Beyond Bowen's Specification Property
BBEYOND BOWEN’S SPECIFICATION PROPERTY
VAUGHN CLIMENHAGA AND DANIEL J. THOMPSON
Abstract.
A classical result in thermodynamic formalism is that for uniformlyhyperbolic systems, every H¨older continuous potential has a unique equilibriumstate. One proof of this fact is due to Rufus Bowen and uses the fact that suchsystems satisfy expansivity and specification properties. In these notes, we surveyrecent progress that uses generalizations of these properties to extend Bowen’sarguments beyond uniform hyperbolicity, including applications to partially hy-perbolic systems and geodesic flows beyond negative curvature. We include a newcriterion for uniqueness of equilibrium states for partially hyperbolic systems with1-dimensional center.
Contents
1. Introduction 2
Part I. Main ideas: uniqueness of the measure of maximal entropy
42. Entropy and thermodynamic formalism 43. Bowen’s original argument: the symbolic case 64. Relaxing specification: decompositions of the language 115. Beyond shift spaces: expansivity in Bowen’s argument 16
Part II. Non-uniform Bowen hypotheses and equilibrium states
Part III. Geodesic flows
Date : September 22, 2020.2010
Mathematics Subject Classification.
Primary: 37D35. Secondary: 37C40, 37D40.V.C. is partially supported by NSF DMS-1554794. D.T. is partially supported by NSF DMS-1461163 and DMS-1954463. a r X i v : . [ m a t h . D S ] S e p VAUGHN CLIMENHAGA AND DANIEL J. THOMPSON Introduction
We survey recent progress in the study of existence and uniqueness of measuresof maximal entropy and equilibrium states in settings beyond uniform hyperbolicityusing weakened versions of specification and expansivity. Our focus is a long-runningjoint project initiated by the authors in [CT12], and extended in a series of papersincluding [CT16, BCFT18]. This approach is based on the fundamental insights ofRufus Bowen in the 1970’s, who identified and formalized three properties enjoyedby uniformly hyperbolic systems that serve as foundations for the equilibrium statetheory: these properties are specification, expansivity, and a regularity conditionnow known as the Bowen property. We relax all three of these properties in orderto study systems exhibiting various types of non-uniform structure. These notesstart by recalling the basic mechanisms of Bowen, and then gradually build up ingenerality, introducing the ideas needed to move to non-uniform versions of Bowen’shypotheses. The generality is motivated by, and illustrated by, examples: we discussapplications in symbolic dynamics, to certain partially hyperbolic systems, and towide classes of geodesic flows with non-uniform hyperbolicity. This survey has itsroots in the authors’ 6-part minicourse at the
Dynamics Beyond Uniform Hyperbol-icity conference at CIRM in May 2019.Part I describes Bowen’s result for MMEs and the simplest case of our general-ization. It begins by recalling the basic ideas of thermodynamic formalism ( §
2) andoutlining Bowen’s original argument in the simplest case: the measure of maximalentropy (MME) for a shift space with specification ( § §
4, we introduce themain idea of our approach, the use of decompositions to quantify the idea of “ob-structions to specification”, and we give an application to β -shifts. Moving beyondthe symbolic case requires the notion of expansivity, and in § §
6, and an application to partialhyperbolicity (the Ma˜n´e example) is described in §
7. Combining the notions ofobstructions to specification and expansivity leads to the general result for MMEs indiscrete-time in §
8, which is applied in § § §
11, we give an introduction in §
12 to the ideas in the paper [BCFT18],including the main “pressure gap” criterion for uniqueness, and how to decomposethe space of orbit segments using a function λ that measures curvature of horo-spheres. We also outline recent results for manifolds without conjugate points andCAT( −
1) spaces. In §
13, we discuss how to improve ergodicity of the equilibrium
EYOND BOWEN’S SPECIFICATION PROPERTY 3 states in non-positive curvature to the much stronger Kolmogorov K -property. Fi-nally, in §
14, we describe our proof of Knieper’s “entropy gap” for geodesic flow ona rank 1 non-positive curvature manifold.To illustrate the broad utility of the specification-based approach to uniqueness,we mention the following applications of the machinery we describe, which go wellbeyond what we are able to discuss in detail in this survey. • Measures of maximal entropy for symbolic examples: β -shifts, S -gap shifts,and their factors [CT12]; certain shifts of quasi-finite type [Cli18]; S-limitedshifts [MS18]; shifts with “one-sided almost specification” [CP19]; ( − β )-shifts [SY20]; • Equilibrium states for symbolic examples: β -shifts in [CT13], their factors in[Cli18, CC19] (in particular, [CC19] studies general conditions under whichthe “pressure gap” condition holds); S -gap shifts in [CTY17]; certain α - β shifts [CLR18]; applications to Manneville–Pomeau and related intervalmaps [CT13]. • Diffeomorphisms beyond uniform hyperbolicity: Bonatti–Viana examples[CFT18]; Ma˜n´e examples [CFT19]; Katok examples [Wan]; certain partiallyhyperbolic attractors [FO20]. • Geodesic flows: non-positive curvature [BCFT18]; no focal points [CKP20,CKP19]; no conjugate points [CKW20]; CAT( −
1) geodesic flows [CLT20b].We also mention two related results: the machinery we describe has recently beenused to prove “denseness of intermediate pressures” [Sun20]; an approach to unique-ness (and non-uniqueness) for equilibrium states using various weak specificationproperties has been developed by Pavlov [Pav16, Pav19] for symbolic and expansivesystems.The current literature in the field is vibrant and continually growing. The scopeof this article is restricted to the specification approach to equilibrium states, andwe largely do not address the literature beyond that. Other uses for the specifica-tion property that we do not discuss include large deviations properties, multifractalanalysis, and universality constructions. Different variants of the specification prop-erty are sometimes more appropriate for these arguments, see e.g. [Yam09, K(cid:32)LO16].We stress that we do not address the use of other techniques to study existenceand uniqueness of equilibrium states. These approaches include transfer operatortechniques, Margulis-type constructions, symbolic dynamics, and the Patterson-Sullivan approach. We suggest the following recent references as a starting pointto delve into the literature: [PPS15, CP17, BCS18, FH19, CPZ19, Cli20]. Classicreferences include [Bow08, PP90, Kel98].We also do not discuss the large and important area of statistical properties forequilibrium states. If f is a C α Anosov diffeomorphism (or if X is an AxiomA attractor) then the unique equilibrium state for the geometric potential ϕ ( x ) = − log | det Df | E u ( x ) | is the physically relevant Sinai–Ruelle–Bowen (SRB) measure.This provides important motivation and application for thermodynamic formalism,and this general setting is one of the major approaches to studying the statistical VAUGHN CLIMENHAGA AND DANIEL J. THOMPSON properties of the SRB measure. References include [Bow08, PP90, BS93, Bal00b,Bal00a, You02, BDV05, Cha15].We sometimes adopt a conversational writing style. We hope that the informalstyle will be helpful for current purposes; we invite the reader to look at our originalpapers, particularly [CT12, CT16, BCFT18] for a more precise account.
Part I. Main ideas: uniqueness of the measure of maximal entropy
We introduce our main ideas in the case of a discrete-time dynamical system(
X, f ). In this section, we often consider the case when (
X, f ) is a shift space. Wealso consider the general topological dynamics setting where X is a compact metricspace and f : X → X is continuous. In many of our examples of interest, X is asmooth manifold and f is a diffeomorphism.2. Entropy and thermodynamic formalism
For a probability vector (cid:126)p = ( p , . . . , p N ) ∈ [0 , N , where (cid:80) p i = 1, the entropy of (cid:126)p is H ( (cid:126)p ) = (cid:80) i − p i log p i . The following is an elementary exercise: • max (cid:126)p H ( (cid:126)p ) = log N ; • H ( (cid:126)p ) = log N ⇔ p i = N for all i ⇔ p i = p j for all i, j .These general principles lie at the heart of thermodynamic formalism for uniformlyhyperbolic dynamical systems, with ‘probability vector’ replaced by ‘invariant prob-ability measure’: • there is a function called ‘entropy’ that we wish to maximize; • it is maximized at a unique measure (variational principle and uniqueness); • that measure is characterized by an equidistribution (Gibbs) property.Now we recall the formal definitions, referring to [DGS76, Wal82, Pet89, VO16] forfurther details and properties.Let X be a compact metric space and f : X → X a continuous map. This givesa discrete-time topological dynamical system ( X, f ). Let M f ( X ) denote the spaceof Borel f -invariant probability measures on X .When f exhibits some hyperbolic behavior, M f ( X ) is typically extremely large– an infinite-dimensional simplex – and it becomes important to identify certain“distinguished measures” in M f ( X ). This includes SRB measures, measures ofmaximal entropy, and more generally, equilibrium measures. Definition 2.1 (Measure-theoretic Kolmogorov–Sinai entropy) . Fix µ ∈ M f ( X ).Given a countable partition α of X into Borel sets, write(2.1) H µ ( α ) := (cid:88) A ∈ α − µ ( A ) log µ ( A ) = (cid:90) − log µ ( α ( x )) dµ ( x )for the static entropy of α , where we write α ( x ) for the element of α containing x .One can interpret H µ ( α ) as the expected amount of information gained by observingwhich partition element a point x ∈ X lies in. Given j ≤ k , the corresponding EYOND BOWEN’S SPECIFICATION PROPERTY 5 dynamical refinement of α records which elements of α the iterates f j x, . . . , f k x liein:(2.2) α kj = k (cid:95) i = j f − i α ⇔ α kj ( x ) = k (cid:92) i = j f − i ( α ( f i x )) . A standard short argument shows that(2.3) H µ ( α n + m − ) ≤ H µ ( α n − ) + H µ ( α n + m − n ) = H µ ( α n − ) + H µ ( α m − ) , so that the sequence c n = H µ ( α n − ) is subadditive: c n + m ≤ c n + c m . Thus, byFekete’s lemma [Fek23], lim c n n exists, and equals inf c n n . We can therefore define the dynamical entropy of α with respect to f to be(2.4) h µ ( f, α ) := lim n →∞ n H µ ( α n − ) = inf n ∈ N n H µ ( α n − ) . The measure-theoretic (Kolmogorov–Sinai) entropy of (
X, f, µ ) is(2.5) h µ ( f ) = sup α h µ ( f, α ) , where the supremum is taken over all partitions α as above for which H µ ( α ) < ∞ .The variational principle [Wal82, Theorem 8.6] states that(2.6) sup µ ∈M f ( X ) h µ ( f ) = h top ( X, f ) , where h top ( X, f ) is the topological entropy of f : X → X , which we will define morecarefully below (Definition 5.2). Now we define a central object in our study. Definition 2.2 (MMEs) . A measure µ ∈ M f ( X ) is a measure of maximal entropy(MME) for ( X, f ) if h µ ( f ) = h top ( X, f ); equivalently, if h ν ( f ) ≤ h µ ( f ) for every ν ∈ M f ( X ).The following theorem on uniformly hyperbolic systems is classical. Theorem 2.3 (Existence and Uniqueness) . Suppose one of the following is true.(1) ( X, f = σ ) is a transitive shift of finite type (SFT).(2) f : M → M is a C diffeomorphism and X ⊂ M is a compact f -invarianttopologically transitive locally maximal hyperbolic set. Then there exists a unique measure of maximal entropy µ for ( X, f ) .Remark . The unique MME can be thought of as the ‘most complex’ invariantmeasure for a system, and often encodes dynamically relevant information such asthe distribution and asymptotic behavior of the set of periodic points. In particular, this holds if X = M is compact and f is a transitive Anosov diffeomorphism. VAUGHN CLIMENHAGA AND DANIEL J. THOMPSON Bowen’s original argument: the symbolic case
The specification property in a shift space.
Following Bowen, we outline aproof of Theorem 2.3 in the first case, when (
X, σ ) is a transitive SFT. The originalconstruction of the MME in this setting is due to Parry and uses the transitionmatrix. Bowen’s proof works for a broader class of systems, which we now describe.Fix a finite set A (the alphabet ), let σ : A N → A N be the shift map σ ( x x . . . ) = x x . . . , and let X ⊂ A N be closed and σ -invariant: σ ( X ) = X . Here A N (andhence X ) is equipped with the metric d ( x, y ) = 2 − min { n : x n (cid:54) = y n } . We refer to X asa one-sided shift space . One could just as well consider two-sided shift spaces byreplacing N with Z (and using | n | in the definition of d ); all the results below wouldbe the same, with natural modifications to the proofs. Note that so far we do notassume that X is an SFT or anything of the sort.Given x ∈ A N and i < j , we write x [ i,j ] = x i x i +1 · · · x j for the word that appearsin positions i through j . We use similar notation to denote subwords of a word w ∈ A ∗ := (cid:83) n A n . Given w ∈ A n , we write | w | = n for the length of the word, and[ w ] = { x ∈ X : x [1 ,n ] = w } for the cylinder it determines in X . We write(3.1) L n := { w ∈ A n : [ w ] (cid:54) = ∅} , L := (cid:91) n ≥ L n , and refer to L as the language of X . Definition 3.1.
The topological entropy of X is h top ( X ) = lim n →∞ n log L n . Weoften write h ( X ) for brevity. The limit exists by Fekete’s lemma using the fact thatlog L n is subadditive, which we prove in Lemma 3.6 below.It is a simple exercise to verify that every transitive SFT has the following prop-erty: there is τ ∈ N such that for every v, w ∈ L there is u ∈ L with | u | ≤ τ suchthat vuw ∈ L . Iterating this, we see that(3.2) for every w , . . . , w k ∈ L there are u , . . . , u k − ∈ L such that | u i | ≤ τ for all i, and w u w u · · · u k − w k ∈ L . We say that a shift space whose language satisfies (3.2) has the specification property .There are a number of different variants of the specification property, some of whichwe will meet later in these notes. The property above, where the connecting words u ∈ L satisfy | u | ≤ τ , is sometimes called weak specification . A stronger specificationproperty is to ask that the connecting words u have length | u | exactly τ , which isa property enjoyed by mixing SFTs. In this section, the word specification refers tothe weaker property. Theorem 3.2 (Shift spaces with specification) . Let ( X, σ ) be a shift space with thespecification property. Then there is a unique measure of maximal entropy on X . EYOND BOWEN’S SPECIFICATION PROPERTY 7
In the remainder of this section, we outline the two main steps in the proof ofTheorem 3.2: proving uniqueness using a Gibbs property ( § § Remark . As mentioned above, the original proof that a transitive SFT has aunique MME is due to Parry [Par64]. Parry constructed the MME using eigendataof the transition matrix for the SFT, and proved uniqueness by showing that anyMME must be a Markov measure, then showing that there is only one MME amongMarkov measures.A different proof of uniqueness in the SFT case was given by Adler and Weiss,who gave a more flexible argument based on showing that if µ is the Parry measure,then every ν ⊥ µ must have smaller entropy. The argument is described in [AW67],with full details in [AW70]. A key step in the proof is to consider an arbitrary set E ⊂ X and relate µ ( E ) to the number of n -cylinders intersecting E . In extendingthe uniqueness result to sofic shifts (factors of SFTs), Weiss [Wei73] clarified thecrucial role of what we refer to below as the “lower Gibbs bound” in carrying outthis step. This is essentially the proof of uniqueness that we use in all the results inthis survey.The crucial difference between Theorem 3.2 and the results of Parry, Adler, andWeiss is the construction of the MME using the specification property rather thaneigendata of a matrix. This is due to Bowen, as is the further generalization tonon-symbolic systems and equilibrium states for non-zero potentials [Bow75]. Thuswe often refer informally to the proof below as “Bowen’s argument”.3.2. The lower Gibbs bound as the mechanism for uniqueness.
It followsfrom the Shannon–McMillan–Breiman theorem that if µ is an ergodic shift-invariantmeasure, then for µ -a.e. x we have(3.3) − n log µ [ x [1 ,n ] ] → h µ ( σ ) as n → ∞ . This can be rewritten as(3.4) 1 n log (cid:16) µ [ x [1 ,n ] ] e − nh µ ( σ ) (cid:17) → µ -a.e. x. In other words, for µ -typical x , the measure µ [ x [1 ,n ] ] decays like e − nh µ ( σ ) in the sensethat µ [ x [1 ,n ] ] /e − nh µ ( σ ) is “subexponential in n ”. The mechanism for uniqueness inthe Parry-Adler-Weiss-Bowen argument is to produce an ergodic measure for whichthis subexponential growth is strengthened to uniform boundedness and applies forall x .The next proposition makes this Gibbs property precise and explain how unique-ness follows; then in § The notes at https://vaughnclimenhaga.wordpress.com/2020/06/23/specification-and-the-measure-of-maximal-entropy/ give a slightly more detailed version of this proof. We will encounter this general principle multiple times: many of our proofs rely on ob-taining uniform bounds (away from 0 and ∞ ) for quantities that a priori can grow or decaysubexponentially. VAUGHN CLIMENHAGA AND DANIEL J. THOMPSON argument appears in [Wei73, Lemma 2] (see also [AW67, AW70]); see [Bow74] fora version that works in the nonsymbolic setting, which we will describe in § Proposition 3.4.
Let X ⊂ A N be a shift space and µ an ergodic σ -invariant measureon X . Suppose that there are K, h > such that for every x ∈ X and n ∈ N , wehave the Gibbs bounds(3.5) K − e − nh ≤ µ [ x [1 ,n ] ] ≤ Ke − nh . Then h = h µ ( σ ) = h top ( X, σ ) , and µ is the unique MME for ( X, σ ) .Proof. First observe that by the Shannon–McMillan–Breiman theorem, the upperbound in (3.5) gives h µ ( σ ) ≥ h , while the lower bound gives h µ ( σ ) ≤ h . Moreover,summing (3.5) over all words in L n gives K − e nh ≤ L n ≤ Ke nh , so h top ( X, σ ) = h .The remainder of the proof is devoted to using the lower bound to show that(3.6) h ν ( σ ) < h = h µ ( σ ) for all ν ∈ M σ ( X ) with ν (cid:54) = µ. This will show that µ is the unique MME.Given ν ∈ M σ ( X ), the Lebesgue decomposition theorem gives ν = tν + (1 − t ) ν for some t ∈ [0 ,
1] and ν , ν ∈ M f ( X ) with ν ⊥ µ and ν (cid:28) µ . By ergodicity, ν = µ , and thus if ν (cid:54) = µ we must have t >
0. Since h ν ( σ ) = th ν ( σ ) + (1 − t ) h ν ( σ )and h ν ( σ ) = h µ ( σ ) ≤ h , we see that to prove (3.6), it suffices to prove that h ν ( σ ) < h whenever ν ⊥ µ .Writing α for the (generating) partition into 1-cylinders, we see that for any ν ∈ M σ ( X ) we have(3.7) nh ν ( σ ) = h ν ( σ n ) = h ν ( σ n , α n − ) ≤ H ν ( α n − ) = (cid:88) w ∈L n − ν [ w ] log ν [ w ] . When ν ⊥ µ , there is a Borel set D ⊂ X such that µ ( D ) = 1 and ν ( D ) = 0.Since cylinders generate the σ -algebra, there is D ⊂ L ( X ) such that µ ( D n ) → ν ( D n ) →
0, where µ ( D n ) := µ (cid:0) (cid:83) w ∈D n [ w ] (cid:1) . We break the sum in (3.7) into twopieces, one over D n and one over D cn = L n \ D n . Observe that (cid:88) w ∈D n − ν [ w ] log ν [ w ] = (cid:88) w ∈D n − ν [ w ] (cid:16) log ν [ w ] ν ( D n ) + log ν ( D n ) (cid:17) = (cid:16) ν ( D n ) (cid:88) w ∈D n − ν [ w ] ν ( D n ) log ν [ w ] ν ( D n ) (cid:17) − ν ( D n ) log ν ( D n ) ≤ ( ν ( D n ) log D n ) + 1 , where the last line uses the fact that (cid:80) ki =1 − p i log p i ≤ log k whenever p i ≥ (cid:80) p i = 1, as well as the fact that − t log t ≤ t ∈ [0 , This requires ergodicity of µ ; one can also give a short argument directly from the definitionof h µ ( σ ) that does not need ergodicity. EYOND BOWEN’S SPECIFICATION PROPERTY 9 holds for D cn , and together with (3.7) this gives(3.8) nh ν ( σ ) ≤ ν ( D n ) log D n + ν ( D cn ) log D cn . Using (3.5) and summing over D n gives µ ( D n ) = (cid:88) w ∈D n µ [ w ] ≥ K − e − nh D n ⇒ D n ≤ Ke nh µ ( D n ) , and similarly for D cn , so (3.8) gives nh ν ( σ ) ≤ ν ( D n ) (cid:0) log K + nh + log µ ( D n ) (cid:1) + ν ( D cn ) (cid:0) log K + nh + log µ ( D cn ) (cid:1) = 2 + log K + nh + ν ( D n ) log µ ( D n ) + ν ( D cn ) log µ ( D cn ) . Rewriting this as n ( h ν ( σ ) − h ) ≤ K + ν ( D n ) log µ ( D n ) + ν ( D cn ) log µ ( D cn ) , we see that the right-hand side goes to −∞ as n → ∞ , since ν ( D n ) → µ ( D n ) →
1, so the left-hand side must be negative for large enough n , which impliesthat h ν ( σ ) < h and completes the proof. (cid:3) Building a Gibbs measure.
Now the question becomes how to build anergodic measure satisfying the lower Gibbs bound. There is a standard constructionof an MME for a shift space, which proceeds as follows: let ν n be any measure on X such that ν n [ w ] = 1 / L n for every w ∈ L n , and then consider the measures(3.9) µ n := 1 n n − (cid:88) k =0 σ k ∗ ν n = 1 n n − (cid:88) k =0 ν n ◦ σ − k . A general argument (which appears in the proof of the variational principle, see forexample [Wal82, Theorem 8.6]) shows that any weak* limit point of the sequence µ n is an MME. If the shift space satisfies the specification property, one can provemore. Proposition 3.5.
Let ( X, σ ) be a shift space with the specification property, let µ n be given by (3.9) , and suppose that µ n j → µ in the weak* topology. Then µ is σ -invariant, ergodic, and there is K ≥ such that µ satisfies the following Gibbsproperty : (3.10) K − e − nh top ( X ) ≤ µ [ w ] ≤ Ke − nh top ( X ) for all w ∈ L n . Combining Propositions 3.4 and 3.5 shows that there is a unique MME µ , whichis the weak* limit of the sequence µ n from (3.9). Thus to prove Theorem 3.2 itsuffices to prove Proposition 3.5. We omit the full proof, and highlight only themost important part of the associated counting estimates. Lemma 3.6.
