Vincent Levorato
email: vincent.levorato@ephe.sorbonne.fr

Marc Bui
email: marc.bui@ephe.sorbonne.fr

Cognitions Humaine

Artificielle

Data Structures and Algorithms for Pretopology: The JAVA based software library PretopoLib

PretopoLib is a library of data types and algorithms based on pretopology theory. Its interest is its data structures representing mathematical sets and all the operations related to them. The main features of the library are a clear separation of computation and visualization, efficient sets data types, and its ease of use to build model or application prototypes in the complex systems simulation framework.

1: Introduction

Modeling complex networks is a complicated task. However, this is very important to find appropriate models to analyze the structure and the dynamics of these networks, as there are lot of scientific domains in which they are involved, like social sciences for instance [START_REF] Lazega | Réseaux sociaux et structures relationnelles[END_REF]. Nowadays, there are some existing models, but there are not conclusive to represent all the dynamics of complex networks, as exposed in one of our previous work [START_REF] Levorato | Modeling the complex dynamics of distributed communities of the web with pretopology[END_REF]. To model complex networks, we propose pretopology theory which is a mathematical modeling tool for the concept of proximity. It provides the powerful tools for the structure analysis and classification. It ensures the follow-up of the process's development of dilation, alliance, adherence, closed subset, acceptability. On one hand, some authors with graph theory have provide a large amount of algorithms, while on the other hand nearly nothing exists in pretopology. Some works and programs have been done, but only for specific purposes, never in a general way [START_REF] Largeron | A pretopological approach for structural analysis[END_REF] [START_REF] Van Le | A new clustering method associated pretopological concepts and k-means algorithm[END_REF]. To build and implement models for specific problems, if researchers start from scratch each time they want to build a pretopological application, tools developed cannot be re-used for further purposes. Starting from these observations, it is clear that a need for a standard library with the data structures and algorithms of pretopology theory exists and can be used for differents kind of networks like in social sciences. We started the project PretopoLib to build a small, but growing library of data types and algorithms in a form which allows them to be used by non-experts and to elaborate simulations using pretopology. This library is splitted into two distinct parts:

• the process engine

• the visualization part

This library can be used to develop models prototypes for researchers familiar with pretopology concepts, but also as a teaching tool useful for students. We are able to graphically represent networks differently compared to actual graphs visualizations and new results on known structures taken from the web communities like emails, forums or blogs. This paper presents our contribution with the realisation of the library which main features are:

• Data types and data structures based on pretopology entities. The data types currently available are mathematical set, element, relation between singletons, link, and ball. The most difficult decision which we faced, was the choice of implementation for sets. For the efficiency of many data structures it is crucial that some operations may have one of the best possible complexity.

• A comfortable data type PretopoNodes (mathematical set). Standards operations on sets are implemented like union and intersection. It is possible to add and delete elements, to test if an element is contained, offering hash table design for low complexity.

The data type PretopoNodes permits to write programs for pretopological problems in a form close to the way the algorithms are usually presented in text articles.

• Ease of use and portability: PretopoLib is written in JAVA. All data types and algorithms are stored in a JAR library. All examples given in this paper show their executable codes.

• A visualization module easy to set up for quick rendering application based on the Prefuse package (http://www.prefuse.org) for fast and good visual rendering.

This paper is structured as follows: in section 2, we introduce pretopology theory concepts, in section 3, we explain which data structures we used in the library, in section 4, we show how visualization is set and we end on the section 5 by samples applications followed by a conclusion.

2: Pretopology Concepts

Pretopology permits to define what is a complex network in a general way, providing a specification for elements, groups of elements and their evolution to model the dynamics of the network. We introduce in this section pretopology concepts basing our network analysis on the question what is the definition of a network with a pretopological approach (cf. [START_REF] Dalud | Modèle prétopologique pour une méthodologie d'analyse des réseaux: concepts et algorithmes[END_REF]).

Let us consider a non-empty finite set E, and P(E) which designates all of the subsets of E.

2.1: Pretopological space

Definition 1 A pretopological space is a pair (E, a) where a is a map a(.) : P(E) → P(E) named pseudo-closure (Fig. 1) and defined as follows : ∀A, A ⊆ E the pseudo-closure of A, a(A) ⊆ E such that :

• a(∅) = ∅ (P 1 ) • A ⊆ a(A) (P 2 )
The pseudo-closure is associated to the dilation process. Thus, a(.) can be applied on a set A in sequence, to model expansions : A ⊂ a(A) ⊂ a 2 (A) ⊂ ... . That means we could follow the process step by step. Using the pseudoclosure, we model the proximity concept, very useful for aggregation process and to define the neighbourhood of a set or an element.

Figure 1. Pseudoclosure of A

Pretopology theory defines several space types associated to a corpus of known results [START_REF] Belmandt | Manuel de prétopologie et ses applications: Sciences humaines et sociales, réseaux, jeux, reconnaissance des formes, processus et modèles, classification, imagerie, mathématiques[END_REF], and the library implements some of these results as a set of basic operations.

2.2: V Pretopological space

Definition 2 A V pretopological space (E, a) is defined by :

• ∀A, B, A ⊆ E, B ⊆ E and A ⊂ B then a(A) ⊂ a(B)
This type of space is very powerful to describe complex properties but is based on weak assumptions.

2.3:

V D Pretopological space Definition 3 A V D pretopological space (E, a) is defined by : ∀A, B, A ⊆ E, B ⊆ E a(A ∪ B) = a(A) ∪ a(B)
Concerning the space properties, the library propose a generic framework letting the possibility for the user to redefine all caracteristics space.

2.4: V S Pretopological space

Definition 4 A V S pretopological space (E, a) is defined by :

∀A, A ⊆ E, a(A) = x∈A a({x})
The library can be optimized depending of the space used, especially for V S pretopological space which is finite.

When transformation doesn't evolve anymore, we obtain closure, which is mainly used by classification algorithms.

2.5: Closure

Definition 5 A ∈ P (E) is closed if and only if : A = a(A) The closure of A ∈ P (E) is the smallest closed subset containing A, noted F (A) or F A .
In pretopology, spaces can be of differents natures: metric, binary, or valued.

2.5.1: Metric space

E has a metric, and a closed ball with a center x and a radisu r is defined as follows:

B(x, r) = {y ∈ E, d(x, y) ≤ r}
Pseudoclosure is defined on E:

∀A ∈ P(E), a(A) = {x ∈ E, B(x, r) ∩ A = ∅} 2.

5.2: Binary space

Elements of E are linked by n binary reflexives relations R i with i ∈ {1, ..., n}.

∀i ∈ {1, ..., n}, R i (x) = {y ∈ E, xR i y}
Pseudoclosure is defined on E:

∀A ∈ P(E), a(A) = {x ∈ E, ∀i ∈ {1, ..., n}R i (x) ∩ A = ∅} 2.5.3: Valued space
Elements of E are linked by valued relations (graph theory extension). Let v: E ×E → N, (x, y) → v(x, y). Pseudoclosure is defined on E:

∀A ∈ P(E), a(A) = {y ∈ E -A, x∈A v(x, y) ≥ s} ∪ A

2.6: Network definition

In this paper, we focus on complex networks. In pretopology, we can define a network as a family of pretopologies on a given set E [START_REF] Dalud | Modèle prétopologique pour une méthodologie d'analyse des réseaux: concepts et algorithmes[END_REF].

Definition 6 Let X be a set. Let I a countable family of indexes. Let {a i , i ∈ I} a family of pretopologies on X. The family of pretopological spaces {(X, a i ), i ∈ I} defines a network on X.

Applications based on this definition are presented in section 5.

3: Data Types

Having introduced network formalism with pretopology, we propose to implement this framework which can be compared to graph theory (like LEDA [START_REF] Mehlhorn | Leda : A library of efficient data types and algorithms[END_REF]). PretopoLib is written in JAVA language which is using a virtual machine to launch program, so we don't have to recompiled code to pass from one OS platform to another (like passing from Windows to Linux or MacOS). JAVA provides data abstraction, object based programming, operator overloading and many other features suitable for implementing data types. For details on the JAVA language see Sun documentation or a book like Thinking in Java [START_REF] Eckel | Thinking in Java[END_REF]. A significant improvement would result, if data types we provide were made available as "package" by some users and could then be reused by others. Of course, this requires to specify the properties of the data type independently of the implementation. The package PretopoNotions deals with the processing part and PretopoVisual is used for the visual part, which be discussed in next section. Let's describe the PretopoLib specifications of the data types PretopoNodes, PretopoEdges and PretopoSpace.

3.1: PretopoNodes

3.1.1: Description

A PretopoNodes object represents a set of elements in a pretopological space. Each element of the set is unique, using the name of the element as primary identifier. The name of each element (called label ) is used in a hash table to find elements quickly into a set. Listing 1 shows a small illustration of PretopoLib's set module.

Listing 1. Creating two sets of respectively three and four elements, then making their intersection. The program fills two sets A and B with differents elements and put the result of their intersection in set C. Lines (1) and (2) import PretopoLib class packages. From line (9) to (13), we create elements (singletons) with unique label ; lines (15) to (17) create sets (A, B, and C), and lines from (20) to (27) fill sets A and B. In line (29), we finally make the intersection of A and B, storing the result in C.

3.1.2: Declaration of a PretopoNodes

Constructors are: public PretopoNodes() which creates an empty set. public PretopoNodes(Elmt elt) which creates a set with only one element. public PretopoNodes(PretopoNodes pn) which creates a set by copy.

3.1.3: Operations on a PretopoNodes

Here are the most common operations: boolean contains(Object o) tests if an element is in the set. boolean equals(Object o) tests if two sets contains the same elements. void addElmt(Elmt elmt) add one element into the set if this one is not in this set already. PretopoNodes intersection(PretopoNodes pn) gives the result set after the intersection operation between two sets.

Operation

List HashTable contains O(n) O(1) equals O(n 2 O(n) addElmt O(n) O(1) intersection O(n 2 ) O(n) union O(n 2 ) O(n) substract O(n 2 ) O(n) isSubset O(n 2 ) O(n)
PretopoNodes union(PretopoNodes pn) gives the result set of the union of two sets.

PretopoNodes substract(PretopoNodes pn) gives a new set by removing all the elements of a given set in parameter.

boolean isSubset(PretopoNodes pn) tests is a set is a subset of a given set.

3.1.4: Implementations

There are several implementations of mathematical sets. One possible implementation is based on lists. The problem with that choice is the complexity of the standard operations introduced in the previous paragraph. We propose an efficient data structure called HashSet in JAVA based on a hash table. In the table 3.1.4, operations costs in the worst case are detailed. All elements are unordered.

As shown is the table 3.1.4, for all operations, the gain of complexity is appreciable, which is quite important on large datasets.

3.2: PretopoEdges

This class represents all the possible relations of a pretopological space. It contains a set of relations, which can be of different type (metric, valued, or binary). It contains a hashtable of all the relations between the elements of a pretopological space. A relation object defines all the links between the elements and more important, it defines the neighbourhood function used for building the pseudoclosure process. In listing 2, using the previous variables of listing 1, we first build a binary relation, then another relation which is valued and a last one which is metric: Listing 2. Creating a binary, a valued and a metric relation.

// c r e a t i n g a b i n a r y r e l a t i o n R e l a t i o n r= new R e l a t i o n ( " b i n a r y " ) ; r . addLink ( e l t c , e l t d ) ; r . addLink ( e l t e , e l t d ) ; r . addSymLink ( e l t c , e l t a ) ; r . addLink ( e l t b , e l t a ) ; // c r e a t i n g a v a l u e d r e l a t i o n RelationV rv= new RelationV ( " v a l u e d " ) ; rv . addSymLink ( e l t a , e l t e , 2 ) ; rv . addSymLink ( e l t b , e l t e , 1 ) ; rv . addSymLink ( e l t b , e l t d , 4 ) ; // c r e a t i n g a m e t r i c r e l a t i o n R e l a t i o n B rb= new R e l a t i o n B ( " m e t r i c " ) ;

In line (2), a binary relation is created, which is the basic relation class. From line (3) to [START_REF] Van Le | A new clustering method associated pretopological concepts and k-means algorithm[END_REF], links are added between elements, sometimes in an oriented way addLink( source , target ) and sometimes in an unoriented way addSymLink( source , target ) . In line [START_REF] Newman | The Structure and Dynamics of Networks[END_REF], a valued relation is created, and from line [START_REF] Sampson | A Novitiate in a Period of Change. An Experimental and Case Study of Social Relationships[END_REF] to line (12), valued links are added. In line (15), a metric relation is created. Note that no links are used for this last relation because it's using directly the coordinates of each elements to know if they are close each other or not (with a given radius using ball concept). The neighbourhood function is implemented in Relation class. Here is how a neighbourhood function like R(x) = x ∈ E, xRy is coded into the Relation class.

Listing 3. Implemented neighbourhood function. public boolean NeighbFunction ( Elmt e l t , PretopoNodes A)

{ boolean b=f a l s e ; A r r a y L i s t <Link> l =( A r r a y L i s t <Link >) r e l l i s t . g e t ( e l t ) ; i f ( l != null ) f or ( int i =0; i <l . s i z e ( ) ; i ++) i f (A. c o n t a i n s ( l . g e t ( i ) . t a r g e t ) ) b=true ; return b ; }

The listing 3 shows how the algorithm is working: it examines the table of list of links between the element given in parameter ( line ( 5) ) and checks if the tested element has a link with an element of A, the set on which we are applying the pseudoclosure on line [START_REF] Mehlhorn | Leda : A library of efficient data types and algorithms[END_REF]. It returns true if the element actually is into the neighbourhood of A, false else ( line (12) ).

