Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms [2 ed.] 9783540776055, 3540776052, 3540776958

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

470

Rufus Bowen

Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Second revised edition Jean-René Chazottes Editor

Preface by David Ruelle

ABC

Author Robert Edward (Rufus) Bowen (1947-1978) University of California at Berkeley USA

Preface by David Ruelle

Editor Jean-René Chazottes Centre de Physique Théorique CNRS-École Polytechnique 91128 Palaiseau Cedex France [email protected]

IHÉS 35, Route de Chartres 91440 Bures-sur-Yvette France [email protected]

ISBN: 978-3-540-77605-5 e-ISBN: 978-3-540-77695-6 DOI: 10.1007/978-3-540-77695-6 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008922889 Mathematics Subject Classification (2000): Primary: 37D20, Secondary: 37D35 c 2008, 1975 Springer-Verlag Berlin Heidelberg ° 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH

Printed on acid-free paper 987654321 springer.com

Preface

The Greek and Roman gods, supposedly, resented those mortals endowed with superlative gifts and happiness, and punished them. The life and achievements of Rufus Bowen (1947–1978) remind us of this belief of the ancients. When Rufus died unexpectedly, at age thirty-one, from brain hemorrhage, he was a very happy and successful man. He had great charm, that he did not misuse, and superlative mathematical talent. His mathematical legacy is important, and will not be forgotten, but one wonders what he would have achieved if he had lived longer. Bowen chose to be simple rather than brilliant. This was the hard choice, especially in a messy subject like smooth dynamics in which he worked. Simplicity had also been the style of Steve Smale, from whom Bowen learned dynamical systems theory. Rufus Bowen has left us a masterpiece of mathematical exposition: the slim volume Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer Lecture Notes in Mathematics 470 (1975)). Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen’s monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems. The area discussed by Bowen came into existence through the merging of two apparently unrelated theories. One theory was equilibrium statistical mechanics, and specifically the theory of states of infinite systems (Gibbs states, equilibrium states, and their relations as discussed by R.L. Dobrushin, O.E. Lanford, and D. Ruelle). The other theory was that of hyperbolic smooth dynamical systems, with the major contributions of D.V. Anosov and S. Smale. The two theories came into contact when Ya.G. Sinai introduced Markov partitions and symbolic dynamics for Anosov diffeomorphisms. This allowed the poweful techniques and results of statistical mechanics to be applied to smooth dynamics, an extraordinary development in which Rufus Bowen played a major role. Some of Bowen’s ideas were as follows. First, only one-dimensional statistical mechanics is discussed: this is a richer theory, which yields what is

VI

Preface

needed for applications to dynamical systems, and makes use of the powerful analytic tool of transfer operators. Second, Smale’s Axiom A dynamical systems are studied rather than the less general Anosov systems. Third, Sinai’s Markov partitions are reworked to apply to Axiom A systems and their construction is simplified by the use of shadowing. The combination of simplifications and generalizations just outlined led to Bowen’s concise and lucid monograph. This text has not aged since it was written and its beauty is as striking as when it was first published in 1975. Jean-Ren´e Chazottes has had the idea to make Bowen’s monograph more easily available by retyping it. He has scrupulously respected the original text and notation, but corrected a number of typos and made a few other minor corrections, in particular in the bibliography, to improve usefulness and readability. In his enterprise he has been helped by Jerˆ ome Buzzi, Pierre Collet, and Gerhard Keller. For this work of love all of them deserve our warmest thanks.

Bures sur Yvette, mai 2007

David Ruelle

Contents

0

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

1

1

Gibbs Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Gibbs Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Ruelle’s Perron-Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . C. Construction of Gibbs Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 8 13 16 22 26

2

General Thermodynamic Formalism . . . . . . . . . . . . . . . . . . . . . . . A. Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 32 36 42 43

3

Axiom a Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Markov Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 47 51 56 58

4

Ergodic Theory of Axiom a Diffeomorphisms . . . . . . . . . . . . . . A. Equilibrium States for Basic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Case φ = φ(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Attractors and Anosov Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 64 69 72

VIII

Contents

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

These notes came out of a course given at the University of Minnesota and were revised while the author was on a Sloan Fellowship.

0 Introduction

The main purpose of these notes is to present the ergodic theory of Anosov and Axiom A diffeomorphisms. These diffeomorphisms have a complicated orbit structure that is perhaps best understood by relating them topologically and measure theoretically to shift spaces. This idea of studying the same example from different viewpoints is of course how the subjects of topological dynamics and ergodic theory arose from mechanics. Here these subjects return to help us understand differentiable systems. These notes are divided into four chapters. First we study the statistical properties of Gibbs measures. These measures on shift spaces arise in modern statistical mechanics; they interest us because they solve the problem of determining an invariant measure when you know it approximately in a certain sense. The Gibbs measures also satisfy a variational principle. This principle is important because it makes no reference to the shift character of the underlying space. Through this one is led to develop a “thermodynamic formalism” on compact spaces; this is carried out in chapter two. In the third chapter Axiom A diffeomorphisms are introduced and their symbolic dynamics constructed: this states how they are related to shift spaces. In the final chapter this symbolic dynamics is applied to the ergodic theory of Axiom A diffeomorphisms. The material of these notes is taken from the work of many people. I have attempted to give the main references at the end of each chapter, but no doubt some are missing. On the whole these notes owe most to D. Ruelle and Ya. Sinai. To start, recall that (X, B, µ) is a probability space if B is a σ-field of subsets of X and µ is a nonnegative measure on B with µ(X) = 1. By an automorphism we mean a bijection T : X → X for which (i) (ii)

E∈B

iff

T −1 E ∈ B,

µ(T −1 E) = µ(E) for E ∈ B .

2

0 Introduction

If T : X → X is a homeomorphism of a compact metric space, a natural σ-field B is the family of Borel sets. A probability measure on this σ-field is called a Borel probability measure. Let M (X) be the set of Borel probability measures on X and MT (X) the subset of invariant ones, i.e. µ ∈ MT (X) if µ(T −1 E) = µ(E) for all Borel sets E. For any µ ∈ M (X) one can define T ∗ µ ∈ M (X) by T ∗ µ(E) = µ(T −1 E). Remember that the real-valued continuous functions C (X) on the compact metric space X form a Banach space under f  = maxx∈X |f (x)|. The weak ∗-topology on the space C (X)∗ of continuous linear functionals α : C (X) → R is generated by sets of the form U (f, ε, α0 ) = {α ∈ C (X)∗ : |α(f ) − α0 (f )| < ε} with f ∈ C (X), ε > 0, α0 ∈ C (X)∗ . ∗
Riesz Representation. For each µ ∈ M (X) define αµ ∈ C (X) by αµ (f ) = f dµ. Then µ ↔ αµ is a bijection between M (X) and

{α ∈ C (X)∗ : α(1) = 1 and α(f ) ≥ 0 whenever f ≥ 0} .

We identify αµ with µ, often writing µ when we mean α(µ). The weak ∗topology on C (X)∗ carries over by this identification to a topology on M (X) (called the weak topology). Proposition. M (X) is a compact convex metrizable space. Proof. Let {fn }∞ n=1 be a dense subset of C (X). The reader may check that the weak topology on M (X) is equivalent to the one defined by the metric d(µ, µ′ ) =

∞ 

n=1

     2−n fn −1  fn dµ − fn dµ′  .

⊓ ⊔

Proposition. MT (X) is a nonempty closed subset of M (X). Proof. Check that T ∗ : M (X) → M (X) is a homeomorphism and note that MT (X) = {µ ∈ M (X) : T ∗ µ = µ}. Pick µ ∈ M (X) and let µn = n1 (µ + T ∗ µ + · · · + (T ∗ )n−1 µ). Choose a subsequence µnk converging to µ′ ∈ M (X). Then µ′ ∈ MT (X). ⊓ ⊔ Proposition. µ ∈ MT (X) if and only if   (f ◦ T ) dµ = f dµ for all f ∈ C (X) . Proof. This is just what the Riesz Representation Theorem says about the statement T ∗ µ = µ. ⊓ ⊔

1 Gibbs Measures

A. Gibbs Distribution Suppose a physical system has possible states 1, . . . , n and the energies of these states are E1 , . . . , En . Suppose that this system is put in contact with a much larger “heat source” which is at temperature T . Energy is thereby allowed to pass between the original system and the heat source, and the temperature T of the heat source remains constant as it is so much larger than our system. As the energy of our system is not fixed any of the states could occur. It is a physical fact derived in statistical mechanics that the probability pj that state j occurs is given by the Gibbs distribution e−βEj pj = n −βE , i i=1 e

1 and k is a physical constant. where β = kT We shall not attempt the physical justification for the Gibbs distribution, but we will state a mathematical fact closely connected to the physical reasoning.

1.1. Lemma. Let real numbers a1 , . . . , an be given. Then the quantity F (p1 , . . . , pn ) =

n  i=1

−pi log pi +

n 

pi ai

i=1

n has maximum value log i=1 eai as (p1 , . . . , pn ) ranges over the simplex {(p1 , . . . , pn ) : pi ≥ 0, p1 + · · · + pn = 1} and that maximum is assumed only by  −1  pj = eaj eai . i

4

1 Gibbs Measures

n This is proved by calculus. The quantity H(p1 , . . . , pn ) = i=1 −pi log pi is called the entropy of the distribution (p1 , . . . , pn ) (note: n ϕ(x) = −x log x is continuous on [0, 1] if we set ϕ(0) = 0.) The term i=1 pi ai is of course the average value of the function a(i) = ai . In the statistical mechanics case ai = −βEi , entropy is denoted S and average energy E. The Gibbs distribution then maximizes S − βE = S −

1 E, kT

or equivalently minimizes E − kT S. This is called the free energy. The principle that “nature minimizes entropy” applies when energy is fixed, but when energy is not fixed “nature minimizes free energy.” We will now look at a simple infinite system, the one-dimensional lattice. Here one has for each integer a physical system with possible states 1, 2, . . . , n. A configuration of the system consists of assigning an xi ∈ {1, . . . , n} for each i:

·

·

x−2

·

x−1

·

Thus a configuration is a point

x0

x1

·

x = {xi }+∞ i=−∞ ∈

·

x2

·

x3

·

·

·

 {1, . . . , n} = Σn . Z

We now make assumptions about energy: (1) associated with the occurrence of a state k is a contribution Φ0 (k) to the total energy of the system independent of which position it occurs at; (2) if state k1 occurs in place i1 , and k2 in i2 , then the potential energy due to their interaction Φ∗2 (i1 , i2 , k1 , k2 ) depends only on their relative position, i.e., there is a function Φ2 : Z × {1, . . . , n} × {1, . . . , n} → R so that Φ∗2 (i1 , i2 , k1 , k2 ) = Φ2 (i1 − i2 ; k1 , k2 ) (also: Φ2 (j; k1 , k2 ) = Φ2 (−j; k2 , k1 )). (3) all energy is due to contributions of the form (1) and (2). Under these hypotheses the energy contribution due to x0 being in the 0th place is 1 φ∗ (x) = Φ0 (x0 ) + Φ2 (j; xj , x0 ). 2 j =0

(We “give” each of x0 and xk half the energy due to their interaction). We now assume that Φ2 j = supk1 ,k2 |Φ(j; k1 , k2 )| satisfies ∞  j=1

Φ2 j < ∞ .

A. Gibbs Distribution

5

Then φ∗ (x) ∈ R and depends continuously on x when {1, . . . , n} is given the discrete topology and Σn = Z {1, . . . , n} the product topology. If we just look at x−m . . . x0 . . . xm we have a finite system (n2m+1 possible configurations) and an energy Em (x−m , . . . , xm ) =

m 

Φ0 (xj )

j=−m

+



−m≤j 0, α ∈ (0, 1) so that vark φ ≤ cαk for all k. Then there is a unique µ ∈ Mσ (Σn ) for which one can find constants c1 > 0, c2 > 0, and P such that

6

1 Gibbs Measures

c1 ≤

µ{y : yi = xi ∀ i = 0, . . . , m}   ≤ c2 m−1 exp −P m + k=0 φ(σ k x)

for every x ∈ Σn and m ≥ 0.

This measure µ is written µφ and called Gibbs measure of φ. Up to constants in [c1 , c2 ] the relative probabilities of the x0 . . . xm ’s are given m−1 by exp k=0 φ(σ k x). For the physical system discussed above one takes φ = −βφ∗ . In statistical mechanics Gibbs states are not defined by the above theorem. We have ignored many subtleties that come up in more complicated systems (e.g., higher dimensional lattices), where the theorem will not hold. Our discussion was a gross one intended to motivate the theorem; we refer to Ruelle [9] or Lanford [6] for a refined outlook. For later use we want to make a small generalization of Σn before we prove the theorem. If A is an n × n matrix of 0’s and 1’s, let ΣA = {x ∈ Σn : Axi xi+1 = 1 ∀i ∈ Z} . That is, we consider all x in which A says that xi xi+1 is allowable for every i. One easily sees that ΣA is closed and σΣA = ΣA . We will always assume that A is such that each k between 1 and n occurs at x0 for some x ∈ ΣA . (Otherwise one could have ΣA = ΣB with B an m × m matrix and m < n.)

1.3. Lemma. σ : ΣA → ΣA is topologically mixing (i.e., when U, V are nonempty open subsets of ΣA , there is an N so that σ m U ∩ V = ∅ ∀m ≥ N ) if and only if AM > 0 (i.e., AM i,j > 0 ∀i, j) for some M . Proof. One sees inductively that Am i,j is the number of (m + 1)-strings a0 a1 . . . am of integers between 1 and n with

(a) Aak ak+1 = 1 ∀k, (b) a0 = i, am = j. Let Ui = {x ∈ ΣA : x0 = i} =  ∅. Suppose ΣA is mixing. Then ∃Ni,j with Ui ∩ σ n Uj = ∅ ∀n ≥ Ni,j . If a ∈ Ui ∩ σ n Uj , then a0 a1 . . . an satisfies (a) and (b); so Am i,j > 0 ∀i, j when m ≥ maxi,j Ni,j . Suppose AM > 0 for some M . As each number between 1 and n occurs as x0 for some x ∈ ΣA , each row of A has at least one positive entry. From this it follows by induction that Am > 0 for all m ≥ M . Consider open subsets U , V of ΣA with a ∈ U , b ∈ V . There is an r so that U ⊃ {x ∈ ΣA : xk = ak ∀|k| ≤ r} V ⊃ {x ∈ ΣA : xk = bk ∀|k| ≤ r} .

For t ≥ 2r + M , m = t − 2r ≥ M and Am > 0. Hence find c0 , . . . , cm with c0 = br , cm = a−r , Ack ck+1 = 1 for all k. Then x = · · · b−2 b−1 b0 · · · br c1 · · · cm−1 a−r · · · a0 a1 · · ·

is in ΣA and x ∈ σ t U ∩ V . So ΣA is topologically mixing. ⊓ ⊔

A. Gibbs Distribution

7

Let FA be the family of all continuous φ : ΣA → R for which vark φ ≤ bαk (for all k ≥ 0) for some positive constants b and α ∈ (0, 1). For any β ∈ (0, 1) one can define the metric dβ on ΣA by dβ (x, y) = β N where N is the largest nonnegative integer with xi = yi for every |i| < N . Then FA is just the set of functions which have a positive H¨ older exponent with respect to dβ . The theorem we are interested in then reads 1.4. Existence of Gibbs measures. Suppose ΣA is topologically mixing and φ ∈ FA . There is unique σ-invariant Borel probability measure µ on ΣA for which one can find constants c1 > 0, c2 > 0 and P such that c1 ≤

µ{y : yi = xi for all i ∈ [0, m)}   ≤ c2 m−1 exp −P m + k=0 φ(σ k x)

for every x ∈ ΣA and m ≥ 0.

This theorem will not be proved for some time. The first step is to reduce the φ’s one must consider. Definition. Two functions ψ, φ ∈ C (ΣA ) are homologous with respect to σ (written ψ ∼ φ) if there is a u ∈ C (ΣA ) so that ψ(x) = φ(x) − u(x) + u(σx) . 1.5. Lemma. Suppose φ1 ∼ φ2 and Theorem 1.4 holds for φ1 . Then it holds for φ2 and µφ1 = µφ2 . Proof.     m−1 m−1  m−1     k k  k  k+1 φ1 (σ x) − φ2 (σ x) =  u(σ x) − u(σ x)      k=0

k=0

k=0

= |u(σ m x) − u(x)| ≤ 2u .

The exponential in the required inequality changes by at most a factor of e2u when φ1 is replaced by φ2 . Thus the inequality remains valid with c1 , c2 changed and P , µ unchanged. ⊓ ⊔ 1.6. Lemma. If φ ∈ FA , then φ is homologous to some ψ ∈ FA with ψ(x) = ψ(y) whenever xi = yi for all i ≥ 0. Proof. For each 1 ≤ t ≤ n pick {ak,t }∞ k=−∞ ∈ ΣA with a0,t = t. Define r : ΣA → ΣA by r(x) = x∗ where  xk for k ≥ 0 ∗ xk = ak,x0 for k ≤ 0 . Let

8

1 Gibbs Measures

u(x) =

∞  j=0

j

(φ(σ j x) − φ(σ j r(x))) .

j

Since σ x and σ r(x) agree in places from −j to +∞, |φ(σ j x) − φ(σ j r(x))| ≤ varj φ ≤ bαj . ∞ As j=0 bαj < ∞, u is defined and continuous. If xi = yi for all |i| ≤ n, then, for j ∈ [0, n], |φ(σ j x) − φ(σ j y)| ≤ varn−j φ ≤ bαn−j and |φ(σ j r(x)) − φ(σ j r(y))| ≤ bαn−j . Hence |u(x) − u(y)| ≤

[ n2 ]  j=0

|φ(σ j x) − φ(σ j y) + φ(σ j r(x)) − φ(σ j r(y))| + 2



αj

j>[ n 2]

⎞

⎛

[ n2 ] n  4b α[ 2 ] ⎜ n−j j⎟ α ⎠≤ · ≤ 2b ⎝ α + 1−α j=0 j>[ n ] 2

This shows that u ∈ FA . Hence ψ = φ − u + u ◦ σ is in FA also. Furthermore ψ(x) = φ(x) +

∞  

∞     φ(σ j+1 r(x)) − φ(σ j+1 x) + φ(σ j+1 x) − φ(σ j r(σx))

j=−1 ∞ 

= φ(r(x)) +

j=0

j=0



 φ(σ j+1 r(x)) − φ(σ j r(σx)) .

The final expression depends only on {xi }∞ i=0 , as we wanted. D. Lind cleaned up the above proof for us. ⊓ ⊔ Lemmas 1.5 and 1.6 tell us that in looking for a Gibbs measure µφ for φ ∈ FA (i.e., proving Theorem 1.4) we can restrict our attention to functions φ for which φ(x) depends only on {xi }∞ i=0 .

B. Ruelle’s Perron-Frobenius Theorem We introduce now one-sided shift spaces. One writes x for {xi }∞ i=0 (we will but never for both things at the same time). continue to write x for {xi }∞ i=−∞ Let   ∞  + ΣA = x ∈ {1, . . . , n} : Axi ,xi+1 = 1 for all i ≥ 0 i=0

B. Ruelle’s Perron-Frobenius Theorem

9

+ + by σ(x)i = xi+1 . σ is a finite-to-one continuous → ΣA and define σ : ΣA + + map of ΣA onto itself. If φ ∈ C (ΣA ) we get φ ∈ C (ΣA ) by φ({xi }∞ i=−∞ ) = φ({xi }∞ = φ(y) whenever xi = yi ). Suppose φ ∈ C (Σ ) satisfies φ(x) A i=0 + for all i ≥ 0. Then one can think of φ as belonging to C (ΣA ) as follows: ∞ φ({xi }∞ i=0 ) = φ({xi }i=−∞ ) where xi for i ≤ 0 are chosen in any way subject + ∞ ) are thus identified with a certain to {xi }i=−∞ ∈ ΣA . The functions C (ΣA subclass of C (ΣA ). We saw in Lemmas 1.5 and 1.6 that one only needs to get + Gibbs measures for φ ∈ C (ΣA ) ∩ FA in order to get them for all φ ∈ FA . In this section we will prove a theorem of Ruelle that will later be used + ) define the operator to construct and study Gibbs measures. For φ ∈ C (ΣA + L = Lφ on C (ΣA ) by  eφ(y) f (y) . (Lφ f )(x) = y ∈σ −1 x

+ It is the fact that σ is not one-to-one on ΣA that will make this operator useful.