Let ( X, σ ) be a shift space with the specification property, with gapsize τ . Then for every n ∈ N , we have (3.11) e nh top ( X ) ≤ L n ≤ Qe nh top ( X ) , where Q = ( τ + 1) e τh top ( X ) . Proof.
For every m, n ∈ N , there is an injective map L m + n → L m × L n defined by w (cid:55)→ ( w [1 ,m ] , w [ m +1 ,m + n ] ), so L m + n ≤ L m L n . Iterating this gives L kn ≤ ( L n ) k ⇒ kn log L kn ≤ n log L n , and sending k → ∞ we get h top ( X ) ≤ n log L n for all n , which proves the lowerbound. For the upper bound we observe that specification gives a map L m × L n →L m + n + τ defined by mapping ( v, w ) to vuwu (cid:48) , where u = u ( v, w ) ∈ L with | u | ≤ τ is the ‘gluing word’ provided by the specification property, and u (cid:48) is any word oflength τ − | u | that can legally follow vuw . This map may not be injective because w can appear in different positions, but each word in L m + n can have at most ( τ + 1)preimages, since v, w are completely determined by vuwu (cid:48) and the length of u . Thisshows that L m + n + τ ≥ τ + 1 L m L n ⇒ L k ( n + τ ) ≥ (cid:16) L n τ + 1 (cid:17) k . Taking logs and dividing by k ( n + τ ) gives1 k ( n + τ ) L k ( n + τ ) ≥ n + τ (cid:0) log L n − log( τ + 1) (cid:1) . Sending k → ∞ and rearranging gives log L n ≤ log( τ +1)+( n + τ ) h top ( X ). Takingan exponential proves the upper bound. (cid:3) With Lemma 3.6 in hand, the idea of Proposition 3.5 is to first prove the boundson µ [ w ] by estimating, for each n (cid:29) | w | and k ∈ { , . . . , n − | w |} , the numberof words u ∈ L n for which w appears in position k ; see Figure 3.1. By consider-ing the subwords of u lying before and after w , one sees that there are at most( L k )( L n − k −| w | ) such words, as in the proof of Lemma 3.6, and thus the boundsfrom that lemma give ν n ( σ − k [ w ]) ≤ ( L k )( L n − k −| w | ) L n ≤ Qe kh top ( X ) Qe ( n − k −| w | ) h top ( X ) e nh top ( X ) = Q e −| w | h top ( X,σ ) ;averaging over k gives the upper Gibbs bound, and the lower Gibbs bound followsfrom a similar estimate that uses the specification property. w L n L k L n − k −| w | Figure 3.1.
Estimating ν n ( σ − k [ w ]).Next, one can use similar arguments to produce c > v, w , there are arbitrarily large j ∈ N such that µ ([ v ] ∩ σ − j [ w ]) ≥ cµ [ v ] µ [ w ]; EYOND BOWEN’S SPECIFICATION PROPERTY 11 this is once again done by counting the number of long words that have v, w in theappropriate positions.Since any measurable sets V and W can be approximated by unions of cylinders,one can use this to prove that lim n µ ( V ∩ σ − n W ) ≥ cµ ( V ) µ ( W ). Considering thecase when V = W is σ -invariant demonstrates that µ is ergodic.4. Relaxing specification: decompositions of the language
Decompositions.
There are many shift spaces that can be shown to have aunique MME despite not having the specification property; see § X be a shift space on a finite alphabet, and L itslanguage. We consider the following more general version of (3.2). Definition 4.1.
A collection of words
G ⊂ L has specification if there exists τ ∈ N such that for every finite set of words w , . . . , w k ∈ G , there are u , . . . , u k − ∈ L with | u i | ≤ τ such that w u w u · · · u k − w k ∈ L .The only difference between this definition and (3.2) is that here we only requirethe gluing property to hold for words in G , not for all words. Remark . In particular, G has specification if there is τ ∈ N such that for every v, w ∈ G , there is u ∈ L with | u | ≤ τ and vuw ∈ G , because iterating this propertygives the one stated above. The property above, which is sufficient for our uniquenessresults, is a priori more general because the concatenated word is not required to liein G .Now we need a way to say that a collection G on which specification holds issufficiently large. Definition 4.3. A decomposition of the language L consists of three collections ofwords C p , G , C s ⊂ L with the property that(4.1) for every w ∈ L , there are u p ∈ C p , v ∈ G , u s ∈ C s such that w = u p vu s . Given a decomposition of L , we also consider for each M ∈ N the collection of words(4.2) G M := { u p vu s ∈ L : u p ∈ C p , v ∈ G , u s ∈ C s , | u p | , | u s | ≤ M } . If each G M has specification, then the set C p ∪ C s can be thought of as the set of obstructions to the specification property. Definition 4.4.
The entropy of a collection of words
C ⊂ L is(4.3) h ( C ) = lim n →∞ n log C n . Theorem 4.5 (Uniqueness using a decomposition [CT12]) . Let X be a shift spaceon a finite alphabet, and suppose that the language L of X admits a decomposition C p GC s such that (I) every collection G M has specification, and (II) h ( C p ∪ C s ) < h ( X ) .Then ( X, σ ) has a unique MME µ .Remark . Note that L = (cid:83) M ∈ N G M ; the sets G M play a similar role to the regularlevel sets that appear in Pesin theory. The gap size τ appearing in the specificationproperty for G M is allowed to depend on M , just as the constants appearing in thedefinition of hyperbolicity are allowed to depend on which regular level set a pointlies in. Similarly, for the unique MME µ one can prove that lim M →∞ µ ( G M ) = 1,which mirrors a standard result for hyperbolic measures and Pesin sets. Remark . In fact we do not quite need every w ∈ L to admit a decompositionas in (4.1). It is enough to have C p , G , C s ⊂ L such that h ( L \ ( C p GC s )) < h ( X ), inaddition to the conditions above [Cli18].We outline the proof of Theorem 4.5. The idea is to mimic Bowen’s proof usingPropositions 3.4 and 3.5 by completing the following steps.(1) Prove uniform counting bounds as in Lemma 3.6.(2) Use these to establish the following non-uniform Gibbs property for any limitpoint µ of the sequence of measures in (3.9): there are constants K, K M ≥ K − M e −| w | h top ( X ) ≤ µ [ w ] ≤ Ke −| w | h top ( X ) for all M ∈ N and w ∈ G M . We emphasize that the Gibbs property is non-uniform in the sense that thelower Gibbs constant depends on M . The upper bound that we will obtainfrom our hypotheses is uniform in M . On a fixed G M , we have uniform Gibbsestimates.(3) Give a similar argument for ergodicity, and then prove that the non-uniformlower Gibbs bound in (4.4) still gives uniqueness as in Proposition 3.4.Once the uniform counting bounds are established, the proof of (4.4) follows thesame approach as before. We do not discuss the third step at this level of generalityexcept to emphasize that it follows the approach given in Proposition 3.4.For the counting bounds in the first step, we start by observing that the bound L n ≥ e nh top ( X ) did not require any hypotheses on the symbolic space X and thuscontinues to hold. The argument for the upper bound in Lemma 3.6 can be easilyadapted to show that there is a constant Q such that G n ≤ Qe nh top ( X ) for all n .Then the desired upper bound for L n is a consequence of the following. Lemma 4.8.
For any r ∈ (0 , , there is M such that G Mn ≥ r L n for all n . Since G M corresponds to a collection of orbit segments rather than a subset of the space, themost accurate analogy might be to think of G M as corresponding to orbit segments that start andend in a given regular level set. EYOND BOWEN’S SPECIFICATION PROPERTY 13
Proof.
Let a i = C pi ∪ C si ) e − ih top ( X ) , so that in particular (cid:80) a i < ∞ by (II). Sinceany w ∈ L n can be written as w = u p vu s for some u ∈ C pi , v ∈ G j , and w ∈ C sk with i + j + k = n , we have L n ≤ G Mn + (cid:88) i + j + k = n max( i,k ) >M ( C pi )( G j )( C sk ) ≤ G Mn + (cid:88) i + j + k = n max( i,k ) >M a i a k Qe nh top ( X ) , where the second inequality uses the upper bound G j ≤ Qe jh top ( X ) . Since (cid:80) a i < ∞ , there is M such that (cid:88) i + j + k = n max( i,k ) >M a i a k Qe nh top ( X ) < (1 − r ) e nh top ( X ) ≤ (1 − r ) L n , where the second inequality uses the lower bound L n ≥ e nh top ( X ) . Combining theseestimates gives L n ≤ G Mn + (1 − r ) L n , which proves the lemma. (cid:3) The same specification argument that gives the upper bound on G n gives acorresponding upper bound on G Mn (with a different constant), and thus we deducethe following consequence of Lemma 4.8. Corollary 4.9.
There are constants a, A > and M ∈ N such that e nh top ( X ) ≤ L n ≤ Ae nh top ( X ) and G Mn ≥ ae nh top ( X ) for all n ∈ N . Remark . In fact, the proof of Lemma 4.8 can easily be adapted to show astronger result: given any γ > r ∈ (0 , M such that if D n ⊂ L n has D n ≥ γe nh top ( X ) , then D n ∩ G Mn ) ≥ r D n . These types of estimates are whatlie behind the claim in Remark 4.6 that the (non-uniform) Gibbs property implies µ ( G M ) → M → ∞ .4.2. An example: beta shifts.
Given a real number β >
1, the corresponding β -transformation f : [0 , → [0 ,
1) is f ( x ) = βx (mod 1). Let A = { , , . . . , (cid:100) β (cid:101) − } ;then every x ∈ [0 ,
1) admits a coding y = π ( x ) ∈ A N defined by y n = (cid:98) βf n − ( x ) (cid:99) ,and we have π ◦ f = σ ◦ π , where σ : A N → A N is the left shift. Observe that π ( x ) n = a if and only if f n − ( x ) ∈ I a , where the intervals I a are as shown in Figure4.1. Given n ∈ N and w ∈ A n , let I ( w ) := n (cid:92) k =1 f − ( k − ( I w k )be the interval in [0 ,
1) containing all points x for which the first n iterates are codedby w . The figure shows an example for which f n ( I ( w )) is not the whole interval[0 , f n ( I ( w )) is equal to the whole interval. Observe that if β is an integer thenthis is true for every word. Formally, I a = { x ∈ [0 ,
1) : (cid:98) βx (cid:99) = a } , so I a = [ aβ ,
1) if a = (cid:100) β (cid:101) −
1, and [ aβ , a +1 β ) otherwise. I I I I (21) f ( I (21)) f ( I (21)) Figure 4.1.
Coding a β -transformation. Definition 4.11.
The β -shift X β is the closure of the image of π , and is σ -invariant.Equivalently, X β is the shift space whose language L is the set of all w ∈ A ∗ suchthat I ( w ) (cid:54) = ∅ ; thus y ∈ A N is in X β if and only if I ( y · · · y n ) (cid:54) = ∅ for all n ∈ N .For further background on the β -shifts, see [R´en57, Par60, Bla89]. We summarizethe properties relevant for our purposes.Write (cid:22) for the lexicographic order on A N and observe that π is order-preserving.Let z = lim x (cid:37) π ( x ) denote the supremum of X β in this ordering. It will be conve-nient to extend (cid:22) to A ∗ , writing v (cid:22) w if for n = min( | v | , | w | ) we have v [1 ,n ] (cid:22) w [1 ,n ] . Remark . Observe that on A ∗ ∪ A N , (cid:22) is only a pre-order, because there are v (cid:54) = w such that v (cid:22) w and w (cid:22) v ; this occurs whenever one of v, w is a prefix ofthe other.The β -shift can be described in terms of the lexicographic ordering, or in termsof the following countable-state graph: • the vertex set is N = { , , , , . . . } ; • the vertex n has 1 + z n +1 outgoing edges, labeled with { , , . . . , z n +1 } ; theedge labeled z n +1 goes to n + 1, and the rest go to the ‘base’ vertex 0.Figure 4.2 shows (part of) the graph when z = 2102001 . . . , as in Figure 4.1. Proposition 4.13.
Given n ∈ N and w ∈ A n , the following are equivalent.(1) I ( w ) (cid:54) = ∅ (which is equivalent to w ∈ L ( X β ) by definition).(2) w [ j,n ] (cid:22) z for every ≤ j ≤ n .(3) w labels the edges of a path on the graph that starts at the base vertex .Idea of proof. Using induction, check that the following are equivalent for every n ∈ N , 0 ≤ k ≤ n , and w ∈ A n .(1) f n ( I ( w )) = f k ( I ( z [1 ,k ] ), where we write I ( z [1 , ) := [0 , w [ j,n ] (cid:22) z for every 1 ≤ j ≤ n , and k is maximal such that w [ n − k +1 ,n ] = z [1 ,k ] .(3) w labels the edges of a path on the graph that starts at the base vertex 0and ends at the vertex k . (cid:3) EYOND BOWEN’S SPECIFICATION PROPERTY 15
Figure 4.2.
A graph representation of X β . Corollary 4.14.
Given x ∈ A N , the following are equivalent.(1) x ∈ X β .(2) σ n ( x ) (cid:22) z for every n .(3) x labels the edges of an infinite path of the graph starting at the vertex .Exercise . Prove that X β has the specification property if and only if z does notcontain arbitrarily long strings of 0s.In fact, Schmeling showed [Sch97] that for Lebesgue-a.e. β >
1, the β -shift X β does not have the specification property. Nevertheless, every β -shift has a uniqueMME. This was originally proved by Hofbauer [Hof78] and Walters [Wal78] usingtechniques not based on specification. Theorem 4.5 gives an alternate proof: writing G for the set of words that label a path starting and ending at the base vertex, and C s for the set or words that label a path starting at the base vertex and neverreturning to it , one quickly deduces the following. • GC s is a decomposition of L . • G M is the set of words labeling a path starting at the base vertex and endingsomewhere in the first M vertices; writing τ for the maximum graph distancefrom such a vertex to the base vertex, G M has specification with gap size τ . • C sn = 1 for every n , and thus h ( C s ) = 0 < h top ( X β ) = log β .This verifies the conditions of Theorem 4.5 and thus provides another proof ofuniqueness of the MME. Remark . Because the earlier proofs of uniqueness did not pass to subshift factorsof β -shifts, it was for several years an open problem (posed by Klaus Thomsen)whether such factors still had a unique MME. The inclusion of this problem in MikeBoyle’s article “Open problems in symbolic dynamics” [Boy08] was our originalmotivation for studying uniqueness using non-uniform versions of the specificationproperty, which led us to formulate the conditions in Theorem 4.5; these can beshown to pass to factors, providing a positive answer to Thomsen’s question [CT12]. Remark . Theorem 4.5 can be applied to other symbolic examples as well, in-cluding S -gap shifts [CT12]. The S -gap shifts are a family of subshifts of { , } Z defined by the property that the number of 0’s that appear between any two 1’s isan element of a prescribed set S ⊂ Z . A specific example is the prime gap shift,where S is taken to be the prime numbers. The theorem also admits an extensionto equilibrium states for nonzero potential functions along the lines described in § β -shifts [CT13], S -gap shifts [CTY17], shifts ofquasi-finite type [Cli18], and α - β shifts (which code x (cid:55)→ α + βx (mod 1)) [CLR18].4.3. Periodic points.
It is often the case that one can prove a stronger version ofspecification, for example, when X is a mixing SFT. Definition 4.18.
Say that
G ⊂ L has periodic specification if there exists τ ∈ N suchthat for all w , . . . , w k ∈ G , there are u , . . . , u k ∈ L τ such that v := w u · · · w k u k ∈L , and moreover x = vvvvv · · · ∈ X .There are two strengthenings of specification here: first, we are assuming that thegap size is equal to τ , not just ≤ τ , and second, we are assuming that the “gluedword” can be extended periodically after adding τ more symbols.If we replace specification in Theorem 4.5 with periodic specification for each G M , then the counting estimates in Lemma 3.6 immediately lead to the followingestimates on the number of periodic points: writing Per n = { x ∈ X : σ n x = x } , wehave(4.5) C − e nh top ( X ) ≤ n ≤ Ce nh top ( X ) . Using this fact and the construction of the unique MME given just before Proposition3.5, one can also conclude that the unique MME µ is the limiting distribution ofperiodic orbits in the following sense:(4.6) 1 n (cid:88) x ∈ Per n δ x weak* −−−→ µ as n → ∞ . This argument holds true in the classical Theorem 3.2, and for β -shifts. It alsoextends beyond the symbolic setting, and a natural analogue of the argument holdsfor regular closed geodesics on rank one non-positive curvature manifolds.5. Beyond shift spaces: expansivity in Bowen’s argument
Now we move to the non-symbolic setting and describe how Bowen’s approachworks for a continuous map on a compact metric space. In particular, his assump-tions apply to and were inspired by the case when X is a transitive locally maximalhyperbolic set for a diffeomorphism f . First we recall some basic definitions.5.1. Topological entropy.Definition 5.1.
Given n ∈ N , the n th dynamical metric on X is(5.1) d n ( x, y ) := max { d ( f k x, f k y ) : 0 ≤ k < n } . EYOND BOWEN’S SPECIFICATION PROPERTY 17
The
Bowen ball of order n and radius (cid:15) > centered at x ∈ X is(5.2) B n ( x, (cid:15) ) := { y ∈ X : d n ( x, y ) < (cid:15) } . A set E ⊂ X is called ( n, (cid:15) ) -separated if d n ( x, y ) > (cid:15) for all x, y ∈ E with x (cid:54) = y ;equivalently, if y / ∈ B n ( x, (cid:15) ) for all such x, y .We define entropy in a more general way than is standard, reflecting our focuson the space of finite-length orbit segments X × N as the relevant object of study;this replaces the language L that we used in the symbolic setting. We interpret( x, n ) ∈ X × N as representing the orbit segment ( x, f x, f x, . . . , f n − x ). Then theanalogy is that a cylinder [ w ] for a word in the language corresponds to a Bowenball B n ( x, (cid:15) ) associated to an orbit segment ( x, n ) ∈ X × N . Given a collection oforbit segments D ⊂ X × N , for each n ∈ N we write(5.3) D n := { x ∈ X : ( x, n ) ∈ D} for the collection of points that begin a length- n orbit segment in D . Definition 5.2 (Topological entropy) . Given a collection of orbit segments
D ⊂ X × N , for each (cid:15) > n ∈ N we write(5.4) Λ( D , (cid:15), n ) := max { E : E ⊂ D n is ( n, (cid:15) )-separated } . The entropy of D at scale (cid:15) > h ( D , (cid:15) ) := lim n →∞ n log Λ( D , (cid:15), n ) , and the entropy of D is(5.6) h ( D ) := lim (cid:15) → h ( D , (cid:15) ) . When D = Y × N for some Y ⊂ X , we write Λ( Y, (cid:15), n ) = Λ( Y × N , (cid:15), n ), h top ( Y, (cid:15) ) = h ( Y × N , (cid:15) ) and h top ( Y ) = lim (cid:15) → h top ( Y, (cid:15) ). In particular, when D = X × N wewrite h top ( X, f ) = h top ( X ) = h ( X × N ) for the topological entropy of f : X → X .When different orbit segments in D are given weights according to their ergodicsum w.r.t. a given potential ϕ , we obtain a notion of topological pressure , which wewill discuss in § Theorem 5.3 (Variational principle) . Let X be a compact metric space and f : X → X a continuous map. Then (5.7) h top ( X, f ) = sup µ ∈M f ( X ) h µ ( f ) . The following construction forms one half of the proof of the variational principle.
Proposition 5.4 (Building a measure of almost maximal entropy) . With
X, f asabove, fix (cid:15) > , and for each n ∈ N , let E n ⊂ X be an ( n, (cid:15) ) -separated set. Considerthe Borel probability measures (5.8) ν n := 1 E n (cid:88) x ∈ E n δ x , µ n := 1 n n − (cid:88) k =0 f k ∗ ν n = 1 n n − (cid:88) k =0 ν n ◦ f − k . Let µ n j be any subsequence that converges in the weak*-topology to a limiting measure µ . Then µ ∈ M f ( X ) and (5.9) h µ ( f ) ≥ lim j →∞ n j log E n j . In particular, for every δ > there exists µ ∈ M f ( X ) such that h µ ( f ) ≥ h top ( X, f, δ ) .Proof. See [Wal82, Theorem 8.6]. (cid:3)
Corollary 5.5.
Let
X, f be as above, and suppose that there is δ > such that h top ( X, f, δ ) = h top ( X, f ) . Then there exists a measure of maximal entropy for ( X, f ) . Indeed, given any sequence { E n ⊂ X } ∞ n =1 of maximal ( n, δ ) -separated sets,every weak*-limit point of the sequence µ n from (5.8) is an MME. In our applications, it will often be relatively easy to verify that h top ( X, f, δ ) = h top ( X, f ) for some δ >
0, and so Corollary 5.5 establishes existence of a measure ofmaximal entropy. Thus the real challenge is to prove uniqueness, and this will beour focus.5.2.
Expansivity.
In Bowen’s general result, the assumption that X is a shift spaceis replaced by the following condition. Definition 5.6 (Expansivity) . Given x ∈ X and (cid:15) >
0, let(5.10) Γ + (cid:15) ( x ) := { y ∈ X : d ( f n y, f n x ) < (cid:15) for all n ≥ } = (cid:92) n ∈ N B n ( x, (cid:15) )be the forward infinite Bowen ball. If f is invertible, let(5.11) Γ − (cid:15) ( x ) := { y ∈ X : d ( f n y, f n x ) < (cid:15) for all n ≥ } be the backward infinite Bowen ball, and let(5.12) Γ (cid:15) ( x ) := Γ + (cid:15) ( x ) ∩ Γ − (cid:15) ( x ) = { y ∈ X : d ( f n y, f n x ) < (cid:15) for all n ∈ Z } be the bi-infinite Bowen ball. The system ( X, f ) is positively expansive at scale (cid:15) > + (cid:15) ( x ) = { x } for all x ∈ X , and (two-sided) expansive at scale (cid:15) > (cid:15) ( x ) = { x } . The system is (positively) expansive if there exists (cid:15) > (cid:15) .It is an easy exercise to check that one-sided shift spaces are positively ex-pansive. A system ( X, f ) is uniformly expanding if there are (cid:15), λ > d ( f y, f x ) ≥ e λ d ( y, x ) whenever x, y ∈ X have d ( x, y ) < (cid:15) . Iterating this propertygives diam B n ( x, (cid:15) ) ≤ (cid:15)e − λn for all n , and thus Γ + (cid:15) ( x ) = { x } , so ( X, f ) is positivelyexpansive.Two-sided shift spaces can easily be checked to be (two-sided) expansive, and wealso have the following.
