proof: Fix n E N arbitrarily. Let k E N. y 9 B ( X , E0) with IIx - yll - co/k we have lie(n, ~ ) y - r

-

We write w E f2k if for all x,

T~:C(,~,,~)(y

-

and

I6,(T:r

1

- 6~(Tur

<_ -~,

~)[I --- IIY - ~11

(2.5)

1< i < d

which clearly implies

1 < max{1,6i(T:r

< 2,

2 - max{l, 6,(Tyr

1 < i < d,

(2.6)

w))}

where 6i(A), 1 < i < d are as introduced at the beginning of Section II.2. One can check that each ftk is measurable and, since every r w) is C 1 and B ( X , co) is compact, for every w there exists k 9 N such that w 9 f2k. Consequently, limk~+o~ P(f2~) = 1 since f ~ C f2k+l for all k 9 N. For each k 9 N, take a maximal E0/k-seperated set Ek of X. We then define a measurable partition ~k = {~k(x) : x 9 Ek) of X such that, with respect to the topology of X as a subspace of R d, ~k(x) C int (~k(x)) and int (~k(x)) = {y 9 X : I l y - x l l < I l y - x i l l i f x :/: xi c Ek} for every x 9 E~. Clearly ~:(x) C B(x,r for all x 9 E~ and d i a m ~ < 2eo/k. Then by L e m m a 1.1

~h,(r

= h,(r

=

lira h ~ ( r

(2.7)

Note that for each finite partition ~ of X one has

hu(r =

)

r

1lim /H"~m--,~ \k=O

<

=

dP(a~) dP(w)

lim--1/[~lH.~(r m--*oo m Lk=l

lim--lf[~lH,0k_..(r

r n ---~c ~ m

-

f H.~(r

)

kk=l

~- 1

H..(r

H.~(()d(~o

~o)-~ I~) d P(~o).

Thus from (2.7) it follows that nhu(r

<

lim

--

k - - - * OO

< --

lim [

Hu~(8(n,w)

-1

~kl~k) d P(w)

H..(e(~,~)-l~klCk)dP(~)

k ---* o o J ~ ' ~ k

212

+ ~

[

k--,oo jf~\12k

deal

H~,~(r

lim Mk + lira ink. k--*oo

(2.8)

k---* co

In what follows we estimate Mk and ink. Let w E f~k. We first estimate the number, written N,~,k(w, x), of elements of~k which intersect r for z E Ek. By (2.5),

,(n,~)~(~)

c ,(n,~)B(~,So/k) C B($(n,w)x+T~$(n,o~)B(O, eo/k), so/k) = r w)x + B (T~r w)B(O, co~k), so/k).

Hence,

&(~') n r

# r

implies

B(x', e0/2k) A [r

w )x + B(T~:r

w)B( O,eo/k ), 2e0/k)] r 0.

Since B(x',so/2k), x' r E~ are disjoint, we know that the number N,~,k(w,x) can not exceed d K

, ntw, x)x d=e f Cl(d) I I max{6i(T~r

1}

i----1

where Ca(d) is a constant depending only on d. Therefore, n~,~ (r

<

E

15k)

#~o(~k(x))logK,~(w,x)

xE Ek

= xEEk ~ f

k(z)

log K~(~, x) ~.~(v).

By (2.6) we havefor y E~k(x) log + 6i(T~:r

< log 2 + log + 6i(Tv~(n,w))

and hence d

log K~(w, x) _< log C1 (d) + d log 2 + Z l~

6i (Try(n, w)).

/=1

Consequently, we obtain Uu~ (r

w)-l~k I~k) d

<

lOgCl(d)+dlog2+[

Jx

Z I~ i----i

213

6i(Tvr

dpw(y)

and hence d

Mk _< log C l ( d ) + dlog2 + / n i x

El~

6i(Tyr

i=1

Now let w E ~ \ ~k. Again we estimate the number Nn,k(w, x) of elements of ~k which intersect r for x E EL. We can not make use of (2.5) in the present case, but we can apply the mean value theorem. For this purpose we define for k > 2 Ln,k (w) =

sup

IITzr

zEB(X,2eo/k)

Let now k > 2. By the mean value theorem we know that for all y,z E B(X,~otk) with IIY- zll -< ~olk

IIr

- r

< Ln,~(w)llY - zll.

From this it follows that

r

c

r

c0/k)

C U(r

L,~,k(w)eo/k).