1.7. Ruelle’s Perron-Frobenius Theorem [10, 11]. Let ΣA be topolog+ ically mixing, φ ∈ FA ∩ C (ΣA ) and L = Lφ as above. There are λ > 0, + + ) for which Lh = λh, L∗ ν = λν, h ∈ C (ΣA ) with h > 0 and ν ∈ M (ΣA ν(h) = 1 and + ). lim λ−m Lm g − ν(g)h = 0 for all g ∈ C (ΣA

m→∞

Proof. Because L is a positive operator and L1 > 0, one has that G(µ) = + + + ) for µ ∈ M (ΣA ). There is a ν ∈ M (ΣA ) with (L∗ µ(1))−1 L∗ µ ∈ M (ΣA G(ν) = ν by the Schauder-Tychonoff Theorem (see Dunford and Schwartz, Linear Operators I, p. 456): Let E be a nonempty compact convex subset of a locally convex topological vector space. Then any continuous G : E → E has a fixed point. In our case G(ν) = ν gives L∗ ν = λν with λ > 0. We will prove 1.7 via a sequence of lemmas. Let b > 0 and α ∈ (0, 1) be any ∞ k constants so that vark φ ≤ bαk for all k ≥ 0. Set Bm = exp k=m+1 2bα and define + ) : f ≥ 0, ν(f ) = 1 , f (x) ≤ Bm f (x′ ), Λ = {f ∈ C (ΣA whenever xi = x′i for all i ∈ [0, m]} .

1.8. Lemma. There is an h ∈ Λ with Lh = λh and h > 0. Proof. One checks that λ−1 Lf ∈ Λ when f ∈ Λ. Clearly λ−1 Lf ≥ 0 and ν(λ−1 Lf ) = λ−1 L∗ ν(f ) = ν(f ) = 1 . Assume xi = x′i for i ∈ [0, m]. Then Lf (x) =

 j

eφ(jx) f (jx)

10

1 Gibbs Measures

where the sum ranges over all j with Ajx0 = 1. For x′ the expression runs over the same j; as jx and jx′ agree in places 0 to m + 1 m+1

′

eφ(jx) f (jx) ≤ eφ(jx ) ebα and so

′

Bm+1 f (jx′ ) ≤ Bm eφ(jx ) f (jx′ )

Lf (x) ≤ Bm Lf (x′ ) .

+ Consider any x, z ∈ ΣA . Since AM > 0 there is a y ′ ∈ σ −M x with y0′ = z0 . For f ∈ Λ M −1    M k exp φ(σ y)f (y) L f (x) = k=0

y∈σ −M x

−M φ

≥e

′

f (y ) .

Let K = λM eM φ B0 . Then 1 = ν(λ−M LM f ) ≥ K −1 f (z) gives f  ≤ K as z is arbitrary. As ν(f ) = 1, f (z) ≥ 1 for some z and we get inf λ−M LM f ≥ K −1 . If xi = x′i for i ∈ [0, m] and f ∈ Λ, one has |f (x) − f (x′ )| ≤ (Bm − 1)K → 0 as m → ∞, since Bm → 1. Thus Λ is equicontinuous and compact by the Arzela-Ascoli Theorem. Λ = ∅ as 1 ∈ Λ. Applying Schauder-Tychonoff Theorem to λ−1 L : Λ → Λ gives us h ∈ Λ with Lh = λh. Furthermore inf h = inf λ−M LM h ≥ K −1 . ⊓ ⊔ 1.9. Lemma. There is an η ∈ (0, 1) so that for f ∈ Λ one has λ−M LM f = ηh + (1 − η)f ′ with f ′ ∈ Λ. Proof. Let g = λ−M LM f − ηh where η is to be determined. Provided ηh ≤ K −1 we will have g ≥ 0. Assume xi = x′i for all i ∈ [0, m]. We want to pick η so that g(x) ≤ Bm g(x′ ), or equivalently (⋆)

η(Bm h(x′ ) − h(x)) ≤ Bm λ−M LM f (x′ ) − λ−M LM f (x) . m

m+1

We saw above that Lf1 (x) ≤ Bm+1 ebα Lf1 (x′ ) ≤ Bm+1 ebα Lf1 (x′ ) for −M +1 M −1 any f1 ∈ Λ. Applying this to f1 = λ L f one has m

λ−M LM f (x) ≤ Bm+1 ebα λ−M LM f (x′ ) . −1 Now h(x) ≥ Bm h(x′ ) because h ∈ Λ. To get (⋆) it is therefore enough to have m −1 )h(x′ ) ≤ (Bm − Bm+1 ebα ) λ−M LM f (x′ ) η(Bm − Bm

or

m

−1 η(Bm − Bm )h ≤ (Bm − Bm+1 ebα )K −1 .

B. Ruelle’s Perron-Frobenius Theorem

11

m

−1 There is an L so that the logarithms of Bm , Bm and Bm+1 ebα are in [−L, L] for all m. Let u1 , u2 be positive constants such that

u1 (x − y) ≤ ex − ey ≤ u2 (x − y) for all x, y ∈ [−L, L], x > y . For (⋆) to hold it is enough for η > 0 to satisfy m

ηhu1 (log Bm + log Bm ) ≤ K −1 u2 (log Bm − log(Bm+1 ebα )) or ηhu1 or



4bαm+1 1−α



≤ K −1 u2 bαm

η ≤ u2 (1 − α)(4αu1 hK)−1 .

⊓ ⊔

1.10. Lemma. There are constants A > 0 and β ∈ (0, 1) so that λ−n Ln f − h ≤ Aβ n for all f ∈ Λ, n ≥ 0. Proof. Let n = M q + r, 0 ≤ r < M . Inductively one sees from Lemma 1.9 and Lh = λh that, for f ∈ Λ, λ−M q LM q f = (1 − (1 − η)q )h + (1 − η)q fq′ where fq′ ∈ Λ. As fq′  ≤ K one has λ−M q LM q f − h ≤ (1 − η)q (h + K) and λ−n Ln f − h = λ−r Lr (λ−M q LM q f − h) ≤ A(1 − η)q+1 ≤ Aβ n , where

A = (1 − η)−1 (h + K) sup λ−r Lr  0≤r 0. Reasoning as in the proof of 1.8 (with Lr (f F ) in place of f ) we get λr ν(f F ) = ν(λ−M LM +r (f F )) ≥ K −1 Lr (f F )(z), for any z. But (f F )(w) > 0 gives Lr (f F )(σ r w) > 0 and so ν(f F ) > 0.

⊓ ⊔

1.12. Lemma. For f ∈ Cr , F ∈ Λ and n ≥ 0, λ−n−r Ln+r (f F ) − ν(f F )h ≤ Aν(|f F |)β n . + For g ∈ C (ΣA ) one has limm→∞ λ−m Lm g − ν(g)h = 0.

Proof. Write f = f + − f − with f + , f − ≥ 0 and f + , f − ∈ Cr . Then λ−n−r Ln+r (f ± F ) − ν(f ± F )h ≤ Aν(|f ± F |)β n . For f ± F ≡ 0, this is obvious; for f ± F ≡ 0 we apply Lemmas 1.11 and 1.10. These inequalities add up to give us the first statement of the lemma. Given g and ε > 0 one can find r and f1 , f2 ∈ Cr so that f1 ≤ g ≤ f2 and 0 ≤ f2 − f1 ≤ ε. As |ν(fi ) − ν(g)| < ε, the first statement of the lemma with F = 1 gives λ−m Lm (fi ) − ν(g)h ≤ ε(1 + h) for large m. Since λ−m Lm f1 ≤ λ−m Lm g ≤ λ−m Lm f2 ,

λ−m Lm g − ν(g)h ≤ ε(1 + h) for large m.

⊓ ⊔

The proof of 1.7 is finished.

C. Construction of Gibbs Measures

13

C. Construction of Gibbs Measures + ) and ν, h, λ are as in Ruelle’s We continue to assume that φ ∈ FA ∩ C (ΣA + ; Perron-Frobenius
Theorem. Then µ = hν is a probability measure on ΣA µ(f ) = ν(hf ) = f (x)h(x) dν(x). + + 1.13. Lemma. µ is invariant under σ : ΣA . → ΣA

+ Proof. We need to show that µ(f ) = µ(f ◦ σ) for f ∈ C (ΣA ). Notice that  ((Lf ) · g)(x) = eφ(y) f (y)g(x) y∈σ −1 x

=



eφ(y) f (y)g(σy)

y∈σ −1 x

= L(f · (g ◦ σ))(x) . Using this µ(f ) = ν(hf ) = ν(λ−1 Lh · f ) = λ−1 ν(L(h · (f ◦ σ)))

= λ−1 (L∗ ν)(h · (f ◦ σ)) = ν(h · (f ◦ σ)) = µ(f ◦ σ) .

⊓ ⊔

+ Because µ is σ-invariant on ΣA there is a natural way to make µ into a + measure on ΣA . For f ∈ C (ΣA ) define f ∗ ∈ C (ΣA ) by

f ∗ ({xi }∞ i=0 ) = min{f (y) : y ∈ ΣA , yi = xi for all i ≥ 0} . Notice that for m, n ≥ 0 one has (f ◦ σ n )∗ ◦ σ m − (f ◦ σ n+m )∗  ≤ varn f . Hence |µ((f ◦ σ n )∗ ) − µ((f ◦ σ n+m )∗ )| = |µ((f ◦ σ n )∗ ◦ σ m ) − µ((f ◦ σ n+m )∗ )| ≤ varn f which approaches 0, as n → ∞, since f is continuous. Hence µ ˜(f ) = limn→∞ µ((f ◦ σ n )∗ ) exists by the Cauchy criterion. It is straightforward to check that µ ˜ ∈ C (ΣA )∗ . By the Riesz Representation Theorem we see that µ ˜ defines a probability measures on ΣA , which we will denote by µ despite the possible ambiguity. Note that µ ˜(f ◦ σ) = lim µ((f ◦ σ n+1 )∗ ) = µ ˜ n→∞

14

1 Gibbs Measures

proving that µ ˜ is σ-invariant. Also µ ˜(f ) = µ(f ) for f ∈ C (ΣA )∗ . Recall that µ is ergodic if µ(E) = 0 or 1 whenever E is a Borel set with σ −1 E = E. One calls µ mixing if lim µ(E ∩ σ −n F ) = µ(E)µ(F ),

n→∞

for all Borel sets E and F . It is clear that mixing implies ergodicity and a standard argument shows that the mixing condition only need be checked for E and F in a basis for the topology. 1.14. Proposition. µ is mixing for σ : ΣA → ΣA . m−1 k Proof. Writing Sm φ(x) = k=0 φ(σ x) one checks inductively that for f , + g ∈ C (ΣA ) one has  eSm φ(y) f (y) . (Lm f )(x) = y∈σ −m x

Then ((Lm f ) · g)(x) =



eSm φ(y) f (y)g(σ m y)

y∈σ −m x

= Lm (f · (g ◦ σ m )) . Let E = {y ∈ ΣA : yi = ai , r ≤ i ≤ s} , F = {y ∈ ΣA : yi = bi , u ≤ i ≤ v} . In checking the mixing condition for E and F we may assume r = u = 0 because µ is σ-invariant. We want to calculate µ(E ∩ σ −n F ) = µ(χE · χσ−n F ) = µ(χE · (χF ◦ σ n ))

= ν(hχE · (χF ◦ σ n )) = λ−n L∗n ν(hχE · (χF ◦ σ n ))

= ν(λ−n Ln (hχE · (χF ◦ σ n ))) = ν(λ−n Ln (hχE ) · χF ) .

Now |µ(E ∩ σ −n F ) − µ(E)µ(F )| = |µ(E ∩ σ −n F ) − ν(hχE )ν(hχF )|   = |ν (λ−n Ln (hχE ) − ν(hχE )h)χF | ≤ λ−n Ln (hχE ) − ν(hχE )h ν(F ) . Because χE ∈ Cs Lemma 1.12 gives, for n ≥ s,

C. Construction of Gibbs Measures

15

λ−n Ln (hχE ) − ν(hχE )h ≤ Aµ(E)β n−s where β ∈ (0, 1). One then has |µ(E ∩ σ −n F ) − µ(E)µ(F )| ≤ A′ µ(E)µ(F )β n−s for n ≥ s where A′ = A (inf h)−1 . Thus µ(E ∩ σ −n F ) → µ(E)µ(F ). ⊓ ⊔ ∞ 1.15. Lemma. Let a = k=0 vark φ < ∞. If x, y ∈ ΣA with xi = yi for i ∈ [0, m), then |Sm φ(x) − Sm φ(y)| ≤ a . Proof. Define y ′ by yi′ =



yi for i ≥ 0 xi for i ≤ 0 .

+ ), φ(σ k y ′ ) = φ(σ k y) for k ≥ 0. Hence Since φ ∈ C (ΣA

|Sm φ(x) − Sm φ(y)| ≤ ≤

m−1  k=0

m−1 

|φ(σ k x) − φ(σ k y ′ )| varm−1−k φ

k=0

≤ a.

⊓ ⊔

We now complete the proof of 1.4. + 1.16. Theorem. µ is a Gibbs measure for φ ∈ FA ∩ C (ΣA ). + Proof. Let E = {y ∈ ΣA : yi = xi for i ∈ [0, m)}. For any z ∈ ΣA there is ′ −m ′ at most one y ∈ σ z with y ∈ E. Thus, using 1.15,  eSm φ(y) h(y)χE (y) Lm (hχE )(z) = y∈σ −m z

≤ eSm φ(x) ea h and so µ(E) = ν(hχE ) = λ−m ν(Lm (hχE )) ≤ λ−m eSm φ(x) ea h . + Thus take c2 = ea h. On the other hand, for any z ∈ ΣA there is at least ′ −m−M ′ z with y ∈ E. Then one y ∈ σ ′

Lm+M (hχE )(z) ≥ eSm+M φ(y ) h(y ′ )

≥ e−M φ−a (inf h) eSm φ(x)

16

1 Gibbs Measures

and

µ(E) = λ−m−M ν(Lm+M (hχE )) ≥ c1 λ−m eSm φ(x)

where c1 = λ−M e−M φ−a . We have verified the desired inequalities on measures of cylinder sets given in 1.4 with P = log λ. Suppose now that µ′ is any other measure satisfying the inequalities in Theorem 1.4 with constants c′1 , c′2 , P ′ . For x ∈ ΣA let Em (x) = {y ∈ ΣA : yi = xi for all i ∈ [0, m)}. Let Tm be a finite subset of ΣA so that ΣA = x∈Tm Em (x) disjointly. Then c′1 e−P

′

m



x∈Tm

eSm φ(x) ≤



µ′ (Em (x))

x∈Tm

=1  ′ ≤ c′2 e−P m eSm φ(x) . x∈Tm

From this one sees that P ′ = limm→∞

1 m

log



x∈Tm

 eSm φ(x) . One can

apply the same reasoning to µ; hence P ′ = P as they equal the same limit. The estimates on µ′ (Em (x)) and µ(Em (x)) give us µ′ (Em (x)) ≤ dµ(Em (x)) ′ where d = c′2 c−1 1 . Taking limits this extends to µ (E) ≤ dµ(Em ) for all Borel sets E. In particular µ′ (E) = 0 when µ(E) = 0. By the Radon-Nikodym Theorem µ′ = f µ for some µ-integrable f . Applying σ one has µ′ = σ ∗ µ′ = (f ◦ σ) σ ∗ µ = (f ◦ σ)µ .

a.e.

As the Radon-Nikodym derivative is unique up to µ-equivalence, f ◦ σ = f . Because µ is ergodic
this gives f equivalent to some constant c. ⊔ 1 = µ′ (ΣA ) = c dµ = c and µ = µ′ . ⊓

D. Variational Principle We will describe Gibbs measures as those maximizing a certain quantity, in a way analogous to Lemma 1.1. If C = {C1 , . . . , Ck } is a partition of a measure k space (X, B, µ) (i.e., the Ci ’s are pairwise disjoint and X = i=1 Ci ), one defines the entropy Hµ (C) =

k 

(−µ(Ci ) log µ(Ci )) .

i=1

If D is another (finite) partition, C ∨ D = {Ci ∩ Dj : Ci ∈ C , Dj ∈ D}.

D. Variational Principle

17

1.17. Lemma. Hµ (C ∨ D) ≤ Hµ (C) + Hµ (D). Proof. Hµ (C ∨ D)−Hµ (C) =

 i,j

 (−µ(Ci ∩ Dj ) log µ(Ci ∩ Dj ))− (−µ(Ci ) log µ(Ci )) i

µ(Ci ∩ Dj ) = −µ(Ci ∩ Dj ) log µ(Ci ) i,j    −µ(Ci ∩ Dj ) µ(Ci ∩ Dj ) = log µ(Ci ) · µ(Ci ) µ(Ci ) j i 

ϕ 1.0

The function ϕ(x) = −x log x (ϕ(0) = 0) is concave on [0, 1] as ϕ′′ (x) < 0 for x ∈ (0, 1). From this it follows that 0.5

ϕ(a1 x1 + a2 x2 ) ≥ a1 ϕ(x1 ) + a2 ϕ(x2 ) [0, 1], a1 + a2= 1. Inductively when x1 , x2 ∈  n n one sees that ϕ( i=1 ai ϕ(xi ) when i=1 ai xi ) ≥ n a = 1 and a ≥ 0. i i i=1

0 0

0.5

1.0

x

µ(C ∩D )

So

i j Applying this to ai = µ(Ci ) and xi = µ(C we get i)       µ(Ci ∩ Dj ) µ(Ci ∩ Dj ) = ϕ(µ(Dj )) . µ(Ci ) ϕ ≤ϕ µ(Ci ) i i

Hµ (C ∨ D) − Hµ (C) ≤



ϕ(µ(Dj )) = Hµ (D) .

j

⊓ ⊔

am 1.18. Lemma. Suppose {am }∞ m=1 is a sequence satisfying inf m > −∞, am am+n ≤ am + an for all m, n. Then limm→∞ m exists and equals inf m amm .

Proof. Fix m > 0. For j > 0, write j = km + n with 0 ≤ n < m. Then aj akm+n akm an kam an = ≤ + ≤ + · j km + n km km km km

Letting j → ∞, k → ∞ one gets lim sup j a lim supj jj

Thus exists and

≤ inf m amm . As equals inf m amm · ⊔ ⊓

aj am ≤ · j m

lim inf j

aj j

≥ inf m

am m ,

one gets that limj

aj j

18

1 Gibbs Measures

1.19. Lemma. If D is a (finite) partition of (X, B, µ) and T an automorphism of (X, B, µ), then 1 Hµ (D ∨ T −1 D ∨ · · · ∨ T −m+1 D) m→∞ m

hµ (T, D) = lim exists.

Proof. Let am = Hµ (D ∨ T −1 D ∨ · · · ∨ T −m+1 D). Then am+n ≤ Hµ (D∨T −1 D∨· · ·∨T −m+1 D)+Hµ (T −m D∨· · ·∨T −m−n+1 D) ≤ am +an since

T −m D ∨ · · · ∨ T −m−n+1 D = T −m (D ∨ · · · ∨ T −n+1 D)

and µ is T -invariant. ⊓ ⊔ Definition. Let µ ∈ Mσ (ΣA ) and U = {U1 , . . . , Un } where Ui = {x ∈ ΣA : x0 = i}. Then s(µ) = hµ (σ, U) is called the entropy of µ. Suppose now that φ ∈ C (ΣA ) and that a0 a1 · · · am−1 are integers between 1 and n satisfying Aak ak+1 = 1. Write sup

a0 a1 ···am−1

Sm φ = sup

m−1  k=0



k

φ(σ x) : x ∈ ΣA , xi = ai for all 0 ≤ i < m

and Zm (φ) =



exp

a0 a1 ···am−1



sup

a0 a1 ···am−1

1.20. Lemma. For φ ∈ C (ΣA ), P (φ) = limm→∞ the pressure of φ).

1 m

Sm φ



.

log Zm (φ) exists (called

Proof. Notice that sup

a0 a1 ···am+n−1

Sm+n φ ≤

sup

a0 a1 ···am−1

Sm φ +

sup

am ···am+n−1

Sn φ .

From this one gets Zm+n (φ) ≤ Zm (φ)Zn (φ); the terms in Zm+n (φ) are bounded by terms in Zm (φ)Zn (φ) and Zm (φ)Zn (φ) may have more terms, all positive. Apply Lemma 1.18 to am = log Zm (φ) (notice am ≥ −mφ). ⊓ ⊔ 1.21. Proposition. Suppose φ ∈ C (ΣA ) and µ ∈ Mσ (ΣA ). Then  s(µ) + φ dµ ≤ P (φ) .