Proposition 5.7. If X is a hyperbolic set for a diffeomorphism f , then ( X, f ) isexpansive. EYOND BOWEN’S SPECIFICATION PROPERTY 19
Sketch of proof.
Choose (cid:15) > x, y ∈ X with d ( x, y ) < (cid:15) ,the local leaves W s ( x ) and W u ( y ) intersect in a unique point [ x, y ] (we do not requirethat this point is in X ). Write d u ( x, y ) = d ( x, [ x, y ]) and d s ( x, y ) = d ( y, [ x, y ]) . Passing to an adapted metric if necessary, hyperbolicity gives λ > d u ( f n x, f n y ) ≥ e λn d u ( x, y ) if d ( f k x, f k y ) < (cid:15) for all 0 ≤ k ≤ n, (5.13) d s ( f − n x, f − n y ) ≥ e λn d s ( x, y ) if d ( f − k x, f − k y ) < (cid:15) for all 0 ≤ k ≤ n. (5.14)In particular, if y ∈ Γ (cid:15) ( x ) then d u ( f n x, f n y ) is uniformly bounded for all n , so d u ( x, y ) = 0, and similarly for d s , which implies that x = [ x, y ] = y . (cid:3) One important consequence of expansivity is the following.
Proposition 5.8. If ( X, f ) is expansive at scale (cid:15) , then h top ( X, f, (cid:15) ) = h top ( X, f ) .Two proof ideas. We outline two proofs in the positively expansive case.One argument uses a compactness argument to show that for every 0 < δ < (cid:15) ,there is N ∈ N such that B N ( x, (cid:15) ) ⊂ B ( x, δ ) for all x ∈ X . This implies that B n + N ( x, (cid:15) ) ⊂ B n ( x, δ ) for all x , and then one can show that the definition of topo-logical entropy via ( n, (cid:15) )-separated sets gives the same value at δ as at (cid:15) .Another method, which is better for our purposes, is to observe that since (cid:15) -expansivity gives (cid:84) n B n ( x, (cid:15) ) = { x } for all x , one can easily show that for every ν ∈ M f ( X ), we have:(5.15) if β is a partition with d n -diameter < (cid:15) , then β is generating for ( f n , ν ).Given a maximal ( n, (cid:15) )-separated set E n , we can choose a partition β n such thateach element of β n is contained in B n ( x, (cid:15) ) for some x ∈ E n , so β n has exactly E n elements. Then we have(5.16) h µ ( f ) = 1 n h µ ( f n ) = 1 n h µ ( f n , β n ) ≤ n H µ ( β n ) ≤ n log E n . Sending n → ∞ gives h µ ( f ) ≤ h top ( X, f, (cid:15) ), and taking a supremum over all µ ∈M f ( X ) proves that h top ( X, f, (cid:15) ) = h top ( X, f ). (cid:3) Specification.
The following formulation of the specification property is givenfor a collection of orbit segments
D ⊂ X × N , and thus is not quite the classicalone, but reduces to (a version of) the classical definition when we take D = X × N .Observe that when X is a shift space and we associate to each ( x, n ) the word x [1 ,n ] ∈ L ( X ), the following agrees with the definition from (3.2). Definition 5.9 (Specification) . A collection of orbit segments
D ⊂ X × N has the specification property at scale δ > τ ∈ N (the gap size or transitiontime ) such that for every ( x , n ) , . . . , ( x k , n k ) ∈ D , there exist 0 = T < T < · · ·
0. We say that (
X, f ) has the specification property if X × N does. Wesay that D has periodic specification if y can be chosen to be periodic with periodin [ s k , s k + τ ]. . . .. . .T T T T k s s s s k x x x x k y n n n n k ≤ τ ≤ τ Figure 5.1.
Bookkeeping in the specification property.First we explain how specification (for the whole system) is established in theuniformly hyperbolic case. Recall from (3.2) and the paragraph preceding it that inthe symbolic case, one can establish specification by verifying it in the case k = 2 andthen iterating. In the non-symbolic case, the proof of specification usually followsthis same approach, but one needs to verify a mildly stronger property for k = 2 toallow the iteration step; one possible version of this property is formulated in thenext lemma. Lemma 5.10.
Given f : X → X , suppose that δ > , δ ≥ , χ ∈ (0 , , and τ ∈ N are such that for every ( x , n ) , ( x , n ) ∈ X × N , there are t ∈ { , , . . . , τ } and y ∈ X such that (5.17) d ( f k y, f k x ) ≤ δ χ n − k for all ≤ k < n and d n ( f n + t y, x ) ≤ δ . Then ( X, f ) has the specification property at scale δ + δ / (1 − χ ) with gap size τ .Proof. Given ( x , n ) , . . . , ( x k , n k ) ∈ X × N , we will apply (5.17) iteratively to pro-duce y , . . . , y k and T , . . . , T k ∈ N such that writing δ (cid:48) = δ + δ / (1 − χ ), we have(5.18) f T i ( y j ) ∈ B n i ( x i , δ (cid:48) ) for all 1 ≤ i ≤ j ≤ k. Once this is done, y k is the desired shadowing point. See Figure 5.2 for an illustrationof the following procedure and estimates.Along with y i , T i , we will produce s i = T i + n i and t i ∈ { , . . . , τ } such that T i +1 = s i + t i . Start by putting y = x , T = 0, and s = n . Then apply(5.17) to ( y , s ) and ( x , n ) to get y ∈ X and t ∈ { , , . . . , τ } such that writing T = s + t , we have d ( f k y , f k y ) ≤ δ χ s − k for all 0 ≤ k < s and d n ( f T y , x ) ≤ δ . In general, once y i , s i are determined (with T i = s i − n i ), we apply (5.17) to ( y i , s i )and ( x i +1 , n i +1 ) to get t i ∈ { , , . . . , τ } and y i +1 ∈ X such that writing T i +1 = s i + t i , EYOND BOWEN’S SPECIFICATION PROPERTY 21 we have d ( f k y i +1 , f k y i ) ≤ δ χ s i − k for all 0 ≤ k < s i and d n i +1 ( f T i +1 y i +1 , x i +1 ) ≤ δ . Now we can verify (5.18) by observing that for all 1 ≤ i ≤ j ≤ k , we have(5.19) d n i ( f T i ( y j ) , f T i ( y i )) ≤ j − (cid:88) (cid:96) = i d n i ( f T i ( y (cid:96) +1 ) , f T i ( y (cid:96) )) ≤ j − (cid:88) (cid:96) = i δ χ s (cid:96) − s i < δ − χ , (the last inequality uses the fact that s (cid:96) − s i ≥ (cid:96) − i ), and also d n i ( f T i ( y i ) , x i ) ≤ δ ,so d n i ( x i , f T i y j ) ≤ d n i ( x i , f T i y i ) + d n i ( f T i y i , f T i y j ) ≤ δ + δ − χ = δ (cid:48) . (cid:3) x f n x x f n x x f n x x f n x y y y f s y f s y f T y f s y f s y < δ < δ χ< δ χ ≤ δ < δ < δ χ ≤ δ < δ ≤ δ Figure 5.2.
Proving specification using a one-step property.
Proposition 5.11. If X is a topologically transitive locally maximal hyperbolic setfor a diffeomorphism f , then ( X, f ) has the specification property.Proof. By Lemma 5.10, it suffices to show that for every sufficiently small δ > χ ∈ (0 ,
1) and τ ∈ N such that for every ( x , n ) , ( x , n ) ∈ X × N , thereare t ∈ { , , . . . , τ } and y ∈ X such that (5.17) holds. To prove this, let δ, ρ > • every x ∈ X has local stable and unstable leaves W sδ ( x ) and W uδ ( x ) withdiameter < δ , and • for every x, y ∈ X with d ( x, y ) < ρ , the intersection W sδ ( x ) ∩ W uδ ( y ) is asingle point, which lies in X .By topological transitivity and compactness, there is τ ∈ N such that for every x, y ∈ X there is t ∈ { , , . . . , τ } with d ( f t x, y ) < ρ , and thus f t ( W uδ ( x )) ∩ W sδ ( y ) (cid:54) = ∅ .Using this fact, given ( x , n ) , ( x , n ) ∈ X × N , we can let t ∈ { , , . . . , τ } besuch that f t ( W uδ ( f n x )) intersects W sδ ( x ). Choosing z in this intersection andputting y = f − ( t + n ) ( z ), we see that y satisfies (5.17) with δ = δ = δ , and thusLemma 5.10 proves the proposition. (cid:3) Remark . Uniform contraction of f along W s is not used; to prove specificationat scale δ (cid:48) , it would suffice to know that if x, y lie on the same local stable leaf and d ( x, y ) ≤ δ , then the same is true of f ( x ) , f ( y ), which still gives the second half of(5.17). In particular, this follows as soon as (cid:107) Df | E s (cid:107) ≤
1. The same idea can alsobe applied to obtain specification on suitable collections
G ⊂ X × N , and can beextended naturally to the continuous-time case.We also emphasize that the exponential contraction asked for in the first halfof (5.17), which is obtained from uniform backwards contraction along W u , canbe significantly weakened. What is really essential for the argument is backwardscontraction in the local unstables by a fixed amount in each of the orbit segments(not necessarily proportional to length), and this is enough to obtain a uniformdistance estimate analogous to (5.19). We carried out the details of this argumentin [BCFT18, §
4] in the non-uniformly hyperbolic setting of rank one geodesic flowfor the family of orbit segments C ( η ), which are defined in this survey in § Proposition 5.13.
Suppose that f : X → X is topologically transitive and has thefollowing properties. • Uniformly expanding: d ( f x, f y ) ≥ e λ d ( x, y ) whenever d ( x, y ) < δ . • Locally onto:
For every x ∈ X , we have f ( B ( x, δ )) ⊃ B ( f x, δ ) . Then ( X, f ) has the specification property at scale δ/ (1 − e − λ ) .Proof. It suffices to verify (5.17) with δ = δ , δ = 0, and χ = e − λ ; then we canapply Lemma 5.10. We need the following consequence of the locally onto property:(5.20) for every x ∈ X and n ∈ N , we have f n ( B n ( x, δ )) ⊃ B ( f n x, δ ) . As in the previous proposition, we use the following consequence of topologicaltransitivity and compactness: given δ >
0, there is τ ∈ N such that for every x, y ∈ X there is t ∈ { , , . . . , τ } with f t ( x ) ∈ B ( y, δ ). Now given ( x , n ) , ( x , n ) ∈ X × N , there is t ∈ { , , . . . , τ } such that f t ( f n ( x )) ∈ B ( x , δ ), and thus (5.20)gives f t ( f n B n ( x , δ )) ⊃ f t B ( f n x , δ ) ⊃ B ( f n + t x , δ ) (cid:51) x . Thus there is y ∈ B n ( x , δ ) such that f n + t ( y ) = x , which verifies (5.17); Lemma5.10 completes the proof. (cid:3) Bowen’s proof revisited.
Bowen’s original uniqueness result [Bow75], whichwe outlined in §
3, was actually given not for shift spaces, but for more generalexpansive systems.
Theorem 5.14 (Expansivity and specification (Bowen)) . Let X be a compact metricspace and f : X → X a continuous map. Suppose that (cid:15) > δ > are such that f has expansivity at scale (cid:15) and the specification property at scale δ . Then ( X, f ) hasa unique measure of maximal entropy. In the symbolic setting, this corresponds to X being a subshift of finite type. EYOND BOWEN’S SPECIFICATION PROPERTY 23
Remark . Bowen’s original paper assumed expansivity and periodic specificationat all scales. We relax the proof mildly so that it does not use periodic orbits andonly uses specification at a fixed scale, small relative to an expansivity constant. We will see examples later where this additional generality is beneficial.The proof of Theorem 5.14 extends the strategy in the symbolic case:(1) establish uniform counting bounds;(2) show that the usual construction of an MME gives an ergodic Gibbs measure;(3) prove that an ergodic Gibbs measure must be the unique MME.Now we examine the role played by expansivity.5.4.1.
Uniform counting bounds.
In the symbolic setting, the first step was to provethe counting bounds on L n given in (3.11). In the general setting, L n is replacedwith Λ( X, (cid:15), n ) from Definition 5.2, and mimicking the arguments in Lemma 3.6leads to the estimates(5.21) e nh top ( X,f, δ ) ≤ Λ( X, δ, n ) ≤ Qe nh top ( X,f,δ ) , where Q = ( τ + 1) e τh top ( X,f,δ ) . Observe that the lower and upper bounds in (5.21) involve the entropy of f atdifferent scales, a phenomenon which did not appear in (3.11). To see why thisoccurs, recall that in the proof of Lemma 3.6 we used an injective map(5.22) L m + n → L m × L n , w (cid:55)→ ( w [1 ,m ] , w [ m +1 ,m + n ] ) , as well as an at-most-( τ + 1)-to-1 map given by specification:(5.23) L m × L n → L m + n + τ , ( v, w ) (cid:55)→ vuwu (cid:48) . In a general metric space, to generalize (5.22) one might first attempt the following: • fixing ρ >
0, let E ρk ⊂ X be a maximal ( k, ρ )-separated set for each k ∈ N ; • by maximality, for every x ∈ X and k ∈ N there is π k ( x ) ∈ E ρk such that x ∈ B k ( π k ( x ) , ρ ); • then consider the map E ρm + n → E ρm × E ρn given by x (cid:55)→ ( π m ( x ) , π n − m ( f m x )).The problem is that injectivity may fail: there could be z ∈ E m such that B m ( z, ρ )contains two distinct points x, y ∈ E m + n , even though d m ( x, y ) ≥ d m + n ( x, y ) ≥ ρ .This possibility can be ruled out by considering a map E ρm + n → E ρm × E ρn ; note theuse of two different scales. With ρ = 3 δ , this leads to the lower bound in (5.21).See [CT14, § E ρk as above, specification (at scale δ ) gives a “gluing map” π : E ρm × E ρn → X . Aslong as ρ ≥ δ , the multiplicity of this map is at most τ + 1 for the same reasonsas in Lemma 3.6. However, since the gluing process in specification can move orbitsegments by up to δ , the image set π ( E ρm × E ρn ) can only be guaranteed to be( ρ − δ, m + n + τ )-separated. Again, taking ρ = 3 δ gives (5.21); see [CT14, § The statements in [CT14] used (cid:15) ≥ δ but this must be corrected to (cid:15) > δ ; see [CT16, § Remark . The reason that these issues do not arise in the symbolic setting isthat there, if δ = and y ∈ B n ( x, δ ), then B n ( y, δ ) = [ y [1 ,n ] ] = [ x [1 ,n ] ] = B n ( x, δ ).In other words, in a shift space, each d n is an ultrametric , for which the triangleinequality is strengthened to d n ( x, z ) ≤ max { d n ( x, y ) , d n ( y, z ) } . In the non-symbolicsetting, if y ∈ B n ( x, δ ) then the most we can say is that B n ( y, δ ) ⊂ B n ( x, δ ), andvice versa. This leads to the “changing scales” aspect of the arguments above, whichappears at several other places in the general proofs.5.4.2. Construction of a Gibbs measure.
With the counting bounds established asin (3.11) and (5.21), the next step in the symbolic proof was to consider measures ν n giving equal weight to every n -cylinder, and prove a Gibbs property for any limitpoint of the measures µ n = n (cid:80) n − k =0 σ k ∗ ν n . For non-symbolic systems, one replacesthe collection of n -cylinders with a maximal ( n, δ )-separated set, and proves thefollowing. Proposition 5.17.
Let X be a compact metric space and f : X → X a continuousmap with the specification property at scale δ > and expansivity at scale (cid:15) , with (cid:15) > δ , and let ρ ∈ (5 δ, (cid:15)/ . Let E n ⊂ X be a maximal ( n, ρ − δ ) -separated set foreach n , and consider the measures (5.24) µ n := 1 E n (cid:88) x ∈ E n n n − (cid:88) k =0 δ f k x . Then there is K ≥ such that every weak* limit point µ of the sequence µ n is f -invariant and satisfies the Gibbs property (5.25) K − e − nh top ( X,f ) ≤ µ ( B n ( x, ρ )) ≤ Ke − nh top ( X,f ) for all x ∈ X, n ∈ N . This statement is a mild extension of the argument in [Bow75], which is simplifiedby having periodic specification at all scales and constructing µ n using periodicorbits. Proposition 5.17 is proven, with the same level of detail on the choice ofscales in [CT16, § § h top ( X, f, cρ ), with c ∈ { , , , } . Itis thus crucial that h top ( X, f, cρ ) = h top ( X, f ), which is provided in this statementby the expansivity assumption. This is the only way in which expansivity is usedin the above proposition.5.4.3.
Ergodicity.
Observe that we have not yet claimed anything about ergodicityof the Gibbs measure µ . In the symbolic case, the argument for the Gibbs propertycan be used to deduce that there is c > k ∈ N such that for every v, w ∈ L and (cid:96) ≥ | v | , there is j ∈ [ (cid:96), (cid:96) + k ) such that µ ([ v ] ∩ σ − j [ w ]) ≥ cµ [ v ] µ [ w ] . EYOND BOWEN’S SPECIFICATION PROPERTY 25
Since any Borel set can be approximated (w.r.t. µ ) by unions of cylinders, this canbe used to deduce that lim j →∞ µ ( V ∩ σ − j W ) ≥ ck µ ( V ) µ ( W )for all V, W ⊂ X , which gives ergodicity. In the non-symbolic setting, one canstill mimic the Gibbs argument to produce c > k ∈ N such that for every( x, n ) , ( y, m ) ∈ X × N and any (cid:96) ≥ n , there is j ∈ [ (cid:96), (cid:96) + k ) such that(5.26) µ ( B n ( x, ρ ) ∩ f − j B m ( y, ρ )) ≥ cµ ( B n ( x, ρ )) µ ( B m ( y, ρ )) . To establish ergodicity from this one needs to approximate arbitrary Borel sets bysets whose µ -measure we control; this can be done by using a sequence of adaptedpartitions β n , for which each element of β n contains a Bowen ball B n ( x, ρ ) and iscontained inside a Bowen ball B n ( x, ρ ). Expansivity implies that this sequence ofpartitions is generating w.r.t. µ , so the rest of the argument goes through as before,and establishes ergodicity. As we saw in the proof of Proposition 5.8, this is alsoenough to guarantee that h top ( X, f, (cid:15) ) = h top ( X, f ). We summarize our conclusionsas follows.
Proposition 5.18.
Let
X, f, δ, µ be as in Proposition 5.17. Suppose that f is ex-pansive at scale δ . Then µ is ergodic and satisfies the Gibbs property (5.25) . Adapted partitions and uniqueness.
The proof that an ergodic Gibbs measureis the unique MME (Proposition 3.4) has the following generalization to the non-symbolic setting.
Proposition 5.19.
Let X be a compact metric space, f : X → X a continuous map,and µ an ergodic f -invariant measure on X . Suppose ρ > is such that • f is expansive (or positively expansive) at scale ρ ; • there are K, h > such that µ satisfies the Gibbs bound (5.27) K − e − nh ≤ µ ( B n ( x, ρ )) ≤ Ke − nh for every x ∈ X and n ∈ N . Then h = h µ ( f ) = h top ( X, f ) , and µ is the unique MME for ( X, f ) .Outline of proof. As before, one starts by using general arguments to prove that h = h µ ( f ) = h top ( X, f ) and to reduce to the case of considering an invariant measure ν ⊥ µ , for which we must show h ν ( f ) < h µ ( f ); this is unchanged from the symboliccase. The next step there was to choose D ⊂ X with µ ( D ) = 1 and ν ( D ) = 0, andapproximate D by a union of cylinders; then similar to (5.16), writing(5.28) nh ν ( f ) = h ν ( f n ) = h ν ( f n , α n − ) ≤ H ν ( α n − ) = (cid:88) w ∈L n − ν [ w ] log ν [ w ] , and splitting the sum between cylinders in D n and those in D cn , one eventually provesthat h ν ( f ) < h µ ( f ) by using the Gibbs bound µ [ w ] ≥ K − e −| w | h top ( X ) .In the non-symbolic setting, the approximation of D follows just as in the para-graph after (5.26). Moreover, we can obtain an analogue of (5.28) by replacing α n − with a partition β n such that every element of β n is contained in B n ( x, ρ ) for some point x in a maximal ( n, ρ )-separated set E n . Finally, as long as we also arrangethat each element of β n contain B n ( x, ρ ), we can use the lower Gibbs bound tocomplete the proof just as in the symbolic case. (cid:3) Remark . The partition β n which appears in the above proof is called an adaptedpartition for E n . Adapted partitions exist for any ( n, ρ )-separated set of maximalcardinality since the sets B n ( x, ρ ) are disjoint and the sets B n ( x, ρ ) cover X . Remark . In the two-sided expansive case, the same argument works, providedwe replace d n and B n with their two-sided versions. That is, we consider balls inthe metric d [ − n,n ] ( x, y ) = max { d ( f k x, f k y ) : − n ≤ k ≤ n } in place of B n . Then oneuses adapted partitions and proceeds as in the positively expansive case. Part
II.
Non-uniform Bowen hypotheses and equilibrium states In §
6, we recall the role played by expansivity in Bowen’s proof of uniqueness,and formulate a uniqueness result using a weaker version of expansivity. Thenin § § § §
10 we describe how this theory extends to equilibrium statesfor nonzero potential functions.6.
