J Ér

Ôme Buzzi
email: jerome.buzzi@math.u-psud.fr

MAXIMAL ENTROPY MEASURES FOR PIECEWISE AFFINE SURFACE HOMEOMORPHISMS

Keywords: Dynamical systems, ergodic theory, topological entropy, variational principle, measures with maximal entropy, Markov structure, Markov tower, symbolic dynamics, coding

We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected "good" returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.

Introduction

Introduced by Anosov and Smale in the 1960's, uniform hyperbolicity is at the core of dynamical system theory. The corresponding systems are well-understood since in particular the works of Sinai, Bowen and Ruelle (see, e.g., [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]) and it is now a central challenge to try to extend our understanding beyond these systems [START_REF] Bonatti | Beyond Uniform Hyperbolicity[END_REF]. We propose that robust entropy conditions provide a way to do this for new open sets of dynamical systems, by implying a non-uniform but global hyperbolic structure, especially with respect to measures maximizing entropy (see section 1.1 for definitions).

Such invariant probability measures can be thought as describing the "complexity" of the dynamics. These measures exist as soon as the dynamics is compact and C ∞ [START_REF] Newhouse | Continuity properties of entropy[END_REF] or somewhat hyperbolic [START_REF] Buzzi | Existence of measures with maximal entropy for interval maps[END_REF], though they are known to fail to exist in finite smoothness for interval maps [START_REF] Buzzi | Intrinsic ergodicity of smooth interval maps[END_REF][START_REF] Ruette | Mixing C r maps of the interval without maximal measure[END_REF] and diffeomorphisms of four dimensional tori [START_REF] Misiurewicz | Diffeomorphism without any measure with maximal entropy[END_REF]. Uniqueness problems are usually much more delicate and can be solved only after a global analysis of the dynamics which we propose to do under entropy conditions.

Entropy-expansion is such a condition. It requires the topological entropy (see section 1.1) to be strictly larger than the supremum of the topological entropies of all smooth hypersurfaces. It is robust in the sense that it is open in the C ∞ topology. Entropy-expanding C ∞ maps T : M → M have a finite number of ergodic and invariant probability measures maximizing the entropy. Their periodic points satisfy a multiplicative lower bound:

(1.1) lim inf n→∞,p|n e -nhtop(T ) #{x ∈ M : T n x = x} > 0 for some integer p ≥ 1 (a period) (see [START_REF] Buzzi | On entropy-expanding maps[END_REF][START_REF] Buzzi | Puzzles of quasi-finite type, zeta functions and symbolic dynamics for multi-dimensional maps[END_REF] for precise definitions and statements including other results). Entropy-expansion is satisfied by plane maps of the type (x, y) → (1.9x 2 + ǫy, 1.8y 2 + ǫx) for small ǫ [START_REF] Buzzi | Ergodicité intrinsèque d'applications chaotiques fibrées[END_REF]. On the interval, the condition reduces to nonzero topological entropy. In fact, entropy-expansion can be understood as a generalization of some aspects of one-dimensional dynamics. Indeed, the previous results were first proved by Hofbauer [START_REF] Hofbauer | On intrinsic ergodicity of piecewise monotonic transformations with positive entropy[END_REF][START_REF] Hofbauer | Periodic points for piecewise monotonic transformations[END_REF] for piecewise monotone maps on the interval and our approach has built on his techniques.

The techniques used in the above mentioned papers do not apply to diffeomorphisms (e.g., a diffeomorphism is never entropy-expanding). However, many properties of interval maps generalize to surface diffeomorphisms so the following is generally expected: Conjecture 1. Consider a C 1+ǫ diffeomorphism (ǫ > 0) of a compact surface with nonzero topological entropy.

The collection of ergodic and invariant probability measures with maximal entropy is countable (possibly finite or empty) and the periodic points satisfy a multiplicative lower bound if there exists at least one measure with maximum entropy.

Conjecture 2. Consider a C ∞ diffeomorphism of a compact surface with nonzero topological entropy.

The collection of ergodic and invariant probability measures with maximum entropy is finite and the periodic points satisfy a multiplicative lower bound.

By a result of S. Newhouse [START_REF] Newhouse | Continuity properties of entropy[END_REF], all C ∞ maps of compact manifolds have at least one measure of maximum entropy. Also a classical theorem of A. Katok [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF] states that, if T is a C 1+ǫ , ǫ > 0, diffeomorphism of a compact surface M , then the number of periodic points satisfies a logarithmic lower bound:

lim sup n→∞ 1 n # log{x ∈ M : T n x = x} ≥ h top (T ),
i.e., a weak version of (1.1). This paper presents the proof of the analogue of Conjecture 2 in the easier case of piecewise affine homeomorphisms. This replaces distortion of smooth diffeomorphisms by the singularities of the piecewise affine maps. However this preserves substantial difficulties. Indeed, there exist piecewise affine maps on surfaces without a maximum measure (see Appendix C, though the examples I know are discontinuous -or continuous but piecewise quadratic) or with infinitely many maximum measures (see also Appendix C). Thus this setting, beyond its own interest as a simple and rather natural class of dynamics, is challenging enough to allow the development of new tools which we hope will apply to diffeomorphisms.

1.1. Definitions and Statements. Let M be a compact two-dimensional manifold possibly with boundary, affine in the following sense. There exists a distinguished atlas of charts:

• identifying the neighborhood of any point of M with an open subset of {(x, y) ∈ R 2 : x ≥ 0 & y ≥ 0}; • inducing affine changes of coordinates.

These charts are called the affine charts. The phenomena we are considering are independent of the global topology, so we could in fact restrict ourselves to the special cases M = T 2 or M = [0, 1] 2 .

A continuous map T : M → M is said to be piecewise affine if there exists a finite partition P of M such that for every A ∈ P , A and T (A) are contained in affine charts which map them to polygons of R 2 with non-empty interiors and T : A → T (A) is affine w.r.t. these affine charts. It is convenient to replace the partition P by the collection P of the interiors of the elements of P . Such a partition P (a partition up to the boundaries of its elements) is called an admissible partition with respect to T . We drop the tilde in the sequel.

Let us recall some facts about entropy (we refer to [START_REF] Walters | An introduction to ergodic theory[END_REF][START_REF]Entropy[END_REF] for further information). The entropy of a non-necessarily invariant subset K ⊂ M is a measure of the "number of orbits" starting from K. Recall that the ǫ, n-ball at x ∈ M is: {y ∈ M : ∀k = 0, 1, . . . , n -1 d(T k y, T k x) < ǫ}. The entropy of K is, according to the Bowen-Dinaburg formula [START_REF] Bowen | Entropy for group endomorphisms and homogeneous spaces[END_REF]: where r(ǫ, n, K) is the minimal number of ǫ, n-balls with union containing K. The topological entropy is h top (T ) := h(T, M ). The entropy of an ergodic and invariant probability measure µ can be defined similarly, according to [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF]: where r(ǫ, n, µ) is the minimal number of ǫ, n-balls whose union has a µ-measure at least λ, for a constant λ ∈ (0, 1) (h(T, µ) is independent of λ).

h(T, µ) := lim
The variational principle states that for T : M → M (in fact for any continuous self-map of a compact metric space [START_REF] Walters | An introduction to ergodic theory[END_REF]): (1.3) h top (T ) = sup µ h(T, µ)

where µ ranges over the T -invariant and ergodic probability measures.

The following combinatorial expression for topological entropy follows from observations of S. Newhouse and will be the starting point of our investigations. Proposition 1.1. Let T be a piecewise affine homeomorphism of a compact surface. The topological entropy of T is given by:

(1.4) h top (T ) = lim sup n→∞ 1 n log #{A 0 ∩ f -1 A 1 ∩ • • • ∩ f -n+1 A n-1 = ∅ : A i ∈ P }.
Remark 1.2. The above entropy formula was also obtained by D. Sands and Y.

Ishii [START_REF] Ishii | Lap number entropy formula for piecewise affine and projective maps in several dimensions[END_REF] by different methods. Misiurewicz and Szlenk [START_REF] Misiurewicz | Entropy of piecewise monotone mappings[END_REF] established the same formula for piecewise monotone maps of the intervals.

The proof is given in Sec. 2.1.

The variational principle (1.3) brings to the fore the ergodic and invariant probability measures µ such that h(T, µ) = sup ν h(T, ν) = h top (f ). We call them maximum measures. A corollary of the proof of the previous Proposition is the following existence result (compare with the examples in appendix C).

Corollary 1.3. A piecewise affine homeomorphism of a compact surface has at least one maximum measure.

Our main result, finally established in Sec. 5.1, is: Theorem 1. Let T : M → M be a piecewise affine homeomorphism of a compact affine surface. Assume that h top (T ) > 0. Then there are finitely many ergodic, invariant probability measures maximizing the entropy (or maximum measures).

We also obtain as by-products (see Sec. 5.2 and 5.3): Proposition 1.4. Let T : M → M be a piecewise affine homeomorphism of a compact affine surface with nonzero topological entropy. The periodic points satisfies a multiplicative lower bound. Proposition 1.5. Let T : M → M be a piecewise affine homeomorphism of a compact affine surface. Let S be the singularity locus of M , that is, the set of points x which have no neighborhood on which the restriction of T is affine.

For any ǫ > 0, there is a compact invariant set K ⊂ M \ S such that

h top (f |K) > h top (f ) -ǫ.
Moreover f : K → K is topologically conjugate to a subshift of finite type (see [START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF]).

1.2. Outline of the Proof. We use an alternative approach to the explicit construction of Markov partitions. We ask less of geometry and use more combinatorics, ergodic theory and entropy estimates to accommodate the resulting nonuniqueness of representation. More precisely, we build small rectangles admitting many returns with "good properties" which allows the construction of a jump transformation and establish semi-uniform estimates, that is, uniform estimates holding on subsets of measures that are lower bounded w.r.t. any large entropy measure. The finite number of maximum measures for the jump transformation follows from results of B.M. Gurevič on countable state Markov shifts. However, the jump transformation is not a first return map to an a priori defined good set. Hence, a careful study of the relation between the jump transformation and the original dynamics is needed to conclude that the maximum measures of both systems can be identified. In fact, we analyze more generally large entropy measures, i.e., invariant and ergodic probability measures with entropy close enough to the supremum.

Let us outline the proof. We start in section 2 by introducing the natural symbolic dynamics of the piecewise affine map using the partition defined by the singularities of T . We first show that this symbolic dynamics has the same entropy as T . This is both a significant result and a fundamental step in our approach. We then establish that the (local) stable W s (x) and unstable W u (x) manifolds of points x ∈ M , i.e., the sets of points with the same past or future w.r.t. the partition P , are line segments outside of an entropy-negligible subset. These line segments can be arbitrarily short and their length may vary discontinuously. However, we prove semi-uniform lower bounds for their lengths and angles, using the conditional entropy w.r.t. the past or the future. A corollary of these bounds is that the boundary of the partition is negligible w.r.t. all ergodic invariant probability measures with nonzero entropy.

At this point, one would like to conclude by an argument of the following type. If there was a large number of maximum measures, then one could find two of them, both giving positive measure to a set S of points with local stable and unstable manifolds much larger than the diameter of S. Hence, one could "jump" back and forth between typical orbits of each of the two measures. But one could expect such mixing to allow the construction of a measure with greater entropy, a contradiction. However, establishing the increase in entropy seems to require too fine estimates (of a multiplicative, rather than logarithmic type). We are thus lead to build a jump transformation with Markov properties which will reduce the problem to loop counting on a graph, for which these fine estimates exist (and, indeed, the uniqueness of the maximum measure has been established in this setting by B.M. Gurevič, see below). Section 3 is devoted to building a Markov structure representing the large entropy dynamics. We first build arrays of Markov rectangles which contain a significant proportion of the dynamics. These are approximate rectangles in the sense they contain open subsets of points with local manifolds that don't cross ("holes" in the local product structure). However we can ensure that the relative measure of such points in very small. Our techniques require however replacing T by some high power T L .

We then define hyperbolic strips following the geometric picture of Markov rectangles usual in uniformly hyperbolic dynamics. We provide tools to build many such strips around typical orbits of large entropy measures, using visits to the Markov rectangles while the stable and unstable manifolds are both "rather large".

These hyperbolic strips are Markov in the sense that they can be freely concatenated as soon as they end and begin in the same rectangle. However, to get a useful Markov representation from this, one needs an invariant way of "cutting" typical orbits into concatenations of such hyperbolic strips. A fundamental difficulty arises here: there exist incompatible decompositions, i.e., which do not admit a common refinement. There does not seem to be an a priori natural set the visits to which could be used to defined invariantly the above "cutting". 1 We conclude section 3 by selecting among hyperbolic strips a subset of admissible ones to get uniqueness in the decomposition of forward orbits (this weak uniqueness will require a more detailed ergodic analysis in section 5). We obtain a notion of good return times and a Markov structure.

The more technical section 4 relates the good return times to geometric and combinatorial properties involving the visits to the Markov rectangles and their holes. It is shown that large entropy measures cannot have too large average good return time.

Finally, section 5 proves the main results. We lift large entropy measures of T to the jump transformations as finite extensions. Using that the latter is isomorphic to a countable state Markov shift, a result of Gurevic [START_REF] Gurevič | Shift entropy and Markov measures in the space of paths of a countable graph[END_REF] allows to conclude the proof of the Theorem.

