Romain Picot-Clémente
email: romain.picot-clemente@u-bourgogne.fr

Christophe Cruz
email: christophe.cruz@u-bourgogne.fr

Christophe Nicolle
email: christophe.nicolle@u-bourgogne.fr

AN ONTOLOGY-BASED APPROACH TO PROVIDE PERSONNALIZED RECOMMENDATIONS USING A STOCHASTIC ALGORITHM

Keywords: Recommender systems, information filtering, semantic web, adaptive hypermedia systems, user modelling, stochastic processes

The use of personalized recommender systems to assist users in the selection of products is becoming more and more popular and wide-spread. The purpose of a recommender system is to provide the most suitable items from an knowledge base, according the user knowledge, tastes, interests, ... These items are generally proposed as ordered lists. In this article, we propose to combine works from adaptive hypermedia systems, semantic web and combinatory to create a new kind of recommender systems suggesting combinations of items corresponding to the user.

INTRODUCTION

In the first years after its creation, Web was the same for everybody. Websites was presenting the same information and the same links to each visitor, without taking into account their goals and their knowledge.

Given the important growth of the number of available information, the diversity of users and the complexity of Web applications, researchers started to question the "one-size-fits-all" approach. Therefore researches have emerged to answer these problems, to propose a Web where the appearance, the behaviour, the resources would be ideally adapted to each individual. This Web would offer a different experience to each user. This Web is called the Adaptive Web.

A first step to reach the Adaptive Web is to reason on local adaptation problems.

This adaptation to users is currently widely found into the so-called domain of recommender systems [START_REF] Tintarev | A survey of explanations in recommender systems[END_REF] [START_REF] Mcsherry | Explanation in recommender systems[END_REF]. These systems have the role to provide recommendations to users. This domain has become important since the mid-year 90. It is yet a big domain of research due to its richness in terms of problem to solve, and given the great number of possible applications to deal with information overload and provide adaptive recommendations. Recommender systems allow companies to filter information, and then to recommend products to their customers, according their preferences. Recommending products or services can consolidate relations between the seller and the customer, and thus, enhance the benefits [START_REF] Zhang | An associative classification-based recommendation system for personalization in B2C e-commerce application[END_REF]. Many systems developed as TrustWalker [START_REF] Jamali | TrustWalker : a Random Walk Model for Combining Trust-Based and Item-Based Recommendation[END_REF] emphasize the current interest of recommendation.

Content-based (CB) and collaborative filtering (CF) methods are two of the main approaches used to form recommendations. Hybrid techniques integrating these two different approaches have also been proposed. The CB method has been developed basing on the textual filtering model described in [START_REF] Oard | The State of the Art in Text Filtering User Modeling and User Adapted Interaction[END_REF]. Generally, in CB systems, the user profile is inferred automatically from documents' content that the user has seen and rated. The profiles and domain documents are then used as input of a classification algorithm. The documents which are similar (in content) to the user profile are considered interesting and are recommended to the user.

CF systems [START_REF] Goldberg | Using collaborative filtering to weave an information tapestry[END_REF] are an alternative to CB systems. The basic idea is to go beyond the experience of an individual user profile and instead to use the experiences of a population or community of users. These systems are designed with the assumption that a good way to find interesting content is to find people with similar tastes and to propose items they like. Typically, each user is associated to a set of nearest-neighbour users, comparing profiles' information. With this method, objects are recommended basing on similarities of users rather than the similarities of objects.

Both CF and CB systems have strengths and weaknesses. In CF systems, the main problem is that the new objects with no rate cannot be recommended. CB systems suffer from deficiencies in the way of selecting items for recommendation. Indeed, the objects are recommended if the user has seen and liked similar objects in the past. Consequently, a variety of hybrid systems have recently been developed:  Some use other users' ratings as additional features in a CB system.  Some use CB methods for the creation of bots producing additional data for "pseudo-users". These data are combined with real users' data using CF methods.  Others use CB predictions to "fill out" the probable user-items' ratings in order to allow CF techniques to produce more accurate recommendations.

In this paper, we propose a content-based recommender system. Its architecture is inspired from adaptive hypermedia researches. Unlike most recommender systems which provide a list of items to users, we suggest to generate combinations of items corresponding to them. A first part will explain what an adaptive hypermedia system is. Then, our proposition is presented. The final part shows a application to tourism domain.