D. Variational Principle

19

1 Sm φdµ = φdµ where Sm φ(x) = Proof. As φ ◦ σ k dµ = φdµ, m m−1 k k=0 φ(σ x). Hence     1 s(µ) + φ dµ ≤ lim Hµ (U ∨ · · · ∨ σ −m+1 U) + Sm φ dµ . m→∞ m













Now U ∨ · · · ∨ σ −m+1 U partitions points x ∈ ΣA according to x0 x1 · · · xm−1 . Thus  Hµ (U ∨ · · · ∨ σ −m+1 U) + Sm φ dµ  ≤ Sm φ µ(a0 · · · am−1 )(− log µ(a0 · · · am−1 )) + sup a0 a1 ···am−1

a0 ···am−1

≤ log Zm (φ) by Lemma 1.1 .

Now let m → ∞.

⊓ ⊔

1.22. Theorem. Let φ ∈ FA , ΣA topologically mixing and µφ the Gibbs measure of φ. Then µφ is the unique µ ∈ Mσ (ΣA ) for which  s(µ) + φ dµ = P (φ) . Proof. Given a0 · · · am−1 , pick x with xi = ai (i = 0, . . . , m − 1) and Sm φ(x) =

sup

a0 a1 ···am−1

Sm φ .

Now, as µ = µφ is the Gibbs measure, µ{y ∈ ΣA : yi = ai ∀ 0 ≤ i < m} ∈ [c1 , c2 ] . exp(−P m + Sm φ(x)) Summing the measure of these sets over all possible a0 · · · am−1 ’s gives 1; so c1 exp(−P m)Zm (φ) ≤ 1 ≤ c2 exp(−P m)Zm (φ) or

Zm (φ) ∈ [c2−1 , c−1 1 ]. exp(P m)

1 It follows that P (φ) = limm→∞ m log Zm (φ) = P . If yi = xi for all i = 0, . . . , m − 1, then

|Sm φ(y) − Sm φ(x)| ≤

m−1  k=0

|φ(σ k y) − φ(σ k y)|

≤ var0 φ + var1 φ + · · · + var[ m ] φ + varm−[ m ] φ + · · · + var0 φ 2

≤ 2c

[ m2 ] 

k=0

αk ≤

2c = d. 1−α

2

20

1 Gibbs Measures

Hence, if B = {y ∈ ΣA : yi = ai for all i = 0, . . . , m − 1}, then for x ∈ B,  Sm φ dµ ≥ −µ(B) [ log µ(B) − Sm φ(x) + d ] −µ(B) log µ(B) + B   ≥ −µ(B) [ log c2 e−P m+Sm φ(x) − Sm φ(x) + d ] ≥ µ(B)(P m − log c2 − d) .

Since Hµ (U ∨ · · · ∨ σ

−m+1

U) + =



Sm φ dµ

 B

≥ we get s(µ) + ≥ lim

m→∞



 B



Sm φ dµ

B



µ(B)(P m − log c2 − d) = P m − log c2 − d,

1 m→∞ m

φ dµ = lim

− µ(B) log µ(B) +



Hµ (U ∨ · · · ∨ σ −m+1 U) +

1 (P m − log c1 − d) = P = P (φ) . m



Sm φ dµ



The reverse inequality was in Proposition 1.21. So  s(µφ ) + φ dµφ = P (φ) . To prove uniqueness we will need a couple of lemmas. 1.23. Lemma. Let X be a compact metric space, µ ∈ M (X), and D = {D1 , . . . , Dn } a Borel partition of X. Suppose {Cm }∞ m=1 is a sequence of partitions so that diam(Cm ) = maxC∈Cm diam(C) → 0 as m → ∞. Then there are partitions {E1m , . . . , Enm } so that

1. each Eim is a union of members of Cm , 2. limm→∞ µ(Eim ∆Di ) = 0 for each i.

Proof. Pick compacts K1 , . . . , Kn with Ki ⊂ Di and µ(Di \Ki ) < ε. Let δ = inf i=j d(Ki , Kj ) and consider m with diam(Cm ) ≤ 2δ . Divide the elements C ∈ Cm into groups whose unions are E1m , . . . , Enm so that C ⊂ Eim

if C ∩ Ki = ∅ .

As diam(Cm ) ≤ 2δ any C ∈ Cm can intersect at most one Ki . Put a C hitting no Ki in any Eim you like. Then Eim ⊃ Ki and   n  m m m µ(Ei ∆Di ) = µ(Di \Ei )+µ(Ei \Di ) ≤ ε + µ X\ ⊓ ⊔ Ki ≤ (n+1) ε . i=1

D. Variational Principle

21

1.24. Lemma. Suppose 0 ≤ p1 , . . . , pm ≤ 1, s = p1 + · · · + pm ≤ 1 and a1 , . . . , am ∈ R. Then   m m   ai e − log s . pi (ai − log ai ) ≤ s log i=1

i=1

Proof. This generalizes 1.1. One shows by calculus that the left side is maxiai ⊓ mized at pi = seeaj · ⊔ j


Proof of Theorem 1.22 (continued). Let ν ∈ Mσ (ΣA ) satisfy s(ν)+ φ dν = P . First suppose ν is singular with respect to µ. Then there is a Borel set B with m m σ(B) = B, µ(B) = 0 and ν(B) = 1. Let Cm = σ −[ 2 ]+1 U∨· · ·∨U∨· · ·∨σ [ 2 ] U. Then diam(Cm ) → 0 (use dβ metric). Applying 1.23 to {B, X\B} one finds sets E m which are unions of elements of Cm and satisfy (µ + ν)(B∆E m ) → 0. m As µ + ν is σ-invariant and σ −m+[ 2 ] B = B one has (µ + ν)(B∆F m ) → 0 m where F m = σ −m+[ 2 ] E m is a union of members of U ∨ · · · ∨ σ −m+1 U. Since 1 s(ν) = inf m Hν (U ∨ · · · ∨ σ −m+1 U) one has     1 −m+1 P = P (φ) = s(ν) + φ dν ≤ U) + Sm φ dν Hν (U ∨ · · · ∨ σ m or mP ≤



B∈U∨···∨σ −m+1 U

   Sm φ dν . −ν(B) log ν(B) + B

Picking xB ∈ B one has Sm φ ≤ Sm φ(xB ) + d on B and so  mP ≤ d + ν(B)(Sm φ(xB ) − log ν(B)) B

≤ d+



B⊂F m

+

ν(B)(Sm φ(xB ) − log ν(B))



B⊂X\F m

ν(B)(Sm φ(xB ) − log ν(B)).

Applying 1.24 mP − d ≤ ν(F m ) log +



exp(Sm φ(xB ))

B⊂F m

ν(X\F m ) log



exp(Sm φ(xB )) + 2K ∗ .

B⊂X\F m

where K ∗ = sup0≤s≤1 (−s log s). Rearranging terms:

22

1 Gibbs Measures

−2K ∗ −d ≤ ν(F m ) log +

ν(X\F m ) log



B⊂F m



B⊂X\F m

≤

ν(F m ) log

≤

log c−1 2



  exp Sm φ(xB ) − mP

  exp Sm φ(xB ) − mP

m c−1 2 µ(B) + ν(X\F ) log

B⊂F m m



c−1 2 µ(B)

B⊂X\F m m

m

+ ν(F ) log µ(F ) + ν(X\F ) log µ(X\F m ) .

Letting m → ∞, ν(F m ) → 1, µ(F m ) → 0 and the above inequality is contradictory. In general, for ν ′ ∈ Mσ (ΣA ), write ν ′ = βν + (1 − β)µ′ where β ∈ (0, 1), ν ∈ Mσ (ΣA ) is singular w.r.t. µ and µ′ ∈ Mσ (ΣA ) is absolutely continuous w.r.t. µ. As ν and µ′ are supported on disjoints sets Pν ′ (φ) = βPν (φ) + (1 − β)Pµ′ (φ),
where Pν (φ) = s(ν) + φ dν. Suppose Pν ′ (φ) = P . Since Pν (φ) ≤ P and Pµ′ (φ) ≤ P (Prop. 1.21), we have Pν (φ) = P or β = 0. We just saw that ′ dν ′ Pν (φ) = P . Thus ν ′ = µ′ and write ν ′ = dν dµ µ. Then the dµ is σ-invariant up to equivalence as ν ′ , µ are both invariant and the Radon-Nikodym derivative ′ ′ is unique up to equivalence. So dν ⊓ ⊔ dµ is constant and ν = µ.

E. Further Properties In this section we look at more examples of the good behavior of Gibbs measures. Throughout we assume µ = µφ with φ ∈ FA and σ|ΣA topologically mixing. Two partitions P and Q are called ε-independent if  |µ(P ∩ Q) − µ(P )µ(Q)| < ε . P ∈P,Q∈Q

Let U = {U1 , . . . , Un } be the partition of ΣA with Uj = {x ∈ ΣA : x0 = j} . The partition U is called weak-Bernoulli (for σ and µ) if for every ε > 0 there is an N (ε) so that P = U ∨ σ −1 U ∨ · · · ∨ σ −s U and Q = σ −t U ∨ · · · ∨ σ −t−r U are ε-independent for all s ≥ 0, r ≥ 0, t ≥ s + N (ε). A well-known theorem of Friedman and Ornstein [4] states that if U is weak-Bernoulli, then (σ, µ) is conjugate to a Bernoulli shift.

E. Further Properties

23

1.25. Theorem. U is weak-Bernoulli for the Gibbs measure µ = µφ . + ) as before. For P ∈ P we have χP ∈ Proof. We may assume that φ ∈ C (ΣA Cr . As in the proof of 1.14, for Q ∈ Q

|µ(P ∩ Q) − µ(P )µ(Q)| ≤ A′ µ(P )µ(Q)β t−s where β ∈ (0, 1). Summing over P, Q  |µ(P ∩ Q) − µ(P )µ(Q)| ≤ A′ β t−s ≤ ε P,Q

when t − s is large.

⊓ ⊔

Because µ is mixing, µ(f · (g ◦ σ n )) → µ(f )µ(g) as n → ∞ for f, g continuous (in fact L2 ) functions. The above proof used that this convergence was exponentially fast for characteristic functions of cylinder sets. This exponential convergence will now be carried over to functions in FA . For α ∈ (0, 1) let Hα be the family of f ∈ C (ΣA ) with vark f ≤ cαk for some c. Hα is a Banach space under the norm f α = f  + sup(α−k vark f ) . k≥0

1.26. Exponential Cluster Property. For fixed α ∈ (0, 1) there are constants D and γ ∈ (0, 1) so that |µ(f · (g ◦ σ n )) − µ(f )µ(g)| ≤ Df α gα γ n for all f, g ∈ Hα , n ≥ 0. Proof. For k ≥ 0 and x ∈ ΣA , let Ek (x) = {y ∈ Σa : yi = xi for all |i| ≤ k} .
Define fk (x) = µ(Ek (x))−1 Ek (x) f dµ. Then µ(fk ) = µ(f ) and f − fk  ≤ f α αk . Hence |µ(f · (g ◦ σ n )) − µ(f )µ(g)| ≤ |µ(fk · (gk ◦ σ n )) − µ(fk )µ(gk )| + |µ((f − fk ) · (g ◦ σ n ))| + |µ(fk · ((g − gk ) ◦ σ n ))| ≤ |µ(fk · (gk ◦ σ n )) − µ(fk )µ(gk )| + 2αk f α gα .

Now  fk is measurable with  respect to the partition P = {Ek (x)}x ; i.e., fk = a χ . Also g = k P ∈P P P P ∈P bP χP . Hence  |µ(fk · (gk ◦ σ n )) − µ(fk )µ(gk )| ≤ |aP bP | |µ(P ∩ σ −n Q) − µ(P )µ(Q)| P,Q∈P

≤ f gA′ β n−2k ≤ f α gα A′ β n−2k .

Letting k = [n/3] we get the result with γ = max(α1/3 , β 1/3 ). ⊓ ⊔

24

1 Gibbs Measures

1.27. Central Limit Theorem. For ψ ∈ FA there is a ξ = ξ(ψ) ∈ [0, ∞) so that    r 2 2 1 1 n→∞ e−x /2ξ dx . µ x ∈ ΣA : √ (Sn ψ(x) − nµ(ψ)) < r −→ √ n ξ 2π −∞ For ξ = 0, convergence is asserted for r = 0 and the expression on the right is taken to be 0 for r < 0 and 1 for r > 0. Remark. We omit the proof, referring the reader to M. Ratner [13]. It is interesting to know when ξ(ψ) = 0. This happens (see [13]) precisely when ψ − µ(ψ) = u ◦ σ − u has a solution u ∈ L2 (µ). It is interesting that in case such u can be found one can find u ∈ FA and so ψ is homologous to a constant. The reasoning for this is very roundabout and it would be good to find a nice direct proof. 1.28. Theorem. Let ΣA be topologically mixing and φ, ψ ∈ FA . The following are equivalent: (i) µφ = µψ . (ii) There is a constant K so that Sm φ(x)−Sm ψ(x) = mK whenever σ m x = x. (iii)There are a constant K and a u ∈ FA so that φ(x) = ψ(x) + K + u(σx) − u(x) for all x ∈ ΣA . (iv)There are constants K and L so that |Sm φ(x) − Sm ψ(x) − mK| ≤ L for all x and all m > 0. If these conditions hold, then K = P (φ) − P (ψ). Proof. (iii) ⇒ (iv) is obvious and (iv) ⇒ (i) is just like Lemma 1.5. Assume µφ = µψ and σ m x = x. From the definition of a Gibbs measure and µφ = µψ one sees that exp(−P (φ)j + Sj φ(x)) ∈ [d1 , d2 ], exp(−P (ψ)j + Sj ψ(x)) where d1 > 0, d2 > 0 are independent of x and j. This is equivalent to |Sj φ(x) − Sj ψ(x) − j(P (φ) − P (ψ))| ≤ M for some M independent of j, x. If σ m x = x, letting j = km, Sj φ(x) = kSm φ(x) and M > k |Sm φ(x) − Sm ψ(x) − m(P (φ) − P (ψ))| . Letting k → ∞ we get (ii) with K = P (φ) − P (ψ). Now assume (ii). In proving (iii) we will need the following standard lemma.

E. Further Properties

25

1.29. Lemma. If T : X → X is a topologically transitive continuous map of a compact metric space, then there is a point x ∈ X so that if U = ∅ is open, N > 0, then T n x ∈ U for some n ≥ N . Proof. As X is 2nd countable, let U1 , U2 , . . . be a basis for the topology. By transitivity, the open set  Vi,N = T −n Ui n≥N

is dense in X. By Baire Category Theorem there is an x ∈



i,N

Vi,N . ⊓ ⊔

Continuing the proof of (iii) from (ii), let x be as in the lemma for X = ΣA , T = σ (topological mixing is stronger than transitivity). Let η = φ − ψ − K ∈ FA . Let Γ = {σ k x : k ≥ 0} and define u : Γ → R by u(σ k x) =

k−1 

η(σ j x) .

j=0

As Γ is dense in ΣA , Γ must be infinite (except in the trivial case of ΣA =one point) and x is not periodic. Thus σ k x = σ m x for m = k and u is well defined on Γ . We will estimate varr (u|Γ ). Suppose y = σ k x, and z = σ m x (m > k) agree in places −r to r. Then xk+s = xm+s for all |s| ≤ r. Define w ∈ ΣA by wi = xt

for i ≡ t (mod m − k) , k ≤ t ≤ m .

Then σ m−k w = w and w, x agree in places k − r to m + r; hence σ j x, σ j w agree in places k − r − j through m + r − j. Now u(w) − u(y) =

m−1 

η(σ j x) .

j=k

Since (ii) gives m−1 

η(σ j w) = 0 ,

j=k

|u(z) − u(y)| ≤

m−1  j=k

|η(σ j x) − η(σ j w)|

≤ varr η + varr+1 η + · · · + varr+1 η + varr η ∞  ≤2 vars η. s=r

Since η ∈ FA , vars η ≤ cαs for some α ∈ (0, 1) and

26

1 Gibbs Measures

varr (u|Γ ) ≤ 2c

∞  s=r

αs =

2c αr . 1−α

So u is uniformly continuous on Γ and therefore extends uniquely to a continuous u : ΣA = Γ → R. Because varr u = varr (u|Γ ), u ∈ FA . For z ∈ Γ , u(σz) − u(z) = η(z) and this equation extends to ΣA by continuity. ⊓ ⊔

References For discussions of Gibbs’ states and statistical mechanics we refer to Ruelle’s book [9] and Lanford [6]. The definition of Gibbs state in statistical mechanics does not coincide with what we gave in Section 1. There Gibbs states are defined for more general φ and then our theorem corresponds to one in statistical mechanics about uniqueness of Gibbs states [3, 10, 6]. In those papers one deal with Σn instead of slightly more general ΣA . Proofs of Theorem 1.4 for ΣA appear in [2, 11, 12]. The variational characterization (Theorem 1.22) of the Gibbs measure µφ is due to Lanford-Ruelle [7] in the Σn case. For ΣA one can find it in [2] or [11]. We have followed the proof in [2], which in turn was based on Adler and Weiss’ proof [1] of a theorem of Parry (Theorem 1.22 with φ = 0). The weak Bernoulli condition (Theorem 1.25) was verified for the Σn case first by Gallavotti [5]. It was extended to ΣA independently by each of Bunimovich, Ratner, Ruelle and this author. Theorem 1.28 is taken from Livˇsic [8] and Sinai [12]. Theorem 1.26 is from [11]. Ruelle gave two proofs of his Perron-Frobenius Theorem, for different circumstances [10, 11]. We have followed parts of each. 1. R. L. Adler, B. Weiss: Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society 98, American Mathematical Society, Providence, R.I. 1970. 2. R. Bowen: Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. 3. R. L. Dobrushin: The problem of uniqueness of a Gibbsian random field and the problem of phase transitions, Func. Anal. and its Appl. 2 (1968) no. 4, 44–57. 4. N. A. Friedman, D. S. Ornstein: On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970) 365–394. 5. G. Gallavotti: Ising model and Bernoulli schemes in one dimension, Comm. Math. Phys. 32 (1973), 183–190. 6. O. E. Lanford: Entropy and equilibrium states in classical statistical mechanics. In “statistical mechanics and mathematical problems” (ed. A. Lenard), Lecture Notes in Physics 20, pp. 1–113. Springer, Berlin, 1973. 7. O. E. Lanford, D. Ruelle: Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys. 13 (1969) 194–215.

References

27

8. A. N. Livˇsic: Certain properties of the homology of Y-systems, Mat. Zametki 10 (1971), 555–564. English translation: Math. Notes Acad. Sci. USSR 10 (1971), 758–763. 9. D. Ruelle: Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam 1969. (1 ) 10. D. Ruelle: Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys. 9 (1968) 267–278. 11. D. Ruelle: A measure associated with axiom-A attractors, Amer. J. Math. 98 (1976), no. 3, 619–654. 12. Ya. Sinai: Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4, 21–64. English translation: Russian Math. Surveys 27 (1972), no. 4, 21–69. 13. M. Ratner: The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math. 16 (1973), 181–197.

1

Reprint: World Scientific Publishing Co., Inc., River Edge, NJ; Imperial College Press, London, 1999 (note of the editor).

2 General Thermodynamic Formalism

A. Entropy In Section D of Chapter 1, we defined the number hµ (T, D) when T is an endomorphism of a probability space and D a finite measurable partition. We now define the entropy of µ w.r.t. T by hµ (T ) = sup hµ (T, D), D

where D ranges over all finite partitions. We will now turn to some computational lemmas. We define Hµ (C|D) = Hµ (C ∨ D) − Hµ (D)    µ(Cj ∩ Di )  µ(Cj ∩ Di ) log =− µ(Di ) µ(Di ) µ(Di ) i j ≥ 0.

Lemma 1.17 says that Hµ (C|D) ≤ Hµ (C). We write C ⊂ D if each set in C is a union of sets in D. 2.1. Lemma. (a) Hµ (C|D) ≤ Hµ (C|E) if D ⊃ E. (b) Hµ (C|D) = 0 if D ⊃ C. (c) Hµ (C ∨ D|E) ≤ Hµ (C|E) + Hµ (D|E). (d) Hµ (C) ≤ Hµ (D) + Hµ (C|D). Proof. Letting ϕ(x) = −x log x, Hµ (C|D) = Since E ⊂ D, one can rewrite this as Hµ (C|D) =

 j

E∈E

µ(E)

  j

i µ(Di ) ϕ



 µ(Di )  µ(Cj ∩ Di )  ϕ · µ(E) µ(Di )

Di ⊂E

µ(Cj ∩Di ) µ(Di )



.

30

2 General Thermodynamic Formalism

 By  the concavity of ϕ (see the proof of Lemma 1.17) one has ϕ( ai xi ) ≥ ai ϕ(xi ) where µ(Cj ∩ Di ) µ(Di ) , xi = · ai = µ(E) µ(Di ) Hence Hµ (C|D) ≤

 j

µ(E) ϕ

E∈E



µ(Cj ∩ E) µ(E)



= Hµ (C|E) .

