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Lecture Notes in Mathematics 470Editors:
J.-M. Morel, CachanF. Takens, GroningenB. Teissier, Paris
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Rufus Bowen
Equilibrium Statesand the Ergodic Theoryof Anosov Diffeomorphisms
Second revised edition
Jean-Ren ChazottesEditor
Preface by David Ruelle
ABC
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AuthorRobert Edward (Rufus) Bowen(1947-1978)University of California at BerkeleyUSA
Preface byDavid RuelleIHS35, Route de Chartres91440 [email protected]
EditorJean-Ren ChazottesCentre de Physique ThoriqueCNRS-cole Polytechnique91128 Palaiseau [email protected]
ISBN: 978-3-540-77605-5 e-ISBN: 978-3-540-77695-6DOI: 10.1007/978-3-540-77695-6
Lecture Notes in Mathematics ISSN print edition: 0075-8434ISSN electronic edition: 1617-9692
Library of Congress Control Number: 2008922889
Mathematics Subject Classification (2000): Primary: 37D20, Secondary: 37D35
c 2008, 1975 Springer-Verlag Berlin HeidelbergThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, 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 publicationor 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 areliable 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 lawsand regulations and therefore free for general use.
Printed on acid-free paper
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Cover design: WMXDesign GmbH
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Preface
The Greek and Roman gods, supposedly, resented those mortals endowed withsuperlative gifts and happiness, and punished them. The life and achievementsof Rufus Bowen (19471978) remind us of this belief of the ancients. WhenRufus died unexpectedly, at age thirty-one, from brain hemorrhage, he was avery 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 hehad lived longer. Bowen chose to be simple rather than brilliant. This was thehard choice, especially in a messy subject like smooth dynamics in which heworked. Simplicity had also been the style of Steve Smale, from whom Bowenlearned dynamical systems theory.
Rufus Bowen has left us a masterpiece of mathematical exposition: the slimvolume Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms(Springer Lecture Notes in Mathematics 470 (1975)). Here a number of resultswhich were new at the time are presented in such a clear and lucid style thatBowens monograph immediately became a classic. More than thirty yearslater, many new results have been proved in this area, but the volume is asuseful as ever because it remains the best introduction to the basics of theergodic theory of hyperbolic systems.
The area discussed by Bowen came into existence through the merging oftwo apparently unrelated theories. One theory was equilibrium statistical me-chanics, 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 dy-namical systems, with the major contributions of D.V. Anosov and S. Smale.The two theories came into contact when Ya.G. Sinai introduced Markov par-titions and symbolic dynamics for Anosov diffeomorphisms. This allowed thepoweful techniques and results of statistical mechanics to be applied to smoothdynamics, an extraordinary development in which Rufus Bowen played a ma-jor role. Some of Bowens ideas were as follows. First, only one-dimensionalstatistical mechanics is discussed: this is a richer theory, which yields what is
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VI Preface
needed for applications to dynamical systems, and makes use of the powerfulanalytic tool of transfer operators. Second, Smales Axiom A dynamical sys-tems are studied rather than the less general Anosov systems. Third, SinaisMarkov partitions are reworked to apply to Axiom A systems and their con-struction is simplified by the use of shadowing. The combination of simpli-fications and generalizations just outlined led to Bowens concise and lucidmonograph. This text has not aged since it was written and its beauty is asstriking as when it was first published in 1975.
Jean-Rene Chazottes has had the idea to make Bowens monograph moreeasily available by retyping it. He has scrupulously respected the originaltext and notation, but corrected a number of typos and made a few otherminor corrections, in particular in the bibliography, to improve usefulness
and readability. In his enterprise he has been helped by Jerome Buzzi, PierreCollet, and Gerhard Keller. For this work of love all of them deserve ourwarmest thanks.
Bures sur Yvette, mai 2007 David Ruelle
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Contents
0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Gibbs Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3A. Gibbs Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B. Ruelles Perron-Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 8C. Construction of Gibbs Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13D. Variational Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16E. Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 General Thermodynamic Formalism . . . . . . . . . . . . . . . . . . . . . . . 29
A. Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B. Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32C. Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D. Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Axiom a Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B. Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47C. Markov Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51D. Symbolic Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Ergodic Theory of Axiom a Diffeomorphisms . . . . . . . . . . . . . . 61A. Equilibrium States for Basic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61B. The Case = (u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64C. Attractors and Anosov Diffeomorphisms. . . . . . . . . . . . . . . . . . . . . . 69References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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VIII Contents
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
These notes came out of a course given at the University of Minnesota andwere revised while the author was on a Sloan Fellowship.
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0
Introduction
The main purpose of these notes is to present the ergodic theory of Anosov andAxiom A diffeomorphisms. These diffeomorphisms have a complicated orbitstructure that is perhaps best understood by relating them topologically andmeasure theoretically to shift spaces. This idea of studying the same examplefrom different viewpoints is of course how the subjects of topological dynamicsand ergodic theory arose from mechanics. Here these subjects return to helpus understand differentiable systems.
These notes are divided into four chapters. First we study the statisticalproperties of Gibbs measures. These measures on shift spaces arise in modernstatistical mechanics; they interest us because they solve the problem of de-termining an invariant measure when you know it approximately in a certain
sense. The Gibbs measures also satisfy a variational principle. This princi-ple is important because it makes no reference to the shift character of theunderlying space. Through this one is led to develop a thermodynamic for-malism on compact spaces; this is carried out in chapter two. In the thirdchapter Axiom A diffeomorphisms are introduced and their symbolic dynam-ics constructed: this states how they are related to shift spaces. In the finalchapter this symbolic dynamics is applied to the ergodic theory of Axiom Adiffeomorphisms.
The material of these notes is taken from the work of many people. I haveattempted to give the main references at the end of each chapter, but no doubtsome are missing. On the whole these notes owe most to D. Ruelle and Ya.Sinai.
To start, recall that (X,B, ) is a probability space if B is a -field ofsubsets of X and is a nonnegative measure on B with (X) = 1. By anautomorphism we mean a bijection T : X X for which
(i) EB iff T1EB,
(ii) (T1E) = (E) for E B .
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2 0 Introduction
If T : X X is a homeomorphism of a compact metric space, a natural-field B is the family of Borel sets. A probability measure on this -field iscalled a Borel probability measure. Let M(X) be the set of Borel probabilitymeasures on X and MT(X) the subset of invariant ones, i.e. MT(X) if(T1E) = (E) for all Borel sets E. For any M(X) one can defineT M(X) by T(E) = (T1E).
Remember that the real-valued continuous functions C(X) on the compactmetric space X form a Banach space under f = maxxX |f(x)|. The weak-topology on the space C(X) of continuous linear functionals : C(X) Ris generated by sets of the form
U(f,,0) = { C(X) : |(f) 0(f)| < }
with f C(X), > 0, 0 C(X).
Riesz Representation. For each M(X) define C(X) by (f) =
f d. Then is a bijection between M(X) and
{ C(X) : (1) = 1 and (f) 0 whenever f 0} .
We identify with , often writing when we mean (). 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 ofC(X). The reader may check thatthe weak topology on M(X) is equivalent to the one defined by the metric
d(, ) =n=1
2n fn1
fnd
fnd
.
Proposition. MT(X) is a nonempty closed subset ofM(X).
Proof. Check that T : M(X) M(X) is a homeomorphism and note thatMT(X) = { M(X) : T = }. Pick M(X) and let n =
1n
( +T + + (T)n1). Choose a subsequence nk converging to
M(X).Then MT(X).
Proposition. MT(X) if and only if(f T) d =
f d for all f C(X) .