ADAPTIVE HYPERMEDIA SYSTEMS

The research domain of adaptive hypermedia has been very prolific these 10 last years. Some systems [START_REF] Cristea | LAOS: Layered WWW AHS authoring model and their corresponding algebraic operators[END_REF] [START_REF] Henze | Adaptive hyperbooks: Adaptation for project-based learning resources[END_REF] have been developed, giving principally solutions for e-Learning which is considered as the first application domain. Each system brings its own architecture and methods. Moreover, few attempts have been made to define reference models [START_REF] De Bra | AHAM: A dexter-based reference model for adaptive hypermedia[END_REF]) [START_REF] Hendrix | A. Meta-levels of adaptation in education[END_REF] but without success because of being not enough generic to take account of the new trends and innovations. Nevertheless, most of the systems and models are based on a set of layers, also called models, which separate clearly the different tasks. Then, we can see that there are at least three models in common, necessary and sufficient to achieve adaptive hypermedia systems according Brusilovsky [START_REF] Brusilovsky | Adaptive hypermedia, User modeling and user-adapted interaction[END_REF] [START_REF] Liberman | Letizia: An Agent that Assists Web Browsing[END_REF] [START_REF] Kamba | Antagonomy: A Personalized Newspaper on the World Wide Web[END_REF]. This method considers that each document and user profile is described by a set of weighted keywords vectors. At the adaptation model, the weights are used to calculate the similarity degree between two vectors and then to propose relevant document to the user. The keywords representation is popular because of its simplicity and its efficiency. Nevertheless, the main drawback is that a lot of information is lost during the representation phase.

 Semantic networks: In semantic networks, each node represents a concept. Minio and Tasso [START_REF] Minio | User Modeling for Information Filtering on INTERNET Services: Exploiting an Extended Version of the UMT Shell[END_REF] present a semantic networks based approach where each node contains a particular word of a corpus and arcs are created following the cooccurrences of the words from connected nodes into the documents. Each domain document is represented like that. In simple systems using only one semantic network to model the user, each node contains only one keyword. The keywords are extracted from pages which the user gives its taste. Then, they are treated to keep only the most relevant ones and are weighted in order to remove those with a weight lesser than a predefined threshold. The selected keywords are then added to the semantic network where each node represents a keyword and each arc their co-occurrence into the documents. With this method, it is possible to evaluate the relevance of a document compared to the user profile. Indeed, it suffices to construct a semantic network of a document and compare it to the semantic network of the user to classify it to interesting, uninteresting or indifferent documents.

 Ontology: This approach is similar to the semantic network approach in the sense that both are represented using nodes and relations between nodes. Nevertheless, in concepts based profiles, nodes represent abstract subjects and not word or set of words. Moreover, links are not only co-occurrence relations between words, they have several significations. The use of ontology can keep a maximum of information. In QuickStep [START_REF] Middleton | Exploiting synergy between ontologies and recommender systems[END_REF], the ontology is used for the research domain and has been created by domain experts. The ontology concepts are represented as vectors of article examples. The users' papers from their publication list are modelled as characteristic' vectors and are linked to concepts using the nearest neighbour algorithm. These concepts are then used to form a user profile. Each concept is weighted by the number of papers linked to it. Recommendations are then made from the correlations between the current interests of the user to topics and papers that are related to these topics. In [START_REF] Cantador | A multilayered ontology-based user profiles and semantic social networks for recommender systems[END_REF] and [START_REF] Sieg | Leaning Ontology-Based User Profiles: A Semantic Approach to Personalized Web Search[END_REF], a predefined ontology is used to model the domain. User profiles are represented with a set of weighted concepts where weight represents the user's interest for a concept. Its interests are determined by analyzing its behaviour.

Three types of adaptation have been highlighted in the researches on adaptive hypermedia systems: content, navigation and presentation adaptation. The content adaptation consists in hiding/showing or highlighting some information. The adaptation model makes the decision of which content has to be adapted and how to display it. The navigation adaptation consists in modifying the link structure suggesting links or forcing the user to follow a destination. There is URLs' adaptation or destinations adaptation. In the first, the adaptation model provides destination links to the presentation model; these links are displayed at the page generation. Whereas, in the second one, the adaptation model provides links without fixed destination to the presentation model; the destination is decided by the adaptation model when the link is accessed by the user. The presentation adaptation consists in insisting (or not) on the content parts or on the links. It consists also in adapting the preferences setting to the device or the page. The adaptation model process makes the decision of which content or links to insist in following the presentation context.