To see (b) one notes that C ∨ D = D when D ⊃ C. For (c) one writes Hµ (C ∨ D|E) = Hµ (C ∨ D ∨ E) − Hµ (D ∨ E) + Hµ (D ∨ E) − Hµ (E) = Hµ (C|D ∨ E) + Hµ (D|E) ≤ Hµ (C|E) + Hµ (D|E)

by (a). Finally Hµ (C) = Hµ (C ∨ D) − Hµ (D|C)

≤ Hµ (C ∨ D) = Hµ (D) + Hµ (C|D) .

⊓ ⊔

2.2. Lemma. Let T be an endomorphism of a probability space (X, B, µ), C and D finite partitions. Then (a) Hµ (T −k C|T −k D) = Hµ (C|D) for k ≥ 0, (b) hµ (T, C) ≤ hµ (T, D) + Hµ (C|D), (c) hµ (T, C ∨ · · · ∨ T −n C) = hµ (T, C). Proof. As µ is T -invariant, Hµ (T −k C|T −k D) = Hµ (T −k C ∨ T −k D) − Hµ (T −k D)

= Hµ (C ∨ D) − Hµ (D) = Hµ (C|D) .

Using Lemma 2.1 Hµ (C ∨ · · · ∨ T −m+1 C) ≤ Hµ (D ∨ · · · ∨ T −m+1 D) + Hµ (C ∨ · · · ∨ T −m+1 C|D ∨ · · · ∨ T −m+1 D) ≤ Hµ (D ∨ · · · ∨ T −m+1 D) +

m−1  k=0

Hµ (T −k C|D ∨ · · · ∨ T −m+1 D)

≤ Hµ (D ∨ · · · ∨ T −m+1 D) +

m−1 

Hµ (T −k C|T −k D)

k=0

= Hµ (D ∨ · · · ∨ T −m+1 D) + mHµ (C|D) . Dividing by m and letting m → ∞,

A. Entropy

31

hµ (T, C) ≤ hµ (T, D) + Hµ (C|D) . Set D = C ∨ · · · ∨ T −n C. Then

1 1 Hµ (D ∨ · · · ∨ T −m+1 D) = Hµ (C ∨ · · · ∨ T −m−n+1 C) . m m

Letting m → ∞, (as

m m+n

→ 1) we get hµ (T, D) = hµ (T, C) .

⊓ ⊔

2.3. Lemma. Let X be a compact metric space, µ ∈ M (X), ε > 0 and C a finite Borel partition. There is a δ > 0 so that Hµ (C|D) < ε whenever D is a partition with diam(D) < δ. Proof. Let C = {C1 , . . . , Cn }. In Lemma 1.23 we showed that, for any α > 0, one could find δ > 0 such that whenever D satisfies diam(D) < δ there is a E = {E1 , . . . , En } ⊂ D with µ(Ei ∆Ci ) < α . The expression Hµ (C|E) =



µ(Ej ) ϕ

i,j



µ(Cj ∩ Ei ) µ(Ei )



depends continuously upon the numbers µ(Cj ∩ Ei )

and µ(Ei ) =

 j

µ(Cj ∩ Ei )

and vanishes when µ(Cj ∩ Ei ) = δij µ(Ei ). Hence, for α small, Hµ (C|E) < ε. Then Hµ (C|D) ≤ Hµ (C|E) < ε by 2.1 (a). ⊓ ⊔ 2.4. Proposition. Suppose T : X → X is a continuous map of a compact metric space, µ ∈ MT (X) and that Dn is a sequence of partitions with diam(Dn ) → 0. Then hµ (T ) = lim hµ (T, Dn ) . n→∞

Proof. Of course hµ (T ) ≥ lim supn hµ (T, Dn ). Consider any partition C. By Lemmas 2.2 (b) and 2.3 hµ (T, C) ≤ lim inf hµ (T, Dn ) . n

Varying C, hµ (T ) ≤ lim inf n hµ (T, Dn ). ⊓ ⊔ A homeomorphism T : X → X is called expansive if there exists ε > 0 so that d(T k x, T k y) ≤ ε for all k ∈ Z ⇒ x = y .

32

2 General Thermodynamic Formalism

2.5. Proposition. Suppose T : X → X is a homeomorphism with expansive constant ε. Then hµ (T ) = hµ (T, D) whenever µ ∈ MT (X), and diam(D) ≤ ε. Proof. Let Dn = T n D ∨ · · · ∨ D ∨ · · · ∨ T −n D. Then diam(Dn ) → 0 using expansiveness. Hence hµ (T ) = limn hµ (T, Dn ). But hµ (T, Dn ) = hµ (T, D) by Lemma 2.2 (c). ⊓ ⊔ Consider the case of σ : ΣA → ΣA and standard partition U = {U1 , . . . , Un } where Ui = {x ∈ ΣA : x0 = i}. Then σ is expansive and 2.5 gives that hµ (σ) = hµ (σ, U) for µ ∈ Mσ (ΣA ). Now hµ (σ, U) is what we denoted by s(µ) in Chapter 1. That s(µ) = hµ (σ) implies that the number s(µ) does not depend on the homeomorphism σ and partition U, but only on σ as an automorphism of the probability space (ΣA , B, µ) (because of the definition of hµ (σ)). 2.6. Lemma. hµ (T n ) = nhµ (T ) for n > 0. Proof. Let C be a partition and E = C ∨ · · · ∨ T −n+1 C. Then n Hµ (C ∨ · · · ∨ T −nm+1 C) nm 1 Hµ (E ∨ T −n E ∨ · · · ∨ T (−m+1)n E) = lim m→∞ m = hµ (T n , E) ≤ hµ (T n ) = nhµ (T ) .

nhµ (T, C) = lim

m→∞

Varying C, nhµ (T ) ≤ hµ (T n ). On the other hand hµ (T n , C) ≤ hµ (T n , E) by 2.2 (b) and 2.1 (b). Hence hµ (T n ) = sup hµ (T n , C) ≤ n sup hµ (T, C) = nhµ (T ) . C

C

⊓ ⊔

B. Pressure Throughout this section T : X → X will be a fixed continuous map on the compact metric space X. We will define the pressure P (φ) of φ ∈ C (X) in a way which generalizes Section D in Chapter 1. Let U be a finite open cover of X, Wm (U) the set of all m-strings U = Ui0 Ui1 · · · Uim−1 of members of U. One writes m = m(U), X(U) = {x ∈ X : T k x ∈ Uik for k = 0, . . . , m − 1}

B. Pressure

Sm φ(U) = sup

m−1  k=0

33



φ(T k x) : x ∈ X(U) .

In case  X(U) = ∅, we let Sm φ(U) = −∞. We say that Γ ⊂ Wm (U) covers X if X = U∈Γ X(U). Finally one defines Zm (φ, U) = inf Γ



exp(Sm φ(U)),

U∈Γ

where Γ runs over all subsets of Wm (U) covering X. 2.7. Lemma. The limit P (φ, U) = lim

m→∞

1 log Zm (φ, U) m

exists and is finite. Proof. If Γm ⊂ Wm (U) and Γn ⊂ Wn (U) each cover X, then Γm Γn = {UV : U ∈ Γm , V ∈ Γn } ⊂ Wm+n (U) covers X. One sees that Sm+n φ(UV) ≤ Sm φ(U) + Sn φ(V) and so 

UV∈Γm Γn

exp(Sm+n φ(UV)) ≤



exp(Sm φ(U))

U∈Γm



exp(Sn φ(V)).

V∈Γn

Thus Zm+n (φ, U) ≤ Zm (φ, U) Zn (φ, U)

and Zm (φ, U) ≥ e−mφ . Hence am = log Zm (φ, U) satisfies the hypotheses of Lemma 1.18. ⊓ ⊔ 2.8. Proposition. The limit P (φ) =

lim

diam(U)→0

P (φ, U)

exists (but may be +∞). Proof. Suppose V is an open cover refining U, i.e., every V ∈ V lies in some U (V ) ∈ U. For V ∈ Wm (V) let U (V) = U (Vi0 ) · · · U (Vim−1 ). If Γm ⊂ Wm (V) covers X, then U (Γm ) = {U (V) : V ∈ Γm } ⊂ Wm (U) covers X. Let γ = γ(φ, U) = sup{|φ(x) − φ(y)| : x, y ∈ U for some U ∈ U}. Then Sm φ(U (V)) ≤ Sm φ(V) + mγ and so Zm (φ, U) ≤ emγ Zm (φ, V), which gives

34

2 General Thermodynamic Formalism

P (φ, U) ≤ P (φ, V) + γ . Now for any U, all V with small diameter refine U and so P (φ, U) − γ(φ, U) ≤ lim inf P (φ, V) . diam(V)→0

Letting diam(U) → 0, γ(φ, U) → 0 and lim sup P (φ, U) ≤ lim inf P (φ, V) .

diam(U)→0

diam(V)→0

We are done. ⊓ ⊔ In cases where confusion may arise we write the topological pressure P (φ) as PT (φ). n−1 2.9. Lemma. Let Sn φ(x) = k=0 φ(T k x). Then PT n (Sn φ) = nPT (φ) for n > 0 .

Proof. Let V = U ∨ · · · ∨ T −n+1 U. Then Wm (V) and Wmn (U) are in oneto-one correspondence; for U = Ui0 Ui1 · · · Uimn−1 let V = Vi0 · · · Vim−1 where Vik = Uikn ∩ T −1 Uikn+1 ∩ · · · ∩ T −n+1 Uikn+n−1 . One sees that X(U) = X(V) Tn T and Smn (Sn φ)(V). Thus one gets φ(U) = Sm n

T T Zmn (φ, U) = Zm (Sn φ, V)

and nPT (φ, U) = PT n (Sn φ, V) .

As diam(U) → 0, diam(V) → 0 and so nPT (φ) = PT n (Sn φ). ⊓ ⊔ We now come to our first interesting result about the pressure P (φ). 2.10. Theorem. Let T : X → X be a continuous map on a compact metric space and φ ∈ C (X). Then  hµ (T ) + φ dµ ≤ PT (φ), for any µ ∈ MT (X). We will first need a couple of lemmas. 2.11. Lemma. Suppose D is a Borel partition of X such that each x ∈ X is in the closures of at most M members of D. Then  hµ (T, D) + φ dµ ≤ PT (φ) + log M .

B. Pressure

35

Proof. Let U be a finite open cover of X each member of which intersects at most M members of D. Let Γm ⊂
Wm (U) cover X. For each B ∈ Dm = D ∨ · · · ∨ T −m+1 D pick xB ∈ B with B Sm φ dµ ≤ µ(B) Sm φ(xB ). Now     1 hµ (T, D) + φ dµ ≤ Hµ (Dm ) + Sm φ dµ m 1  ≤ µ(B)(− log µ(B) + Sm φ(xB )) m B  1 log exp(Sm φ(xB )) ≤ m B

by Lemma 1.1. For each xB pick UB ∈ Γm with xB ∈ X(UB ). This map B → UB is at most M m to one. As Sm φ(xB ) ≤ Sm φ(UB ), one has   1 hµ (T, D) + φ dµ ≤ log M m exp(Sm φ(U)) m U∈Γm

≤ log M +

1 log Zm (φ, U) . m

Letting m → ∞ and then diam(U) → 0, we obtain the desired inequality. ⊓ ⊔ 2.12. Lemma. Let A be a finite open cover of X. For each n > 0 there is a Borel partition Dn of X so that (a) D lies inside some member of T −k A for each D ∈ Dn and k = 0, . . . , n−1, (b) at most n|A| sets in Dn can have a point in all their closures. Proof. Let A = {A1 , . . . , Am } and g1 , . . . , gm be a partition of unity subordinate to A. Then G = (g1 , . . . , gm ) : X → sm−1 ⊂ Rm where sm−1 is an m − 1 dimensional simplex. Now U = {U1 , . . . , Um } is an open cover where Ui = {x ∈ sm−1 : xi > 0} and G−1 Ui ⊂ Ai . Since (sm−1 )n is nm − n dimensional, there is a Borel partition D∗n of (sm−1 )n so that (a’) each member of D∗n lies in some Ui1 × · · · × Uin , and (b’) at most nm members of D∗n can have a common point in all their closures. Then Dn = L−1 D∗n works where L = (G, G ◦ T, . . . , G ◦ T n−1 ) : X → (sm−1 )n .

⊓ ⊔

Proof of 2.10. Let C be a Borel partition and ε > 0. By Lemma 2.3 find an open cover A so that Hµ (C|D) < ε whenever D is a partition every member of which is contained in some member of A. Fix n > 0, let E = C ∨ · · · ∨ T −n+1 C and Dn as in Lemma 2.12. Then (see the proof of 2.6)

36

2 General Thermodynamic Formalism

hµ (T, C) +



   1 n φ dµ ≤ hµ (T , E) + Sn φ dµ n    1 1 n ≤ hµ (T , Dn ) + Sn φ dµ + Hµ (E|Dn ) n n 1 1 ≤ (PT n (Sn φ) + log(n|A|)) + Hµ (E|Dn ) n n

by Lemma 2.11. Now Hµ (E|Dn ) ≤

n−1 

Hµ (T −k C|Dn ) .

k=0

Since Dn refines T −k A for each k, one has Hµ (T −k C|Dn ) < ε (since µ is T -invariant, T −k A bears the same relation to T −k C as A to C). Hence, using 2.9,  1 hµ (T, C) + φ dµ ≤ PT (φ) + log(n|A|) + ε . n

Now let n → ∞ and then ε → 0.

⊓ ⊔

2.13. Proposition. Let T1 : X1 → X1 , T2 : X2 → X2 be continuous maps on compact metric spaces, π : X1 → X2 continuous and onto satisfying π ◦ T1 = T2 ◦ π. Then PT2 (φ) ≤ PT1 (φ ◦ π) for φ ∈ C (X2 ). Proof. For U an open cover of X2 one sees that PT2 (φ, U) = PT1 (φ ◦ π, π −1 U) . As in the proof of 2.8 PT1 (φ ◦ π, π −1 U) ≤ PT1 (φ ◦ π) + γ(φ ◦ π, π −1 U) . But γ(φ ◦ π, π −1 U) = γ(φ, U) → 0 as diam(U) → 0. Hence, letting diam(U) → ⊔ 0 we get PT2 (φ) ≤ PT1 (φ ◦ π). ⊓

C. Variational Principle  Let U be a finite opencover of X. We say that Γ ⊂ W ∗ (U) = m>0 Wm (U) covers K ⊂ X if K ⊂ U∈Γ X(U). For λ > 0 and Γ ⊂ W ∗ (U) define Z(Γ, λ) =



U∈Γ

λm(U) exp(Sm(U) φ(U)).

C. Variational Principle

37

2.14. Lemma. Let P = P (φ, U) and λ > 0. Suppose that Z(Γ, λ) < 1 for some Γ covering X. Then λ ≤ e−P . M Proof. As X is compact we may take Γ finite and Γ ⊂ m=1 Wm (U). Then n Z(Γ ≤ Z(Γ, λ)n where Γ n = {U1 U2 · · · Un : Ui ∈ Γ }. Letting Γ ∗ = ∞ , λ) n n=1 Γ , one has ∞  Z(Γ n , λ) < ∞ . Z(Γ ∗ , λ) = n=1

Fix N and consider x ∈ X. Since Γ covers X, one can find U = U1 U2 · · · Un ∈ Γ ∗ with

(a) x ∈ X(U), and (b) N ≤ m(U) < N + M .

Let U∗ be the first N symbols of U. Then SN φ(U∗ ) ≤ Sm(U) φ(U) + M φ . For Γ N the set of U∗ so obtained,   λN exp SN φ(U∗ ) ≤ max 1, λ−M

eM φ Z(Γ ∗ , λ),

ΓN

or λN ZN (φ, U) ≤ constant. It follows that λ ≤ e−P . ⊓ ⊔ Let δx be the unit-measure concentrated on the point x. Define

and

δx,n = n−1 (δx + δT x + · · · + δT n−1 x )

V (x) = {µ ∈ M (X) : δx,nk → µ for some nk → ∞} .

V (x) = ∅ as M (X) is a compact metric space. Now T ∗ δx,n = δT x,n and for f ∈ C (X), |T ∗ δx,n (f )−δx,n (f )| = n−1 |f (T n x)−f (x)| ≤ 2n−1 f . This shows V (x) ⊂ MT (X). Let E be a finite set, a = (a0 , . . . , ak−1 ) ∈ E k . One defines the distribution µa on E by

and

µa (e) = k −1 (number of j with aj = e)  H(a) = − µa (e) log µa (e) . e∈E

2.15. Lemma. Let x ∈ X, µ ∈ V (x), U a finite open cover of X and ε > 0. There are m and arbitrarily large N for which one can find U ∈ WN (U) satisfying the following (a) x ∈ X(U),
(b) SN φ(U) ≤ N ( φdµ + γ(U) + ε),

38

2 General Thermodynamic Formalism

(c) U contains a subword of length km ≥ N − m which, when viewed as a = a0 , . . . , ak−1 ∈ (Um )k satisfies 1 H(a) ≤ hµ (T ) + ε. m Proof. Let U = {U1 , . . . , Uq }. Recall that γ(U) = sup{|φ(y) − φ(z)| : y, z ∈ Ui for some i} . Pick a Borel partition C = {C1 , . . . , Cq } with C i ⊂ Ui . There is an m so that 1 ε ε Hµ (C ∨ · · · ∨ T −m+1 C) ≤ hµ (T, C) + ≤ hµ (T ) + · m 2 2 Let δx,nj → µ. For n′ > n one has δx,n′ =

n′ − n n δx,n + δT n x,n′ −n . ′ n n′

If we replaced nk by the nearest multiple of m, this formula shows that µ would still be the limit. Thus we assume nj = mkj . Let D1 , . . . , Dt be the nonempty members of C∨· · ·∨T −m+1 C and for each Di find a compact Ki ⊂ Di with µ(Di \Ki ) < β (β > 0 small). Each Di is contained in some member of U ∨ · · · ∨ T −m+1 U and one can find an open set Vi ⊃ Ki for which this is still true. Furthermore we may assume Vi ∩ Vj = ∅ for i = j. Now enlarge each Vi to a Borel set Vi∗ still contained in a member of U ∨ · · · ∨ T −m+1 U and so that {V1∗ , . . . , Vt∗ } is a Borel partition of X. Now fix nj = mkj . Let Mi be the number of s ∈ [0, nj ) with T s x ∈ Vi∗ and Mi,r the number of such s ≡ r (mod m). Define pi,r = Mi,r /kj and pi = Mi /nj =

1 m (pi,0

+ · · · + pi,m−1 ) . As δx,nj → µ, one has

lim inf pi ≥ µ(Ki ) ≥ µ(Di ) − β, j→∞

and lim supj→∞ pi ≤ µ(Ki ) + tβ ≤ µ(Di ) + tβ. For β small enough and j large enough one has       1 1 ε − − pi log pi ≤ µ(Di ) log µ(Di ) + m m 2 i i ≤ hµ (T ) + ε .

By the concavity of ϕ(x) = −x log x (see 1.17) ϕ(pi ) ≥

m−1  r=0

1 ϕ(pi,r ) m

C. Variational Principle

and so

39

m−1

1  ϕ(pi,r ) . m r=0 i i   For some r ∈ [0, m) one must have i ϕ(pi,r ) ≤ i ϕ(pi ) and so 

ϕ(pi ) ≥

1  ϕ(pi,r ) ≤ hµ (T ) + ε . m i

For N = nj + r with j large we form U = U0 U1 · · · UN −1 ∈ UN as follows. For s < r pick Us ∈ U containing T s x. For each Vi∗ we choose U0,i ∩ T −1 U1,i ∩ · · · ∩ T −m+1 Um−1,i ⊃ Vi∗ . For s > r we write s = r + mp + q with p ≥ 0, m > q ≥ 0, pick i with T r+mp x ∈ Vi∗ and let Us = Uq,i . Letting ap = U0,i U1,i · · · Um−1,i we have U = U0 · · · Ur−1 a0 a1 · · · akj −1 .

Now a = (a0 a1 · · · akj −1 ) has its distribution µa on Um given by the probabilities {pi,r }ti=1 and some zeros. So 1 1  H(a) = ϕ(pi,r ) ≤ hµ (T ) + ε. m m i 1  We  yet to check (b). Since
have
δx,nj → µ, for j large we will have N δx,N (φ)− φ dµ < ε or SN φ(x) ≤ N ( φ dµ + ε). As x ∈ X(U), SN φ(U) ≤ SN φ(x) + N γ(U). ⊓ ⊔

2.16. Lemma. Fix a finite set E and h ≥ 0. Let R(k, h) = {a ∈ E k : H(a) ≤ h}. Then 1 lim sup log |R(k, h)| ≤ h . k→∞ k

Proof. For any distribution ν on E and α ∈ (0, 1) consider Rk (ν) = {a ∈ E k : |µa (e) − ν(e)| < α ∀e ∈ E} . ∞ Let µ be the Bernoulli measure on Σ = i=0 E with the distribution µ(e) = (1 − α)ν(e) + α/|E| .