3.3: PretopoSpace

This class represents a pretopological space itself. It is composed of a set of elements and a set of relations. It contains the pseudoclosure method and all the algorithms which are relying on it, like closure or open set. Three types of pretopological spaces are usually manipulated. By default, binary space, valued space and metric space are supported. One of the fondamental operation in the library is the pseudoclosure. . . . i f ( param . g e t C l a s s ( ) . e q u a l s ( Double . c l a s s ) ) // v a l u e d r e l a t i o n s s p a c e { RelationV r =( RelationV ) R e l a t i o n s . g e t R e l a t i o n ( r e l l a b e l ) ; I t e r a t o r <Elmt> i t r =( I t e r a t o r <Elmt>) E . getNodeSet ( ) . i t e r a t o r ( ) ; while ( i t r . hasNext ( ) ) { Elmt e l t=i t r . next ( ) ; i f ( r . NeighbFunction ( e l t , s e t , ( Double ) param ) ) pseudoCset . addElmt ( e l t ) ; } } . . . return pseudoCset ; }

In the listing 4, we can see the generic overload flexibility of the implementation. It takes three input parameters: the set on which we want to apply the pseudoclosure, the name of the relation, and a parameter used for the neighbourhood function. In this listing, only the part about the valued space is given as example. On line(12), we extract the desired relation, and we look through E the elements which are neighbours of the given set ( lines(13) to (19) ). Finally, it returns the set representing the pseudoclosure of the given set line(24). They are several interesting caracteristics with the library implementation:

• It is possible to constitute neighbourhoods and groups.

• Inter-relations can be visualized.

• User can interact by manipulating elements and groups.

4: Visualization

The Visualization Part is based on Prefuse: it has shown some very good visual rendering results. Being used to represent graphs, it also allows us to represent convex polygons that we use to represent sets.

To display a pretopological space, we have to use the object PretopoVisualization. One can see how to make a display window into a JPanel JAVA structure (display window). void showlinks() shows links between elements. void rearrange_display(int type) rearranges display view with a chosen type: NOR-MAL, GRID, RANDOM, or FIXED.

void resetgroup() remove all the visual sets. void setgroup(java.util.HashSet <java.lang.String> group_nodes, int igrp) creates a visual set of elements using a palette color.

5: Examples of application

To illustrate the use of our library, let us consider two samples of data. The first is about a community of monks studied by the sociologist Samuel F. Sampson [START_REF] Sampson | A Novitiate in a Period of Change. An Experimental and Case Study of Social Relationships[END_REF], who described the various social relations among the novices who were preparing to enter the clerical orders. Although small, this network has a large number of different relationships, which enable us to test the various features of our tool. The second sample, from Valdis Krebs (http://www.orgnet.com/) concerns the Web which is describing a network of books on American policy sold on Amazon.com. The relationship between books represent copurchase done by the same customers, as indicated on the site by "people who bought this product also purchased."

The results in Fig. 2 show a pseudoclosure on an element (in this case AMBROSE 9), with green relations representing friendship and red relations representing esteem. Here, the threshold of the neighbourhood is set in such a way that only those who have the greater esteem and friendship are concerned. Regardless of relations, the final set which is the pseudoclosure of {AMBROSE 9} in this case is equal to {AMBROSE 9 LOUIS 11 ELIAS 17}. Using visualization facilities of the library, we show that the user can move the elements in his own way, as well as sets while retaining the coherence of visualization.

For the second sample data, there is only one type of binary relation. However, even if there are many more elements (105) and connections (441), it is a small network, nevertheless sufficient for the demonstration. In this case, we use one of the provided algorithm of the library to isolate the group which is locally most connected. This group of books are client links to discover all the other books, and moreover, the group is the most popular books. Once identified, it has become clear that the group consists of 38 books and has a degree of 164, contains a large number of books dealing which is a hot topic of discussion in the time period the experiment was conducted. Concerning visualization, if display becomes fuzzy even after we have extracted components out of other elements (Fig. 3), simply hide the relationship to clarify the display.

6: Conclusion

PretopoLib is a library of efficient data types and algorithms. At present, its strength relies on pretopological algorithms and the data structures related to them, which continue to evolve. This is the first project which aim for the goals described in the introduction. The advantages of PretopoLib are indeed:

• a clear separation between notions and visualization • a straight forward implementation of the specification of the algorithms

• the inclusion of many data structures and algorithms

Listing 4 .

 4 Part of the pseudoclosure code concerning the valued space type. public PretopoNodes p s e u d o c l o s u r e ( PretopoNodes s e t , S t r i n g r e l l a b e l , Object param ) { PretopoNodes pseudoCset=new PretopoNodes ( s e t ) ; PretopoNodes E=new PretopoNodes ( Nodes ) ; E=E . s u b s t r a c t ( s e t ) ;

Figure 2 .

 2 Figure 2. Visualization sample

Figure 3 .

 3 Figure 3. Search for the books group having the higher degree

Table 1 .

 1 Simple operations costsboolean intersects(PretopoNodes pn) tests if a given set instersects the concerned set.


David Burguet

C 2 surface diffeomorphisms have symbolic extensions

come

C 2 surface diffeomorphisms have symbolic extensions

Introduction

Given a dynamical system (X, T ) (i.e. a continuous map T on a compact metrizable space X) one can try to encode it with a finite alphabet. More precisely we wonder if it admits a topological extension which is a subshift over a finite alphabet. Such an extension is called a symbolic extension. When a dynamical system has symbolic extensions we are interested in minimizing their entropy. The topological symbolic extension entropy h sex (T ) = inf{h top (Y, S) : (Y, S) is a symbolic extension of (X, T )} estimates how the dynamical system (X, T ) differs from a symbolic extension from the point of view of entropy. These questions lead to a deep theory which was developped mainly by M.Boyle and T.Downarowicz, who related the existence of symbolic extensions and their entropy with the convergence of the entropy of (X, T ) computed at finer and finer scales [START_REF] Boyle | The entropy theory of symbolic extension[END_REF].

Dynamical systems with symbolic extensions have necessarily finite topological entropy, but the converse is false : dynamical systems with finite topological entropy may not have symbolic extensions. Nonetheless it was proved by M.Boyle, D.Fiebif, U.Fiebig [START_REF] Boyle | Residual entropy, conditional entropy and subshift covers[END_REF] that asymptotically hexpansive dynamical systems with finite topological entropy admit principal symbolic extensions, i.e. which preserve the entropy of invariant measures. Following Y.Yomdin [START_REF] Yomdin | Volume growth and entropy[END_REF], J.Buzzi showed that C ∞ maps on a compact manifold are asymptotically h-expansive [START_REF] Buzzi | Intrinsic ergodicity for smooth interval maps[END_REF]. In particular such maps admit principal symbolic extensions. On the other hand C 1 maps without symbolic extensions have been built in several works [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF], [START_REF] Asaoka | Hyperbolic set exhibing C 1 -persistent homoclinic tangency for higher dimensions[END_REF], [START_REF] Burguet | Examples of C r interval map with large symbolic extension entropy[END_REF]. T.Downarowicz and A.Maass have recently proved that C r maps of the interval f : [0, 1] → [0, 1] with 1 < r < +∞ have symbolic extensions [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF]. More precisely they showed that h sex (f ) ≤ r log f ′ ∞ r-1

. The author built explicit examples [START_REF] Burguet | Examples of C r interval map with large symbolic extension entropy[END_REF] proving this upper bound is sharp. Similar C r examples with large symbolic extension entropy have been previously built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF]. The results of T.Downarowicz and A.Maass has been extended by the author in any dimension to nonuniformly entropy expanding maps (i.e. C 1 maps whose ergodic measures with positive entropy have nonnegative Lyapunov exponent) of class C r with 1 < r < +∞ [START_REF] Burguet | Symbolic extensions for C r nonuniformly entropy expanding maps[END_REF]. The existence of symbolic extensions for general C r maps with 1 < r < +∞ is still an open question.

We pointed out in the previous paragraph the known links between the regularity of the map and the existence of symbolic extensions. But symbolic extensions can also be obtained by using geometrical arguments. For example it is well known that for uniformly hyperbolic dynamical systems, a Markov partition gives rise to a principal symbolic extension [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]. The elements of this partition are rectangles which are union of stable and unstable local manifolds. One can also just refer to the expansiveness of uniformly hyperbolic dynamical system and to the previously mentioned result of M.Boyle, D.Fiebif, U.Fiebig [START_REF] Boyle | Residual entropy, conditional entropy and subshift covers[END_REF] to prove the existence of principal extensions.

In fact h-expansiveness is also checked under a weaker assumption of hyperbolicity : partially hyperbolic dynamical system with a center bundle splitting into one dimensional subbundles [13] [26].

In the following we adapt such geometrical arguments to control the local dynamical complexity of hyperbolic measures. More precisely we introduce finite time rectangles which are pieces of finite time stable and unstable manifolds. Under a condition of small oscillation of the derivative these rectangles allows us to control the growth of the cardinality of separated sets. This condition can be ensured by the C 2 regularity of the map as in Yomdin's theory. In this way this paper can be considered as a first attempt to combine the theory of hyperbolic dynamical systems and Yomdin's theory.

T.Downarowicz and S.Newhouse have conjectured in [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF] that C 2 diffeomorphisms on a compact manifold have symbolic extensions. The following theorem answers affirmatively to this conjecture in the two dimensional case.

Theorem 1 Let T : M → M be a C 2 surface diffeomorphism, then T admits symbolic extensions and moreover h sex (T ) ≤ h top (T ) + 2R(T )

where R(T ) is the dynamical Lipschitz 1 constant of T , that is R(T ) := lim n→+∞ log + DT n n .

We recall in the second section the background of the theory of symbolic extensions. Then we state our main results and we reduce them to a theorem of reparametrization of Bowen balls by contracting maps from the square in a similar (but finer) approach of Yomdin's C 2 theory. In the fourth section we introduce the finite time stable field which is a natural generalization at finite time of the usual stable field. Assuming the oscillation of the derivative is small compared to the size of the derivative we prove the finite time stable field has bounded derivative. In the fifth section we define the new notion of finite time rectangle. Under the previous assumption on the oscillation of the derivative and some assumption of hyperbolicity we show these rectangles do not carry entropy. We also compare rectangles at successive times. The last section is devoted to the proof of the reparametrization of Bowen balls by such rectangles.

Preliminaries

In the following we denote M(X, T ) the set of invariant Borel probability measures of the dynamical system (X, T ) and M e (X, T ) the subset of ergodic measures. We endow M(X, T ) with the weak star topology. Since X is a compact metric space, this topology is metrizable. We denote dist a metric on M(X, T ). It is well known that M(X, T ) is compact and convex and its extreme points are exactly the ergodic measures. Moreover if µ ∈ M(X, T ) there exists an unique Borel probability measure M µ on M(X, T ) supported by M e (X, T ) such that for all Borel subsets B of X we have µ(B) = ν(B)dM µ (ν). This is the so called ergodic decomposition of µ. A Borel map f : M(X, T ) → R is said to be harmonic if f (µ) = Me(X,T ) f (ν)dM µ (ν) for all µ ∈ M(X, T ). It is a well known fact that affine upper semicontinuous maps are harmonic. In general 2 the measure theoretical entropy h : M(X, T ) → R + is not upper semicontinuous but it is always harmonic [START_REF] Walters | An introduction to ergodic theory[END_REF].

If f is a real Borel map defined on M e (X, T ), the harmonic extension f of f is the function defined on M(X, T ) by :

f (µ) := Me(X,T ) f (ν)dM µ (ν)
It is easily seen that f coincides with f on M e (X, T ) and that f is harmonic.

1 R(T ) does not depend on the Riemannian metric on M 2 It may not be upper semicontinuous for C r map for any r ∈ R + [START_REF] Misiurewicz | Diffeomorphism without any measure with maximal entropy[END_REF]. However it is for C ∞ maps [START_REF] Newhouse | Continuity properties of entropy[END_REF].

Entropy structure

The measure theoretical entropy function can be computed in many ways as limits of a nondecreasing sequence of nonnegative functions defined on M(X, T ) (with deacreasing sequence of partitions, formula of Brin-Katok,...). The entropy structures are such particular sequences whose convergence reflect the topological dynamic : they allow for example to compute the tail entropy [START_REF] Burguet | A direct proof of the tail variational principle and its extension to maps[END_REF] [START_REF] Downarowicz | Entropy structure[END_REF], but also especially the symbolic extension entropy [START_REF] Boyle | The entropy theory of symbolic extension[END_REF] [START_REF] Downarowicz | Entropy structure[END_REF].

We skip the formal definition of entropy structures. In the present paper we just use that Newhouse local entropy (See subsection 2.4) is an entropy structure.

Symbolic extension entropy function

A symbolic extension of (X, T ) is a subshift (Y, S) of a full shift on a finite number of symbols, along with a continuous surjection π : Y → X such that T • π = π • S. Given a symbolic extension π : (Y, S) → (X, T ) we consider the extension entropy h π ext : M(X, T ) → R + defined for all µ ∈ M(X, T ) by :

h π ext (µ) = sup π * ν=µ h(ν)
Then the symbolic entropy function, h sex : M(X, T ) → R + , is :

h sex = inf h π ext
where the infimum holds over all the symbolic extensions of (X, T ). By convention, if (X, T ) does not admit any symbolic extension we simply put h sex ≡ +∞. Recall we define in the introduction the topological symbolic extension entropy h sex (T ) as the infimum of the topological entropy of the symbolic extensions of (X, T ) (as previously we put h sex (T ) = +∞ when there are no such extensions). M.Boyle and T.Downarowicz proved that these two quantities are related by the following variational principle :

h sex (T ) = sup µ∈M(X,T ) h sex (µ) (1)

The Estimate Theorem

One of the main tools introduced in [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF] is the so called Estimate Theorem. We can roughly resume the statement as follows : in order to estimate the symbolic entropy function one only needs to bound the local entropy of an ergodic measure near an invariant one by the difference of the values of some upper semicontinuous affine function on M(X, T ) at these two measures.