Relaxing the expansivity hypothesis
In this section, we describe how we relax the expansivity property. Our motivatingexamples are diffeomorphisms for which expansivity fails, but for which the failure ofexpansivity is “invisible” to the MME. In these examples, the failure of expansivityis a lower entropy phenomenon, and this leaves room for us to develop a version ofBowen’s argument for the MME.As explained in the previous section, Bowen’s proof of uniqueness uses expansivityto guarantee that certain sequences of partitions are generating with respect to everyinvariant ν . In fact, in every place where this property is used, it is enough to knowthat this holds for all ν with sufficiently large entropy.More precisely, at the end of the proof, in (the analogue of) (5.28), it suffices toknow that α n − is generating for ( f n , ν ) when ν is an arbitrary MME, because if ν is not an MME then we already have h ν < h µ , which was the goal. This is alsosufficient for the approximation of D by elements of the partitions β n , and thusProposition 5.19 remains true if we replace expansivity with the assumption thatfor every MME ν , we have Γ (cid:15) ( x ) = { x } for ν -a.e. x .In Proposition 5.17, the argument for ergodicity required a similar generatingproperty. Finally, in Proposition 5.8, it suffices to have this generating propertyw.r.t. a family of measures ν over which sup ν h ν ( f ) = h top ( X, f ).With these observations in mind, we make the following definitions.
EYOND BOWEN’S SPECIFICATION PROPERTY 27
Definition 6.1 ([BF13]) . An f -invariant measure µ is almost expansive at scale (cid:15) if Γ (cid:15) ( x ) = { x } for µ -a.e. x ; equivalently, if the non-expansive set NE( (cid:15) ) = { x ∈ X :Γ (cid:15) ( x ) (cid:54) = { x }} has µ (NE( (cid:15) )) = 0. Replacing Γ (cid:15) by Γ + (cid:15) gives NE + and a notion of almost positively expansive . Definition 6.2 ([CT14]) . The entropy of obstructions to expansivity at scale (cid:15) is h ⊥ exp ( X, f, (cid:15) ) := sup { h µ ( f ) : µ ∈ M ef ( X ) is not almost expansive at scale (cid:15) } = sup { h µ ( f ) : µ ∈ M ef ( X ) and µ (NE( (cid:15) )) > } . We write h ⊥ exp ( X, f ) = lim (cid:15) → h ⊥ exp ( X, f, (cid:15) ) for the entropy of obstructions to expan-sivity , without reference to scale. The entropy of obstructions to positive expansivity h ⊥ exp + is defined analogously.From the discussion after Proposition 5.17, we see that we can replace the as-sumption of expansivity with the assumption that h ⊥ exp ( X, f, ρ ) < h top ( X, f ), sincethen every ergodic ν with h ν ( f ) > h ⊥ exp ( X, f, ρ ) is almost expansive, so the Propo-sition goes through. Similarly in Proposition 5.18and Proposition 5.19, it sufficesto assume that h ⊥ exp ( X, f, ρ ) < h top ( X, f ).Now we have all the pieces for a uniqueness result using non-uniform expansivity.
Theorem 6.3 (Unique MME with non-uniform expansivity [CT14]) . Let X be acompact metric space and f : X → X a continuous map. Suppose that (cid:15) > δ > are such that h ⊥ exp ( X, f, (cid:15) ) < h top ( X, f ) , and that f has the specification property atscale δ . Then ( X, f ) has a unique measure of maximal entropy. Derived-from-Anosov systems
We describe a class of smooth systems for which expansivity fails but the entropyof obstructions to expansivity is small. The following example is due to Ma˜n´e[Ma˜n78]; we primarily follow the discussion in [CFT19], and refer to that paper forfurther details and references.7.1.
Construction of the Ma˜n´e example.
Fix a matrix A ∈ SL (3 , Z ) withsimple real eigenvalues λ u > > λ s > λ ss >
0, and corresponding eigenspaces F u,s,ss ⊂ R . Let f : T → T be the hyperbolic toral automorphism defined by A ,and let F u,s,ss be the corresponding foliations of T . Define a perturbation f of f as follows.Fix ρ > ρ (cid:48) > f is expansive at scale ρ . Let q ∈ T be a fixed point of f , and set f = f outside of B ( q, ρ ). Inside B ( q, ρ ), perform a pitchfork bifurcationin the center direction as shown in Figure 7.1, in such a way that • the foliation W c := F s remains f -invariant, and we write E c = T W c ; • the cones around F u and F ss remain invariant and uniformly expanding for Df and Df − , respectively, so they contain Df -invariant distributions E u,ss that integrate to f -invariant foliations W u,ss ; See [CT14, Proposition 2.7] for a detailed proof that h top ( X, f, ρ ) = h top ( X, f ) in this case. F s F ss F u qB ( q, ρ ) f W c W ss W u qpf Figure 7.1.
Ma˜n´e’s construction. • E cs = E c ⊕ E ss integrates to a foliation W cs ; • outside of B ( q, ρ (cid:48) ), we have (cid:107) Df | E cs (cid:107) ≤ λ s < f is partially hyperbolic with T T = E u ⊕ E c ⊕ E ss = E u ⊕ E cs . Observe that(7.1) λ c ( f ) := sup {(cid:107) Df | E cs ( x ) (cid:107) : x ∈ T } > q .Now consider a diffeomorphism g : T → T that is C -close to f . Such a g remains partially hyperbolic, with(7.2) λ c ( g ) > > λ s ( g ) := sup {(cid:107) Df | E cs ( x ) (cid:107) : x ∈ T \ B ( q, ρ (cid:48) ) } . Existence of a unique MME was proved for such g by Ures [Ure12] and by Buzzi,Fisher, Sambarino, and V´asquez [BFSV12], using the fact that there is a semiconju-gacy from g back to the hyperbolic toral automorphism f . We outline an alternateproof using Theorem 6.3, which has the benefit of extending to class of nonzeropotential functions [CFT19].7.2. Estimating the entropy of obstructions.
Although the map g behaves asif it is uniformly hyperbolic outside of B ( q, ρ ), the presence of fixed points withdifferent indices inside this ball causes expansivity to fail. Indeed, let p denote oneof the two fixed points created via the pitchfork bifurcation, and let x be any pointon the leaf of W c that connects p to q . Then for every (cid:15) >
0, the bi-infinite Bowenball Γ (cid:15) ( x ) is a non-trivial curve in W c , rather than a single point. However, we cangive a simple mild criterion on the orbit of a point x which rules out Γ (cid:15) ( x ) beingnon-trivial, and we can argue that this criterion is satisfied for most points in ourexamples. Lemma 7.1.
Let g be a partially hyperbolic diffeomorphism with a splitting E u ⊕ E c ⊕ E s such that E c is 1-dimensional and integrable. Then there is (cid:15) > suchthat Γ (cid:15) ( x ) ⊂ W c ( x ) for every x . Moreover, for every λ > there is (cid:15) > such that (7.3) lim n →∞ n log (cid:107) Dg − n | E c ( x ) (cid:107) > λ ⇒ Γ (cid:15) ( x ) = { x } . EYOND BOWEN’S SPECIFICATION PROPERTY 29
Sketch of proof.
Following the argument for expansivity in the uniformly hyperbolicsetting, we choose (cid:15) such that whenever d ( x, y ) < (cid:15) , we can get from x to y bymoving a distance d s along a leaf of W s , then a distance d c along a leaf of W c , thena distance d u along a leaf of W u . The argument given there shows that if y ∈ Γ (cid:15) ( x )then we must have d s ( x, y ) = d u ( x, y ) = 0, which implies that y ∈ W c ( x ). For (7.3),we observe that if the condition on Dg − n is satisfied, then there are arbitrarily large n such that(7.4) (cid:107) Dg − n | E c ( x ) (cid:107) > ce λn . Choosing (cid:15) > | log (cid:107) Dg | E c ( z ) (cid:107) − log (cid:107) Dg | E c ( z (cid:48) ) (cid:107)| < λ/ d ( z, z (cid:48) ) < (cid:15) , we see that any y ∈ Γ (cid:15) ( x ) satisfies(7.5) d ( g − n x, g − n y ) ≥ ce λn/ d ( x, y )for all n satisfying (7.4). Since n can become arbitrarily large, this implies that d ( x, y ) = 0. (cid:3) Remark . Replacing backwards time with forwards time, the analogous resultfor positive Lyapunov exponents is also true: lim n log (cid:107) Dg n | E c ( x ) (cid:107) > λ implies thatΓ (cid:15) ( x ) = { x } .For the Ma˜n´e examples, we can use (7.2) to control (cid:107) Dg − n | E c ( x ) (cid:107) in terms ofhow much time the orbit of x spends outside B ( q, ρ ); together with Lemma 7.1, thisallows us to estimate the entropy of NE( (cid:15) ). To formalize this, we write χ = T \ B ( q,ρ ) and observe that by the definition of λ c ( g ) and λ s ( g ) in (7.1) and (7.2), we have (cid:107) Dg − n | E c ( x ) (cid:107) ≥ λ s ( g ) − s n ( x ) λ c ( g ) − ( n − s n ( x )) where s n ( x ) := n − (cid:88) k =0 χ ( g − k x ) . It follows that(7.6) lim n →∞ n log (cid:107) Dg − n | E c ( x ) (cid:107) ≥ − ( r ( x ) log λ s ( g ) + (1 − r ( x )) log λ c ( g ))where we write r ( x ) = lim n →∞ n s n ( x ) = lim n →∞ n n − (cid:88) k =0 χ ( g − k x ) . Fix λ ∈ (0 , − log λ s ( g )) and let r > − ( r log λ s ( g ) + (1 − r ) log λ c ( g )) > λ .Then Lemma 7.1 and (7.6) show that for a sufficiently small (cid:15) >
0, we have(7.7) NE( (cid:15) ) ⊂ { x : r ( x ) < r } . Since f is Anosov, the uniform counting bounds in (5.21) give a constant Q suchthat Λ( X, f , (cid:15), n ) ≤ Qe nh top ( X,f ) for all n . Using this together with (7.7) one canprove the following. Lemma 7.3 ([CFT18, § . Writing H ( t ) = − t log t − (1 − t ) log(1 − t ) for theusual bipartite entropy function, the Ma˜n´e examples satisfy h ⊥ exp ( g, (cid:15) ) < r ( h top ( X, f ) + log Q ) + H (2 r ) . Idea of proof.
Given an ergodic measure µ that satisfies µ (NE( (cid:15) )) and thus satisfieslim n S n χ ( g − n x ) ≤ r for µ -a.e. x , the Katok entropy formula [Kat80] can be used toshow that h µ ( f ) ≤ h ( C ), where(7.8) C := { ( x, n ) ∈ T × N : S n χ ( x ) ≤ rn } . To estimate h ( C ), the idea is to partition an orbit segment ( x, n ) ∈ C into pieceslying entirely inside or outside of B ( q, ρ ). There can be at most rn pieces lyingoutside, so the number of transition times between inside and outside is at most2 rn . The number of ways of choosing these transition times is thus at most (cid:18) n rn (cid:19) = n !(2 rn )!((1 − r ) n )! ≈ e H (2 r ) n , where the approximation can be made more precise using Stirling’s formula or arougher elementary integral estimate. This contributes the H (2 r ) term to the esti-mate; the remaining terms are roughly due to the observation that given a pattern oftransition times for which the segments lying outside B ( q, ρ ) have lengths k , . . . , k m ,the number of (cid:15) -separated orbit segments in C associated to this pattern is at most m (cid:89) j =1 Λ( X, f , (cid:15), k i ) ≤ m (cid:89) j =1 Qe k i h top ( X,f ) ≤ Q m e rnh top ( X,f ) ≤ ( Qe h top ( X,f ) ) rn , since no entropy is produced by the sojourns inside B ( q, ρ ). (cid:3) Since there is a semi-conjugacy from g to f , we have h top ( X, g ) ≥ h top ( X, f ).Thus we have h ⊥ exp ( g ) < h top ( g ) whenever r satisfies(7.9) r ( h top ( X, f ) + log Q ) + H (2 r ) < h top ( X, f ) . Recall that r must be chosen large enough such that λ s ( g ) r λ c ( g ) − r <
1. Equiva-lently, for a given value of r , the perturbation must be chosen small enough for thisto hold (that is, λ c must be close enough to 1). Thus given f , we can find r smallenough such that (7.9) holds, and then for any sufficiently small perturbation theabove argument guarantees that h ⊥ exp ( X, g ) < h top ( X, g ). Remark . Since Γ (cid:15) ( x ) ⊂ W c ( x ), which is one-dimensional, it is not hard toshow that h top ( W c ( x )) = 0, and thus h top (Γ (cid:15) ( x )) = 0 [CY05, CFT19]; in otherwords, f is entropy expansive . Entropy expansivity implies that h top ( X, f, (cid:15) ) = h top ( X, f ) [Bow72], which for systems with (coarse) specification is sufficient for theconstruction of a Gibbs measure in Proposition 5.17. However, there does not seemto be any way to use entropy expansivity to carry out the arguments for ergodicityand uniqueness. The issue is that we need to use Bowen balls to construct adaptedpartitions which approximate Borel sets. When Γ (cid:15) ( x ) is a point, the two-sidedBowen ball at x is a neighborhood of the point, which is key to the approximationargument. The analysis is significantly more difficult even when Γ (cid:15) ( x ) (cid:54) = x has asimple explicit characterization, see § for more details in the flow case. If all we knowabout Γ (cid:15) ( x ) is that h (Γ (cid:15) ( x )) = 0 it is unclear how to proceed. On the other hand,for the Bonatti–Viana examples introduced in [BV00], entropy expansivity can fail EYOND BOWEN’S SPECIFICATION PROPERTY 31 [BF13] even while the condition h ⊥ exp < h top is satisfied [CFT18]. The Bonatti-Viana examples are 4-dimensional analogues of the Ma˜n´e examples that involve twoseparate perturbations and have a dominated splitting T T = E cu ⊕ E cs but arenot partially hyperbolic. We were able to study their thermodynamic formalism in[CFT18] despite these difficulties.7.3. Specification for Ma˜n´e examples.
In order to apply Theorem 6.3 to theMa˜n´e examples, one must investigate the specification property. Globally, specifi-cation at all scales certainly fails. Two approaches to deal with this are possible,and it is instructive to consider both - our choice is to work with a coarse specifi-cation property globally, or specification at all scales on a ‘good collection of orbitsegments’.The key ingredient we are missing from the uniformly hyperbolic case is uniformcontraction along W cs , which is replacing W s . We explain why we can obtaincoarse specification globally. As explained in Remark 5.12, uniform contraction isnot needed for the proof of specification; it suffices to know that(7.10) W csδ ( x ) ⊂ B n ( x, δ ) for all x. Since contraction in W cs can fail for the Ma˜n´e example only in B ( q, ρ (cid:48) ), one can easilyshow that (7.10) continues to hold as long as δ > ρ (cid:48) , and thus g has specificationat these scales. Choosing ρ (cid:48) to be small enough relative to ρ , Theorem 6.3 appliesand establishes existence of a unique MME.To see that the Ma˜n´e example does not have the specification property at allscales, observe that for sufficiently small δ >
0, the forward infinite Bowen ballΓ + δ ( q ) is the 1-dimensional local stable leaf W ssδ ( q ). Suppose that g has specificationat scale δ with gap size τ , and let x be any point whose orbit never enters B ( q, ρ ).Specification gives y ∈ W uδ ( x ) and 0 ≤ k ≤ τ such that f k ( y ) ∈ W ssδ ( q ); In otherwords, f − τ ( W ssδ ( q )) intersects every local unstable leaf associated to an orbit thatavoids B ( q, ρ ). But this is impossible because the dimensions are wrong. Thus, if we want a global specification property, we must work at a fixed coarsescale, as described above. We explore the other option of returning to the ideasfrom § The general result for MMEs in discrete-time
Now we formulate a general result that combines the symbolic result using de-compositions with Theorem 6.3 by allowing both expansivity and specification tofail, provided the obstructions have small entropy. This allows us to cover some Use specification to get y n ∈ f n ( B n ( x, δ )) ∩ f − k n ( B n ( q, δ )) for 0 ≤ k n ≤ τ , choose k such that k n = k for infinitely many values of n , and let y be a limit point of the corresponding y n . Note that f − τ ( W ssδ ( q )) intersects a local leaf of W cu in at most finitely many points, andthus thus intersects at most finitely many of the corresponding local leaves of W u ; however, thereare uncountably many of these corresponding to points that never enter B ( q, ρ ). new classes of examples, as we will see later, and is also important in dealing withnonzero potential functions.Recall from § L of a shift space consistsof C p , G , C s ⊂ L such that every w ∈ L can be written as w = u p vu s where u p ∈ C p , v ∈ G , and u s ∈ C s . As discussed in § L with the space of orbit segments X × N , where ( x, n ) corresponds to the orbitsegment x, f ( x ) , f ( x ) , . . . , f n − ( x ). Definition 8.1. A decomposition for X × N consists of three collections C p , G , C s ⊂ X × N for which there exist three functions p, g, s : X × N → N such that forevery ( x, n ) ∈ X × N , the values p = p ( x, n ), g = g ( x, n ), and s = s ( x, n ) satisfy p + g + s = n , and ( x, p ) ∈ C p , ( f p x, g ) ∈ G , ( f p + s x, s ) ∈ C s . Given a decomposition, for each M ∈ N we write G M := { ( x, n ) ∈ X × N : p ( x, n ) ≤ M and s ( x, n ) ≤ M } . Theorem 8.2 (Non-uniform Bowen hypotheses for maps (MME case)) . Let X be acompact metric space and f : X → X a continuous map. Suppose that (cid:15) > δ > are such that h ⊥ exp ( X, f, (cid:15) ) < h top ( X, f ) , and that the space of orbit segments X × N admits a decomposition C p GC s such that (I) every collection G M has specification at scale δ , and (II) h ( C p ∪ C s , δ ) < h top ( X, f ) .Then ( X, f ) has a unique measure of maximal entropy. The proof of Theorem 8.2 requires an extension of the counting arguments fordecompositions ( § § § §§ G M . As in § Remark . If G has specification at all scales, then a short continuity argument[CT16, Lemma 2.10] proves that every G M does as well, which establishes (I).9. Partially hyperbolic systems with one-dimensional center
Theorem 8.2 can be applied to a broad class of partially hyperbolic systems, whichincludes the Ma˜n´e examples. This result has not previously appeared elsewhere. Wegive an outline of the proof. Further details are analogous to the case of the Ma˜n´eexamples, and we emphasize the key new points.
Theorem 9.1.
Let f : M → M be a partially hyperbolic diffeomorphism with T M = E u ⊕ E c ⊕ E s . Assume that dim E c = 1 and that every leaf of the foliations W s and W u is dense in M . EYOND BOWEN’S SPECIFICATION PROPERTY 33
Let ϕ c ( x ) = log (cid:107) Df | E c ( x ) (cid:107) , and given µ ∈ M ef ( M ) , let λ c ( µ ) = (cid:82) ϕ c dµ be the center Lyapunov exponent of µ . Consider the quantities (9.1) h + := sup { h µ ( f ) : µ ∈ M ef ( M ) , λ c ( µ ) ≥ } ,h − := sup { h µ ( f ) : µ ∈ M ef ( M ) , λ c ( µ ) ≤ } . Suppose that h + (cid:54) = h − . Then f has a unique MME.Remark . Since h top ( X, f ) = max( h + , h − ), the condition h + (cid:54) = h − is equivalent tothe condition that either h + < h top ( X, f ) or h − < h top ( X, f ). The only way for thiscondition to fail is if there is an ergodic MME with λ c = 0, or if there are (at least)two ergodic MMEs for which λ c takes both signs. See § Remark . For 3-dimensional partially hyperbolic diffeomorphisms homotopic toAnosov, Ures [Ure12] showed that there is a unique measure of maximal entropy.In this setting, Crisostomo and Tahzibi [CT19b] gave some interesting criteria foruniqueness (and in some case finiteness) of equilibrium states.First observe that arguments similar to those given for the Ma˜n´e example inLemma 7.1 and Remark 7.2 show that h ⊥ exp ( f ) ≤ min( h + , h − ), so the condition h ⊥ exp ( f ) < h top ( f ) is satisfied whenever h + (cid:54) = h − . Remark . The upper bound on h ⊥ exp for the Ma˜n´e examples in Lemma 7.3 isactually an upper bound on h + in that setting, verifying that h + ( g ) < h top ( g )whenever the perturbation is small enough. Moreover, the leaves of W u are alldense for these examples [PS06], so Theorem 9.1 applies to the Ma˜n´e examples.The rest of the proof of Theorem 9.1 consists of finding a decomposition C p , G , C s for X × N such that G has specification at all scales and h ( C p ∪ C s ) < h top ( X, f ).We describe the general argument in the case when h + < h top ( f ), so intuitively, allof the large entropy parts of the system have negative central Lyapunov exponents.9.1. A small collection of obstructions.
We take C s = ∅ . To describe C p , wefirst observe that the condition h + < h top ( f ) implies thatsup { h µ ( f ) : µ ∈ M f , λ c ( µ ) ≥ } < h top ( f ) , where the difference is that now the supremum allows non-ergodic measures as well,and then a weak*-continuity argument gives r > { h µ ( f ) : µ ∈ M f , λ c ( µ ) ≥ − r } < h top ( f ) . We can relate the left-hand side of (9.2) to h ( C p ), where C p := { ( x, n ) ∈ M × N : S n ϕ c ( x ) ≥ − rn } . One relationship between these was mentioned when we bounded h ⊥ exp for the Ma˜n´eexample (though the function being summed there was different). Here we want togo the other way and obtain an upper bound on h ( C p ). For this we observe that if we let E n ⊂ C pn be any ( n, (cid:15) )-separated set, ν n the equidistributed atomic measureon E n , and µ n = n (cid:80) n − k =0 f k ∗ ν n , then half of the proof of the variational principle[Wal82, Theorem 8.6] shows that any limit point of µ n is f -invariant and has h µ ( f ) ≥ h ( C p , (cid:15) ) . Moreover, λ c ( µ ) = (cid:82) ϕ c dµ ( x ) ≥ − r by weak*-convergence and the definition of C p .Together with (9.2), we conclude that h ( C p ) < h top ( f ).9.2. A good collection with specification.