Each a ∈ Rk (ν) corresponds to a cylinder set Ca of Σ. Since each e ∈ E occurs in a at most k(ν(e) + α) times,  µ(Ca ) ≥ µ(e)k(ν(e)+α) . e

As the Ca are disjoint and have total measure 1,

40

2 General Thermodynamic Formalism

1 ≥ |Rk (ν)|

or



µ(e)k(ν(e)+α) ,

e

 1 log |Rk (ν)| ≤ −(ν(e) + α) log µ(e) k e  ≤ H(µ) + 3α| log µ(e)| . e

As µ(e) ≥ α/|E|, we get 1 log |Rk (ν)| ≤ H(µ) + 3α|E|(log |E| − log α) . k When α → 0, the second term on the right approaches 0 and H(µ) → H(ν) uniformly in ν. Hence, for any ε > 0 one can find α small enough that 1 log |Rk (ν)| ≤ H(µ) + ε, k for all k and ν. Once α is so chosen, let N be a finite set of distributions on E so that (a) H(ν) ≤ h for ν ∈ N , and (b) if H(ν ′ ) ≤ h then for some ν ∈ N one has |ν ′ (e) − ν(e)| < α Then R(k, h) ⊂



ν∈N

and

for all e.

Rk (ν), 1 1 log |R(k, h)| ≤ log |N | + h + ε k k 1 lim sup log |R(k, h)| ≤ h + ε. k→∞ k

Now let ε → 0. ⊓ ⊔ 2.17. Variational Principle. Let T : X → X be a continuous map on a compact metric space and φ ∈ C (X). Then    PT (φ) = sup hµ (T ) + φ dµ µ

where µ runs over MT (X). Proof. Let U be a finite cover of X and ǫ > 0. For each m > 0 let Xm be the set of points  x ∈ X for which 2.15 holds with this m and some µ ∈ V (x). By 2.15 X = m Xm since V (x) = ∅. For u ∈ R
let Ym (u) be the set of x ∈ Xm for which 2.15 holds for some µ ∈ V (x) with φdµ ∈ [u − ε, u + ε]. Set

C. Variational Principle

41

   c = sup hµ (T ) + φ dµ . µ

For x ∈ Ym (u) the µ satisfies hµ (T ) ≤ c − u + ε. The a ∈ (Um )k appearing in 2.15 (c) for x ∈ Ym (u) lie in R(k, m(c − u + 2ε), Um ). The number of possibilities for U for any fixed N = km is hence at most b(N ) = |E|m |R(k, m(c − u + 2ε), Um )| . By 2.16

1 log b(N ) ≤ c − u + 2ε . N Let Γ = Γm,u be the collection of all U showing up in the present situation for some N greater than a fixed N0 . Then Γ covers Ym (u) and lim sup N →∞

Z(Γ, λ) ≤

∞ 

λN b(N ) exp(N (u + 2ε + γ(U))) .

N =N0

For large enough N0 , b(N ) ≤ exp(N (c − u + 3ε)) and Z(Γ, λ) ≤ ≤

∞ 

N =N0 ∞ 

λN exp(N (c + 5ε + γ(U))) . βN =

N =N0

β N0 , 1−β

where β = λ exp(c + 5ε + γ(U)) < 1. We have seen that for λ < exp(−(c + 5ε + γ(U))) any Ym (u) ∞can be covered by Γ ⊂ W ∗ (U) with Z(Γ, λ) as small as desired. As X = m=1 Xm and Xm = Ym (u1 ) ∪ · · · ∪ Ym (ur ) where u1 , . . . , ur are ε-dense in [−φ, φ], taking unions of such Γ ′ s we obtain a Γ covering X with Z(Γ, λ) < 1. By Lemma 2.14, λ ≤ e−P (φ,U) or P (φ, U) ≤ c + 5ε + γ(U) . As ε was arbitrary, P (φ, U) ≤ c + γ(U). Finally P (φ) ≤

diam(U)→0

≤

diam(U)→0

lim

P (φ, U)

lim

(c + γ(U)) = c .

The inequality c ≤ P (φ) follows from Theorem 2.10.

⊓ ⊔

2.18. Corollary. Suppose {Xα }α∈Λ is a family of compact subsets of X and T Xα ⊂ Xα for each α. Then PT (φ) = sup PT |Xα (φ|Xα ) . α

42

2 General Thermodynamic Formalism

Proof. If µ ∈ MT (Xα ), then µ ∈ MT (X) and  PT (φ) ≥ hµ (T ) + φ dµ . Hence PT (φ) ≥

sup µ∈MT (Xα )



hµ (T ) +



φ dµ



= PT |Xα (φ|Xα ) .

If x ∈ Xα , then V (x) ⊂ MT (Xα ) and so     c′ = sup hµ (T ) + φ dµ : µ ∈ V (x) x∈X

≤ sup PT |Xα (φ|Xα ) . α

In the proof of 2.17 what was actually used about the number c was c ≥
hµ (T ) + φ dµ for µ ∈ V (x). So c′ would work just as well there to yield PT (φ) ≤ c′ . ⊓ ⊔

D. Equilibrium States
If µ ∈ MT (X) satisfies hµ (T )+ φ dµ = PT (φ), then µ is called an equilibrium state for φ (w.r.t. T ). The Gibbs state µφ of φ ∈ FA in Chapter 1 was shown to be the unique equilibrium state for such a φ. 2.19. Proposition. Suppose that for some ε > 0 one has hµ (T, D) = hµ (T ) whenever µ ∈ MT (X) and diam(D) < ε. Then every φ ∈ C (X) has an equilibrium state. Proof. We show
that µ → hµ (T ) is upper semi-continuous on MT (X). Then µ → hµ (T ) + φdµ will be also, and the proposition follows from 2.17 and the fact that an u.s.c. function on a compact space assumes its supremum. Fixing µ ∈ MT (X), α > 0, and D = {D1 , . . . , Dn } with diam(D) < ε, one 1 Hµ (D ∨ · · · ∨ T −m+1 D) ≤ hµ (T ) + α for some m. Let β > 0 and pick a has m m−1 compact set Ki0 ,...,im−1 ⊂ k=0 T −k Dik with   ! " −k µ T Dik Ki0 ,...,im−1 < β . k

m−1 

Then Di ⊃ Li = j=0 {T j Ki0 ,...,im−1 : ij = i}. As the Li are disjoint compact sets, one can find a partition D′ = {D1′ , . . . , Dn′ } with diam(D′ ) < ε and Li ⊂ int(Di′ ). One then has   ! −k ′ Ki0 ,...,im−1 ⊂ int T Dik . k

References

43

If ν is close to µ in the weak topology, one will have   ! −k ′ T Dik ≥ µ(Ki0 ,...,im−1 ) − β ν k

  −k ′    Dik − µ k T −k Dik  ≤ 2βnm . For β small enough, this and ν kT implies 1 Hν (D′ ∨ · · · ∨ T −m+1 D′ ) m 1 ≤ Hµ (D ∨ · · · ∨ T −m+1 D) + α ≤ hµ (T ) + 2α . ⊓ ⊔ m

hν (T ) = hν (T, D′ ) ≤

2.20. Corollary. If T is expansive, every φ ∈ C (X) has an equilibrium state. Proof. Recall 2.5.

⊓ ⊔

One notices that the condition in 2.19 has nothing to do with φ. Taking φ = 0, one defines the topological entropy of T by h(T ) = PT (0) . The motivation for this chapter comes from two places: the theory of Gibbs states from statistical mechanics and topological entropy from topological dynamics (see references). Conditions on φ become important for the uniqueness of equilibrium state and then only after stringent conditions have been placed on the homeomorphism T . The Axiom A diffeomorphisms will be close enough to the subshifts σ : ΣA → ΣA so that one can prove uniqueness statements.

References The definition of hµ (T ) is due to Kolmogorov and Sinai (see [2]). For expansive T Ruelle [15] defined PT (φ) and proved Theorems 2.10, 2.17 and 2.20. For general T the definition and results are due to Walters [16]. In the transition from ΣA to a general compact metric space X, most of the work has to do with the more complicated topology of X. Walters’ paper is therefore closely related to earlier work on the topological entropy h(T ), i.e., the case φ = 0. The definition of h(T ) was made by Adler, Konheim and McAndrew [1]. The theorems for this case are due to Goodwyn [10] (Theorem 2.10), Dinaburg [6] (X finite dimensional, 2.17), Goodman [8] (general X, 2.17), and Goodman [9] (2.20). For subshifts these results were proved earlier by Parry [14]. The proofs we have given in these notes are adaptations of [4]. Gureviˇc [11] gives a T where φ = 0 has no equilibrium states and Misiurewicz [13] gives such a T which is a diffeomorphism. The condition in 2.19 is satisfied by a class of maps which includes all affine maps on Lie groups [3] and Misiurewicz [13] showed that equilibrium states exist under a somewhat weaker condition.

44

2 General Thermodynamic Formalism Ruelle [15] showed that for expansive T a Baire set of φ have unique equilibrium states. Goodman [9] gives a minimal subshift where φ = 0 has more than one equilibrium state. I believe mathematical physicists know of specific φ on Σn which do not have unique equilibrium states; in statistical mechanics one looks at Zm actions instead of just homeomorphisms and gets nonuniqueness for m ≥ 2 even with simple φ’s. Uniqueness was proved in [5] for certain φ when T satisfies expansiveness and a very restrictive condition called specification; this result has been carried over to flows by Franco-Sanchez [7]. Finally we mention a very interesting result in a different direction. Let T : M → M be a continuous map on a compact manifold and λ an eigenvalue of the map T∗ : H1 (M ) → H1 (M ) on one-dimensional homology. Then Manning [12] showed h(T ) ≥ log |λ|. It is conceivable that this inequality is true for λ for any Hk (M ) (not just k = 1) provided T is a diffeomorphism.

1. R. L. Adler, A. G. Konheim and M. H. McAndrew: Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. 2. P. Billingsley: Ergodic Theory and Information, Wiley (1965). (1 ) 3. R. Bowen: Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972), 323331. 4. R. Bowen: Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125-136. 5. R. Bowen: Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193-202. 6. E. I. Dinaburg: On the relations among various entropy characteristics of dynamical systems, Math. USSR Izvestia 5 (1971), 337-378. 7. E. Franco-Sanchez: Flows with unique equilibrium states, Ph.D. thesis. Univ. of California, Berkeley, 1974. 8. T. N. T. Goodman: Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971), 176-180. 9. T. N. T. Goodman: Maximal measures for expansive homeomorphisms, J. London Math. Soc. (2) 5 (1972), 439-444. 10. L. W. Goodwyn: Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679-688. 11. B. M. Gureviˇc: Topological entropy of denumerable Markov chains, Soviet Math. Dokl. 10 (1969), 911-915. 12. A. Manning: Topological entropy and the first homology group, in “Dynamical Systems” - Warwick 1974 (Proc. Symp. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974), pp. 185-190. Lect. Notes in Math. 468, Springer, Berlin, 1975. 13. M. Misiurewicz: Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys. 21 (1973), 903–910. 14. W. Parry: Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55-66. 15. D. Ruelle: Statistical mechanics on a compact set with Zν action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 185 (1973), 237-251. 16. P. Walters: A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1975), no. 4, 937-971. 1

Reprint: Robert E. Krieger Publishing Co., Huntington, N.Y., 1978 (note of the editor).

3 Axiom a Diffeomorphisms

A. Definition We now suppose that f : M → M is a diffeomorphism of a compact C ∞ Riemannian manifold M . Then  the derivative of f can be considered a map df : T M → T M where T M = x∈M Tx M is the tangent bundle of M and dfx : Tx M → Tf (x) M . Definition. A closed subset Λ ⊂ M is hyperbolic if f (Λ) = Λ and each tangent space Tx M with x ∈ Λ can be written as a direct sum Tx M = Exu ⊕ Exs of subspaces so that (a) Df (Exs ) = Efs (x) , Df (Exu ) = Efu(x) ; (b) there exist constants c > 0 and λ ∈ (0, 1) so that Df n (v) ≤ cλn v and

Df −n (v) ≤ cλn v

(c) Exs , Exu vary continuously with x.

when v ∈ Exs , n ≥ 0 when v ∈ Exu , n ≥ 0;

 Remark. Condition (c) actually follows from the others. Eu = x∈Λ Exu  and E s = x∈Λ Exs are continuous subbundles of TΛ M = x∈Λ Tx M and TΛ M = E u ⊕ E s . A point x ∈ M is non-wandering if  U∩ f n U = ∅, n>0

for every neighborhood U of x. The set Ω = Ω(f ) of all non-wandering points is seen to be closed and f -invariant. A point x is periodic if f n x = x for some n > 0; clearly such an x is in Ω(f ).

46

3 Axiom a Diffeomorphisms

Definition. f satisfies Axiom A if Ω(f ) is hyperbolic and Ω(f ) = {x : x is periodic}. This definition is due to Smale [14]. A type of f studied extensively by Russian mathematicians is the Anosov diffeomorphism: f is Anosov if M is hyperbolic [2]. We shall see a little later that such diffeomorphisms always satisfy Axiom A. Right now we mention that it is unknown whether Ω(f ) = M for every Anosov f . The reader should study the examples in [14]. The Riemannian metric on M is used to state condition (b) in the definition of hyperbolic set. The truth of this condition does not depend upon which metric is used although the constants c and λ do. A metric is adapted (to an Axiom A f ) if Ω(f ) is hyperbolic with respect to it with c = 1. 3.1. Lemma. Every Axiom A diffeomorphism has an adapted metric. Proof. This lemma is due to Mather. See [8] for a proof. ⊓ ⊔ We will always use an adapted metric. This will keep various estimates a bit simpler. For x ∈ M define W s (x) = {y ∈ M Wεs (x) = {y ∈ M W u (x) = {y ∈ M Wεu (x) = {y ∈ M

: : : :

d(f n x, f n y) → 0 as n → ∞} d(f n x, f n y) ≤ ε for all n ≥ 0} d(f −n x, f −n y) → 0 as n → ∞} d(f −n x, f −n y) ≤ ε for all n ≥ 0} .

The following stable manifold theorem is the main analytic fact behind the behavior of Axiom A diffeomorphisms. 3.2. Theorem. Let Λ be a hyperbolic set for a C r diffeomorphism f . For small ε > 0 (a) Wεs (x), Wεu (x) are C r disks for x ∈ Λ with Tx Wεs (x) = Exs , Tx Wεu (x) = Exu ; (b) d(f n x, f n y) ≤ λn d(x, y) for y ∈ Wεs (x), n ≥ 0, and d(f −n x, f −n y) ≤ λn d(x, y) for y ∈ Wεu (x), n ≥ 0; (c) Wεs (x), Wεu (x) vary continuously with x (in C r topology). Proof. See Hirsch and Pugh [8]. ⊓ ⊔ One consequence of 3.2 is that Wεs (x) ⊂ W s (x) for x ∈ Λ. One then sees that  f −n Wεs (f n x) W s (x) = n≥0

for x ∈ Λ. Similarly

W u (x) =



n≥0

f n Wεu (f −n x) .

B. Spectral Decomposition

47

3.3. Canonical Coordinates. Suppose f satisfies Axiom A. For any small ε > 0 there is a δ > 0 so that Wεs (x) ∩ Wεu (y) consists of a single point [x, y] whenever x, y ∈ Ω(f ) and d(x, y) ≤ δ. Furthermore [x, y] ∈ Ω(f ) and [·, ·] : {(x, y) ∈ Ω(f ) × Ω(f ) : d(x, y) ≤ δ} −→ Ω(f ) is continuous. Proof. See Smale [14]. The first statement follows because the intersection Wεs (x) ∩ Wεu (x) = {x} is transversal and such intersections are preserved under small perturbation. To get [x, y] ∈ Ω(f ) uses that the periodic points are dense in Ω(f ). In the Anosov case one of course has [x, y] ∈ M and thus canonical coordinates on M (instead of Ω(f )) without any assumption on periodic points. ⊓ ⊔ 3.4. Lemma. Let Λ be a hyperbolic set. Then there is an ε > 0 so that Λ is expansive in M , i.e., if x ∈ Λ and y ∈ M with y = x, then d(f k x, f k y) > ε for some k ∈ Z . Proof. Otherwise y ∈ Wεs (x) ∩ Wεu (x). x is also in this intersection and so y = x by 3.3. ⊓ ⊔

B. Spectral Decomposition From now on f will always be an Axiom A diffeomorphism. 3.5. Spectral Decomposition. One can write Ω(f ) = Ω1 ∪ Ω2 ∪ · · · ∪ Ωs where the Ωi are pairwise disjoint closed sets with (a) f (Ωi ) = Ωi and f |Ωi is topologically transitive; (b) Ωi = X1,i ∪· · ·∪Xni ,i with the Xj,i ’s pairwise disjoint closed sets, f (Xj,i ) = Xj+1,i (Xnj +1,i = X1,i ) and f ni |Xj,i topologically mixing. Proof ([14, 4]). For p ∈ Ω periodic let Xp = W u (p) ∩ Ω. Let δ be as in 3.3. We claim that Xp = Bδ (Xp ) = {y ∈ Ω : d(y, Xp ) < δ} . As periodic points are dense in Ω, it is enough to see that a periodic q ∈ Bδ (Xp ) is in Xp . Pick x ∈ W u (p) ∩ Ω with d(x, q) < δ and consider x′ = [x, q] ∈ W u (p) ∩ W s (q) ∩ Ω. Letting f m p = p and f n q = q, one has and So q ∈ Xp .

f kmn x′ ∈ f kmn W u (p) = W u (f kmn p) = W u (p) d(f kmn x′ , q) = d(f kmn x′ , f kmn q) → 0

as k → ∞ .

