The degree of Bowen factors and injective codings of diffeomorphisms
TTHE DEGREE OF BOWEN FACTORS AND INJECTIVE CODINGS OFDIFFEOMORPHISMS
J´ER ˆOME BUZZI
Abstract.
We show that symbolic finite-to-one extensions of the type constructed byO. Sarig for surface diffeomorphisms induce H¨older-continuous conjugacies on large sets.We deduce this from their
Bowen property . This notion, introduced in a joint work with M.Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on thenotion of degree from finite equivalence theory and magic word isomorphisms.As an application, we give lower bounds on the number of periodic points first for surfacediffeomorphisms (improving a result of Sarig) and for Sina¨ı billiards maps (building on aresult of Baladi and Demers). Finally we characterize surface diffeomorphisms admittinga H¨older-continuous coding of all their aperiodic hyperbolic measures and give a slightlyweaker construction preserving local compactness. Introduction
In this text, a dynamical system is an automorphism of a standard Borel space and allmeasures are understood to be ergodic, invariant, Borel probability measures. A Markovshift is a “subshift defined by a countable oriented graph”. It is equipped with a standarddistance (see Sec. 2 for precise definitions and references).For a smooth diffeomorphism f of a compact manifold, a measure is called hyperbolic ifit has no zero Lyapunov exponent and both a positive and a negative exponent (see, e.g.,[16, Chap. S] for background on smooth ergodic theory). A measure is called χ -hyperbolic for some χ >
0, if it is hyperbolic and has no exponent in the interval [ − χ, χ ].We build conjugacies from the finite-to-one extensions of surface diffeomorphisms of Sarig[24] making them injective while preserving the H¨older-continuity and discarding only asubset negligible with respect to all (invariant probability) measures: Theorem 1.1.
Let f be a diffeomorphism with H¨older-continuous differential mapping acompact boundaryless C ∞ surface M to itself. For any numbers < χ (cid:48) < χ , there exist aMarkov shift S : X → X and a H¨older continuous map π : X → M such that: • π ◦ S = f ◦ π ; • π : X → M is injective; • π ( X ) has full measure for any χ -hyperbolic measure; • for any periodic x ∈ X , the periodic point π ( x ) defines a χ (cid:48) -hyperbolic measure. Previous injectivity results [4, 25, 7] were only with respect to a single measure, a restrictedclass of measures, or by jettisoning the continuity and discarding periodic orbits.
Mathematics Subject Classification (2010):
Primary 37B10; Secondary 37C25; 37C40; 37E30
Keywords: symbolic dynamics; Markov shifts; degree of codings; magic isomorphisms; periodic points;smooth ergodic theory; surface diffeomorphisms.
Date : December 10, 2019. a r X i v : . [ m a t h . D S ] D ec J´ER ˆOME BUZZI
As an application, we deduce from well-known results on Markov shifts estimates on theperiodic counts of surface diffeomorphisms. Consider the hyperbolic periodic points withgiven minimal period and Lyapunov exponents (defined by identifying a periodic orbit withthe obvious measure) bounded away from zero by a number χ > χ ( f, n ) := { x ∈ M : { f k ( x ) : k ∈ Z } has cardinality n and is χ -hyperbolic } . We denote the cardinality of a set by | · | . Theorem 1.2.
Let f be a C ∞ -diffeomorphism of a closed surface M . Assume that itstopological entropy h top ( f ) is positive. Then there is some integer p ≥ such that: (1.3) ∀ χ < h top ( f ) lim inf n → ∞ p | n e − n · h top ( f ) | per χ ( f, n ) | ≥ p. If the diffeomorphism is topologically mixing, one can take p = 1 . This improves the previous estimate due to Sarig [24]:(1.4) ∃ p ≥ n → ∞ p | n e − n · h top ( f ) |{ x : x = f n x and is χ -hyperbolic }| > . Indeed, not only do we have an explicit constant, but we control the minimal period. Bycomparison, the estimate (1.4) is compatible, e.g., with per χ ( f, n ) = ∅ for infinitely many n a multiple of p .Thanks to works of Baladi and Demers [1] and Lima and Matheus [19], our general resultscan be applied to the classical collision map T B of any two-dimensional Sina¨ı billiard B (see,e.g., [8] for background) satisfying the following two conditions:(BD1) all trajectories have a nontangential collision (see [1, strong finite horizon propertybefore Rem. 1.1]);(BD2) some combinatorial entropy (denoted by h ∗ and introduced in [1, Def. 2.1]) is abovesome threshold defined in [1, eqs. (1.2) and (1.2)].We denote by Λ B the hyperbolicity constant of T B from eqs. (2.2)-(2.3) in [1]. We remarkthat Baladi and Demers also prove that h ∗ = sup { h ( T B , ν ) : ν ∈ Prob erg ( T B ) } . Theorem 1.5. If T B is the collision map of a two-dimensional Sina¨ı billiard B satisfyingconditions (BD1) and (BD2), then: lim inf n →∞ e − n · h ∗ | per Λ B ( T B , n ) | ≥ . This strengthens [1, Cor. 2.7] by eliminating the possibility of a period, counting theperiodic orbits by their minimal periods, and replacing the positive lower bound by theinteger 1.We derive these results by proving a general result about a large class of symbolic dynamics,see our Main Theorem below.Theorem 1.1 improves on Sarig’s coding by making it injective. One would also like tohave an image as large as possible. The following shows that, in some sense, one cannotmuch improve on Sarig’s result in this direction:
EGREES OF BOWEN FACTORS 3
Theorem 1.6.
Let f ∈ Diff r ( M ) be a diffeomorphism of a closed surface with r > . Thenthere exist a Markov shift S : X → X and a map π : ( S, X ) → ( f, M ) such that f ◦ π = π ◦ S and:(i) for all µ ∈ Prob erg ( f ) with positive entropy, there is ν ∈ Prob( S ) with π ∗ ( ν ) = µ ;(ii) π is H¨older-continuous for the standard metric on X ;if and only if the exponents of ergodic invariant probability measures with positive entropyare bounded away from zero. General theorem.
The above will be a consequence of an abstract theorem aboutfactors of Markov shifts. A symbolic system ( S, X ) is some shift-invariant subset of A Z where A , the alphabet , is a countable (possibly finite) set, together with the action of the shift S : ( x n ) n ∈ Z (cid:55)→ ( x n +1 ) n ∈ Z . We equip X with the standard distance: d ( x, y ) := exp( − inf {| n | : x n (cid:54) = y n } ). A semiconjugacy π : ( S, X ) → ( T, Y ) between dynamical systems (
S, X ) and(
T, Y ) is a map π : X → Y such that π ◦ S = T ◦ π and π ( X ) ⊂ Y . A one-block code is asemiconjugacy π : X → Y between symbolic systems such that π ( x ) = (Π( x n )) n ∈ Z for somemap between the alphabets (this map Π : A → B is the code ). A map φ between two metricspaces is 1 -Lipschitz if d ( φ ( x ) , φ ( y )) ≤ d ( x, y ) for all pairs of points x, y .Recall the following definition from [4], taken from R. Bowen’s analysis of Markov parti-tions [3]. Definition 1.7 (Boyle-Buzzi) . A semiconjugacy π : ( S, X ) → ( T, Y ) satisfies the Bowenproperty if X is symbolic and if there is a reflexive and symmetric relation ∼ on its alphabet A such that, for all x, y ∈ X , (1.8) π ( x ) = π ( y ) ⇐⇒ ∀ n ∈ Z x n ∼ y n , in which case we say that x, y ∈ X are Bowen equivalent and write x ≈ y .The relation ∼ on A is called a Bowen relation for π (or admitted by π ). It is said to be locally finite if { b ∈ A : b ∼ a } is finite for each a ∈ A . We note that the Bowen property generalizes both:- D. Fried’s finitely presented systems [12] which are exactly the continuous Bowenfactors of subshifts of finite type;- one-block codes which admits as a transitive Borel relations the equivalence relationsdefined by their codes. There is a partial converse: any Bowen semiconjugacy π witha transitive Bowen relation ∼ can be written as π = ψ ◦ Π where Π is the one-blockcode defined by a (cid:55)→ { b : b ∼ a } and ψ : Π( X ) → π ( X ) is a Borel conjugacy.We show: Main Theorem.
Let ( S, X ) be a Markov shift on some alphabet A . Let X be its regularpart , i.e., the set of sequences x ∈ X such that, for some u, v ∈ A , u occurs infinitely many times in ( x n ) n ≤ and v occurs infinitely many times in ( x n ) n ≥ . Let π : ( S, X ) → ( T, Y ) be a Borel semiconjugacy such that: - ( T, Y ) is a Borel automorphism; - π is finite-to-one, i.e., π − ( y ) is finite for every y ∈ Y ; - π has the Bowen property with respect to a locally finite relation on A . J´ER ˆOME BUZZI
Then there are a Markov shift ( ˜ S, ˜ X ) and a -Lipschitz map φ : ˜ X → X such that π ◦ φ :˜ X → Y defines an injective semiconjugacy and π ◦ φ ( ˜ X ) carries all invariant measures of π ( X ) . Remark that if π is continuous or H¨older-continuous, then so is π ◦ φ .1.2. Further results, comments, and questions.
Note that a map φ : ˜ X → X is 1-Lipschitz if and only if there is a length-preserving map Φ : (cid:83) n ≥ L n +1 ( ˜ X ) → (cid:83) n ≥ L n +1 ( X )such that φ ( x ) [ − n,n ] = Φ( x [ − n,n ] ) for all n ≥
0. The 1-Lipschitz map in the above theoremmay fail to be a one-block code because it may fail to commute with the shift.
Ingredients of the proof.
We adapt classical ideas from the theory of subshifts of finite type.The first part builds on tools from finite equivalence theory and more specifically jointwork with Mike Boyle [4]. This leads to Theorem 3.3 which is an abstract version of anunpublished result of Sarig [25]. The second part of the proof involves ideas from magicword isomorphisms and the degree of almost conjugacies [21, chap. 9]. It allows to partitionaccording to the number of preimages while preserving the Markov structure. We concludeby injectively coding subsets with larger and larger numbers of preimages.
Good coding for given measures.
We can specialize our results to a given measure of inter-est. For instance, given a surface diffeomorphism with positive topological entropy and adistinguished ergodic measure maximizing the entropy µ , we obtain an irreducible Markovshift X and a H¨older-continuous conjugacy π : X → M such that π ( X ) has full µ -measure.This was implicit in [4, Prop. 6.3]. We refer to [7] for further results in this direction. Bounds for periodic points.
Kaloshin [15] has shown that, C r -generically (1 ≤ r < ∞ ), thenumber of periodic points grows arbitrarily fast with the period. However, these periodicpoints have Lyapunov exponents going to zero. In fact, Burguet [5] has shown the followinglogarithmic estimate, for any C ∞ surface diffeomorphism: ∀ χ < h top ( f ) lim n →∞ ,p | n n log | per χ ( f, n ) | = h top ( f ) . Question 1.
Is there a C ∞ surface diffeomorphism f with positive entropy such that, forsome χ > : lim sup n →∞ e − n · h top ( f ) | per χ ( f, n ) | = ∞ ? Beyond surface diffeomorphisms.
Ben Ovadia’s higher-dimensional generalization [2] of Sarig’scoding also yields finite-to-one semiconjugacies that are Bowen with respect to a locally finiterelation. Hence our abstract theorem also applies in this setting.
Better symbolic representations.
For a topologically transitive surface diffeomorphism, S. Cro-visier, O. Sarig, and the author [7] have shown that, for any given parameter χ >
0, there isa finite-to-one, H¨older-continuous transitive symbolic dynamics coding a subset carrying all χ -hyperbolic measures. Applying our main theorem makes this coding injective but destroysthe transitivity. We ask: EGREES OF BOWEN FACTORS 5
Question 2.
For a topologically transitive C ∞ diffeomorphism of a closed surface and anynumber χ > , can one get a H¨older-continuous injective coding by a transitive Markov shifta subset carrying all χ -hyperbolic measures? Our Theorems 5.2 and 5.3 below provide partial solutions. We build a H¨older-continuous,finite-to-one coding by a transitive Markov shift whose injectivity set is “large” in a weakersense than above: it has full measure with respect to a given measure or for all fully supportedmeasures. These theorems are applied to surface diffeomorphisms in [7].To capture all hyperbolic measures, one can apply Sarig’s construction countably manytimes with a parameter χ decreasing to 0. One obtains a sequence of semiconjugacies withlarger and larger images but smaller and smaller H¨older exponents. In [4], together withM. Boyle, we were able to ”fuse” all these semiconjugacies by using a Borel construction. Question 3.