Theorem 2 (Downarowicz, Maass) [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF] Let (X, T ) be a dynamical system with finite topological entropy. Let H = (h k ) k be an entropy structure. Let g be an upper semicontinuous affine function on M(X, T ) which dominates the entropy function at all ergodic measures and such that for every γ > 0 and µ ∈ M(X, T ) there exists k µ ∈ N and δ µ > 0 such that for every ergodic measure ν satisfying dist(ν, µ) < δ µ it holds that :

(h -h kµ )(ν) ≤ g(µ) -g(ν) + γ
Then, for every invariant measure µ on X,

h sex (µ) ≤ h(µ) + g(µ)
It is worth noting that the Estimate Theorem holds in a pure topological setting. This result is in fact a consequence of the major Theorem of Symbolic Extension [START_REF] Boyle | The entropy theory of symbolic extension[END_REF] which allows to compute the symbolic extension entropy function from the properties of convergence of any entropy structure.

Newhouse local entropy

Let us first recall some usual notions relating to the entropy of dynamical systems (we refer to [START_REF] Walters | An introduction to ergodic theory[END_REF] for a general introduction to entropy). Consider a continuous map T : X → X with (X, d) a compact metric space. Let n ∈ N and δ > 0. A subset F of X is called a (n, δ) separated set when for all x, y ∈ F there exists 0 ≤ k < n such that d(f k x, f k y) ≥ δ. Let Y be a subset of X. A subset F of Y is called a (n, δ) spanning set of Y when for all y ∈ Y there exists z ∈ F such that d(f k (x), f k (y)) < δ for all 0 ≤ k < n. Given a point x ∈ X we denote by B(x, n, δ) the Bowen ball centered at x of radius δ and length n :

B(x, n, δ) := {y ∈ X, d(T k y, T k x) < δ for k = 0, ..., n -1}
We recall now the "Newhouse local entropy". Let x ∈ X, ǫ > 0, δ > 0, n ∈ N and F ⊂ X a Borel set, we define :

H(n, δ|x, F, ǫ) := log max ♯E : E ⊂ F B(x, n, ǫ) and E is a (n, δ) separated set H(n, δ|F, ǫ) := sup x∈F H(n, δ|x, F, ǫ) h(δ|F, ǫ) := lim sup n→+∞ 1 n H(n, δ|F, ǫ) h(X|F, ǫ) := lim δ→0 h(δ|F, ǫ)
Then for any ergodic measure ν we put :

h N ew (X|ν, ǫ) := lim σ→1 inf ν(F )>σ h(X|F, ǫ)
Given a nonincreasing sequence (ǫ k ) k∈N converging to 0, we consider the sequence H N ew = (h N ew k ) k∈N where h N ew k := h -h N ew (X|., ǫ k ) is the harmonic extension of h -h N ew (X|., ǫ k ) for all integers k. T.Downarowicz proved this sequence defines an entropy structure [START_REF] Downarowicz | Entropy structure[END_REF] for homeomorphisms and the author extends the result in the noninvertible case [START_REF] Burguet | Entropy and local complexity of differentiable dynamical sytems[END_REF]. In particular h N ew (X|., ǫ k ) converges pointwise to zero when k goes to infinity.

We will use the following technical inequality :

Lemma 1 For all integers k > 0 and for all ergodic measures ν :

h N ew T (X|ν, ǫ) ≤ h N ew T k (X|ν, ǫ) k
Proof : Clearly we have the inclusion B T (x, nk, ǫ) ⊂ B T k (x, n, ǫ). Moreover for all δ > 0 there exists δ ′ > 0 such that any (nk, δ) separated set for T is (n, δ ′ ) separated for T k . The conclusion of the lemma follows easily from these two facts.

Lyapunov exponents for surface diffeomorphisms

Let (M, ) be a compact Riemannian surface and let T : M → M be a diffeomorphism and ν be an ergodic T -invariant measure. We denote

D x T = sup v∈TxM-{0} DxT (v) v
the induced norm of the differential D x T of T at x and DT = sup x∈M D x T the supremum norm of the differential of T . According to Oseledet's theorem [START_REF] Oseledets | A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems[END_REF], there exist two real numbers χ + (ν) ≥ χ -(ν), a measurable splitting of the tangent bundle T M = E 1 E 2 into two invariant subbundles E 1 and E 2 and a Borel set F with ν(F ) = 1 such that for all x ∈ F and all (v

1 , v 2 ) ∈ E 1 × E 2 : lim |n|→+∞ 1 n log D x T n (v 1 ) = χ + (ν) lim |n|→+∞ 1 n log D x T n (v 2 ) = χ -(ν) Remark that χ + (ν) = lim n→+∞ 1 n log D x T n and χ -(ν) = lim n→-∞ 1
n log D x T n for all x ∈ F . The real numbers χ + (ν) and χ -(ν) are the well-known Lyapunov exponents of ν (we use also the notations χ + (ν, T ) and χ -(ν, T ) to avoid ambiguities). We denote χ + 0 (ν) = max(χ + (ν), 0) and χ - 0 (ν) = min(χ -(ν), 0). According to the subadditive ergodic theorem, we have χ + 0 (ν) = inf n∈N

1 n M log + D x T n dν(x) = lim n→+∞ 1 n M log + D x T n dν(x) and similarly -χ - 0 (ν) = inf n∈N 1 n M log + D x T -n dν(x) = lim n→+∞ 1 n M log + D x T -n dν(x).
Another application of the subadditive ergodic theorem leads to the following lemma : Lemma 2 Let ν be an ergodic measure, then 1 n log + (D x T n ) -1 converges almost everywhere to -χ - 0 (ν) when n goes to +∞.

Proof : By the subadditive ergodic theorem, the sequence (

1 n log + (D x T n ) -1 ) n∈N converges ν almost everywhere to inf n∈N log + (DxT n ) -1 dν(x) n
when n goes to +∞. Then we have by invariance of ν :

log + (D x T n ) -1 dν(x) = log + D T n x T -n dν(x) = log + D x T -n dν(x)
We prove now elementarily that the harmonic extension of χ + 0 is upper semicontinuous. The following lemma is in fact valid in any dimension. In the one dimensional case it was proved by T.Downarowicz and A.Maass by using a clever argument of convexity (See Fact 2.5 of [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF]).

Lemma 3 For all µ ∈ M(M, T ), we have :

χ + 0 (µ) = inf n∈N 1 n M log + D x T n dµ(x)
In particular χ + 0 : M(M, T ) → R + is upper semicontinuous.

Proof : For all integers n > 0 we consider the function f n : M(M, T ) → R + defined by :

∀µ ∈ M(M, T ), f n (µ) = log + D x T n dµ(x)
This function is clearly continuous and affine, and therefore harmonic. Also (f n (µ)) n∈N is a subadditive sequence for all µ ∈ M(M, T ). We already observe that χ + (ν) = lim n→+∞ fn(ν) n for all ergodic measures ν. Consider now a general measure µ ∈ M(M, T ). We have :

χ + 0 (µ) := Me(M,T ) χ + 0 (ν)dM µ (ν) = Me(M,T ) lim n→+∞ f n (ν) n dM µ (ν)
Obvously f n (ν) ≤ log + DT for all ergodic measures ν. Therefore by applying the theorem of dominated convergence we get :

χ + 0 (µ) = lim n→+∞ Me(M,T ) f n (ν) n dM µ (ν)
and by harmonicity of f n :

χ + 0 (µ) = lim n→+∞ f n (µ) n
But the sequence (f n (µ)) n∈N is subadditive so that :

χ + 0 (µ) = inf n∈N f n (µ) n
We conclude that χ + 0 is an upper semicontinuous function as an infimum of a family of continuous functions.

In the following we are interesting in the entropy of ergodic measures. Recall the Ruelle-Margulis inequality states that for a C 1 map T : M → M on a compact manifold M the entropy h T (ν) of an ergodic measure ν is bounded from above by the sum of its positive Lyapunov exponents. When T is a surface diffeomorphism it is easily seen by applying the Ruelle-Margulis inequality to T and its inverse T -1 and by using the equality h T (ν) = h T -1 (ν) that any ergodic measure ν ∈ M(M, T ) with non zero entropy has exactly one positive and one negative Lyapunov exponent, i.e., with the previous notations, χ + (ν) > 0 and χ -(ν) < 0 and moreover

h(ν) ≤ min(χ + (ν), -χ -(ν)) 3 Statements Theorem 3 (Main Theorem) Let T : M → M be a C 2 surface diffeomorphism. Then for all µ ∈ M(M, T ), h sex (µ) ≤ h(µ) + 2χ + 0 (µ)
The Main Theorem follows easily from the following theorem by applying the Estimate Theorem to the upper semicontinuous affine function g 0 = 2χ + 0 (which dominates the entropy function at all ergodic measures according to Ruelle-Margulis inequality) and to the entropy structure H N ew . Remark that Theorem 1 stated in the introduction is the topological version of the Main Theorem : it is deduced by the usual variational principle for the entropy and the variational principle for the symbolic extension entropy (Equation ( 1)).

Theorem 4 Let T : M → M be a C 2 surface diffeomorphism. Let µ be an invariant measure and fix some γ > 0. Then there exist δ µ > 0 and ǫ µ > 0 such that for every ergodic measure ν with dist(ν, µ) < δ µ it holds that :

h N ew (M |ν, ǫ µ ) ≤ 2χ + 0 (µ) -2χ + 0 (ν) + γ
In the general case (any dimension, any intermediate regularity, noninvertible maps) we conjecture that Theorem 4 can be extended in the following way : 3 . Let µ be an invariant measure and fix some γ > 0. Then there exist δ µ > and ǫ µ > 0 such that for every ergodic measure ν with dist(ν, µ) < δ µ it holds that :

Conjecture 1 Let r > 1. Let M be a compact manifold of dimension d and T : M → M a C r map
h N ew (M |ν, ǫ µ ) ≤ Σ + χ(µ) -Σ + χ(ν) r -1 + γ
where Σ + χ denotes the sum of the positive Lyapunov exponents.

Remark 1 By adapting the proof of Lemma 3 one easily shows that

Σ + χ(µ) = inf n∈N 1 n max k=1,...,d log + Λ k D x T n k dµ(x)
and therefore Σ + χ is also an upper semicontinuous function on M(M, T ).

3 for r / ∈ N we mean that T ∈ C [r] and

D [r] T is r -[r]-Hölder
This conjecture was proved by T.Downarowicz and A.Maass in dimension one [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF]. In [START_REF] Burguet | Symbolic extensions for C r nonuniformly entropy expanding maps[END_REF] we prove the conjecture in any dimension up to a factor d and an additional term corresponding to the sum of the negative Lyapunov exponents of ν (then to apply the Estimate Theorem we are reduced to the class of nonuniformly entropy expanding maps because of this remaining term). Theorem 4 corresponds to the case of C 2 surface diffeomorphisms up to a factor 2.

Remark 2 By the Passage theorem of T.Downarowicz and A.Maass [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF] the inequality in Theorem 4 holds in fact for general invariant measures ν close to µ. Then following the proof of the Estimate Theorem we get that u 1 (µ)

≤ 2χ + 0 (µ) 3
for all invariant measures µ. Taking the supremum over all invariant measures µ we have according to the tail variational principle the following bound on the tail4 entropy : h tail (T ) ≤ 2R(T ) 3 . By using Yomdin's theory, J.Buzzi [START_REF] Buzzi | Intrinsic ergodicity for smooth interval maps[END_REF] proved that h tail (S) ≤ d r R(S) when S is a C r map with r ≥ 1 on a compact manifold of dimension d. Here we consider C 2 surface diffeomorphism (d = r = 2) so that our upper bound is better than Buzzi's one. It is known [START_REF] Buzzi | Intrinsic ergodicity for smooth interval maps[END_REF] that Buzzi's inequality is sharp for noninvertible maps but it is reasonnable to think that only the expanding directions are involved in the creation of tail entropy as in the Ruelle-Margulis inequality for the entropy (see also the previous conjecture) and thus that h tail (T ) ≤ R(T ) 2 for a C 2 surface diffeomorphism T .

We reduce now Theorem 4 to a result of reparametrization of Bowen's balls by contracting maps as in Yomdin's theory. Let us first introduce the notion of finite time hyperbolic sets. We fix a Riemannian metric on the surface M and we denote by d the induced distance on M .