48

3 Axiom a Diffeomorphisms

Notice that f Xp = Xf (p) since f W u (p) = W u (f (p)). If q ∈ Xp as above, then Wδu (q) ⊂ Xp and  W u (q) = f kmn Wδu (q) k≥0

⊂



f kmn Xp = Xp .

k≥0

(Note that y ∈ W u (q) iff f −kmn y → q as k → ∞.) It follows that Xq ⊂ Xp . If x′ is as above, then f kmn x′ ∈ Xq for large k as Xq = Bδ (Xq ) is open in Ω. As f imn Xq = Xf imn q = Xq , one has f jmn x′ ∈ Xq for all j and p = lim f jmn x′ ∈ X q = Xq . j→−∞

The above argument with the roles of p and q reversed gives Xp ⊂ Xq . In summary, if q ∈ Xp with p, q periodic, Xp = Xq . Now any two Xp , Xq are either disjoint or equal. For if Xq ∩ Xq = ∅, then this intersection is open in Ω and hence contains a periodic point r; then Xp = Xr = Xq . Now   Ω= Bδ (Xp ) = Xp , p periodic

p

and so by compactness (the Xp are open) let Ω = Xp1 ∪ · · · ∪ Xpt with the Xpj ’s pairwise disjoint. Then f (Xpj ) = Xf pj intersects and hence equals some Xpi . So f permutes the Xpj ’s and the Ωi are just the union of the Xpj ’s in the various cycles of the permutation. The transitivity in (a) is implied by the mixing in (b). We finish by showing f N : Xr → Xr is mixing whenever r is periodic and N positive with f N Xr = Xr . Suppose U, V are nonempty subsets of Xr open in Xr (i.e., in Ω). Pick periodic points p ∈ U and q ∈ V , say f m p = p, f n q = q. For each 0 ≤ j < mn with f j p ∈ Xr one can find a point x′j as in the beginning of this proof so that x′j ∈ f j U and f kmn x′j ∈ V for large k . Writing tN = kmn + j, 0 ≤ j ≤ mn, we have f j p = f tN p ∈ Xr and f kmn x′j = f tN (f −jN x′j ) ∈ f tN U ∩ V provided k is large. Then f tN U ∩ V = ∅ for large t and f N |Xr is topologically mixing. ⊓ ⊔ The Ωi in the spectral decomposition of Ω(f ) are called the basic sets of f . Notice that if g = f n and n is a multiple of every ni , then the basic sets

B. Spectral Decomposition

49

of g are the Xj,i ’s and g|Xj,i is mixing. We will at times restrict our attention to mixing basic sets and recover the general case by considering f n . A sequence x = {xi }bi=a (a = −∞ or b = +∞ is permitted) of points in M is an α-pseudo-orbit if d(f xi , xi+1 ) < α for all i ∈ [a, b − 1) . A point x ∈ M β-shadows x if d(f i x, xi ) ≤ β for all i ∈ [a, b] . 3.6. Proposition. For every β > 0 there is an α > 0 so that every α-pseudoorbit {xi }bi=a in Ω (i.e., every xi ∈ Ω) is β-shadowed by a point x ∈ Ω. Proof. Let ε > 0 be a small number to be determined later and choose δ ∈ (0, ε) as in 3.3, i.e., Wεs (x)∩Wεu (y)∩Ω = ∅ whenever x, y ∈ Ω with d(x, y) ≤ δ. Pick M so large that λM ε < δ/2 and then α > 0 so that: if {yi }M i=0 is an α−pseudo-orbit in Ω, then j d(f y0 , yj ) < δ/2 for all j ∈ [0, M ] . M Consider first an α-pseudo-orbit {xi }ri=0 with r > 0. Define x′kM recursively ′ for k ∈ [0, r] by x0 = x0 and

x′(k+1)M = Wεu (f M x′kM ) ∩ Wεs (x(k+1)M ) ∈ Ω . This makes sense: d(f M x′kM , f M xkM ) ≤ λM ε < δ/2 and d(f M xkM , x(k+1)M ) < δ/2 by the choice of α; so d(f M x′kM , x(k+1)M ) < δ and x′(k+1)M exists. Now set x = f −rM x′rM . For i ∈ [0, rM ] pick s with i ∈ [sM, (s + 1)M ), then d(f i x, f i−sM x′sM ) ≤ ≤

r 

t=s+1 r 

t=s+1

d(f i−tM x′tM , f i−tM +M x(t−1)M ) ε λtM −i ≤

ελ 1−λ

where we used x′tM ∈ Wεu (f M x′(t−1)M ). Since x′sM ∈ Wεs (xsM ) d(f i−sM x′sM , f i−sM xsM ) ≤ ε . By the choice of α one has d(f i−sM xsM , xi ) < δ/2 . By the triangle inequality d(f i x, xi ) ≤

δ ελ +ε+ · 1−λ 2

50

3 Axiom a Diffeomorphisms

For small ε this is less that the given β. rM Now any α-pseudo-orbit {xi }ni=0 in Ω extends to {xi }i=0 when rM ≥ n by setting xi = f i−n xn for i ∈ (n, rM ]. An x ∈ Ω shadowing this extended pseudo-orbit will shadow the original one. If {xi }bi=a is a finite α-pseudo orbit, −a then so is {xj+a }b−a x shadowing the j=0 and x shadowing this one yields f original. Thus the proposition holds for finite pseudo-orbits. If {xi }+∞ i=−∞ is an α-pseudo-orbit in Ω, then find x(m) ∈ Ω β-shadowing {xi }m and let x i=−m +∞ (m) ⊔ be a limit point of the sequence x . Then x ∈ Ω β-shadows {xi }i=−∞ . ⊓ 3.7. Corollary. Given any β > 0 there is an α > 0 so that the following holds: if x ∈ Ω and d(f n x, x) < α, then there is an x′ ∈ Ω with f n x′ = x′ and d(f k x, f k x′ ) ≤ β for all k ∈ [0, n] . Proof. Let xi = f k x for i ≡ k (mod n), k ∈ [0, n). Then {xi }∞ i=−∞ is an α-pseudo-orbit. Let x′ ∈ Ω β-shadow it. Then d(f i x′ , f i f n x′ ) ≤ d(f i x′ , xi ) + d(xi , f i+n x′ ) ≤ 2β and by expansiveness (Lemma 3.4) f n x′ = x′ . ⊓ ⊔ 3.8. Anosov’s Closing Lemma. If f is an Anosov diffeomorphism, then f satisfies Axiom A. Proof. We must show that the periodic points are dense in Ω(f ). We have been assuming f satisfies Axiom A; however 3.3 is true also for Anosov diffeomorphisms and so then is 3.6 and 3.7, using M in place of Ω(f ). If y is a non-wandering point for an Anosov f , then for any γ one can find x with d(x, y) < γ and d(f n x, x) < γ for some n. The periodic points x′ constructed in 3.7 for such x converge to y. ⊓ ⊔ 3.9. Fundamental Neighborhood. Let f satisfy Axiom A. There is a neighborhood U of Ω(f ) so that ! f n U = Ω(f ) . n∈Z

Proof. Let β be small and α as in 3.6. Pick γ < α/2 so that ∀x, y ∈ M , d(x, y) < γ

implies

d(f x, f y) < α/2 .  Let U = {y ∈ M : d(y, Ω) < γ}. If y ∈ n∈Z f n U , pick xi ∈ Ω with d(f i y, xi ) < γ. Then d(f i y, f i x) < β + γ

for all i .

For small β and γ this implies y = x ∈ Ω by 3.4. ⊓ ⊔ For Ωj a basic set of an Axiom A diffeomorphism one let and

W s (Ωj ) = {x ∈ M : d(f n x, Ωj ) → 0 as n → ∞}

W u (Ωj ) = {x ∈ M : d(f −n x, Ωj ) → 0 as n → ∞} .

C. Markov Partitions

51

Using the definition of non-wandering sets it is easy to check that f n x → Ω and f −n x → Ω as n → ∞. As Ω = Ω1 ∪ · · · ∪ Ωs is a disjoint union of closed invariant sets one then sees that M=

s 

W s (Ωj ) =

j=1

s 

W u (Ωj )

j=1

and that there are disjoint unions.   3.10. Proposition. W s (Ωj ) = x∈Ωj W s (x) and W u (Ωj ) = x∈Ωj W u (x). For ε > 0 there is a neighborhood Uj of Ωj so that  ! Wεs (x) f −k Uj ⊂ Wεs (Ωj ) = x∈Ωj

k≥0

and

!

k≥0

k

f Uj ⊂

Wεu (Ωj )

=



Wεu (x) .

x∈Ωj

Proof. Suppose f n y → Ωj as n → ∞; say d(f n y, Ωj ) < γ for all n ≥ N . Pick xn ∈ Ωj for n ≥ N with d(xn , f n y) ≤ γ; for n < N let xn = f n−N xN . The {xn }∞ n=−∞ is a pseudo-orbit in Ωj . Letting x ∈ Ωj shadow it, one gets f N y ∈ Wεs (f N x) ⊂ W s (f N x) (provided γ was small enough). Then y ∈ f −N W s (f N x) = W s (x). The reverse s inclusion, W (Ωj ) ⊃ x∈Ωj W s (x), is clear. The proof for W u (Ωj ) is similar and we have proved the second statement with Uj = {y ∈ M : d(y, Ωj ) < γ}. ⊓ ⊔

C. Markov Partitions A subset R ⊂ Ωs is called a rectangle if it has small diameter and [x, y] ∈ R whenever x, y ∈ R . R is called proper if R is closed and R = int(R) (int(R) is the interior of R as a subset of Ωs ). For x ∈ R, let W s (x, R) = Wεs (x) ∩ R

and W u (x, R) = Wεu (x) ∩ R

where ε is small and the diameter of R is small compared to ε. 3.11. Lemma. Let R be a closed rectangle. As a subset of Ωs , R has boundary ∂R = ∂ s R ∪ ∂ u R where

52

3 Axiom a Diffeomorphisms

∂ s R = {x ∈ R : x ∈ / int(W u (x, R))} ∂ u R = {x ∈ R : x ∈ / int(W s (x, R))} and the interiors of W u (x, R), W s (x, R) are as subsets of Wεu (x)∩Ω, Wεs (x)∩ Ω. Proof. If x ∈ int(R), then W u (x, R) = R∩(Wεu (x)∩Ω) is a neighborhood of x in Wεu (x)∩Ω since R is a neighborhood of x in Ω. Similarly x ∈ int(W u (x, R)). Suppose x ∈ int(W u (x, R)) and x ∈ int(W s (x, R)). For y ∈ Ωs near x the points [x, y] ∈ Wεs (x) ∩ Ω and [x, y] ∈ Wεu (x) ∩ Ω depend continuously on y. Hence for y ∈ Ωs close enough to x, [x, y] ∈ R and [y, x] ∈ R. Then y ′ = [[y, x], [x, y]] ∈ R ∩ Wεs (y) ∩ Wεu (y) and y ′ = y as Wεs (y) ∩ Wεu (y) = {y}. Thus x ∈ int(R). ⊓ ⊔ Definition. A Markov partition of Ωs is a finite covering R = {R1 , . . . , Rm } of Ωs by proper rectangles with (a) int(Ri ) ∩ int(Rj ) = ∅ for i = j, (b) f W u (x, Ri ) ⊃ W u (f x, Rj ) and f W s (x, Ri ) ⊂ W s (f x, Rj ) when x ∈ int(Ri ), f x ∈ int(Rj ). 3.12. Theorem. Let Ωs be a basic set for an Axiom A diffeomorphism f . Then Ωs has Markov partitions R of arbitrarily small diameter. Proof. Let β > 0 be very small and choose α > 0 small as in Proposition 3.6, i.e., every α-pseudo-orbit in Ωs is β-shadowed in Ωs . Choose γ < α/2 so that d(f x, f y) < α/2 when

d(x, y) < γ .

Let P = {p1 , . . . , pr } be a γ-dense subset of Ωs and   ∞  Σ(P ) = q ∈ P : d(f qj , qj+1 ) < α for all j . −∞

For each q ∈ Σ(P ) there is a unique θ(q) ∈ Ωs which β-shadows q; for each x ∈ Ωs there are q with x = θ(q). For q, q ′ ∈ Σ(P ) with q0 = q0′ we define q ∗ = [q, q ′ ] ∈ Σ(P ) by qj∗ =



qj qj′

for for

j≥0 j ≤ 0.

Then d(f j θ(q ∗ ), f j θ(q)) ≤ 2β for j ≥ 0 and d(f j θ(q ∗ ), f j θ(q)) ≤ 2β for j ≤ 0. s u So θ(q ∗ ) ∈ W2β (θ(q)) ∩ W2β (θ(q ′ )), i.e.,

C. Markov Partitions

53

θ[q, q ′ ] = [θ(q), θ(q ′ )] . We now see that Ts = {θ(q) : q ∈ Σ(P ), q0 = ps } is a rectangle. For x, y ∈ Ts we write x = θ(q), y = θ(q ′ ) with q0 = ps = q0′ . Then [x, y] = θ[q, q ′ ] ∈ Ts .

Suppose x = θ(q) with q0 = ps and q1 = pt . Consider y ∈ W s (x, Ts ), y = θ(q ′ ), q0 = ps . Then y = [x, y] = θ[q, q ′ ] and f y = θ(σ[q, q ′ ]) ∈ Tt as σ[q, q ′ ] has q ′ = pt in its zeroth position. Since f y ∈ Wεs (f x) (diam(Ts ) ≤ 2β is small compared to ε), f y ∈ Wεs (f x, Tt ). We have proved (i) f W s (x, Ts ) ⊂ W s (f x, Tt ). A similar proof shows f −1 W u (f x, Tt ) ⊂ W u (x, Ts ), i.e., (ii) f W u (x, Ts ) ⊃ W u (f x, Tt ). Each Ts is closed; this follows from the following lemma. 3.13. Lemma. θ : Σ(P ) → Ωs is continuous.

Proof. Otherwise there is a γ > 0 so that for every N one can find q N , q ′N ∈ ′ Σ(P ) with qj,N = qj, for all j ∈ [−N, N ] but d(θ(q N ), θ(q ′N )) ≥ γ. If xN = N ′ θ(q N ) , yN = θ(q ) one has d(f j xN , f j yN ) ≤ 2β

∀j ∈ [−N, N ] .

Taking subsequences we may assume xN → x and yN → y as N → ∞. Then d(f j x, f j y) ≤ 2β for all j and d(x, y) ≥ γ; this contradicts expansiveness of ⊔ f |Ωs . ⊓ Now T = {T1 , . . . , Tr } is a covering by rectangles and (i) and (ii) above are like the Markov condition (b). However the Tj ’s are likely to overlap and not be proper. For each x ∈ Ωs let

T(x) = {Tj ∈ T : x ∈ Tj } and T ∗ (x) = {Tk ∈ T : Tk ∩Tj = ∅ for some Tj ∈ T(x)}.  As T is a closed cover of Ωs , Z = Ωs \ j ∂Tj is an open dense subset of Ωs . In fact, using arguments similar to 3.11, one can show that Z ∗ = {x ∈ Ωs : Wεs (x) ∩ ∂ s Tk = ∅ and Wεu (x) ∩ ∂ u Tk = ∅ for all Tk ∈ T ∗ (x)}

is open and dense in Ωs . For Tj ∩ Tk = ∅, let 1 Tj,k = {x ∈ Tj : W u (x, Tj ) ∩ Tk = ∅, W s (x, Tj ) ∩ Tk = ∅} = Tj ∩ Tk

2 Tj,k = {x ∈ Tj : W u (x, Tj ) ∩ Tk = ∅, W s (x, Tj ) ∩ Tk = ∅}

3 = {x ∈ Tj : W u (x, Tj ) ∩ Tk = ∅, W s (x, Tj ) ∩ Tk = ∅} Tj,k

4 = {x ∈ Tj : W u (x, Tj ) ∩ Tk = ∅, W s (x, Tj ) ∩ Tk = ∅}. Tj,k

54

3 Axiom a Diffeomorphisms Ws

Wu 4 Tj,k

2 Tj,k

3 Tj,k

1 Tj,k

Tj Tk

If x, y ∈ Tj , then W s ([x, y], Tj ) = W s (x, Tj ) and W u ([x, y], Tj ) = n W u (y, Tj ); this implies Tj,k is a rectangle open in Ωs and each x ∈ Tj ∩ Z ∗ n lies in int(Tj,k ) for some n. For x ∈ Z ∗ define R(x) =

!

n n ) : x ∈ Tj , Tk ∩ Tj = ∅ and x ∈ Tj,k }. {int(Tj,k

Now R(x) is an open rectangle (x ∈ Z ∗ ). Suppose y ∈ R(x) ∩ Z ∗ . Since R(x) ⊂ T(x) and R(x) ∩ Tj = ∅ for Tj ∈ / T(x), one gets T(y) = T(x). For n Tj ∈ T(x) = T(y) and Tk ∩ Tj = ∅, y lies in the same Tj,k as x does since n ′ Tj,k ⊃ R(x); hence R(y) = R(x). If R(x) ∩ R(x ) = ∅ (x, x′ ∈ Z ∗ ), there is a y ∈ R(x) ∩ R(x′ ) ∩ Z ∗ ; then R(x) = R(y) = R(x′ ). As there are only finitely n many Tj,k ’s there are only finitely many distinct R(x)’s. Let R = {R(x) : x ∈ Z ∗ } = {R1 , . . . , Rm } . For x′ ∈ Z ∗ , R(x′ ) = R(x) or R(x′ ) ∩ R(x) = ∅; hence (R(x) \R(x)) ∩ Z ∗ = ∅. As Z ∗ is dense in Ωs , R(x) \R(x) has no interior (in Ωs ) and R(x) = int(R(x)). For R(x) = R(x′ )     int R(x) ∩ int R(x′ ) = R(x) ∩ R(x′ ) = ∅ .

To show that R is Markov we are left to verify condition (b). Suppose x, y ∈ Z ∗ ∩ f −1 Z ∗ , R(x) = R(y) and y ∈ Wεs (x). We will show R(f x) = R(f y). First T(f x) = T(f y). Otherwise assume f x ∈ Tj , f y ∈ / Tj . Let f x = θ(σq) with q1 = pj and q0 = ps . Then x = θ(q) ∈ Ts and by inclusion (i) above

C. Markov Partitions

55

f y ∈ f W s (x, Ts ) ⊂ W s (f x, Tj ) , contradicting f y ∈ / Tj . Now let f x, f y ∈ Tj and Tk ∩ Tj = ∅. We want to show n that f x, f y belong to the same Tj,k . As f y ∈ Wεs (f x) we have W s (f y, Tj ) = s W (f x, Tj ). We will derive a contradiction from W u (f y, Tj ) ∩ Tk = ∅ , f z ∈ W u (f x, Tj ) ∩ Tk . Recall that f x = θ(σq), q1 = pj , q0 = ps . Then by inclusion (ii) f z ∈ W u (f x, Tj ) ⊂ f W u (x, Ts )

or z ∈ W u (x, Ts ) .

Let f z = θ(σq ′ ); q1′ = pk and q0 = pt . Then z ∈ Tt and f W s (z, Tt ) ⊂ W s (f z, Tk ). Now Ts ∈ T(x) = T(y) and z ∈ Tt ∩ Ts = ∅. Now z ∈ W u (x, Ts ) ∩ Tt and so there is some z ′ ∈ W u (y, Ts ) ∩ Tt as x, y are n . Then in the same Ts,t z ′′ = [z, y] = [z, z ′ ] ∈ W s (z, Tt ) ∩ W u (y, Ts ), and f z ′′ = [f z, f y] ∈ W s (f z, Tk )∩W u (f y, Tj ) (using f z, f y ∈ Tj a rectangle), a contradiction. So R(f x) = R(f y). For small δ > 0 the sets ⎧ ⎧ ⎫ ⎫ ⎬ ⎬   ⎨   ⎨ Y1 = Wδs (z) : z ∈ Wδu (z) : z ∈ ∂ s Tj and Y2 = ∂ u Tj ⎩ ⎩ ⎭ ⎭ j

j

are closed and nowhere dense (like in the proof of 3.11). Now Z ∗ ⊃ Ωs \(Y1 ∪Y2 ) is open and dense. Furthermore if x ∈ / (Y1 ∪ Y2 ) ∩ f −1 (Y1 ∪ Y2 ) then x ∈ ∗ −1 ∗ s Z ∩ f Z and the set of y ∈ W (x, R(x)) with y ∈ Z ∗ ∩ f −1 Z ∗ is open and dense in W s (x, R(x) ) (as a subset of Wεs (x) ∩ Ω). By the previous paragraph R(f y) = R(f x) for such y; by continuity f W s (x, R(x) ) ⊂ R(f x) . As f W s (x, R(x) ) ⊂ Wεs (f x), f W s (x, R(x) ) ⊂ W s (f x, R(f x) ). If int(Ri ) ∩ f −1 int(Rj ) = ∅, then this open subset of Ωs contains some x satisfying the above conditions, Ri = R(x) and Rj = R(f x). For any x′ ∈ Ri ∩ f −1 Rj one has W s (x′ , Ri ) = {[x′ , y] : y ∈ W s (x, Ri )} and f W s (x′ , Ri ) = {[f x′ , f y] : y ∈ W s (x, Ri )} ⊂ {[f x′ , z] : z ∈ W s (f x, Rj )} ⊂ W s (f x′ , Rj ) .

This completes the proof of half of the Markov conditions (b). The other half is proved similarly and the proof is omitted. Alternatively one could apply the above to f −1 , noting that Wfu = Wfs−1 . ⊓ ⊔

56

3 Axiom a Diffeomorphisms

D. Symbolic Dynamics Throughout this section R = {R1 , . . . , Rm } will denote a Markov partition of a basic set Ωs . One defines the transition matrix A = A(R) by  1 if int(Ri ) ∩ f −1 int(Rj ) = ∅ Aij = 0 otherwise . 3.14. Lemma. Suppose x ∈ Ri , f x ∈ Rj , Aij = 1. Then f W s (x, Ri ) ⊂ W s (f x, Rj ) and f W u (x, Ri ) ⊃ W u (f x, Rj ). Proof. This is just the same as the last part of the proof of 3.12. ⊓ ⊔   Definition. ∂ s R = j ∂ s Rj and ∂ u R = j ∂ u Rj .

3.15. Proposition. f (∂ s R) ⊂ ∂ s R and f −1 (∂ u R) ⊂ ∂ u R.  Proof. The set j (int(Ri ) ∩ f −1 int(Rj )) is dense in Ri . For any x ∈ Ri one can therefore find some j and xn ∈ int(Ri ) ∩ f −1 int(Rj ) with limn→∞ xn = x. Then Aij = 1, x ∈ Rij and f x ∈ Rj . Hence f W u (x, Ri ) ⊃ W u (f x, Rj ). If fx ∈ / ∂ s R, then W u (f x, Rj ) is a neighborhood of f x in Wεs (f x) ∩ Ω and so u W (x, Ri ) is a neighborhood of x in Wεs (x) ∩ Ωs – that is x ∈ / ∂ s Ri . We have s s −1 u u shown f (∂ R) ⊂ ∂ R. One gets f (∂ R) ⊂ ∂ R by a similar argument or by applying the first argument to f −1 in place of f . ⊓ ⊔ 3.16. Lemma. Let D ⊂ Wδs (x) ∩ Ω and C ⊂ Wδu (x) ∩ Ω. Then the rectangle [C, D] is proper iff D = int(D) and C = int(C) as subsets of Wδs (x) ∩ Ω and Wδu (x) ∩ Ω respectively. Proof. This is like 3.11.