Proposition 1.4 rests on a classical estimate of Vere-Jones [START_REF] Vere-Jones | Ergodic properties of nonnegative matrices[END_REF] on the number of loops of countable oriented graphs together with a combinatorial argument to transfer this estimate to periodic points of T . Proposition 1.5, the possibility of approximating in entropy of the whole map by a compact set away from the singularity set, follows from a similar property 1 Notice that shadowing lemmas à la Katok [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF] give comparable results for surface diffeomorphisms. The problem is to find invariant decompositions.

of countable state Markov shifts: they are approximated in entropy by finite state Markov shifts according to Gurevič.

There are three appendices: (A) recalls some facts about measure-theoretic entropy, (B) proves a lifting theorem for the tower defined by a return time and (C) gives some examples of piecewise affine maps. 1.3. Some Comments. The results presented here allow the analysis of the maximal entropy measures of a simple and natural class of dynamics by representing them by countable state Markov shifts. Along the same lines, one can probably make the representation more precise to get further results, for instance:

• classification by the topological entropy and periods up to isomorphisms modulo entropy-negligible subsets [START_REF] Boyle | Almost isomorphism of countable state Markov shifts[END_REF]; • precise counting of isolated periodic points, e.g., existence of a meromorphic extension of the Artin-Mazur zeta function like in [START_REF] Buzzi | Puzzles of quasi-finite type, zeta functions and symbolic dynamics for multi-dimensional maps[END_REF]; • uniqueness of the maximal entropy measure under a transitivity assumption and/or a bound on the number of maximum measures in terms of the cardinality of the partition P .

In a slightly different direction, one would like to understand the nature of the symbolic dynamics of piecewise affine surface homeomorphisms (see Sec. 2.1). Is there a tractable class of subshifts containing these symbolic dynamics, i.e., a class that would do for piecewise affine surface homeomorphisms what subshifts of quasifinite type [START_REF] Buzzi | Subshifts of quasi-finite type[END_REF] do for piecewise monotone interval maps? It would also be natural to apply the techniques of this paper to more general dynamics. First, piecewise homographic surface homeomorphisms can be analyzed in the same way. Then one could try to analyze other general classes of piecewise affine maps, especially in higher dimensions (e.g., uniformly expanding maps or entropy-expanding maps or entropy-hyperbolic homeomorphisms [START_REF] Buzzi | Hyperbolicity through entropies[END_REF]). Most questions are still open despite some partial results (see, e.g., [START_REF] Buzzi | Piecewise isometries have zero topological entropy[END_REF][START_REF] Tsujii | Absolutely continuous invariant measures for expanding piecewise linear maps[END_REF][START_REF] Kruglikov | Entropy via multiplicity[END_REF]) and we should stress that new problems immediately appear. From the point of view of entropy alone:

• There exist piecewise affine continuous maps on surfaces and piecewise affine homeomorphisms in dimension 3 for which the right hand side of (1.4) For diffeomorphisms, the main difficulty with our approach is to find a link between short stable/unstable manifolds and small entropy, e.g., one would need to relate small Lyapunov charts to entropy bounds for a smooth diffeomorphism. An analysis of C 1 surface diffeomorphisms with nonzero topological entropy with dominated splitting is in preparation.

1.4. Acknowledgements. I would like to thank the anonymous referees for many remarks which have significantly improved the exposition.

Pointwise Estimates

2.1. Symbolic Dynamics. We define a symbolic dynamics for the map T using some admissible partition P , that is, a finite collection of disjoint open polygons with dense union. A key step is showing that ∂P has zero measure w.r.t. any µ ∈ P erg 0 (T ), where P erg h (T ) denotes the set of ergodic, invariant probability measures of T with h(T, µ) > h. Definition 2.1. Let P be an admissible partition. x ∈ M is nice if for every n ∈ Z, T n x belongs to an element A n of the admissible partition P . The sequence A ∈ P Z thus defined is the P -itinerary of x.

The symbolic dynamics of T, P is:

Σ := {A ∈ P Z : ∃x ∈ M ∀n ∈ Z T n x ∈ A n }
endowed with the shift map:

σ(A) = (A n+1 ) n∈Z .
A standard result (see, e.g., [START_REF] Walters | An introduction to ergodic theory[END_REF]) states that since Σ is a subshift, it admits at least one maximal entropy measure. Hence, a "close enough" relation between the invariant measures of Σ and T will imply existence of a maximum measure also for T . By the variational principle, we shall also get that T and Σ have the same topological entropy. The Misiurewicz-Szlenk formula for T will then follow. Indeed, for a subshift like Σ, the topological entropy is computed by counting the cylinders,

[A 0 . . . A n-1 ] := {x ∈ Σ : x 0 . . . x n-1 = A 0 . . . A n-1 }, that is: h top (Σ) = lim n→∞ 1 n log #{[A 0 . . . A n-1 ] = ∅ : A ∈ P n }.
Neither T nor its symbolic dynamics is an extension of the other in general, hence it is convenient to introduce the following common extension:

Σ ⋉ M := {(A, x) ∈ P Z × x : ∀n ∈ Z T n x ∈ A n } with T (A, x) = (σA, T x).
The close relation between the measures of T and Σ alluded to above is: Lemma 2.2. Both maps π 1 : Σ⋉M → Σ and π 2 : Σ⋉M → M are entropy preserving: for every invariant probability measure µ on Σ ⋉ M , h(σ, π 1 µ) = h(T, π 2 µ) = h( T , µ). Moreover, π 1 and π 2 induce onto maps between the sets of (ergodic) invariant probability measures.

In particular, the topological entropies of the three systems are equal by the variational principle recalled in eq. (1.3).

The proof of the above Lemma rests on two geometric/combinatorial properties. The first is the following observation by S. Newhouse, very specific of our setting (it is false in higher dimensions or without the invertibility assumption, see the Appendix):

Lemma 2.3. The multiplicity entropy [START_REF] Buzzi | Intrinsic ergodicity of Affine Maps in [0, 1] d[END_REF]:

h mult (T ) := lim sup n→∞ 1 n log max x∈M mult(P n , x) with mult(Q, x) := #{A ∈ Q : x ∈ Q}
is zero for any piecewise affine homeomorphism of a surface.

The second is a property of linear maps:

∀x = y ∈ S ∃0 ≤ k < n T k-1 . . . T 1 T 0 (x -y) > ǫ} = 0.
We leave the easy proofs of Lemmas 2.3 and 2.4 to the reader.

Proof of Lemma 2.2: Lemma 2.3, resp. Lemma 2.4, implies that for all x ∈ M , resp. Σ, for i = 2, resp. i = 1, h( T , π -1 i {x}) = 0 (this is the entropy of a subset as recalled in (1.2)). Now, π 1 : Σ ⋉ M → Σ and π 2 : Σ ⋉ M → M are both compact topological extensions. Hence, one can apply Bowen's result [START_REF] Bowen | Entropy for group endomorphisms and homogeneous spaces[END_REF]:

h( T , μ) = h(σ, π 1 µ) = h(T, π 2 µ)
for all invariant probability measures μ of Σ ⋉ M .

2.2. Invariant Manifolds and Lyapunov Exponents. For a fixed partition P , we have the following:

Definition 2.5. The stable manifold at A ∈ Σ is the following set (convex in the affine charts):

W s (A) := n≥1 T -n A n
The unstable manifolds W u (A) are defined by replacing n ≥ 1 by n ≤ -1 in the above equation.

The Lyapunov exponents along the stable or unstable direction at A ∈ Σ are:

λ u (A) := lim n→±∞ 1 n log (T n A ) ′ ±1 ±1 and λ s (A) := lim n→±∞ 1 n log (T n A ) ′ ∓1 ∓1
where T n A is the affine composition

(T |A n-1 )•• • ••(T |A 0 ) (if n ≥ 0) or [(T |A -1 ) • • • • • (T |A n )] -1 (if n < 0).
Any nice x ∈ M defines a unique itinerary A and we write W s (x) for W s (A), λ u + (x) for λ u + (A) and so on. The first goal of this section is the following "non-singularity" result: Proposition 2.6. Let µ ∈ P erg 0 (T ). The following holds:

• µ(∂P ) = 0 (in particular, µ-a.e. x ∈ M is nice);

• The Lyapunov exponents exist and satisfy

λ s (x) ≤ -h(T, µ) < 0 < h(T, µ) ≤ λ u (x) for µ-a.e. x ∈ M ; • W s (x) and W u (x) are line segments containing x in their relative interiors int W s (x) and int W u (x) for µ-a.e. x ∈ M .
Proof. Let µ ∈ P erg 0 (T ). As we have not yet proved that a.e. x ∈ M is nice, we have to work in the extension Σ ⋉ M to be able to speak of itineraries, invariant manifolds and so on. By compactness, there exists an invariant and ergodic probability measure μ of T : Σ ⋉ M ←֓ such that π 2 μ = µ. We have h( T , μ) > 0 by Lemma 2.2.

We first consider the invariant manifolds.

Claim 2.7. μ-a.e. (A, x) ∈ Σ ⋉ M , (i) W u (A) is a line segment; (ii) x is not an endpoint of this segment.
Proof of the claim: To begin with, observe that W u (σA) ⊂ T (W u (A)) so that dim(W u (σA)) ≤ dim(W u (A)). As μ is invariant and ergodic, dim(W u (A)) = d μ-a.e. for some d ∈ {0, 1, 2}. Claim (i) above is that d = 1.

Let P be the natural partition of Σ ⋉ M (coming from the canonical partition of Σ). By Lemma 2.2, ( T , μ) has the same entropy as the corresponding symbolic system. Thus, h( T , μ) = h( T , μ, P ). Using conditional entropy (see Appendix A) we can compute h( T , μ, P ) as H μ( P | P -) where P -:= n≥1 T n P . Observe that

A → W u (A) is P --measurable.
We exclude the cases d = 0, 2 by contradiction. Assume first d = 0, i.e., W u (A) is a single point x ∈ M for μ-a.e. (A, x) ∈ Σ ⋉ M . This implies that:

h( T , μ) = H μ( P | P -) = lim n→∞ 1 n H μ( P n | P -) ≤ h mult (T, P ) = 0, a contradiction, excluding the case d = 0.
Now assume d = 2, so μ-a.e. W u (A) is the closure of the interior of W u (A). By construction, two distinct unstable manifolds have disjoint interiors. Therefore, there can be only countably many of them, after discarding a set of zero μ-measure. In particular, W u (A) = W u (A 0 ) on a set of positive measure for some A 0 . By Poincaré recurrence, there exists an integer n > 0 such that

T n (W u (A 0 )) = W u (A 0 ). This implies that π 1 μ is periodic, hence 0 = h(σ, π 1 μ) = h( T , μ). The contradiction proves (i).
We turn to (ii

). If x ∈ ∂W u (A), then T (x) ∈ ∂W u (σ(A)). Thus if (ii) is false, then x ∈ ∂W u (A) µ-a.e. But this implies that, for any ǫ > 0, any large n, nh( T , μ) = H μ( P n | P -) ≤ log 2+log max x∈M #{A ∈ P n : A ∋ x} ≤ log 2+(h mult (T, P )+ǫ)n.
As h mult (T, P ) = 0, it would follow that h( T , μ) = 0, a contradiction.

We now turn to the exponents. First they do exist by the classical Oseledets Theorem (see, e.g., [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]).

Claim 2.8. For π 1 μ-a.e. A ∈ Σ, the Lyapunov exponents satisfy: λ s (A) < 0 < λ u (A).

Remark 2.9. The above result will be a consequence of the Ruelle-Margulis inequality [22, p. 669] once we shall have proved that µ(∂P ) = 0.

Proof: We establish the existence of a positive Lyapunov exponent µ-a.e. The existence of a negative exponent will follow by considering T -1 . Let • A be some measurable family of norms. Consider the family

• ′ A , A ∈ Σ defined by: v ′ A := v A /|W u (A)| A for v W u (A) | • | A being the length w.r.t. to • A (using the affine structure). As T (W u (A)) ⊃ W u (σA), we have that T ′ |E u (A) ′ A ≥ 1 (where E u (A) is the unstable direction at A -the invariant family of directions defined by W u (A)) for µ-a.e. A ∈ Σ. T (W u (A)) = W u (σA) µ-a.e. would imply h( T , μ) = H μ( P | P -) = 0. Hence, T |E u (A) ′
A > 1 on a set of positive measure and:

λ u (A) = log T ′ |E u (B) ′ B dμ(B) > 0 for µ-a.e. A ∈ Σ.
We finish the proof of Proposition 2.6. Let µ ∈ P erg 0 (T ). Let μ be a lift of µ to Σ ⋉ M . By Lemma 2.2, h( T , μ) = h(T, µ) > 0.

Claims 2.7 and 2.8 prove all the claims of the Proposition except µ(∂P ) = 0. Now, W u (A) and W s (A) are line segments µ-a.e. by Claim 2.7. Their directions carry distinct Lyapunov exponents by Claim 2.8, hence they must make µ-a.e. a non-zero angle. If x ∈ ∂P , then T x or T -1 x would be the end point of at least one of these line segments, a contradiction. Hence µ(∂P ) = 0.

That µ(∂P ) = 0 for all ergodic invariant probability measures with nonzero entropy has the following immediate but important consequence:

Corollary 2.10. The partially defined map π :

Σ ′ → M {π(x)} := n≥0 T k [A -k . . . A k ]
with Σ ′ the subset of Σ where the above intersection is indeed a single point, defines an entropy-preserving bijection between the sets of ergodic, invariant probability measures of T and of Σ with nonzero entropy.

Semi-Uniform Estimates.