Given a surface diffeomorphism, can one get a continuous finite-to-one codingby a Markov shift of a subset carrying all hyperbolic measures ? Can it be done injectively?
Because of Theorem 1.6 one cannot ask for a H¨older-continuous semiconjugacy.
Local compactness.
Our proof of the Main Theorem does not preserve local compactness.We do not know if it can be done. The following very natural question asked by the refereeremains open:
Question 4.
In Theorem 1.1, is it possible to code using a locally compact Markov shift?
In Appendix B, we provide a partial answer: we obtain local compactness but injectivityholds only after restricting to the regular part.1.3.
Outline of the paper.
In Section 2, we recall some basic definitions and make somecomments about Bowen relations. In Section 3, we introduce Bowen quotients inspired bya classical construction of Manning [22] from the theory of Markov partitions. These are anabstract version of a construction of Sarig [25]. The proof rests on Proposition 3.9, whichadapts lemmas from the theory of finite equivalence of subshifts of finite type due to Hedlund[14] and Coven and Paul [9, 10, 11]. In Section 4, we combinatorially characterize the fiberswith minimal cardinality by adapting the notion of magic word from the theory of almostconjugacy of shifts of finite type (see [21, chap. 9]).In Section 5, we show how to a get a coding with a large injectivity set, especially withrespect to a given measure or to all fully supported measures and then deduce the maintheorem from the previous constructions. We proceed by induction on the number of preim-ages. The Bowen quotients make the semiconjugacy injective where its fibers had a givencardinality and then discard these points. The magic word theory preserves the Markovstructure and the Bowen property.In Section 6, we apply our main theorem to Sarig’s coding of surface diffeomorphisms [24]and prove Theorems 1.1 and 1.2 using Newhouse [23] (for smoothness) and Buzzi-Crovisier-Sarig [7] (for transitivity). In Section 7, we prove Theorem 1.6 characterizing surface diffeo-morphisms with H¨older-continuous codings. In the Appendices we further discuss the Bowenrelation, provide a locally compact construction, and deduce Theorem 1.5.
J´ER ˆOME BUZZI
Acknowledgments.
I thank Pierre Berger, Sylvain Crovisier, Yuri Lima, and Omri Sarigfor useful comments. I especially thank Mike Boyle for pointing out mistakes in an earlyversion of this text and simplifying the proof of Proposition 3.9. I am also grateful toSylvain Crovisier for discussions leading to Section 5.1 and to Yuntao Zang for pointing outan imprecision in Lemma 6.1. I finally thank the referee for both corrections and suggestionsthat have improved this paper.2.
Definitions and first properties
Borel systems.
A standard Borel space is a set equipped with the Borel σ -field gen-erated by a Polish topology (i.e., generated by a metric making the space complete andseparable, see [17] for background). A dynamical system is an automorphism S of such aspace X . We denote it by ( S, X ) (or just S or X when convenient).A full subset for S is a subset of X with measure equal to 1 for all measures of S . Asubset is null if its complement is a full subset. We say that a property of points holds almost everywhere (or just a.e.) without reference to a measure, if it holds on such a fullsubset.By the Lusin-Novikov Theorem [17, (18.10)], the direct image of a Polish space by acountable-to-one Borel map is Borel. In fact, there is a countable partition of the Polishspace into Borel subsets on which the map is injective and one can apply the Lusin-SuslinTheorem [17, (15.2)]. Therefore: Lemma 2.1.
Let p : ( S, X ) → ( T, Y ) be a Borel semiconjugacy between dynamical systems.If ν ∈ Prob( T ) satisfies p − ( y ) is finite and nonempty for ν -a.e. y ∈ Y , then there is µ ∈ Prob( S ) with p ∗ ( µ ) = ν . In particular, if p is finite-to-one and onto: - p ∗ : Prob( S ) → Prob( T ) is onto; - for any Borel subsets U ⊂ X , V ⊂ Y : U is a null subset ⇐⇒ p ( U ) is a null subset; V is a null subset ⇐⇒ p − ( V ) is a null subset. Symbolic dynamics.
Let A be a countable (possibly finite) discrete set, called thealphabet. The shift is ( x n ) n ∈ Z (cid:55)→ ( x n +1 ) n ∈ Z . A symbolic system ( S, X ) is the restric-tion S of the shift to some invariant subset X of A Z with the usual distance: d ( x, y ) :=exp ( − inf {| n | : x n (cid:54) = y n } ). A symbolic system needs not be closed.For x ∈ X and integers a ≤ b , x [ a,b ] = x a x a +1 . . . x b ∈ A b − a +1 and x [ a,b ) = x [ a,b − . Theset of X, n -words is L n ( X ) := { x [0 ,n ) : x ∈ X } and the language is L ( X ) := (cid:83) n ≥ L n ( X ).A word w ∈ L n ( X ) has length | w | := n . The word w occurs at n ∈ Z in some x ∈ X if x [ n,n + | w | ) = w . An n -word w defines a cylinder:[ w ] X := { y ∈ X : y [0 ,n ) = w } . As usual, words differing by an integer translation of their indices are identified.Sarig’s regular set is the subset X ⊂ X of sequences x ∈ X such that there are u, v ∈ A satisfying: { n ≥ x − n = u } , { n ≥ x n = v } are both infinite . Recall that all measures in this paper are understood to be ergodic and invariant Borel probabilitymeasures.
EGREES OF BOWEN FACTORS 7
If the alphabet is finite, then X = X . If π : X → Y is a semiconjugacy with X a symbolicsystem, its regular part is the restriction π : X → Y . We also write π or π whenconvenient.A sequence x ∈ X is word recurrent if any word w that occurs in x occurs infinitely oftenin both x ( −∞ , and x [0 , ∞ ) . We say that w occurs i.o. in x or that x sees i.o. w . Wedenote by X rec ⊂ X the set of such sequences. Note that it carries all invariant probabilitymeasures on X (in particular, it contains all periodic orbits). We have the obvious inclusion X rec ⊂ X .A Markov shift ( S, X ) is a symbolic system over some alphabet A such that X can becharacterized as the set of bi-infinite paths on some simple, directed graph G , that is, X := { x ∈ A Z : ∀ n ∈ Z x n G → x n +1 } . The graph G describes the Markov shift X .2.3. Remarks about the Bowen property.
We comment on related notions, the (non)-uniqueness of the symmetric relation involved in its characterization and its eventual exten-sion from the regular part to the whole of a factor.1) A semiconjugacy may have the Bowen property without being Borel. Indeed, if π isBowen, then so is ϕ ◦ π for any self-conjugacy ϕ : Y → Y (i.e., a bijection that commuteswith the dynamics).2) Being Bowen and finite-to-one are independent properties of semiconjugacies (neitherimplies the other) as shown by the following examples: (i) π : { , , } Z → { , } Z given bythe block code a (cid:55)→ b mod 2 is Bowen and infinite-to-one; (ii) π : { , } Z → S given by π ( x ) = exp 2 iπ (cid:80) n ≥ − n − x n which is at most 2-to-1 but cannot be Bowen since it is neitherinjective nor constant.3) On the one hand, many Bowen semiconjugacies do not admit any transitive Bowen re-lations. On the other hand, the equivalence between sequences must be transitive. ThusBowen relations are special reflexive and symmetric relations on the alphabet. Bowen [3,p. 13] asked:
Problem (Bowen).
Let
A, B be two n × n -matrices with entries zero or one. Let ≈ be therelation defined on X := { x ∈ { , . . . , n } Z : ∀ p ∈ Z A ( x p , x p +1 ) = 1 } by x ≈ y ⇐⇒ ∀ n ∈ Z B ( x n , y n ) = 1 . Decide whether this relation is transitive. If so, decide whether the shift on X/ ≈ is topologically conjugate to the non-wandering set of a uniformly hyperbolic system.
4) A given Bowen semiconjugacy can admit distinct Bowen relations. See Appendix A forthe canonical relation defined by a semiconjugacy.5) In our main examples the Bowen property hold in the regular part of the symbolic system.Even when the semiconjugacy has a unique uniformly continuous extension to the wholesymbolic system, the latter may fail to satisfy the Bowen property, see Appendix A. A simple directed graph is a graph with oriented arrows and such that for any vertices a, b there is atmost one arrow from a to b . For instance, a continuous Bowen factor of a compact symbolic system with a transitive Bowen relationmust be zero dimensional.
J´ER ˆOME BUZZI Bowen Quotients
We introduce our basic construction: the Bowen quotient. Given an integer N ≥ π , we are going to build another semiconjugacy π N whose preimagesare the sets of N preimages of a common point. This is a purely combinatorial constructionwhich:– preserves the class of finite-to-one, Bowen semiconjugacies of regular parts of Markovshifts;– produces π N which is one-to-one above the points where the first semiconjugacy was N -to-1;– has in its image the points with at least N preimages by π , up to a null set.This construction is closely related to previous work with Boyle [4, Prop. 6.3]. Similarconstructions go back to Hedlund [14] and Coven and Paul [9, 10, 11] (for subshifts offinite type, see [21, chap. 8 and 9]); Manning [22] and Bowen [3] (for coding of Axiom-Adiffeomorphisms).3.1. Definition and statement.
Let (
S, X ) be a Markov shift with alphabet A and under-lying graph G . Let π : X → Y be a Borel semiconjugacy on the regular part X . Assumethat it admits a Bowen relation ∼ . Given an integer N ≥
1, let:- A N be the collection of subsets A ⊂ A with cardinality N whose elements are pairwiserelated by ∼ ;- G N be the simple directed graph over A N with arrows: A G N → B if and only if there isa bijection φ : A → B such that a G → b ⇐⇒ b = φ ( a ) for all ( a, b ) ∈ A × B . Definition 3.1.
Let π : X → Y be a semiconugacy with a locally finite Bowen relation ∼ .Given an integer N ≥ , the Bowen quotient of order N of ( π : X → Y, ∼ ) is ( π N : X N → Y, N ∼ ) where: (BQ1) X N is the regular part of the Markov shift X N defined by the graph G N ; (BQ2) π N : X N → Y satisfies for all ˆ x ∈ X N , π N (ˆ x ) = π ( x ) for any x ∈ X s.t. x n ∈ ˆ x n ( ∀ n ∈ Z ) ; (BQ3) A N ∼ B if and only if ∀ ( a, b ) ∈ A × B a ∼ b . The following definition will be convenient for our purposes.
Definition 3.2.
A semiconjugacy π : X → Y is excellent for some relation ∼ if X is asymbolic system, Y is a dynamical system and: (EX1) ∼ is a locally finite, reflexive and symmetric relation on the alphabet of X ; (EX2) π is Bowen with respect to the relation ∼ ; (EX3) π is Borel and finite-to-one. In the next statement and throughout this paper, (cid:0) pq (cid:1) = p ! q !( q − p )! and is zero if q > p . Theorem 3.3.
Let X be a Markov shift with regular part X . Let π : X → Y be anexcellent semiconjugacy for some Bowen relation ∼ . Then, for any integer N ≥ , theBowen quotient ( π N : X N → Y, N ∼ ) of order N is well-defined and excellent. EGREES OF BOWEN FACTORS 9
Moreover there is a finite-to-one, -Lipschitz map q N : X N → X such that:(1) if X is locally compact, then so is X N ;(2) π N = π ◦ q N : X → Y ;(3) | π − N ( y ) | ≤ (cid:0) | π − ( y ) | N (cid:1) for all y ∈ Y with equality except on a null set;(4) q N : X N → X is proper, i.e., for any compact set K ⊂ X , q − N ( K ) ∩ X N iscompact. Definition 3.4.
The degree spectrum of a Borel semiconjugacy π : Z → Y is: ∆( π ) := { k ≥ { y ∈ Y : | π − ( y ) | = k } is not a null set } . Corollary 3.5.