⊓ ⊔

Definition. Let R, S be two rectangles. S will be called a u-subrectangle of R if (a) S = ∅, S ⊂ R, S is proper, and (b) W u (y, S) = W u (y, R) for y ∈ S. 3.17. Lemma. Suppose S is a u-subrectangle of Ri and Aij = 1. Then f (S)∩ Rj is a u-subrectangle of Rj . Proof. Pick x ∈ Ri ∩ f −1 Rj and set D = W s (x, Ri ) ∩ S. Because S is a u-subrectangle (condition (b)) one has  W u (y, Ri ) = [W u (x, Ri ), D] . S= y∈D

As S is proper and nonempty, by 3.16 D = ∅ and D = int(D). Now

D. Symbolic Dynamics

f (S) ∩ Rj =



y∈D

57

(f W u (y, Ri ) ∩ Rj ) .

By 3.14, f (y) ∈  Rj and f W u (y, Ri ) ∩ Rj = W u (f y, Rj ). So f (S) ∩ Rj = y′ ∈f (D) W u (y ′ , Rj ) = [W u (f x, Rj ), f (D)]. Since Rj = [W u (f x, Rj ), W s (f x, Rj )] is proper, one has W u (f x, Rj ) proper. As f maps Wεs (x)∩Ω homeomorphically onto a neighborhood in Wεs (f x)∩Ω, f (D) = int(f (D)) and so f (S) ∩ Rj is proper by 3.16. f (S) ∩ Rj = ∅ as f (D) = ∅; if y ′′ ∈ f (S) ∩ Rj , then y ′′ ∈ W u (y ′ , Rj ) for some y ′ ∈ f (D) and W u (y ′′ , Rj ) = W u (y ′ , Rj ) ⊂ f (S) ∩ Rj . So f (S) ∩ Rj is a u-subrectangle of Rj . ⊓ ⊔  3.18. Theorem. For each a ∈ ΣA the set j∈Z f −j Raj consists of a single point, denoted π(a). The map π : ΣA → Ωs is a continuous  surjection, π ◦σ = f ◦ π, and π is one-to-one over the residual set Y = Ωs \ j∈Z f j (∂ s R ∪ ∂ u R).

Proof. If a1 a2 · · · an is a word with Aaj aj+1 = 1, then inductively using 3.17 one sees that ⎞ ⎛ n−1 n ! ! f n−1−j Raj ⎠ f n−j Raj = Ran ∩ f ⎝ j=1

j=1

is a u-subrectangle of Ran . From this one gets that Kn (a) =

n !

f −j Raj

j=−n

is nonempty and the closure of its interior. As Kn (a) ⊃ Kn+1 (a) ⊃ · · · we have ∞ ∞ ! ! K(a) = Kn (a) = ∅ . f −j Raj = j=−∞

n=1

If x, y ∈ K(a), then f j x, f j y ∈ Raj are close for all j ∈ Z and so x = y by expansiveness. As ⎞ ⎛ ! ! −j −j K(σa) = f Raj+1 = f ⎝ f Raj ⎠ j

j

= f K(a),

one has π ◦ σ = f ◦ π. That π is continuous is proved like 3.13. As ∂ s R ∪ ∂ u R is nowhere dense, Y is residual. For x ∈ Y pick aj with f j x ∈ Raj . As x ∈ Y , f j x ∈ int(Raj ) and so Aaj aj+1 = 1. Thus a = {aj } ∈ ΣA and x = π(a). If / ∂ s R ∪ ∂ u R; so π is x = π(b), then f j x ∈ Rbj and bj = aj because f j x ∈ injective over Y . As π(ΣA ) is a compact subset of Ωs containing a dense set Y , π(ΣA ) = Ωs . ⊓ ⊔

58

3 Axiom a Diffeomorphisms

3.19. Proposition. σ : ΣA → ΣA is topologically transitive. If f |Ωs is topologically mixing so is σ : ΣA → ΣA . Proof. Let U, V be nonempty open in ΣA . For some a, b ∈ ΣA and N one has U ⊃ U1 = {x ∈ ΣA : xi = ai ∀ i ∈ [−N, N ]} V ⊃ V1 = {x ∈ ΣA : xi = bi ∀ i ∈ [−N, N ]} . Now ∅=  int(KN (a)) = and

∅=  int(KN (b)) =

N !

f −j int(Raj ) = U2

j=−N N !

f −j int(Rbj ) = V2 .

j=−N

Also, if x = π(x) ∈ U2 , then f j x ∈ Rxj and f j x ∈ int(Raj ) imply xj = aj ; so π −1 (U2 ) ⊂ U1 . Similarly π −1 (V2 ) ⊂ V1 . Since f |Ωs is transitive, f n U2 ∩ V2 = ∅ for various large n. Then ∅=  π −1 (f n U2 ∩ V2 ) = π −1 (f n U2 ) ∩ π −1 (V2 ) ⊂ f nU ∩ V . This same argument shows that σ|ΣA is mixing if f |Ωs is.

⊓ ⊔

References The basic references are Anosov [2] for Anosov diffeomorphisms and Smale [14] for Axiom A diffeomorphisms. The stable manifold theorem for hyperbolic sets is due to Hirsch and Pugh [8]. Canonical coordinates and spectral decomposition are from [14], with the mixing part of 3.5 from [4]. The idea of pseudo-orbit has probably occurred to many people. Proposition 3.6 is explicitly proved in [6], though earlier similar statements are in [4] and for Anosov diffeomorphisms in [2]. Sinai [12] stated 3.6 explicitly for Anosov diffeomorphisms. Corollary 3.7 is in [2] for Anosov diffeomorphisms and [3] for Axiom A diffeomorphisms. As the name implies, 3.8 is due to Anosov. Results 3.9 and 3.10 are from [7], [15] and for their proofs we have followed [6]. Symbolic dynamics for certain geodesic flows goes back to Hadamard and was developed by Morse [9]. It was carried out for the horseshoe by Smale [13] and for automorphisms of the torus by Adler and Weiss [1]. Sinai [10], [11] did Sections 3 and 3 for Anosov diffeomorphisms and this was carried over to Axiom A diffeomorphisms in [5]. 1. R. Adler, B. Weiss: Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society 98, American Mathematical Society, Providence, R.I. 1970.

References

59

2. D. Anosov: Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). 3. R. Bowen: Topological entropy and Axiom A, Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968, pp. 23–41, Amer. Math. Soc., Providence, R.I., 1970. 4. R. Bowen: Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397. 5. R. Bowen: Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. 6. R. Bowen: ω-limit sets for Axiom A diffeomorphisms, J. Differential Equations 18 (1975), no. 2, 333–339. 7. M. Hirsch, J. Palis, C. Pugh and M. Shub: Neighborhoods of hyperbolic sets, Invent. Math. 9 1969/1970 121–134. 8. M. Hirsch and C. Pugh: Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968, pp. 133–163, Amer. Math. Soc., Providence, R.I., 1970. 9. M. Morse: A One-to-One Representation of Geodesics on a Surface of Negative Curvature, Amer. J. Math. 43 (1921), no. 1, 33–51. 10. Ya. Sinai: Markov partitions and C-diffeomorphisms, Func. Anal. and its Appl. 2 (1968), no. 1, 64-89. 11. Ya. Sinai: Construction of Markov partitions, Func. Anal. and its Appl. 2 (1968), no. 2, 70-80. 12. Ya. Sinai: Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4, 21–64. English translation: Russian Math. Surveys 27 (1972), no. 4, 21–69. 13. S. Smale: Diffeomorphisms with many periodic points, in “Differential and Combinatorial Topology”. A Symp. in Honor of Marston Morse, pp. 63-80, Princeton Univ. Press, Princeton, N.J., 1965. 14. S. Smale: Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. 15. S. Smale: The Ω-stability theorem, Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968, pp. 289–298, Amer. Math. Soc., Providence, R.I.

4 Ergodic Theory of Axiom a Diffeomorphisms

A. Equilibrium States for Basic Sets Recall that a function φ is H¨ older continuous if there are constants a, θ > 0 so that |φ(x) − φ(y)| ≤ a d(x, y)θ . 4.1. Theorem. Let Ωs be a basic set for an Axiom A diffeomorphism f and φ : Ωs → R H¨ older continuous. Then φ has a unique equilibrium state µφ (w.r.t. f |Ωs ). Furthermore µφ is ergodic; µφ is Bernoulli if f |Ωs is topologically mixing. 4.2. Lemma. There are ε > 0 and α ∈ (0, 1) for which the following are true: if x ∈ Ωs , y ∈ M , and d(f k x, f k y) ≤ ε for all k ∈ [−N, N ], then d(x, y) < αN . Proof. See p. 140 of [12]. ⊓ ⊔ Proof of 4.1. Let R be a Markov partition for Ωs of diameter at most ε, A the transition matrix for R and π : ΣA → Ωs as in 3.D. Let φ∗ = φ ◦ π. If x, y ∈ ΣA have xk = yk for k ∈ [−N, N ], then f k π(x) , f k π(y) ∈ Rxk = Ryk for k ∈ [−N, N ] . This gives d(π(x), π(y)) < αN , |φ∗ (x) − φ∗ (y)| ≤ a (αθ )N and φ∗ ∈ FA . First we assume f |Ωs is mixing. Then σ|ΣA is mixing by 3.19 and we have a Gibbs measure µφ∗ as in Chapter 1. Let Ds = π −1 (∂ s R) and Du = π −1 (∂ u R). Then Ds and Du are closed subsets of ΣA , each smaller than ΣA , and σDs ⊂ Ds , σ −1 Du ⊂ Du . As µφ∗ is σ-invariant, µφ∗ (σ n Ds ) = µφ∗ (Ds ); using σ n+1 Ds ⊂ σ n Ds one has ⎞ ⎛ ! µφ∗ ⎝ σ n Ds ⎠ = µφ∗ (Ds ) . n≥0

62

4 Ergodic Theory of Axiom a Diffeomorphisms

 Now n≥0 σ n Ds has measure 0 or 1 as it is σ-invariant and µφ∗ is ergodic (see 1.14); since its complement (a nonempty open set) has positive measure by 1.4, one gets µφ∗ (Ds ) = 0. Similarly one sees µφ∗ (Du ) = 0. Now let µφ = π ∗ µφ∗ , i.e., µφ (E) = µφ∗ (π −1 E). Then µφ is f -invariant and the automorphisms of the measure spaces  (σ, µφ∗ ), (f, µφ ) are conjugate since π is one-to-one except on the null set n∈Z σ n (Ds ∪ Du ). In particular hµφ (f ) = hµφ∗ (σ) and so (1 )   hµφ (f ) + φ dµφ = hµφ∗ (σ) + φ∗ dµφ∗ = Pσ (φ∗ ) ≥ Pf (φ) .

Hence (2 ) Pσ (φ∗ ) = Pf (φ) and µφ is an equilibrium state for φ by Chapter 2. Now µφ is Bernoulli because of 1.25. 4.3. Lemma. For any µ ∈ Mf (Ωs ) there is a ν ∈ Mσ (ΣA ) with π ∗ ν = µ.
Proof. This well-known fact is proved as follows. F (g ◦ π) = µ(g) = gdµ defines a positive linear functional on a subspace of C (ΣA ). By a modification of the Hahn-Banach Theorem F extends to C (ΣA ), still positive. As F (1) = F (1 ◦ π) = 1, F is identified with some β ∈ M (ΣA ). By compactness let ν = limk→∞ n1k (β + σ ∗ β + · · · + (σ nk −1 )∗ β). Then σ ∗ ν = ν and π ∗ ν = µ (using π ∗ (σ k )∗ β = (f k )∗ π ∗ β = (f k )∗ µ = µ). ⊓ ⊔ Proof of 4.1 (continued). Suppose µ is any equilibrium state of φ and pick ν ∈ Mσ (ΣA ) with π ∗ ν = µ. Then hν (σ) ≥ hµ (f ) and so   ∗ hν (σ) + φ dν ≥ hµ (f ) + φ dµ = P (φ) = P (φ∗ ) . Thus ν is an equilibrium state for φ∗ and ν = µφ∗ by 1.22. Then µ = π ∗ µφ∗ = µφ . We have left the case Ωs = X1 ∪· · ·∪Xm with f Xk = Xk+1 and f m |X1 mix1 and so µ′ = m µ|X1 ∈ Mf m (X1 ). ing. For µ ∈ Mf (Ωs ), one has µ(X1 ) = m ′ Conversely, if µ ∈ Mf m (X1 ), then µ ∈ Mf (Ωs ) where µ(E) =

m−1 1  ′ µ (X1 ∩ f k E) . m k=0

′ m One checks that
µ ↔ µ ′ defines
a bijection Mf (Ωs ) ↔ Mf m (X1 ), hµ′ (f
)= mhµ (f ), and Sm φ dµ = m φ dµ. Finding µ maximizing
hµ (f ) + φ dµ is equivalent therefore to finding µ′ maximizing hµ′ (f m ) + Sm φ dµ′ . For φ older on X1 and therefore one is done since X1 H¨ older on Ωs , Sm φ will be H¨ is a mixing basic set of f m . ⊓ ⊔ 1

2

The second equality follows from Theorem 1.22; The inequality comes from Proposition 2.13 (note of the editor). Using the variational principle 2.17 (note of the editor).

A. Equilibrium States for Basic Sets

63

4.4. Proposition. Let φ : Ωs → R be H¨ older continuous and P = Pf |Ωs (φ). For small ε > 0 there is a bε > 0 so that, for any x ∈ Ωs and for all n,  µφ y ∈ Ωs : d(f k y, f k x) < ε ∀ k ∈ [0, n] ≥ bε exp(−P n + Sn φ(x)) .

Proof. Choose the Markov partition R above to have diam(R) < ε. Assume first f |Ωs is mixing. Pick x ∈ ΣA with π(x) = x. Then  B = y ∈ Ωs : d(f k y, f k x) < ε ∀ k ∈ [0, n]  ⊃ π y ∈ ΣA : yk = xk ∀ k ∈ [0, n] .

Applying 1.4 and P (φ∗ ) = P (φ) = P one gets

µφ (B) ≥ c1 exp(−P n + Sn φ(x)) . We leave it to the reader to reduce the general case to the mixing one as in the proof of 4.1. ⊓ ⊔ 4.5. Proposition. Let φ, ψ : Ωs → R be two H¨ older continuous functions. Then the following are equivalent: (i) µφ = µψ . (ii) There are constants K and L so that |Sm φ(x) − Sm ψ(x) − Km| ≤ L for all x ∈ Ωs and all m ≥ 0. (iii) There is a constant K so that Sm φ(x) − Sm ψ(x) = Km when x ∈ Ωs with f m x = x. (iv) There is a H¨ older function u : Ωs → R and a constant K so that φ(x) − ψ(x) = K + u(f x) − u(x). If these conditions hold, K = P (φ) − P (ψ). Proof. Let φ∗ = φ ◦ π and ψ ∗ = ψ ◦ π. We assume f |Ωs is mixing and leave the reduction to this case to the reader. If µφ = µψ , then µφ∗ = µψ∗ and by Theorem 1.28 there are K and L so that |Sm φ∗ (x) − Sm ψ ∗ (x) − Km| ≤ L for x ∈ ΣA . For x ∈ Ωs , picking x ∈ π −1 (x), this gives us (ii). Assume (ii) and f m x = x. Then L ≥ |Smj φ(x) − Smj ψ(x) − mjK| = j|Sm φ(x) − Sm ψ(x) − mK| . Letting j → ∞ we get (iii). If (iv) is true, then φ∗ (x) − ψ ∗ (x) = K + u(π(σx)) − u(π(x)) and µφ∗ = µψ∗ by Theorem 1.28. One then has µφ = π ∗ µφ∗ = π ∗ µψ∗ = µψ . Now we assume (iii) and prove (iv). Let η(x) = φ(x) − ψ(x) − K and pick x ∈ Ωs with dense forward orbit (Lemma 1.29). Let A = {f k x : k ≥ 0} and define u : A → R by

64

4 Ergodic Theory of Axiom a Diffeomorphisms k

u(f x) =

k−1 

η(f j x) .

j=0

For z ∈ A one has u(f z) − u(z) = η(z). Pick ε and α as in 4.2. By 3.7 there is δ > 0 so that if y ∈ Ωs and d(f n y, y) < δ, then there is a y ′ ∈ Ωs with f n y ′ = y ′ and d(f k y, f k y ′ ) < 2ε for all k ∈ [0, n]. Let R be the maximum ratio that f expands any distance. Suppose y = f k x, z = f m x with k < m and d(y, z) < ǫ/2RN . Providing N is large one has z = f n y, n = m − k, and d(f n y, y) < δ. Then find y ′ as above with f n y ′ = y ′ . Then, as Sn η(y ′ ) = 0,   |u(z) − u(y)| = Sn η(y) − Sn η(y ′ ) ≤

n−1  j=0

|η(f j y) − η(f j y ′ )| .

By the choice of R and y ′ one sees that d(f j y, f j y ′ ) < ε for all j ∈ [−N, n + N ] . For j ∈ [0, n) Lemma 4.2 gives d(f j y, f j y ′ ) < αmin{j+N,N +n−j} . Because η is H¨older, |η(f j y) − η(f j y ′ )| ≤ a αθmin{j+N,N +n−j} |u(y) − u(z)| ≤ 2a N +1

∞ 

r=N

αθr ≤ a′ αθN .

Pick N so that d(y, z) ∈ [ε/2R , ε/2RN ]. Taking γ > 0 so that (1/R)γ ≥ αθ one has |u(y) − u(z)| ≤ a′′ d(y, z)γ . Thus u is H¨older on A and extends uniquely to a H¨ older function on A¯ = Ωs . The formula η(z) = u(f z) − u(z) extends to Ωs by continuity. ⊓ ⊔

B. The Case φ = φ(u) Recall that M has a Riemannian structure and this induces a volume measure m on M . We will assume for the remainder of this chapter that f : M → M is a C 2 Axiom A diffeomorphism and Ωs is a basic set for f . For x ∈ Ωs let φ(u) (x) = − log λ(x) where λ(x) is the Jacobian of the linear map Df : Exu → Efux using inner products derived from the Riemannian metric.

B. The Case φ = φ(u)

65

4.6. Lemma. If Ωs is a C 2 basic set, then φ(u) : Ωs → R is H¨ older continuous. Proof. The map x → Exu is H¨older (see 6.4 of [12]) and Exu → φ(u) (x) is differentiable, so the composition x → φ(u) (x) is H¨ older. ⊓ ⊔ By Theorem 4.1 the function φ(u) has a unique equilibrium state which we denote µ+ = µφ(u) . While φ(u) depends on the metric used, when f m x = x Sm φ(u) (x) = − log Jac(Df m : Exu → Exu ) does not depend on the metric (this Jacobian is the absolute value of the determinant). By 4.5 one sees that the measure µ+ on Ωs and P (φ(u) ) do not depend on which metric is used. 4.7. Volume Lemma. Let  Bx (ε, m) = y ∈ M : d(f k x, f k y) ≤ ε for all k ∈ [0, m) .

If x ∈ Ωs is a C 2 basic set and ε > 0 is small, then there is a constant Cε so that m(Bx (ε, m)) ∈ [Cε−1 , Cε ] exp(Sm φ(u) (x)) for all x ∈ Ωs . Proof. See 4.2 of [9]. ⊓ ⊔ 4.8. Proposition. Let Ωs be a C 2 basic set.  (a) Letting B(ε, n) = x∈Ωs Bx (ε, n), one has (for small ε > 0) Pf |Ωs (φ(u) ) = lim

n→∞

1 log m(B(ε, n)) ≤ 0 . n

Wεs (x). If m(Wεs (Ωs )) > 0, then  (u) Pf |Ωs (φ ) = 0 and hµ+ (f ) = − φ(u) dµ+ .

(b) Let Wεs (Ωs ) =



x∈Ωs

Proof. Call E ⊂ M (n, ε)-separated if whenever y, z are two distinct points in E, one can find k ∈ [0, n) with d(f k y, f k z) > δ. Choose En (δ) maximal among the (n, ε)-separated subsets of Ωs . For x ∈ Ωs one has x ∈ By (ε, n) for some y ∈ En (δ); otherwise  En (δ) ∪ {x} is (n, ε)-separated. Then Bx (ε, n) ⊂ By (δ + ε, n), B(ε, n) ⊂ y∈En (δ) By (δ + ε, n) and by 4.7

(⋆)

m(B(ε, n)) ≤ Cδ+ε

For δ ≤ ε,



y∈En (δ)



y∈En (δ)

exp(Sn φ(u) (y)).