We obtain now more quantitative estimates, which we call semi-uniform in the sense that they are uniform on a set of uniformly lower-bounded mass for all large entropy measures. To state these results, we need the following "distortion" estimate. By compactness of M and invertibility of T ,

d(T ) := sup log T ′ (x).u T ′ (x).v : x ∈ M, u, v ∈ R 2 \ {0}, u = v < ∞.
Proposition 2.11. For any µ 0 < htop(T ) d(T ) , there exist h 0 < h top (T ), θ 0 > 0 and ℓ 0 > 0 such that for any µ ∈ P erg h0 (T ), the following properties occur jointly on a set of measure at least µ 0 :

ρ(x) := min σ=s,u d(x, ∂W σ (x)) ≥ ℓ 0 (2.1) α(x) := ∠(W s (x), W u (x)) > θ 0 (2.2)
Here ∠(W s (x), W u (x)) is the angle between the two lines defined by W s (x) and W u (x). We declare α(x) = ρ(x) = 0, if W s (x) or W u (x) fail to be line segments.

Remark 2.12. In fact we obtain a number µ 0 < 1 arbitrarily close to 1 satisfying (2.1). However this is not the case for (2.2). We believe that this cannot be done. Indeed, one can easily build a smooth surface diffeomorphism with nonzero entropy such that for some µ * > 0 and h * > 0, there are invariant probability measures with entropy at least h * such that the stable and unstable directions make an arbitrarily small angle on a set of measure at least µ * .

We do not know if these measures can be taken to have entropy arbitrarily close to the topological entropy or if piecewise affine examples exist.

We first prove the lower bound on angles by comparing the distortion with the entropy.

Claim 2.13. For any 0 < h 1 < h top (T ), there exists θ 1 > 0 such that the set where α(x) > θ 1 has measure at least h 1 /d(T ) for all measures µ ∈ P erg h1 .

The Ruelle-Margulis inequality applied to (T, µ) and (T -1 , µ) (which is valid as T ′ is constant on each element of P and µ(∂P ) = 0) yields:

(2.3) h top (T ) ≤ λ u (µ) -λ s (µ) 2 = 1 2 M log T ′ (x)|E u (x) T ′ (x)|E s (x) dµ(x).
By continuity there exists

θ 1 > 0 such that, for all u, v ∈ R 2 \ {0} with ∠(u, v) ≤ θ 1 , ∀x ∈ M \ ∂P log T ′ (x).u T ′ (x).v ≤ h 1 .
Therefore, setting m := µ({x

∈ M : α(x) > θ 1 }): 2h(T, µ) ≤ m • d(T ) + (1 -m) • h 1 so that, assuming h(T, µ) > h 1 : m ≥ 2h(T, µ) -h 1 d(T ) -h 1 ≥ 2h(T, µ) -h 1 d(T ) > h 1 d(T ) .
This proves Claim 2.13.

Claim 2.14. For any µ 3 < 1, there exists ℓ 0 > 0 such that

(2.4) ∀µ ∈ P erg h3 (T ) µ ({x ∈ M : d(x, ∂W u (x)) > ℓ 0 }) > µ 3 for h 3 = h top (T )(1 -(1 -µ 3 )/2).
To prove (2.4) let ǫ = (1µ 3 )h top (T )/2 > 0, Λ be a Lipschitz constant for T , n ≥ log 2/ǫ be a large integer and r = r(ǫ, n) > 0 be a small number such that max x∈M #{A ∈ P n : B(x, r) ∩ A = ∅} ≤ 1 2 e (h mult (T,P )+ǫ)n and #P n ≤ e (htop(T )+ǫ)n .

Let µ ∈ P erg h3 (T ), X 0 := {x ∈ M : d(x, ∂W u (x)) ≤ Λ -n r} and denote by µ|X 0 the normalized restriction of µ to X 0 . Using standard facts about entropy (see Appendix A) we get

nh(T, µ) = H µ (P n |P -) ≤ H µ (P n ∨ {X 0 , M \ X 0 }|P -) ≤ H µ ({X 0 , M \ X 0 }) + µ(X 0 )H µ|X0 (P n |P -) + (1 -µ(X 0 ))H µ|M\X0 (P n |P -) ≤ log 2 + µ(X 0 ) sup x∈X0 log #{A ∈ P n : A ∩ X 0 ∩ W u (x) = ∅} + (1 -µ(X 0 )) log #P n ≤ log 2 + µ(X 0 )(h mult (T, P ) + ǫ)n + (1 -µ(X 0 ))(h top (T ) + ǫ)n. Hence h(T, µ) ≤ (1 -µ(X 0 ))h top (T ) + µ(X 0 )h mult (T, P ) + ǫ + 1 n log 2 = h top (T ) + 2ǫ -µ(X 0 )(h top (T ) -h mult (T, P )).
implying that:

µ(X 0 ) ≤ h top (T ) -h(T, µ) -ǫ h top (T ) -h mult (T, P ) ≤ h top (T ) -h(T, µ) + ǫ h top (T ) < 1 -µ 3
using h mult (T, P ) = 0 and h(T, µ) > h 3 . The claim is proved.

Proof of Proposition 2.11: Claim 2.13 gives θ 0 > 0 such that (2.2) holds on a set of measure at least h top (T )/2d(T ) wrt all measures in P erg htop(T )/2 (T ). Claim 2.14 applied to T and T -1 with µ 3 = 1h top (T )/8d(T ), shows that for

h 0 = h top (T ) 1 - 1 16
h top (T ) d(T ) ≥ h top (T )/2,
(2.2) and (2.1) hold jointly on a set of measure at least h top (T )/4d(T ) w.r.t. all measures in P erg h0 (T ).

Construction of the Markov Structure

Roughly speaking, the estimates of the previous section will allow us to build a collection of (non-uniform) "Markov rectangles" which will "control enough" of the dynamics to analyze all measures of large entropy.

3.1. Markov Rectangles. Definition 3.1. A (Markov) rectangle is a closed topological disk R contained in an affine chart and bounded by four line segments, alternatively included in stable and unstable manifolds, making respectively the unstable boundary,

∂ u R = ∂ u 1 R ∪ ∂ u 2 R
, and the stable one,

∂ s R = ∂ s 1 R ∪ ∂ s 2 R. See Fig. 2

. A Markov array is a finite collection of Markov rectangles with disjoint interiors.

Not every passage of an orbit inside a rectangle is useful. We need the following properties. Definition 3.2. A point x is controlled by a rectangle R if x is nice, belongs to R and if W s (x) and W u (x) each intersects ∂R in two points. Note that control depends on the partition P used to define W u (x) and W s (x). If necessary we speak of control with respect to P .

x ∈ R is 10-controlled if, moreover, ρ(x) > 10diamR where ρ was defined in (2.1).

x ∈ R is s-controlled if x is nice, x ∈ R and W s (x) intersects ∂R in two points.
The sets of controlled (resp. 10-controlled, s-controlled) points is denoted by

κ(R) (resp. κ 10 (R), κ s (R)).
A point is controlled by a Markov array R if it is controlled by one of the rectangles of the array. We define κ(R), κ 10 (R), κ s (R) in the obvious way.

Using the previous lower bounds on the lengths and angles of invariant manifolds we shall first prove: Lemma 3.3. There exist numbers h 0 < h top (T ) and µ 0 > 0 and a Markov array R such that for all µ ∈ P erg h0 (T ),

µ(κ 10 (R)) > µ 0 .
Our analysis requires the following slightly stronger statement (i.e., we only tolerate "small holes"): Lemma 3.4. There is µ 0 > 0 such that for any ǫ 0 > 0, there exist a number h 0 < h top (T ) and a Markov array R such that for any µ ∈ P erg h0 (T ), This will be obtained by subdividing the rectangles in the Markov array from Lemma 3.3 into sub-rectangles much smaller than most stable/unstable manifolds.

• µ(κ 10 (R)) > µ 0 ; • µ(R \ κ 10 (R)) < ǫ 0 µ 0 .
The final twist is that as we replace the partition P by the convex partition P R generated by P and the Markov array R (see Fig. 1), some invariant manifolds may shrink, say W u (x) :

= n≥1 T n P R (T -n x) W u (x), diminishing the set of controlled points. Indeed, W u (f (x))
W u (f (x)) when W u (x) crosses the boundary of a rectangle from R before crossing ∂P (x). We shall see however that if these intersections are sufficiently separated in time, then W u (x) = W u (x) for most points x ∈ R w.r.t. large entropy measures. To guarantee that large separation, we use the following construction: Definition 3.5. If R is an array of Markov rectangles contained in an element of P and L is a positive integer, then the (R, L)-extension of (M, T, P, R) is (M + , T + , P + , R + ), defined in the following way:

• M + = M × {0, . . . , L -1}; • T + (x, k) = (T x, k + 1 mod L);
• P + is the finite partition of M + which coincides with a copy of P on each M × {k} for k = 0 and coincides on M × {0} with a copy of P R ;

• R + = {R × {0} : R ∈ R}.
The conclusion of this section is: Proposition 3.6. Let (M, T, P ) be a piecewise affine surface homeomorphism with nonzero entropy. There exist µ 0 > 0, h 0 < h top (T ) such that for any ǫ 0 > 0, there is a Markov array R and a positive integer L 0 with the following properties. Fix any L + ≥ L 0 and let (M + , T + , P + , R + ) be the (R, L + )-extension of (M, T, P, R).

For each µ ∈ P erg h0 (T ), there exists an ergodic invariant probability measure µ + of T + with π(µ + ) = µ (where π(x, k) = x) such that:

(i) L + • µ + (κ 10 (R + )) > µ 0 ; (ii) L + • µ + (R + \ κ 10 (R + )) < ǫ 0 • µ 0 .
The above controlled sets are defined w.r.t. to the invariant manifolds relative to P + (which contains the Markov array R + ).

Notice that it is enough to prove our results (Theorem 1 and Propositions 1.5 and 1.4) for some periodic extension.

We now prove Lemmas 3.3 and 3.4 and Proposition 3.6. We begin by the following: Lemma 3.7. Given ℓ 0 > 0 and 0 < θ 0 < 2π, there exists a finite collection of rectangles R (1) , . . . , R (Q) such that:

(1) diam(R (i) ) < ℓ 0 /10;

(2) any x ∈ M with ρ(x) > ℓ 0 and ∠(W u (x), W s (x)) > θ 0 belongs to at least one R (i) .

This easily implies Lemma 3.3 using Proposition 2.11 and observing that the finite collection of rectangles above can be subdivided by boundary lines like those of Fact 3.9 below so that their interiors become disjoint, defining the required Markov array R.

Proof of Lemma 3.7: Let

K * := {x ∈ M : ρ(x) > ℓ 0 and ∠(W u (x), W s (x)) > θ 0 }. Let {K j } Q
j=1 be a finite partition of K * whose elements have diameter less than θ 0 ℓ 0 /100 and lie within an affine chart of M . We fix j.

Recall that the collection of closed subsets K of the compact metric space M is a compact space w.r.t. the Hausdorff metric:

d(A, B) = inf{ǫ > 0 : A ⊂ B(B, ǫ) and B ⊂ B(A, ǫ)} where B(A, ǫ) is the ǫ-neighborhood of the set A, i.e., {x ∈ M : d(A, x) < ǫ}.
The easy proofs of the following two facts are left to the reader. Fact 3.9. Assume that K j is as above. Then there exist two points x 1 , x 2 ∈ K j , two non-trivial line segments L 1 , L 2 and two itineraries A 1 , A 2 ∈ Σ(T, P ) with the following properties.

• L i is contained in the boundary of W s (A i ) as a subset of M ;

• in some affine chart, K j lies between the two lines supporting L 1 and L 2 . We call L 1 , L 2 a pair of stable boundary lines of K j .

We now prove Lemma 3.7. Consider two distinct one-dimensional stable manifolds W s (A) and W s (B) which intersect in a single point p. p must the endpoint of at least one of them: otherwise, if A n = B n , then p ∈ ∂A n ∩ ∂B n and both W s (A) and W s (B) are parallel to the same segment of ∂A n ∩ ∂B n . Thus their intersection contains a non-trivial line segment.

Observe that if x, y ∈ K j , W u (x) and W u (y), which are line segments, must have disjoint relative interiors or be parallel and overlapping. Thus

∠(W u (x), W u (y)) ≤ θ 0 /50.
As ∠(W u (z), W s (z)) > θ 0 for all z ∈ K, we get:

∠(W u (x), W s (y)) > θ 0 /2.
Consider a pair of "stable boundary lines", resp. "unstable boundary lines", given by Fact 3.9 applied to (T, K j ), resp. applied to (T -1 , K j ). Let R (j) be the rectangle bounded by these four line segments. R (j) is contained in the intersection of two strips with almost parallel sides of width ≤ diamK j and making an angle at least θ 0 /2. Hence diam(R (j) ) < 5diam(K j )/θ 0 < ℓ 0 /20.

On the other hand, R (j) ⊃ K j , hence j R (j) ⊃ K * .

Proof of Lemma 3.4: Apply Lemma 3.3 to get R, µ 0 > 0 and h 0 < h top (T ). Recall that ρ(x) is the distance between x and the endpoints of its invariant manifolds (or 0 if one of those is not a line segment). By Claim 2.14 applied with µ 3 := 1ǫ 0 µ 0 /2 to T and T -1 , there exist h 1 < h top (T ) and

ℓ 1 = ℓ 1 (ǫ 0 µ 0 ) such that ∀µ ∈ P erg h1 (T ) µ({x ∈ M : ρ(x) < ℓ 1 }) < ǫ 0 µ 0 .
Let us cut each big rectangle R from R into sub-rectangles R ′ with diameter at most ℓ 1 /10, obtaining a new Markov array R ′ . Using Fact 3.9 again, we can do it by finitely many stable and unstable manifolds (or line segments bounding those).