Proof. This is just what the Riesz Representation Theorem says about thestatement T = .
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1
Gibbs Measures
A. Gibbs Distribution
Suppose a physical system has possible states 1, . . . , n and the energies of thesestates are E1, . . . , E n. Suppose that this system is put in contact with a muchlarger heat source which is at temperature T. Energy is thereby allowed topass between the original system and the heat source, and the temperature Tof 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 isa physical fact derived in statistical mechanics that the probability pj thatstate j occurs is given by the Gibbs distribution
pj =eEjni=1 e
Ei,
where = 1kT and k is a physical constant.We shall not attempt the physical justification for the Gibbs distribution,
but we will state a mathematical fact closely connected to the physical rea-soning.
1.1. Lemma. Let real numbers a1, . . . , an be given. Then the quantity
F(p1, . . . , pn) =ni=1
pi logpi +ni=1
piai
has maximum value logn
i=1 eai as (p1, . . . , pn) ranges over the simplex
{(p1, . . . , pn) : pi 0, p1 + + pn = 1} and that maximum is assumedonly by
pj = eaj
i
eai
1.
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4 1 Gibbs Measures
This is proved by calculus. The quantity H(p1, . . . , pn) =n
i=1 pi logpi iscalled the entropy of the distribution (p1, . . . , pn) (note: (x) = x log x iscontinuous on [0, 1] if we set (0) = 0.) The term
ni=1piai is of course
the average value of the function a(i) = ai. In the statistical mechanicscase ai = Ei, entropy is denoted S and average energy E. The Gibbsdistribution then maximizes
S E = S 1kT
E,
or equivalently minimizes E kT S. This is called the free energy. The prin-ciple 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 in-teger a physical system with possible states 1, 2, . . . , n. A configuration of the
system consists of assigning an xi {1, . . . , n} for each i:
x2 x1 x0 x1 x2 x3 Thus a configuration is a point
x = {xi}+i= Z
{1, . . . , n} = n .
We now make assumptions about energy:
(1) associated with the occurrence of a state k is a contribution 0(k) to thetotal 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 totheir interaction 2(i1, i2, k1, k2) depends only on their relative position,i.e., there is a function 2 : Z {1, . . . , n} {1, . . . , n} R so that
2(i1, i2, k1, k2) = 2(i1 i2; k1, k2)
(also: 2(j; k1, k2) = 2(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 0thplace is
(x) = 0(x0) +j=0
1
2 2(j; xj , x0).
(We give each of x0 and xk half the energy due to their interaction). Wenow assume that 2j = supk1,k2 |(j; k1, k2)| satisfies
j=1
2j < .
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A. Gibbs Distribution 5
Then (x) R and depends continuously on x when {1, . . . , n} is given thediscrete topology and n =
Z{1, . . . , n} the product topology.
If we just look at xm . . . x0 . . . xm we have a finite system (n2m+1 possible
configurations) and an energy
Em(xm, . . . , xm) =m
j=m
0(xj) +
mj 0, (0, 1) so thatvark ck for allk. Then there is a unique M(n) for which one canfind constants c1 > 0, c2 > 0, and P such that
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6 1 Gibbs Measures
c1 {y : yi = xi i = 0, . . . , m}exp
P m + m1k=0 (kx) c2
for every x n and m 0.This measure is written and called Gibbs measure of . Up to
constants in [c1, c2] the relative probabilities of the x0 . . . xms are given
by expm1
k=0 (kx). For the physical system discussed above one takes
= . In statistical mechanics Gibbs states are not defined by the abovetheorem. We have ignored many subtleties that come up in more complicatedsystems (e.g., higher dimensional lattices), where the theorem will not hold.Our discussion was a gross one intended to motivate the theorem; we refer toRuelle [9] or Lanford [6] for a refined outlook.
For later use we want to make a small generalization of n before we provethe theorem. If A is an n n matrix of 0s and 1s, let
A = {x n : Axixi+1 = 1 i Z} .That is, we consider all x in which A says that xixi+1 is allowable for every i.One easily sees that A is closed and A = A. We will always assumethat A is such that each k between 1 and n occurs at x0 for some x A.(Otherwise one could have A = B with B an m m matrix and m < n.)1.3. Lemma. : A A is topologically mixing (i.e., when U, V are non-empty open subsets of A, there is an N so that
mU V = m N) ifand only if AM > 0 (i.e., AMi,j > 0 i, j) for some M.Proof. One sees inductively that Am
i,jis the number of (m + 1)-strings
a0a1 . . . am of integers between 1 and n with
(a) Aakak+1 = 1 k,(b) a0 = i, am = j.
Let Ui = {x A : x0 = i} = .Suppose A is mixing. Then Ni,j with Ui nUj = n Ni,j . If
a Ui nUj, then a0a1 . . . an satisfies (a) and (b); so Ami,j > 0 i, j whenm maxi,j Ni,j .
Suppose AM > 0 for some M. As each number between 1 and n occurs asx0 for some x A, each row of A has at least one positive entry. From thisit follows by induction that Am > 0 for all m M.
Consider open subsets U, V of A with a U, b V. There is an r sothat U {x A : xk = ak |k| r}
V {x A : xk = bk |k| r} .For t 2r + M, m = t 2r M and Am > 0. Hence find c0, . . . , cm withc0 = br, cm = ar, Ackck+1 = 1 for all k. Then
x = b2b1b0 brc1 cm1ar a0a1 is in A and x tU V. So A is topologically mixing.
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A. Gibbs Distribution 7
Let FA be the family of all continuous : A R for which vark bk(for all k 0) for some positive constants b and (0, 1). For any (0, 1)one can define the metric d on A by d(x, y) =
N where N is the largestnonnegative integer with xi = yi for every |i| < N. Then FA is just the setof functions which have a positive Holder exponent with respect to d . Thetheorem we are interested in then reads
1.4. Existence of Gibbs measures. Suppose A is topologically mixing and FA. There is unique -invariant Borel probability measure on A forwhich one can find constants c1 > 0, c2 > 0 and P such that
c1 {y : yi = xi for all i [0, m)}
expP m + m1
k=0(kx) c2
for every x A and m 0.This theorem will not be proved for some time. The first step is to reduce
the s one must consider.
Definition. Two functions , C(A) are homologous with respect to (written ) if there is a u C(A) so that
(x) = (x) u(x) + u(x) .1.5. Lemma. Suppose 1 2 and Theorem 1.4 holds for 1. Then it holdsfor 2 and 1 = 2 .
Proof. m1k=0
1(kx)
m1k=0
2(kx)
=m1k=0
u(k+1x) u(kx)
= |u(mx) u(x)| 2u .The exponential in the required inequality changes by at most a factor ofe2u when 1 is replaced by 2. Thus the inequality remains valid with c1,c2 changed and P, unchanged. 1.6. Lemma. If FA, then is homologous to some FA with (x) =(y) whenever xi = yi for all i 0.Proof. For each 1 t n pick {ak,t}k= A with a0,t = t. Definer : A A by r(x) = x where
xk =
xk for k 0ak,x0 for k 0 .
Let
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8 1 Gibbs Measures
u(x) =j=0
((jx) (jr(x))) .
Since jx and jr(x) agree in places from j to +,|(jx) (jr(x))| varj bj .
As
j=0 bj < , u is defined and continuous. Ifxi = yi for all |i| n, then,
for j [0, n],|(jx) (jy)| varnj bnj
and|(jr(x)) (jr(y))| bnj .