Definition 1 For any χ + > 0 > χ -, min(χ + , -χ -) > γ > 0 and C > 1, we denote for all integers n :

H n T (χ + , χ -, γ, C) := {x ∈ M : ∀1 ≤ k ≤ n, C -1 e (χ + -γ)k ≤ D x T k ≤ Ce (χ + +γ)k , C -1 e (-χ --γ)k ≤ D T k x T -k ≤ Ce (-χ -+γ)k }
This set captures the hyperbolicity with logarithmic expansion χ + and contraction χ -with error γ at finite time. The following proposition bounds the local dynamical complexity of finite time hyperbolic sets. We denote H : [1, +∞[→ R the function defined by

H(t) = -1 t log( 1 t ) -(1 - 1 t ) log(1 -1 t ). Moreover [x]
is the integer part of x if x > 0 and zero if not. Finally we say that a map from the unit square

[0, 1] 2 to a surface M is C 1 if it can be extended in a C 1 map on an open neighborhood of [0, 1] 2 .
Proposition 1 Let T : M → M be a surface diffeomorphism and let χ + > 0 > χ -, min(χ + ,-χ -,1) 3 > γ > 0 and C > 1. Then there exist ǫ > 0 depending only on D 2 T , D 2 T -1 , DT and DT -1 , a real number D depending only on χ + , χ -, γ, C, DT and DT -1 and a universal constant A > 0 with the following properties. For all x ∈ M and for all positive integers n there exists a family

F n of C 1 maps from [0, 1] 2 to M such that : (i) ∀φ ∈ F n ∀0 ≤ l ≤ n, D(T l • φ) ≤ 1 (ii) H n T (χ + , χ -, γ, C) ∩ B(x, n + 1, ǫ) ⊂ φ∈Fn φ([0, 1] 2 ) (iii) log ♯F n ≤ 2 + H([λ + n (x, T ) -χ + ] + 3) λ + n (x, T ) -χ + n + An + D with λ + n (x, T ) := 1 n n-1 l=0 log + D T l x T .
When the size ǫ of the Bowen ball at x is small, λ + n (y, T ) is close to λ + n (x, T ) for all y in the Bowen ball. Then the term λ + n (x, T ) -χ + is (up to a small error term) the logarithm of the defect of multiplicativity of the norm of the composition D y T n = D T n-1 y T • ... • D y T when y belongs also to the finite hyperbolic set H n T (χ + , χ -, γ, C).

We deduce now Theorem 4 from the above statement. In the proof the terms λ + n (x, T ) for typical ν points x and χ + will be respectively related with the Lyapunov exponents of µ and ν where ν is an ergodic measure near an invariant measure µ. Moreover the quantity H([λ + n (x, T ) -χ + ] + 3) will be negligible.

Proof of Theorem 4 assuming Proposition 1 :

Let µ ∈ M(M, T ). By Lemma 3 we choose k µ ∈ N s.t.

χ + 0 (µ, T ) = inf n∈N log + D x T n dµ(x) n ≥ log + D x T kµ dµ(x) k µ -γ
One can also assume k µ large enough s.t. 2H(k µ γ)R(T ) < γ and A kµ < γ. Then by continuity of µ → log + D x T kµ dµ(x) and by upper semicontinuity of χ + 0 one can choose the parameter δ µ > 0 such that for all ergodic measures ν with dist(ν, µ) < δ µ we have :

log + D x T kµ dν(x) -log + D x T kµ dµ(x) < γ χ + 0 (µ, T ) > χ + 0 (ν, T ) -γ
We fix some ergodic measure ν with dist(µ, ν) < δ µ . One can assume h(ν) > 3γ. By Ruelle-Margulis inequality we have therefore χ

+ (ν) = χ + 0 (ν) > 3γ and χ -(ν) = χ - 0 (ν) < -3γ.
According to the subadditive ergodic theorem (see Lemma 2), there exists for any 0

< σ < 1 a Borel set F σ of ν measure larger than σ s.t. ∃n σ ∀n > n σ , χ + (ν) -1 kµn log + D x T kµn < γ and χ -(ν) -1 kµn log + D T kµ n x T -kµn < γ.
Remark that these inequalities can be rewritten as :

∃C > 1 ∀n ∈ N, F σ ⊂ H n T kµ (χ + (ν), χ -(ν), γ, C
) One can also assume by the ergodic theorem that (λ + n (x, T kµ )) n∈N are converging uniformly in x ∈ F σ to log + D y T kµ dν(y).

We apply now Proposition 1 to T kµ and to a given point x ∈ F σ : there exists ǫ µ > 0 depending only on D 2 T kµ , D 2 T -kµ , DT kµ and DT kµ , a real number D depending only on χ + (ν), χ -(ν), γ, C, DT kµ and DT -kµ and families (F n ) n∈N of C 1 maps from [0, 1] 2 to M satisfying the properties (i),(ii),(iii) of Proposition 1.

For all 0 < δ <

1 let E δ ⊂ [0, 1] 2 denote the subset of points of the square of the form kδ with k ∈ N 2 . Consider φ ∈ F n . It follows from the first item (i) that φ(E δ ) is a (n + 1, δ) spanning set of φ([0, 1] 2 ) for T kµ . Indeed given a point y = φ(z) with z ∈ [0, 1] 2 there exists t ∈ E δ such that
z -t < δ and then by using the first item (i), we get d T lkµ y, T lkµ (φ(t)) < δ for l = 0, ..., n, that is y ∈ B T kµ (φ(t), n + 1, δ). Since two points lying in the same Bowen ball of radius δ and length n + 1 are not (n + 1, 2δ) separated, we get :

max ♯E : E ⊂ F σ ∩ φ([0, 1] 2 ) and E is a (n + 1, 2δ) separated set for T kµ ≤ 1 δ 2
and then

max ♯E : E ⊂ F σ ∩ B(x, n + 1, ǫ) and E is a (n + 1, 2δ) separated set for T kµ ≤ ♯F n δ 2
By Proposition 1 (iii) it follows that :

h T kµ (M |F, ǫ µ ) ≤ lim n→+∞ sup x∈Fσ 2 + H([λ + n (x, T kµ ) -χ + (ν, T kµ )] + 3) λ + n (x, T kµ ) -χ + (ν, T kµ ) + A
According to the definition of F σ we have :

lim n→+∞ inf x∈Fσ λ + n (x, T kµ ) -χ + (ν, T kµ ) = lim n→+∞ sup x∈Fσ λ + n (x, T kµ ) -χ + (ν, T kµ ) = log + D y T kµ dν(y) -χ + (ν, T kµ ) ≥ 0
Moreover we deduce from the choice of δ µ that :

log + D y T kµ dµ(y) -χ + (ν, T kµ ) -γ ≤ log + D y T kµ dν(y) -χ + (ν, T kµ ) ≤ log + D y T kµ dµ(y) -χ + (ν, T kµ ) + γ
and then by the choice of k µ and since χ + 0 (µ, T ) > χ + 0 (ν, T ) -γ we get :

χ + 0 (µ, T kµ ) -χ + 0 (ν, T kµ ) -2k µ γ ≤ log + D y T kµ dν(y) -χ + 0 (ν, T kµ ) ≤ χ + 0 (µ, T kµ ) -χ + 0 (ν, T kµ ) + 2k µ γ ≤ χ + 0 (µ, T kµ ) -χ + 0 (ν, T kµ ) + 4k µ γ
Now we distinguish cases :

• either χ + 0 (µ, T ) -χ + 0 (ν, T ) < 3γ, then the term lim n→+∞ sup x∈Fσ H([λ + n (x, T kµ )-χ + 0 (ν, T kµ )]+ 3) λ + n (x, T kµ ) -χ + 0 (ν, T kµ
) is bounded above by 5 log(2)k µ γ.

• or χ + 0 (µ, T ) ≥ χ + 0 (ν, T )+3γ, then we have lim n→+∞ inf x∈Fσ λ + n (x, T kµ )-χ + 0 (ν, T kµ ) ≥ k µ γ. But recall we choose k µ large enough so that 2R(T )H(k µ γ) < γ. It follows that :

lim n→+∞ sup x∈Fσ H([λ + n (x, T kµ ) -χ + 0 (ν, T kµ )] + 3) λ + n (x, T kµ ) -χ + 0 (ν, T kµ ) ≤ H(k µ γ)2k µ R(T ) ≤ k µ γ
We get finally in both cases :

h T kµ (M |F, ǫ µ ) ≤ 2 χ + 0 (µ, T kµ ) -χ + 0 (ν, T kµ ) + (5 log(2) + 8)k µ γ + A
Then by letting σ go to 1 we obtain since A kµ < γ :

h N ew T kµ (M |ν, ǫ µ ) k µ ≤ 2 χ + 0 (µ, T ) -χ + 0 (ν, T ) + (5 log(2) + 9)γ Moreover according to Lemma 1, we have h N ew T (M |ν, ǫ µ ) ≤ h N ew T kµ (M|ν,ǫµ) kµ
. Therefore :

h N ew T (M |ν, ǫ µ ) ≤ 2 χ + 0 (µ, T ) -χ + 0 (ν, T ) + (5 log(2) + 9)γ
This concludes the proof of Theorem 3 because γ can be chosen arbitrarily small.

Finite time stable fields

We endow R 2 with the Euclidian norm . The induced norm on the spaces L(R 2 , R 2 ) and L(L(R 2 , R 2 ), R 2 ) will be also denoted . Also we write DT := sup x∈U D x T and

D φ T := sup x∈[0,1] 2 D φ(x) T where φ : [0, 1] 2 → R 2 and T is a C 1 map defined on an open subset U of R 2 containing the image of φ.
In the two following sections we consider a sequence

T := (T n ) n∈N of C 2 diffeomorphisms from B(0, 2) ⊂ R 2 to R 2 with T n (0) = 0 for all n ∈ N. For each n ∈ N we denote T n the composition T n • ... • T 1 defined on B(n, 2) := {x ∈ R 2 , 0 ≤ ∀k < n, T k x < 2}.
We extend the notion of finite time hyperbolic set to this background by defining for all real numbers χ + > 0 > χ -, min(χ + , -χ -) > γ > 0 and C > 1 the set :

H n T (χ + , χ -, γ, C) := {x ∈ B(n, 2) : ∀1 ≤ k ≤ n, C -1 e (χ + -γ)k ≤ D x T k ≤ Ce (χ + +γ)k , C -1 e (-χ --γ)k ≤ D T k x T -k ≤ Ce (-χ -+γ)k } Recall a matrix A ∈ Gl 2 (R) is said hyperbolic if A , A -1 > 1.
Such a matrix sends a circle to an ellipse. This defines two specific orthogonal directions with orthogonal images : the most contracted and the most expanded one. Let e A and f A be unit vectors with arbitrary sense and with directions coinciding respectively with the most contracted and the most expanded directions, i.e. e A = f A = 1, Ae A = A -1 -1 and Af A = A . If u, v ∈ R 2 we will denote by ∠u, v the oriented angle of the two directed lines generated by u and v, that is the angle belonging to ] -π, π] determined up to integral multiples of 2π. We also write |∠|u, v := |∠u, v|.

Let us denote by

U n the open subset of R 2 defined by U n := {x ∈ B(n, 2), D x T n is hyperbolic}, e n : U n → R 2 the
most contracted field (also called the n finite time stable field), that is e n (x) = e DxT n for all x ∈ U n , and f n : U n → R 2 the most expanded one (also called the n finite time unstable field), that is f n (x) = f DxT n . One can choose the vector fields e n and f n to be continuous (an open subset of R 2 is orientable) and ∠e n (x), f n (x) = π 2 for all x ∈ U n . We also consider the vector fields e n ) are orthogonal so that, the angles ∠e n (x), e n (y) (resp. ∠e

(n) n (x), e (n) n (y)) and ∠f n (x), f n (y) (resp. ∠f (n) n (x), f (n)
n (y)) are equal. As e n is a unit vector field, the image of D x e n is orthogonal to e n (x) for all x ∈ U n (by derivating the relation e n (x) 2 = 1). The same remark apply to e

(n) n , f n , f (n)
n . Also by derivating the orthoganility relations e n (x).f n (x) = 0 and e

(n) n (x).f (n) n (x) = 0 we get D x e n (u).f n (x) = -D x f n (u).e n (x) and D x e (n) n (u).f n = -D x f (n) n (u).e (n) n (x). It follows that D x e n (u) = D x f n (u) and D x e (n) n (u) = D x f (n) n (u) for all u ∈ R 2 and for all x ∈ U n . If φ : [0, 1] 2 → R 2 is a C 1 map, we say that φ satisfies (H n ) when : (H n ) : D(x → D φ(x) T n ) ≤ K D φ T n ; D(x → D T n •φ(x) T -n ) ≤ K D T n •φ T -n .
The real number K is a very small universal constant (K = 10 -10 is suitable). This constant will be slightly changed at finitely many steps of the proof but we always denote it by K to simplify the computations. For example we will write 2K = K, e K = 1 + K, etc.

Observe that (H n ) is symmetrical by inversing the dynamical system, i.e. if φ

: [0, 1] 2 → R 2 is a C 1 map satisfying (H n ) for the map T n , then so does T n • φ for the map T -n .
This assumption is the main key point of this paper. For interval maps it follows easily from the total order on R that the maximal cardinality of (n, δ) separated sets lying in a monotone branch of f n is less than n/δ. Then to estimate the Newhouse local entropy one only needs to count the number of monotone branches (which coincide with the invertible branches) [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF]. Contrarily to the one dimensional case where positive entropy requires noninvertibility, surface diffeomorphisms may have positive entropy. The Smale horseshoe is a key model of such maps : indeed any surface diffeomorphism admits an invariant compact hyperbolic set in the closure of periodic points with topological entropy arbitrarily close to the global topological entropy [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF] and horseshoes then arise near homoclinic points according to Smale-Birkhoff homoclinic theorem [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields[END_REF]. In this context entropy results in some sense from a combination of expansion, contraction and bending. One can try to define in dimension two a notion analagous to the notion of monotone branches in dimension one. The natural idea is to consider pieces of the surface where the effect of bending is small and this corresponds exactly to the assumption (H n ) (see the following proposition). For example the Smale horseshoe clearly does not satisfy (H n ). Observe also that if T is an interval map and φ :

[0, 1] → [0, 1] is a C 1 map satisfying D(x → D φ(x) T n ) ≤ K D φ T n then the image of φ lies in a monotone branch of T n .
Another approach was developped by Y.Yomdin in the eighties [28] [29]. He used semialgebraic tools to bound the local dynamical complexity of C r maps. In Yomdin's theory which applies in any dimension and in any intermediate regularity one reparametrizes a Bowen ball of order n by C r maps φ defined on the unit square such that the C r norm (i.e. the maximum over 1 ≤ s ≤ r of the supremum norm of the s th derivative) of T k • φ is less than 1 for 0 ≤ k ≤ n -1 (in particular the maps T k • φ are contracting and therefore the reparametrization maps φ do not separate points). Remark that these conditions for r ≥ 2 do not lead to the assumption (H n ). Indeed the reparametrization φ can be so contracting that the ocillation of DT n on the image of φ is still not small compared to its supremum norm DT n . Conversely the assumption (H n ) do not imply that D 2 (T • φ) < 1. However to bound the entropy we do not need to control the second derivative of T • φ : as a naive example an invertible C 2 interval map do not separate points and may have a large C 2 norm but the oscillation of the derivative is less than the supremum norm of the derivative to ensure the invertibility.