Even if recommender systems are often differentiated from adaptive hypermedia systems, a lot of similarity between these two types of systems can be highlighted. Indeed, the recommender systems provide recommendations using different algorithms, as it is done in the adaptation model. Moreover, we can see that they model also users' tastes and domain's items, as it is done in adaptive hypermedia systems with the user model and domain model. Nevertheless, recommender systems perform only adaptation of the content whereas adaptive hypermedia systems realize two more adaptation types.

Following these observations, a recommendation system appears to be a constrained adaptive hypermedia system. Thus, it seems clear that recommender systems can be defined as a subset of adaptive hypermedia systems, whatever its type (CB or CF).

The use of an adaptive hypermedia architecture for the creation of recommender systems is interesting because we can clearly define the tasks associated with each part of the application, and it gives the opportunity to evolve the system adding modules and/or other types of adaptation without difficult modifications of parts already implemented. For instance, a CB recommender system could be improved with features of CF systems, adding a group model where clusters of users can be defined.

For the creation of our CB recommender system, we base on adaptive hypermedia architecture. Beyond the use of the three main ones (domain, user and adaptation model), a goal model has been added.

It allows the modelling of users' goals. A description of the proposition is explained in the following part.

PROPOSITION

This section describes the architecture of the proposed recommender system based on an adaptive hypermedia architecture. This system aims to generate a combination of necessary items for the purpose of the application according the items types and constraints (semantics, localization, ...). An item is a piece of information provided by the information system transformed into the recommender system. The architecture of the system is composed of four components: The domain model component, the goal model component, the user model component, the adaptation model component (Figure . 1).

The domain model component

To fully express the knowledge of a domain, the domain model component uses an OWL ontology. An item will be an individual of this domain ontology which is generated from data provided by data sources like relational databases, XML documents, Excel files, etc. Actually, the ontology is enriched by the schema of the data sources and populated by the data of the sources. This construction is carried out using expert's knowledge. Three steps have been identified to achieve this goal. The first step consists in using the database as a starting point for the basis of the domain ontology. The second step consists in enriching this basic ontology by domain experts adding new axioms usually not present in the data source. Usually, this process comprises the addition of semantic links between concepts which are often unqualified in the relational schema. Moreover, the hierarchical structure of concepts is refined with the help of professionals. The last step consists in populating the ontology with instances selected from the data source.

The ontology creation process is partly manual and automatic. The first step and the second step require the investment of domain experts. The third step is automatic and, fortunately, it is the only one that is brought to be repeated regularly to update the data of the ontology. The structural part of the ontology which is set up by the first two steps will not often be changed.

The goal model component

Our objective is to use the domain ontology within a specific user-centered application. In this context, an adjustment of the ontology must be made to adapt it to the usage and, especially, to the user modeling. Hence, the domain ontology is enriched with goal concepts. These concepts define the user possible objectives into the system.

A goal concept is composed of first order logic rules. A rule makes a link between individuals of the ontology and the related goal. Each rule has a weight representing the importance of it within a goal concept.

For example, we could have the goal "culturalActivity " defined by the weighted rules:

-(Cinema(x) ^ haveCulturalAspect(x,y) -> culturalActivity (x), 5) -(Theater(x) ^ haveCulturalAspect(x,y) -> culturalActivity (x), 9) -(Museum(x) ^ haveCulturalAspect(x,y) -> culturalActivity (x), 10)

The set of the goal concepts is called the goal model.

The user model component

From there, we have an ontology oriented to a particular application by a set of goal concepts. We have now to model the user within the system in order to offer an adapted environment. The user is modeled in two distinct parts. The first one concerns the characteristics that are independent of the domain, also called the static part. This may be the age, size, gender, etc. They are simply modeled using <attribute, value> vectors. The second part represents the user features directly related to the domain. It consists of goal concepts, representing the goals of the user with weights which allow the position of a discriminating index in terms of preferences. This part is said dynamic and is an overlay on the domain model. Together, these two parts represent the user model and the instantiation of this model represents a profile of a user. For instance, a user profile can be: User1 (<age, 20>, <gender, male>, …) (<physicalActivity,5>, <culturalActivity,1>, …)

The adaption model component

This component is responsible to provide a combination of items regarding the domain ontology. This set of item is adapted to the user profile which is an instance of the user model. In order to enclose the solution to provide, a pattern of combination is defined. The pattern constrains the elements to be returned to the user by filtering to the type, the number, the item order or the geolocalisation for example. Using the goals weights into the user profile and the weighted connections between the individuals and goal concepts, the weights are combined linearly to be propagated to the individuals.