Thus, N,,k (w, x) can not exceed the maximal number of disjoint balls with radius eo/2k which intersect B(r w)x, L,~,k(w)Eo/k + Go~k). Hence N~,~(w, x) <_C2(d) max{L,,k(w), 1} d, where C2(d) is a constant depending only on d. From this it follows that -1

g~.(r

~kl~)

<

)--:~,~(~k(x))logg.,~(~,~) xf: E~

< logC2(d) + dlog + Lmk(w ) < logC2(d) + dlog + Ln,2(w). Then we obtain f

mk

_< log C2(d) + d ] log + Ln,~(w)dP(w). .in \nk

(2.9)

From (2.3)it follows that log + L,~,2(w) E Ll(fl, P). Then by (2.9) we have lim mk < log C~(d) k--* oo

since P(Q \ fl~) (2.8) yields

,Oask

, +cr

This together with the estimate of Mk and d

~h.(r i=1

<_ c + fa

xX

l~176 214

(2.10)

where C = l o g C l ( d ) + d l o g 2 + logC2(d). Dividing (2.10) by n and letting n ~ + o % we obtain (2.4). 12 P r o o f o f T h e o r e m 2.1. W i t h o u t loss of generality, we assume t h a t (f2,r P, 0) is a Polish system. Let h be a Coo embedding of X into R 2d+1. We will identify X and T~X, x C X respectively with their images under h and Th. Let , ( X ) = {(x, v) 9 X x R 2d+1 : v_l_T:vM} be the normal bundle to X. T h e exponential map exp: T R 2a+1 = R 2d+1 x R 2d+l > R 2d+l is defined by exp:v v = x + v. Let G be the restriction of exp to v(X). T h e n there is an E > 0 such t h a t {(x,v) E u(X) : Ilvll < r is C ~ diffeomorphically m a p p e d by G onto {y 9 R 2d+l : d(y,X) < c} ~f N~(X) (N~(X) is called a tubular n e i g h b o u r h o o d of X , see [Elw]l). We extend r to N~(X) in the following way: Define r = j o r o 7r, where rr: N~(X) ~ X is the projection onto X and j : X ---* Nr is the inclusion into N~(X). In particular, we have r = r for all x 9 X and for all aJ E [2. Choose a number 0 < ~0 < r and p u t X = B(X,60). The chain rule and the compactness of X and )( imply t h a t there is a constant C, which depends only on X , such t h a t

sup IIT (1, )II

~E)(

c

sup

:vEX

IIT r

for all w E f L Since X is compact, all Riemannian metrics on X are equivalent and therefore the L y a p u n o v exponents of r (with invariant measure #) are independent of the choice of the Riemannian metric. Note also t h a t no new L y a p u n o v exponents other t h a n -cx~ are created for r (with invariant measure #) since for z E X and v E T:VN~(X) with v_l_T:vM

0 T h u s T h e o r e m 2.1 follows from T h e o r e m 2.2.

215

[]

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216

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218

Subject Index absolutely continuous 2, 3, 85, 196 .A-conditional entropy 10 A-conditional entropy with respect to adapted Riemannian metric 183 angle 40 aperture 73 Brownian motion

10

116

C2-norm 23, 47 canonical system of conditional measures 6 center unstable set 146 Chapman-Kolmogorov equation 110 conditional entropy 7 conditional expectation 3 conditional expectation operator 3 continuous family of C a embedded k-dimensional discs convolution semigroup 110 density 92, 190 density point 4 disintegration 208 elliptic 118 entropy 7, 31, 133, 187, 208 ergodic 21, 28, 132 ergodic decomposition 21, 29 factor-space 6 family ofsamplemeasures Feller semigroup 113 fixed 19

131, 187

(global) stable manifold 73, 121 (global) unstable manifold 136, 184, 189 HSlder continuous 73, 75 hyperbolic 183 hyperbolic attractor 184 increasing 158 invariant measure

24, 111, 130, 207

219

64

Jacobian

3

Lebesgue space 5 local stable manifold 64, 122 local unstable manifold 146, 195 Lyapunov exponents 37, 119, 134, 187, 209 Lyapunov metric 59 mean conditional entropy 7 measurable partition 5 measurable partition subordinate to We-manifolds measurable partition subordinate to W~-manifolds multiplicity 37, 119, 134, 187, 209

93 136, 184, 189

n-point motion 111 non-degenerate SDE 118 p-th exterior power space 40 Pesin's entropy formula 91, 127, 137, 184, 190 Polish space 2 Polish system 207 probability measure having absolutely continuous conditional measures on WS-manifolds 93 probability measure having absolutely continuous conditional measures on W~-manifolds 136, 184, 189 Radon-Nikodym derivative 2 random dynamical system 207 relation number 49 Ruelle's inequality 45, 127, 201 SBR measure 185 SBR property 136 semigroup of linear operators 112 separable 1 skew product transformation 207 stochastic flow of C ~ diffeomorphisms 110 stopping (Markov) time 113 Stratonovich stochastic differentiM equation strong Markov process 113 t0-time-step entropy 124, 125 transition probabilities 92, 190 transitivity 7 transversal 84 transverse metric 163 220

116

z~+ (M, v)-invariant 27 X(U, v)-invariant 190

221