Observe that κ 10 (R ′ ) ⊃ κ 10 (R) and that the points in R ′ \ κ 10 (R ′ ) which have line segments as invariant manifolds, are ℓ 1 -close to an endpoint of their stable/unstable manifold. Hence

µ(κ 10 (R ′ ))µ(κ 10 (R)) ≥ µ 0 and µ(R ′ \ κ 10 (R ′ )) ≤ ǫ 0 µ 0 .
for all µ ∈ P erg h1 (T ).

Proof of Proposition 3.6: We apply Lemma 3.4 with ǫ 0 /2 obtaining µ 0 > 0 (independent of ǫ 0 ), h 0 < h top (T ) and a Markov array R. Let P R be the convex partition previously defined. We go to the (R, L + )-extension (M + , T + , P + ) of (M, T, P ) for some large integer L + to be specified. As we observed, there always exists an ergodic, T + -invariant measure µ + extending µ. Maybe after replacing it by its image under (x, i) → (T j x, i + j mod L + ) for some constant j ∈ {0, 1, . . . , L + -1}, we get an ergodic extension µ + such that

L + • µ + (κ 10 (R) × {0}) ≥ µ(κ 10 (R)) ≥ µ 0 .
As the extension is finite-to-one, µ + has the same entropy as µ.

Let x ∈ M . If the unstable manifold for T + , W u + (x, 0) := n≥1 T p erext n P + (T -n + (x, 0)) is strictly shorter than W u (x) × {0}, then it is bounded by T kL+ + (y, 0) with y an intersection point of W u (T -kL+ x) for some k ≥ 1, with one of the new boundary segments, I, of P R . Hence P kL+ (T -L x) is determined by the past of T -kL+ x and I picked among finitely many choices. Note that this number of choices depends only on P and R but not on L + .

A standard counting argument shows that if this happened on a subset of M ×{0} with µ + -measure at least

1 2 ǫ 0 µ 0 • L -1 + , then h(T, µ) = h(T + , µ + ) ≤ (1 - 1 2 ǫ 0 µ 0 )h top (T ) + ǫ(L + )
where ǫ(L) → 0 as L → ∞. This is strictly less than h top (T ) if L + is large enough (which we ensure by taking L 0 large). So it is excluded for large entropy measures. The (R, L)-periodic extension (M + , T + , P + , R + ) has the required properties for all large integers L + . 3.2. Hyperbolic strips. (M, T, P ) is some piecewise affine surface homeomorphism with nonzero entropy and R is some Markov array with R ⊂ P (eventually (M, T, P, R) will be the previously built periodic extension (M + , T + , P + , R + )). We shall use the following picture to define finite itineraries that can be freely concatenated. This is adapted from uniformly hyperbolic dynamics.

Definition 3.10. A quadrilateral Q u-crosses a rectangle R ∈ R if Q ⊂ R
and its boundary is the union of two subsegments of the stable boundary of R (the stable boundary of Q) and two line segments (the unstable boundary of Q), these four segments being pairwise disjoint, except for their endpoints. s-crossing is defined similarly.

A u-rectangle is a quadrilateral which u-crosses some rectangle R ∈ R and whose unstable boundary is made of two segments of unstable manifolds. A srectangle is defined similarly (see Figure 2).

For n ≥ 1, a hyperbolic n-strip (or just n-strip) is an s-rectangle S such that int T k (S) is included in some element of P for each k = 0, . . . , n -1 and T n (S) is a u-rectangle. A hyperbolic strip is an n-strip for some n ≥ 1.

We write P b a (x) for b k=a T -(k-a) P (T k x) (we assume implicitly that x is nice -this fails only on an entropy-negligible set by Proposition 2.6). The following is immediate.

Facts 3.11.

1. A hyperbolic n-strip is necessarily of the form P n 0 (x) for some x ∈ R.

Two hyperbolic strips are either nested or have disjoint interiors.

We now give some tools to build hyperbolic strips. Lemma 3.12. For 0 < m < n, if P m 0 (x) and P n m (x) are both hyperbolic strips, then so is P n 0 (x). This is easy to show using Fig. 3. Sufficiently long invariant manifolds allow the construction of hyperbolic strips from scratch: Lemma 3.13. Let x ∈ κ 10 (R) and n ≥ 1 such that T n x ∈ κ 10 (R). Then P n 0 (x) is a hyperbolic strip.

Observe that the weaker condition x ∈ κ(R) ∩ T -n κ(R) does not imply that P n 0 (x) is a hyperbolic strip.

T n P n 0 (x) ⊂ T m P n m (x) ⊂ R ′′ (R, R ′ , R ′′ ∈ R).
Proof: See Fig. 4. Let R, R ′ be the elements of R containing x and T n x. Consider the diamond (the quadrilateral) L generated by W s (x) and

T -n W u (T n x). By convexity, L is contained in T -1 P n 1 (x). Consider one side [uv] of T n L with u / ∈ R ′ and d(x, u) > 10 • diam(R ′ ). Let {a} := [uv] ∩ ∂ s R ′ . We have d(u, v) ≥ d(u, T n x) -diam(R ′ ) ≥ 9diam(R ′ ). Hence d(v, a) ≤ diam(R ′ ) ≤ (1/9)d(u, v).
Let (abcd) be the quadrilateral defined by the points of ∂T n L ∩ ∂R ′ . It u-crosses R ′ . It remains to prove that L s-crosses R so that R∩T -n Q is the desired hyperbolic strip and must be P Proof: By Lemma 3.13, P m n (x) is a hyperbolic strip. Apply Lemma 3.12 to conclude.

We need the following technical fact. Lemma 3.15. Let µ be an atomless invariant probability measure. For µ-a.e. x ∈ κ s (R), for some R ∈ R and all n ≥ 1, the intersection of W s (x) with ∂ u R is disjoint from all the vertices of P n , n ≥ 1. In particular, ∂ u R ∩ ∂P n (x) is the union of two non-trivial segments.

Proof: We proceed by contradiction assuming that the above fails: on a subset of κ s (R i ) with positive measure at least one of these intersection points coincides with a vertex z of the polygon ∂P n (x) (so W s (z) = W s (x)). Reducing this subset, we assume the vertex z to be a fixed one, say z + .

By Poincaré recurrence, there must exist infinitely many n ≥ 0 such that T n x ∈ W s (z + ). Considering two such integers n 1 < n 2 , we get that T n2-n1 (W s (z + )) ⊂ W s (z + ). This implies that all points of W s (z + ) converge to a periodic orbit. Thus, the ergodic decomposition of µ has an atom, a contradiction. We show that if x ∈ κ s (R), then subsequent visits to κ 10 (R) either give a hyperbolic strip or a shadowing property which will lead to an entropy bound. Lemma 3.16. Let x ∈ κ s (R) and 0 ≤ m < n be such that T m x, T n x ∈ κ 10 (R). Excluding a set of zero measure of points x, if P n 0 (x) is not a hyperbolic strip then P m 0 (x) determines P n 0 (x) up to a choice of multiplicity 4.

Proof: W s (x) crosses the rectangle R ∈ R containing x. Hence Lemma 3.15 implies that ∂P m 0 (x)∩∂ u R is the union of two unstable, non-trivial segments:

[a, b], [c, d]. Let [a ′ , b ′ ], [c ′ , d ′ ]
be their images by T m |P m 0 (x). Let Q ′ be the quadrilateral generated by them. By convexity Q ′ ⊂ T m (P m 0 (x)). H := P n m (x) is a hyperbolic strip by Lemma 3.13.

Q ′ and H intersect. If int H ∩ {a ′ , b ′ , c ′ , d ′ } = ∅, then Q ′ would
go across H, and P m 0 (x) would be a hyperbolic strip, contrary to assumption.

Thus, at least one of the four vertices a, b, c, d determined by P m 0 (x) is contained in int H, this point determines H and therefore P n 0 (x) as claimed.

3.3. Admissible Strips and Good Returns. In this section, R is some Markov array with R ⊂ P . Hyperbolic strips defined above have no uniqueness property: a point x ∈ κ s (R) sits in an infinite sequence of nested hyperbolic strips. This motivates the following notion.

Definition 3.17. For n ≥ 1, the admissible n-strips are defined by induction on n. A 1-strip is always admissible. For n > 1, an admissible n-strip S is an n-strip such that for all 1 ≤ m < n such that S is included in an admissible m-strip, T m (S) meets the interior of no hyperbolic strip. An admissible strip is an admissible n-strip for some n ≥ 1.

Figure 5 shows a hyperbolic n-strip S (hatched) which is not admissible.

Definition 3.18. Let (M, T, P ) be a piecewise affine surface homeomorphism with a Markov array contained in P . For a point x ∈ M , the (good) return time is τ = τ (x), the minimal integer τ ≥ 1 such that both following conditions hold:

• x belongs to an admissible τ -strip;

• T τ (x) ∈ κ s (R).
These conditions are defined with respect to some partition containing some Markov array.

If there is no such integer τ , then we set τ (x) = ∞.

Remark 3.19. Note that, at this point, we break the symmetry between the future and the past.

We shall use repeatedly the following obvious observation: Fact 3.20. If n is the smallest integer such that P n 0 (x) is a hyperbolic strip (equivalently: P n 0 (x) is an n-strip which is not contained in a k-strip for any k < n; P n 0 (x) does not meet a k-strip distinct from P k 0 (x) for any k < n; P n 0 (x) is a hyperbolic strip which is maximum w.r.t. inclusion) then P n 0 (x) is an admissible n-strip. Remark 3.21. One could consider the following changes in the definition of admissibility:

1) replacing "T m (S) meets no hyperbolic strip" by "T m (S) meets no admissible strip" would not change the notion. Indeed, suppose that T m (S) meets a hyperbolic strip H. Let k ≥ 1 be the smallest integer such that H is contained in a k-strip, say H k . The minimality of k implies that H k is admissible and

H k ⊃ H so that T m (S) meets H k .
ii) replacing "S is included in an admissible m-strip" by "S is included in a hyperbolic m-strip" would exclude some admissible strips and so would cause a problem in the proof of the (key) Claim 4.2 (for the proof that k = n i in the notations there).

Admissibility gives the following uniqueness property. Denote the one-sided symbolic dynamics by: Σ + (T,

P ) = {A 0 A 1 A 2 • • • ∈ P N : A ∈ Σ(T, P )}.

Lemma 3.22. A positive itinerary A ∈ Σ + (T, P ) can be decomposed in at most one way as an infinite concatenation of admissible strips.

Proof: Consider two distinct decompositions of A into admissible strips, that is, n 0 = 0 < n 1 < n 2 < . . . and m 0 = 0 < m 1 < m 2 < . . . , such that A ni . . . A ni+1 and A mi . . . A mi+1 are admissible strips for all i ≥ 0. By deleting the identical initial segments, we can assume that the decompositions differ from the beginning, say n 1 < m 1 . It follows that the admissible m 1 -strip H := [A 0 . . . A m1 ] is contained into the n 1 -admissible strip [A 0 . . . A n1 ]. Thus T n1 (H) meets [A n1 . . . A n2 ] which is another admissible strip, contradicting admissibility.

Analysis of Large Return Times

In this section (M, T, P ) is a piecewise affine homeomorphism with positive topological entropy. In the first two subsections, we analyze the implications of a long return time τ (x)(x) from a geometric and then a combinatorial point of view. We then apply this to invariant measures with very large "average" return times to bound the entropy of these measures. Let 0 ≤ N 1 ≤ N 2 ≤ N 0 < N be defined as follows:

• 0 < N 0 < N be the smallest integer such that T N0 x ∈ κ 10 (R) and P N0 0 (x) is a hyperbolic strip (we set N 0 := N if there is no such integer);

• 0 ≤ N 1 < N be the smallest integer such that T N1 x ∈ κ 10 (R) (we set N 1 := N 0 = N if there is no such integer); • 0 ≤ N 2 < N 0 be the largest integer such that T N2 x ∈ κ 10 (R) (we set N 2 := N 1 = N 0 if there is no such integer).
• n 1 , . . . , n r (r ≥ 0) be the admissible times, that is the successive integers in: {0 ≤ k < N : P k 0 (x) is an admissible strip } with the convention n r+1 = N ; • m i1 , . . . , m is(i) (s(i) ≥ 0) be the hyperbolic times, that is, for each i, the successive integers:

{n i < m < n i+1 : P m 0 (x) is a m-strip and T m x ∈ κ s (R)}.
with the convention m is(i) := m i1 := n i+1 and s(i) = 0 if the above set is empty. Then P N (x) is determined, up to a choice of multiplicity 4 • 2 r , by:

(1) the integers N 1 , N 2 , r and n i , m i1 , m is(i) for 1 ≤ i ≤ r;

(2) ([[a,b[[ denotes the integer interval with a included and b excluded, etc.).

P (T k x) for k ∈ [[0, N 1 ]] ∪ [[N 2 , n 1 [[; (3) P (T k x) for k ∈ r i=1 [[n i , m i1 ]] ∪ [[m is(i) , n i+1 ]] \ [[0, N 1 ]].
Proof: The inequality 0 ≤ N 1 ≤ N 2 ≤ N 0 is easily checked.
We can assume that N 1 < N , as otherwise there is nothing to show. We first claim that P N1 0 (x) determines P N2 0 (x) up to a choice of multiplicity 4. If N 1 ≥ N 0 , then N 1 = N 2 = N 0 and there is nothing to show. Otherwise N 1 ≤ N 2 < N 0 and T N1 x, T N2 x ∈ κ 10 (R). As P N2 0 (x) is not hyperbolic this implies, by Lemma 3.16, the above claim.