We now describe a ‘good’ collectionof orbit segments G , and define a decomposition. To this end, take an arbitrary orbitsegment ( x, n ) ∈ M × N , and remove the longest possible element of C p from itsbeginning. That is, let p = p ( x, n ) be maximal with the property that ( x, p ) ∈ C p .Then we have S p ϕ c ( x ) ≥ − rp and S k ϕ c ( x ) < − rk for all p < k ≤ n. Subtracting the first from the second gives S k − p ϕ c ( f p x ) = S k ϕ c ( x ) − S p ϕ c ( x ) < − r ( k − p ) , which we can rewrite as S j ϕ c ( f j x ) < − rj for all 0 ≤ j ≤ n − p. In other words, as shown in Figure 9.1, we have (9.3) ( f p x, n − p ) ∈ G := { ( y, m ) : S j ϕ c ( y ) < − rj for all 0 ≤ j ≤ m } .x f n ( x ) ∈ C p ∈ G f p ( x ) ⇓ S k ϕ c < − krS k ϕ c ≥ − kr ⇓ Figure 9.1.
A decomposition C p G of the space of orbit segments.Moreover, by choosing δ > | ϕ c ( y ) − ϕ c ( z ) | < r/ d ( y, z ) < δ , we see that if ( y, m ) ∈ G and z ∈ B m ( y, δ ), then(9.4) (cid:107) Df j | E cs ( z ) (cid:107) ≤ e − rj/ for all 0 ≤ j ≤ m. There is a clear analogy between what we are doing here and the notion of hyperbolic time introduced by Alves [Alv00, ABV00].
EYOND BOWEN’S SPECIFICATION PROPERTY 35
This is enough to prove the specification property for G . If E cs is integrable, thenone can simply use the proof from the uniformly hyperbolic case verbatim, using(9.4) to guarantee that(9.5) W csδ ( x ) ⊂ B n ( x, δ ) whenever ( x, n ) ∈ G . Since questions of integrability in partial hyperbolicity can be subtle [RHRHU16],we point out that one can still establish the specification property without assumingintegrability of E cs . To do this, fix θ > center-stable cone K cs ( x ) := { v + w : v ∈ E cs , w ∈ E u , (cid:107) w (cid:107) < θ (cid:107) v (cid:107)} ⊂ T x M ;then when establishing the “one-step specification” property in (5.17), one can takean admissible manifold W (cid:51) f n ( x ) that has T y W ⊂ K cs ( x ) at each y ∈ W , andreplace W csδ ( x ) with f − n ( W ) ∩ B ( x , δ ) in the argument. As long as θ > x , n ) for vectors in K cs to guarantee that (9.5) holds.10. Unique equilibrium states
For the sake of simplicity, we have so far restricted our attention to measuresof maximal entropy. However, the entire apparatus developed above works equallywell for equilibrium states associated to “sufficiently regular” potential functions.10.1.
Topological pressure.
First we recall the notion of topological pressure . Aswith topological entropy in § D ⊂ X × N ; our definition reducesto the standard one when D = X × N . Definition 10.1.
Given a continuous potential function ϕ : X → R and a collectionof orbit segments D ⊂ X × N , for each (cid:15) > n ∈ N we consider the partitionsum (10.1) Λ( D , ϕ, (cid:15), n ) := sup (cid:110) (cid:88) x ∈ E e S n ϕ ( x ) : E ⊂ D n is ( n, (cid:15) )-separated (cid:111) , where S n ϕ ( x ) = (cid:80) n − k =0 ϕ ( f k x ) is the n th Birkhoff sum. The pressure of ϕ on thecollection D at scale (cid:15) > P ( D , ϕ, (cid:15) ) := lim n →∞ n log Λ( D , ϕ, (cid:15), n ) , and the pressure of ϕ on the collection D is(10.3) P ( D , ϕ ) := lim (cid:15) → P ( D , ϕ, (cid:15) ) . As with entropy, in the case when D = Y × N we write Λ( Y, ϕ, (cid:15), n ), etc.The variational principle for topological pressure states that(10.4) P ( X, ϕ ) = sup µ ∈M f ( X ) (cid:16) h µ ( f ) + (cid:90) ϕ dµ (cid:17) . A measure that achieves the supremum is called an equilibrium state for (
X, f, ϕ ).As was the case with the MME, there is a standard construction from the proof ofthe variational principle that establishes existence of an equilibrium state in manycases: we have the following generalization of Proposition 5.4 and Corollary 5.5.
Proposition 10.2 (Building approximate equilibrium states) . With
X, f, ϕ as above,fix (cid:15) > , and for each n ∈ N , let E n ⊂ X be an ( n, (cid:15) ) -separated set. Consider theBorel probability measures (10.5) ν n := 1 (cid:80) x ∈ E n e S n ϕ ( x ) (cid:88) x ∈ E n δ x e S n ϕ ( x ) , µ n := 1 n n − (cid:88) k =0 f k ∗ ν n = 1 n n − (cid:88) k =0 ν n ◦ f − k . Let µ n j be any subsequence that converges in the weak*-topology to a limiting measure µ . Then µ ∈ M f ( X ) and (10.6) h µ ( f ) + (cid:90) ϕ dµ ≥ lim j →∞ n j log (cid:88) x ∈ E nj e S nj ϕ ( x ) . In particular, for every δ > there exists µ ∈ M f ( X ) such that h µ ( f ) + (cid:82) ϕ dµ ≥ P ( X, f, ϕ, δ ) .Proof. See [Wal82, Theorem 9.10]. (cid:3)
Corollary 10.3.
Let
X, f be as above, and suppose that there is δ > such that P ( X, ϕ, δ ) = P ( X, ϕ ) . Then there exists an equilibrium state for ( X, f, ϕ ) . Indeed,given any sequence { E n ⊂ X } ∞ n =1 of maximal ( n, δ ) -separated sets, every weak*-limitpoint of the sequence µ n from (10.5) is an equilibrium state. There is an analogue of Proposition 5.8 for pressure: if (
X, f ) is expansive atscale (cid:15) , then P ( X, ϕ, (cid:15) ) = P ( X, ϕ ), so Corollary 10.3 establishes existence of anequilibrium state, as well as a way to construct one. Then the goal becomes toprove uniqueness.10.2.
Regularity of the potential function: the Bowen property.
Even foruniformly hyperbolic systems, one should not expect every continuous potentialfunction to have a unique equilibrium state. Indeed, for the full shift it is possible toshow that given any finite set E of ergodic measures, there is a continuous potentialfunction ϕ whose set of equilibrium states is precisely the convex hull of E ; see[Isr79, p. 117] and [Rue78, p. 52].For expansive systems ( X, f ) with specification, uniqueness of the equilibriumstate can be guaranteed by the following regularity condition on the potential.
Definition 10.4.
A continuous function ϕ : X → R has the Bowen property atscale (cid:15) >
V > x, n ) ∈ X × N and y ∈ B n ( x, (cid:15) ), we have | S n ϕ ( y ) − S n ϕ ( x ) | ≤ V .The following generalization of Theorems 3.2 and 5.14 is the full statement ofBowen’s original result from [Bow75], with the slight modification that we make thescales explicit. EYOND BOWEN’S SPECIFICATION PROPERTY 37
Theorem 10.5.
Let X be a compact metric space and f : X → X a continuous map.Suppose that there are (cid:15) > δ > such that f is expansive or positively expansive atscale (cid:15) and has the specification property at scale δ . Then every continuous potentialfunction ϕ : X → R with the Bowen property at scale (cid:15) has a unique equilibriumstate. The proof of Theorem 10.5 follows the argument outlined earlier for Theorems 3.2and 5.14 in § § § h µ ( σ ) + (cid:90) ϕ dµ = lim n →∞ n (cid:0) − log µ [ x [1 ,n ] ] + S n ϕ ( x ) (cid:1) . For an equilibrium state, the left-hand side is P ( ϕ ), and this can be rewritten as P ( ϕ ) + lim n →∞ n (log µ [ x [1 ,n ] ] − S n ϕ ( x )) = 0, or equivalently,lim n →∞ n log (cid:16) µ [ x [1 ,n ] ] e − nP ( ϕ )+ S n ϕ ( x ) (cid:17) = 0 . As with the Gibbs property for the MME, uniqueness of the equilibrium state canbe guaranteed by requiring that the quantity inside the logarithm be bounded awayfrom 0 and ∞ . Generalizing to arbitrary compact metric spaces by replacingcylinders with Bowen balls, we say that a measure µ has the Gibbs property for apotential ϕ at scale (cid:15) if there are constants K > P ∈ R such that for every x ∈ X and n ∈ N , we have(10.7) K − e − nP + S n ϕ ( x ) ≤ µ ( B n ( x, (cid:15) )) ≤ Ke − nP + S n ϕ ( x ) . If it is known that every equilibrium measure is almost expansive at scale (cid:15) (recallDefinition 6.1) – in particular, if (
X, f ) is expansive at scale (cid:15) – and if µ is anergodic Gibbs measure for ϕ , then the analogue of Proposition 3.4 holds: we have P = P ( ϕ ) = h µ ( f ) + (cid:82) ϕ dµ , and µ is the unique equilibrium state for ( X, f, ϕ ).The proof is essentially the same, although now the computations involve Birkhoffsums.Similarly, in the proof of the uniform counting bounds and the construction ofan ergodic Gibbs measure using the procedure in Proposition 10.2, one encountersmultiple steps where a Birkhoff sum S n ϕ ( x ) must be replaced with S n ϕ ( y ) for some y in the Bowen ball around x , and the Bowen property is required at these steps toguarantee “bounded distortion” in the estimates.Recalling that topologically transitive locally maximal hyperbolic sets have ex-pansivity and specification, it is natural to ask which potential functions have theBowen property: how much does Theorem 10.5 extend Theorem 2.3? Observe that this is impossible if ϕ does not satisfy the Bowen property. Proposition 10.6. If X is a locally maximal hyperbolic set for a diffeomorphism f ,then every H¨older continuous function ϕ : X → R has the Bowen property at scale (cid:15) , where (cid:15) is the scale of the local product structure.Proof. Recalling the estimates (5.13) and (5.14) in the proof of Proposition 5.7, wesee that for every y ∈ B n ( x, (cid:15) ) and every k ∈ { , , . . . , n − } , we have d u ( f k x, f k y ) ≤ e − λ ( n − k ) (cid:15) and d s ( f k x, f k y ) ≤ e − λk (cid:15). Writing C for the H¨older constant and γ for the H¨older exponent, we obtain | ϕ ( f k x ) − ϕ ( f k y ) | ≤ Cd ( f k x, f k y ) γ ≤ C (cid:0) d u ( f k x, f k y ) , d s ( f k x, f k y )) (cid:1) γ ≤ C (2 (cid:15) ) γ max( e − λ ( n − k ) γ , e − λkγ ) , and summing over 0 ≤ k < n gives | S n ϕ ( x ) − S n ϕ ( y ) | ≤ n − (cid:88) k =0 C (2 (cid:15) ) γ max( e − λ ( n − k ) γ , e − λkγ ) ≤ C (2 (cid:15) ) γ n − (cid:88) k =0 e − λγ ( n − k ) + e − λγk ≤ C (2 (cid:15) ) γ ∞ (cid:88) k =0 e − λγk =: V. This last quantity is finite and independent of x, y, n , which establishes the Bowenproperty for ϕ . (cid:3) Remark . The theorem “H¨older potentials for uniformly hyperbolic systems haveunique equilibrium states” is well-entrenched enough that it is worth stressing thefollowing point: it is the dynamical Bowen property (bounded distortion), ratherthan the metric H¨older property, that is truly important here. In particular, ifwe consider a non-uniformly hyperbolic system that is conjugate to a uniformlyhyperbolic one, such as the Manneville–Pomeau interval map or Katok map of thetorus, then every potential with the Bowen property continues to have a uniqueequilibrium state, but there may be H¨older potentials with multiple equilibriumstates. However, determining which potentials have the Bowen property may be anontrivial task.10.3.
The most general discrete-time result.
Recalling the weakened versionsof expansivity and specification used in Theorem 8.2, it is natural to ask for auniqueness result for equilibrium states that uses a weakened version of the Bowenproperty. Observe that the Bowen property can be formulated for a collection oforbit segments (rather than the entire system) by replacing X × N in Definition 10.4with G ⊂ X × N . Definition 10.8.
A continuous function ϕ : X → R has the Bowen property at scale (cid:15) >
G ⊂ X × N if there is a constant V > x, n ) ∈ G and y ∈ B n ( x, (cid:15) ), we have | S n ϕ ( y ) − S n ϕ ( x ) | ≤ V . EYOND BOWEN’S SPECIFICATION PROPERTY 39
To formulate our most general discrete-time result on uniqueness of equilibriumstates, we replace the entropy of obstructions to expansivity from Definition 6.2with the pressure of obstructions to expansivity at scale (cid:15) :(10.8) P ⊥ exp ( φ, (cid:15) ) := sup (cid:110) h µ ( f ) + (cid:90) ϕ dµ : µ ∈ M ef ( X ) and µ (NE( (cid:15) )) > (cid:111) . Theorem 10.9 ([CT16, Theorem 5.6]) . Let X be a compact metric space, f : X → X a homeomorphism, and ϕ : X → R a continuous potential function. Suppose thatthere are (cid:15) > δ > such that P ⊥ exp ( ϕ, (cid:15) ) < P ( ϕ ) and there exists a decomposition ( C p , G , C s ) for X × N with the following properties: (I) every collection G M has specification at scale δ , (II) ϕ has the Bowen property on G at scale (cid:15) , and (III) P ( C p ∪ C s , ϕ, δ ) < P ( ϕ ) .Then ( X, f, ϕ ) has a unique equilibrium state.Remark . In applications to non-uniformly hyperbolic systems, it is very oftenthe case that there is a natural collection of orbit segments G along which thedynamics is uniformly hyperbolic; this is the most common way of establishingspecification for G , as we saw in §
7. In this case the proof of Proposition 10.6 showsthat every H¨older potential ϕ has the Bowen property on G . Then the question ofuniqueness boils down to determining which H¨older potentials have the pressure gapproperties (III) and P ⊥ exp ( ϕ, (cid:15) ) < P ( ϕ ). It is often the case that one or both of theseconditions fails for some H¨older potentials, as in the Manneville–Pomeau example.10.4. Partial hyperbolicity.
For partially hyperbolic systems with one-dimensionalcenter as in §
9, Theorem 10.9 can be used to extend Theorem 9.1.
Theorem 10.11.
Let
M, f, ϕ c be as in Theorem 9.1. Given a H¨older continuouspotential function ϕ : M → R , consider the quantities P + := sup (cid:110) h µ ( f ) + (cid:90) ϕ dµ : µ ∈ M ef ( M ) , λ c ( µ ) ≥ (cid:111) ,P − := sup (cid:110) h µ ( f ) + (cid:90) ϕ dµ : µ ∈ M ef ( M ) , λ c ( µ ) ≤ (cid:111) . If P + (cid:54) = P − , then ( M, f, ϕ ) has a unique equilibrium state. Beyond the properties from §
9, the only additional ingredient required for The-orem 10.11 is the fact that ϕ has the Bowen property on the collection of orbitsegments G defined in (9.3), which follows from Remark 10.10 and the hyperbolicityestimate in (9.4); then uniqueness follows from Theorem 10.9.It is worth noting that the condition P + (cid:54) = P − (and thus the condition h + (cid:54) = h − )can be formulated in terms of the topological pressure function. The function t (cid:55)→ P ( ϕ + tϕ c ) is convex, being the supremum of the affine functions P µ : t (cid:55)→ h µ ( f ) + (cid:90) ϕ dµ + tλ c ( µ ) tP ( ϕ + tϕ c ) Figure 10.1.
Some possible graphs of t (cid:55)→ P ( ϕ + tϕ c ).over all µ ∈ M ef ( M ). Some of its possible shapes are shown in Figure 10.1.Suppose there is t > P ( ϕ + tϕ c ) < P ( ϕ ), as in the third graph inFigure 10.1. Then given any µ ∈ M ef ( M ) with λ c ( µ ) ≥
0, we have(10.9) h µ ( f ) + (cid:90) ϕ dµ = P µ (0) ≤ P µ ( t ) ≤ P ( ϕ + tϕ c ) < P ( ϕ ) , and taking a supremum over all such µ gives P + ≤ P ( ϕ + tϕ c ) < P ( ϕ ), so that thecondition of Theorem 10.11 is satisfied and ( M, f, ϕ ) has a unique equilibrium state,which has negative center Lyapunov exponent.A similar argument holds if there is t < P ( ϕ + tϕ c ) < P ( ϕ ), as inthe first graph in Figure 10.1; (10.9) applies to all µ ∈ M ef ( M ) with λ c ( µ ) ≤
0, sothat P − < P ( ϕ ) = P + , and there is a unique equilibrium state, which has positivecenter Lyapunov exponent.We see that the only way to have P + = P − is if the function t (cid:55)→ P ( ϕ + tϕ c ) hasa global minimum at t = 0. Thus one could restate the last line of Theorem 10.11as the conclusion that ( M, f, ϕ ) has a unique equilibrium state if there is t (cid:54) = 0 suchthat P ( ϕ + tϕ c ) < P ( ϕ ). In particular, returning to Theorem 9.1, f has a uniqueMME if there is t (cid:54) = 0 such that P ( tϕ c ) < P (0) = h top ( f ). Part
III.
Geodesic flows
In this part, we focus on our geometric applications. In §
11, we introduce somegeometric background, and in §
12 we describe the main results and some of the keyideas from the paper [BCFT18]. In §
13, we discuss our approach to the Kolmogorov K -property. In §
14, we give the main ideas of proof for the “pressure gap” for a wideclass of potentials for geodesic flow on a rank 1 non-positive curvature manifold.11.
Geometric preliminaries
Overview.
Let M = ( M n , g ) be a closed connected C ∞ Riemannian manifoldwith dimension n , and F = ( f t ) t ∈ R denote the geodesic flow on the unit tangentbundle X = T M . The geodesic flow is defined by picking a point and a direction(i.e. an element of T M ), and walking at unit speed along the geodesic determinedby that data. More precisely, f t ( v ) = ˙ c v ( t ), where c v : R → M is the unique unitspeed geodesic with ˙ c v (0) = v . Geodesic flows are of central importance in thetheory of dynamical systems, and encode many important features of the geometry EYOND BOWEN’S SPECIFICATION PROPERTY 41 and topology of the underlying manifold M . For general background on geodesicflows, we refer to [Lee18, BG05].If all sectional curvatures of M are negative at every point, then F is a transitiveAnosov flow. In particular, the thermodynamic formalism is very well understood.To go beyond negative curvature, one generally needs the tools of non-uniformhyperbolicity. There are three further classes of manifolds that generally exhibitsome kind of non-uniformly hyperbolic behaviour: nonpositive curvature; no focalpoints; and no conjugate points. The relationships are as follows:negative curv. ⇒ nonpositive curv. ⇒ no focal points ⇒ no conjugate points . The reverse implications all fail in general.The definition of nonpositive curvature is easy: all sectional curvatures are ≤ (cid:102) M and consider arbitrary geodesics c , c with c (0) = c (0),then non-positive curvature implies that t (cid:55)→ d ( c ( t ) , c ( t )) is convex, while no focalpoints is equivalent to the condition that t (cid:55)→ d ( c ( t ) , c ( t )) be nondecreasing for allsuch c , c , and no conjugate points is equivalent to the condition that this functionnever vanish for t >
0; in other words, there is at most one geodesic connecting anytwo points in (cid:102) M . In § −
1) spaces (which generalizenegative curvature) and CAT(0) spaces (which generalize non-positive curvature).For intuition, negative curvature has the effect of spreading out geodesics whichpass through the same point (think of a saddle), while positive curvature has theeffect of bringing them back together after a finite amount of time (think of a sphere).As described in [Gul75], one can imagine starting with a negatively curved surfaceand then “raising a bump of positive curvature”; at first the positive curvature effectis weak enough that the geodesic flow remains Anosov, but eventually the Anosovproperty is destroyed, and raising the bump far enough creates conjugate points.In these notes, we focus on the case of equilibrium states for manifolds with non-positive curvature using specification-based techniques as in [BCFT18]; this relieson a continuous-time version of Theorem 10.9, which we formulate in § . We also state and sketch recent results by the first-namedauthor, Knieper and War for the MME to surfaces with no conjugate points, andsurvey some relevant recent results for CAT( −
1) and CAT(0) spaces.In the remainder of this section we collect some geometric preliminaries. Someof the definitions are taken verbatim from [BCFT18] for notational consistency.For more details, we recommend recent works [BCFT18, GS14], and more classicalreferences [Bal95, Ebe01, Ebe96]. Another specification-based proof of uniqueness of the MME on surfaces without focal pointswas given by Gelfert and Ruggiero [GR19]
Surfaces.
For purposes of exposition, we will often think about the surfacecase n = 2, although our approach applies in higher dimension too. By the Gauss–Bonnet theorem, the sphere has no metric of nonpositive curvature, and the onlysuch metrics on the torus are flat everywhere; it can be easily verified that thecorresponding geodesic flows have zero topological entropy and are not topologicallytransitive. Thus we are interested in studying surfaces of genus at least 2.As a first example, we can think about a surface of genus 2 with an embeddedflat cylinder, and negative curvature elsewhere. We could also consider the casewhere the flat cylinder collapses to a single closed geodesic on which the curvaturevanishes, with strictly negative curvature elsewhere. In higher dimensions, muchmore complicated examples exist, such as the 3-dimensional Gromov example thatwe describe in § M of genus at least 2 with non-positive curvature, we let K : M → ( −∞ , π : T M → M the natural projection of a tangentvector to its footpoint. Then we define the singular set to be(11.1) Sing := { v ∈ T M : K ( π ( f t v )) = 0 for all t ∈ R } . That is, Sing is the set of v for which the corresponding geodesic γ v experiences 0curvature for all time. All other vectors are called regular :(11.2) Reg := T M \ Sing = { v ∈ T M : K ( π ( f t v )) < t ∈ R } . Although the negative curvature encountered along regular geodesics guaranteessome expansion/contraction, this may be arbitrarily weak because the geodesic canbe arranged to experience 0 curvature for a long time (e.g., wrapping round anembedded flat cylinder) before hitting any negative curvature.The set Sing is closed and flow-invariant, while the set Reg is open. The regularset is nonempty because M has genus at least 2, and in fact Reg is dense in T M .In higher dimensions one has a similar dichotomy between singular and regularvectors, which we will describe in the next section. This gives a partition of T M asReg (cid:116) Sing, where Sing is closed and flow-invariant. As with surfaces, we will restrictour attention to the case when Reg (cid:54) = ∅ ; this rank 1 assumption rules out examplessuch as direct products, and is the typical situation, as demonstrated by the higherrank rigidity theorem of Ballmann and Burns–Spatzier [Bal85, BS87b, BS87a].11.3. Invariant foliations via horospheres.