For instance, if a goal is weighted of 3 in a profile and is constituted of two rules with the respective weights 5 and 1, then the individuals linked to the goal by these rules receive the respective weights ) 5 3 ( 15 and ) 1 3 ( 3 which can be cumulated if individuals are associated to the two goals at once. This weight propagation into the ontology is not done with the whole ontology. Indeed, it is not necessary to weight individuals which cannot be involved in the final solution. Only individuals that instantiate a concept from the domain model specified into the pattern of combination are weighted. In other words, the individuals which the type (the concept) is not reflected in the pattern are not weighted. Thus, we have a subset of weighted individuals from the domain ontology. We now wish to find a "good" combination of these individuals depending on the pattern. The notion of quality of a combination is defined by one or more relevance functions that will take into account the individual weights and other attributes (geographic coordinates, volume, etc.).

For example, we could have this pattern < (Activity, Activity, Activity, Activity), 1km > which constrains the combination to have 4 activities in a neighborhood of 1km.

The purpose consists then in optimizing the relevance function, in order to find the optimal solution. Searching an optimal solution means browsing the set of possible solutions and returns the best one. However, the number of possible solutions grows very rapidly according to the number of individuals, and it becomes quickly impossible to find the best solution in a reasonable time. Thus, to resolve this problem, a stochastic algorithm is used. This latter takes as input all the weighted individuals and the pattern of combination to provide. This algorithm will return "good" items approaching the best one. However, the results will depend on the type of the used stochastic algorithm which optimize the time for real-time application. Finally, the user is offered a combination of items supposed to correspond to their profile. The adaptation model is composed of all these processes. This modeling is being applied to the tourism domain in the region of Côte-d'Or in France for the company Côte-d'Or Tourisme. The aim is to create a tourism application that should provide a combination of tourism products from Côte-d'Or according to a user profile. At the beginning, a domain ontology has been created with all the concepts and the individuals related to the application domain. This ontology was supplied from a database composed of more than 4000 tourism products. Then, a goal model has been defined using goal concepts like "Week end", "Going out with friends", "with a baby", etc. This knowledge was generated from the specialists of the domain represented by people working for the company Côte-d'Or Tourisme.

An empirical pattern is defined to determine what kind of combination the adaptation model has to return. The relevance function giving the relevance of a combination is based on the interest weights and the coordinates of the tourism products, because it is not relevant to propose an activity in the morning and a restaurant for lunch with a distance of more than 50 kilometers. The traveling time required to reach the restaurant after the activity ending is inappropriate.

A variance threshold needs to be set in order to define the maximum preferred variance between the individuals coordinates. This variance characterized the value dispersion regarding the average, in this case the threshold. Subsets in this pattern are possible. For instance, we can define a pattern like "Accommodation, Restaurant1, Activity, Restaurant2" in which "Hotel and Restaurant1" are the first subset, and "Activity and Restaurant2" the second subset. In addition, a variance threshold is defined for each one. Thereby, the system can use more complex patterns for the combinations.

The variance of a combination is defined as follow:

1 0 2 2 1 0 2 1 0 1 1 1 ) var( N i N i iy iy N i ix ix C N C C N C N C (1)
where C is a combination, N the number of elements into the combination, C i the i th element of the combination, and C ix and C iy the x and y coordinates of the i th element.

The weight of a combination is defined as follow:

1 0 1 ) ( N i iweight C N C weight (2)
where C iweight is the weight of the i th element.

Using the variance and the weight function, the relevance of a combination is: Then, a stochastic algorithm is performed. We use a simulated annealing algorithm [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF] which try to optimise the relevance function, so that it can provide a combination of tourism products matching the user interests and with close coordinates.

BENCHMARK

Some tests of the algorithm for the generation of combinations have been done on a set of four thousand tourism products. In these tests, the average time required for the generation of a combination of six products was around 3 seconds. But, this time depends on the different parameters (temperature decrease rate, the number of iteration per level of temperature) necessary to perform the simulated annealing algorithm. For example, the faster is the temperature decrease and the lower is the number of iterations, the faster is the generation, but the worse is the resulting combination. In any case, this time is better than the time required to find the best combination by browsing all the possibilities. For example, in our test, finding the best combination needed around 3 hours against 3 seconds using the simulated annealing algorithm. These times are only given to have orders of magnitude, more tests need to be performed to have exacts results and to prove the interest of our proposition. Nevertheless, given these few results, the algorithm seems to give a relevant solution according a predefined relevance function with a lesser cost (in terms of time) than calculating the optimal solution.