In the setting of the above theorem, π N ( X N ) ⊂ π ( X ) . In fact, (3.6) ∆( π N ) = (cid:26)(cid:18) rN (cid:19) : r ∈ ∆( π ) and r ≥ N (cid:27) and, for each r ∈ ∆( π ) with r ≥ N , (cid:8) y ∈ Y : | π − N ( y ) | = (cid:0) rN (cid:1)(cid:9) is contained in (cid:8) y ∈ Y : | π − ( y ) | ≥ r (cid:9) and equal to (cid:8) y ∈ Y : | π − ( y ) | = r (cid:9) up to a null set. All the claims above are straightforward consequences of the theorem, except possiblyfor eq. (3.6), which we now prove. Let Z := { y ∈ Y : | π − N ( y ) | (cid:54) = (cid:0) | π − ( y ) | N (cid:1) } . It is anull set by item (3) above. If r ∈ ∆( π ) and r ≥ N , let s := (cid:0) | π − ( y ) | N (cid:1) ≥
1. Note that { y ∈ Y : | π − ( y ) | = r } \ Z is not a null set and is included in { y ∈ Y : | π − N ( y ) | = s } , so s ∈ ∆( π N ). For the converse, let s ∈ ∆( π N ) so { y ∈ Y : | π − N ( y ) | = s } \ Z is not a null set.Hence, by item (3), s = (cid:0) rN (cid:1) for some r ≥ N such that { y ∈ Y : | π − ( y ) | = r } is not a nullset. Hence r ∈ ∆( π ). Eq. (3.6) is proved. Remarks 3.7.
1. The proof of the theorem will give an explicit null set Y where the inequality in item(3) may be strict.2. The following example shows that X N may fail to be irreducible even if X is irreducible.However, using the magic word theory of Section 4 we will show in Theorem 5.3 that onecan restrict the semiconjugacy to an irreducible component of X N without diminishing theimage. Example 3.8.
Let G be a simple, directed graph with set of vertices A and let p be a positive integer.Define G over A := A × ( Z /p Z ) by: ( a, i ) G → ( b, j ) def ⇐⇒ i + 1 = j and a G → b . Define a symmetric relation ∼ on A by ( a, i ) ∼ ( b, j ) def ⇐⇒ a = b and consider a Bowen semiconjugacy with this relation. Hence, for any N ≥ , A N = {{ a } × I : a ∈ A and I ⊂ Z /p Z with | I | = N } and { a } × I → { b } × J if and only if a → b and I + 1 = J .Observe that G N may fail to be irreducible even when G is irreducible. For instance, if p = 4 and N = 2 ,for any a ∈ G , { ( a, , ( a, } and { ( a, , ( a, } belong to distinct irreducible components of G N . Resolving property.
We begin by studying the combinatorics of finite fibers over anorbit with some recurrence. Let (
S, X ) be a Markov shift with alphabet A and ( T, Y ) besome dynamical system. Let π : X → T be a semiconjugacy with some Bowen relation ∼ .Denote the Bowen equivalence by ≈ and the equivalence class of x ∈ X by (cid:104) x (cid:105) := { y ∈ X : y ≈ x } . Let us call a point x ∈ X recurrent for some function φ : X → Z if, for each n ∈ Z , { k ∈ Z : φ ( S k x ) = φ ( S n x ) } is neither lower bounded nor upper bounded. Poincar´e recurrenceimplies that the set of recurrent points for any given measurable function is a full set. Proposition 3.9.
Let ( π, ∼ ) be an excellent semiconjugacy defined on a Markov shift X described by some graph G . For x ∈ X and n ∈ Z , let A ( x, n ) := { y n : y ∈ (cid:104) x (cid:105)} . Let X ∗ bethe set of points x ∈ X which are simultaneously recurrent for the three following functions: R ( x ) := |A ( x, | , n + ( x ) := |{ y [0 , ∞ ) : y ∈ (cid:104) x (cid:105)}| , n − ( x ) := |{ y ( −∞ , : y ∈ (cid:104) x (cid:105)}| . Then for any x ∈ X ∗ and n ∈ Z ,(a) for each n ∈ Z , the restriction x (cid:55)→ x n defines a bijection between (cid:104) x (cid:105) and A ( x, n ) ;(b) for each n ∈ Z , |A ( x, n ) | = |A ( x, n + 1) | and, for every a ∈ A ( x, n ) there is a unique b ∈ A ( x, n + 1) such that a G → b ;(c) (cid:104) x (cid:105) ⊂ X ∗ . This proposition is related to finite equivalence theory and especially some classical resultsof Coven and Paul [9] (see [21, Thm 8.1.16]).
Proof.
Step 1.
Given x ∈ X recurrent for the function n + and a ∈ Z , there is an integer (cid:96) ≥ (which can be taken arbitrarily large) such that, among y ∈ (cid:104) x (cid:105) , y [ a,a + (cid:96) ) determines y . In the above situation, we say that [ a, a + (cid:96) ) is an admissible interval for x .Let x, a be as above. Since (cid:104) x (cid:105) is finite, n + ( S − n x ) = |(cid:104) x (cid:105)| for n large enough. Observethat the function n + is monotone along orbits. By recurrence, it is constant along the orbits.Hence |(cid:104) x (cid:105)| = n + ( S a x ) = |{ y [ a,a + (cid:96) ) : y ∈ (cid:104) x (cid:105)}| for all large integers (cid:96) . Fixing such an integer (cid:96) , the obvious surjectivity of the map y (cid:55)→ y [ a,a + (cid:96) ) implies its bijectivity. Step 2.
Item (a): for any n ∈ Z , y n determines y for y ∈ (cid:104) x (cid:105) . Given n ∈ Z , Step 1 provides an admissible interval [ a, b ] ⊂ [ n +1 , ∞ ). If there were distinct y, y (cid:48) ∈ (cid:104) x (cid:105) such that y n = y (cid:48) n but y ( −∞ ,n ) (cid:54) = y (cid:48) ( −∞ ,n ) , the spliced sequence y (cid:48) ( −∞ ,n ) y [ n, ∞ ) of X coinciding with y on [ a, b ], would contradict the admissibility of [ a, b ]. Thus y n determines y ( −∞ ,n ] for y ∈ (cid:104) x (cid:105) .Symmetric arguments (using n − ) show that y n determines the whole sequence y ∈ (cid:104) x (cid:105) ,proving item (a). Step 3.
Item (b): R := { ( a, b ) ∈ A ( x, n ) × A ( x, n + 1) : a G → b } is a bijection Observe first that for every a ∈ A ( x, n ), a = y n for some y ∈ (cid:104) x (cid:105) so that a → y n +1 with y n +1 ∈ A ( x, n + 1). Assume a G → b and a G → b (cid:48) with a ∈ A ( x, n ) and b, b (cid:48) ∈ A ( x, n + 1).Therefore there are y, z, z (cid:48) ∈ (cid:104) x (cid:105) such that y n = a , z n +1 = b , and z (cid:48) n +1 = b (cid:48) . Consideringthe splicings y ( −∞ ,n ] z [ n +1 , ∞ ) and y ( −∞ ,n ] z (cid:48) [ n +1 , ∞ ) , Step 2 implies that b = b (cid:48) . Thus R definesa unique map A ( x, n ) → A ( x, n + 1). A symmetric argument gives an inverse map A ( x, n +1) → A ( x, n ), hence R is bijective: item (b) is proved. Step 4.
Item (c): if x ∈ X ∗ , then (cid:104) x (cid:105) ⊂ X ∗ This is clear from the definition of X ∗ . The proposition is proved. (cid:3) EGREES OF BOWEN FACTORS 11
Bowen quotients.
We prove Theorem 3.3. To begin with, we let G N , A N , X N , and N ∼ as in Definition 3.1. Items (BQ1) and (BQ3) are then satisfied by construction.We now define π N to satisfy (BQ2) and item (2) of the theorem. We note the followingeasy consequence of the definition of G N : Fact 3.10.
For −∞ ≤ i < < j ≤ ∞ , let ˆ x = (ˆ x n ) i 0. For those indices n , ˆ x Jn is contained in the finite set { b : b ∼ a } , hence must take some value infinitely many times.The same holds for negative indices, proving that ˆ x J ∈ X N . Thus the inclusion in eq. (3.11)is an equality and: ∀ y ∈ π ( X ) \ Y | π − N ( y ) | = (cid:18) rN (cid:19) . Item (3) of the theorem is proved. We note for future reference the following consequence: Fact 3.12. Let π : X → Y be an excellent semiconjugacy with Bowen quotient π N : X N → Y . For all x ∈ X outside a null set, if | π − ( π ( x )) | ≥ N then ∃ ˆ x ∈ X N s.t. ∀ n ∈ Z x n ∈ ˆ x n . We check that N ∼ is a Bowel relation for π N . First, let ˆ x, ˆ y ∈ X N with π N (ˆ x ) = π N (ˆ y ). Let a ∈ ˆ x and b ∈ ˆ y . Fact. 3.10 gives (unique) sequences x ∈ [ a ] X , y ∈ [ b ] X with π ( x ) = π N (ˆ x ), π N (ˆ y ) = π ( y ). Thus π ( x ) = π ( y ) and x ∼ y . It follows that ˆ x N ∼ ˆ y and then ˆ x N ≈ ˆ y , byequivariance.Conversely, let ˆ x, ˆ y ∈ X N with ˆ x N ≈ ˆ y . Picking a ∈ ˆ x and b ∈ ˆ y , Fact 3.10 gives x ∈ [ a ] X , y ∈ [ b ] X such that x n ∈ ˆ x n and y n ∈ ˆ y n for all n ∈ Z . Thus π ( x ) = π N (ˆ x ) and π N (ˆ y ) = π ( y ).From the definition of N ∼ , we have x ≈ y . The Bowen property for ∼ implies π ( x ) = π ( y )hence π N (ˆ x ) = π N (ˆ y ). The Bowen property (EX2) is established.Finally, we prove that q N is proper. Note that a subset K of a symbolic system is relativelycompact if and only if, for each n ∈ Z , { x n : x ∈ K } is finite. Fix a relatively compact K ⊂ X and n ∈ Z . By construction, q N (ˆ x ) ∈ K implies that ˆ x n , a set of N symbols from A , contains only symbols that are Bowen related to elements of { x n : x ∈ K } . Since ∼ islocally finite, it follows that { ˆ x n : q N (ˆ x ) ∈ K } is finite and q − ( K ) is relatively compact.Item (4) is proved. (cid:3) Combinatorial degree We are going to characterize the subset of a Bowen semiconjugacy where the cardinalityof the fibers is minimal by the recurrence of some words. To this end, we adapt the notionsof degree and magic word from the classical theory of one-block codes between subshifts offinite type (see Hedlund [14] and more generally [21, chap. 9]).It is convenient to disregard the factor map π : X → Y and to focus on the symbolicsystem X and the Bowen relation. Definition 4.1. Given a symbolic system ( S, Z ) on some alphabet A , an (abstract) Bowenrelation is a reflexive, symmetric relation ∼ on A such that the relation on Z defined by x ≈ y def ⇐⇒ ∀ n ∈ Z x n ∼ y n is an equivalence relation. In this section, ∼ is a Bowen relation on the regular part X of a Markov shift X . Recallthat X rec ⊂ X is the set of word recurrent sequences in X (i.e., any word that occurs onceis seen i.o. –see p. 7). Recall also that the Bowen equivalence class of any x ∈ X is denotedby: (cid:104) x (cid:105) := { y ∈ X : y ≈ x } . EGREES OF BOWEN FACTORS 13 Degree of Bowen relations. The relation ∼ on the alphabet of X induces anotherreflexive and symmetric relation on L ( X ) (also denoted by ∼ ) according to v ∼ w def ⇐⇒| v | = | w | and v ∼ w , . . . , v | v |− ∼ w | v |− .We will consider the languages L ( X rec ) ⊂ L ( X ) ⊂ L ( X ). In general, they are distinct.However they are equal when X is the disjoint union of its irreducible components. Definition 4.2. Given a Bowen relation ∼ , the degree of a word w ∈ L ( X ) at some index ≤ i < | w | is: δ ∼ ( w, i ) := |{ v i : v ∈ L ( X ) , v ∼ w }| δ ∼ ( w ) := min { δ ∼ ( w, i ) : 0 ≤ i < | w |} . The degree of ∼ is: δ rec ( ∼ ) := inf { δ ∼ ( w ) : w ∈ L ( X rec ) } . A magic word is a word w ∈ L ( X rec ) realizing this infimum. A couple ( w, i ) that realizes itis called a magic couple. Observe