Hence
|u(x) u(y)| [n2 ]j=0
|(jx) (jy) + (jr(x)) (jr(y))| + 2j>[n2 ]
j
2b
[
n2 ]
j=0
nj +j>[n2 ]
j
4b [n2 ]
1
This shows that u FA. Hence = u + u is in FA also. Furthermore
(x) = (x) +
j=1
(
j+1
r(x)) (j+1
x)
+
j=0
(
j+1
x) (j
r(x))
= (r(x)) +
j=0
(j+1r(x)) (jr(x)) .
The final expression depends only on {xi}i=0, as we wanted. D. Lind cleanedup the above proof for us.
Lemmas 1.5 and 1.6 tell us that in looking for a Gibbs measure for FA (i.e., proving Theorem 1.4) we can restrict our attention to functions for which (x) depends only on {xi}i=0.
B. Ruelles Perron-Frobenius Theorem
We introduce now one-sided shift spaces. One writes x for {xi}i=0 (we willcontinue to write x for {xi}i= but never for both things at the same time).Let
+A =
x
i=0
{1, . . . , n} : Axi,xi+1 = 1 for all i 0
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B. Ruelles Perron-Frobenius Theorem 9
and define : +A +A by (x)i = xi+1. is a finite-to-one continuousmap of +A onto itself. If C(+A) we get C(A) by ({xi}i=) =({xi}i=0). Suppose C(A) satisfies (x) = (y) whenever xi = yifor all i 0. Then one can think of as belonging to C(+A) as follows:({xi}i=0) = ({xi}i=) where xi for i 0 are chosen in any way subjectto {xi}i= A. The functions C(+A) are thus identified with a certainsubclass ofC(A). We saw in Lemmas 1.5 and 1.6 that one only needs to getGibbs measures for C(+A ) FA in order to get them for all FA.
In this section we will prove a theorem of Ruelle that will later be usedto construct and study Gibbs measures. For C(+A) define the operatorL = L on C(+A ) by
(Lf)(x) = y 1x
e(y)
f(y) .
It is the fact that is not one-to-one on +A that will make this operatoruseful.
1.7. Ruelles Perron-Frobenius Theorem [10, 11]. Let A be topolog-ically mixing, FA C(+A ) and L = L as above. There are > 0,h C(+A ) with h > 0 and M(+A ) for which Lh = h, L = ,(h) = 1 and
limm
mLmg (g)h = 0 for all g C(+A) .
Proof. BecauseL
is a positive operator andL
1 > 0, one has that G() =(L(1))1L M(+A ) for M(+A). There is a M(+A ) withG() = by the Schauder-Tychonoff Theorem (see Dunford and Schwartz,Linear Operators I, p. 456): Let E be a nonempty compact convex subset ofa locally convex topological vector space. Then any continuous G : E Ehas a fixed point. In our case G() = gives L = with > 0.
We will prove 1.7 via a sequence of lemmas. Let b > 0 and (0, 1) be anyconstants so that vark bk for all k 0. Set Bm = exp
k=m+1 2b
k
and define
= {f C(+A ) : f 0, (f) = 1 , f(x) Bmf(x),whenever xi = x
i for all i [0, m]} .
1.8. Lemma. There is an h
with
Lh = h and h > 0.
Proof. One checks that 1Lf when f . Clearly 1Lf 0 and(1Lf) = 1L(f) = (f) = 1 .
Assume xi = xi for i [0, m]. Then
Lf(x) =j
e(jx)f(jx)
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10 1 Gibbs Measures
where the sum ranges over all j with Ajx0 = 1. For x the expression runsover the same j; as jx and jx agree in places 0 to m + 1
e(jx)f(jx) e(jx)ebm+1Bm+1f(jx) Bme(jx)f(jx)
and soLf(x) BmLf(x) .
Consider any x, z +A . Since AM > 0 there is a y Mx with y0 = z0.For f
LMf(x) =yMx
exp
M1
k=0(ky)f(y)
eMf(y) .
Let K = MeMB0. Then 1 = (MLMf) K1f(z) gives f K as z
is arbitrary. As (f) = 1, f(z) 1 for some z and we get infMLMf K1.If xi = x
i for i [0, m] and f , one has
|f(x) f(x)| (Bm 1)K 0
as m , since Bm 1. Thus is equicontinuous and compact bythe Arzela-Ascoli Theorem. = as 1 . Applying Schauder-TychonoffTheorem to 1L : gives us h with Lh = h. Furthermoreinfh = infM
LMh
K1.
1.9. Lemma. There is an (0, 1) so that for f one has MLMf =h + (1 )f with f .Proof. Let g = MLMf h where is to be determined. Provided h K1 we will have g 0. Assume xi = xi for all i [0, m]. We want to pick so that g(x) Bmg(x), or equivalently
() (Bmh(x) h(x)) BmMLMf(x) MLMf(x) .
We saw above that Lf1(x) Bm+1 ebm+1Lf1(x) Bm+1 ebmLf1(x) forany f1 . Applying this to f1 = M+1LM1f one has
M
LM
f(x) Bm+1 ebm
M
LM
f(x
) .
Now h(x) B1m h(x) because h . To get () it is therefore enough tohave
(Bm B1m )h(x) (Bm Bm+1ebm
) MLMf(x)or
(Bm B1m )h (Bm Bm+1ebm
)K1 .
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B. Ruelles Perron-Frobenius Theorem 11
There is an L so that the logarithms ofBm, B1m and Bm+1ebm
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 > 0 to satisfy
hu1(log Bm + log Bm) K1u2(log Bm log(Bm+1ebm
))
or
hu1
4bm+1
1
K1u2bm
or
u2(1 )(4u1hK)1 . 1.10. Lemma. There are constants A > 0 and (0, 1) so that
nLnf h An
for all f , n 0.Proof. Let n = M q + r, 0 r < M. Inductively one sees from Lemma 1.9and Lh = h that, for f ,
MqLMqf = (1 (1 )q)h + (1 )qfqwhere fq
. As
fq
K one has
MqLMqf h (1 )q(h + K)
and
nLnf h = rLr(MqLMqf h) A(1 )q+1 An,
whereA = (1 )1(h + K) sup
0r
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12 1 Gibbs Measures
Proof. Assume xi = xi for i [0, m]. Then
Lr(f F)(x) =
j1jrx
exp
r1k=0
(k(j1 jrx))
f(j1 jrx)F(j1 jrx)
where j1 jr runs over all r-strings of symbols for which j1 jrx +A .In the expression for Lr(f F)(x) one has j1 jr running over the samer-strings. Now f(j1 jrx) = f(j1 jrx) as f Cr, F(j1 jrx) Bm+rF(j1 jrx), and (k(j1 jrx)) (k(j1 jrx)) + varm+rk.Since
Bm+r expr1k=0var
m+rk Bm+r exp
m+rj=m+1 b
j Bm,each term in the above expression for Lr(f F)(x) is bounded by Bm times thecorresponding term for Lr(f F)(x). Hence Lr(f F)(x) BmLr(f F)(x).
One must still show (f F) > 0. Reasoning as in the proof of 1.8 (withLr(f F) in place of f) we get
r(f F) = (MLM+r(f F)) K1Lr(f F)(z),
for any z. But (f F)(w) > 0 gives Lr(f F)(rw) > 0 and so (f F) > 0. 1.12. Lemma. For f Cr, F and n 0,
nrLn+r(f F) (f F)h A(|f F|)n .For g C(+A ) one has limm mLmg (g)h = 0.Proof. Write f = f+ f with f+, f 0 and f+, f Cr. Then
nrLn+r(fF) (fF)h A(|fF|)n.