CONCLUSIONS

This paper presents a new content based recommender system. The idea consists in taking advantages of the semantic Web technologies, the properties of adaptive hypermedia systems, and in combining them with combinatory algorithms in order to provide recommendations, adapted combinations of items. The simulated annealing algorithm is used in order to solve the problem of the polynomial time search required to generate a combination of tourism products. It gives a solution which approaches the best solution in a short time. The few results seem to be good considering the time required to obtain them and comparing to the best solutions. Nevertheless, for the future, we need to make more tests and benchmarks to quantify more precisely the relevance of our system. Moreover, we could improve the quality of the propositions by taking into account some group of users as it is done in collaborative filtering recommender systems. It is possible by adding a group model into the architecture. Thus, the recommender system would become a hybrid recommender system.

Figure 1 .

 1 Figure 1. The components of the architecture

  of the geographic coordinates for the j th subset of C, and L the number avec subsets.

ACKNOWLEDGEMENTS

This research is done in collaboration with Côted'Or Tourisme, a company which aims to promote tourism in its region.


H B Nielsen

N S Mankoč Borštnik

K Nagao

G Moultaka
email: gilbert.moultaka@lpta.univ-montp2.fr

8 Complex Action Functioning as Cutoff and De Broglie-Bohm Particle

.

Introduction

In the contribution by M. Ninomiya and H.B. Nielsen to this workshop [START_REF] Nielsen | Nonexistence of irreversible processes in compact space-time[END_REF] it were suggested that the model of Ninomiya and Nielsen [3] would lead to improvement in the sense of interpretation of quantum mechanics. In the present discussion contribution we shall estimate how does this model with a complex action lead to that a very narrow range of paths come to dominate the Wentzel-Dirac-Feynman-path integral, since we use a path integral with integration over the whole phase space and not only over the configuration space alone as can also be chosen. This may at first looks as to be in contradiction with the Heisenberg uncertainty relation, but it should be stressed that the metaphysical way in which the result of Ninomiya and Nielsen about the dominance of some narrow classes or some discrete sets of classes of paths is not in contradiction with what one can achieve with wave functions or rather cannot achieve. In fact, Heisenberg's result is the information we can have about the quantum system and still be able to use it, whereas the information one can obtain out of the model with the extremely narrow region dominating the dynamics in the phase space according to the pathway dominance range is completely useless to work with. In fact this information is what one can claim one has about a particle in the time in between preparation and observation, say if one prepares its momentum and measures its position. Then one could metaphysically claim that in the period between the preparation and the measurement one has both the prepared momentum and the observed position at the same time. This claim is however totally useless and could fundamentally not be tested by further experiments because it would, if such an experiment is performed, lead to a disturbance of conditions and thus would spoil the correctness of the claim. In spite of being useless one could however still with good metaphysical right uphold that indeed in such a time interval a particle has both momentum and position at much more accurate values than allowed by the Heisenberg uncertainty principle. If one really takes seriously that in the complex action model everything is calculable just from the expression for the action, i.e. mainly the coupling constants and the form of the action, then one could in principle calculate (but it would be exceedingly hard and in practise totally impossible) this narrow range of dominating paths, meaning essentially an up to a very little uncertainty classical path. So if we could use such unrealistic but possible calculations we would indeed have the Heisenberg uncertainty violating prediction! In this sense we must say that in principle in our model there is no Heisenberg uncertainty principle at the metaphysical level. Such a metaphysical classical state of the system is extremely reminiscent of the Bohm-DeBroglie interpretation of quantum mechanics about which G. Moultaka has talked at this workshop [4].

Can one find, using the metaphysical way of treating our universe (or any system with extremely many degrees of freedom), with the complex action assuming the phase space of coordinates and momenta, the way for cutting away most of the space in a consistent way? Does the narrow range of dominating paths (meaning almost a classical path) help to make the theories of the Kaluza-Klein-kind renormalizable or at least trustable?

One of the open problems of the Kaluza-Klein[like] theories is, namely, the renormalizability of these theories. Even if one studies properties of a system of fermions interacting with the gauge gravity fields far bellow the quantum gravity regime, yet is the consistent treatment of the cutoff and correspondingly the renormalizability of the approach questionable. We suggest that the complex action as proposed by Ninomia and Nielsen [3] play a role of a cut of in loops diagrams for the Kaluza-Klein[like] theories. The spin-charges-family-theory [5], proposed by N.S. Mankoč Borštnik and presented and discussed in this workshop as a promising theory for explaining open questions of the standard model, is besides proposing the mechanism for generating families (and consequently possibly explaining the appearance of the masses and mixing matrices of fermions), unifying the spin and the charges into only the spin. This spin-charges-family-theory is namely sharing many a difficulty with the Kaluza-Klein[like] theories. One of these difficulties is also the cut off problem. We propose in this contribution that the imaginary action might help to make a choice of a trustable cut off.