that for any w ∈ L ( X rec ), deg ∼ ( w ) ≥ ∼ is reflexive). As soon as deg rec ( ∼ )is finite (e.g., if ∼ is locally finite), there always exist magic words.Given a word W ∈ L ( X ), X W denotes the set of sequences that see i.o. W : X W := { x ∈ X : ∃ m k , n k → ∞ such that W occurs in x at − m k and at n k } . Note that X W is an invariant, possibly empty, subset of X . We start with two simplelemmas. Lemma 4.3. Assume that the Bowen equivalence classes: (cid:104) x (cid:105) := { y ∈ X : y ≈ x } arefinite for all x ∈ X . Let W ∈ L ( X ) with δ ∼ ( W ) = 1 . If x ∈ X sees i.o. W , then x isonly equivalent to itself, that is: ∀ x ∈ X W (cid:104) x (cid:105) = { x } . Proof. Let 0 ≤ I < | W | such that δ ∼ ( W, I ) = 1 and x ∈ X W . Pick an increasing sequence ofintegers ( n k ) k ∈ Z such that x [ n k ,n k + | W | ) = W . If there is a distinct y ∈ X with x ≈ y , one canfind k < l such that x [ n k + I,n l + I ] (cid:54) = y [ n k + I,n l + I ] . However, y [ n k ,n k + | W | ) ∼ W implies that y n k + I = W I since δ ∼ ( W, I ) = 1. Consider the infinitely many distinct arbitrary concatenations ofthe two words x [ n k + I,n l + I ) , y [ n k + I,n l + I ) . They belong to the Markov shift X and in fact to X since they see i.o. the symbol W I . Moreover, they belong to a single Bowen equivalenceclass which is infinite, a contradiction. (cid:3) Lemma 4.4. Assume that ∼ is a locally finite Bowen relation. For any x ∈ X , (4.5) | (cid:104) x (cid:105) | ≥ δ ∼ ( x ) := min { δ ∼ ( x p . . . x p + (cid:96) − ) : p ∈ Z , (cid:96) ≥ } . Proof. Fix x ∈ X . For each n ≥ 1, let A n := { y : y ∈ (cid:10) x [ − n,n ] (cid:11) } . This defines a non-increasing sequence of sets contained in { b ∈ A : b ∼ x } which is finite. Hence there is n such that A n = A n for n ≥ n . Note that |A n | ≥ δ ∼ ( x ). Now, fix a ∈ A n and, foreach n ≥ 0, pick y n ∈ (cid:10) x [ − n,n ] (cid:11) with y n = a . For each k ∈ Z , { y nk : n ≥ | k |} is finite (sincethe Bowen relation is locally finite). Thus one can find an accumulation point y ∈ A Z (i.e.,there is n j ↑ ∞ such that, for each k ∈ Z , y k = y n j k for all large j ). It is easy to check that y ∈ X and y ∈ (cid:104) x (cid:105) . Varying a ∈ A n , eq. (4.5) follows. (cid:3) Figure 1. The subshift of finite type in Example 4.114.2. Magic semiconjugacies. We relate the combinatorial degree of the Bowen relationwith the cardinality of the fibers of the semiconjugacy. Theorem 4.6. Let X be a Markov shift and let ∼ be a locally finite Bowen relation for X .Let x ∈ X rec with (cid:104) x (cid:105) finite. The following are equivalent:(a) (cid:104) x (cid:105) has exactly δ rec ( ∼ ) elements;(b) x sees some magic word W for ∼ . The following example shows that the implication ( b ) = ⇒ ( a ) in Theorem 4.6 may failwhen (cid:104) x (cid:105) is infinite or when x / ∈ X rec . Example 4.7. Let X = { , , } Z and for a, b ∈ { , , } , let a ∼ b ⇐⇒ | b − a | = 0 , . Note that X = X , L ( X rec ) = L ( X ) , and δ rec ( ∼ ) = δ ∼ (1) = 1 . For x ∈ X rec distinct from ∞ such as x = (10) ∞ , (cid:104) x (cid:105) is infinite.For y = 1 ∞ k ∞ with k ≥ , y ∈ X \ X rec sees i.o. the magic word , however: |(cid:104) y (cid:105)| = 2 k > δ rec ( ∼ ) . The degree has a geometric meaning: Corollary 4.8. Let X be a Markov shift such that X rec (cid:54) = ∅ and let ∼ be a locally finiteBowen relation for X . If (cid:104) x (cid:105) is finite for each x ∈ X rec , then (4.9) δ rec ( ∼ ) = min {|(cid:104) x (cid:105)| : x ∈ X rec } = min { k ≥ { x ∈ X : | (cid:104) x (cid:105) | = k } is not null } . In particular, δ rec ( ∼ ) only depends on the Bowen equivalence relation ≈ .Proof. The inequality δ rec ( ∼ ) ≤ min {|(cid:104) x (cid:105)| : x ∈ X rec } follows from Lemma 4.4 since δ rec ( ∼ ) ≤ δ ∼ ( x ) for x ∈ X rec . Conversely, let W be a magic word for ∼ over X rec . By definition,there is x ∈ X rec that sees i.o. W . By Theorem 4.6, | (cid:104) x (cid:105) | = δ rec ( ∼ ), proving the firstequality.We show that δ rec ( ∼ ) is equal to d := min { k ≥ { x ∈ X : | (cid:104) x (cid:105) | = k } is not null } .Since { x ∈ X : | (cid:104) x (cid:105) | = d } has positive measure for some invariant probability measure, itcontains a recurrent point, so δ rec ( ∼ ) = min {| (cid:104) x (cid:105) | : x ∈ X rec } ≤ d .Conversely, there is x ∈ X rec such that | (cid:104) x (cid:105) | = δ rec ( ∼ ). By Theorem 4.6, x sees i.o. somemagic word W . Since X is a Markov shift, one can find y ∈ X that sees i.o. W and isperiodic. Its orbit is a non-null set, hence d ≤ δ rec ( ∼ ). (cid:3) Remark 4.10. The next example shows that there is no simple analogue of Corollary 4.8for X , even if one replaces the degree deg rec ( ∼ ) by min { deg ∼ ( w ) : w ∈ L ( X ) } . Example 4.11. Let X be the subshift of finite type defined by the directed graph in Fig. 1. Define π : X →{− , , +1 } Z as the projection on the first coordinate with Bowen relation ( a, b, c ) ∼ ( a (cid:48) , b (cid:48) , c (cid:48) ) ⇐⇒ a = a (cid:48) . EGREES OF BOWEN FACTORS 15 Note that X = X and π ( X ) is the union of two fixed points (+1) ∞ , ( − ∞ and a heteroclinic orbit: { σ k ((+1) ∞ · ( − ∞ ) : k ∈ Z } . Note also that X rec is the union of: - four -periodic orbits mapped to the two fixed points, defining Bowen equivalence classes with elements each; - four heteroclinic orbits, each mapped to the heteroclinic orbit, defining Bowen equivalence classeswith elements each.The following is easily checked: inf w ∈L ( X ) δ ∼ ( w ) = δ ∼ ((0 , , , 0) = 1 < inf {| (cid:104) x (cid:105) | : x ∈ X } = 4 < inf {| (cid:104) x (cid:105) | : x ∈ X rec } = 6= inf w ∈L ( X rec ) , ≤ i< | w | |{ v i : v ∈ L ( X rec ) , v ∼ w } = inf w ∈L ( X rec ) , ≤ i< | w | δ ∼ ( w ) = δ ∼ ((1 , , , . To prepare for the proof of Theorem 4.6, we fix a magic couple ( W, I ) in X over X rec .Since ∼ is locally finite, M = | (cid:104) W (cid:105) | is finite. We enumerate its elements and the symbolsat index I : (cid:104) W (cid:105) = { W , . . . , W M } and { a , . . . , a d } = { W I , . . . , W MI } where d = δ rec ( ∼ ). Obviously, d ≤ M . We can assume: a i = W iI for i = 1 , . . . , d .To any word that can be written as a concatenation W uW , we associate: T ij ( W uW ) := { a i ¯ τ : va i ¯ τ a j w ∈ L ( X ) , va i ¯ τ a j w ∼ W uW for some | v | = I, | w | = | W |− I − } for 1 ≤ i, j ≤ d . We call the words a i ¯ τ ∈ T ij ( W uW ) transitions . Note that these words havethe same length as W u . Matching transitions can be concatenated: Claim 4.12. For any ≤ i, j, k ≤ d , any W uW u (cid:48) W ∈ L ( X rec ) , there is an injection: T ij ( W uW ) × T jk ( W u (cid:48) W ) → T ik ( W uW u (cid:48) W ) , ( τ, τ (cid:48) ) (cid:55)−→ τ τ (cid:48) . Proof. Let τ, τ (cid:48) be as above. We can write τ = a i ¯ τ , τ (cid:48) = a j ¯ τ (cid:48) . By definition, va i ¯ τ a j w ∼ W uW and v (cid:48) a j ¯ τ (cid:48) a k w (cid:48) ∼ W u (cid:48) W where v, w, v (cid:48) , w (cid:48) are words of lengths | v | = | v (cid:48) | = I and | w (cid:48) | = | w | = | W | − I − 1. Thus, the concatenation va i ¯ τ a j ¯ τ (cid:48) a k w (cid:48) belongs to L ( X ) and isrelated to W uW u (cid:48) W . Hence a i ¯ τ a j ¯ τ (cid:48) ∈ T ik ( W uW u (cid:48) W ). The injectivity is obvious. (cid:3) Any transition can be extended to the right and to the left: Claim 4.13. For any ≤ i ≤ d , any W uW ∈ L ( X rec ) , the following two sets are not empty: T i ∗ ( W uW ) := (cid:91) ≤ j ≤ d T ij ( W uW ) and T ∗ i ( W uW ) := (cid:91) ≤ j ≤ d T ji ( W uW ) . Moreover, T ∗ ( W uW ) := (cid:83) ≤ i ≤ d T i ∗ ( W uW ) = (cid:83) ≤ j ≤ d T ∗ j ( W uW ) has at least d elements.Proof. Obviously, { m I : m ∈ L ( X ) , m ∼ W uW } ⊂ { w I : w ∈ L ( X ) , w ∼ W } = { a , . . . , a d } . Since ( W, I ) is magic, the cardinalities are equal and finite so the inclusion is anequality. Since { m I : m ∼ W uW } = { a : aτ ∈ T ∗ ( W uW ) } , it follows that |T ∗ ( W uW ) | ≥ d .It also follows that T ∗ ( W uW ) contains a word beginning with a i so that T i ∗ ( W uW ) is notempty. Likewise T ∗ i ( W uW ) is not empty. (cid:3) A word u will be called special if T i ∗ ( W uW ) has more than one element for some 1 ≤ i ≤ d . Lemma 4.14. Fix a magic word W . Assume that x ∈ X rec is an infinite concatenation (4.15) . . . W u − W u W u W . . . where each u k is some word. More precisely, there is an increasing integer sequence ( n k ) k ∈ Z such that, for all k ∈ Z , x [ n k ,n k +1 − = W u k .If there are (at least) ≤ K ≤ ∞ distinct integers k ∈ Z such that u k is special, then |(cid:104) x (cid:105)| ≥ K + d. Moreover, for each k ∈ Z , { y n k + I : y ∈ (cid:104) x (cid:105)} = { a i : i = 1 , . . . , d } .Proof. We may and do assume that (cid:104) x (cid:105) ∼ is finite and that K is finite (by an easy reduction).To simplify notation, we assume that n = − I and that there are K positive integers k with u k special (using shift invariance). For each n ≥ 0, let K ( n ) be the number of integers0 < k < n with u k special. For each n ≥ 1, let U n := u W . . . W u n − .We claim that for every n ≥ T ∗ ( W U n W ) := (cid:91) ≤ i ≤ d T i ∗ ( W U n W ) has at least K ( n ) + d elements { w : w ∈ T ∗ ( W U n W ) } = { a , . . . , a d } . We proceed by induction. Claim 4.13 implies that |T ∗ ( W u W ) | ≥ d which is eq. (4.16) for n = 1 since K (1) = 0. Assume eq. (4.16) for some n ≥ 1. Claims 4.12 and 4.13, show thateach element of T ∗ ( W U n W ) can thus be extended to an element of T ∗ ( W U n +1 W ). Thus |T ∗ ( W U n +1 W ) | ≥ |T ∗ ( W U n W ) | and { w : w ∈ T ∗ ( W U n +1 W ) } ⊃ { w : w ∈ T ∗ ( W U n W ) } .Eq. (4.16) follows if K ( n + 1) = K ( n ). Otherwise K ( n + 1) = K ( n ) + 1 and W u n W is specialso some element of T ∗ ( W U n W ) has at least two distinct extensions in T ∗ ( W U n +1 W ). Thiscompletes the induction and proves the claim (4.16).Observe that the words in T ∗ ( W U n W ) are the prefixes of length | W U n | of the words in T ∗ ( W U n +1 W ). Hence one can take an inductive limit and obtain Y ⊂ A [0 , ∞ ) such that,for each n ≥ T ∗ ( W U n W ) = { y [0 , | W U n |− : y ∈ Y} . It is easy to see that ( Z − := { , − , − , . . . } ):- Y has at least K + d elements;- each y ∈ Y satisfies: y ∈ { a , . . . , a d } , y n X → y n +1 for all n ∈ N , and y ∼ x [0 , ∞ ) .For each 1 ≤ i ≤ d , an analogous use of Claims 4.13 and 4.12 provides an infinite one-sidedsequence z i ∈ A Z − such that z i = a i and z in X → z in +1 for all n < 0, and z i ∼ x ( −∞ , . Therefore { z y ( −∞ , − y [0 , ∞ ) ) : y ∈ Y} ⊂ (cid:104) x (cid:105) ∼ so that |(cid:104) x (cid:105)| ≥ K + d . (cid:3) Proof of Theorem 4.6. Let x ∈ X rec with finite class (cid:104) x (cid:105) . First, we assume that x seesno magic word i.o. Since x ∈ X rec , no magic word can appear in x . By Lemma 4.4, | (cid:104) x (cid:105) | ≥ δ ∼ ( x ) > δ rec ( ∼ ).Conversely, we assume by contradiction that x sees i.o. some magic word W and that |(cid:104) x (cid:105)| (cid:54) = δ rec ( ∼ ). By Lemma 4.4, this implies that |(cid:104) x (cid:105)| ≥ δ rec ( ∼ ) + 1. We decompose x asin eq. (4.15) (remark that the magic word W can occur inside the fillers u k ). For k largeenough: |{ y [ n − k + I,n k −| W | + I ] : y ∈ (cid:104) x (cid:105)}| ≥ δ rec ( ∼ ) + 1 . EGREES OF BOWEN FACTORS 17 Thus one can find y, y (cid:48) ∈ (cid:104) x (cid:105) such that y n − k + I = y (cid:48) n − k + I but y [ n − k + I,n k −| W | + I ] (cid:54) = y (cid:48) [ n − k + I,n k −| W | + I ] . Hence, writing a i for y n − k + I = y (cid:48) n − k + I , |T i ∗ ( W u − k W . . . W u k W ) | ≥ u − k W . . . W u k is a special word. This contradicts the following claim and thereforeproves the theorem. (cid:3) Claim 4.17. No special word occurs in x .Proof of the claim. Assume by contradiction that there is a special word u ∗ such that W u ∗ W occurs in x . Since x is recurrent, W u ∗ W occurs infinitely often. Select a decomposition asin eq. (4.15) such that W u k W = W u ∗ W for infinitely many integers k . By Lemma 4.14, (cid:104) x (cid:105) must be infinite, a contradiction. (cid:3) Injective codings We use the previous constructions and results to build injective codings on larger andlarger sets. We will first see that a Bowen quotient (Def. 3.1) may produce a coding witha large injectivity set. We will then see how to repeat this construction to capture all theimage through suitable recodings.For convenience, we recall some definitions. An excellent semiconjugacy (Def. 3.2) is aBorel, finite-to-one semiconjugacy which admits a locally finite Bowen relation. Sometimeswe will abuse notation denoting the Bowen relation by the corresponding semiconjugacy(even though the semiconjugacy does not determine the Bowen relation).The degree spectrum of a semiconjugacy π : X → Y is (Def. 3.4):∆( π ) := { n ≥ { y ∈ Y : | π − ( y ) | = n } is not a null set } . A Bowen quotient and its injectivity set. We analyze the Bowen quotient con-struction using the magic word theory from the previous section. Let X magic N := { x ∈ X N : ∃ w ∈ L ( X rec N ) such that w is a magic word for π N and x sees w i.o. } . In this subsection, we say that a function C : A × A → N is a multiplicity bound for π : X → Y if for all x ∈ X , | π − ( π ( x )) | ≤ C ( a, b ) for every ( a, b ) ∈ A × A such that x − n = a , resp. x n = b , for infinitely many positive integers n .We have the following. Lemma 5.1. Let X be a Markov shift and let π : X → Y be an excellent semiconjugacy forsome Bowen relation ∼ . Let N belong to its degree spectrum ∆( π ) and let ( π N : X N → Y, N ∼ ) be the Bowen quotient of ( π, ∼ ) with order N . The following holds:(1) ( π N , N ∼ ) is excellent and has degree δ rec ( π N ) = min ∆( π N ) = 1 . Moreover, if π admitsa multiplicity bound, so does π N ;(2) if X is locally compact, so is X N ;(3) π N = π ◦ q N with q N : X N → X a -Lipschitz map such that q N ( X N ) ⊂ X ;(4) q N : X N → X is proper, i.e., q − N ( K ) ∩ X N is compact for any compact K ⊂ X ;(5) π N ( X N ) ⊂ { y ∈ π ( X ) : | π − ( y ) | ≥ N } and the difference is a null set; (6) ∆( π N ) = { (cid:0) rN (cid:1) : r ∈ ∆( π ) , r ≥ N } and, for r ∈ ∆( π ) with r ≥ N , the set (cid:8) y ∈ Y : | π − N ( y ) | = (cid:0) rN (cid:1)(cid:9) :(i) is included in { y ∈ Y : | π − ( y ) | ≥ r } ;(ii) is equal to { y ∈ Y : | π − ( y ) | = r } up to a null set;(7) X magic N ⊂ { x ∈ X N : | π N − ( π N ( x )) | = 1 } and the difference is a null set.Proof. This is Theorem 3.3 and Corollary 3.5 except for the following points.The computation of the degree δ rec ( π N ) = min ∆( π N ) = (cid:0) NN (cid:1) = 1 follows from Corol-lary 4.8. If there is a multiplicity bound C for π , the following gives a multiplicity boundfor π N : | π − N ( y ) | ≤ (cid:18) | π − ( y ) | N (cid:19) ≤ C N ( A, B ) := sup ( a,b ) ∈ A × B C ( a, b ) N /N ! , completing the proof of item (1).To prove item (7), note first that the inclusion follows from Lemma 4.3 and that, byTheorem 4.6, the difference is included in X N \ X rec N which is a null set. (cid:3) We deduce the following theorems for use in [7]. Theorem 5.2. Let ( X, S ) be a Markov shift, ( Y, T ) be a dynamical system, and let π : X → Y be an excellent semiconjugacy. Let µ ∈ Prob erg ( T ) with µ ( π ( X )) = 1 . Thenthere exist a Markov shift ˆ X and a semiconjugacy ˆ π : ˆ X → Y such that:(1) ˆ π : ˆ X → Y is an excellent semiconjugacy. Moreover, if π admits a multiplicitybound, so does ˆ π ;(2) if X is locally compact, so is ˆ X ;(3) ˆ π = π ◦ q | ˆ X where q : ˆ X → X is a -Lipschitz map with q ( ˆ X ) ⊂ X ;(4) q : ˆ X → X is proper, i.e., q − ( K ) ∩ ˆ X is compact for any compact K ⊂ X ;(5) ˆ π ( ˆ X ) ⊂ π ( X ) and for µ -a.e. y ∈ Y , | ˆ π − ( y ) | = 1 ;(6) there is an invariant measure ˆ µ on ˆ X such that ˆ π : ( S, ˆ µ ) → ( T, µ ) is an isomorphism;(7) X is irreducible.Proof. Observe that y (cid:55)→ | π − ( y ) | is a T -invariant function. By ergodicity, it has a µ -a.e.constant and positive value we denote N . Obviously N ∈ ∆( π ). Let ( π N : X N → Y, N ∼ ) bethe Bowen quotient of ( π, ∼ ) of order N as in Lemma 5.1. Thus π N satisfies all the claimsabove except possibly for items (5)-(7).Item (6)(ii) for r = N of the lemma implies that | π − N ( y ) | = 1 for µ -a.e. y ∈ Y . Therefore,there is a unique ˆ µ ∈ Prob( S | X N ) such that π N : (ˆ µ, S N ) → ( µ, T ) is an isomorphism. Since q N ( X N ) ⊂ X , we have π N ( X N ) ⊂ π ( X ). These remarks yield items (5) and (6). As ¯ µ isergodic, it is carried by an irreducible component ˆ X of X N . It is now clear that q := q N | ˆ X and ˆ π := π ◦ q have all the claimed properties. (cid:3) Theorem 5.3. Let X be a Markov shift and let π : X → Y be an excellent semiconjugacy.Then there exist another Markov shift ˆ X and a semiconjugacy ˆ π : ˆ X → Y such thatproperties (1)-(4) in Theorem 5.2 hold and, moreover:(5’) ˆ π ( ˆ X ) ⊂ π ( X ) and the difference is a null set; EGREES OF BOWEN FACTORS 19 (6’) there is a word ˆ W ∈ L ( ˆ X rec ) s.t. for any x ∈ ˆ X that sees i.o. ˆ W , ˆ π − (ˆ π ( x )) = { x } ;(7’) if X is irreducible, then so is ˆ X . Remark 5.4. As noted in the proof below, our argument gives a stronger result than statedin item (5’). If X is irreducible, we obtain ˆ π ( ˆ X ) = π ( X ) . In the general case, thespectral decomposition of X into irreducible components X i , i ∈ I , shows that: (cid:83) i ∈ I π ( X i ) ⊂ ˆ π ( ˆ X ) ⊂ π ( X ) . Proof of Theorem 5.3. Let ˆ π : ˆ X → Y with ˆ ∼ be the Bowen quotient of order N =min ∆( π ). Lemma 5.1 yields items (1)-(4) and (5’) and δ rec (ˆ π ) = 1. Theorem 4.6 impliesitem (6’) for any magic word ˆ W for ˆ ∼ .We now assume that X is irreducible and prove that item (7’) can be satisfied whilekeeping the other properties. We first observe that items (1)-(4) and the inclusion in item(5) are obviously preserved by restriction to any irreducible component. We are going toselect an irreducible component for which the second half of item (5’) and item (6’) aresatisfied.During this proof, we will say that a sequence ˆ x ∈ ˆ X is related to some X -word w , ifthere are infinitely many p ≥ p ≤ ∀ ≤ i < | w | w i ∈ ˆ x p + i . Observation. For any periodic x ∈ X , | π − ( π ( x )) | ≥ N := min ∆( π ) and therefore byFact 3.12, there exists ˆ x ∈ ˆ X with x n ∈ ˆ x n for all n ∈ Z . Claim 5.5. There is an irreducible component ˆ Z of ˆ X that contains any ˆ x ∈ X related tosome magic word for ∼ over X rec . Moreover, δ rec (ˆ π | ˆ Z ) = 1 .Proof of the claim. Let ( w, i ) be a magic couple for π | X over X rec . Define the set A w := { v i : v ∈ L ( X ) s.t. v ∼ w } with cardinality | A w | = δ rec ( π ). Note that since w ∈ L ( X rec ),there is a periodic point x ∈ X that sees w . By the observation, this implies the existenceof ˆ x ∈ ˆ X related to w .Now let ˆ x ∈ ˆ X be related to w . By Fact 3.10, for all n ∈ Z ˆ x n = { z an : a ∈ ˆ x } wherefor each a ∈ ˆ x , z a = Q (ˆ x, x ). Hence, for all a ∈ ˆ x , z ap . . . z ap + | w |− ∼ w and ˆ x p + i ⊂ A w .Since these sets have equal cardinalities, we have: ˆ x p + i = A w . Therefore, ˆ x belongs to theirreducible component ˆ Z w of ˆ X containing the symbol A w .If v is another magic word, there is a periodic orbit x ∈ X that sees i.o. v and also seesi.o. w ( X is transitive). The observation yields some ˆ x ∈ ˆ X which is related to both v and w so ˆ Z v = ˆ Z w . Thus there is an irreducible component ˆ Z that contains all ˆ x ∈ ˆ X relatedto any magic word for π .To show that δ rec (ˆ π | ˆ Z ) = δ rec (ˆ π ) = 1, it suffices to find a magic word for ˆ π in L ( ˆ Z rec ).Let w be a magic word for π . Given a periodic x ∈ X with w occuring at index 0, theobservation yields a periodic, hence word recurrent ˆ x ∈ ˆ X with w n ∈ ˆ x n for all 0 ≤ n < | w | .Let ˆ w := ˆ x . . . ˆ x | w |− . Obviously ˆ w ∈ L ( ˆ Z rec ). We check that ˆ w is a magic word for ˆ π .Let ˆ v ∈ L ( ˆ X rec ) such that ˆ v ˆ ∼ ˆ w . By Fact 3.10, ˆ v n = { v an : a ∈ ˆ v } where v a ∈ L ( X )for all 0 ≤ n < | ˆ w | . In particular v an ∼ w n for all 0 ≤ n < | w | . As above, it follows thatˆ v i = { v ai : a ∈ ˆ v } = A w . This implies that