It remains to prove the following claim.

Claim 4.2. Except for an entropy-negligible subset of points x ∈ M , the following holds. Given some 1 ≤ i ≤ r with s(i) > 0, Q := P mi1 ni (x) and integers n i , m i1 , m is , there are only two possibilities for P mis ni (x) (s denotes s(i)).

Proof: Let R, R ′ ∈ R be the rectangles containing T ni x, T mi1 x and let ℓ be the line segment through T ni x, directed by W s (T ni x) and bounded by ∂R. We first show that ℓ ⊂ Q. T ni x / ∈ κ s (R) as τ (x) > n i . Thus W s (T ni x) does not s-cross R: ℓ ⊂ W s (T ni x). There exists k > n i such that T k-ni ℓ is not contained in the closure of an element of P . Take k ≥ 1 minimal. If one had k > m i1 , then T mi1-ni ℓ ⊂ W s (T mi1 x) (recall that T mi1 x ∈ κ s (R ′ )) so that for all k ≥ m i1 , T k-ni ℓ would be contained in an element of P , implying ℓ ⊂ W s (T ni x), a contradiction. Thus, k ≤ m i1 and ℓ ⊂ Q as claimed.

Disregarding an entropy-negligible set of points x, we can assume that ℓ divides Q into two subsets with non-empty interiors, say Q + , Q -. There cannot exist stable manifolds that s-cross R both in Q + and Q -: by convexity this would imply that W s (T ni x) also s-crosses. Thus there is an s-rectangle in R disjoint from κ s (R) which contains at least one of Q + , Q -. Let W s (B + ) ("above") and W s (B -) be the stable manifolds bounding this gap (recall Fact 3.8). Also at least one of W s (B ± ) (say W s (B + )) is not contained in Q so the interior of Q does not meet W s (B + ). Thus Q determines W s (B + ): it is the "lowermost" stable manifold "above" Q which crosses R. Likewise W s (B -) is the "uppermost" stable manifold "below" W s (B + ) which crosses R.

By definition, S := P mis 0 (x) is hyperbolic. Also m is ∈]]n i , n i+1 [[ is not admissible hence there exists 0 < k < m is < n i+1 such that S is included in an admissible k-strip and T k (S) meets an admissible strip. If k < n i , then the same admissible strip would preclude the admissibility of P ni 0 (x) is admissible. The contradiction shows that k ≥ n i . As n i+1 is the smallest admissible time after n i , k = n i .

Thus T ni (S) meets an admissible strip which must be either "above" W s (B + ) or "below" W s (B -). This implies that P mis ni (x) meets and therefore contains W s (B ± ) ∩ R (for one of the signs ±). It follows that Q determines P mis ni (x), up to a binary choice.

Combinatorial Estimates.

Remark 4.3. In the remainder of this section, the controlled sets and return times are understood to be with respect to the partition P + and the Markov array R + ⊂ P + of some periodic extension as defined in Definition 3.5.

We extract from Proposition 4.1 the following complexity bound. For each positive integer L, let (M + , T + , P + ) be the (L, R)-extension of (M, T, P ).

Let L, N ≥ 1, M, R, S ≥ 0 be some integers. Consider I = I(N, L, M, R, S) the set of cylinders (P + ) N (x) for x ∈ κ s (R + ) such that, in the notations of Proposition 4.1 applied to the periodic (L, R)-extension: (1) n 1 is the smallest integer such that (P + ) n1 0 (x) is hyperbolic and

• τ (x) ≥ N ; • r = R and #{m ij ≥ N 0 : 1 ≤ i ≤ r, 1 ≤ j ≤ s(i)} = S; • N 2 -N 1 = M . Let ρ > R/N . Then:
n 1 ≤ N 0 ; (2) {n i : 1 ≤ i ≤ r} ⊂ {0 ≤ k < N : T k + x ∈ R + \ κ(R + )}; (3) {N 0 ≤ k < N : T k + x ∈ κ 10 (R + )} ⊂ {m ij : 1 ≤ i ≤ r and 1 ≤ j ≤ s(i)}.
Proof: By definition n 1 is the smallest integer such that (P + ) n1 0 (x) is an admissible strip so (1) is just Fact 3.20.

(P + ) ni 0 (x) being an admissible strip,

T ni + x ∈ R + . n i < N so T ni + x / ∈ κ(R + ), proving (2).
The

m ij are the times m ∈]]n 1 , N [[ (or, equivalently, m ∈ [[0, N [[ by property (1)) such that T m + x ∈ κ s (R +
) and (P + ) m 0 (x) is a hyperbolic, but not admissible, strip. As (P + ) N0 0 (x) is a hyperbolic strip and T N0 + x ∈ κ 10 (R + ), Corollary 3.14 gives that N 0 ≤ k < N and T k + x ∈ κ 10 (R + ) implies that (P + ) k 0 (x) is a hyperbolic strip. This strip cannot be admissible as

T k + x ∈ κ s (R + ) and k < N , hence such k is some m ij , proving (3).
Proof of Proposition 4.4: According to Proposition 4.1, given N , M , R and S, to determine an element of I(N, L, M, R, S) we need to specify:

(1) the integers N 1 , N 2 , n 1 , . . . , n R and m i1 , m is(i) for i = 1, . . . , R;

(2) the itineraries (P + ) N1 (x), (P

+ ) n1-1 N2 (x) (if n 1 > N 2 ) and (P + ) N -1 nr +1 (x); (3) (P + ) mi1 ni (x), (P + ) ni+1 m is(i) (x) for each i = 1, . . . , r; (4) a choice among 4 • 2 R . Observe that N 1 ≤ N 2 ≤ N 0 . Using property (3) of Lemma 4.5 (in particular N 0 is some m ij ), it follows that: # r i=1 ]]m i1 , m is(i) [[ \ ]]N 1 , N 2 [[ ≥ # r i=1 ]]m i1 , m is(i) [[ \ ]]0, N 0 [[ ≥ # i = 1, . . . , r 1 ≤ j < s(i) m ij ≥ N 0 ]]m ij , m ij + L[[ ≥ (S -R)(L -1) ≥ (S -R)L -S
recalling the definitions of L and S. Hence, the number of choices for those items is bounded by:

(1) N 2 N R 3 where a b = a!/b!(a -b)! is the binomial coefficient; (2-3) C R+2 * exp ((h top (T ) + ǫ * )(N -(S -R)L -M + S))); (4) 4 • 2 R . Recalling that 3 n αn ∼ 1 √ 2πα(1-α) n -1/2 e H(α)n as n → ∞, i.e., log n αn ≤ H(α)n + C(α) and that k ∈ [[0, (n-1)/2]] → n
k is increasing, the stated bound follows with: K * (ρ, N ) = 3H(ρ) + ρ log 2 + 3N -1 log N + N -1 log 4C(ρ). 4.3. Large Average Return Times and Entropy. We are going to apply the previous estimates linking long return times either to visits to the holes (R \ κ(R)) or to low entropy. We will show that for a suitable choice of the parameters of our constructions, large entropy measures have finite average return time.

Recall the good return time τ : κ s (R) → N (possibly infinite) of Definition 3.18. We define τ n (x), n ≥ 1, inductively by τ 1 (x) = τ (x) and

τ n+1 (x) = τ (T τn(x) (x)) (τ n+1 (x) = ∞ if τ n (x) = ∞).
The essential supremum of a function f over a subset X with respect to a measure µ is:

µ-sup x∈X f (x) := inf X ′ =X[µ] sup x∈X ′ f (x)
where X ′ ranges over the measurable subsets of X such that µ(X \ X ′ ) = 0 (X and f are assumed to be measurable). Our key estimate is the following: Proposition 4.6. There exist h 2 < h top (T ) and L 2 < ∞ with the following property. Consider the Markov array R defined by Proposition 3.6. For any integer L + ≥ L 2 , let (M + , T + , P + , R + ) be the L + -periodic extension of Definition 3.5.

3 f (t) ∼ g(t) iff limt→∞ f (t)/g(t) = 1 and H(t) = -t log t -(1 -t) log(1 -t).
Then, for each µ ∈ P erg h2 (T + ), the good return time with respect to P + and R + , satisfies:

τ * (µ) = µ-sup x∈R+ τ * (x) < ∞ where τ * (x) := lim sup n→∞ 1 n τ n (x).
Remark 4.7.

(1) τ * (x) < ∞ a.e. already ensures that almost every point in κ(R) has a good return. But we shall need more.

(2) The proof below does not provide a semi-uniform bound on τ * as our estimates below depend on the speed of convergence of some ergodic averages (see Remark 4.8

below).

Proof of Proposition 4.6: The first step of the proof fixes a Markov array R and a periodic extension of T and finds a candidate upper bound for τ * (µ). The second step defines a language (a collection of words of increasing lengths) with small entropy. The final step shows that large average return times imply that this language is enough to describe the measure. A large average can therefore happen only for low entropy measures.

Step 1: The Markov Array We apply Proposition 3.6 and obtain first numbers µ 0 > 0 and h 0 < h top (T ). We let 0 < ǫ 0 < min(h top (T ), 1)/200 be small enough so that in the notations of Proposition 4.4:

(4.1) K * (ǫ 0 µ 0 ) < µ 0 100 h top (T ).
We pick L * to be so large that, for all ℓ ≥ L * (4.2) K * (ǫ 0 µ 0 , ℓ) < µ 0 100 h top (T ).

Proposition 3.6 now gives an integer L 2 (T, ǫ 0 , L * ) ≥ L * and a Markov array R such that the following holds.

For each L + ≥ L 2 , any µ ∈ P erg h0 (T ) can be lifted to an ergodic invariant probability measure µ + on the periodic extension (M + , T + , P + , R + ) satisfying:

L + µ + (κ 10 (R + )) > µ 0 and L + µ + (R + \ κ(R + )) < ǫ 0 µ 0 .
Recall that P R denotes the convex partition of M generated by P and R.

We fix ǫ * := (µ 0 /100)h top (T ) and define C * = C * (P R , ǫ * ) < ∞ as in Proposition 4.4. Note that C * does not depend on L + . Hence, possibly after increasing L + , we may assume that

L + > log C * (µ 0 /100)h top (T )
.

Fix ℓ * ≥ L * so large that 3 log C * /ℓ * < (µ 0 /100)h top (T ).

We omit the sharp subscript in the sequel so that M, T, P, µ, µ 0 will denote in fact M + , T + , P + , µ + , µ 0+ (in particular L + µ 0 is the original µ 0 ). To refer to the original µ or µ 0 , we shall write µ ♭ or µ 0♭ . It will be a convenient exception to continue to write P R for P R ♭ ♭ .

We let

h 2 := max(h 0 , h top (T ) (1 -0.9L + µ 0 )) < h top (T )
and fix some µ ∈ P erg h2 (T ) together with µ + as above. According to the Birkhoff ergodic theorem, one can find K 1 ⊂ M and L 1 < ∞ such that (4.3) µ(M \ K 1 ) < ǫ 0 µ 2 0 /(10 6 log #P R ) and, for all x ∈ K 1 : (4.4)

∀n ≥ L 0 := ǫ 0 µ 0 1000 L 1 1 n #{0 ≤ k < n : T k x ∈ κ 10 (R)} -µ(κ 10 (R)) < µ 0 1000 2 . ∀n ≥ L 1 1 n #{0 ≤ k < n : T k x ∈ R \ κ(R)} < ǫ 0 µ 0 .
Remark 4.8. The above L 1 is the only estimate in the proof of this proposition which does not seem semi-uniform.

Increasing ℓ * if necessary, we assume

ℓ * ≥ (1000/µ 0 )L 1
and ℓ ℓ ℓ/ℓ * ≤ e (L+µ0/100)htop (T )•ℓ for all ℓ ≥ ℓ * . We set:

(4.5) τ max := 1000 log #P R µ 0 ℓ * .
To prove that τ * (µ) ≤ τ max , we assume by contradiction that (4.6)

M 1 := {x ∈ κ(R) : lim sup n→∞ 1 n τ n (x) > τ max } has positive µ-measure.
Step 2: Low entropy language

For each integer ℓ ≥ 1 we define a set C(ℓ) of P R -words of length ℓ as

C(ℓ) := ℓ1 + • • • + ℓ k = ℓ k ≤ ℓ/τmax C(ℓ 1 , . . . , ℓ k )
Here C(ℓ 1 , . . . , ℓ k ) is the set of all concatenations γ 1 . . . γ k where each γ m (1 ≤ m ≤ k) is a word (i.e., a finite sequence) of length |γ m | = ℓ m satisfying:

• type 1 requirement: γ m is an itinerary from I(ℓ m , L + , M, R, S) (in the notation of Proposition 4.4) with: (4.7)

ℓ m ≥ ℓ * , L + (S -R) + M -S ≥ 98 100 L + µ 0 ℓ m and R < min(h top (T ), 1) 200 L + µ 0 ℓ m ;
Recall that L + µ 0 is the original µ 0 , independent on L + . • type 2 requirement: the sum of the lengths of these segments is less than

(4.8) (h top (T ) + ǫ * ) (ℓ k -(L + (S -R) + M -S)) + K(ǫ 0 µ 0 , ℓ k )ℓ k + (2R + 3) log C * ≤ h top (T ) 1 + L + µ 0 100 1 - 98 100 L + µ 0 ℓ k + 2L + µ 0 100 h top (T )ℓ k + 3 log C * ≤ h top (T ) 1 - 94 100 L + µ 0 ℓ k + 3 log C * ≤ h top (T ) (1 -0.93L + µ 0 ) ℓ k , Hence, (4.9) #C(ℓ) ≤ exp ((L + µ 0 /100)h top (T )ℓ) × exp (h top (T ) (1 -0.93L + µ 0 ) ℓ) × (#P R ) (L+µ0/500 log #P R )ℓ ≤ e h2ℓ .
Step 3: Consequence of Large Return Times

We are going to show that, for all x ∈ κ(R), all large enough integers n:

(4.10) τ n (x) > τ max • n =⇒ P τn(x) (x) ∈ C(τ n (x)).
Observe that this will imply that, any ergodic and invariant measure µ such that τ * (µ) > τ max has entropy at most h 2 using Proposition A.2 with (4.9) and:

• M 0 := {x ∈ M : {n ≥ 0 : T -n x ∈ M 1 } is infinite} (recall eq. (4.6)); • a i (x) := min{j ≥ i : T -j x ∈ M 1 } for all i ≥ 1; • b i (x) := τ n (T -ai(x) x) with n a positive integer such that τ ni (x) ≥ max(a i (x), τ max • n i ),
concluding the proof of Proposition 4.6. We now prove (4.10).