For fF 0, this is obvious; for fF 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 > 0 one can find r and f1, f2 Cr so that f1 g f2 and0 f2 f1 . As |(fi) (g)| < , the first statement of the lemma withF = 1 gives
mLm(fi) (g)h (1 + h)for large m. Since mLmf1 mLmg mLmf2,
mLmg (g)h (1 + h)
for large m. The proof of 1.7 is finished.
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C. Construction of Gibbs Measures 13
C. Construction of Gibbs Measures
We continue to assume that FA C(+A ) and , h, are as in RuellesPerron-Frobenius Theorem. Then = h is a probability measure on +A ;(f) = (hf) =
f(x)h(x) d(x).
1.13. Lemma. is invariant under : +A +A .Proof. We need to show that (f) = (f ) for f C(+A ). Notice that
((Lf) g)(x) =
y1x
e(y)f(y)g(x)
=
y1xe(y)
f(y)g(y)
= L(f (g ))(x) .
Using this
(f) = (hf)
= (1Lh f)= 1(L(h (f )))= 1(L)(h (f ))= (h (f ))= (f
) .
Because is -invariant on +A there is a natural way to make into a
measure on A. For f C(A) define f C(+A) by
f({xi}i=0) = min{f(y) : y A, yi = xi for all i 0} .
Notice that for m, n 0 one has
(f n) m (f n+m) varnf .
Hence
|((f
n))
((f
n+m))
|=
|((f
n)
m)
((f
n+m))
| varnf
which approaches 0, as n , since f is continuous.Hence (f) = l i mn ((f n)) exists by the Cauchy criterion. It isstraightforward to check that C(A). By the Riesz RepresentationTheorem we see that defines a probability measures on A, which we willdenote by despite the possible ambiguity. Note that
(f ) = limn
((f n+1)) =
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14 1 Gibbs Measures
proving that is -invariant. Also (f) = (f) for f C(A).Recall that is ergodic if (E) = 0 or 1 whenever E is a Borel set with
1E = E. One calls mixing if
limn
(E nF) = (E)(F),
for all Borel sets E and F. It is clear that mixing implies ergodicity and astandard argument shows that the mixing condition only need be checked forE and F in a basis for the topology.
1.14. Proposition. is mixing for : A A.Proof. Writing Sm(x) =
m1k=0 (
kx) one checks inductively that for f,
g C(+A) one has
(Lmf)(x) =
ymx
eSm(y)f(y) .
Then
((Lmf) g)(x) =
ymx
eSm(y)f(y)g(my)
= Lm(f (g m)) .Let
E =
{y
A : yi = ai, r
i
s
},
F = {y A : yi = bi, u i v} .In checking the mixing condition for E and F we may assume r = u = 0because is -invariant. We want to calculate
(E nF) = (E nF)= (E (F n))= (hE (F n))= nLn(hE (F n))= (nLn(hE (F n)))= (nLn(hE) F) .
Now
|(E nF) (E)(F)| = |(E nF) (hE)(hF)|= |(nLn(hE) (hE)h)F | nLn(hE) (hE)h (F) .
Because E Cs Lemma 1.12 gives, for n s,
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C. Construction of Gibbs Measures 15
nLn(hE) (hE)h A(E)nswhere (0, 1). One then has
|(E nF) (E)(F)| A(E)(F)ns
for n s where A = A (infh)1. Thus (E nF) (E)(F). 1.15. Lemma. Let a =
k=0 vark < . If x, y A with xi = yi for
i [0, m), then|Sm(x) Sm(y)| a .
Proof. Define y by
y
i = yi for i
0
xi for i 0 .Since C(+A), (ky) = (ky) for k 0. Hence
|Sm(x) Sm(y)| m1k=0
|(kx) (ky)|
m1k=0
varm1k
a . We now complete the proof of 1.4.
1.16. Theorem. is a Gibbs measure for FA C(+A ).Proof. Let E = {y A : yi = xi for i [0, m)}. For any z +A there isat most one y mz with y E. Thus, using 1.15,
Lm(hE)(z) =
ymz
eSm(y)h(y)E(y)
eSm(x)ea hand so
(E) = (hE)= m(
Lm(h
E
))
meSm(x)ea h .Thus take c2 = e
ah. On the other hand, for any z +A there is at leastone y mMz with y E. Then
Lm+M(hE)(z) eSm+M(y)h(y)
eMa (infh) eSm(x)
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16 1 Gibbs Measures
and(E) = mM(Lm+M(hE)) c1meSm(x)
where c1 = MeMa. We have verified the desired inequalities on mea-
sures of cylinder sets given in 1.4 with P = log .Suppose now that is any other measure satisfying the inequalities in
Theorem 1.4 with constants c1, c2, P
. For x A let Em(x) = {y A :yi = xi for all i [0, m)}. Let Tm be a finite subset of A so that A =
xTmEm(x) disjointly. Then
c1ePm
xTm
eSm(x) xTm
(Em(x))
= 1
c2ePmxTm
eSm(x) .
From this one sees that P = limm1m log
xTm
eSm(x)
. One can
apply the same reasoning to ; hence P = P as they equal the same limit.The estimates on (Em(x)) and (Em(x)) give us
(Em(x)) d(Em(x))where d = c2c
11 . Taking limits this extends to
(E) d(Em) for all Borelsets E. In particular (E) = 0 when (E) = 0. By the Radon-NikodymTheorem = f for some -integrable f. Applying one has
=
= (f
)
= (f ) .As the Radon-Nikodym derivative is unique up to -equivalence, f a.e.= f.Because is ergodic this gives f equivalent to some constant c.
1 = (A) =
c d = c and = .
D. Variational Principle
We will describe Gibbs measures as those maximizing a certain quantity, in away analogous to Lemma 1.1. IfC = {C1, . . . , C k} is a partition of a measurespace (X,B, ) (i.e., the Cis are pairwise disjoint and X =
ki=1 Ci), one
defines the entropy
H(C) =k
i=1
((Ci)log (Ci)) .
IfD is another (finite) partition,
C D = {Ci Dj : Ci C , Dj D}.
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18 1 Gibbs Measures
1.19. Lemma. IfD is a (finite) partition of (X,B, ) and T an automor-phism of (X,B, ), then
h(T,D) = limm
1
mH(D T1D Tm+1D)
exists.
Proof. Let am = H(D T1D Tm+1D). Then
am+nH(DT1DTm+1D)+H(TmDTmn+1D)am+ansince
TmD
Tmn+1D = Tm(D
Tn+1D)
and is T-invariant. Definition. Let M(A) andU = {U1, . . . , U n} where Ui = {x A :x0 = i}. Then s() = h(,U) is called the entropy of .
Suppose now that C(A) and that a0a1 am1 are integers between1 and n satisfying Aakak+1 = 1. Write
supa0a1am1
Sm = sup
m1k=0
(kx) : x A, xi = ai for all 0 i < m
and
Zm() =
a0a1am1
exp
supa0a1am1
Sm
.
1.20. Lemma. For C(A), P() = limm 1m log Zm() exists (calledthe pressure of ).
Proof. Notice that
supa0a1am+n1
Sm+n supa0a1am1
Sm + supamam+n1
Sn .
From this one gets Zm+n() Zm()Zn(); the terms in Zm+n() arebounded by terms in Zm()Zn() and Zm()Zn() may have more terms,
all positive. Apply Lemma 1.18 to am = log Zm() (notice am m). 1.21. Proposition. Suppose C(A) and M(A). Then
s() +
d P() .
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D. Variational Principle 19
Proof. As
kd = d, 1m Smd = d where Sm(x) =m1k=0 (
kx). Hence
s() +
d lim
m
1
m
H(U m+1U) +
Sm d
.