δ rec (ˆ π | ˆ Z ) = δ ˆ ∼ ( ˆ w, i ) = 1. (cid:3) Let x ∈ X . We are going to show that x ∈ ˆ π ( ˆ Z ) by finding y ∈ X Bowen equivalentto x and which can be approximated by q (ˆ x n ) with periodic ˆ x n ∈ ˆ Z . Fix a magic word w for π | X . There are symbols a, b of X and integers m k , n k ≥ k such that x − m k = a and x n k = b for all k ≥ 1. There is an X -word u . . . u (cid:96) +1 , (cid:96) ≥ 1, with u = b and u (cid:96) +1 = a andcontaining w as a subword (since X is irreducible). For each k ≥ 1, let x k ∈ X be theperiodic sequence with period τ k := n k + m k + (cid:96) + 1 defined by: ∀ i = − m k , . . . , n k + (cid:96) x ki = (cid:26) x i if − m k ≤ i ≤ n k ,u i − n k +1 if n k ≤ i < n k + (cid:96). Note that for all i ∈ Z , A i := { x ki : k ≥ } ⊂ { x i : k ≤ | i |} is finite. The local finitenessof the Bowen relation implies that, for all i ∈ Z , the set of symbols B i := { s : ∃ t ∈ A i s.t. s ∼ t } is finite.Since x k ∈ X is periodic, the observation gives ˆ x k ∈ ˆ X such that x ki ∈ ˆ x ki for all i ∈ Z .In particular, ˆ x k is related to w so it belongs to ˆ Z by the claim. Note that ˆ x ki ⊂ { s : s ∼ x ki } hence ˆ x ki ⊂ B i for all i ∈ Z and k ≥ 1. Thus there is a point of accumulation ˆ x = lim n ˆ x k ( n ) ∈ ˆ Z for some sequence k ( n ) ↑ ∞ . If x i = a (resp. b ), then, for all large k , ˆ x ki ⊂ { c : c ∼ a (resp. c ∼ b ) } which is finite and independent of i ∈ Z , hence ˆ x ∈ ˆ Z .Let y k := q (ˆ x k ) for k ≥ 1. As q is continuous, y k ( n ) = q (ˆ x k ( n ) ) converges to the sequence y := q (ˆ x ). Since ˆ x ∈ ˆ X , we have y ∈ X . For all i ∈ Z , y ki ∈ ˆ x ki by construction, hence y ki ∼ x ki . Recalling that x ki = x i for all k ≥ | i | , we get y ≈ x . By the Bowen property: π ( x ) = π ( y ) = π ( q (ˆ x )) = ˆ π (ˆ x ) . Thus π ( X ) ⊂ ˆ π ( ˆ Z ) = π ( q ( ˆ Z )) ⊂ π ( X ), so ˆ π ( ˆ Z ) = π ( X ), yielding item (5’). Sincedeg rec ( π | ˆ Z ) = 1, Theorem 4.6 yields item (6’). (cid:3) Remark 5.6. The periodic approximation argument in the last part of the proof of Theo-rem 5.3 is partly inspired by some geometric construction of [7] . Preparations. We turn to the proof of the Main Theorem. Let π : X → Y bean excellent semiconjugacy for some Bowen relation ∼ . We are going to build an injectivecoding of the image π ( X ). We start with the following simple fact about partially orderedsets. In this paper N is the set of nonnegative integers. Fact 5.7. Let ( O , (cid:22) ) be a countable (possibly finite) set together with a partial order (cid:22) .There is a bijection σ : { n ∈ N : n < |O|} → O which is nondecreasing, i.e., (5.8) ∀ i, j : σ ( i ) (cid:22) σ ( j ) = ⇒ i ≤ j if and only if all initial segments { b ∈ O : b (cid:22) a } , a ∈ O , are finite.Proof. If σ : N → O is a bijection satisfying eq. (5.8), then any initial segment { b ∈ O : b (cid:22) σ ( i ) } is finite as a subset of σ ( { , , . . . , i } ). We now assume that all initial segmentsare finite and proceed to build the bijection σ .If O is finite, then one can define σ : { , . . . , n − } → O inductively by choosing, for each0 ≤ k < n , σ ( k ) to be some minimal element α among O \ σ ( { , . . . , k − } ), i.e., such that: ∀ β ∈ O \ σ ( { , . . . , k − } ) β (cid:22) α = ⇒ β = α. EGREES OF BOWEN FACTORS 21 We assume now that O is infinite so there is a bijection s : N → O . We define integers N < N < . . . and σ |{ , . . . , N n − } inductively by setting N = 0 and, for each n ≥ { b ∈ O : b (cid:22) s ( n ) } \ σ ( { , . . . , N n − } ) as { b n, , . . . , b n,(cid:96) n } where i (cid:55)→ b n,i is injective and non-decreasing;(2) set N n +1 := N n + (cid:96) n and σ ( N n + i ) = b i for i = 0 , . . . , (cid:96) n − σ is a nondecreasing bijection. (cid:3) We will apply the following elementary construction to an enumeration of the magic wordsfor the relation ∼ in X over X rec . Lemma 5.9. Let X be a Markov shift. Let W := ( W j ) ≤ j Since the set of subwords of a given word is finite, Fact 5.7 allows us to assume (maybeafter a permutation) that:(5.10) W i subword of W j = ⇒ i ≤ j For 1 ≤ j < J , consider the following subset of X W : X j := { x ∈ X : x sees i.o. W j , none of W , . . . , W j − occurs in x } . The injective code we are going to build will have image (cid:83) ≤ j 1. Now the loop graph G is defined by taking as vertices the couples ( v, (cid:96) )where v is a first return loop at W i and 0 ≤ (cid:96) < | v | , and as arrows: ( v, (cid:96) ) → ( v, (cid:96) + 1) if v is a first return loop and 1 ≤ (cid:96) + 1 < | v | ( v, | v | − → ( w, 0) if v, w are first return loops . The corresponding shift is mapped into X by the one-block code ( v, (cid:96) ) (cid:55)→ v (cid:96) (i.e., the firstsymbol of the word v (cid:96) ). Contrary to usual practice, we do not identify all ( v, 0) vertices with a single distinguished vertex. We define G by keeping from G only the vertices ( v, (cid:96) ) where v is a first return loop( y , . . . , y k − ) that is good , i.e., whose extensions :( y , . . . , y k − , W i )map to an X -word of length k + | W i | that does not contain any of the words W , . . . , W i − .Let S i be the Markov shift defined by this loop graph G and define p i : S i → X i to bethe restriction of the previous map. Claim 5.11. The map p i : S i → X i is a topological conjugacy defined by a one-block code.Proof of the claim. It is obvious that p i is a one-block code. We have to check that it definesa bijection and that its inverse is continuous.Consider some x ∈ X i . It can be lifted to a concatenation of first return loops since x sees i.o. W i . These first return loops must be good since x avoids W , . . . , W i − . Thus x belongs to the image of G . Conversely, let x ∈ X be the image of some ˆ x on G , i.e., aninfinite concatenation of good first return loops. Assume by contradiction that some W j , j < i occurs in x . By (5.10), this occurrence may overlap but cannot contain any occurrenceof W i . Thus W j occurs in the image of some extended first return loop, so the first returnloop is not good. This contradicts the definition of G , proving that p i ( S i ) = X i .Note that p i is invertible with inverse defined by: ∀ x ∈ X i p i (ˆ x ) = x = ⇒ ˆ x = (( x [ j,j + N − ) − n ≤ j The proof of the above lemma does not provide a locally compact Markovshift S , even if X is compact. Proof of the Main Theorem. Let π : X → Y be an excellent semiconjugacy witha Bowen relation ∼ . We are going to divide the image π ( X ) according to the number ofpreimages and then successively reduce each of these numbers to one (ignoring null sets).We assume that π ( X ) is not null as otherwise there is nothing to show.Let (∆( i )) ≤ i
For any ≤ i < I , there are a Markov shift S i and a one-block code p i : S i → p i ( S i ) such that: p i ( S i ) ⊂ (cid:101) Z i with the difference a null set, π i ◦ p i ( S i ) = Y i up to a null set,and π i ◦ p i is injective.Proof of the claim. Lemma 5.9 provides an injective one-block code p i of some Markov shift S i into ˜ Z i with p i ( S i ) ⊂ (cid:101) Z i with the difference a null set.By item (d) above and Lemma 2.1, π i ◦ p i ( S i ) = Y i up to a null set. By item (c), π i ◦ p i isinjective. (cid:3) To conclude, let S be the disjoint union (cid:70) ≤ i For any ≤ j < i < I , π i ( Z i ) ∩ π j ( (cid:101) Z j ) = ∅ . In particular, the images π i ( (cid:101) Z i ) , ≤ i < I , are pairwise disjoint. To prove this claim, note that y ∈ π i ( Z i ) implies that | π − i − ( y ) ∩ Z i − | ≥ ∆ i − ( i ) > | π − j ( y ) ∩ Z j | ≥ ∆ j ( i ) > 1. However, y ∈ π j ( (cid:101) Z j ) implies | π − j ( y ) ∩ Z i − | =1. Hence π i ( Z i ) ∩ π j ( (cid:101) Z j ) = ∅ as claimed. The last assertion follows from (cid:101) Z i ⊂ Z i . Remark 5.15. The Bowen quotient is used for two seemingly distinct purposes: first, toremove points whose images have already been taken care of; second, to lower the minimaldegree to . Applications to surface diffeomorphisms We prove Theorem 1.1 and a more precise version of Theorem 1.2.Let f be a C α -diffeomorphism, α > 0, of a smooth closed surface M with h top ( f ) > χ > χ < χ (arbitrarily close to χ , see below). Recall that a measure is χ -hyperbolicif it has one positive exponent larger than χ and one negative exponent less than − χ .As observed in [4, Sec. 8], Sarig [24] provides a Markov shift ˆΣ and a H¨older-continuoussemiconjugacy ˆ π : ˆΣ → M such that, ˆΣ denoting its regular part:(P1) ˆ π | ˆΣ admits a Bowen relation (called affiliation in [24, Sec. 12.3]) which is locallyfinite (see [4, Summary 8.1(4)(5)]);(P2) ˆ π | ˆΣ is finite-to-one (as explained in [20] the claim that ˆ π is finite-to-one on Σ itselfwas made erroneously in [24]);(P3) µ (ˆ π ( ˆΣ )) = 1 for any χ -hyperbolic measure µ ∈ Prob erg ( f ).