Now let the dimension of M be any n ≥
2. We describe invariant stable and unstable foliations W s and W u of X = T M that are tangent to invariant subbundles E s and E u in T X = T T M along which wewill eventually obtain the contraction and expansion estimates necessary to studyuniqueness of equilibrium states. EYOND BOWEN’S SPECIFICATION PROPERTY 43
We must be a little careful in defining these foliations: we cannot ask that W s ( v )is the set of w ∈ T M so that d ( f t v, f t w ) → t → ∞ like we can in the uniformlyhyperbolic setting. We must allow points that stay bounded distance apart (in theuniversal cover) for all forward time. However, this does not work as the definitionof W s because it does not distinguish the stable from the flow direction. To dothings properly, there are two approaches. • Local approach:
Use stable and unstable orthogonal Jacobi fields to define E s and E u locally; see § • Global approach:
Define stable and unstable horospheres H s and H u in theuniversal cover (cid:102) M (this is typically done using Busemann functions) and usethese to get W s , W u .We outline this second approach here. Given v ∈ T M , let ˜ v ∈ T (cid:102) M be a lift of v ,and construct H s (˜ v ) as follows: for each r > S r (˜ v, +) = { x ∈ (cid:102) M : d (cid:102) M ( x, π ( f r ˜ v )) = r } denote the set of points at distance r from π ( f r ˜ v ) = c ˜ v ( r ), and let H s (˜ v ) be thelimit of S r ( v, +) as r → ∞ . This defines a hypersurface that contains the point π ˜ v . Writing W s (˜ v ) for the unit normal vector field to H s (˜ v ) on the same side as˜ v , the stable manifold W s ( v ) is the image of W s (˜ v ) under the canonical projection T (cid:102) M → T M .The unstable horosphere H u (˜ v ) and the unstable manifold W u ( v ) are definedanalogously, replacing S r (˜ v, +) with S r (˜ v, − ) = { x ∈ (cid:102) M : d (cid:102) M ( x, π ( f − r ˜ v )) = r } . The horospheres are C manifolds, so W s ( v ) and W u ( v ) are C manifolds, andwe can define the stable and unstable subspaces E s ( v ) , E u ( v ) ⊂ T v T M to be thetangent spaces of W s ( v ) , W u ( v ) respectively. The bundles E s , E u , which are bothglobally defined in this way, are respectively called the stable and unstable bundles.They are invariant and depend continuously on v ; see [Ebe01, GW99].The following is equivalent to the standard definition of the regular set via Jacobifields, which we will give in the next section. Definition 11.1.
A vector v ∈ T M is regular if E s ( v ) ∩ E u ( v ) is trivial (containsonly the 0 vector in T v T M ), and singular otherwise. Write Reg ⊂ T M for the setof regular vectors, and Sing ⊂ T M for the set of singular vectors.On Reg, we obtain the expected splitting T v T M = E s ( v ) ⊕ E u ( v ) ⊕ E c ( v ), where E c ( v ) is the flow direction. This splitting degenerates on Sing. Definition 11.2.
The manifold M is rank 1 if Reg (cid:54) = ∅ .Finally, we define a function which is of great importance in thermodynamic for-malism. The geometric potential is the function that measures infinitesimal volume growth in the unstable distribution: ϕ u ( v ) = − lim t → t log det( df t | E u ( v ) ) = − ddt (cid:12)(cid:12)(cid:12) t =0 log det( df t | E u ( v ) ) . The potential ϕ u is continuous and globally defined. When M has dimension 2, ϕ u is H¨older along unstable leaves [GW99]. It is not known whether ϕ u is H¨older alongstable leaves. In higher dimensions, it is not known whether ϕ u is H¨older continuouson either stable or unstable leaves. An advantage of our approach is that we sidestepthe question of H¨older regularity for ϕ u .11.4. Jacobi fields and local construction of stables/unstables.
Now we givean alternate description of the stable and unstable subbundles and foliations, whichcan be shown to agree with the definitions in the previous section.A
Jacobi field along a geodesic γ is a vector field along γ obtained by taking a one-parameter family of geodesics that includes γ and differentiating in the parametercoordinate; equivalently, it is a vector field along γ satisfying(11.3) J (cid:48)(cid:48) ( t ) + R ( J ( t ) , ˙ γ ( t )) ˙ γ ( t ) = 0 , where R is the Riemannian curvature tensor on M and (cid:48) represents covariant differ-entiation along γ .We often want to remove the variations through geodesics in the flow directionfrom consideration. If J ( t ) is a Jacobi field along a geodesic γ and both J ( t ) and J (cid:48) ( t ) are orthogonal to ˙ γ ( t ) for some t , then J ( t ) and J (cid:48) ( t ) are orthogonal to ˙ γ ( t )for all t . Such a Jacobi field is an orthogonal Jacobi field .A Jacobi field J ( t ) along a geodesic γ is parallel at t if J (cid:48) ( t ) = 0. A Jacobi field J ( t ) is parallel if it is parallel for all t ∈ R . Definition 11.3.
A geodesic γ is singular if it admits a nonzero parallel orthogonalJacobi field, and regular otherwise.If γ is singular in the sense of Definition 11.3, then every ˙ γ ( t ) ∈ T M is singularin the sense of Definition 11.1, and similarly for regular.We write J ( γ ) for the space of orthogonal Jacobi fields for γ ; given v ∈ T M there is a natural isomorphism ξ (cid:55)→ J ξ between T v T M and J ( γ v ), which has theproperty that(11.4) (cid:107) df t ( ξ ) (cid:107) = (cid:107) J ξ ( t ) (cid:107) + (cid:107) J (cid:48) ξ ( t ) (cid:107) . An orthogonal Jacobi field J along a geodesic γ is stable if (cid:107) J ( t ) (cid:107) is bounded for t ≥
0, and unstable if it is bounded for t ≤
0. The stable and the unstable Jacobifields each form linear subspaces of J ( γ ), which we denote by J s ( γ ) and J u ( γ ),respectively. The corresponding stable and unstable subbundles of T T M are E u ( v ) = { ξ ∈ T v ( T M ) : J ξ ∈ J u ( γ v ) } ,E s ( v ) = { ξ ∈ T v ( T M ) : J ξ ∈ J s ( γ v ) } . EYOND BOWEN’S SPECIFICATION PROPERTY 45
The bundle E c is spanned by the vector field that generates the flow F . We alsowrite E cu = E c ⊕ E u and E cs = E c ⊕ E s . The subbundles have the followingproperties (see [Ebe01] for details): • dim( E u ) = dim( E s ) = n −
1, and dim( E c ) = 1; • the subbundles are invariant under the geodesic flow; • the subbundles depend continuously on v , see [Ebe01, GW99]; • E u and E s are both orthogonal to E c ; • E u and E s intersect non-trivially if and only if v ∈ Sing; • E σ is integrable to a foliation W σ for each σ ∈ { u, s, cs, cu } .It is proved in [Bal82, Theorem 3.7] that the foliation W s is minimal in the sensethat W s ( v ) is dense in T M for every v ∈ T M . Analogously, the foliation W u isalso minimal. 12. Equilibrium states for geodesic flows
The general uniqueness result for flows.
We recall the general definitionsof topological pressure, variational principle, and equilibrium states for flows, whichare analogous to the discrete-time definitions from § X and a continuous flow F = ( f t ) on X , we write M F ( X ) = (cid:84) t ∈ R M f t ( X ) for the space of flow-invariant Borel probability measureson X , and M eF ( X ) ⊂ M F ( X ) for the set of ergodic measures.For (cid:15) > t >
0, and x ∈ X , the Bowen ball of radius (cid:15) and order t is B t ( x, (cid:15) ) = { y ∈ X | d ( f s x, f s y ) < (cid:15) for all 0 ≤ s ≤ t } . A set E ⊂ X is ( t, (cid:15) ) -separated if for all distinct x, y ∈ E we have y / ∈ B t ( x, (cid:15) ).Given a continuous potential function ϕ : X → R , we write Φ( x, t ) = (cid:82) t ϕ ( f s x ) ds for the integral of ϕ along an orbit segment of length t . We interpret D ⊂ X × [0 , ∞ )as a collection of finite-length orbit segments by identifying ( x, t ) with the orbitsegment starting at x and lasting for time t . Writing D t := { x ∈ X : ( x, t ) ∈ D} ,the partition sums associated to D and ϕ are(12.1) Λ( D , ϕ, (cid:15), t ) = sup (cid:110) (cid:88) x ∈ E e Φ( x,t ) : E ⊂ D t is ( t, (cid:15) )-separated (cid:111) . The pressure of ϕ on the collection D is given by (10.2)–(10.3), replacing n with t : P ( D , ϕ ) = lim (cid:15) → P ( D , ϕ, (cid:15) ) , P ( D , ϕ, (cid:15) ) = lim t →∞ t log Λ( D , ϕ, (cid:15), t ) . We continue to write P ( Y, ϕ ) = P ( Y × [0 , ∞ ) , ϕ ) for Y ⊂ X , and often abbreviate P ( ϕ ) = P ( X, ϕ ). The variational principle for pressure states that P ( ϕ ) = sup µ ∈M F ( X ) (cid:16) h µ ( f ) + (cid:90) ϕ dµ (cid:17) . A measure that achieves the supremum is an equilibrium state for (
X, f, ϕ ). When ϕ = 0, we recover the topological entropy h ( F ), and an equilibrium state for ϕ = 0is called a measure of maximal entropy . Remark . As in the discrete-time case, if the entropy map µ (cid:55)→ h µ is uppersemi-continuous then equilibrium states exist for each continuous potential function.Geodesic flows in non-positive curvature are entropy-expansive due to the flat striptheorem [Kni98]; this guarantees upper semi-continuity and thus existence.In light of Remark 12.1, the real question is once again uniqueness. Our maintool will be a continuous-time analogue of Theorem 10.9, which gives non-uniformversions of specification, expansivity, and the Bowen property that are sufficient togive uniqueness.The main novelty compared with the discrete-time case is the expansivity condi-tion. For an expansive map, the set of points that stay close to x for all time is onlythe point x itself. For an expansive flow, this set is an orbit segment of x . Our setof non-expansive points for a flow is defined accordingly. For x ∈ X and (cid:15) >
0, welet the bi-infinite Bowen ball beΓ (cid:15) ( x ) = { y ∈ X : d ( f t x, f t y ) ≤ (cid:15) for all t ∈ R } . The set of non-expansive points at scale (cid:15) is (compare this to Definition 6.1)(12.2) NE( (cid:15), F ) := { x ∈ X | Γ (cid:15) ( x ) (cid:54)⊂ f [ − s,s ] ( x ) for any s > } , where f [ a,b ] ( x ) = { f t x : a ≤ t ≤ b } . The pressure of obstructions to expansivity is P ⊥ exp ( ϕ ) := lim (cid:15) → P ⊥ exp ( ϕ, (cid:15) ) , where P ⊥ exp ( ϕ, (cid:15) ) = sup µ ∈M eF ( X ) (cid:110) h µ ( f ) + (cid:90) ϕ dµ : µ (NE( (cid:15), F )) = 1 (cid:111) . Remark . For rank 1 geodesic flow, a simple argument using the flat strip the-orem guarantees that NE( (cid:15), F ) ⊂ Sing, so we have P ⊥ exp ( ϕ ) ≤ P (Sing , ϕ ).The definitions of specification and the Bowen property are completely analogousto Definitions 5.9 and 10.8 from the discrete-time case. Definition 12.3.
A collection of orbit segments
G ⊂ X × [0 , ∞ ) has the specificationproperty at scale δ > τ > x , t ) , . . . , ( x k , t k ) ∈G , there exist 0 = T < T < · · · < T k and y ∈ X such that f T i ( y ) ∈ B t i ( x i , δ ) forall i , and moreover, writing s i = T i + t i , we have s i ≤ T i +1 ≤ s i + τ for all i .We say that G has the specification property if it has the specification property atscale δ for every δ > We note that the original formulation of expansivity for flows by Bowen and Walters [BW72]allows reparametrizations, which suggests that one might consider a potentially larger set in place ofΓ (cid:15) for expansive flows. The main motivation for allowing reparametrizations is to give a definitionthat is preserved under orbit equivalence. However, this is not relevant for our purposes. In oursetup, the natural notion of expansivity would be to ask that there exists (cid:15) so that NE( (cid:15), F ) = ∅ .This definition is sufficient for the uniqueness results, and strictly weaker than Bowen–Waltersexpansivity, although it is not an invariant under orbit equivalence. See the discussion of kinematicexpansivity in [FH19]. EYOND BOWEN’S SPECIFICATION PROPERTY 47
Definition 12.4.
A continuous function ϕ : X → R has the Bowen property at scale (cid:15) >
G ⊂ X × [0 , ∞ ) if there is V > x, t ) ∈ G and y ∈ B t ( x, (cid:15) ), we have | Φ( y, t ) − Φ( x, t ) | ≤ V .We say that ϕ has the Bowen property on G if there exists (cid:15) > ϕ hasthe Bowen property at scale (cid:15) on G .An argument following the proof of Proposition 10.6 shows that for uniformlyhyperbolic flows, any H¨older continuous function has the Bowen property. Moregenerally, Remark 10.10 applies here as well: if the flow is uniformly hyperbolicalong a collection of orbit segments G ⊂ X × [0 , ∞ ), then every H¨older ϕ has theBowen property on G .As in Definition 8.1 for discrete time, a decomposition for X × [0 , ∞ ) consistsof three collections P , G , S ⊂ X × [0 , ∞ ) for which there exist three functions p, g, s : X × [0 , ∞ ) → [0 , ∞ ) such that for every ( x, t ) ∈ X × [0 , ∞ ), the values p = p ( x, t ), g = g ( x, t ), and s = s ( x, t ) satisfy t = p + g + s , and( x, p ) ∈ P , ( f p ( x ) , g ) ∈ G , ( f p + g ( x ) , s ) ∈ S . The conditions we are interested in depend only on the collections ( P , G , S ) ratherthan the functions p , g , s . However, we work with a fixed choice of ( p, g, s ) for theproof of the abstract theorem to apply.One small difference from the discrete-time case is that we need to “fatten up” P and S slightly before imposing the smallness condition in the general uniquenesstheorem. To this end, for a collection D ⊂ X × [0 , ∞ ), we define[ D ] := { ( x, k ) ∈ X × N : ( f − s x, k + s + t ) ∈ D for some s, t ∈ [0 , } . Theorem 12.5 (Non-uniform Bowen hypotheses for flows [CT16]) . Let ( X, F ) be acontinuous flow on a compact metric space, and ϕ : X → R be a continuous potentialfunction. Suppose that P ⊥ exp ( ϕ ) < P ( ϕ ) and X × [0 , ∞ ) admits a decomposition ( P , G , S ) with the following properties: (I) G has specification; (II) ϕ has the Bowen property on G ; (III) P ([ P ] ∪ [ S ] , ϕ ) < P ( ϕ ) . Then ( X, F, ϕ ) has a unique equilibrium state µ ϕ .Remark . The reason that in general we control the pressure of [ P ] ∪ [ S ] ratherthan the collection P ∪ S is a consequence of a technical step in the proof of theabstract result in [CT16] that required a passage from continuous to discrete time.This distinction does not matter for the λ -decompositions described in the nextsection, which cover all the applications we discuss here; see [CT19a, Lemma 3.5].12.2. Geodesic flows in non-positive curvature.
Now we return to the specificsetting of geodesic flow in non-positive curvature. In § X = T M – cannot occur unless there is a pressure gap P (Sing , ϕ )
13, and the pressure gap itself isdiscussed in § Uniqueness can fail without a pressure gap.
For uniformly hyperbolic flowsand H¨older continuous potentials, there is a unique equilibrium state, and this equi-librium state gives positive weight to every open set; it is fully supported . Forgeodesic flow in nonpositive curvature, this conclusion cannot hold unless there is a pressure gap , which we now describe.Since the singular set Sing is closed and flow-invariant, we can apply the varia-tional principle to the restriction of the flow to Sing, and obtain P (Sing , ϕ ) = sup (cid:110) h µ ( f ) + (cid:90) ϕ dµ : µ ∈ M F (Sing) (cid:111) . As discussed in Remark 12.1, the geodesic flow is entropy-expansive and thus theentropy map µ (cid:55)→ h µ ( f ) is upper semi-continuous. This guarantees that there exists ν ∈ M F (Sing) with h ν ( f ) + (cid:82) ϕ dµ = P (Sing , ϕ ).If P (Sing , ϕ ) = P ( ϕ ), then ν is an equilibrium state for ( T M, F, ϕ ), and even ifit happens that ν is the unique equilibrium state (which can be arranged, but is notgenerally expected), it is not fully supported. Thus in order to obtain the classicalconclusion of unique equilibrium state and full support, we require a pressure gap P (Sing , ϕ ) < P ( ϕ ).To see that the case P (Sing , ϕ ) = P ( ϕ ) can actually occur, we observe that thereis a natural ( f t )-invariant volume measure µ L on X = T M called the Liouvillemeasure . Locally, µ L is the product of the Riemannian volume on M and Haarmeasure on the unit sphere of dimension n −
1. Using the Ruelle–Margulis inequality,the Pesin entropy formula, and the fact that − (cid:82) ϕ u dµ is the sum of the positiveLyapunov exponents for µ (where ϕ u is the geometric potential), one can show that P ( ϕ u ) = 0 and that µ L is an equilibrium state for ϕ u .In negative curvature, ϕ u is H¨older and µ L is the unique equilibrium state. Innon-positive curvature, however, µ L often fails to be the unique equilibrium state. For example, in the surface case, it is easily checked that P (Sing , ϕ u ) = P ( ϕ u ) = 0,and any closed geodesic in Sing defines two equilibrium states for ϕ u (one for eachdirection of travel around the geodesic).Since a general uniqueness result for ϕ u is impossible, we often turn our attentionto the one-parameter family of potentials qϕ u , where q ∈ R . Equilibrium states forthese potentials are geometrically relevant, and a natural question is to identify therange of values for q so that uniqueness holds. We mention that µ L (Reg) > µ L | Reg is known to be ergodic. Ergodicity of µ L ,which is a major open problem, is thus equivalent to the question of whether µ L (Sing) = 0. EYOND BOWEN’S SPECIFICATION PROPERTY 49
Uniqueness given a pressure gap.
Our main result on uniqueness of equilib-rium states for geodesic flow in non-positive curvature is the following.
Theorem 12.7 (Uniqueness of equilibrium states for rank 1 geodesic flow [BCFT18]) . Let ( f t ) be the geodesic flow over a closed rank 1 manifold M and let ϕ : T M → R be ϕ = qϕ u or be H¨older continuous. If ϕ satisfies the pressure gap(12.3) P (Sing , ϕ ) < P ( ϕ ) , then ϕ has a unique equilibrium state µ . This equilibrium state is hyperbolic, fullysupported, and is the weak ∗ limit of weighted regular closed geodesics in the sense of § . Knieper used a Patterson–Sullivan type construction on the boundaryat infinity to prove uniqueness of the MME (the case ϕ = 0) and deduce the entropygap h (Sing) < h ( T M ) from this [Kni98]. This construction has recently beenextended to manifolds with no focal points by Fei Liu, Fang Wang, and WeishengWu [LWW20]. We work in the other direction: we need to first establish the gap(see Theorem 12.10 below), and then use this to prove uniqueness.In §
13 we discuss the following result on strengthened ergodic properties for theequilibrium states in Theorem 12.7, due to Ben Call and the second-named author.
Theorem 12.9 (K and Bernoulli properties [CT19a]) . Any unique equilibrium stateprovided by Theorem 12.7 has the K-property. The unique MME has the Bernoulliproperty.
In dimension 2, the Margulis–Ruelle inequality gives h (Sing) = 0, from which thepressure gap (12.3) follows when sup ϕ − inf ϕ < h ( X ), via a soft argument basedon the variational principle. In higher dimensions we may have h (Sing) > § h (Sing) < h ( X ) established byKnieper is nontrivial. In §
14 we outline a direct proof of this gap that uses thespecification property, and that generalizes to some nonzero potentials as follows.
Theorem 12.10 (Direct proof of entropy/pressure gap) . For geodesic flow on aclosed rank 1 manifold M , every continuous potential ϕ that is locally constant ona neighbourhood of Sing satisfies the pressure gap condition (12.3) .Remark . When Sing is a finite union of periodic orbits, which is the case forreal analytic surfaces of non-positive curvature, Theorem 12.10 can be used to provethat the pressure gap holds for a C -open and dense set of potential functions.For surfaces, the fact that ϕ u | Sing = 0 and h (Sing) = 0 implies that P (Sing , qϕ u ) =0 for all q ∈ R . It is an easy consequence of the Margulis–Ruelle inequality andPesin’s entropy formula that P ( qϕ u ) > q < , and thus qϕ u has a unique equilibrium state for all q <
1. We obtain the classicpicture of the pressure function in non-uniform hyperbolicity, shown in Figure 12.1.
This is analogous to the familiar picture in the case of non-uniformly expandinginterval maps with indifferent fixed points, e.g., the Manneville-Pomeau map [PS92,Urb96, Sar01]. qP ( qϕ u )( −∞ , , ∞ ): everyeq. st. singular q = 1: both µ L and singular eq. st. Figure 12.1.
Pressure for surfaces with non-positive curvature.12.2.3.
Pressure and periodic orbits.
We describe the sense in which the uniqueequilibrium state is the limit of periodic orbits, analogously to § a < b ,let Per R ( a, b ] denote the set of closed regular geodesics with length in the interval( a, b ]. For each such geodesic γ , let Φ( γ ) be the value given by integrating ϕ around γ ; that is, Φ( γ ) := Φ( v, | γ | ) = (cid:82) | γ | ϕ ( f t v ) dt , where v ∈ T M is tangent to γ and | γ | is the length of γ . Given T, δ >
0, letΛ ∗ Reg ( ϕ, T, δ ) = (cid:88) γ ∈ Per R ( T − δ,T ] e Φ( γ ) . For a closed geodesic γ , let µ γ be the normalized Lebesgue measure around theorbit. We consider the measures µ Reg
T,δ = 1Λ ∗ Reg ( ϕ, T, δ ) (cid:88) γ ∈ Per R ( T − δ,T ] e Φ( γ ) µ γ . We say that regular closed geodesics weighted by ϕ equidistribute to a measure µ iflim T →∞ µ Reg
T,δ = µ in the weak* topology for every δ > Main ideas of the proof of uniqueness.