Now U m+1U partitions points x A according to x0x1 xm1.Thus
H(U m+1U) +
Sm d
a0am1
(a0 am1)( log (a0 am1)) + supa0a1am1
Sm
log Zm() by Lemma 1.1 .Now let m . 1.22. Theorem. Let FA, A topologically mixing and the Gibbs mea-sure of . Then is the unique M(A) for which
s() +
d = P() .
Proof. Given a0 am1, pick x with xi = ai (i = 0, . . . , m 1) andSm(x) = sup
a0a1am1
Sm .
Now, as = is the Gibbs measure,{y A : yi = ai 0 i < m}
exp(P m + Sm(x)) [c1, c2] .
Summing the measure of these sets over all possible a0 am1s gives 1; soc1 exp(P m)Zm() 1 c2 exp(P m)Zm()
orZm()
exp(P m) [c12 , c11 ] .
It follows that P() = limm1m log Zm() = P.
If yi = xi for all i = 0, . . . , m
1, then
|Sm(y) Sm(x)| m1k=0
|(ky) (ky)|
var0 + var1 + + var[m2 ] + varm[m2 ] + + var0
2c[m2 ]k=0
k 2c1 = d .
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20 1 Gibbs Measures
Hence, if B = {y A : yi = ai for all i = 0, . . . , m 1}, then for x B,(B)log (B) +
B
Sm d (B) [ log (B) Sm(x) + d ]
(B) [ log
c2ePm+Sm(x)
Sm(x) + d ]
(B)(P m log c2 d) .Since
H(U m+1U) +
Sm d
= B (B)log (B) +
B
Sm dB
(B)(P m log c2 d) = P m log c2 d,
we get
s() +
d = lim
m
1
m
H(U m+1U) +
Sm d
limm
1
m(P m log c1 d) = P = P() .
The reverse inequality was in Proposition 1.21. So
s() + d = P() .To prove uniqueness we will need a couple of lemmas.
1.23. Lemma. Let X be a compact metric space, M(X), and D ={D1, . . . , Dn} a Borel partition of X. Suppose {Cm}m=1 is a sequence of par-titions so that diam(Cm) = maxCCm diam(C) 0 as m . Then thereare partitions {Em1 , . . . , E mn } so that1. each Emi is a union of members ofCm,2. limm (E
mi Di) = 0 for each i.
Proof. Pick compacts K1, . . . , K n with Ki Di and (Di\Ki) < . Let = infi=j d(Ki, Kj) and consider m with diam(Cm) 2 . Divide the elementsC
Cm into groups whose unions are E
m1 , . . . , E
mn so that
C Emi if C Ki = .As diam(Cm) 2 any C Cm can intersect at most one Ki. Put a C hittingno Ki in any E
mi you like. Then E
mi Ki and
(Emi Di) = (Di\Emi )+(Emi \Di) +
X\ni=1
Ki
(n+1) .
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References 27
8. A. N. Livsic: Certain properties of the homology ofY-systems, Mat. Zametki10(1971), 555564. English translation: Math. Notes Acad. Sci. USSR 10 (1971),758763.
9. D. Ruelle: Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., NewYork-Amsterdam 1969. (1)
10. D. Ruelle: Statistical mechanics of a one-dimensional lattice gas, Comm. Math.Phys. 9 (1968) 267278.
11. D. Ruelle: A measure associated with axiom-A attractors, Amer. J. Math. 98(1976), no. 3, 619654.
12. Ya. Sinai: Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no.4, 2164. English translation: Russian Math. Surveys 27 (1972), no. 4, 2169.
13. M. Ratner: The central limit theorem for geodesic flows on n-dimensional man-ifolds of negative curvature, Israel J. Math. 16 (1973), 181197.
1 Reprint: World Scientific Publishing Co., Inc., River Edge, NJ; Imperial CollegePress, London, 1999 (note of the editor).
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48 3 Axiom a Diffeomorphisms
Notice that f Xp = Xf(p) since f Wu(p) = Wu(f(p)). If q Xp as above,then Wu (q) Xp and
Wu(q) =k0
fkmnWu (q)
k0
fkmnXp = Xp .
(Note that y Wu(q) iff fkmny q as k .) It follows that Xq Xp.If x is as above, then fkmnx Xq for large k as Xq = B(Xq) is open in .As fimnXq = Xfimnq = Xq, one has f
jmnx Xq for all j and
p = limj f
jmn
x
Xq = Xq .
The above argument with the roles of p and q reversed gives Xp Xq. Insummary, 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 and hence contains a periodic point r; thenXp = Xr = Xq. Now
=
p periodic
B(Xp) =p
Xp,
and so by compactness (the Xp are open) let
= Xp1 Xpt
with the Xpj s pairwise disjoint. Then f(Xpj ) = Xfpj intersects and henceequals some Xpi . So f permutes the Xpj s and the i are just the union ofthe Xpj s in the various cycles of the permutation.
The transitivity in (a) is implied by the mixing in (b). We finish by showingfN : Xr Xr is mixing whenever r is periodic and N positive with f
NXr =Xr. Suppose U, V are nonempty subsets of Xr open in Xr (i.e., in ). Pickperiodic points p U and q V, say fmp = p, fnq = q. For each 0 j < mnwith fjp Xr one can find a point x
j as in the beginning of this proof so
thatxj f
jU and fkmnxj V for large k .
Writing tN = kmn +j, 0 j mn, we have f
j
p = f
tN
p Xr andfkmnxj = f
tN(fjNxj) ftNU V
provided k is large. Then ftNU V = for large t and fN|Xr is topologicallymixing.
The i in the spectral decomposition of (f) are called the basic sets off. Notice that if g = fn and n is a multiple of every ni, then the basic sets
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52 3 Axiom a Diffeomorphisms
sR = {x R : x / int(Wu(x, R))}uR = {x R : x / int(Ws(x, R))}
and the interiors ofWu(x, R), Ws(x, R) are as subsets of Wu (x), Ws (x)
.
Proof. Ifx int(R), then Wu(x, R) = R(Wu (x)) is a neighborhood ofxin Wu (x) since R is a neighborhood ofx in . Similarly x int(W
u(x, R)).Suppose x int(Wu(x, R)) and x int(Ws(x, R)). For y s near x thepoints
[x, y] Ws (x) and [x, y] Wu (x)
depend continuously on y. Hence for y s close enough to x, [x, y] R and
[y, x] R. Then
y = [[y, x], [x, y]] R Ws (y) Wu (y)
and y = y as Ws (y) Wu (y) = {y}. Thus x int(R).
Definition. A Markov partition of s is a finite coveringR = {R1, . . . , Rm}of s by proper rectangles with
(a) int(Ri) int(Rj) = for i = j,(b) f Wu(x, Ri) W
u(fx,Rj) andf Ws(x, Ri) W
s(fx,Rj) when x int(Ri), f x int(Rj).
3.12. Theorem. Let s be a basic set for an Axiom A diffeomorphism f.
Then s has Markov partitions R of arbitrarily small diameter.
Proof. Let > 0 be very small and choose > 0 small as in Proposition 3.6,i.e., every -pseudo-orbit in s is -shadowed in s. Choose < /2 so that
d(fx,fy) < /2 when d(x, y) < .
Let P = {p1, . . . , pr} be a -dense subset of s and
(P) =
q
P : d(f qj , qj+1) < for all j
.
For each q (P) there is a unique (q) s which -shadows q; for each
x s there are q with x = (q).For q, q (P) with q0 = q
0 we define q
= [q, q] (P) by
qj =
qj for j 0qj for j 0 .