(P4) any ergodic ν on ˆΣ, ˆ π ∗ ( ν ) is χ/ G whose vertices Ψ p s ,p u are double charts , that is, local charts Ψ : ( − r, r ) → M centered at some point x ∈ M together with two numbers p s , p u > r = min( p s , p u ).The charts Ψ are defined by Pesin theory as exp x ◦ C χ ( x ) where exp x is the exponential mapcentered at x and C χ ( x ) is the Oseledets-Pesin reduction matrix. These charts make “thehyperbolicity of f uniform”: for any arrow Ψ p s ,p u → Φ q s ,q u in G , the map Φ − ◦ f ◦ Ψ is closeto a linear map ( x , x ) (cid:55)→ ( λx , κx ) with λ > e χ and κ < e − χ .Each Markov shift has its cylinders. For Σ, they are: Z − n (Ψ n , . . . , Ψ n ) := π { x ∈ Σ : ∀| k | ≤ n x k = Ψ k } ⊂ M where each Ψ k is a double chartwhile those in ˆΣ are: − n [ R − n , . . . , R n ] := ˆ π { x ∈ ˆΣ : ∀| k | ≤ n x k = R k } ⊂ M where each R k is a rectangle . In this way, two H¨older-continuous semiconjugacies π : Σ → M and ˆ π : ˆΣ → M are definedby some shadowing properties. (Contrarily to ˆ π | ˆΣ , the map π | Σ is not finite-to-one.)To prove the last claim of Theorem 1.1, we use a strengthening of property (P4) above: wecan replace χ/ χ , at least in the case of periodic orbits. We freelyuse the terminology and notations from [24], including the two semiconjugacies ˆ π : ˆΣ → M and π : Σ → M . Lemma 6.1. Given ˜ χ < χ , there is a coding ˆ π : ˆΣ → M with (P1)-(P4) as above thatadditionally satisfies the following property: for any periodic sequence ˆ x ∈ ˆΣ , ˆ π (ˆ x ) is ˜ χ -hyperbolic.Proof. Let x := ˆ π (ˆ x ). Lemma 12.2 from [24] yields a sequence of double charts (Ψ n ) n ∈ Z ∈ Σsuch that, for all n ≥ − n [ˆ x − n . . . ˆ x n ] ⊂ Z − n (Ψ − n , . . . , Ψ n ). It follows from Proposition 4.11in [24] that all points in π ([Ψ ] ∩ Σ ) can be written Ψ ( t ) with t ∈ R close to 0 ∈ R . Hence EGREES OF BOWEN FACTORS 25 ˆ π (ˆ x ) lift to t in the domain of the chart Ψ : that is, for each n ∈ Z , t n := Ψ − n ◦ f n (ˆ π (ˆ x )) iswell-defined. Letting f k := Ψ − k ◦ f ◦ Ψ k − , we have t k = f k ( t k − ) and: Df n ( x ) = D Ψ n ◦ Df n ◦ · · · ◦ Df ◦ D Ψ − ( x ) . Thus (cid:107) Df n ( x ) (cid:107) ≥ (cid:107) D Ψ n ( t n − ) − (cid:107) − · (cid:107) D Ψ ( t ) (cid:107) − · (cid:107) Df n ◦ . . . Df (cid:107) By Proposition 3.4 of [24], choosing the parameter (cid:15) > (cid:107) Df n ◦ · · · ◦ Df (cid:107) ≥ ( e χ − (cid:15) ) n ≥ e n ˜ χ . Since Ψ k = exp x k ◦ C χ ( x k ) where x k is the center of Ψ k [24, eq. (2.2)], we have: (cid:107) D Ψ − n (cid:107) ≤ C · (cid:107) C χ ( x n ) − (cid:107) for some constant C (depending only on f ). Since ˆ x is periodic, [24,Theorem 10.2] shows that Ψ n takes only finitely many values as n ranges over Z . It followsthat setting C ( x ) := inf n ≥ Lip(Ψ − n ) − . Lip(Ψ ) − > 0, we get: ∀ n ≥ (cid:107) Df n ( x ) (cid:107) ≥ C ( x ) e n ˜ χ . Hence the periodic orbit O ( x ) has a positive exponent larger than or equal to ˜ χ . A symmetricargument shows that O ( x ) is ˜ χ -hyperbolic. (cid:3) Proof of Theorem 1.1. We consider Sarig’s coding with the addition of the property fromLemma 6.1. The previous discussion shows that our Main Theorem applies. It produces anew coding of the form ˆ π ◦ q , with q H¨older-continuous, and whose image can be smaller,but only by a null set. The new coding is easily seen to satisfy our claims. (cid:3) We turn to the counting of hyperbolic periodic orbits. This requires the following estimatein eq. (6.3). It is folklore, but since we did not find a reference we will deduce it from [18,chap. 7], using freely its terminology and notations. A measure maximizing the entropy (or: m.m.e.) of some Borel automorphism is aninvariant Borel probability measure which realizes the supremum of the Kolmogorov-Sinaientropy over all invariant probability measures. Lemma 6.2. If ( X, σ ) is an irreducible Markov which is positively recurrent (i.e., it hassome m.m.e. and its entropy is finite) with period p , then: (6.3) lim n → ∞ p | n e − nh top ( f ) · |{ x ∈ X : |{ σ k x : k ∈ Z }| = n }| ≥ p. Proof. We freely use results and notations from Kitchens’ book [18] and in particular thegenerating functions L ab ( z ) and R ab ( z ). First suppose that X is mixing (i.e., p = 1). Fixsome symbol a ∈ A occuring in X and set λ := e h top ( f ) . Since X is recurrent, Theorem7.1.18 implies: lim n →∞ λ − n |{ x ∈ X : x = a, σ n x = x }| = 1 µ ( a )where µ ( a ) := (1 /λ ) L (cid:48) aa (1 /λ ). Since X is positive recurrent, Lemma 7.1.21 yields µ ( a ) = (cid:96) ( a ) · r ( b ) where (cid:96) ( a ) := ( L aj (1 /λ )) j ∈A and r ( b ) := ( R jb (1 /λ )) j ∈A . We also have L aa (1 /λ ) = We note that a similar estimate was obtained, e.g., in [6] but with a stronger assumption (the SPRproperty) and stronger conclusion (an error estimate). R aa (1 /λ ) = 1 for all a ∈ A as X is recurrent (see the proof of Lemma 7.1.8, recalling that,by definition, T aa (1 /λ ) = ∞ if and only if X is recurrent) . Thus,1 µ ( a ) = 1 (cid:80) j ∈A L aj (1 /λ ) R ja (1 /λ ) = L aa (1 /λ ) R aa (1 /λ ) (cid:80) j ∈A L aj (1 /λ ) R ja (1 /λ ) . Now Lemma 7.2.15, implies that ν ([ b ]) = (cid:96) ( a ) b r ( a ) b (cid:80) j (cid:96) ( a ) j r ( a ) j for any a, b ∈ A . Thus,lim n →∞ λ − n |{ x ∈ X : x = a, σ n x = x }| = ν ([ a ]) . For p > 1, the cyclic decomposition from [18, p. 223] yields:lim n →∞ λ − n |{ x ∈ X : x = a, σ n x = x }| = pν ([ a ]) . Using the decomposition: { x ∈ X : x = a, σ n x = x } = (cid:71) k | n { x ∈ X : x = a, |{ σ j ( x ) : j ∈ Z }| = k } and noting that k | n implies k = n or k ≤ n/ 2, we get:lim n →∞ λ − n |{ x ∈ X : x = a, |{ σ j ( x ) : j ∈ Z }| = n }| = pν ([ a ]) . Since ν ( X ) = 1, a routine argument shows eq. (6.3). (cid:3) We are going to obtain the following relation between periodic points and measures max-imizing the entropy: Theorem 6.4. Let f ∈ Diff α ( M ) where M is a closed surface and α > . Assumethat there are distinct ergodic measures maximizing the entropy: µ , . . . , µ r with periods p , . . . , p r ≥ . Fix ˜ χ < h top ( f ) . Then (6.5) lim inf n → ∞ p , . . . , p r | n e − nh top ( f ) · | per ˜ χ ( f, n ) | ≥ p + . . . p r . When f is C ∞ smooth, Newhouse’s Theorem [23] shows that there is at least one m.m.e.If, additionally, f is topologically mixing, [7] shows that there is a m.m.e. with period equalto 1. Therefore: Corollary 6.6. In the setting of the above theorem, assuming additionally that f is C ∞ weobtain: • for some integer p ≥ , lim inf n →∞ ,p | n e − nh top ( f ) · | per ˜ χ ( f, n ) | ≥ p ; • if f is topologically mixing, lim inf n →∞ e − nh top ( f ) · | per ˜ χ ( f, n ) | ≥ . This implies Theorem 1.2. Proof of Theorem 6.4. We fix ˜ χ < h top ( f ) and consider a coding ˆ π : ˆΣ → M as in Lemma 6.1.If µ , . . . , µ r ∈ Prob erg ( f ) are distinct m.m.e.’s, Sarig’s [24] shows that each µ i is isomorphicto the product of a Bernoulli scheme and a circular permutation of some order p i . Being anm.m.e. is invariant under Borel conjugacy, hence ˆΣ carries distinct m.m.e.’s ν , . . . , ν r with π ∗ ( ν i ) = µ i . EGREES OF BOWEN FACTORS 27 By general results about Markov shifts and their m.m.e.’s [13], ˆΣ contains disjoint irre-ducible components X , . . . , X r , where each X i carries a distinct m.m.e. ν i . In particular,each X i is positive recurrent and has period equal to p i .By Lemma 6.1, the following map is well-defined and injective: π : { x ∈ ˆΣ : |{ σ j ( x ) : j ∈ Z }| = n } → per ˜ χ ( f, n ) . The claim (6.5) now follows from Lemma 6.2. (cid:3) An obstruction to H¨older-continuous coding We prove Theorem 1.6 characterizing surface diffeomorphisms with H¨older-continuoussymbolic dynamics. Recall that a map π : X → M is H¨older-continuous with some positiveexponent α if there is a constant C < ∞ such that, for all x, y ∈ X , d ( π ( x ) , π ( y )) ≤ C exp ( − α inf {| n | : x n (cid:54) = y n } ) . Let f be a diffeomorphism of a compact d -dimensional manifold M and let µ ∈ Prob erg ( f ).Write its Lyapunov exponents as λ ( f, µ ) > · · · > λ u ( f, µ ) > ≥ λ u +1 ( f ) > · · · > λ r ( f, µ ).This measure has saddle type if λ u +1 < < u < d . LetProb hyp ( f ) := { µ ∈ Prob erg ( f ) : µ is aperiodic and of saddle type } . Recall that for µ ∈ Prob hyp ( f ), χ ( µ ) := min( λ u ( f, µ ) , − λ u +1 ( f, µ )) . Sarig’s theorem [24]and its higher dimensional generalization by Benovadia [2] yield a global coding if χ ( f ) :=inf { χ ( µ ) : µ ∈ Prob erg ( f ) } is positive. We prove: Proposition 7.1. Let f ∈ Diff ( M ) with M a closed manifold. Let ( S, X ) be a Markovshift and let π : ( S, X ) → ( f, M ) be a semiconjugacy. Assume that π is H¨older-continuouswith exponent α > . Given any ν ∈ Prob erg (Σ) , if µ := π ∗ ( ν ) is hyperbolic and atomless,then: (cid:98) χ ( µ ) := min( λ ( f, µ ) , − λ d ( f, µ )) ≥ α. This shows that inf { (cid:98) χ ( f, µ ) : µ ∈ Prob hyp ( f ) } ≥ α is a necessary condition for the exis-tence of a H¨older-continuous coding with exponent α . Since (cid:98) χ ( f, µ ) = χ ( f, µ ) in dimension2, Theorem 1.6 is established. Proof. Let ν ∈ Prob erg (Σ) such that π ∗ ( ν ) ∈ Prob hyp ( f ). For µ -a.e. x ∈ Σ, the Pesin stablemanifold of y := π ( x ) W s ( y ) := { z ∈ M : lim n →∞ n log d ( f n ( z ) , f n ( y )) < } satisfies:(7.2) W s ( y ) = { z ∈ M : z = y or lim n →∞ n log d ( f n ( z ) , f n ( y )) ∈ [ λ r ( f, µ ) , } . Since ν is not carried by a periodic orbit, it is carried by a nontrivial irreducible componentof the Markov shift. Hence there is z ∈ Σ such that z (cid:54) = x and z n = y n for all n ≥ The period of an irreducible Markov shift is the greatest common divisor of the periods of its periodicpoints. Therefore d ( σ n x, σ n z ) = Ce − n for some C > n ≥ 0. Now the H¨older-continuity of π gives C (cid:48) > ∀ n ≥ d ( f n ( y ) , f n ( π ( z ))) ≤ C (cid:48) e − αn . This exponential convergence implies that π ( z ) ∈ W s ( y ). By eq. (7.2), λ r ( f, π ∗ ( ν )) ≤ − α .By considering ( f − , µ ), we obtain λ ( f, π ∗ ( ν )) ≥ α . Thus (cid:98) χ ( π ∗ ( ν )) ≥ α . (cid:3) Appendix A. Further remarks A.1. Canonical Bowen relation. A semiconjugacy π : X → Y of a symbolic system X can admit several Bowen relations. However, one can define its canonical relation over itsalphabet A by: ∀ a, b ∈ A a π ∼ b def ⇐⇒ π ([ a ] X ) ∩ π ([ b ] X ) (cid:54) = ∅ (recall that [ · ] Z denotesthe cylinder in Z defined by some word). The above relation is obviously reflexive andsymmetric. We denote by π ≈ the induced Bowen equivalence on X . Lemma A.1. For an arbitrary semiconjugacy π : X → Y , the following implication holds: (A.2) ∀ x, y ∈ X π ( x ) = π ( y ) = ⇒ x π ≈ y. If the semiconjugacy π is Bowen, then the canonical relation is a Bowen relation and it isthe minimal one: if ∼ is any Bowen relation for π , then a π ∼ b = ⇒ a ∼ b for any a, b ∈ A .Proof. The implication (A.2) is immediate. Now assume that π has some Bowen relation ∼ and let a, b ∈ A with a π ∼ b : there are x ∈ [ a ] X and y ∈ [ b ] X with π ( x ) = π ( y ). The Bowenproperty for ∼ gives a ∼ b so we have proved a π ∼ b = ⇒ a ∼ b . Now it is obvious that π ∼ isa Bowen relation for π . (cid:3) Remark A.3. We do not know whether the reflexive and symmetric relation that appearsin Sarig’s construction (called affiliation) is canonical. Additionally, we do not know if theBowen quotient (Theorem 3.3) of a canonical