Let x ∈ κ(R). We consider a large integer n such that τ n (x) > τ max • n. (4.3) and the ergodic theorem give:

1 τ n (x) #{0 ≤ k < τ n (x) : T k x / ∈ K 1 } < ǫ 0 µ 2 0 /(10 6 log #P R ).
Let N := τ n (x) and, for k = 0, . . . , n-1, let I k be the integer interval

[[τ k (x), τ k+1 (x)[[ and ℓ k := #I k . Let B 1 ⊂ [[0, n[[ be the set of those integers 0 ≤ k < n such that #{m ∈ I k : T m x / ∈ K 1 } ≥ ǫ 0 µ 0 1000 ℓ k .
The union of those segments I k occupies only a small proportion of [[0, N [[:

k∈B1 ℓ k ≤ 1000 ǫ 0 µ 0 × ǫ 0 µ 2 0 10 6 log #P R N ≤ µ 0 1000 log #P R • N. Let B 2 ⊂ [[0, n[[ be the set of k's such that ℓ k ≤ ℓ * .
They also occupy a small proportion:

k∈B2 ℓ k ≤ ℓ * n ≤ ℓ * N τ max ≤ µ 0 1000 log #P R • N,
by the choice of τ max .

Therefore the segments I k for k ∈ B 1 ∪ B 2 satisfy the type 2 requirement in the definition of C(ℓ). It is enough to prove that the remaining I k s satisfy the type 1 requirement.

For such segments I k , p 1 := min{p ≥ 0 : T p+τ k (x) x ∈ K 1 } satisfies:

ℓ k ≥ ℓ * ≥ 1000 µ 0 L 1 = 10 6 ǫ 0 µ 2 0 L 0 .
This fulfills the first requirement of (4.7). Hence, in the notations of Proposition 4.4:

(4.13)

N 1 := min{j ≥ 0 : T τ k (x)+j x ∈ κ 10 (R)} ≤ p 1 + L 0 ≤ ǫ 0 µ 0 500 ℓ k .
Also, by eq. ( 4.4) and ℓ k -

p 1 ≥ L 1 : #{j ∈ [[τ k (x) + p 1 , τ k+1 (x)[[: T j x ∈ R \ κ(R)} < ǫ 0 µ 0 (ℓ k -p 1 ).
Hence, using point (2) of Lemma 4.5:

(4.14) R ≤ r ′ := #{j ∈ I k : T j x ∈ R \ κ(R)} < 2ǫ 0 µ 0 ℓ k .
Note that this implies R ≤ ℓ k • µ 0 min(h top (T ), 1)/200, a part of the type 1 requirement. It remains to show the lower-bond on L + (S -R) + M . First, similarly to (4.14):

#{j ∈ I k : T j x ∈ κ 10 (R)} -µ(κ 10 (R))ℓ k < µ 0 500 ℓ k .
Setting, again as in Proposition 4.4:

N 0 := min{j ≥ 0 : T τ k (x)+j (x) ∈ κ 10 (R) and P j (T τ k (x) x) is hyperbolic} (observe that N 0 might be large) and S := #{m ij > N 0 : i, j} we get, using point (3) of Lemma 4.5,

S ≥ s ′ := #{j ∈]]τ k (x) + N 0 , τ k+1 (x)[[: T j x ∈ κ 10 (R)}. Also, M := N 2 -N 1 = [N 0 ] κ -N 1 where: [t] κ := max{n ∈ [[0, t[[: T n x ∈ κ 10 (R)}
(we define [t] κ := t if there is no such integer). To complete our estimate, we consider two cases.

-First case: N 0 < p 1 + L 1 . We use the trivial bound M ≥ 0, (4.11), (4.12) and (4.4) to get:

s ′ ≥ #{j ∈ [[τ k (x)+p 1 +L 1 , τ k+1 (x)[[: T j x ∈ κ 10 (R)} ≥ 499 500 µ 0 (ℓ k -p 1 -L 1 ) ≥ 498 500 µ 0 ℓ k Hence, (4.15) L + S + M ≥ 498 500 L + µ 0 ℓ k ≥ 99 100 L + µ 0 ℓ k .
-Second case: N 0 ≥ p 1 + L 1 . Using the definition of p 1 , K 1 and L 0 :

s ′ ≥ µ(κ 10 (R)) (1 -10 -3 )(ℓ k -p 1 ) -(1 + 10 -3 )(N 0 -p 1 )
.

(the case of improper orbits is similar and easier and left to the reader). It is now enough to apply Proposition A.2 using eqs. (4.9-4.10) to get that h(f, µ) ≤ h 2 , a contradiction.

Proof of the Main Results

We finally prove the main results by building a Markov system from the arbitrary concatenations of admissible strips and relating it to the dynamics of the piecewise affine homeomorphism. This is done in the analysis of large entropy measures and then used for the other claims.

5.1. Maximal Entropy Measures. We prove Theorem 1 about the finite number of maximum measures.

Step 1: Tower Fix a Markov array R as in Proposition 3.6, defining a (R, L + )-periodic extension (M + , T + , P + , R + ) of (M, T, P, R). This may only increase the number of maximum measures as it is a finite topological extension. Let Σ := Σ(T + , P + ) be its symbolic dynamics (see Definition 2.1). Corollary 2.10 shows that it is enough to prove the results for Σ.

We now build an invertible tower (see Appendix B) Σ over Σ. This is done by defining a return time τ : Σ τ → N * for some Σ τ ⊂ Σ. Definition 5.1. An extended admissible P + -word is a word w 0 . . . w n over P + such that [w 0 . . . w n ] is an admissible strip. w 0 . . . w n-1 is the associated admissible P + -word.

For a sequence A ∈ Σ, we define inductively t 1 (A) := sup{n ≥ 1 : A 0 . . . A n is an extended admissible word} ∈ N * ∪ {-∞, ∞} and t n+1 (A) = t 1 (σ tn(A) (A)) (or t n (A) if it was infinite). Let Σ τ := {A ∈ Σ : ∀n ≥ 1 t n (A) ∈ N * }.

We tacitly exclude entropy-negligible subsets of points of M + and of Σ (these correspond by Lemma 2.10). Claim 5.2. Σ τ coincides with the set of P + -itineraries of the points x ∈ κ s (R + ).

Proof of the Claim: x has a finite good return time m := τ (x) by Proposition 4.6. Let A ∈ Σ be its itinerary. Observe that S 0 := A 0 . . . A m-1 is admissible and T m + x ∈ κ s (R + ). By induction, A splits into a concatenation of admissible words S 0 S 1 S 2 . . . . The cylinders defined by the finite concatenations S 0 S 1 . . . S k A τ k+1 (x) , k ≥ 0, are hyperbolic strips.

Clearly t 1 (A) ≥ τ (x). Now, if H := (P + ) n 0 (x) is hyperbolic with n > m, then H is contained in S 0 (an m-admissible strip) and T m H meets the hyperbolic strip S 1 . Hence H cannot be admissible, proving that t

1 (A) = τ (x) < ∞. t n (A) < ∞ for all n ≥ 1 follows from invariance. Hence, A ∈ Σ τ .
For the converse, let A ∈ Σ τ and denote by x the point with itinerary A.

[A 0 . . . A m ] is an admissible strip for m = t 1 (A) so A 0 ∈ R. If k = t n (A), then [A 0 . . . A k ]
is a concatenation of admissible strips, hence a k-strip. k being arbitrarily large, it follows that W s (x), the intersection of the previous strips, must cross A 0 , proving x ∈ κ s (R).

By construction, for all A ∈ Σ τ , σ t1(A) (A) ∈ Σ τ . Hence t 1 is a return time and defines an invertible tower T : Σ → Σ in the sense of Appendix B. Moreover, Corollary 4.10 shows that any large entropy measure µ on Σ(T + , P + ) has eventually bounded gaps in the sense of Definition B.2. By Proposition B.3, any such measure can be lifted to Σ and any invariant probability measure of Σ is a finite extension of one in Σ(T + , P + ) (in particular both measures have the same entropy).

It follows also that h( T ) := sup µ h(σ|hS, µ) = h top (Σ) so that maximum measures of Σ lift to maximum measures of T .

To prove the theorem it is therefore enough to show that the tower Σ has finitely many ergodic measures of maximal entropy.

Step 2: Markov Structure Recall the following definition (see [START_REF] Gurevič | Thermodynamic formalism for symbolic Markov chains with a countable number of states, (Russian)[END_REF] for background):

Definition 5.3. A Markov shift is a space of sequences Σ(G) := {x ∈ V Z : ∀n ∈ Z x n → x n+1 in G}
where G is an oriented graph with a countable set of vertices V together with the left shift σ.

We also recall that a graph G as above is (strongly) irreducible if for every (A, B) ∈ V 2 , there is a path from A to B on G. An (strongly) irreducible component is a subgraph with this property maximum w.r.t. inclusion.

It will be convenient to say that w ′ is a follower of w if the two are admissible words w, w ′ in the sense of Definition 5.1, and s being the first symbol of w ′ , the concatenation ws is an extended admissible word.

(w, i) → (w, i + 1) if i + 1 < |w|, (w, |w| -1) → (w ′ , 0) if w ′ is a follower of w.
The tower Σ is measurably conjugate 4 to the Markov shift Σ(G).

Proof of the Claim: Define p : Σ(G) → Σ by p(α) = (A, ω) ∈ Σ × {0, 1} Z where, if α n =: (w 0 . . . w ℓ-1 , k), then A n = w k and ω n = 1 if and only if k = 0. Observe that the sequence A thus obtained is a concatenation of admissible words so A ∈ Σ. Also, whenever m and n are two successive integers with ω m = ω n = 1, A n A n+1 . . . A m is an admissible word. Finally ω m = 1 for infinitely many positive and negative integers m. The proof of Claim 5.2 shows that σ m A ∈ Σ τ for all such m, so that (A, ω) ∈ Σ. Thus p is well-defined and is clearly a measurabke conjugacy.

Step 3: Conclusion Proof of Theorem 1: G has at most one irreducible component for each vertex of the type (w 0 , 0). These corrrespond to the finitely many rectangles in the Markov array R. Hence, by a classical result of Gurevič [START_REF] Gurevič | Shift entropy and Markov measures in the space of paths of a countable graph[END_REF], Σ(G) with h(Σ(G)) < ∞ has finitely many maximum measures, proving the Main Theorem.

Number of Periodic

Points. We prove Proposition 1.4 about the number of periodic points. We use the construction of the proof of Theorem 1. To prove the lower bound (1.1). It is enough to prove it for T + , as T + is a finite extension of T . We assume by contradiction that the number of points fixed under T n + is such that for any integer p ≥ 1, there is a sequence n k → ∞ of multiples of p such that (5.1) lim

n k →∞ N T+ (n k )
e n k htop(T ) = 0 In the following we denote (M + , T + , P + , R + ) by (M, T, P, R).

The starting point is the following estimate for Σ(G). Consider a maximum measure. It is carried on one irreducible component. For simplicity, we replace G by that irreducible component. By Gurevič [START_REF] Gurevič | Shift entropy and Markov measures in the space of paths of a countable graph[END_REF], the existence of a maximum measure for Σ(G) implies that G is positive recurrent with parameter R = e -htop(T ) . By Vere-Jones [START_REF] Vere-Jones | Ergodic properties of nonnegative matrices[END_REF], this implies that the number N G (n) of loops of length n based at a given vertex satisfies, for some positive integer p:

(5.2) lim n→∞, p|n N G (n)e -htop(T )n = p Each n-periodic sequence A in the symbolic dynamics Σ is associated to a closed, convex set j≥0 T j + [A -j . . . A j ]
invariant under T n + . This set contains at least one point fixed by T n + which we denote by π(A). It remains to show that π does not identify too much points. Consider π from the set Σ(n) of n-periodic sequences to the set M + (n) of nperiodic T + -orbits. Our assumption (5.1) implies that, for some sequence m n → ∞ (where p|n) this map is at least m n -to-1 on a subset Σ ′ (n) of Σ(n) with cardinality at least e nhtop(T ) /3. We use the following observation: Lemma 5.5. Let A 1 , . . . , A m ∈ Σ be such that π(A 1 ) = • • • = π(A m ) =: x. If the finite words A i 0 . . . A i n-1 , i = 1, . . . , m, are pairwise distinct, then: (1) either T k x is a vertex of P + for some 0 ≤ k < n;

(2) or there exist r ≥ (m -1)/2 distinct integers 0 ≤ n 1 < • • • < n r < n such that for all (k, l) with 1 ≤ k < ℓ < n, T n k x lies on the interior of an edge of P + . Moreover, if v k is the direction of the open edge containing T n k x, then the image by

T n ℓ -n k of v k is transverse to v ℓ . Proof: Let A i 0 . . . A i n-1 , i = 1, .
. . , m be finite words as in the above statement. We show that the failure of (1) implies [START_REF] Bonatti | Beyond Uniform Hyperbolicity[END_REF].