Then d(fj(q), fj(q)) 2 for j 0 and d(fj(q), fj(q)) 2 for j 0.So (q) Ws2((q)) W
u2((q
)), i.e.,
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C. Markov Partitions 53
[q, q] = [(q), (q)] .
We now see that Ts = {(q) : q (P), q0 = ps} is a rectangle. Forx, y Ts we write x = (q), y = (q
) with q0 = ps = q0. Then
[x, y] = [q, q] Ts .
Suppose x = (q) with q0 = ps and q1 = pt. Consider y Ws(x, Ts), y = (q
),q0 = ps. Then
y = [x, y] = [q, q] and
f y = ([q, q]) Tt
as [q, q] has q = pt in its zeroth position. Since f y Ws (f x) (diam(Ts)
2 is small compared to ), f y Ws
(fx,Tt). We have proved(i) f Ws(x, Ts) W
s(fx,Tt).A similar proof shows f1Wu(fx,Tt) Wu(x, Ts), i.e.,(ii) f Wu(x, Ts) W
u(fx,Tt).Each Ts is closed; this follows from the following lemma.
3.13. Lemma. : (P) s is continuous.
Proof. Otherwise there is a > 0 so that for every N one can find qN
, qN
(P) with qj,N = q
j,N for all j [N, N] but d((qN), (q
N)) . If xN =
(qN
) , yN = (q) one has
d(fjxN , fjyN) 2 j [N, N] .
Taking subsequences we may assume xN x and yN y as N . Thend(fjx, fjy) 2 for all j and d(x, y) ; this contradicts expansiveness off|s .
Now T = {T1, . . . , T r} is a covering by rectangles and (i) and (ii) aboveare like the Markov condition (b). However the Tj s are likely to overlap andnot be proper. For each x s let
T(x) = {Tj T : x Tj} and T(x) = {Tk T : TkTj = for some Tj T(x)}.
As T is a closed cover of s, Z = s\j Tj is an open dense subset of s.
In fact, using arguments similar to 3.11, one can show that
Z = {x s : Ws (x)
sTk = and Wu (x)
uTk = for all Tk T(x)}
is open and dense in s.For Tj Tk = , let
T1j,k = {x Tj : Wu(x, Tj) Tk = , W
s(x, Tj) Tk = } = Tj Tk
T2j,k = {x Tj : Wu(x, Tj) Tk = , W
s(x, Tj) Tk = }
T3j,k = {x Tj : Wu(x, Tj) Tk = , W
s(x, Tj) Tk = }
T4j,k = {x Tj : Wu(x, Tj) Tk = , W
s(x, Tj) Tk = }.
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54 3 Axiom a Diffeomorphisms
T4j,k T3j,k
T2j,k T1j,k
Tk
Tj
W
s
Wu
If x, y Tj, then Ws([x, y], Tj) = W
s(x, Tj) and Wu([x, y], Tj) =
Wu(y, Tj); this implies Tnj,k is a rectangle open in s and each x Tj Z
lies in int(Tnj,k) for some n. For x Z define
R(x) = {int(Tnj,k) : x Tj , Tk Tj = and x T
nj,k} .
Now R(x) is an open rectangle (x Z). Suppose y R(x) Z. SinceR(x) T(x) and R(x) Tj = for Tj / T(x), one gets T(y) = T(x). ForTj T(x) = T(y) and Tk Tj = , y lies in the same T
nj,k as x does since
Tnj,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 finitelymany Tnj,ks 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 s, R(x) \R(x) has no interior (in s) 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 f1Z, R(x) = R(y) and y Ws (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 = (q) with q1 = pj and q0 = ps. Then x = (q) Ts and byinclusion (i) above
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C. Markov Partitions 55
f y f Ws(x, Ts) Ws(fx,Tj) ,
contradicting f y / Tj . Now let fx,fy Tj and Tk Tj = . We want to showthat fx,fy belong to the same Tnj,k. As f y W
s (f x) we have W
s(f y , T j) =Ws(fx,Tj). We will derive a contradiction from
Wu(f y , T j) Tk = , f z Wu(fx,Tj) Tk .
Recall that f x = (q), q1 = pj , q0 = ps. Then by inclusion (ii)
f z Wu(fx,Tj) f Wu(x, Ts) or z W
u(x, Ts) .
Let f z = (q); q1 = pk and q0 = pt. Then z Tt and f Ws(z, Tt)
W
s
(f z , T k). Now Ts T
(x) =T
(y) and z Tt Ts = .Now z Wu(x, Ts) Tt and so there is some z Wu(y, Ts) Tt as x, y are
in the same Tns,t. Then
z = [z, y] = [z, z] Ws(z, Tt) Wu(y, Ts),
and f z = [f z , f y] Ws(f z , T k)Wu(f y , T j) (using f z , f y Tj a rectangle),
a contradiction. So R(f x) = R(f y).For small > 0 the sets
Y1 =
Ws (z) : z
j
sTj
and Y2 =
Wu (z) : z
j
uTj
are closed and nowhere dense (like in the proof of 3.11). Now Z s\(Y1Y2)is open and dense. Furthermore if x / (Y1 Y2) f
1(Y1 Y2) then x Z f1Z and the set of y Ws(x, R(x)) with y Z f1Z is open anddense in Ws(x, R(x) ) (as a subset of Ws (x) ). By the previous paragraphR(f y) = R(f x) for such y; by continuity
f Ws(x, R(x) ) R(f x) .
As f Ws(x, R(x) ) Ws (f x), f Ws(x, R(x) ) Ws(fx,R(f x) ).
If int(Ri) f1int(Rj) = , then this open subset of s contains some x
satisfying the above conditions, Ri = R(x) and Rj = R(f x). For any x
Ri f1Rj one has W
s(x, Ri) = {[x, y] : y Ws(x, Ri)} and
f Ws(x, Ri) = {[f x, f y] : y Ws(x, Ri)} {[f x, z] : z Ws(fx,Rj)}
Ws(f x, Rj) .
This completes the proof of half of the Markov conditions (b). The otherhalf is proved similarly and the proof is omitted. Alternatively one could applythe above to f1, noting that Wuf = W
sf1
.
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4
Ergodic Theory of Axiom a Diffeomorphisms
A. Equilibrium States for Basic Sets
Recall that a function is Holder continuous if there are constants a, > 0so that
|(x) (y)| a d(x, y) .
4.1. Theorem. Let s be a basic set for an Axiom A diffeomorphism f and : s R Holder continuous. Then has a unique equilibrium state (w.r.t. f|s). Furthermore is ergodic; is Bernoulli iff|s is topologicallymixing.
4.2. Lemma. There are > 0 and (0, 1) for which the following are true:ifx s, y M, andd(f
kx, fky) for allk [N, N], thend(x, y) < N.
Proof. See p. 140 of [12].
Proof of 4.1. Let R be a Markov partition for s of diameter at most , Athe transition matrix for R and : A s as in 3.D. Let
= . Ifx, y A have xk = yk for k [N, N], then
fk(x) , fk(y) Rxk = Ryk for k [N, N] .
This gives d((x), (y)) < N, |(x) (y)| a ()N and FA.First we assume f|s is mixing. Then |A is mixing by 3.19 and we have a
Gibbs measure as in Chapter 1. Let Ds = 1(sR) and Du = 1(uR).Then Ds and Du are closed subsets of A, each smaller than A, andDs Ds,
1Du Du. As is -invariant, (nDs) = (Ds);
using n+1Ds nDs one has
n0
nDs
= (Ds) .
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A. Equilibrium States for Basic Sets 63
4.4. Proposition. Let : s R be Holder continuous and P = Pf|s ().For small > 0 there is a b > 0 so that, for any x s and for all n,
y s : d(fky, fkx) < k [0, n]
b exp(P n + Sn(x)) .