Theorem 12.7 is proved using thegeneral result in Theorem 12.5. As observed in Remark 12.2, we have P ⊥ exp ( ϕ ) ≤ P (Sing , ϕ ), so the condition P ⊥ exp ( ϕ ) < P ( ϕ ) follows immediately from the pressuregap assumption (12.3), and it remains to find a decomposition of the space of orbitsegments satisfying (I)–(III). We will do this using a function λ : X → [0 , ∞ ) thatmeasures ‘hyperbolicity’. We want this function to be such that: Here, we are following a notation convention of Katok: when we say a geodesic, we meanoriented geodesic, and we are considering γ as a periodic orbit living in T M . EYOND BOWEN’S SPECIFICATION PROPERTY 51 (1) λ vanishes on Sing;(2) λ uniformly positive implies uniform hyperbolicity estimates.There is a convenient geometrically-defined function which has the desired proper-ties, whose definition in dimension 2 is simple: we let λ ( v ) be the minimum of thecurvature of the stable horosphere H s ( v ) and the unstable horosphere H u ( v ). If v ∈ Sing, then λ ( v ) = 0 due to the presence of a parallel orthogonal Jacobifield. The set { v ∈ Reg : λ ( v ) = 0 } may be non-empty, but it has zero measure forany invariant measure [BCFT18, Corollary 3.6].If λ ( v ) ≥ η >
0, then we have various uniform estimates at the point v , forexample on the angle between E u ( v ) and E s ( v ), and on the growth of Jacobi fieldsat v . Thus, the function λ serves as a useful ‘measure of hyperbolicity’. In particular,we get the following distance estimates: given η > δ = δ ( η ) > v ∈ T M , and w, w (cid:48) ∈ W sδ ( v ), we have(12.4) d s ( f t w, f t w (cid:48) ) ≤ d s ( w, w (cid:48) ) e − (cid:82) t ( λ ( f τ v ) − η/ dτ for all t ≥ , where d s is the distance on W s . We get similar estimates for w, w (cid:48) ∈ W uδ ( v ).Now we use λ to define a decomposition. We give a general definition since theprocedure here applies not just to geodesic flows, but to other examples includingthe partially hyperbolic systems in § § § Definition 12.12.
Let X be a compact metric space and F = ( f t ) a continuousflow on X . Let λ : X → [0 , ∞ ) be a bounded lower semicontinuous function andfix η >
0. The λ -decomposition (with constant η ) of X × [0 , ∞ ) is given by defining B ( η ) = (cid:110) ( x, t ) | t (cid:90) t λ ( f s ( x )) ds < η (cid:111) , G ( η ) = (cid:110) ( x, t ) | ρ (cid:90) ρ λ ( f s ( x )) ds ≥ η and 1 ρ (cid:90) ρ λ ( f − s f t ( x )) ds ≥ η for all ρ ∈ [0 , t ] (cid:111) and then putting P = S = B ( η ) and G = G ( η ). We decompose an orbit segment( x, t ) by taking the longest initial segment in P as the prefix, and the longest terminal For manifolds M with Dim( M ) ≥
2, we define λ : T M → [0 , ∞ ) as follows. Let H s , H u be the stable and unstable horospheres for v . Let U sv : T πv H s → T πv H s be the symmetric linearoperator defined by U ( v ) = ∇ v N , where N is the field of unit vectors normal to H on the sameside as v . This determines the second fundamental form of the stable horosphere H s . We define U uv : T πv H u → T πv H u analogously. Then U uv and U sv depend continuously on v , U u is positivesemidefinite, U s is negative semidefinite, and U u − v = −U sv . For v ∈ T M , let λ u ( v ) be the minimumeigenvalue of U uv and let λ s ( v ) = λ u ( − v ). Let λ ( v ) = min( λ u ( v ) , λ s ( v )).The functions λ u , λ s , and λ are continuous since the map v (cid:55)→ U u,sv is continuous, and we have λ u,s ≥
0. When M is a surface, the quantities λ u,s ( v ) are just the curvatures at πv of the stableand unstable horocycles, and we recover the definition of λ stated above. This allows us to use indicator functions of open sets, which is helpful in some applications. segment in S as the suffix : that is, p ( x, t ) = sup { p ≥ x, p ) ∈ P} and s ( x, t ) = sup { s ≥ f t − s x, s ) ∈ S} . The good core is what is left over; see Figure 12.2. v f t ( x ) ∈ P ∈ S f p ( x ) f t − s ( x ) ⇓∈ G average( λ ) ≥ η average( λ ) < η Figure 12.2. A λ -decomposition.For rank 1 geodesic flow, the decompositions associated to the horosphere curva-ture function λ have the following useful properties:(1) we can relate P ([ P ] ∪ [ S ] , ϕ ) to P (Sing , ϕ );(2) the specification and Bowen properties hold for G and ϕ .For the first of these, one can show that when η > P ( P ∪ S , ϕ ) is closeto the pressure of the set of orbit segments along which the integral of λ vanishes;this in turn can be shown to equal P (Sing , ϕ ). Thus the pressure gap assumption(12.3) gives us P ([ P ] ∪ [ S ] , ϕ ) < P ( X, ϕ ) for sufficiently small η , which is (III) inTheorem 12.5.For the second of these, one can in fact prove the specification property for thelarger collection(12.5) C ( η ) = { ( v, t ) : λ ( v ) > η, λ ( f t v ) > η } ;this will be useful in §
14. Observe that G ( η ) ⊂ C ( η ). The proof of the specificationproperty is essentially the one from the uniformly hyperbolic case, as described in § §
4] for the full proof.The key ingredient is uniformity of the local product structure at the end points ofthe orbit segments. This is provided by the condition that λ is uniformly positive atthese points. Then we use uniform density of unstable leaves to transition betweenorbit segments. We additionally need some definite expansion along the unstable ofeach orbit segment, which follows from the uniformity of λ at the endpoints. We could also define the class of one-sided λ -decompositions by taking the longest initialsegment in B ( η ), declaring what is left over to be good, and setting S = ∅ , or conversely byputting S = B ( η ) and P = ∅ . This formalism is defined in [Cal20]: the decompositions in § λ -decompositions. EYOND BOWEN’S SPECIFICATION PROPERTY 53
Remark . In fact, C ( η ) satisfies a stronger version of specification than the oneformulated in Definition 12.3: one can replace the conclusion that the shadowingcan be accomplishedfor some T < T < · · · < T k satisfying T i +1 − T i − t i ∈ [0 , τ ]with the stronger conclusion that it can be accomplishedfor every T < T < · · · < T k satisfying T i +1 ≥ T i + τ . That is, we are able to take all the transition times to be exactly τ , or any length atleast τ that we choose. This stronger conclusion is important in both the K -propertyresult in §
13 and the entropy gap result discussed in § v, t ) ∈ G ( η ) and w, w (cid:48) ∈ W sδ ( v ), we have d s ( f τ w, f τ w (cid:48) ) ≤ d s ( w, w (cid:48) ) e − τη/ for all τ ∈ [0 , t ] , with a similar estimate along the unstables (going backwards from the end of theorbit segment). Together with the local product structure, this allows the Bowenproperty on G for H¨older continuous potentials to be deduced from the same argu-ment used in Proposition 10.6. Remark . Since it is not known whether the geometric potential ϕ u is H¨oldercontinuous, an alternate proof is required to show that it satisfies the Bowen propertyon G . This is one of the hardest parts of the analysis of [BCFT18], and relies ondetailed estimates involving the Riccati equation.Combining the ideas described above verifies the hypotheses of the abstract resultin Theorem 12.5, so that the pressure gap (12.3) yields a unique equilibrium state.12.3. Unique MMEs for surfaces without conjugate points.
When M ismerely assumed to have no conjugate points, life is substantially harder becausemany of the geometric tools used in the previous section are no long available, suchas convexity of horospheres, monotonicity of the distance function, and continuity ofthe stable and unstable foliations of T M (cf. the “dinosaur” example of Ballmann,Brin, and Burns [BBB87]).Under the additional (strong) assumption that the flow is expansive, uniquenessof the MME was proved by Aur´elien Bosch´e, a student of Knieper, in his Ph.D.thesis [Bos18]. The following result says that at least in dimension 2, we can removethe assumption of expansivity. Theorem 12.15 ([CKW20]) . Let M be a closed manifold of dimension 2, withgenus ≥ , equipped with a smooth Riemannian metric without conjugate points.Then the geodesic flow on T M has a unique measure of maximal entropy.Remark . A higher-dimensional version of Theorem 12.15 is available [CKW20],but requires additional assumptions on M : existence of a ‘background’ metric withnegative curvature; the divergence property; residually finite fundamental group; and a certain ‘entropy gap’ condition. All of these can be verified for every metricwithout conjugate points on a surface of genus 2.Theorem 12.15 is proved using a coarse-scale expansivity and specification result.Issues of coarse scale did not arise in our non-positive curvature result, where weobtained the specification property at arbitrarily small scales. This removed a greatdeal of technicality from the analysis. We will not discuss the general coarse-scaleanalogue of Theorem 12.5, since we do not use it. Instead, we state the special casewhere ϕ = 0 and G = X × [0 , ∞ ), which suffices for Theorem 12.15. This is thecontinuous-time analogue of Theorem 6.3. Theorem 12.17 ([CT16]) . Let X be a compact metric space and ( f t ) : X → X acontinuous flow. Suppose that (cid:15) > δ > are such that h ⊥ exp ( X, ( f t ) , (cid:15) ) < h ( X, ( f t )) ,and that the system has the specification property at scale δ . Then ( X, ( f t )) has aunique measure of maximal entropy. Note that Theorem 12.17 is stated using the hypothesis of specification for theentire system, without passing to a subcollection of orbit segments. The key tool inproving this fact for surfaces without conjugate points is the
Morse Lemma , whichstates that if g, g are two metrics on M such that g has no conjugate points and g has negative curvature, then there is a constant R > c, α aregeodesic segments w.r.t. g, g , respectively, in the universal cover (cid:102) M that agree attheir endpoints, then they remain within a distance R for along their entire length.Since M is a surface of genus ≥
2, it admits a metric of negative curvature. Givenan orbit segment ( v, t ) ∈ T M × (0 , ∞ ) for the g -geodesic flow, let p, q be the startand end points of some lift of the corresponding g -geodesic segment to the universalcover. Let w ∈ T M × (0 , ∞ ) lift to the unique unit tangent vector that begins a g -geodesic segment starting at p and ending at q , and let s be the g -length of thissegment. Then E : ( v, t ) (cid:55)→ ( w, s ) defines a map from the space of g -orbit segmentsto the space of g -orbit segments with the property that ( v, t ) and E ( v, t ) remainwithin R for their entire lengths.Using this correspondence, one can take a finite sequence of g -orbit segments( v , t ) , . . . , ( v k , t k ), find g -orbit segments E ( v i , t i ) that remain within R , and usethe specification property for the (Anosov) g -geodesic flow to shadow these (w.r.t. g ) by a single orbit segment ( y, T ). Then E − ( y, T ) is a shadowing orbit (w.r.t. g ) for the original segments ( x i , t i ), for which the transition times are uniformlybounded.Writing down the details of the scales involved, one finds that the geodesic flowfor g , has specification at scale δ = 100 A R , where A ≥ A − ≤(cid:107) v (cid:107) g / (cid:107) v (cid:107) g ≤ A for all v ∈ T M . (Existence of A follows from compactness.)To apply Theorem 12.17, it remains to prove that obstructions to expansivity atsome scale (cid:15) > δ have small entropy. The problem with this is that R itself, andespecially 40 δ = 4000 A R , is likely much larger than the diameter of M . So at In fact one can improve this estimate, but the formula is more complicated [CKW20].
EYOND BOWEN’S SPECIFICATION PROPERTY 55 this point, it looks like the previous paragraph is completely vacuous – any orbitsegment of the appropriate length shadows the ( v i , t i ) segments to within δ .The solution is to pass to a finite cover. By gluing together enough copies ofa fundamental domain for M , one can find a finite covering manifold N whoseinjectivity radius is > (cid:15) . Observe that • the geodesic flow on T M is a finite-to-1 factor of the geodesic flow on T N ,so there is an entropy-preserving bijection between their spaces of invariantmeasures, and in particular there is a unique MME for the geodesic flow over M if and only if there is a unique MME over N ; • the argument for specification that we gave above still works for the geodesicflow on N , with the same scale, because this scale comes from the MorseLemma and is given at the level of the universal cover.So it only remains to argue that h ⊥ exp ( (cid:15) ) < h top for the geodesic flow on N . Thisis done by observing that if d ( f t v, f t w ) < (cid:15) for all t ∈ R but w does not lie on theorbit of v , then lifting to geodesics on (cid:102) M and using the fact that we are below theinjectivity radius of N allows us to conclude that the lifts of v, w are tangent todistinct geodesics between the same pair of points on the ideal boundary ∂ (cid:102) M . Thusif µ is any ergodic invariant measure that is not almost expansive at scale (cid:15) , then µ gives full weight to the set of vectors tangent to such “non-unique geodesics”.On the other hand, if h µ >
0, then µ is a hyperbolic measure by the Margulis–Ruelle inequality, and thus by Pesin theory, µ -a.e. v has transverse stable and un-stable leaves. These leaves are the normal vector fields to the stable and unstablehorospheres, and thus these horospheres meet at a single point, meaning that thegeodesic through v is the unique geodesic between its endpoints on the ideal bound-ary. By the previous paragraph, this means that µ is almost expansive. It followsthat h ⊥ exp ( (cid:15) ) = 0 < h top , and so there is a unique MME by the coarse-scale resultTheorem 12.17.We remark that the proof technique sketched here does not extend to non-zeropotentials, and a theory of equilibrium states for surfaces with no conjugate pointsbeyond the MME case is currently not available.12.4. Geodesic flows on metric spaces.
Another natural direction to extend theclassical case of geodesic flow on a negative curvature manifold is to generalize be-yond the Riemannian case. The geodesic flow on a compact locally CAT( −
1) metricspace is one such generalization. Here, a geodesic is a curve that locally minimizesdistance, and the flow acts on the space of bi-infinite geodesics parametrized withunit speed. In the Riemannian case this space is naturally identified with T M . TheCAT( −
1) property is a negative curvature condition which roughly says that a geo-desic triangle is thinner than a comparison geodesic triangle in the model hyperbolicspace with curvature −
1. While one expects these flows to exhibit similar behavior Formally, one needs to take a finite index subgroup of π ( M ) that avoids all non-identityelements corresponding to a large ball in (cid:102) M ; this is possible because π ( M ) is residually finite . to the classical case, branching phenomena and the lack of smooth structure areobstructions to some of the usual techniques.More generally, one can study geodesic flow on a compact locally CAT(0) metricspace, in which geodesic triangles are thinner than Euclidean triangles. This is ageneralization of geodesic flow in Riemannian non-positive curvature.We survey some recent results in this direction. In the CAT( −
1) case (allowingcusps), the MME has been well-studied using the boundary at infinity approach, see[Rob03]. Constantine, Lafont and the second-named author studied the compactlocally CAT( −
1) case using the specification approach [CLT20b], and later using asymbolic dynamics approach [CLT20a], proving that every H¨older continuous po-tential has a unique equilibrium state, and obtaining many of the strong stochasticproperties one expects from the classical case (e.g., Central Limit Theorem, Bernoul-licity, Large Deviations). Broise-Alamichel, Paulin and Parkonnen [BAPP19] haveextended the equilibrium state constructions and results of Paulin, Pollicott andSchapira [PPS15] to the CAT( −
1) case for a restricted class of potentials whichincludes the locally constant ones. (See § § −
1) spaces, andparticularly for trees, which is the focus of their work.The CAT(0) case has seen substantial recent advances in the MME case, notablyby Ricks [Ric19], who has proved uniqueness of the MME by extending Knieper’sconstruction. A theory of equilibrium states for translation surfaces, which is animportant class of CAT(0) examples, is currently being developed by Call, Constan-tine, Erchenko, Sawyer and Work. A theory of equilibrium states for the generalCAT(0) setting is currently open.13.
Kolmogorov property for equilibrium states
Moving up the mixing hierarchy.
We describe results of Ben Call andthe second-named author on the Kolmogorov and Bernoulli properties [CT19a].A flow-invariant measure µ is said to have the Kolmogorov property , or K -property ,if every time- t map has positive entropy with respect to any non-trivial partition ξ : that is, for every partition ξ that does not contain a set of full measure, and forevery t (cid:54) = 0, we have h µ ( f t , ξ ) > Theorem 13.1.
Let F = ( f t ) be the geodesic flow over a closed rank 1 manifold M and let ϕ : T M → R be ϕ = qϕ u or be H¨older continuous. If P (Sing , ϕ ) < P ( ϕ ) ,then the unique equilibrium state µ ϕ has the Kolmogorov property. In the case ϕ = 0, the mixing property for the unique MME was known due towork of Babillot [Bab02]. Theorem 13.1 strengthens this. We recall the hierarchy This can also be formulated in terms of the
Pinsker σ -algebra for µ , which can be thoughtof as the biggest σ -algebra with entropy 0: the measure µ has the K -property if and only if thePinsker σ -algebra for µ is trivial. EYOND BOWEN’S SPECIFICATION PROPERTY 57 of mixing properties (this is an “express train” version of the hierarchy):Bernoulli ⇒ K ⇒ mixing of all orders ⇒ mixing ⇒ weak mixing ⇒ ergodic . When dim( M ) = 2, it was shown by Ledrappier, Lima, and Sarig [LLS16] thatequilibrium states are Bernoulli; their proof uses countable-state symbolic dynamicsfor 3-dimensional flows. In higher dimensions, Theorem 13.1 gives the strongestknown results.The implications in the mixing hierarchy are not “if and only if”s in general.However, in smooth settings with some hyperbolicity, a classic strategy for provingthe Bernoulli property is to move up the hierarchy, establishing K , and then provingthat K implies Bernoulli. This approach was notably carried out by Ornstein andWeiss [OW73, OW98], Pesin [Pes77], and Chernov and Haskell [CH96]. In particular,a major success of Pesin theory is his proof that the Liouville measure restricted tothe regular set is Bernoulli. We refer to the recent book of Ponce and Var˜ao [PV19]for more details on this process. Here we simply mention that this approach can becarried out for the unique MME of rank 1 geodesic flow, and this is done in [CT19a]. Theorem 13.2 (Bernoulli property [CT19a]) . Let ( f t ) be the geodesic flow over aclosed rank 1 manifold M . The unique measure of maximal entropy is Bernoulli. Ledrappier’s approach.
The main tool in the proof of Theorem 13.1 is afantastic result of Ledrappier [Led77], which deserves to be more widely known.Ledrappier’s proof is about one page long, and gives criteria for the K -property interms of thermodynamic formalism. The original result is for discrete-time systems.We state here a version of it for flows; the proof is given in [CT19a], and in moredetail in [Cal20].Given a flow F = ( f t ) on a compact metric space X , the idea is to consider theproduct flow ( X × X, F × F ), i.e., the flow ( f s × f s ) s ∈ R given by(13.1) ( f s × f s )( x, y ) = ( f s x, f s y ) for s ∈ R . Theorem 13.3 (Criteria for K -property) . Let ( X, F ) be a flow such that f t isasymptotically entropy expansive for all t (cid:54) = 0 , and let ϕ be a continuous functionon X . Let ( X × X, F × F ) be the product flow (13.1) , and define Φ : X × X → R by Φ( x , x ) = ϕ ( x ) + ϕ ( x ) .If Φ has a unique equilibrium measure in M F × F ( X × X ) , then the unique equi-librium state for ϕ in M F ( X ) has the Kolmogorov property. The fact that (
X, F, ϕ ) has a unique equilibrium state when ( X × X, F × F, Φ)does is a consequence of the following simple lemma.
Lemma 13.4.
Let µ be an equilibrium state for ( X, F, ϕ ) . Then µ × µ is an equi-librium state for ( X × X, F × F, Φ) .Proof. Observe that h µ × µ ( f × f ) = h µ ( f ) + h µ ( f ) and (cid:90) Φ d ( µ × µ ) = (cid:90) ϕ dµ + (cid:90) ϕ dµ. Therefore, h µ × µ ( f × f ) + (cid:82) Φ d ( µ × µ ) = 2 P ( X, F, ϕ ) = P ( X × X, F × F, Φ) . (cid:3) From Lemma 13.4 we see that if µ, ν are distinct equilibrium states for ( X, F , ϕ ),then µ × µ and ν × ν are both equilibrium states for Φ. If Φ has a unique equilibriumstate, then this means that µ × µ = ν × ν and hence µ = ν ; thus, we get uniquenessof the equilibrium state downstairs, and we see that if Φ has a unique equilibriumstate, it must have the form µ × µ where µ is the unique equilibrium state for ϕ .Now the main idea of Ledrappier’s argument can be stated quite quickly: By theargument above, if Φ has a unique equilibrium state, then so does ϕ . Write µ forthis measure; then µ × µ is the unique equilibrium state for Φ . Now assume that µ is not K . Then µ has a non-trivial Pinsker σ -algebra. This can be used to defineanother equilibrium state for Φ . Contradiction. Decompositions for products.