By (δ/2, n) ⊂ B(ε, n) is a disjoint union and so

66

(⋆⋆)

4 Ergodic Theory of Axiom a Diffeomorphisms −1 m(B(ε, n)) ≥ Cδ/2

Since φ(u) is H¨older, we have



y∈En (δ)

exp(Sn φ(u) (y)).

|φ(u) (x) − φ(u) (y)| ≤ a d(x, y)θ for some a, θ > 0 and all x, y ∈ Ωs . Suppose x ∈ By (ε, n) ∩ Ωs . Then for j ∈ [0, n) d(f j x, f j y) < αmin{j,n−j−1} by Lemma 4.2. Hence |Sn φ(x) − Sn φ(y)| ≤

n−1  j=0

≤ 2a

|φ(u) (f j x) − φ(u) (f j y)| ∞ 

αkθ = γ .

k=0

Fix δ ≤ ε and let U be an open cover of Ωs with diam(U) < δ. Let Γ ⊂ Un cover Ωs . For each y ∈ En (δ) pick Uy ∈ Γ with y ∈ X(Uy ). Then Sn φ(u) (Uy ) ≥ Sn φ(u) (y). If Uy = Uy′ , then d(f k y, f k y ′ ) ≤ diam(U) < δ and y = y ′ as En (δ) is (n, δ)-separated. Thus   exp(Sn φ(u) (U)) ≥ exp(Sn φ(u) (y)). U∈Γ

y∈En (δ)

Using this together with (⋆) above one gets P (φ(u) , U) = lim

n→∞

 1 log inf exp(Sn φ(u) (U)) Γ n U∈Γ

1 ≥ lim sup log m(B(ε, n)) . n→∞ n

Letting diam(U) → 0, one replaces P (φ(u) , U) with P (φ(u) ). Now let U be an open cover and let δ be a Lebesgue number for U. For each y ∈ En (δ) one can pick Uy ∈ Un with By (δ, n) ∩ Ωs ⊂ X(Uy ). Let Γ = {Uy : y ∈ En (δ)}. Then Γ covers Ωs since every x ∈ Ωs lies in some By (δ, n) with y ∈ En (δ). Also Sn φ(u) (Uy ) ≤ Sn φ(u) (y) + γ and so Zn (φ(u) , U) ≤



exp(Sn φ(u) (Uy ))

Uy ∈Γ n

≤ eγ



y∈En (δ)

exp(Sn φ(u) (y)) .

B. The Case φ = φ(u)

67

Using (⋆⋆) we get 1 log Zn (φ(u) , U) n 1 ≤ lim inf log m(B(ε, n)) . n→∞ n

P (φ(u) , U) = lim

n→∞

Noting that m(B(ε, n)) ≤ m(M ), the proof of (a) is complete. Now m(B(ε, n)) ≥ m(Wεs (Ωs )) as Wεs (Ωs ) ⊂ B(ε, n). If m(Wεs (Ωs )) > 0, the formula from (a) yields P (φ(u) ) = 0. Since µ+ is an equilibrium state for φ(u) ,  ⊔ hµ+ (f ) + φ(u) dµ+ = P (φ(u) ) = 0 . ⊓ A basic set Ωs is an attractor if it has small neighborhood U with f (U ) ⊂ U . By Proposition 3.10 this is equivalent to Wεs (Ωs ) being a neighborhood of Ωs . 4.9. Lemma. Let Ωs be a C 1 basic set. If Wεu (x) ⊂ Ωs for some x ∈ Ωs , then Ωs is an attractor. If Ωs is not an attractor there exists γ > 0 such that for every x ∈ Ωs , there is y ∈ Wεu (x) with d(y, Ωs ) > γ. Proof. If Wεu (Ωs ) ⊂ Ωs , then  Ux = {Wεs (y) : y ∈ Wεu (x)}

is a neighborhood of x in M (see Lemma 4.1 of [11]). Choose a periodic point p ∈ Ux , say f m x = x. For some small β one has Wβu (p) ⊂ Ux ; if z ∈ Wβu (p) lies in Wεs (y) (y ∈ Wεu (Ωs )), then d(f n z, f n y) < ε and d(f −n z, f −n p) < β for n ≥ 0. By Theorem 3.9 one has z ∈ Ωs and Wβu (p) ⊂ Ωs . Then also  W u (p) = k≥0 f mk Wβu (p) ⊂ Ωs . Now Xp = W u (p) and Ωs = Xp ∪ f Xp ∪ · · · ∪ f N Xp for some N . For each N x ∈ k=0 f k W u (p) = Y one has Wεu (x) ⊂ Ωs and so Ux as defined above is a neighborhood of x in M . Since Wεs (x), Wεu (x) depend continuously on x ∈ Ωs , one can find a δ > 0 independent of x so that Ux contains the 2δ-ball Bx (2δ) about x in M for all x ∈ Y (see [11, Lemma 4.1]). Then  BΩs (δ) ⊂ {Ux : x ∈ Y } ⊂ Wεs (Ωs ) and Ωs is an attractor. To prove (b) notice that the set

Vγ = {x ∈ Ωs : d(y, Ωs ) > γ for some y ∈ Wεu (x)} is open becauseWεu (x) varies continuously with x. Also Vγ increases when γ decreases and γ>0 Vγ = Ωs by statement (a). By compactness Vγ = Ωs for some γ > 0. ⊓ ⊔

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4 Ergodic Theory of Axiom a Diffeomorphisms

4.10. Second Volume Lemma. Let Ωs be a C 2 basic set. For small ε, δ > 0 there is a d = d(ε, δ) > 0 so that m(By (δ, n)) ≥ d m(Bx (ε, n)) whenever x ∈ Ωs and y ∈ Bx (ε, n). Proof. See 4.3 of [9]. ⊓ ⊔

4.11. Theorem. Let Ωs be a C 2 basic set. The following are equivalent: (a) Ωs is an attractor. (b) m(W s (Ωs )) > 0. (c) Pf |Ωs (φ(u) ) = 0. ∞ Proof. As W s (Ωs ) = n=0 f −n Wεs (Ωs ), (b) is equivalent to m(Wεs (Ωs )) > 0. If Ωs is an attractor, then (b) is true since Wεs (Ωs ) is a neighborhood of Ωs . (c) follows from (b) by Proposition 4.8 (b). We finish by assuming Ωs is not an attractor and showing P (φ(u) ) < 0. Given a small ε > 0 choose γ as in 4.9. Pick N so that u f N Wγ/4 (x) ⊃ Wεu (f N x)

for all x ∈ Ωs . Let E ⊂ Ωs be (γ, n)-separated. For x ∈ E there is a y(x, n) ∈ Bx (γ/4, n) with d(f n+N y(x, n), Ωs ) > γ u u (since f n Bx (γ/4, n) ⊃ Wγ/4 (f n x) and f N Wγ/4 (f n x) ⊃ Wεu (f N +n x)). Choose N N δ ∈ (0, γ/4) so that d(f z, f y) < γ/2 whenever d(z, y) < δ. Then

By(x,n) (δ, n) ⊂ Bx (γ/2, n), f n+N By(x,n) (δ, n) ∩ BΩs (γ/2) = ∅ .

Hence By(x,n) (δ, n) ∩ B(γ/2, n + N ) = ∅. Using the Second Volume Lemma  m(B(γ/2, n)) − m(B(γ/2, n + N )) ≥ m(By(x,n) (δ, n)) x∈E

≥ d(3γ/2, δ)



m(Bx (3γ/2, n))

x∈E

≥ d(3γ/2, δ) m(B(γ/2, n)) . Therefore, setting d = d(3γ/2, δ) m(B(γ/2, n + N )) ≤ (1 − d) m(B(γ/2, n)) and by Proposition 4.8 (a) Pf |Ωs (φ(u) ) ≤

1 log(1 − d) < 0 . N

⊓ ⊔

Remark. It is possible to find a C 1 basic set (a horseshoe) which is not an attractor but nevertheless has m(W s (Ωs )) > 0 [8].

C. Attractors and Anosov Diffeomorphisms

69

C. Attractors and Anosov Diffeomorphisms r Because M = k=1 W s (Ωk ), Theorem 4.11 implies that m-almost all x ∈ M approach an attractor under the action of a C 2 Axiom A diffeomorphism f . This leads us next to the following result. 4.12. Theorem. Let Ωs be a C 2 attractor. For m-almost all x ∈ W s (Ωs ) one has  n−1 1 lim g(f k x) = g dµ+ n→∞ n k=0

for all continuous g : M → R (i.e., x is a generic point for µ+ ).
n−1 Proof. Let us write g(n, x) = n1 k=0 g(f k x) and g = g dµ+ . Fix δ > 0 and define the sets Cn (g, δ) = {x ∈ M : |g(n, x) − g| > δ} E(g, δ) = {x ∈ M : |g(n, x) − g| > δ for infinitely many n} ∞ ∞  ! Cn (g, δ) . = N =1 n=N

Choose ε > 0 so that |g(x) − g(y)| < δ when d(x, y) < ε. Now fix N > 0 and choose sets RN , RN +1 , . . . successively as follows. Let Rn (n ≥ N ) be a maximal subset of Ωs ∩ Cn (g, 2δ) satisfying the conditions: (a) Bx (ε, n) ∩ By (ε, k) = ∅ for x ∈ Rn , y ∈ Rk , N ≤ k < n, (b) Bx (ε, n) ∩ Bx′ (ε, n) = ∅ for x, x′ ∈ Rn , x = x′ .

If y ∈ Wεs (Ωs ) ∩ Cn (g, 3δ) (n ≥ N ) and y ∈ Wεs (z) with z ∈ Ωs , then z ∈ Cn (g, 2δ) by the choice of ε. By the maximality of Rn one has Bz (ε, n) ∩ Bx (ε, k) = ∅ for some x ∈ Rk , N ≤ k ≤ n . Then y ∈ Bz (ε, n) ⊂ Bz (ε, k) ⊂ Bx (2ε, k) and so Wεs (Ωs ) ∩

∞ 

n=N

Cn (g, 3δ) ⊂

∞  

Bx (2ε, k) .

k=N x∈Rk

Using the Volume Lemma 4.7 one gets   ∞ ∞    s m Wε (Ωs ) ∩ exp(Sk φ(u) (x)). Cn (g, 3δ) ≤ c2ε n=N

k=N x∈Rk

∞  The definition of Rn shows that VN = k=N x∈Rk Bx (ε, k) is a disjoint union. The choice of ε gives Bx (ε, k) ⊂ Ck (g, δ) for x ∈ Rk ⊂ Ck (g, 2δ) and so

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4 Ergodic Theory of Axiom a Diffeomorphisms

∞ VN ⊂ k=N Ck (g, δ). Since the measure µ+ is ergodic, the Ergodic Theorem implies  ∞   0 = µ+ (E(g, δ)) = lim µ+ Cn (g, δ) n→∞

n=N

and thus limN →∞ µ (VN ) = 0. By 4.8 (b) one has Pf |Ωs (φ(u) ) = 0 and then by 4.4 ∞   + exp(Sk φ(u) (x)). µ (VN ) ≥ bε +

k=M x∈Rk

+

As µ (VN ) → 0, the sum on the right approaches 0 as N → ∞. Using the inequality of the preceding paragraph one sees   ∞  s Cn (g, 3δ) = 0 . lim m Wε (Ωs ) ∩ N →∞

n=N

This in turn implies m(Wεs (Ωs ) ∩ E(g, 3δ)) = 0. For δ ′ > 3δ the set E(g, δ ′ ) ∩ f −n Wεs (Ωs ) ⊂ f −n (E(g, 3δ) ∩ Wεs (Ωs )) has measure 0 since f preserves measure (w.r.t. m). Thus m(E(g, δ ′ ) ∩ W s (Ωs )) ≤

∞ 

n=0

  m E(g, δ ′ ) ∩ f −n Wεs (Ωs ) = 0 .

1 Fixing g still but letting δ ′ = m → 0 one gets limn g(n, x) = g for all x ∈ s W (Ωs ) outside"an m-null set A(g). Let {gk }∞ k=1 be a dense subset of C (M ); ∞ s A(g ) one gets that lim for x ∈ W (Ωs ) k n g(n, x) = g for all g ∈ C (M ). k=1 ⊓ ⊔

4.13. Corollary. Suppose f : M → M is a transitive C 2 Anosov diffeomorphism. If f leaves invariant a probability measure µ which is absolutely continuous with respect to m, then µ = µ+ . Proof. In this case M = Ω = Ω1 is the spectral decomposition. Let g ∈ C (M ). By the Ergodic Theorem there is a function g ∗ : M → R so that n−1 1 g(f k x) = g ∗ (x) n→∞ n

lim

k=0

for µ-almost all x. Let A be the set of x ∈ M with  n−1 1 k g(f x) = gdµ+ . lim n→∞ n k=0

Because m(A) = 1 and µ ≪ m, µ(A) = 1. It follows that g ∗ (x) = µ-almost all x. Then




gdµ+ for

C. Attractors and Anosov Diffeomorphisms



gdµ =



g ∗ dµ =



71

gdµ+ .

As this holds for all g ∈ C (M ), µ = µ+ . ⊓ ⊔ Remark. If M is connected, then M = Ω1 = X1 and f is mixing. So µ above is Bernoulli. 4.14. Theorem. Let f be a transitive C 2 Anosov diffeomorphism. The following are equivalent: (a) f admits an invariant measure of the form dµ = hdm with h a positive H¨ older function. (b) f admits an invariant measure µ absolutely continuous w.r.t. m. (c) Df n : Tx M → Tx M has determinant 1 whenever f n x = x. Proof. Clearly (a) implies (b). Assume (b) holds. Let λ(s) (x) be the Jacobian of Df : Efs −1 x → Exs and φ(s) (x) = log λ(s) (x). Now f −1 is Anosov with s s u u Ex,f −1 = Ex,f an Ex,f −1 = Ex,f . Also (u)

λf −1 (x) = Jacobian Df −1 : Exs → Efs −1 x = λ(s) (x)−1

(u)

(u)

and so φf −1 (x) = − log λf −1 (x) = φ(s) (x). There is an invariant measure µ− so that  n−1 1 lim g(f −k x) = g dµ− n→∞ n k=0

(u)

for m-almost all x; µ− is the unique equilibrium state for φf −1 w.r.t. f −1 . Notice that equilibrium states w.r.t. f −1 are the same as those w.r.t. f ; for Mf (M ) = Mf −1 (M ) and hν (f ) = hν (f −1 ). So µ− = µφ(s) . Applying 4.13 to both f and f −1 we see µφ(u) = µ+ = µ = µ− = µφ(s) . By 4.8 (b), P (φ(u) ) = 0 = P (φ(s) ). By 4.5 one has, for f n x = x, n−1  k=0

φ(u) (f k x) −

n−1 

φ(s) (f k x) = 0.

k=0

Exponentiating, 1 = (det Df n |Exu ) (det Df n |Exs ) = (det Df n |Tx M ) . Now assume (c) and let φ(x) = log Jac (Df : Tx M → Tf x M ). Then

72

4 Ergodic Theory of Axiom a Diffeomorphisms

Sn φ(x) = log

n−1  k=0

Jac (Df : Tf k x M → Tf k+1 x M )

= log Jac (Df n : Tx M → Tf k x M ) = 0 when f k x = x. By Proposition 4.5 (with ψ = 0, K = 0) there is a H¨ older u : M → R with φ(x) = u(f x) − u(x). Let h(x) = eu(x) . Thinking of µ = hdm as the absolute value of a form f ∗ (hdm)(f x) = h(x) eφ(x) dm(f x) = h(f x) dm(f x) = (h dm)(f x) . So µ is f -invariant. ⊓ ⊔ Remark. Actually h above will be C 1 . See [13, 14]. 4.15. Corollary. Among the C 2 Anosov diffeomorphisms the ones that admit no invariant measures µ ≪ m are open and dense. Proof. [18], page 36. We use condition 4.14 (c). Suppose f n x = x and det(Df n |Tx M ) = 1. For f1 near f , f1 will be Anosov and have a periodic point x1 near x with f1n x1 = x1 . Then det(Df1n |Tx1 M ) will be near det(Df n |Tx M ) and not equal to one. We are using stability theory not covered in these notes. On the other hand, if f n x = x with det(Df n |Tx M ) = 1, this condition is destroyed by the right small perturbation of f . ⊓ ⊔ In the case where f admits no invariant µ ≪ m, the measure m is actually dissipative [10].

References This chapter contains the main theorems of these notes. The notes as a whole constitute a version of Sinai’s program [18] for applying statistical mechanics to diffeomorphisms. It was Ruelle who carried Sinai’s work on Anosov diffeomorphisms over to Axiom A attractors [11] and brought in the formalism of equilibrium states [15]. Theorem 4.1 is from [6, 7]; see [18] for the Anosov case. Results 4.5, 4.13, 4.14 and 4.15 came from [13, 14, 18]. Section 4 is taken verbatim from [9]. Theorem 4.12 is due to Ruelle [16] (we followed the proof in [9]). Ruelle [16] along these lines also proved that f n µ → µ+ when µ ≪ m has support in a neighborhood of an attractor; for the Anosov case this result is due to Sinai [17]. That µ+ is Bernoulli in the transitive Anosov case is due to Azencott [4]. The ergodic theory of Anosov diffeomorphisms with µ+ ≪ m has been studied in many papers; see for instance [2] or [3]. In case φ(u) is a constant function µ+ is the unique invariant measure which maximizes entropy (φ = 0 and φ(u) have the same equilibrium state). For hyperbolic automorphisms of the 2-torus µ+ is Haar measure and construction

References

73

in 4.1 is due to Adler and Weiss [1]. This paper is very important in the development of the subject and is good reading. When φ(u) = C the measure µ which maximizes entropy still has the following geometrical significance: the periodic points of Ωs are equidistributed with respect to µ [5]. K. Sigmund [19] studied the generic properties of measures on Ωs . 1. R. Adler, B. Weiss: Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society 98, American Mathematical Society, Providence, R.I. 1970. 2. D. Anosov: Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). 3. D. Anosov, Ya. Sinai: Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967) no. 5 (137), 107–172. English translation: Russ. Math. Surveys 22 (1967), no. 5, 103-167. 4. R. Azencott: Diff´eomorphismes d’Anosov et sch´emas de Bernoulli, C. R. Acad. Sci. Paris, S´ er. A-B 270 (1970), A1105-A1107. 5. R. Bowen: Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397. 6. R. Bowen: Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no.3, 193-202. 7. R. Bowen: Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory 8 (1974/75), no. 4, 289–294. 8. R. Bowen: A horseshoe with positive measure, Invent. Math. 29 (1975), no. 3, 203-204. 9. R. Bowen, D. Ruelle: The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181-202. 10. B. Gureviˇc, V. Oseledec: Gibbs distributions, and the dissipativity of C-diffeomorphisms, Soviet Math. Dokl. 14 (1973), 570–573. 11. M. Hirsch, J. Palis, C. Pugh and M. Shub: Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/1970), 121–134. 12. M. Hirsch and C. Pugh: Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968, pp. 133–163, Amer. Math. Soc., Providence, R.I., 1970. 13. A. N. Livˇsic: Cohomology of dynamical systems, Math. USSR-Izv. 6 (1972), 1278-1301. 14. A. N. Livˇsic, Ya. Sinai: Invariant measures that are compatible with smoothness for transitive C-systems, Soviet Math. Dokl. 13 (1972), 1656-1659. 15. D. Ruelle: Statistical mechanics on a compact set with Zν action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 185 (1973), 237-251. 16. D. Ruelle: A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976), no. 3, 619-654. 17. Ya. Sinai: Markov partitions and C-diffeomorphisms, Func. Anal. and its Appl. 2 (1968), no. 1, 64-89. 18. Ya. Sinai: Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4, 21–64. English translation: Russian Math. Surveys 27 (1972), no. 4, 21–69. 19. K. Sigmund: Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math. 11 (1970), 99-109.

Index

Anosov, 46, 71, 72 Attractor, 67–69 Axiom A, 46 Basic set, 48

Markov partition, 52 Mixing, 6, 14 Non-wandering, 45

Canonical coordinates, 47 Central limit theorem, 24

Pressure, 18, 33, 34 Pseudo-orbit, 49

Entropy, 4, 16, 18, 29, 43 Equilibrium state, 42, 61 Ergodic, 14 Exponential clustering, 23

Rectangle, 51

FA , 7 Free energy, 4 Gibbs measure, 6, 7, 15 Homologous, 7 Hyperbolic, 45

ΣA , 6 Separated, 65 Shift, 5, 9 Transition matrix, 56 Variational principle, 19, 40 Weak Bernoulli, 22

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