Proof. Choose the Markov partition R above to have diam(R) < . Assumefirst f|s is mixing. Pick x A with (x) = x. Then
B =
y s : d(fky, fkx) < k [0, n]
y A : yk = xk k [0, n]
.
Applying 1.4 and P() = P() = P one gets
(B) c1
exp(P n + Sn
(x)) .
We leave it to the reader to reduce the general case to the mixing one as inthe proof of 4.1.
4.5. Proposition. Let , : s R be two Holder continuous functions.Then the following are equivalent:
(i) = .(ii) There are constants K and L so that |Sm(x) Sm(x) Km| L for
all x s and all m 0.(iii) There is a constant K so that Sm(x) Sm(x) = Km when x s
with fmx = x.(iv) There is a Holder function u : s R and a constant K so that (x)
(x) = K+ u(f x) u(x).If these conditions hold, K = P() P().
Proof. Let = and = . We assume f|s is mixing and leavethe reduction to this case to the reader. If = , then = and byTheorem 1.28 there are K and L so that
|Sm(x) Sm
(x) Km| L
for x A. For x s, picking x 1(x), this gives us (ii).
Assume (ii) and fmx = x. Then
L |Smj(x) Smj(x) mjK| = j|Sm(x) Sm(x) mK| .
Letting j we get (iii). If (iv) is true, then
(x) (x) = K+ u((x)) u((x))
and = by Theorem 1.28. One then has = =
= .Now we assume (iii) and prove (iv). Let (x) = (x) (x) K and pick
x s with dense forward orbit (Lemma 1.29). Let A = {fkx : k 0} and
define u : A R by
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B. The Case = (u) 65
4.6. Lemma. If s is aC2 basic set, then (u) : s R is Holder continu-ous.
Proof. The map x Eux is Holder (see 6.4 of [12]) and Eux
(u)(x) isdifferentiable, so the composition x (u)(x) is Holder.
By Theorem 4.1 the function (u) has a unique equilibrium state which wedenote + = (u) . While
(u) depends on the metric used, when fmx = x
Sm(u)(x) = log Jac(Dfm : Eux E
ux )
does not depend on the metric (this Jacobian is the absolute value of thedeterminant). By 4.5 one sees that the measure + on s and P(
(u)) do not
depend on which metric is used.
4.7. Volume Lemma. Let
Bx(, m) =
y M : d(fkx, fky) for all k [0, m)
.
If x s is aC2 basic set and > 0 is small, then there is a constant C sothat
m(Bx(, m)) [C1 , C] exp(Sm
(u)(x))
for all x s.
Proof. See 4.2 of [9].
4.8. Proposition. Let s be aC2
basic set.(a) Letting B(, n) =
xs
Bx(, n), one has (for small > 0)
Pf|s ((u)) = lim
n
1
nlog m(B(, n)) 0 .
(b) LetWs (s) =xs
Ws (x). If m(Ws (s)) > 0, then
Pf|s ((u)) = 0 and h+(f) =
(u)d+ .
Proof. Call E M (n, )-separated if whenever y, z are two distinct pointsin E, one can find k [0, n) with d(fky, fkz) > . Choose En() maximal
among the (n, )-separated subsets ofs. For x s one has x By(, n) forsome y En(); otherwise En() {x} is (n, )-separated. Then Bx(, n) By( + , n), B(, n)
yEn()
By( + , n) and by 4.7
() m(B(, n)) C+
yEn()exp(Sn(
u)(y)).
For ,yEn()
By(/2, n) B(, n) is a disjoint union and so
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66 4 Ergodic Theory of Axiom a Diffeomorphisms
() m(B(, n)) C1/2
yEn() exp(Sn(u)(y)).
Since (u) is Holder, we have
|(u)(x) (u)(y)| a d(x, y)
for some a, > 0 and all x, y s. Suppose x By(, n) s. Then forj [0, n)
d(fjx, fjy) < min{j,nj1}
by Lemma 4.2. Hence
|Sn(x) Sn(y)| n1
j=0
|(u)(fjx) (u)(fjy)|
2ak=0
k = .
Fix and let U be an open cover of s with diam(U) < . Let Un cover s. For each y En() pick Uy with y X(Uy). Then
Sn(u)(Uy) Sn
(u)(y). If Uy = Uy , then d(fky, fky) diam(U) < and
y = y as En() is (n, )-separated. ThusU
exp(Sn(u)(U))
yEn()
exp(Sn(u)(y)).
Using this together with () above one gets
P((u),U) = limn
1
nlog inf
U
exp(Sn(u)(U))
limsupn
1
nlog m(B(, n)) .
Letting diam(U) 0, one replaces P((u),U) with P((u)).Now let U be an open cover and let be a Lebesgue number for U. For
each y En() one can pick Uy Un with By(, n) s X(Uy).
Let = {Uy : y En()}. Then covers s since every x s lies in someBy(, n) with y En(). Also
Sn(u)(Uy) Sn(u)(y) +
and so
Zn((u),U)
Uy
n
exp(Sn(u)(Uy))
e
yEn()
exp(Sn(u)(y)) .
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68 4 Ergodic Theory of Axiom a Diffeomorphisms
4.10. Second Volume Lemma. Let s be aC2 basic set. For small , > 0there is a d = d(, ) > 0 so that
m(By(, n)) d m(Bx(, n))
whenever x s and y Bx(, n).
Proof. See 4.3 of [9].
4.11. Theorem. Let s be aC2 basic set. The following are equivalent:
(a) s is an attractor.(b) m(Ws(s)) > 0.(c) Pf|s (
(u)) = 0.
Proof.As Ws(
s) =
n=0fnWs
(
s), (b) is equivalent to m(Ws
(
s)) > 0.
If s is an attractor, then (b) is true since Ws (s) is a neighborhood of s.
(c) follows from (b) by Proposition 4.8 (b). We finish by assuming s is notan attractor and showing P((u)) < 0.
Given a small > 0 choose as in 4.9. Pick N so that
fNWu/4(x) Wu (f
Nx)
for all x s. Let E s be (, n)-separated. For x E there is a y(x, n) Bx(/4, n) with
d(fn+Ny(x, n), s) >
(since fnBx(/4, n) Wu/4(f
nx) and fNWu/4(fnx) Wu (f
N+nx)). Choose
(0, /4) so that d(fNz, fNy) < /2 whenever d(z, y) < . Then
By(x,n)(, n) Bx(/2, n),
fn+NBy(x,n)(, n) Bs(/2) = .
Hence By(x,n)(, n) B(/2, n + N) = . Using the Second Volume Lemma
m(B(/2, n)) m(B(/2, n + N)) xE
m(By(x,n)(, n))
d(3/2, )xE
m(Bx(3/2, n))
d(3/2, ) m(B(/2, n)) .
Therefore, setting d = d(3/2, )
m(B(/2, n + N)) (1 d) m(B(/2, n))
and by Proposition 4.8 (a)
Pf|s ((u))
1
Nlog(1 d) < 0 .
Remark. It is possible to find aC1 basic set (a horseshoe) which is not anattractor but nevertheless has m(Ws(s)) > 0 [8].
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C. Attractors and Anosov Diffeomorphisms 69
C. Attractors and Anosov Diffeomorphisms
Because M =rk=1 W
s(k), Theorem 4.11 implies that m-almost all x Mapproach an attractor under the action of a C2 Axiom A diffeomorphism f.This leads us next to the following result.
4.12. Theorem. Let s be aC2 attractor. For m-almost allx Ws(s) one
has
limn
1
n
n1k=0
g(fkx) =
g d+
for all continuous g : M R (i.e., x is a generic point for +).