Each word A i 0 . . . A i n-1 defines an element of P n + containing x in its closure so mult(x, P n + ) ≥ m. We assume that (1) fails. Observe that (i) mult(x,

P k+1 + ) = mult(x, P k + ) if T k x is in the interior of an element of P + or if for all A ∈ P k + , P k + T k + (A ∩ B(x, ǫ
)) ⊂ B for some B ∈ P + and ǫ > 0. (ii) mult(x, P k+1 + ) ≤ mult(x, P k + ) + 2. (ii) uses that T + is a piecewise affine surface homeomorphism so the preimage of an edge may locally divide in at most two subsets at most two of elements of P k + touching x.

Such lifting requires that τ be "not too large" (see [START_REF] Zweimuller | Invariant measures for general(ized) induced transformations[END_REF] where the classical integrability condition is studied). Our condition is in terms of the following "iterates" of τ : the functions τ m : B → N * , m ≥ 1, are defined, as before, by: τ

1 := τ and τ m+1 (x) := τ (T τm(x) (x)) if τ m (x) < ∞, τ m+1 (x) := ∞ otherwise. Definition B.2. x ∈ X has an improper orbit if (B.3) n(x) := {n ∈ N : T -n x ∈ B and ∀m ≥ 1τ m (T -n x) < ∞} is finite.
x ∈ X has t-gaps for some 0 < t < ∞ if x has an improper orbit or if there exist two integer sequences (n k ) k∈N and (m k ) k∈N , m k > 0 for all k ≥ 0, such that:

∀k ≥ 0 τ m k (T n k x) ≥ max(t•m k , k) and sup k≥1 min{|i| : i ∈ [n k , n k +τ m k (T n k x)]} < ∞.
A measure has eventually bounded gaps, if for some t < ∞, the set of points in X with t-gaps has zero measure.

Note that τ (T n x) = ∞, for a single n, implies that x has t-gaps for any t < ∞.

Proposition B.3. Let T : X → X be a self-map with a return time τ : B → N * . Then:

• every T -invariant ergodic probability measure µ with eventually bounded gaps can be lifted to a T -invariant ergodic probability measure on X; • any T -invariant, ergodic probability measure μ is a finite extension of the T -invariant measure π(μ).

Proof of Proposition B.3: We first prove the existence of a lift for µ like above. We follow the strategy of [START_REF] Zweimuller | Invariant measures for general(ized) induced transformations[END_REF] and [START_REF] Keller | Lifting measures to Markov extensions[END_REF] (which was inspired by constructions of Hofbauer) and define the following non-invertible tower to get a convenient topology:

X := {(x, k, τ ) ∈ X × N × N : ∃y ∈ B τ (y) = τ, k < τ and x = T k y} T (x, k, τ ) := (T (x), k + 1, τ ) if k + 1 < ℓ, (T (x), 0, τ (T (x))) otherwise.
For any integer K, we write XK := {(x, k, τ ) ∈ X : k = K}, X≤K := k≤K Xk and define π(x, k, τ ) = x. Observe that π • T = T • π and that ( X, T ) is a natural extension of ( X, T ) through (x, ω) → (x, k, ℓ) with k ≥ 0 minimal such that ω -k = 1 and ℓ = τ (T -k x). Hence it is enough to lift µ to X. Fix t < ∞ such that the set of points of X with t-gaps has zero µ-measure. Let μ0 be the probability measure defined by μ0 ({(x, 0, τ (x)) : x ∈ A}) = µ(A) for all Borel sets A (sets disjoint from the above ones have zero μ0 -measure). We have π(μ 0 ) = µ but, except in trivial cases, T * μ0 = μ0 so we consider:

μn := 1 n n-1 k=0 T k μ0
and try to take some accumulation point μ. We identify μn with its density with respect to μ∞ , the σ-finite measure defined by μ∞

({(x, k, τ (T -k x)) : x ∈ A and k < τ (T -k x)}) = µ(A ∩ {τ > k})
for all Borel sets A ⊂ B and all k ≥ 0. As π(μ n ) = µ, we must have:

dμ n dμ ∞ ≤ 1.
Using the Banach-Alaoglu theorem, i.e., the weak star compactness of the unit ball of L ∞ (μ ∞ ) as the dual of L 1 (μ ∞ ), we get an accumulation point of the μn , i.e., a measure μ on X with dμ/dμ ∞ ≤ 1 such that, for some subsequence n k → ∞,

(B.4) ∀f ∈ L 1 (μ ∞ ) lim k→∞ f dμ n k = f dμ.
Observe that μ is T -invariant: indeed, eq. (B.4) together with the T -invariance of µ implies that dμ • T -1 /dμ ≤ 1 whereas μ • T -1 ( X) = μ( X) so the previous inequality must be an equality μ-almost everywhere.

This invariance and the ergodicity of µ implies that π μ = αµ for some 0 ≤ α ≤ 1. It remains to prove that μ = 0 so that it can be renormalized into the announced lift of µ. Assume by contradiction that μ = 0. Hence, for any L < ∞: Hence eq. (B.5) implies that x (in fact any of its preimages in the natural extension) has t-gaps for all t > 0, contradicting the assumption on µ.

1 X≤L dμ n k = 1 n k #{0 ≤ k < n k : T k (x,
We now show that any T -invariant, ergodic probability measure μ is a finite extension of µ := π(μ). By definition of X, μ( [START_REF] Billingsley | Ergodic theory and information[END_REF]) > 0 where [START_REF] Billingsley | Ergodic theory and information[END_REF] = {(x, ω) ∈ X : ω 0 = 1}. Assume that there is some positive measure subset S ⊂ X, and some number K of measurable functions: ω 1 , . . . , ω K : S → {0, 1} Z such that, for all x ∈ S, (x, ω i (x)) ∈ X, ω i (x) = ω j (x) for i = j and, for all j = 1, . . . , K:

lim n→∞ 1 n #{0 ≤ k < n : ω j -k (x) = 1} = μ([1]). If K • μ([1]
) > 1, then, for a.e. x ∈ S, there exist two distinct indices j, j ′ ∈ {1, . . . , K} and arbitrarily large integers n k → ∞ such that ω j -n k (x) = ω j ′ -n k (x). But this implies ω j (x) = ω j ′ (x) by (B.2). The contradiction proves K ≤ μ([1]) -1 < ∞: μ is a finite extension of µ.

Appendix C. Examples

Positive Multiplicity Entropy

Example 1 (see [START_REF] Buzzi | Intrinsic ergodicity of Affine Maps in [0, 1] d[END_REF]). There exists a continuous, piecewise affine surface map (M, T, P ) with h mult (T, P ) > 0 and h top (T ) = 0. Example 2 (see Kruglikov and Rypdal [START_REF] Kruglikov | Entropy via multiplicity[END_REF]). There exists a piecewise affine homeomorphism (M, T, P ) with dim M = 3 and h mult (T, P ) = h top (T ) > 0.

Let ([0, 1] 2 , T 2 , P 2 ) be a piecewise affine homeomorphism with nonzero topological entropy. Consider the pyramid M := [0, 1] 2 where A denotes the convex subset of R 3 generated by O := (0, 0, 0) and A × {1}. Define T : M → M as the piecewise affine map with partition P := { A \ {O} : A ∈ P 2 } such that: T (O) = O, T (x, y, 1) = (T 2 (x, y), 1) for each vertex (x, y) of P 2 . Observe that h top (T ) = h top (T 2 ) and that T has an obvious measure of maximal entropy carried by the invariant set [0, 1] 2 × {1}. Finally, considering T n around (0, 0, 0) it is easy to see that h mult (T, P ) = h top (T 2 ) = h top (T ).

Example 3. There exists a piecewise affine homeomorphism (M, S, P ) with dim M = 3, h mult (S, P ) > 0 and h top (S) = 0. Define S from the previous example T by S(x, y, z) := T (x, y, z)/2 on the pyramid M so that 0 is a sink. To make S onto, add a symmetric pyramid M -whose summit is a source. It is easy to see that h top (T ) = log 2. We claim that sup µ h(T, µ) = log 2. Clearly the supremum is bounded by h top (f ). Conversely, for any h < log 2, one can find an invariant measure on the full shift (σ, {0, 1} N ) such that µ([1 K ]) = 0 for some K = K(h) < ∞ with h(σ, µ) > h. It is then easy to construct an isomorphic T -invariant measure (with support included in y ≤ y 0 for any given 0 < y 0 < 1), proving that sup µ h(T, µ) ≥ log 2. The equality follows from h top (T ) = log 2. The same observations allow the construction of the sequence µ n with the claimed properties.

No Maximal Entropy Measure

On the other hand, assume that µ is an invariant and ergodic probability measure with h(T, µ) = 2. µ must be supported on y < 1. Hence the map π that sends a point of R 2 to the ray from the origin that contains it, maps (T, µ) to (f, π * µ) where f : θ → 1 -2|θ| on [-1, 1]. The fibers of π are contained in line segments originating from O on which T is linear, hence they have zero entropy and π is entropy-preserving [START_REF] Bowen | Entropy for group endomorphisms and homogeneous spaces[END_REF]. This implies that π * µ is the (1/2, 1/2)-Bernoulli measure. Using, say the Central Limit Theorem, we get that, for µ-a.e. (x, y) ∈ ABO, there exists a positive integer n such that #{0 ≤ k < n : f k (π(x, y)) < 1/2} < n 2 -| log y| 2 log 2 so that T m (x, y) ∈ XY BA for some m ≤ n, contradicting the invariance of the measure: there is no maximum measure.

Example 5. There exists a continuous, piecewise quadratic surface map T such that for any invariant probability measure µ: h(T, µ) < sup ν h(T, ν).

On the rectangle [1, 2] × [-1, 1], consider T (x, y) := (x, T x (y)) with:

T x (y) = x(2-x) 2 -x|y| if |y| < 2 -x -x(2-x) 2 otherwise For each 1 ≤ x < 2, [-1, 1] is mapped into the T x -forward invariant segment [-x(2-x) 2 , x(2-x)

2

] on which T x has constant slope x. Hence h top (T x ) = log x for x = 2. Clearly, T (2, y) = (2, 0) so h top (T 2 ) = 0.

Infinitely Many Maximal Entropy Measures

Example 6. There is a piecewise affine continuous map, resp. homeomorphism, of [0, 1] 2 , resp. [0, 1] 3 , with uncountably many ergodic invariant probability measures with non-zero, maximal entropy. Indeed, such examples are trivially obtained from piecewise affine maps on [0, 1] or homeomorphisms on [0, 1] 2 with non-zero topological entropy by taking a direct product with the identity on the unit interval. It is the low dimension (one for maps, two for homeomorphisms) that prevents the existence of such indifferent factors and ensures the finite number of maximum measures under the simple condition of nonzero topological entropy.

  (1.2) h(T, K) := lim ǫ→0 h(T, K, ǫ) with h(T, K, ǫ) := lim sup n→∞ 1 n log r(ǫ, n, K)

  ǫ→0 h(T, µ, ǫ) with h(T, µ, ǫ) := lim sup n→∞ 1 n log r(ǫ, n, µ)

Lemma 2 . 4 .lim sup n→∞ 1 n

 241 Let d ≥ 1. For each n ≥ 0, let T n : R d → R d be a linear map. Then lim ǫ→0 log max{#S : ∀0 ≤ k < n diam(T k-1 . . . T 1 T 0 S) ≤ 1 and

Figure 1 .

 1 Figure 1. The convex partition P refining both P (outside lines) and R (the two quadrilaterals at the center).

Fact 3 . 8 .

 38 Let A n ∈ Σ(T, P ) converge to A + . By taking a subsequence, W s (A n ) also converges in the Hausdorff metric, say to H ⊂ M . Then H ⊂ W s (A + ).

Figure 2 .

 2 Figure 2. From left to right: a rectangle R, an s-rectangle H and a u-rectangle V . The (approximately) horizontal, resp. vertical, line segments are segments of stable, resp. unstable manifolds.

Figure 3 .

 3 Figure 3. Proof of Lemma 3.12. From left to right: (a) P n 0 (x) ⊂ P m 0 (x) ⊂ R; (b) the u-rectangle T m (P m 0 (x)) crossing the srectangle P n m (x) (both inside R ′ ); (c)T n P n 0 (x) ⊂ T m P n m (x) ⊂ R ′′ (R, R ′ , R ′′ ∈ R).

  n 0 (x). It is enough to check that T -n {a, b, c, d} lies outside of R. The map being piecewise affine, d(T -n v, T -n a) ≤ (1/9)d(T -n u, T -n v). But d(x, T -n v) ≥ 10 • diam(R). Hence, T -n a is outside of R. The same holds for the pre-images of b, c, d. Hence T -n (abcd) s-crosses R. Therefore Corollary 3.14. Let n ≥ 1 be such that P n 0 (x) is a hyperbolic strip and T n x ∈ κ 10 (R). If m > n satisfies T m x ∈ κ 10 (R), then P m 0 (x) is also a hyperbolic strip.