We introduce now a very weak assumption of hyperbolicity. If φ : [0, 1] 2 → R 2 is a C 1 map, we say that φ satisfies (F n ) when :

(F n ) : ∀t ∈ [0, 1] 2 , D φ(t) T n D T n •φ(t) T -n ≥ 2 ; D φ T n D T n •φ T -n ≥ max k=0,...,n D φ T k .
In the following proposition we prove that under the assumptions (H n ) and (F n ) the finite time stable and unstable fields e n , f n and their image e

(n) n , f (n) n
do not oscillate on the image of φ.

Proposition 2 Let n ∈ N and let

φ : [0, 1] 2 → R 2 be a C 1 map satisfying (H n ) and (F n ) then we have for all y ∈ [0, 1] 2 : D y (e n • φ) ≤ K D y (e (n) n • φ) ≤ K Proof :
Clearly it is enough to prove that for all t ∈ [0, 1] 2 there exists a function ǫ : R + → R + with lim t→0 ǫ(t) = 0 such that |∠|e n • φ(t), e n • φ(s) ≤ K s -t + s -t ǫ( s -t ) and |∠|e

(n) n • φ(t), e (n) n • φ(s) ≤ K s -t + s -t ǫ( s -t ) for all s ∈ [0, 1] 2 , that is with the Laudau notation |∠|e n • φ(t), e n • φ(s) ≤ K s -t + o s→t ( s -t ) and |∠|e (n) n • φ(t), e (n) n • φ(s) ≤ K s -t + o s→t ( s -t ). Let t, s ∈ [0, 1] 2 .
It follows from the triangular inequality that :

|∠|e (n) n • φ(t), e (n) n • φ(s) = |∠|f (n) n • φ(t), f (n) n • φ(s) = |∠|D φ(t) T n f n (φ(t)), D φ(s) T n f n (φ(s)) ≤ |∠|D φ(t) T n f n (φ(t)), D φ(t) T n f n (φ(s)) +|∠|D φ(t) T n f n (φ(s)), D φ(s) T n f n (φ(s))
The first part of the right member can be bounded in the following way :

tan ∠D φ(t) T n f n (φ(t)), D φ(t) T n f n (φ(s)) = tan ∠f n (φ(t)), f n (φ(s)) D φ(t) T n D T n φ(t) T -n Indeed we have f n (φ(s)) = f n (φ(t)) cos ∠f n (φ(s)), f n (φ(t)) + e n (φ(t)) sin ∠f n (φ(s)), f n (φ(t)). Then by applying D φ(t) T n : D φ(t) T n f n (φ(s)) = f (n) n (φ(t)) D φ(t) T n cos ∠f n (φ(t)), f n (φ(s))+ e (n) n (φ(t)) D T n •φ(t) T -n -1 sin ∠f n (φ(t)), f n (φ(s)) Therefore tan ∠D φ(t) T n f n (φ(t)), D φ(t) T n f n (φ(s)) = D T n •φ(t) T -n -1 sin ∠f n (φ(t)), f n (φ(s)) D φ(t) T n cos ∠f n (φ(t)), f n (φ(s)) = tan ∠f n (φ(t)), f n (φ(s)) D φ(t) T n D T n •φ(t) T -n
Consider now the second part |∠|D φ(t) T n f n (φ(s)), D φ(s) T n f n (φ(s)). We have :

D φ(t) T n f n (φ(s)) -D φ(s) T n f n (φ(s)) ≥ |D φ(t) T n f n (φ(s)).e (n) n (φ(s))| ≥ D φ(t) T n f n (φ(s)) × sin |∠|D φ(t) T n f n (φ(s)), D φ(s) T n f n (φ(s)) But we have K s-t D φ T n ≥ D φ(t) T n f n (φ(s))-D φ(s) T n f n (φ(s)) and D φ(t) T n ≥ (1- K) D φ T n by assumption (H n ). Moreover by continuity D φ(t) T n f n (φ(s)) goes to D φ(t) T n when s goes to t. It implies that sin |∠|D φ(t) T n f n (φ(s)), D φ(s) T n f n (φ(s)) ≤ K s-t +o s→t ( s- t ).
Finally we have :

|∠|e (n) n • φ(t), e (n) n • φ(s) ≤ |∠|f n (φ(t)), f n (φ(s)) D φ(t) T n D T n •φ(t) T -n + K s -t + o s→t ( s -t )
By considering T n • φ instead of φ and T -1 instead of T we get symetrically :

|∠|f n • φ(t), f n • φ(s) ≤ |∠|e (n) n (φ(t)), e (n) n (φ(s)) D φ(t) T n D T n •φ(t) T -n + K s -t + o s→t ( s -t )
But by assumption (F n ) we have D φ(t) T n D T n •φ(t) T -n ≥ 2 for all t ∈ [0, 1] 2 and then one concludes the proof of the lemma by combining the two previous inequalities.

Observe that in fact we only need to suppose the first part of the assumption (F n ) (which is symmetrical) in the previous proof ; the second part will be useful in the next sections.

Finite time Rectangle

The vectors fields e n , f n : U n → R 2 are C 1 so that they can be locally integrated. We consider the integral curves E n (z) (resp. F n (z)) through z to the field e n (resp. f n ). These curves are called n finite time stable (resp. unstable) manifolds. This approach comes from the works of M.Benedicks and L.Carleson on the Henon map [START_REF] Benedicks | The dynamics of the Henon map[END_REF] and was formalized by M.Holland and S.Luzzato [START_REF] Holland | A new proof of the Stable Manifold Theorem for hyperbolic fixed points on surfaces[END_REF] [START_REF] Holland | Stable manifolds under very weak hyperbolicity conditions[END_REF]. In particular, under some assumptions of hyperbolicity on surfaces, M.Holland and S.Luzzato [START_REF] Holland | A new proof of the Stable Manifold Theorem for hyperbolic fixed points on surfaces[END_REF] give an alternative proof of the existence of stable manifolds by proving the convergence of finite time stable manifolds. Following the notion of rectangle in the theory of hyperbolic dynamical systems we introduce now finite time rectangles. This new finite time concept is particularly well adapted to the computation of entropy.

Definition 2 A subset R n of U n is called a n-rectangle if there exists a diffeomorphism φ : V → W ⊂ U n defined on an open neighborhood V of [0, 1] 2 with φ([0, 1] 2 ) = R n , such that φ([0, 1] × {0}) and φ([0, 1] × {1}) (resp. φ({0} × [0, 1]
) and φ({1} × [0, 1]) are included in a n finite time stable (resp. unstable) manifold.

In the following the map φ and the couple (φ, R n ) will also refer to a n-rectangle.

Observe that if (φ, R n ) is a n-rectangle for T n then (T n • φ, T n R n ) is a n-rectangle for T -n .
We can give a satisfactory geometrical reparametrization of a n-rectangle R n . For any point x of R n , we reparametrize E n (x) ∩ R n and F n (x) ∩ R n from [0, 1] with constant rate in the same direction as e n (x) and f n (x). We get in this way two maps of the interval φ e

x : [0 ). Remark also that ∂ 1 φ x,n (t, s) (resp. ∂ 2 φ x,n (t, s)) is colinear to e n (φ x,n (t, s)) (resp. to f n (φ x,n (t, s))). In fact φ x,n is a foliation box for the foliations F n and E n simultaneously. Such reparametrizations φ x,n will be called admissible charts.

, 1] → E n (x)∩R n and φ f x : [0, 1] → F n (x) ∩ R n . Then the map φ x,n : [0, 1] 2 → R n defined by φ x,n (t, s) = F n (φ e x (t)) ∩ E n (φ f x (s)) is a diffeomorphism from [0, 1]
To simplify the notations, we write a ≃ λ b for a, b > 0 and λ ≥ 1 which means that λ -1 ≤ a b ≤ λ. Moreover if γ is a C 1 curve its lenght will be denoted by l(γ). We say that a n-rectangle R n satisfies (G n ) when :

(G n ) : ∀x ∈ R n , (1 + K)l(E n (x) ∩ R n ) ≥ l(F n (x) ∩ R n ) ≥ (1 -K) l(E n (x) ∩ R n ) max k=0,...,n D x T k .
The assumption (G n ) is not symmetrical : if R n is a n-rectangle satisfying (G n ) then this assumption is not satisfied in general by the n-rectangle T n R n for T -n . However we prove in the next lemma that (1

+ K)l(T n F n (x) ∩ T n R n ) ≥ l(T n E n (x) ∩ T n R n ) assuming (H n ), (G n ) and (F n ) for R n .

Finite time rectangle with small oscillation of the derivative

We show that under the assumptions (H n ), (G n ) and (F n ) the n-rectangles really look like rectangles : the n finite time stable (resp. unstable) manifolds foliating a n-rectangle have almost the same length (Proposition 3) and more precisely the map φ x,n reparametrizes these manifolds almost with constant rate (Proposition 4).

Lemma 4 Let n ∈ N and let (φ, R n ) be a n-rectangle satisfying (H n ), (G n ) and (F n ). Then we have for all x ∈ R n :

(1 + K)l(T n F n (x) ∩ T n R n ) ≥ l(T n E n (x) ∩ T n R n ) Proof : First remark that the assumption (H n ), which states that D(x → D φ(x) T n ) ≤ K D φ T n and D(x → D T n •φ(x) T -n ) ≤ K D T n •φ T -n , imply respectively that D x T n ≃ 1+K D φ T n and D T n (x) T -n ≃ 1+K D T n •φ T -n for all x ∈ R n . Therefore l(T n F n (x) ∩ T n R n ) ≃ 1+K D φ T n l(F n (x) ∩ R n ) l(T n E n (x) ∩ T n R n ) ≃ 1+K D T n •φ T -n -1 l(E n (x) ∩ R n )
and then according to the hypotheses (G n ) and (F n )

l(T n F n (x) ∩ T n R n ) l(T n E n (x) ∩ T n R n ) ≃ 1+K D φ T n D T n •φ T -n l(F n (x) ∩ R n ) l(E n (x) ∩ R n ) ≥ D φ T n D T n •φ T -n (1 + K) max k=0,...,n D x T k ≥ 1 1 + K
This concludes the proof of the lemma.

Proposition 3 Let (φ, R n ) be a n-rectangle satisfying (H n ), (G n ) and (F n ). Then we have for all x, y ∈ R n : l(E n (x) ∩ R n ) ≃ 1+K l(E n (y) ∩ R n ) l(F n (x) ∩ R n ) ≃ 1+K l(F n (y) ∩ R n )
Proof : Fix x, y ∈ R n and let z ∈ R n be the intersection point of E n (y) and F n (x). One deduces easily from |∠|e n (u), e n (v) ≤ K for all u, v ∈ R n that :

• l(E n (x) ∩ R n ) ≃ 1+K φ e x (1) -x + x -φ e x (0) ; • l(E n (y) ∩ R n ) ≃ 1+K φ e y (1) -z + z -φ e y (0) .
Moreover the quadrilaterals with vertices φ e x (0), φ e y (0), x, z and φ e x (1), φ e y (1), x, z are almost rectangles : their interior angles differ from π 2 by at most K. Then we have by trivial trigonometric arguments x-φ e

x (1) ≤ (1+K) z -φ e y (1) +K x-z and x-φ e x (0) ≤ (1+K) z -φ e y (0) + K x -z . Going back to the length of the finite time stable manifolds we obtain l(

E n (x) ∩ R n ) ≤ (1 + K)l(E n (y) ∩ R n ) + Kl(F n (x) ∩ R n ) and finally l(E n (x) ∩ R n ) ≤ (1 + K)l(E n (y) ∩ R n ) since we have (1 + K)l(E n (x) ∩ R n ) ≥ l(F n (x) ∩ R n ) by assumption (F n ).
This concludes the first point of the lemma by switching the role of x and y.

One can apply the previous proof to the n-rectangle

T n R n for T -1 because (1+K)l(T n F n (x)∩ T n R n ) ≥ l(T n E n (x) ∩ T n R n ) for all x ∈ R n according to the previous lemma. We get then l(T n F n (x) ∩ T n R n ) ≃ 1+K l(T n F n (y) ∩ T n R n ) and finally l(F n (x) ∩ R n ) ≃ 1+K l(F n (y) ∩ R n ) since l(T n F n (u) ∩ T n R n ) ≃ 1+K D φ T n l(F n (u) ∩ R n ) for all u ∈ R n by assumption (H n ).
In Lemma 4 and Proposition 3 we did not require that the reparametrization φ of the n-rectangle are admissible charts. From now we will only consider reparametrizations of nrectangles of this kind.