What are conditions which the system must fulfil that the complex action model start to be efficient or usable in the sense, that it helps to make a choice of a very narrow part of the phase space of momenta and coordinates at least metaphysically? And how could one use it when describing systems, like quarks, hadrons, nuclei, atoms, molecules, scattering of particles on slits, and so on? How such cases come along with both, complex action model and the Bohm-DeBroglie interpretation of quantum mechanics.

A typical shape of the phase space distributions

corresponding to the |A(t) > and < B(t)| states in the complex action model.

In this section we argue using the Lyapunov-exponent or better the Lyapunovmatrix when discussing properties of the universe existing for a very long time that the two states |A(t) > and < B(t)|, defined in the contribution by H.B. Nielsen and M. Ninomiya [START_REF] Nielsen | Nonexistence of irreversible processes in compact space-time[END_REF] or in [3], that the first, |A(t) >, may be considered as a sort of wave functions describing the state of the universe which is favoured by having low action S I up to the time t from the beginning of the time, and the second, < B(t)|, a sort of hidden variable wave function expressing a similar favourite state with respect to S I coming from the time interval between t and the end of the time. In fact we define these two wave functions from the complex action model as a fundamental formulation from the functional path integral

exp( i * S[path]) Dpath, (18.1)
by splitting it up into two factors

< q|A(t) > = with path(t)=q exp( i * S -∞→t [path]) Dpath, < B(t)|q > = with path(t)=q exp( i * S t→∞ [path]) Dpath. (18.2)
Here

S -∞→t [path] = t -∞ L(path(t ′ )) dt, S t→∞ [path] = ∞ t L(path(t ′ )) dt (18.3)
and the subscript "with path(t) = q" means that we only include those paths which end at time t with representing the configuration point q in the path way integration. In the case of < q|A(t) > we only use half paths from the beginning of time -symbolized by -∞ to the finite time t, while in the definition of < B(t)|q > we similarly only use half paths from t to the end of time, symbolized by ∞. We say that we split up the original functional integral (18.1), because we immediately see that

< B(t)|A(t) >= exp( i * S[path])Dpath, (18.4)
where here the time integration region is from the beginning of time to the end of time, although we have delete the index telling this so that we have put indeed,

S[path] = S -∞ to ∞ [path]. (18.5)
The idea is to seek to estimate the shape of the distribution in phase space describing in the best way the wave packet corresponding to the states |A(t) > and < B(t)|. In a classical approximation one should get the state |A(t) > by developing forward to time t a state determined roughly in some time prior to t by "optimising'(minimizing) S I . In thinking of such a development during long times we have to have in mind how does the development of a series of very close (infinitesimally close) classical starting states in phase space develop as time goes on, and this is given by a matrix which is a generalization of the Lyapunov exponent. In fact if one phase space point P 2 deviates from another infinitesimally close one P 1 by an infinitesimal vector in phase space l, this distance vector l(t) will develop with time exponentially in the sense that