Given Ledrappier’s result, our strategy forproving the K property in Theorem 13.1 is now clear. We want to show that theproduct system of two copies of the geodesic flow has a unique equilibrium state forthe class of potentials under consideration.So let’s find a decomposition for the product system. Problem:
Lifting decompositions to products in general does not work well. Onefact we do have in our favor is that if G has good properties, then so does G × G .However, we need
G × G to arise in a decomposition for ( X × X, F × F ). In generalthis does not look at all promising: for example, the reader may try to do it for the S -gap shifts as studied in [CT12], and will quickly see the issue. Idea:
Work with a nice class of decompositions that does behave well underproducts. We claim that the λ -decompositions from Definition 12.12 form such aclass. To see this, suppose we have a λ -decomposition ( P , G , S ) for a flow ( X, F ),and define ˜ λ : X × X → [0 , ∞ ) by(13.2) ˜ λ ( x, y ) = λ ( x ) λ ( y ) . This function inherits lower semicontinuity from λ , and we can consider the ˜ λ -decomposition ( ˜ P , ˜ G , ˜ S ) for ( X × X, F × F ).Given (( x, t ) , ( y, t )) ∈ ˜ G , it follows from (13.2) and boundedness of λ that wehave ( x, t ) , ( y, t ) ∈ G (with an appropriate choice of η ), and thus ˜ G ⊂ G × G . Thismeans that specification and the Bowen property for ˜ G can be deduced from thecorresponding properties for G .But how big are ˜ P and ˜ S ? If λ = 0 on one of the coordinates, then anything isallowed on the other. Roughly, we can show that: P ( ˜ P ∪ ˜ S , Φ) ≈ P ( ϕ ) + P ( P ∪ S , ϕ ) . EYOND BOWEN’S SPECIFICATION PROPERTY 59
Recall that P (Φ) = 2 P ( ϕ ). Thus, if we have P ( P ∪ S , ϕ ) < P ( ϕ ), then we expectto be able to obtain the estimate P ( ˜ P ∪ ˜ S , Φ) < P (Φ). This is the strategy carriedout in [CT19a, Cal20].13.4. Expansivity issues.
Specification and regularity are not the whole story; infact, dealing with continuous time and related expansivity issues is the most difficultpoint in our analysis.Recall from (12.2) that for flows we defineNE( (cid:15), F ) := { x ∈ X | Γ (cid:15) ( x ) (cid:54)⊂ f [ − s,s ] ( x ) for any s > } . For a product flow as in (13.1), the set Γ (cid:15) ( x, y ) always contains f [ − s,s ] x × f [ − s,s ] y .That is, we are considering a flow with a 2-dimensional center. The theory in § (cid:15), F × F ) as defined for a flow is the whole space!We have to build a new theory that uses information about(13.3) NE × ( (cid:15) ) := { ( x, y ) ∈ X × X | Γ (cid:15) ( x, y ) (cid:54)⊂ f [ − s,s ] ( x ) × f [ − s,s ] ( y ) for any s > } . There are no new difficulties with counting estimates, but serious issues arisewhen we build adapted partitions. In the discrete time case, our adapted partitionelements look like pixels and can be used to approximate sets. In the flow case, ouradapted partition elements approach a small piece of orbit, so look like thin cigars.Collections of partition elements can thus be used to approximate flow-invariantsets. In the ‘product of flows’ case, the best we can do is approximate sets invariantunder f s × f t for all s, t ∈ R . This creates new technical obstacles that must beovercome in our uniqueness proof. In particular, to run our ergodicity proof, weneed to be able to approximate sets which are invariant only under f s × f s for all s ∈ R . This disconnect is a fundamental additional difficulty.In [CT19a], this difficulty is overcome by proving weak mixing for µ using a lowerjoint Gibbs estimate which gives a kind of partial mixing for sets that are flowed outby a small time interval. This can be used to prove weak mixing of µ by a spectralargument. This is equivalent to the desired ergodicity of µ × µ .14. Knieper’s entropy gap
Entropy in the singular set.
For the geodesic flow on a rank 1 non-positivecurvature manifold, we have stated and discussed our main results on uniquenessof equilibrium states, and the K property for these equilibrium states. Our resultshold under the hypothesis of the pressure gap P (Sing , ϕ ) < P ( ϕ ). Thus, being ableto verify the pressure gap is of central importance for our results. In this sectionwe outline the proof that the gap holds for ϕ = 0, when it reduces to the entropygap h (Sing) < h ( X ). The argument extends easily to potentials that are locallyconstant on a neighbourhood of Sing, as claimed in Theorem 12.10.Our introduction of rank 1 manifolds in § µ ∈ M eF (Sing) has h µ ( f ) ≤ λ + ( µ ) = (cid:82) − ϕ u dµ = 0 by the Margulis–Ruelle inequality, where the last equality uses the fact that ϕ u | Sing ≡ h (Sing) = 0,and since h ( X ) > and it is not atall clear a priori that the entropy gap should always hold. The Gromov exampledescribed in [Kni98, §
6] demonstrates that starting in dimension 3, we may have h (Sing) >
0. To construct this example, let M be a surface of constant negativecurvature with one infinite cusp. Now cut off the cusp and flatten the end so that itis isometric to a flat cylinder with radius r . Take the product M = M × S , where S is the circle of radius r . This defines a non-positive curvature 3-manifold withboundary, where the boundary is a flat torus ∂M = ∂M × S . Now let M = S × M so that ∂M = S × ∂M . Glue M and M along the boundaries (note that the orderof the factors is reversed) to obtain a 3-manifold M .One can show that the regular set in T M consists of all vectors in T M whosegeodesic enters the non-flat part of both M and M . The singular set is then theset of vectors whose geodesics stay entirely on one side (or in the flat cylinder). Itis not hard to see that h (Sing) >
0. In fact, by defining M using a cut arbitrarilyhigh up the cusp, one can make h ( X ) − h (Sing) arbitrarily close to 0, and indeedit is not immediately obvious that this difference is non-zero. Why should there bean entropy gap at all?Knieper’s work in [Kni98] proved that there is a unique MME for rank 1 geodesicflow, and that this measure is fully supported on T M . This in turn implies theentropy gap, as explained in § § § Warm-up: shifts with specification.
The basic mechanism for using speci-fication to produce entropy is simply to construct exponentially many orbit segments“by hand”. This idea can be seen in its simplest form in the following result, whichhas been known since the 70’s, see [DGS76].
Theorem 14.1.
Let ( X, σ ) be a shift space with the following strong specification property: there is τ ∈ N such that for all v, w ∈ L = L ( X ) , there is u ∈ L τ suchthat vuw ∈ L . If X has more than one point, then the strong specification propertyhas positive entropy.Proof. Fix n ∈ N such that there are w , w ∈ L n with w (cid:54) = w . For each k ≥ { , } k → L k ( n + τ ) byΦ( i ) = w i v w i v · · · v k − w i k v k , In dimension 2, it is in fact an open problem whether Sing can contain non-periodic orbits[BM19], but this does not affect the argument that h (Sing) = 0. EYOND BOWEN’S SPECIFICATION PROPERTY 61 where all the v j have length τ and the expression on the right hand side is chosen tobe in the language of X . The existence of such a word is guaranteed by the strongspecification property.Since w (cid:54) = w , we can see that Φ is injective on { , } k , so L k ( n + τ ) ( X ) ≥ k .Taking logs, dividing by k ( n + τ ), and sending k → ∞ gives h ( X ) ≥ lim k →∞ k ( n + τ ) log 2 k = 1 n + τ log 2 > . (cid:3) We take this basic idea further, and sketch a proof of the following result aboutshifts with specification. The interest here is not so much in the statement, butrather in the fact that the proof contains the main entropy production idea that wewill use for geodesic flow in the next section.
Theorem 14.2.
Consider a shift space ( X, σ ) with the strong specification property.Let Y ⊂ X be a compact invariant proper subset. Then h ( Y ) < h ( X ) .Proof. We use the specification property, words in L ( Y ) and a single word w / ∈ L ( Y )to construct at least e n ( h ( Y )+ (cid:15) ) words in L n ( X ) for large n , giving the desired result.Since Y (cid:54) = X , we can fix w / ∈ L ( Y ). Let t be the length of w , and τ thegap size in the strong specification property. We fix a “window size” n > t + 2 τ ;given N ∈ N , we divide the indices { , , . . . , nN } into N “windows” of the form { kn + 1 , kn + 2 , . . . , ( k + 1) n } for 1 ≤ k ≤ N . In particular, given y ∈ L nN ( Y ),we consider the subwords of y that appear in each window, which have the form u k := y [ kn +1 , ( k +1) n ] for 1 ≤ k ≤ n .Within each window, we can perform the following ‘surgery’ to replace u k with aword that is in L n ( X ) but not L ( Y ): u k (cid:55)→ u k [1 ,n − t − τ ] v wv , where the words v , v of length τ are chosen as needed for the specification property.In each of the N windows of length n , we can decide whether to do surgery or not.Given this choice, we use the specification property to create a new word of length nN ; as long as we performed at least 1 surgery, this new word lies in L ( X ) but notin L ( Y ). In this way, from a single word y [1 ,nN ] , we can create 2 N − nN in L ( X ) \ L ( Y ) by varying over all the possible choices of windows fordoing this surgery procedure. Note that these words are all distinct because withineach window, we can determine whether or not we did surgery by checking whetherthe word w appears.This looks promising; however, it is too naive: we have to be careful as we varyover y [1 ,nN ] ∈ L ( Y ). In any window we selected for surgery, we are losing all theinformation on the last t + 2 τ entries in the window. This means that up to L t +2 τ distinct words could be mapped to the same word for each window we select forsurgery. If we select too many windows, the gain in new words is far outweighed bythe loss coming from this multiplicity estimate. Fix:
Carry out surgery on a small proportion of the windows, and argue that thenumber of new words created beats the loss of multiplicity.
More precisely, fix α > N − N windows,i.e., the set A = { n, n, n, . . . , ( N − n } . Assuming for convenience that αN ∈ N , we declare αN − A tobe “on”, and denote the set of “on” points by J . Let J αN be the set of all such J ,that is: J αN = { J ⊂ A : J = αN − } . Note that since N − kαN − k ≥ α for all 1 ≤ k < αN , we have J αN = (cid:18) N − αN − (cid:19) = αN − (cid:89) k =1 N − kαN − k ≥ (cid:16) α (cid:17) αN − = αe ( − α log α ) N . Fix y = y [1 ,nN ] ∈ L nN ( Y ). Given J ∈ J αN , we carry out our surgery procedureon the windows whose boundaries are determined by J . We obtain a new wordΦ J ( y ) ∈ L nN ( X ) which is definitely not in L ( Y ).The set { Φ J ( y ) : J ∈ J αN } is disjoint because we can recover J from Φ J ( y )by looking at which windows contain the “marker” w . Given J , the maximumnumber of words y ∈ L nN ( Y ) that can have the same image Φ J ( y ) is C αN − , where C = L t +2 τ ( Y ) is independent of α and N . Thus if we carry out this procedure foreach word in L nN ( Y ) and each J ∈ J αN , we obtain (cid:16) (cid:91) y [1 ,nN ] ∈L nN ( Y ) (cid:91) J Φ J ( y ) (cid:17) ≥ ( C − ) αN − (cid:18) N − αN − (cid:19) L nN ( Y ) , which gives L nN ( X ) ≥ αe ( − α log α ) N e − αN log C L nN ( Y ) . Taking logs, dividing by N , and sending N → ∞ , we see that h ( X ) ≥ h ( Y ) + αn ( − log α − log C ) . If α > h ( X ) > h ( Y ). (cid:3) The idea is that we want to split a word y [1 ,nN ] into αN subwords and perform surgeries nearthe points where it was split; these are the “on” points in A . Each such window determined by the set J has length some multiple of n . The surgeryprocedure is to remove the last t + 2 τ symbols from each window and replace with a word ofthe form v wv where the words v j are provided by the specification property to ensure that thisprocedure creates a word in L nN ( X ). EYOND BOWEN’S SPECIFICATION PROPERTY 63
Entropy gap for geodesic flow.
Now we return our attention to the geo-desic flow on X = T M for a closed rank 1 non-positive curvature manifold M andoutline the proof of the entropy gap h ( X ) > h (Sing).We follow the same entropy production strategy described in the previous section.The singular set Sing ⊂ X is a compact invariant proper subset. But how shouldwe construct orbits? We do not expect that orbit segments contained in Sing willhave the specification property. For example, orbit segments which are contained inthe interior of a flat strip definitely do not have the specification property becauseof the flat geometry. If we stay (cid:15) -close inside the flat strip on the time interval [0 , t ],the amount of additional time needed to escape the flat strip grows with t .So we want to use a specification argument on orbit segments without specifica-tion, which does not immediately look promising. Let us recall what kind of orbits do have specification: it suffices to know that both the start and end of the orbitsegment are ‘uniformly’ in the regular set.More precisely, for any η >
0, we have the specification property on the collection C ( η ) = { ( x, t ) : x, f t x ∈ Reg( η ) } , where Reg( η ) = { x : λ ( v ) ≥ η } . See § λ and discussion ofwhy the specification property holds on C ( η ).In order to make use of this fact, we require a reasonable way to approximateorbit segments in Sing by orbit segments in C ( η ). This will be given by a mapΠ t : Sing → Reg, which can be roughly summarized by the following slogan (whichdoesn’t make sense as a rigorous statement):
Move the start of ( v, t ) along its stable into Reg( η ) . Move the endalong an unstable into Reg( η ) . We now explain the construction that makes this idea precise. In our approximationof ( v, t ), we ask that:(1) Π t ( v ) , Π t ( f t v ) ∈ Reg( η ).(2) there exists L so f s (Π t v ) and Sing are close for s ∈ [ L, t − L ].In the second property, one might hope to find L so f s (Π t v ) and f s v are close for s ∈ [ L, t − L ]; however, this is too much to ask for. We can see the issue if ( v, t )is in the middle of a flat strip; the best we can hope for is that the orbit of Π t ( v )approaches the edge of the flat strip; see Figure 14.1, which also illustrates thefollowing “regularizing” procedure.We fix η so Reg( η ) has nonempty interior. Then using density of stable andunstable leaves, together with a compactness argument, we show the following:There exists R > v ∈ T M we have both W sR ( v ) ∩ Reg( η ) (cid:54) = ∅ and W uR ( v ) ∩ Reg( η ) (cid:54) = ∅ .Using this fact, given v ∈ Sing, choose v (cid:48) ∈ W sR ( v ) ∩ Reg( η ). Then for f t ( v (cid:48) ),choose f t ( w ) ∈ W uR ( f t v (cid:48) ) ∩ Reg( η ). Define Π t ( v ) := w .By continuity of λ , we have λ ( w ) ≥ η for an η slightly smaller than η . Wecan argue that the function λ u ( f t w ) is small along all of the orbit segment exceptfor an initial and terminal run of uniformly bounded length. This in turn implies that d ( f t w, Sing) is small, giving us condition (2). The reason λ u ( f t w ) must besmall away from the ends of the orbit segment is that otherwise small local stableand unstable manifolds centered here would get big too fast, contradicting that theendpoints of the orbit segment are in stable and unstable manifolds of size R . Thisis made precise by Proposition 3.13 of [BCFT18], which tells us that on a compactpart of the regular set, for fixed (cid:15) and R , an (cid:15) -stable/unstable manifold grows in auniform amount of time to cover a R -stable/unstable manifold. f L w f t − L wδv ∈ Sing f t v flat strip W sR ( v ) v ′ ∈ Reg( η ) f t v ′ W uR ( f t v ′ ) f t w ∈ Reg( η ) w ∈ Reg( η ) Figure 14.1.
The regularizing function Π t : v (cid:55)→ w .In conclusion, we obtain the following properties: Theorem 14.3.
For every δ > and η ∈ (0 , η ) , there exists L > such that forevery v ∈ Sing and t ≥ L , the image w = Π t ( v ) has the following properties: (1) w, f t ( w ) ∈ Reg( η ) ; (2) d ( f s ( w ) , Sing) < δ for all s ∈ [ L, t − L ] ; (3) for every s ∈ [ L, t − L ] , f s ( w ) and v lie in the same connected component of B (Sing , δ ) := { w ∈ T M : d ( w, Sing) < δ ) } . This result is found in [BCFT18, Theorem 8.1], where the proof of (2) containssome typos: we take this opportunity to correct these typos by providing a completeproof here. (Most of this proof is word-for-word identical to the one in [BCFT18].)
Proof of Theorem 14.3.
Let δ, η, η be as in the statement of the theorem. For prop-erty (1), it is immediate from the definition of Π t that λ ( f t w ) ≥ η . By uniformcontinuity of λ , we can take (cid:15) sufficiently small such that if v ∈ W u(cid:15) ( v ) and λ ( v ) ≥ η , then λ ( v ) ≥ η . By [BCFT18, Corollary 3.14], there exists T > t ≥ T and f t ( w ) ∈ W uR ( f t v (cid:48) ), then w ∈ W u(cid:15) ( v (cid:48) ). Thus, if λ ( v (cid:48) ) ≥ η , then λ ( w ) ≥ η . Thus, item (1) of the theorem holds for any t ≥ T .We turn our attention to item (2). [BCFT18, Proposition 3.4] tells us that thereare η (cid:48) , T > λ u ( f s v ) ≤ η (cid:48) for all | s | ≤ T , then d ( v, Sing) < δ.
Given v ∈ Sing, we have Π s ( v ) = v (cid:48) ∈ W sR ( v ), and λ ( f s v ) = 0 for all s . EYOND BOWEN’S SPECIFICATION PROPERTY 65
By continuity of λ u , we can take (cid:15) sufficiently small such that if v ∈ W s(cid:15) ( v ),then | λ u ( v ) − λ u ( v ) | < η (cid:48) /
2. Applying [BCFT18, Proposition 3.13] to the compactset { v : λ u ( v ) ≥ η (cid:48) / } ⊂ Reg gives T > λ u ( v ) ≥ η (cid:48) / τ ≥ T ,then f − τ W s(cid:15) ( v ) ⊃ W sR ( f − τ v ) and f τ W u(cid:15) ( v ) ⊃ W uR ( f τ v ).Suppose for a contradiction that λ u ( f s v (cid:48) ) ≥ η (cid:48) / s ≥ T . Applying theprevious paragraph with v = f s v (cid:48) gives f s v ∈ f s W sR ( f s v (cid:48) ) ⊂ W s(cid:15) ( f s v (cid:48) ). By ourchoice of (cid:15) , this gives λ u ( f s v ) >
0, contradicting the fact that v ∈ Sing, and weconclude that λ u ( f s v (cid:48) ) < η (cid:48) / s ≥ T .Similarly, if there is s ∈ [ T , t − T ] such that λ u ( f s w ) ≥ η (cid:48) , then the sameargument with v = f s w and τ = t − s gives f s v (cid:48) ∈ f − ( t − s ) W uR ( f t w ) ⊂ W u(cid:15) ( f s w ),and our choice of (cid:15) gives λ u ( f s v (cid:48) ) ≥ λ u ( f s w ) − η (cid:48) / ≥ η (cid:48) /
2, a contradiction since λ u ( f s v (cid:48) ) < η (cid:48) / s ≥ T . Thus λ u ( f s w ) < η (cid:48) for all s ∈ [ T , t − T ].Applying (14.1) gives d ( f s w, Sing) < δ for all s ∈ [ T + T , t − T − T ]. Thus,taking L = max( T , T + T ), assertions (1) and (2) follow for s ≥ L .For item (3) of the theorem, we observe that v and w can be connected by apath u ( r ) that follows first W sR ( v ), then f − t ( W uR ( f t v (cid:48) )) (see Figure), and that thearguments giving d ( f s w, Sing) < δ also give d ( f s u ( r ) , Sing) < δ for every s ∈ [ L, t − L ] and every r . We conclude that f s v and f s w lie in the same connected componentof B (Sing , δ ) for every such s . (cid:3) The collection { (Π t ( v ) , t ) : v ∈ Sing } has the specification property. This isbecause an orbit segment (Π t ( v ) , t ) both starts and ends in Reg( η ). As discussed,the collection C ( η ) of such orbit segments has the specification property.We certainly do not expect the map Π t to preserve separation of orbits. Forexample, in Figure 14.1, we would expect a v ∈ Sing defining a geodesic parallel to γ v (for example the arrow just above v in the picture) to be mapped to the same (orsimilar) point. However, using estimates in the universal cover, which we omit here,we can argue that Π t has bounded multiplicity on a ( t, (cid:15) ) separated set, independentof t , in the following sense. Proposition 14.4.
For every (cid:15) > , there exists C > such that if E t ⊂ Sing is a ( t, (cid:15) ) -separated set for some t > , then for every w ∈ T M , we have { v ∈ E t | d t ( w, Π t v ) < (cid:15) } ≤ C . Now let us return to our entropy production argument. It is basically the argu-ment we saw in § t beforeapplying the specification property, as shown in Figure 14.2.As before, consider a time window [0 , nN ]. Given a subset J of αN − { n, n, n, . . . , ( N − n } , we write (cid:96) , (cid:96) , . . . , (cid:96) αN for the lengths of theintervals (in order) whose endpoints are determined by J .For ( v , v , . . . , v αN ) ∈ Sing αN , we apply the map Π (cid:96) i − T to each coordinate andglue the resulting orbit segments in C ( η ) using specification (where T is the transitiontime in the specification property at a suitable scale).Run this construction over ( g (cid:96) i − T , (cid:15) )-separated sets for Sing in each coordinate,and for each choice of J , we construct exponentially more orbits than there are in . . . n n n n n n n n ( N − n ( N − nN n cut at elements of J (circled) . . . Π (cid:96) − T Π (cid:96) − T regularize B (Sing , δ )Reg( η )“glue” with specification B (Sing , δ )Reg( η ) . . . recover J Figure 14.2.
Gluing singular orbits.Sing. The argument is analogous to our previous entropy production argument:for α > (cid:0) N − αN − (cid:1) term beats the loss coming frommultiplicity in the construction. In particular, we conclude that h ( X ) > h (Sing).14.4. Other applications of pressure production.
The argument for entropyand pressure production described above is quite flexible, and can be used in manyother contexts. For example, in [CT13] we used a variation on this argument toshow that for a continuous potential ϕ with the Bowen property on the β -shift Σ β ,lim n →∞ n n − (cid:88) i =1 ϕ ( σ i w β ) < P (Σ β , ϕ ) , where w β is the lexicographically maximal sequence in Σ β ; this in turn establisheda pressure gap condition leading to a uniqueness result, similar to the proceduredescribed above for geodesic flow.Another variation of the argument can be used to prove that a unique equilib-rium state µ ϕ coming from Bowen’s original theorem (i.e., from the assumptions ofexpansivity, specification and the Bowen property) satisfies P ( ϕ ) > sup µ ∈M f ( X ) (cid:90) ϕ dµ, and thus that the entropy of µ ϕ is positive. Such a potential is often called hyper-bolic . This idea was extended recently in the symbolic setting in [CC19]. https://vaughnclimenhaga.wordpress.com/2017/01/26/entropy-bounds-for-equilibrium-states/ EYOND BOWEN’S SPECIFICATION PROPERTY 67
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