Proposition 4 Let (φ x,n , R n ) be a n-rectangle satisfying (H n ), (F n ) and (G n ). Then we have for all t, s ∈ [0, 1] 2 : ∂ 1 φ x,n (t, s) ≃ 1+K l(E n (x) ∩ R n ) ∂ 2 φ x,n (t, s) ≃ 1+K l(F n (x) ∩ R n ) Proof : We first prove that ∂ 1 φ x,n (t, s) ≃ 1+K l(E n (x) ∩ R n ).
For simplicity we put φ := φ x,n . Let s 0 ∈ [0, 1] such that x ∈ φ([0, 1] × s 0 ). Since the n finite time stable manifold of x is reparametrized with constant rate we have

∂ 1 φ(t, s 0 ) = ∂ 1 φ e x (t) = l(E n (x) ∩ R n ) for all t ∈ [0, 1]. Fix t, s, s ′ ∈ [0, 1], we have : ∂ 1 φ(t, s) -∂ 1 φ(t, s ′ ) = ∂ 1 φ(t, s).e n (φ(t, s)) -∂ 1 φ(t, s ′ ).e n (φ(t, s ′ )) = s s ′ ∂ 2 (∂ 1 φ(t, u).e n (φ(t, u))) du = s s ′ ∂ 2 ∂ 1 φ(t, u).e n (φ(t, u))du + s s ′ ∂ 1 φ(t, s).∂ 2 e n (φ(t, u))du Now ∂ 2 ∂ 1 φ(t, u) = ∂ 1 ∂ 2 φ(t, u) = ∂ 1 ∂ 2 φ(t, u) f n (φ(t, u)) + ∂ 2 φ(t, u) ∂ 1 f n (φ(t, u)) so that by Proposition 2 : s s ′ ∂ 2 ∂ 1 φ(t, u).e n (φ(t, u))du ≤ K s s ′ ∂ 2 φ(t, u) du ≤ Kl(F n (φ(t, s 0 )) ∩ R n )
and then according to Proposition 3 and assumption (G n ) :

s s ′ ∂ 2 ∂ 1 φ(t, u).e n (φ(t, u))du ≤ Kl(F n (x) ∩ R n ) ≤ Kl(E n (x) ∩ R n ) Moreover s s ′ ∂ 1 φ(t, u).∂ 2 e n (φ(t, u))du ≤ K s s ′ ∂ 1 φ(t, u) du
We deduce from the two previous inequalities that :

∂ 1 φ(t, s) ≤ ∂ 1 φ(t, s ′ ) + Kl(E n (x) ∩ R n ) + K s s ′ ∂ 1 φ(t, u) du so that by Grönwall Lemma we get ||∂ 1 φ(t, s) ≤ ( ∂ 1 φ(t, s ′ ) + Kl(E n (x) ∩ R n )) e K|s-s ′ | ≤ (1 + K) ∂ 1 φ(t, s ′ ) + Kl(E n (x) ∩ R n ). Then by choosing s = s 0 or s ′ = s 0 we conclude that ∂ 1 φ(t, s) ≃ 1+K l(E n (x)∩R n ) for all (t, s) ∈ [0, 1] 2 . To get the other relation ∂ 2 φ x,n (t, s) ≃ 1+K l(F n (x) ∩ R n ) we apply the previous proof to the n-rectangle T n R n for T -1 .
We deduce from the two previous propositions that the change of admissible charts of a given n-rectangle is C 1 bounded so that the choice of the admissible chart do not interfere to bound the dynamical complexity of T on R n :

Corollary 1 Let R n a n-rectangle satisfying (H n ), (G n ) and (F n ). Then we have for all x, y ∈ R n : D(φ x,n • φ -1 y,n ) ≤ 1 + K
The following Corollary allows us to bound the derivative of T n • φ x,n as soon as the diameter of T n R n is bounded, i.e. we control the derivative of this map by the size of its image.

Corollary 2 Let (φ x,n , R n ) be a n-rectangle satisfying (H n ), (G n ), (F n ) and such that the di- ameter of T n R n is less than 2.
Then we have for all x ∈ R n :

D(T n • φ x,n ) ≤ 2 + K Proof : It's enough to bound the derivative in the unstable direction, that is ∂ 2 (T n • φ x,n ) ≤ 2 + K.
We have for all t ∈ [0, 1] :

1 0 ∂ 2 (T n • φ x,n )(t, s)ds = T n • φ x,n (t, 1) -T n • φ x,n (t, 0) ≤ 2 Remark now that : (1 -K) 1 0 ∂ 2 (T n • φ x,n )(t, s) ds ≤ 1 0 ∂ 2 (T n • φ x,n )(t, s).f (n) n (x)ds ≤ 1 0 ∂ 2 (T n • φ x,n )(t, s)ds We have also ∂ 2 (T n •φ x,n )(t, s) = ∂ 2 φ x,n (t, s) × D φx,n(t,s) T n ≃ 1+K l(F n (x)∩R n ) D x T n for all (t, s) ∈ [0, 1] 2 by Proposition 4. Therefore ∂ 2 (T n •φ x,n )(t, s) ≤ 2+K for all (t, s) ∈ [0, 1] 2 .

From n-to n + 1-rectangles

The end of this section devoted to the analysis of finite time rectangles deals with the links between n-rectangles and n + 1-rectangles. This point is crucial for the induction step in the proof of Proposition 1 presented in the next section.

In the following we consider sequences of maps T = (T n ) n∈N such that T n and its inverse

T -1 n is uniformly C 1 bounded, that is S(T ) := max(sup n DT n , sup n DT -1 n
) < +∞. To compare nand n + 1-rectangles we also need stronger hyperbolicity assumptions than the weak hypothesis (F n ). Let χ + > 0 > χ -, min(χ + ,-χ -) 3 > γ > 0 and C > 1. We will consider nrectangles (φ, R n ) included in the finite time hyperbolic set H n T (χ + , χ -, γ, C) with "n large". By n large we mean the statements of this subsection hold for n larger than an integer N depending only on χ + , χ -, γ, C and S(T ).

Oberve that such n-rectangles satisfy the assumption (F n ) and even (F n+1 ). Indeed if (φ, R n ) is a n-rectangle with R n ⊂ H n T (χ + , χ -, γ, C), then we have for large n :

D φ T n D T n •φ T -n max k=0,...,n D x T k ≥ C -3 e n(-χ --3γ) ≥ 1 and for all t ∈ [0, 1] 2 : D φ(t) T n D T n •φ(t) T -n ≥ C -2 e n(χ + -χ --2γ) ≥ 2
The next lemma estimates the angle |∠|e n (x), e n+1 (x) under the condition of finite time hyperbolicity :

Lemma 5 For all x ∈ H n T (χ + , χ -, γ, C) and for large n,

tan |∠|e n (x), e n+1 (x) ≤ 2 D T n x T n+1 (D T n x T n+1 ) -1 (D x T n ) -1 D x T n (2)
Proof :

We write e n (x) = e n+1 (x) cos ∠e n (x), e n+1 (x) + f n+1 (x) sin ∠e n (x), e n+1 (x) and we apply D x T n+1 :

D x T n+1 e n (x) = e (n+1) n+1 (x) D T n+1 x T -n-1 -1 cos ∠e n (x), e n+1 (x)+ f (n+1) n+1 (x) D x T n+1 sin ∠e n (x), e n+1 (x)
and then by taking the Euclidian norm :

D x T n+1 e n (x) 2 = D T n+1 x T -n-1 -2 cos 2 ∠e n (x), e n+1 (x) + D x T n+1 2 sin 2 ∠e n (x), e n+1 (x)
We compute :

tan |∠|e n (x), e n+1 (x) = D x T n+1 e n (x) 2 -D T n+1 x T -n-1 -2 D x T n+1 2 -D x T n+1 e n (x) 2 1 2
As x ∈ H n T (χ + , χ -, γ, C) we get for n large enough :

tan |∠|e n (x), e n+1 (x) ≤ D x T n+1 e n (x) / D x T n+1 (1 -D x T n+1 e n (x) 2 / D x T n+1 2 ) 1 2 ≤ 2 D x T n+1 e n (x) D x T n+1 ≤ 2 D T n x T n+1 (D T n x T n+1 ) -1 (D x T n ) -1 D x T n
This concludes the proof of the lemma.

Given a n-rectangle (φ x,n , R n ) we define its saturated set, Sat(R n ), as the smallest subset of R n saturated in n + 1 stable and unstable manifolds containing the middle third of R n , that is 5 :

Sat(R n ) := z ∈ R n , ∃u, v ∈ φ x,n 1 3 , 2 3 2 , s.t. z ∈ E n+1 (u) ∩ F n+1 (v)
5 Note that Sat(Rn) depends on the choice of the admissible chart φx,n

Proposition 5 For any n-rectangle (φ x,n , R n ) with large n included in

H n T (χ + , χ -, γ, C) satisfy- ing (H n ) and (G n ), the saturated set Sat(R n ) of R n is a n+1-rectangle included in φ x,n 1 3 -K, 2 3 + K 2 .
Proof :

Let (t 0 , s 0 ) ∈ [ 1 3 , 2 3 ] 2 . Denote α(t, s 0 ) ∈ [0, 1] the C 1 map such that α(t 0 , s 0 ) = s 0 and φ x,n (t, α(t, s 0 )) = F n (φ x,n (t, α(t, s 0 ))) ∩ E n+1 (φ x,n (t 0 , s 0 )) for all t ∈ [0, 1]. Observe that d dt φ x,n (t, α(t, s 0 )) is colinear to e n+1 (φ x,n (t, α(t, s 0 ))). But d dt φ x,n (t, α(t, s 0 )) = ∂ 1 φ x,n (t, α(t, s 0 )) + ∂ 1 α(t, s 0 )∂ 2 φ x,n (t, α(t, s 0 ))
and then according to Lemma 5 and Proposition 4 we have with z := φ x,n (t, α(t, s 0 )) :

(1 -K) l(F n (x) ∩ R n ) l(E n (x) ∩ R n ) |∂ 1 α(t, s 0 )| ≤ tan |∠|e n (z), e n+1 (z) ≤ (2 + K)S(T ) 2 D x T n D T n x T -n
It follows from the hyperbolicity assumption and the hypothesis (G n ) that for n large enough we have

|∂ 1 α(t, s 0 )| ≤ K and thus α(t, s 0 ) ∈ [ 1 3 -K, 2 3 + K] for all t ∈ [0, 1]. It means that E n+1 (φ x,n (t 0 , s 0 ))∩R n ⊂ φ x,n [0, 1] × [ 1 3 -K, 2 3 + K] . Similarly we prove that F n+1 (φ x,n (t 0 , s 0 ))∩ R n ⊂ φ x,n [ 1 3 -K, 2 3 + K] × [0, 1] .
We show now that the conclusions of Proposition 3 and 4 apply to the saturated n+1-rectangle of a n-rectangle satisfying the assumptions (H n ), (H n+1 ) and (G n ).

Proposition 6 For any n-rectangle (φ x,n , R n ) with n large included in H n T (χ + , χ -, γ, C) satisfying (H n ), (H n+1 ) and (G n ) the saturated n + 1-rectangle Sat(R n ) satisfies for all x ′ ∈ R n and

y ′ ∈ Sat(R n ) : l(F n+1 (y ′ ) ∩ Sat(R n )) ≃ 1+K l(F n (x ′ ) ∩ R n ) 3 (3) l(E n+1 (y ′ ) ∩ Sat(R n )) ≃ 1+K l(E n (x ′ ) ∩ R n ) 3 (4)
Proof : The lemma follows easily from the following relations which hold for large n :

• ∀z, z ′ ∈ R n , ∠e n+1 (z), e n+1 (z ′ ) < K by Proposition 2 ;

• ∀z ∈ R n , ∠e n (z), e n+1 (z) < K by Lemma 5 ;

• φ x,n ([ 1 3 , 2 3 ] 2 ) ⊂ Sat(R n ) ⊂ φ x,n ([ 1 3 -K, 2 3 + K] 2
) by Proposition 5.

Proposition 7 For any n-rectangle (φ x,n , R n ) with n large included in H n T (χ + , χ -, γ, C) satisfying (H n ), (H n+1 ) and (G n ) and for any y ∈ Sat(R n ), the saturated n+1-rectangle (φ y,n+1 , Sat(R n )) has the following properties for all t, s ∈ [0, 1] 2 :

∂ 1 φ y,n+1 (t, s) ≃ 1+K l(E n+1 (y) ∩ Sat(R n )) ∂ 2 φ y,n+1 (t, s) ≃ 1+K l(F n+1 (y) ∩ Sat(R n ))
The proof of Proposition 7 is technical and may be skipped in a first reading.