l(t) = exp(λ * t) l(0) (18.6)
where λ is a matrix with the order being equal to the dimension of the phase space. If the vector l(t start ) at the starting time t start has components along the subspace of positive eigenvalues for the matrix λ, the components in this space will grow up very drastically during sufficiently long time, while on the other hand the components in the subspace of negative eigenvalues will grow smaller and smaller as time passes. If thus at some time the starting state was selected by the S I to be in some not especially elongated region and essentially just one quantum state (we speculate that this is the selection at some close to Big Bang time), then as time goes on this region will be more and more contracted in the λ negative eigenvalue subspace directions, while it will be expanded in the positive eigenvalue directions. After a long time -i.e. when t has become long after the era of the strongest influence of S I -the region representing the most favourite state at time t, that is just |A(t) >, becomes very contracted in the directions corresponding to the negative eigenvalue subspace and very elongated in the directions corresponding to the positive eigenvalue subspace. This means that approximately this region corresponding to the state |A(t) > becomes a surface of dimension as the number of positive eigenvalues of λ lying in the phase space, probably not a flat surface but a curved smooth one. Similarly -but now we can say time reversed -we obtain that the phase space region corresponding to the state < B(t)| will be a very extended surface while strongly contracted in other directions. For < B(t)| we must imagine that the dependence on the minimization on S I on what goes on in the future -of the time t -determines in some presumably far future which state would be most favourable and then we must imagine how to develop backwardly (backward to time t) this most favoured state. The development under such a backward development is again exponential and given by a metric similar to the λ from before. Now however we develop a negative time namely from the presumably far future time back to the time t. Again some eigenvalues shrink under this backward development while others expand drastically. We therefore again obtain that the region in the phase space roughly describing < B(t)| has the shape of a very extended surface. Both surfaces, the surface for |A(t) > and that for < B(t)|, have dimensions presumably about the half dimension of the phase space, since they had their dimension given by number of respectively negative and positive eigenvalues of matrices of order of the dimension of phase space. In case they have really this half dimension of phase space both, their intersection will be point wise. That is to say they will intersect in a discrete set of points typically (if they intersect at all). Such an intersection in one or a few points would mean that our whole model predicts essentially one or a few classical solutions with very little uncertainty to represent the dominant part of the functional integral! If we -metaphysically may be -take this dominant region -the overlap of the |A(t) > region with the < B(t)| one -in the space of paths to represent the realized history of the universe, then we have reached a picture in which the universe runs through a development in which the conjugate variables (i.e. momentum and position) are much more accurately determined than (formally) the Heisenberg uncertainty principle allows for. That is to say: The metaphysical picture put forward in our model turns out to deliver a classical picture in the sense that there is approximately a totally classically development, so that our complex action model makes it approximately as if there really were a true classical development as one would have imagined before quantum mechanics were invented. Well, we have the tiny deviation from this picture that there will typically not be only one such classical development, but rather several although still a discrete set of them.

A proposal for cutting off by means of the complex action model

It is the main purpose of the present discussion and contribution to point out that we have a hope that the complex action of H.B. Nielsen and M. Ninomiya is offering a "physical" mechanism for a cutoff and correspondingly find the "philosophical" support for the higher dimensional Kaluza-Klein-like theories which are not renormalizable. One could namely claim: we know a mechanism that in principle will cut off the divergences and replace them by finite expressions depending on the support of S I effects of the complex action model. Let us show how does such a principal cut off mechanism appear in the complex action model! Let us therefore very shortly remind ourselves how can one get in the complex action the "usual" quantum mechanics. The basic approximation to reproduce the (usual) quantum mechanics in the complex action model is that we approximate the projection operator on the future-determined state |B(t) >, the hidden variable state we could call it, by a unit operator

|B(t) >< B(t)| ≈ N * I (18.7)
where N is an unimportant normalization factor and I is the unit operator. The argumentation for statistically justifying this approximation to be used for making the Born-probability distribution so as to obtain the usual expectation value formula from the one suggested at first in the complex action model goes with an ergodicity-like approximation. The hidden variable state from the future < B(t)| affected by S I is, as we mentioned in the previous section, essentially given by some favourable state in a presumably far future extrapolated backward in time through a large amount of time. Then if this time is long and the system, the universe, is roughly an ergodic system, we will argue that all states have almost the same practical chance for being the state < B(t)|. In this way we count that all states in some basis for the Hilbert space are equally likely to be in the state Thus we have justified the approximation (18.7). We hope that since the two states (as a function of a phase space) of a system, one describing the developing of the system from the very beginning up to the time t (A(t)) and the second describing the system from the very end backward up to time t (B(t)), define as an overlap a very tiny part of the phase space, the idea is that knowing this phase space, that is some almost classical solutions, would help us to make a choice of an appropriate cut off.


Ricardo Coutinho

Bastien Fernandez

Extensive Bounds on the Entropy of Repellers in Expanding Coupled Map Lattices

Keywords: Mathematics Subject Classification (2010). Primary 37L60; Secondary 37D50, 37B40. Keywords. Coupled Map Lattices, Piecewise Expanding Dynamical Systems, Topological Entropy

Beyond the uncoupled regime, the rigorous description of the dynamics of (piecewise) expanding coupled map lattices remains largely incomplete. To address this issue, we study repellers of periodic chains of linearly coupled Lorenz-type maps which we analyze by means of symbolic dynamics. Whereas all symbolic codes are admissible for sufficiently small coupling intensity, when the interaction strength exceeds a chain length independent threshold, we prove that a large bunch of codes is pruned and an extensive decay follows suit for the topological entropy. This quantity however does not immediately drops off to 0. Instead, it is shown to be continuous at the threshold and to remain extensively bounded below by a positive number in a large part of the expanding regime. The analysis is firstly accomplished in a piecewise affine setting where all calculations are explicit and is then extended by continuation to CML based on C 1 -perturbations of the individual map.