relation is itself canonical. A.2. Consequences for continuous extensions. In our most important examples, thesemiconjugacy is continuous over the Markov shift X but the Bowen property is only knownfor the regular part X . It is then natural to consider π | X . It is easy to see that X isa Markov shift: setting A := { x : x ∈ X } ⊂ A , X = X ∩ ( A ) Z . As π is continuous, π | X is determined by its regular part but the Bowen property may fail to extend to X .Denote by π ∼ the canonical relation induced by π | X and by π ≈ the corresponding relationon X . Lemma A.4. Let π : X → Y be a continuous semiconjugacy with X a Markov shift. If therestriction of π to X has the Bowen property, then: (A.5) ∀ x, y ∈ X x π ≈ y = ⇒ π ( x ) = π ( y ) . Proof. Let x, y ∈ X with x π ≈ y and n ≥ 1. As x − n π ∼ y − n , there are x − n ∈ σ n [ x − n ] X , y − n ∈ σ n [ y − n ] X with π ( x − n ) = π ( y − n ). By the Bowen property, this implies x − n π ≈ y − n . EGREES OF BOWEN FACTORS 29 Likewise, there are x n ∈ σ − n [ x n ] X , y n ∈ σ − n [ y n ] X with x n π ≈ y n . Define ˜ x n ∈ X by:˜ x nk = x − nk for k ≤ − nx k for | k | ≤ nx nk for k ≥ n. Define ˜ y n similarly. Observe that ˜ x n , ˜ y n both belong to X and ˜ x n π ≈ ˜ y n so that π (˜ x n ) = π (˜ y n ). Since π is continuous, π ( x ) = lim n π (˜ x n ) = lim n π (˜ y n ) = π ( y ). (cid:3) Still, the Bowen property may fail to extend to X as in the following example. Example A.6. Consider the graph with set of vertices Z ∪ { α, ω } and arrows n → ( n + 1) , α → n , n → ω , α → α , ω → ω ( for all n ∈ Z ) . Let ( S, X ) be the induced Markov shift. Let π : X → Y ⊂ { , , / , . . . } Z bethe semiconjugacy such that ( the vertical bar is immediately to the left of index :(1) α ∞ , ω ∞ (cid:55)→ ∞ ;(2) α ∞ | n · ( n + 1) · · · ( n + (cid:96) − · ω ∞ (cid:55)→ ∞ | (cid:96) · (cid:96) · · · (cid:96) · ∞ (for all n ∈ Z , (cid:96) ∈ N );(3) α ∞ · n · ( n + 1) · ( n + 2) · · · (cid:55)→ ∞ (for all n ∈ Z );(4) . . . ( n − · ( n − · n ω ∞ (cid:55)→ ∞ (for all n ∈ Z );(5) · · · − · − · · · · · · (cid:55)→ ∞ . π is well-defined and continuous. The regular sequences are those in (1) and (2).It is easy to check that π is Bowen on X for the symmetric relation generated by n ∼ m for all n, m ∈ Z and α ∼ ω . If the semiconjugacy π was Bowen on X , (1) and (5) would imply that all symbols would berelated, contradicting (2). Appendix B. A locally compact recoding Our Main Theorem does not preserve local compactness. In this appendix we provide analternate construction which preserves local compactness at the expense of a slightly weakerinjectivity property: Theorem B.1. Let ( S, X ) be a locally compact Markov shift on some alphabet A . Let X be its regular part. Let π : ( S, X ) → ( T, Y ) be a Borel semiconjugacy such that: - ( T, Y ) is a Borel automorphism; - π is finite-to-one, i.e., π − ( y ) is finite for every y ∈ Y ; - π has the Bowen property with respect to a locally finite relation on A .Then there are a locally compact Markov shift ( ˜ S, ˜ X ) and a -Lipschitz map φ : ˜ X → X such that π ◦ φ : ˜ X → Y defines a semiconjugacy satisfying: - π ◦ φ | ˜ X is injective; - π ◦ φ ( ˜ X ) carries all invariant measures of π ( X ) .Proof. We explain the required changes in proof of the Main Theorem. An inspection ofthe proof of the Main Theorem shows that the local compactness is lost in Lemma 5.9. Itsuffices to replace this lemma with the following statement. (cid:3) Lemma B.2. Let X be a locally compact Markov shift. Let W := ( W j ) ≤ j 0) can have indegree large than 1. However ( v, | v | − → ( w, 0) whenever v | v |− → w = W i in X [ N ] . Since they are infinitely many words v (their length beingunbounded), S i is not locally compact.We define a new graph G as follows. Let: V := { ( v, j, L − , L + ) ∈ L ( X [ N ] ) × N × N : ( v, j ) ∈ V with | v | ≤ min( L − , L + ) } ;( v, j, L − , L + ) → ( w, k, M − , M + ) if and only if ( v, j ) G → ( w, k )and M − := max( | v | , L − − , L + := max( | w | , M + − . Let T i be the Markov shift defined by G and define p | T i as p i ◦ q where q ( v, k, L ) = ( v, k ). Step 1. Local compactness. Let ( v, j, L − , L + ) → ( w, k, M − , M + ) on G . Note that M − = max( | v | , L − − 1) and M + ≤ L + + 1 . It follows that, given ( v, j, L − , L + ) there are finitely many possibilities for ( M − , M + ). Inparticular | w | is bounded. Now w = v or w starts by the fixed X -word W i . Since X islocally compact, this gives finitely many possibilities for ( w, k ). Thus the outdegree of anyvertex in G is finite. Likewise the indegree of any vertex is finite. The local compactness,i.e., item (1), is proved.We define the flat part of S i = Σ( G ) to be: S (cid:91)i := { ( v, j ) ∈ S i : lim n →±∞ | v n | − | n | = −∞} where ( v n , j n ) n ∈ Z = ( v, j ) . Step 2. The flat part has full measure for any invariant probability measure µ on S i . We can restrict to µ ergodic. Now, assume by contradiction that lim n →±∞ | v n | − | n | = −∞ fails for a set D of points ( v, j ) ∈ S i with positive µ -measure. Hence, for any ( v, j ) ∈ D ,there are a constant C > n such that | v n | ≥ | n | − C ≥ | n | / n , if σ k ( v, j ) ∈ E n := { ( w, (cid:96) ) ∈ S i : | w | > n } for k = k for some0 ≤ k < n , then it holds for all k in some positive interval segment of length | v k | ≥ n/ k . Therefore it holds for at least n/ ≤ k < (3 / n . Hence, for anyinteger N , ∀ x ∈ D lim sup n →∞ / n { ≤ k < (3 / n : σ k ( x ) ∈ E N } ≥ / . By the pointwise ergodic theorem, this implies that µ ( E N ) ≥ / N , contradictingthe σ -additivity of µ . Hence µ ( S (cid:91)i ) = 1. EGREES OF BOWEN FACTORS 31 Step 3. There is a canonical lift ι : S (cid:91)i → T i which is well-defined with q ◦ ι = id . Given ( v, j ) ∈ S i , we let: L n − ( v ) := max k ≥ | v n − k | − k and L n + ( v ) := max k ≥ | v n + k | − k. and define the canonical lift as: ι : ( v n , j n ) n ∈ Z (cid:55)−→ ( v n , j n , L n − ( v ) , L n + ( v )) n ∈ Z Let ( v, j ) ∈ S (cid:91)i . We check that ι ( v, j ) is well-defined. First, the numbers L n − ( v ) , L n + ( v ) arewell-defined since, by the definition of S (cid:91)i , | v n + k | − k < k ≥ 0. Note also: L n + ( v ) = max( | v n | , max k ≥ | v n + k | − k ) = max( | v n | , max k ≥ | v n +1+ k | − k − | v n | , L + n +1 ( v ) − L n +1 − ( v ) = max( | v n +1 | , L n − ( v ) − v n , j n , L n − ( x ) , L n + ( v )) n ∈ Z ∈ T i . Thus ι : S (cid:91)i → T i is well-defined.The identity q ◦ ι = id is trivial. Step 4. The map q : T i → S (cid:91)i is well-defined and ι ◦ q | T i = id . To see that q is well-defined, it suffices to check that q ( T i ) ⊂ S (cid:91)i . Let z := ( v, j, L − , L + ) ∈ T i . Thus there is some z ∗ := ( v ∗ , j ∗ , L ∗− , L ∗ + ) ∈ V that appears infinitely many times in z n when n ≥ 0. Let n ≥ m ( n ) be the largest index less than n such that z m ( n ) = z ∗ . Observe that L n + ≤ L ∗ + + ( n − m ). Thus | v n | − n ≤ L ∗ + − m ( m ).Thus lim n →∞ | v n | − n = −∞ as lim n →∞ m ( n ) = + ∞ . The limit when n → −∞ is handledsimilarly using L ∗− , proving that q is well-defined.We turn to the identity ι ◦ q | T i = id. Let x ∈ T i . We must show that it coincides withthe canonical lift ˜ x := ι ( q ( x )). Write ( v n , j n , L n − , L n + ) := x n and (˜ v n , ˜ j n , ˜ L n − , ˜ L n + ) := ˜ x n .Since x ∈ T i , there is a symbol a := ( v, j, M − , M + ) which appears infinitely often inthe past of x . Thus there are arbitrarily large integers N such that x − N = a . It followsthat L K − = | v K | for some − N ≤ K ≤ − N + M − + 1. Indeed, otherwise one would have: L − N + M − +1 − = M − − M − − < 0, a contradiction.By an easy induction, the definition of the arrows in G implies that: ∀ n ∈ Z L n − ≥ max k ≥ | v n − k | − k It follows that the canonical lift is as small as possible in the following sense: ∀ n ∈ Z ˜ L n − ≤ L n − . Since L K − = | v K | , then L K − = ˜ L K − and therefore L k − = ˜ L k − for all k ≥ K and in particular, all k > − N + M − . Since N is arbitrarily large, it follows that L n − = ˜ L n − for all n ∈ Z . A similarreasoning applies to the sequence ( L n + ) n ∈ Z , concluding the proof that x = y and therefore ofthe identity.We note that this identity implies that q is injective. The theorem is proved. (cid:3) Appendix C. Application to Sina¨ı billiards collision maps We prove Theorem 1.5, i.e., the lower bound on the periodic points for the billiard mapsconsidered by Baladi and Demers [1]. We fix such a collision map T B defined by a two-dimensional Sina¨ı billiard satisfying conditions (BD1) and (BD2) quoted in our introduction.Theorem 2.4 of [1] yields a strongly mixing measure µ ∗ ∈ Prob erg ( T B ) such that(C.1) h ( T B , µ ∗ ) = sup { h ( T B , ν ) : ν ∈ Prob( T B ) } = h ∗ where h ∗ is a combinatorial entropy from their eq. (1.1).Let π : (Σ , σ ) → ( M, T B ) be the coding built by [19, Thm. 1.3] for some hyperbolicityparameter Λ < χ < χ ( f, µ ∗ ) := min( λ ( f, µ ∗ ) , − λ ( f, µ ∗ )). As in Sarig’s construction fordiffeomorphisms, Σ is a Markov shift and π is a H¨older-continuous semiconjugacy. Notethat M is a two-dimensional compact manifold with boundary and that, writing M for thedomain where both T B and its inverse are well-defined and differentiable, π (Σ ) ⊂ π (Σ) ⊂ (cid:92) n ∈ Z T − nB ( M ) ⊂ M \ ∂M (the middle inclusion is nontrivial but is proved in [19]).An inspection of the proof in [19] shows that the semiconjugacy on Σ admits a Bowenrelation just as in the smooth case of [24]. Indeed, though this is not stated in [19], it followsfrom the same arguments as in the original case, see Section 6. This Bowen relation is locallyfinite thanks to [19, Prop. 7.1(2)].According to [1, Thm. 2.4], µ ∗ is T -adapted in the sense of [19]. Since it is χ -hyperbolicfrom the choice of χ , [19, Thm. 1.3] implies the existence of ˆ µ ∗ ∈ Prob erg ( σ ) such that π ∗ (ˆ µ ∗ ) = µ ∗ . Since π | Σ is finite-to-one, π ∗ : Prob(Σ , σ ) → Prob( T B ) preserves the entropy.Therefore, eq. 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