Proof : Let us first prove that ∂ 1 φ y,n+1 (t, s) ≃ 1+K l(E n+1 (y) ∩ R n ) for all (t, s) ∈ [0, 1] 2 . It follows from Proposition 2 that D(f n+1 • φ x,n ) ≤ K. Then by Proposition 4, we have for all z ∈ R n :

D z f n+1 ≤ Kl(F n (x) ∩ R n ) -1
Let z and z ′ be two points of Sat(R n ) lying on the same n + 1 stable manifold. We reparametrize

F n+1 (z)∩Sat(R n ) and F n+1 (z ′ )∩Sat(R n ) by arclength. We get two curves z n+1 : [-S(z), T (z)] → F n+1 (z) and z ′ n+1 : [-S(z ′ ), T (z ′ )] → F n+1 (z ′ ) with z n+1 (0) = z and z ′ n+1 (0) = z ′ .
We have for allmin(S(z), S(z ′ )) ≤ u ≤ min(T (z), T (z ′ )) :

z n+1 (u) -z ′ n+1 (u) ≤ z -z ′ + t 0 f n+1 (z n+1 (s)) -f n+1 (z ′ n+1 (s)) ds ≤ z -z ′ + t 0 D φx,n f n+1 z n+1 (s) -z ′ n+1 (s) ds
By Grönwall Lemma we have then for allmin(S(z),

S(z ′ )) ≤ u ≤ min(T (z), T (z ′ )) (in particular |u| ≤ l(F n+1 (z) ∩ Sat(R n )) ≤ l(F n (x) ∩ R n )) : z n+1 (u) -z ′ n+1 (u) ≤ z -z ′ e |u| D φx,n fn+1 ≤ z -z ′ e |u|Kl(Fn(x)∩Rn) -1 ≤ (1 + K) z -z ′
Let us denote by Z ′ n+1 (u) ∈ Sat(R n ) the intersection point of E n+1 (z n+1 (u)) and F n+1 (z ′ ). Remark that by continuity of the n + 1 stable field e n+1 there exists w in the finite time unstable piece [z n+1 (u), Z ′ n+1 (u)] such that the vector z n+1 (u)-Z ′ n+1 (u) is colinear to e n+1 (w). Then since

|∠|e n+1 (v), e n+1 (v ′ ) ≤ K for all v, v ′ ∈ R n it follows that F n+1 (z) ∩ Sat(R n ) (resp. F n+1 (z ′ ) ∩ Sat(R n )) lies in the cone {v, |∠|v -z n+1 (u), f n+1 (w) < K} (resp. {v, |∠|v -Z ′ n+1 (u), f n+1 (w) < K}).
In particular the distance d(z n+1 (u), F n+1 (z ′ )) is bounded from below by |z n+1 (u) -Z ′ n+1 (u)| cos K. It follows that :

z n+1 (u) -Z ′ n+1 (u) ≤ d(z n+1 (u), F n+1 (z)) cos K ≤ (1 + K) z n+1 (u) -z ′ n+1 (u) ≤ (1 + K) z -z ′
Let t, t ′ , s, s ′ such that φ y,n+1 (t, s) = z, φ y,n+1 (t ′ , s) = z ′ , φ y,n+1 (t, s ′ ) = z n+1 (u) and φ y,n+1 (t ′ , s ′ ) = Z ′ n+1 (u). When z goes to z ′ we get from the last inequality : ∂ 1 φ y,n+1 (t, s ′ ) ≤ (1 + K) ∂ 1 φ y,n+1 (t, s) and by inverting the roles of s and s ′ , we have for all t, s, s ′ ∈ [0, 1] :

∂ 1 φ y,n+1 (t, s ′ ) ≃ 1+K ∂ 1 φ y,n+1 (t, s)
But the n + 1 stable manifold of y is reparametrized with constant rate. It follows that :

∂ 1 φ y,n+1 (t, s ′ ) ≃ 1+K l(E n+1 (y) ∩ Sat(R n ))
We deal now with the partial derivative ∂ 2 φ y,n+1 . By Proposition 2 applied to T n+1 • φ x,n for T -1 , we have :

D(e (n+1) n+1 • φ x,n ) ≤ K (5) Let us compute D(T n+1 • φ x,n ) -1 = D(φ -1 x,n )DT -n-1 .
In the bases (e

(n+1) n+1 , f (n+1)
n+1 ) → (e n+1 , f n+1 ) the matrix associated to DT -n-1 is given by :

(DT n+1 ) -1 0 0 DT n+1 -1
and by Proposition 4 the matrix associated to

D(φ -1 x,n ) in the bases (e n , f n ) → (∂ t , ∂ s ) is up to a factor 1 + K : l(E n (x) ∩ R n ) -1 0 0 l(F n (x) ∩ R n ) -1
Finally the matrix passage from the basis (e n+1 , f n+1 ) to the basis (e n , f n ) is just the rotation matrix with angle ∠e n+1 , e n . After a simple computation we get for all z ∈ T n+1 R n up to a factor 1 + K :

D z (T n+1 • φ x,n ) -1 = cos(∠en+1,en) (DxT n+1 ) -1
l(En(x)∩Rn) -sin(∠en+1,en) DxT n+1 l(En(x)∩Rn) sin(∠en+1,en) (DxT n+1 ) -1 l(Fn(x)∩Rn) cos(∠en+1,en) DxT n+1 l(Fn(x)∩Rn)

We deduce from the inequalities tan |∠|e n (x), e n+1 (x) <

2S(T ) 2 (DxT n ) -1 DxT n (Lemma 5) and (1 + K)l(E n (x) ∩ R n ) ≥ l(F n (x) ∩ R n ) ≥ (1 -K) l(En(x)∩Rn)
DxT n (Hypothesis (G n )) that for n large enough :

D z (T n+1 • φ x,n ) -1 ≤ (1 + K)l(E n (x) ∩ R n ) -1 (D x T n+1 ) -1
and by Proposition 6 and assumption (H n+1 ) :

D z (T n+1 • φ x,n ) -1 ≤ 1 + K 3 l(E n+1 (y) ∩ Sat(R n )) -1 (D x T n+1 ) -1 ≤ l T n+1 E n+1 (y) ∩ T n+1 Sat(R n ) -1
Then we have by Equation ( 5) :

D z e (n+1) n+1 • T -n-1 ≤ Kl T n+1 E n+1 (y) ∩ T n+1 Sat(R n ) -1
and thus we can apply the first part of the proof to the n + 1-rectangle T n+1 Sat(R n ) for T -1 to get :

∂ 2 (T n+1 • φ y,n+1 ) ≃ 1+K l T n+1 F n+1 (y) ∩ T n+1 Sat(R n )
and therefore again by assumption (H n+1 ) :

∂ 2 φ y,n+1 ≃ 1+K l(F n+1 (y) ∩ Sat(R n ))
We deduce from the previous proposition that the change of coordinates φ -1 x,n • φ y,n+1 from the n + 1 to the n foliations is C 1 bounded.

Corollary 3 For any

n-rectangle (φ x,n , R n ) with n large included in H n T (χ + , χ -, γ, C) satisfying (H n ), (H n+1 ) and (G n ) and for any y ∈ Sat(R n ), the saturated n+1-rectangle (φ y,n+1 , Sat(R n )) satisfies : • D (φ x,n ) -1 • φ y,n+1 ≤ 1 3 + K ; • D (φ y,n+1 ) -1 • (φ x,n ) /[ 1 3 , 2 3 ] 2 ≤ 3 + K.

Proof :

The previous propostion claims that in the bases (∂ t , ∂ s ) → (e n+1 , f n+1 ) the matrix associated to Dφ y,n+1 is up to a factor 1 + K :

l(E n+1 (y) ∩ Sat(R n )) 0 0 l(F n+1 (y) ∩ Sat(R n ))
then as in the previous proof we get after an easy matricial computation that for all z ∈ [0, 1] 2 and up to a factor 1 + K :

D z (φ x,n ) -1 • φ y,n+1 = cos(∠e n+1 , e n ) l(En+1(x)∩Sat(Rn)) l(En(x)∩Rn)
sin(∠e n+1 , e n ) l(Fn+1(x)∩Sat(Rn)) l(En(x)∩Rn)

sin(∠e n+1 , e n ) l(En+1(x)∩Sat(Rn)) l(Fn(x)∩Rn) cos(∠e n+1 , e n ) l(Fn+1(x)∩Sat(Rn)) l(Fn(x)∩Rn)

Then we conclude the proof by Proposition 6, assumption (G n ) and Lemma 5.

6 Reparametrization of (n, ǫ) Bowen balls

This section is devoted to the proof of Proposition 1. We begin with two technical lemmas.

Combinatorial Lemma

We recall first a usual combinatorial lemma which was already used by T.Downarowicz and A.Maass in [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF].

Definition 3 Let S ∈ N and n ∈ N. We say that a sequence of n positive integers

K n := (k 1 , ..., k n ) admits the value S if 1 n n i=1 k i ≤ S.
The number of sequences of n positive integers admitting the value S is exactly the binomial coefficient nS n . By applying Stirling's formula we get easily :

Lemma 6 The logarithm of the number of sequences of n positive integers admitting the value S is at most nSH(S) + 1.

Lemma of Linear algebra

The following basic algebraic lemma allows us to control the oscillation of D T n •φ T -n and D φ T n at the same time. It claims that the defect of multiplicativity of the norm of a product of two matrices of GL 2 (R) is almost equal to this of the product of the inverses.

Lemma 7 Let A, B ∈ GL 2 (R) then AB A B = B -1 A -1 A -1 B -1
Proof : This follows immediately from the formula det(C) = C C -1 which holds for all matrices C of GL 2 (R) and from the multiplicativity of the determinant. This fact is specific to the dimensions 1 and 2. Indeed consider for x = 0 the matrices A(x), B(x) ∈ GL 3 (R) defined by :

A(x) =   1 0 0 0 x -1 0 x 1   and B(x) =   x x 0 1 -1 0 0 0 1   One easily computes : A(x) -1 =   1 0 0 0 1 2x 1 2x 0 -1 2 1 2   and B(x) -1 =   1 2x 1 2 0 1 2x -1 2 0 0 0 1   A(x)B(x) =   x x 0 x -x -1 x -x 1   and B(x) -1 A(x) -1 =   1 2x 1 4x 1 4x 1 2x -1 4x -1 4x 0 -1 2 1 2   In particular we have A(x) F = B(x) F ∼ √ 2x, A(x) -1 F = B(x) -1 F ∼ 3 2 , A(x)B(x) F ∼ √ 6x but B(x) -1 A(x) -1 F ∼ 1 2 when x goes to +∞.
We give now a dynamical interpretation of the previous lemma in our setting. In the proof of Theorem 4 assuming Proposition 1 the quantity λ + n (x, T )-1 n log D x T n is close to χ + (µ)-χ + (ν) when n goes to infinity for typical ν points x (where ν is an ergodic measure near an invariant measure µ). If we define similarly λ

- n (x, T ) = 1 n n-1 l=0 log + (D T l x T ) -1 then we can also ensure that λ - n (x, T ) -1 n log (D x T n ) -1 is close to -χ -(µ) + χ -(ν)
when n goes to infinity. Then by applying the previous algebraic lemma at each time and by averaging along the orbit of x we obtain that χ + (µ) -χ + (ν) ≃ -χ -(µ) + χ -(ν) for ν close to µ. But observe now this fact follows from the continuity in µ ∈ M(X, T ) of χ + + χ -(µ) = M log Jac x T dµ(x).

Proof of Proposition 1

As in Yomdin's theory, we consider the local dynamic at one point. We fix a riemanian structure on the surface M and we denote d the induced distance on M , R inj the radius of injectivity and exp : T M (R inj ) → M the exponential map, where T M (r) :

= {(x, u), u ∈ T x M, u x < r}. There exist R < R ′ < R inj such that T (B(x, R)) ⊂ B(T x, R ′ ) for all x ∈ M . Let x ∈ M and n ∈ N. We consider the map T x n : T T n-1 x M (R) → T T n x M (R ′ ) defined by T x n = exp -1 T n x • T • exp T n-1 x . For all ǫ < R 2 , we put T x n,ǫ = ǫ -1 T x n (ǫ.) : B(0, 2) → T T n x M ≃ R 2 and T x ǫ := (T x n,ǫ ) n∈N .
We choose ǫ > 0 be small enough such that

D 2 T x n,ǫ ≤ inf z∈B(0,2) D z T x n,ǫ and D 2 (T x n,ǫ ) -1 ≤ inf z∈B(0,2) (Dz T x n,ǫ ) -1 DT x n,ǫ
for all integers n and all x ∈ M . One can also assume that D

x T ≃ 1+K D y T and D T x T -1 ≃ 1+K D T y T -1 for all x, y ∈ M with d(x, y) < 2ǫ and therefore D z T x n,ǫ ≃ 1+K D z ′ T x n,ǫ and D T x n,ǫ z (T x n,ǫ ) -1 ≃ 1+K D T x n,ǫ z ′ (T x n,ǫ ) -1
for all n ∈ N and for all z, z ′ ∈ B(0, 2).

We define now subsets of the Bowen ball B(n + 1, 1) where the defect of multiplicativity of the norm of the composition DT i+1 • DT i is prescribed at each step 1 ≤ i ≤ n.

Definition 4 Let T := (T n ) n∈N be a sequence of C 2 maps from B(0, 2) to R 2 and let K n := (k 1 , ..., k n ) be a sequence of n positive integers, we denote H(K n ) the subset of B(n + 1, 1) defined by : H(K n ) := y ∈ B(n + 1, 1) : ∀1 ≤ i ≤ n, log + D y T i max( D T i y T i+1 , 1)

D y T i+1 + 1 = k i
In the next lemma we estimate the numbers of such sets intersecting a finite time hyperbolic set by using the combinatorial argument of Lemma 6. We write λ + n := 1 . We apply finally the combinatorial Lemma 6

to conclude the proof of Lemma 8.

Proposition 1 follows now direcly from Lemma 8 and the following Proposition 8 applied to the sequences T = T x ǫ for ǫ > 0 chosen as previously and for all x ∈ M .