Figure 4 .

 4 Figure 4. Construction of the hyperbolic strip in the proof of Lemma 3.13 (left: time 0 around R, right: time n around R ′ ; approximately (vertical) horizontal lines are segments of (un)stable manifolds; dashed ones are (pre)images of regular ones. The colored "diamond" is L. The rectangle inscribed in L is the hyperbolic strip.

Figure 5 .

 5 Figure 5. An example of a non-admissible n-strip S. For some 0 < m < n, S is contained in some m-strip S ′ (hence T m S ′ ucrosses) and T m S meets the interior of some hyperbolic strip H. The above represents T m S ⊂ T m S ′ and H. Point x is at the crossing of the stable and unstable lines and point o is a little below, on the same unstable line. The four s-crossing rectangles are the maximum hyperbolic strips. In such a situation, it might be possible to then split the itinerary of S into that of S ′ followed by that of H, yielding a choice for representing the itinerary of any point like o above, at each of its visit to S. Compare with Lemma 3.22.

4. 1 .

 1 Geometric Analysis. We analyze geometrically the implications of a large return time. Proposition 4.1. Let (M, T, P ) be a piecewise affine surface homeomorphism and let R ⊂ P be a Markov array. Let x ∈ κ s (R) and let 1 ≤ N ≤ τ (x).

Proposition 4 . 4 .

 44 Let (M, T, P ) be a piecewise affine surface homeomorphism and let R be a Markov array. Let ǫ * > 0 and let C * = C * (ǫ * ) < ∞ be such that ∀n ≥ 0 #(P R ) n ≤ C * e (htop(T )+ǫ * )n

  log #I ≤ (h top (T ) + ǫ * )(N -L(S -R) -M + S) + K * (ρ, N )N + (ρ + 3/N )N log C * where K * (•) and K * (•, •) are universal 2 functions satisfying K * (ρ, N ) ↓ K * (ρ) when N → ∞ and K * (ρ) ↓ 0 when ρ → 0. The proof of the above will use: Lemma 4.5. In the notation of Proposition 4.1,

L+µ0

  500 log #P ℓ. Observe that the union defining C(ℓ) has at most ℓ ℓ [ℓ/τmax] ≤ e (L+µ0/100)htop(T )ℓ terms. It remains to bound #C(ℓ 1 , . . . , ℓ n ). By Proposition 4.4, the logarithm of #I(ℓ m , L + , M, R, S) under condition (4.7) is bounded by

(4. 11 ) p 1 ≤ ǫ 0 µ 0 1000 ℓ k by the definition of B 1 .

 1111 By the definition of B 2 , (4.12)

Claim 5 . 4 .

 54 Let G be the oriented graph with vertices (w, i) where w is any admissible word and 0 ≤ i < |w| (| • | is the length of the word) and arrows:

  0, τ (x)) ∈ X≤L } dµ → 0 So, possibly for a further subsequence:(B.5) 1 n k #{0 ≤ k < n k : T k (x, 0, τ (x)) ∈ X≤L } → 0 µ-a.e. Now, #{0 ≤ k < n : T k (x, 0, τ (x)) ∈ X0 } < ǫn =⇒ τ ǫn (x) ≥ n.

Figure 6 .

 6 Figure 6. Geometry of (left) a continuous piecewise affine map with h mult (T, P ) > htop(T ) = 0; (right) a discontinuous piecewise affine map with no maximum measure. Consider some triangle ABO in R 2 with non-empty interior and let M, A ′ , B ′ be the middle points of [AB], [AO], [BO] (see Fig. 6, left). Let T be affine in each of the triangles τ 0 := AM O and τ 1 := BM O with T (O) = O, T (A) = T (B) = A ′ , T (M ) = B ′ so that T : ABO → ABO is conjugate to (θ, r) → (1 -2|θ|, r/2) on (-1, 1) × (0, 1). Take P = {τ 0 , τ 1 } as the admissible partition. h mult (T, P ) = log 2 (because all words on {τ 0 , τ 1 } appear in the symbolic dynamics and the corresponding cylinders contain O in their closure). On the other hand, the only invariant probability measure is the Dirac supported by O hence h top (T ) = 0 as claimed.

Example 4 .

 4 There exists a piecewise affine surface (M, T, P ) discontinuous map T such that there is no maximum measure. More precisely there exists a sequence of invariant probability measures µ n with lim n→∞ h(T, µ n ) = h top (T ) > 0 but µ n converges weakly to an invariant Dirac measure.Remark. The above formulation excludes trivial examples likeT : [0, 1] → [0, 1]with T (x) = 1/4 + x/2 for x > 1/2 and T (x) = x + 1/2 for x ≤ 1/2 which has no invariant probability measure.Let T be a piecewise affine map defined on the triangle XY O with O = (0, 0), X = (-2, 2) and Y = (2, 2). Let A = (-1, 1), B = (1, 1) and M = (0, 1), and A ′ = A/2, B ′ = B/2 and M ′ = M/2 (see Fig.6, right). We require:(1) T |XY BA is the identity;(2) T : AM O → A ′ B ′ O is affine with A → A ′ , M → B ′ , O → O; (3) T : M BO → Y XO is affine with M → Y , B → X, O → O.

K * does not depend on any of the data T : M → M, N, M, R, S, L, C * , ǫ * .

That is, there is a bimeasurable bijection ψ : Σ → Σ(G) such that ψ • T = σ • ψ.

Hence, using (1001/1000)L + µ(κ 10 (R)) ≤ 1 and M = [N 0 ] κ -N 1 :

In light of (4.13), to prove that eq. (4.15) also holds in this second case it is enough to show: Claim 4.9. For any

Proof of claim 4.9: We distinguish two cases. First assume that [t] κ < p 1 + L 0 . Then t, the first visit to κ 10 (R) after [t] κ , is bounded by the first visit after p 1 + L 0 , i.e., using (4.13) and ℓ k ≥ (1000/µ 2 0 ) which follows from (4.12):

µ 0 ℓ k proving the claim in this case. Second we assume that [t] κ > p 1 + L 0 . Then:

proving the claim.

In both cases, eq. (4.15) together with (4.14) implies:

This establishes the remaining part of the type 1 requirement on P τ k+1 (x) Proof: Given µ, a large entropy measure, we fix τ max as in (4.5) and we proceed by contradiction assuming that for each large t > τ max , there is a set of positive µ-measure S with the following property. For each x ∈ S, there exist sequences of integers n k ∈ Z and m k ∈ N * such that:

This implies that mult(x, P n + ) ≤ 1 + 2#{0 ≤ k < n : T k + x is on an edge of P + }. The Lemma follows.

Proof of Proposition 1.4: Only const • n points of T + can satisfy assertion (1) in the above Lemma. As h mult (T + , P + ) = 0, their preimages in Σ ′ (n) are in subexponential number. The remainder of Σ ′ (n) corresponds to points x whose orbit stays off the vertices of P + and which admit distinct sequences

Given such an x, fix 0

The number of such vertices, for given n j , is bounded by const • #P 2n/(mn-1) + . Taking into account the choice of n j , the number of such xs is bounded by: e (2/mn)(htop(T )+ǫ)n for all large n. Thus, for large multiples n of p,

As m n → ∞, this contradicts the Vere-Jones estimate (5.2), proving the lower bound (1.1) and Proposition 1.4.

5.3.

Entropy away from the singularities. Proposition 1.5 is a corollary of the proof of Theorem 1 using the following result: Proposition 5.6 (B.M. Gurevič [START_REF] Gurevič | Shift entropy and Markov measures in the space of paths of a countable graph[END_REF]). Let G be a countable, oriented graph. Let G 0 ⊂ G 1 ⊂ . . . be a non-decreasing sequence of finite subgraphs exhausting G: any vertex and any arrow of G belong to G n with n large enough.

Then

Proof of Proposition 1.5. In the proof of Theorem 1, one has shown that there is a countable oriented graph G such that the corresponding countable state Markov shift whose maximum measures have entropy h top (T ). Let G n be the finite subgraph defined by keeping only the vertices and arrows of G which are on a loop of length at most n and based at one of the finitely many symbols representing an element of the Markov array. The sequence G n exhausts G. The above proposition therefore ensures that, for any ǫ > 0, there is some G N with topological entropy at least h top (T )ǫ.

The projection of this subshift is a compact invariant subset K of M which contains only points with infinitely many visits to the controlled set κ(R) in the future and in the past. If K met the singularity lines of T , there would be such a point x with the additional property that

But this would prevent any future or past visit to κ(R), a contradiction.

Finally, the above construction makes the isomorphism with Σ(G n ) obvious (it is indeed one-to-one as K does not meet the boundary of P ), but the latter is a subshift of finite type as G n is finite.

Appendix A. Bounds on Metric Entropy

We recall some standard notations and facts about entropy and a few consequences. (X, B, µ) is a probability space. H µ (P ) := -A∈P µ(A) log µ(A) denotes the mean entropy of a partition (we shall leave implicit all measurability assumptions). For Y ⊂ X, (µ|Y )(•) := (µ(Y )) -1 µ(• ∩ Y ) (zero if µ(Y ) = 0). For a sub-σ-algebra A of B, the conditional entropy is:

where E(•|A) is the conditional expectation with respect to µ.

First, if P is a partition, Y ⊂ X and A is a sub-σ-algebra of A, then,

Second, the entropy of a measure can be computed as the average of the entropies given the past. More precisely, we have the following statement: Lemma A.1. Let µ be an invariant probability measure for some bimeasurable bijection T : X → X. Let P be a finite, measurable partition. Then:

where with P -is the past partition generated by T n P , n ≥ 1.

In particular, if N (n, x, P ) = #{P n-1 0 (y) : y ∈ X and P -1 -∞ (y) = P -1 -∞ (x)} where P b a (x) := (A n ) a≤n≤b with T n x ∈ A n , then:

Proof: For eq. (A.2), see, e.g., [START_REF] Petersen | Ergodic theory[END_REF]Ex. 4(b) p.243] for entropy as an average of conditional information.

Observe that

Hence the integrand in (A.2) can be replaced by the above right hand side. Eq. (A.3) now follows from the standard bound:

-

In combination with Rudolph's backward Vitali Lemma [30, Theorem 3.9 p.33], this yields the convenient estimate: Proposition A.2. Let µ be an ergodic, σ-invariant probability measure on A Z with finite alphabet A. Assume that there exist a measurable family of subsets W (A -, ℓ) ⊂ P ℓ (for A -∈ A Z-, ℓ ≥ 1) with cardinality bounded by Ce Hℓ and a subset Σ 0 ⊂ A Z of positive measure such that, for all A ∈ Σ 0 , there are sequences of integers

Condition (2) above means that the intervals [[a i , b i ]] do not escape to infinity: they all intersect some [-R, R] for some R large.

Proof: To apply Rudolph's Backward Vitali Lemma, we need

for all large enough i, for all A ∈ Σ 0 . By passing to subsequences, depending on A, we can assume the existence of the (possibly infinite) limits lim i→∞ a i (A), lim i→∞ b i (A) for all A ∈ Σ 0 . Assume for instance that lim i a i (A) = -∞ and lim i→∞ b i (A) < 0 for a.e. A ∈ Σ 0 , the other cases being similar or trivial. By assumption, inf i b i (A) > -∞ for all A ∈ Σ 0 . Restricting Σ 0 we can assume that this infimum is some fixed number b ∈ Z. Replacing Σ 0 by σ min(b,0) Σ 0 ensures eq. (A.4).

Rudolph Lemma implies that for any ǫ > 0, for µ-a.e. A, for all large enough integers n, one can find a disjoint cover of a fraction at least 1

gives that h(σ, µ) ≤ H + 3ǫ log ǫ + ǫ log #A. We conclude by letting ǫ → 0 + .

Appendix B. Tower Lifts

We study towers from a point of view closely related to that of Zweimuller [START_REF] Zweimuller | Invariant measures for general(ized) induced transformations[END_REF]. Let T be an ergodic invertible transformation of a probability space (X, µ) and let B be a measurable subset of X. A return time is a function τ : B → N * := {1, 2, . . . , ∞} which is measurable and such that T τ (x) (x) ∈ B for all x ∈ B with τ (x) < ∞ (but τ is not necessarily the first return time).

We are interested in lifting T -invariant measures to the following invertible tower:

(B.1) X := {(x, ω) ∈ X × {0, 1} Z : ω n = 1 =⇒ T n x ∈ B and τ (T n x) = min{k ≥ 1 : ω n+k = 1} } \ X * with T (x, ω) = (T x, σ(x)) and X * is the set of (x, ω) with only finitely many 1s either in the future or in the past of ω.

Observe that:

(B.2) (x, ω), (x, ω ′ ) ∈ X and ω n = ω ′ n = 1 for some n =⇒ ∀k ≥ n ω k = ω ′ k .

( X, T ) is an extension of a subset of (X, T ) through π : X → X defined by π(x, ω) = x.

Remark B.1. The jump transformation T τ : {x ∈ B : τ (x) < ∞} → B is defined by T τ (x) := T τ (x) (x). It is closely related to T . Indeed, T τ is isomorphic to the first return map of T on [1] := {(x, ω) ∈ X : ω 0 = 1} so any T -invariant probability measure gives by restriction and normalization a T τ -invariant probability measure (see [START_REF] Zweimuller | Invariant measures for general(ized) induced transformations[END_REF]).