Proof. Let us write g(n, x) = 1n
n1k=0 g(fkx) and g =
g d+. Fix > 0 anddefine the sets
Cn(g, ) = {x M : |g(n, x) g| > }
E(g, ) = {x M : |g(n, x) g| > for infinitely many n}
=
N=1
n=N
Cn(g, ) .
Choose > 0 so that |g(x) g(y)| < when d(x, y) < .Now fix N > 0 and choose sets RN, RN+1, . . . successively as follows.
Let Rn (n N) be a maximal subset of s Cn(g, 2) satisfying the condi-tions:
(a) Bx(, n) By(, k) = for x Rn, y Rk, N k < n,(b) Bx(, n) Bx(, n) = for x, x
Rn, x = x.
If y Ws (s) Cn(g, 3) (n N) and y Ws (z) with z s, then
z Cn(g, 2) by the choice of . By the maximality of Rn one has
Bz(, n) Bx(, k) = for some x Rk, N k n .
Then y Bz(, n) Bz(, k) Bx(2, k) and so
Ws (s)
n=N
Cn(g, 3)
k=N
xRk
Bx(2, k) .
Using the Volume Lemma 4.7 one gets
m
Ws (s)
n=N
Cn(g, 3)
c2
k=N
xRk
exp(Sk(u)(x)).
The definition of Rn shows that VN =k=N
xRk
Bx(, k) is a disjointunion. The choice of gives Bx(, k) Ck(g, ) for x Rk Ck(g, 2) and so
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C. Attractors and Anosov Diffeomorphisms 71
gd =
gd =
gd+ .
As this holds for all g C(M), = +.
Remark. If M is connected, thenM = 1 = X1 andf is mixing. So aboveis Bernoulli.
4.14. Theorem. Let f be a transitive C2 Anosov diffeomorphism. The fol-lowing are equivalent:
(a) f admits an invariant measure of the form d = hdm with h a positiveHolder function.
(b) f admits an invariant measure absolutely continuous w.r.t. m.
(c) Dfn : TxM TxM has determinant 1 whenever fnx = x.
Proof. Clearly (a) implies (b). Assume (b) holds. Let (s)(x) be the Jacobianof Df : Esf1x E
sx and
(s)(x) = log (s)(x). Now f1 is Anosov withEux,f1 = E
sx,f an E
sx,f1 = E
ux,f. Also
(u)f1(x) = Jacobian Df
1 : Esx Esf1x
= (s)(x)1
and so (u)f1(x) = log
(u)f1(x) =
(s)(x). There is an invariant measure
so that
limn
1
n
n1k=0
g(fk
x) =
g d
for m-almost all x; is the unique equilibrium state for (u)f1 w.r.t. f
1.
Notice that equilibrium states w.r.t. f1 are the same as those w.r.t. f;for Mf(M) = Mf1(M) and h(f) = h(f
1). So = (s) . Applying 4.13to both f and f1 we see
(u) = + = = = (s) .
By 4.8 (b), P((u)) = 0 = P((s)). By 4.5 one has, for fnx = x,
n1
k=0
(u)(fkx) n1
k=0
(s)(fkx) = 0.
Exponentiating,
1 = (det Dfn|Eux ) (det Dfn|Esx) = (det Df
n|TxM) .
Now assume (c) and let (x) = log Jac (Df : TxM TfxM). Then
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References 73
in 4.1 is due to Adler and Weiss [1]. This paper is very important in the devel-opment of the subject and is good reading. When (u) = C the measure whichmaximizes entropy still has the following geometrical significance: the periodicpoints ofs are equidistributed with respect to [5]. K. Sigmund [19] studiedthe generic properties of measures on s.
1. R. Adler, B. Weiss: Similarity of automorphisms of the torus, Memoirs of theAmerican Mathematical Society 98, American Mathematical Society, Provi-dence, R.I. 1970.
2. D. Anosov: Geodesic flows on closed Riemann manifolds with negative curva-ture, Proc. Steklov Inst. Math. 90 (1967).
3. D. Anosov, Ya. Sinai: Certain smooth ergodic systems, Uspehi Mat. Nauk 22(1967) no. 5 (137), 107172. English translation: Russ. Math. Surveys 22 (1967),
no. 5, 103-167.4. R. Azencott: Diffeomorphismes dAnosov et schemas de Bernoulli, C. R. Acad.Sci. Paris, Ser. 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 The-ory8 (1974/75), no.3, 193-202.
7. R. Bowen: Bernoulli equilibrium states for Axiom A diffeomorphisms, Math.Systems Theory8 (1974/75), no. 4, 289294.
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. Gurevic, V. Oseledec: Gibbs distributions, and the dissipativity of
C-diffeomorphisms, Soviet Math. Dokl. 14 (1973), 570573.11. M. Hirsch, J. Palis, C. Pugh and M. Shub: Neighborhoods of hyperbolic sets,
Invent. Math. 9 (1969/1970), 121134.12. M. Hirsch and C. Pugh: Stable manifolds and hyperbolic sets, Proc. Sympos.
Pure Math., Vol. XIV, Berkeley, Calif., 1968, pp. 133163, Amer. Math. Soc.,Providence, R.I., 1970.
13. A. N. Livsic: Cohomology of dynamical systems, Math. USSR-Izv. 6 (1972),1278-1301.
14. A. N. Livsic, Ya. Sinai: Invariant measures that are compatible with smoothnessfor transitive C-systems, Soviet Math. Dokl. 13 (1972), 1656-1659.
15. D. Ruelle: Statistical mechanics on a compact set with Z action satisfyingexpansiveness 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, 2164. English translation: Russian Math. Surveys 27 (1972), no. 4, 2169.
19. K. Sigmund: Generic properties of invariant measures for Axiom A diffeomor-phisms, Invent. Math. 11 (1970), 99-109.
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Index
Anosov, 46, 71, 72
Attractor, 6769
Axiom A, 46
Basic set, 48
Canonical coordinates, 47
Central limit theorem, 24
Entropy, 4, 16, 18, 29, 43
Equilibrium state, 42, 61
Ergodic, 14
Exponential clustering, 23
FA, 7
Free energy, 4
Gibbs measure, 6, 7, 15
Homologous, 7
Hyperbolic, 45
Markov partition, 52
Mixing, 6, 14
Non-wandering, 45
Pressure, 18, 33, 34
Pseudo-orbit, 49
Rectangle, 51
A, 6
Separated, 65Shift, 5, 9
Transition matrix, 56
Variational principle, 19, 40
Weak Bernoulli, 22
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Vol. 1803: G. Dolzmann, Variational Methods for Crys-talline Microstructure Analysis and Computation (2003)Vol. 1804: I. Cherednik, Ya. Markov, R. Howe,G. Lusztig,Iwahori-Hecke Algebras and their Representation Theory.Martina Franca, Italy 1999. Editors: V. Baldoni, D. Bar-basch (2003)Vol. 1805: F. Cao, Geometric Curve Evolution and ImageProcessing (2003)Vol. 1806: H. Broer, I. Hoveijn. G. Lunther, G. Vegter,Bifurcations in Hamiltonian Systems. Computing Singu-larities by Grbner Bases (2003)Vol. 1807: V. D. Milman, G. Schechtman (Eds.), Geomet-ric Aspects of Functional Analysis. Israel Seminar 2000-2002 (2003)Vol. 1808: W. Schindler, Measures with Symmetry Prop-erties (2003)Vol. 1809: O. Steinbach, Stability Estimates for HybridCoupled Domain Decomposition Methods (2003)
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