Proposition 8 Let χ + > 0 > χ -, min(χ + ,-χ -,1)

D y (x → D φ ′ n (x) T n+1 ) ≤ Ke -kn D T n •φ ′ n (y) T n+1 D φ ′ n (y) T n + D ψ(y) (x → D φn(x) T n ) ≤ Ke -kn D T n •φ ′ n (y) T n+1 D φ ′ n (y) T n + D φn T n ≤ Ke -kn D T n •φ ′ n (w) T n+1 D φ ′ n (w)
T n and as φ ′ n (w) ∈ H(K n ) we obtain :

D y (x → D φ ′ n (x) T n+1 ) ≤ K D φ ′ n (w) T n+1 ≤ K D φ ′
n T n+1 Similarly we have :

D y (x → D T n+1 •φ ′ n (x) T -(n+1) ) = D T n •φ ′ n (y) T -n D y (x → D T n+1 •φ ′ n (x) T -1 n+1 ) + D y (x → D T n •φ ′ n (x) T -n )D T n+1 •φ ′ n (y) T -1 n+1
The norm of the term D y (x → D T n+1 •φ ′ n (x) T -1 n+1 ) is bounded from above in the following way

D y (x → D T n+1 •φ ′ n (x) T -1 n+1 ) ≤ D 2 T n+1 •φ ′ n T -1 n+1 D y (T n+1 • φ ′ n ) ≤ D 2 T n+1 •φ ′ n T -1 n+1 D T n •φ ′ n (y) T n+1 D y (T n • φ ′ n ) ≤ D 2 T n+1 •φ ′ n T -1 n+1 D T n •φ ′ n (y) T n+1 Ke -kn D ψ(y) (T n • φ n ) ≤ Ke -kn D T n+1 •φ ′ n (y) T -1 n+1
where the last inequality follows from D

D T n+1 •φ ′ n (w) T -1 n+1 D T n •φ ′ n (w) T -n D T n+1 •φ ′ n (w) T -(n+1) = D φ ′ n (w) T n D T n •φ ′ n (w) T n+1 D φ ′ n (w) T n+1 ≤ e kn
Therefore : • φ ′ n+1 ) ≤ 1 and then by the previous step, we get :

D(x → D T n+1 •φ ′ n (x) T -(n+1) ) ≤ K D T n+1 •φ ′ n (w) T -(n+1) ≤ K D T n+1 •φ ′ n T -(n+1)
D(x → D φ ′ n+1 (x) T n+1 ) ≤ D(x → D φ ′ n (x) T n+1 ) D(φ ′ -1 n • φ ′ n+1 ) ≤ K D φ ′ n T n+1 ≤ K D φ ′
n+1 T n+1 and in the same way :

D(x → D T n+1 •φ ′ n+1 (x) T -(n+1) ) ≤ K D T n+1 •φ ′ n+1 T -(n+1)
By the induction hypothesis (ii) and again Corollary 3 (first item) we have also

Dφ ′ n+1 ≤ Dφ ′ n D(φ ′ -1 n • φ ′ n+1 ) ≤ 1 4
and similarly for all 1 ≤ k ≤ n :

D(T k • φ ′ n+1 ) ≤ 1
Obverve finally that φ ′ n ([ We consider only rectangles φ ′ n+1 = φ n+1,y whose image by T n+1 meets the unit ball. Let φ ′ n+1 such a rectangle. We choose a, b ∈ [0, 1] such that the image by T n+1 of the n + 1subrectangle (φ n+1 , R n+1 ) of (φ ′ n+1 , R ′ n+1 ), defined by φ n+1 (t, s) = φ ′ n+1 (t, a + s(b -a)) for all t, s ∈ [0, 1], contains B(0, 3 2 ) ∩ T n+1 R ′ n+1 . This is possible because |∠|f

(n+1) n+1 (x), f (n+1)
n+1 (y) ≤ K according to Lemma 4 for all x, y ∈ φ ′ n+1 ([0, 1] 2 ). Remark that l(E n+1 (x) ∩ R n+1 ) = l(E n+1 (x) ∩ R ′ n+1 ) and l(F n+1 (x) ∩ R n+1 ) ≤ l(F n+1 (x) ∩ R ′ n+1 ) for all x ∈ R n+1 . Moreover either one can

n

  (x) = DxT n en(x) DxT n en(x) and f (n) n (x) = DxT n fn(x) DxT n fn(x) on U n . Observe that e (n) n • T -n and f (n) n • T -n defined on T n U n are respectively the expanded and the contracted fields for T -n . Since T n is a C 2 map, all these vectors fields are in fact C 1 . The vector fields e n and f n (resp. e (n) n and f (n)

  2 to M . One can associate to any subrectangle [a, b] × [c, d] of the unit square the n-rectangle φ x,n ([a, b] × [c, d]

n n- 1 i=0 1 n n- 1 i=0Lemma 8 3 >

 11183 log + D 0 T i . When T = T x ǫ we have λ + n = λ + n (x, T ) = log + D T i x T . Let χ + > 0 > χ -, min(χ + ,-χ -,1) γ > 0 and C > 1 and let T := (T n ) n∈N be a sequence of C 2 maps from B(0, 2) to R 2 such that D z T n ≃ 2 D z ′ T n for all n ∈ N and z, z ′ ∈ B(0,[START_REF] Asaoka | Hyperbolic set exhibing C 1 -persistent homoclinic tangency for higher dimensions[END_REF]. There exists an integer N depending only on C such that for n > N the number of sequences K n-1 such that H(K n-1 ) has a non empty intersection with H n T (χ + , χ -, γ, C)∩B(n, 1) is bounded above by e 3n-2 e (n-1)(λ+ n -χ + )H([λ + n -χ + ]+3) Indeed if y ∈ H n T (χ + , χ -, γ, C) ∩ B(n, 1), then n-1 i=0 log + D T i y T i+1log D y T n ≤ n(λ + n + log 2 -χ + + γ) + log C.Thus the sequence log Dy T i max( D T i y Ti+1 ,1) Dy T i+1 + 1 i=1,...,n-1 admits the value [λ + n -χ + ] + 3 for n > log C 1-log 2-1 3

1 n+1D 1 n+1D 1 n+1D

 111 2 T -1 n+1 ≤ inf z∈B(0,2) (Dz Tn+1)-1 DTn+1and the induction hypothesis (ii). Then we get according to the induction hypothesis (iv) andD Tn+1z T -1 n+1 ≃ 1+K D Tn+1z ′ T -1n+1 for all z, z ′ ∈ B(0, 2) :D(x → D T n+1 •φ ′ n (x) T -(n+1) ) ≤ Ke -kn D T n+1 •φ ′ n (y) T -T n •φ ′ n (y) T -n + D ψ(y) (x → D T n •φn(x) T -n ) ≤ Ke -kn D T n+1 •φ ′ n (y) T -T n •φ ′ n (y) T -n + D T n •φn T -n ≤ Ke -kn D T n+1 •φ ′ n (w) T -T n •φ ′ n (w)T -n Now we have by applying Lemma 7 :

  Second step : we saturate the n-rectangle φ ′ n in a n + 1-rectangle φ ′ n+1 satisfying again (H n+1 ). Let us denote by φ ′ n+1 = φ n+1,y the saturated n + 1-rectangle Sat(φ ′ n ([0, 1] 2 )) of the nrectangle φ ′ n for some y ∈ Sat(φ ′ n ([0, 1] 2 )). According to Corollary 3 (first item) we have D(φ ′ -1 n

4 9 , 5 9 ] 2 )

 92 ⊂ φ ′ n+1 ([ 1 3 -K, 2 3 + K] 2 ) by Corollary 3 (second item) and that R ′ n+1 := φ ′ n+1 ([0, 1] 2 ) satisfies (1 + K)l(E n+1 (y) ∩ R ′ n+1 ) ≥ l(F n+1 (y) ∩ R ′ n+1 ) ≥ (1 -K) l(En+1(y)∩R ′ n+1 )max k=0,...,n DyT k for all y ∈ R ′ n+1 by assumption (G n ) and by Proposition 6.Third step : We cut the n + 1-rectangle φ ′ n+1 to get a new n + 1-rectangle φ n+1 satisfying D(T n+1 • φ n+1 ) ≤ 1.

We refer to[START_REF] Downarowicz | Entropy structure[END_REF] and to[START_REF] Burguet | A direct proof of the tail variational principle and its extension to maps[END_REF] for the definitions of u 1 and h tail (T ) and the tail variational principle.

Note that l(E n+1 (x) ∩ R n+1 ) ≤ 1 4 since Dφ ′ n+1 ≤ 1

> γ > 0 and C > 1 and let T := (T n ) n∈N be a sequence of C 2 maps from B(0, 2) to R 2 with S(T ) < +∞ and such that D 2 T n ≤ inf z∈B(0,2) D z T n , D 2 T -1 n ≤ inf z∈B(0,2) (DzTn) -1 DTn

n for all integers n and for all z, z ′ ∈ B(0, 2). Then there exists a real number D depending only on χ + , χ -, γ, C and S(T ) and an universal constant A such that for all sequences K n-1 = (k 1 , ..., k n-1 ) of n -1 positive integers there exists a family F n of admissible charts of n-rectangles included in H n T (χ + , χ -, γ, 2C) satisfying :

With the previous notations the assumptions (iii)+(iv) and (v) are respectively (H k ) for all 0 ≤ k ≤ n and (G n ).

Proof : We argue by induction on n ≥ N where N is an integer such that all the statements of Subsection 5.2 holds for n > N . The initial step is easily checked. We assume now the existence of the family F n and we built F n+1 . Let φ n ∈ F n .

First step : We divide the n-rectangle φ n in n-subrectangles φ ′ n satisfying (H n+1 ).

We cover the square [ 4 9 , 5 9 ] 2 into ([K -1 e kn ] + 1) 2 subsquares of size 1 [K -1 e kn ]+1 < Ke -kn such that by reparametrizing these subsquares from the unit square by affine contractions ψ we have

2 ). Note that by Proposition 2 the n-rectangles φ ′ n satisfy again the assumption (G n ). From now on we only consider n-rectangles φ ′ n whose middle ninth intersect the set

T n+1 where the last inequality follows from D 2 T n+1 ≤ inf z∈B(0,2) D z T n+1 and the induction hypothesis (ii). Then we get according to the induction hypothesis (iii) and

max k=0,...,n+1 DxT k for all x ∈ R n+1 , that is R n+1 satisfies the assumption (G n+1 ). Furthermore φ ′ n+1 satisfies assumption (H n+1 ) and then so does φ n+1 . By Corollary 2 it follows that D(T n+1 •φ n+1 ) ≤ 2+K. Finally by distinguishing the cases 0 <, = a ≤ b <, = 1 it is easily seen that φ

In summary we have :

2 ) by induction hypothesis (vi) ;

2 ) according to the first step ;

) according to the third step.

Therefore by dividing the n + 1-rectangles (φ n+1 , R n+1 ) into at most 20 subrectangles and by reparametrizing them by affine contractions we can ensure that H n+1

7 Beyond C 2 surface diffeomorphisms

Non-invertible maps

In the previous proof of Proposition 1 one only needs to estimate the entropy locally and therefore only local invertibility is required. Theorem 4 and therefore Theorem 3 are also valid in the context of surface local diffeomorphisms. Indeed in the proof of Theorem 4 global invertiblity was just used to prove by Ruelle-Margulis inequality that ergodic measures with positive Newhouse local entropy and therefore with positive entropy have one positive and one negative Lyapunov exponent. This fact is still true for surface local diffeomorphisms. Indeed the author proved in [START_REF] Burguet | Symbolic extensions for C r nonuniformly entropy expanding maps[END_REF] the following "local invertible" version of Ruelle-Margulis inequality :

Lemma 9 [START_REF] Burguet | Symbolic extensions for C r nonuniformly entropy expanding maps[END_REF] Let M be a compact surface and let T : M → M be a local diffeomorphism. There exists ǫ > 0 such that for any ergodic measure ν, we have :

The lemma remains true in higher dimensions by replacing χ + 0 by the sum of the positive Lyapunov exponents and χ - 0 by the sum of the negative Lyapunov exponents [START_REF] Burguet | Symbolic extensions for C r nonuniformly entropy expanding maps[END_REF].

The conjecture 1 is still open for C 2 noninvertible surface maps with critical points.

Higher regularity

In Yomdin's theory [START_REF] Yomdin | Volume growth and entropy[END_REF], [START_REF] Yomdin | C r -resolution[END_REF], [START_REF] Gromov | Entropy, homology and semi-algebraic geometry[END_REF], [START_REF] Burguet | A Proof of Yomdin-Gromov algebraic Lemma[END_REF] one reparametrizes the Bowen ball by semi-algebraic contracting maps. Here we avoid the semi-algebraic tools involving in this theory but we are then reduced to the C 2 case. Assume T : M → M is a C r map with r ∈ N and r > 2 and φ n : [0, 1] 2 → M is a n-rectangle. To bound the oscillation of the derivative D φn(x) T n one can try to control the r derivative D r (x → D φn(x) T n ) and then approximate the derivative at one point by its Lagrangian polynomial as in Yomdin's theory. Following the proof of Proposition 1 the number of subdivisions at step n we need to bound D r (x → D φn(x) T n ) by D φn T n is of order Ke 2kn r-1 which is consistent with Conjecture 1. However to bound the dynamical complexity of the Lagrangian polynomial of D φn(x) T n as in Yomdin's theory we have to compose φ n with a semi-algebraic map ψ : [0, 1] 2 → [0, 1] 2 and no more a homothety. Then it seems difficult to relate the new map φ ′ n = φ n • ψ with a nor n + 1-rectangle.

Higher dimensions

We used in the proof typical one dimensional arguments to analyse the finite time stable and unstable manifolds (in particular the integrability of finite time stable and unstable fields). Moreover ergodic measures with positive entropy need not to be hyperbolic (i.e. with nonzero Lyapunov exponents) in higher dimensions. Finally the creation of horseshoes is not the only mechanism to create entropy. Indeed a horseshoe generates many periodic points but M.Herman [START_REF] Herman | Construction d'un difféomorphisme minimal d'entropie topologique non nulle[END_REF] built a minimal diffeomorphism on a compact 4-dimensional manifold with positive topological entropy.