Introduction

Coupled Map Lattices (CML) were introduced almost thirty years ago to simulate the time evolution of spatially extended systems [START_REF] Kaneko | Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice[END_REF]. Often designated as "discrete time dynamical systems with discrete space", their dynamics is generated by the iterations of a mapping acting on real sequences. In applications, these sequences usually represent the spatial profile of a physical quantity distributed in an extended domain; e.g. the status of an advection flow in atmospheric circulation [START_REF] Lind | Modeling velocity in gradient flows with coupled-map lattices with advection[END_REF], the distribution of a dispersed population in Ecology [START_REF] Solé | Spiral waves, chaos and multiple attractors in lattice models of interacting populations[END_REF] or the concentration of a chemical oscillator in a large vessel [START_REF] Waller | Spatial and temporal structure in systems of coupled nonlinear oscillators[END_REF].

While easy to implement, CML were specifically designed to preclude numerical instabilities and other asymptotic divergences that materialize in traditional PDE's and coupled ODE's models [START_REF] Oono | Computationally efficient modeling of ordering of quenched phases[END_REF]. Their simplicity and convenience contributed to their success as a simulation tool in many scientific disciplines [START_REF] Kaneko | Theory and Applications of Coupled Map Lattices[END_REF]. This buoyant usage in applications has naturally called for the development of a mathematical theory that could put the observed phenomenology onto a rigorous footing.

Whereas their numerical simulations are easily accessible, the rigorous description of the global dynamics of CML is usually not [START_REF] Chazottes | Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems[END_REF]. The latent existence of large/infinite dimensional attractors brings additional problems to standard difficulties that arise in the analysis of concrete dynamical systems. Moreover, their mappings are typically composed of an individual nonlinearity at each lattice site and a (diffusive-like) interaction between the sites. The relative weight of these components is quantified by a coupling parameter whose intensity has a deep impact on the attractor; an increase of coupling may bring its dimension from a large number down to or below 1.

Not surprisingly, mathematical results on arbitrarily large/infinite lattices have been limited to the cases where one of the ingredients dominates the dynamics. In particular, for (piecewise) expanding individual maps, accomplishments mostly concern the existence of physical measure in the weak interaction regime, when the CML resembles the interaction-free system, see e.g. [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF][START_REF] Bricmont | Coupled analytic maps[END_REF][START_REF] Bunimovich | Space-time chaos in coupled map lattices[END_REF][START_REF] Jiang | Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations[END_REF][START_REF] Keller | Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension[END_REF]. In brief, these results are based on perturbation arguments applied to uniformly hyperbolic subsets. Proofs have however required substantial efforts and the introduction of original objects in order to be achieved for the models employed in simulations. For smooth real individual maps, some rigorous results can be found on synchrony phenomena that take place when the interaction prevails [START_REF] Fernandez | Global synchronisation in translation invariant coupled map lattices[END_REF][START_REF] Jost | Spectral properties and synchronization in coupled map lattices[END_REF]. But to the best of our knowledge, no study exists on the (global) CML dynamics beyond the uncoupled regime when the individual map is (piecewise) expanding. Of note, few papers have considered the entire coupling range by limiting the analysis to small lattices, basically with 2 sites [START_REF] Fernandez | Coupling two unimodal maps of simple kneading sequences[END_REF][START_REF] Keller | Some phase transitions in coupled map lattices[END_REF][START_REF] Koiller | Coupled map networks[END_REF]. Moreover, proofs of the phase transitions, i.e. the co-existence of two 'Gibbs phases' on a unique indecomposable attractor, have been given in specially designed examples [START_REF] Bardet | Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling[END_REF][START_REF] Gielis | Coupled map lattices with phase transitions[END_REF].

The aim of this paper is to provide insights into the dynamics of (standard) CML on large lattices with piecewise expanding individual maps, throughout the domain where the mapping remains expanding. For simplicity, focus is on the description of repellers in periodic chains of coupled piecewise increasing Lorenz-type [START_REF] Guckenheimer | Nonlinear oscillators, dynamical systems and bifurcations of vector fields[END_REF][START_REF] Rand | The topological classification of lorenz attractors[END_REF][START_REF] Robinson | Dynamical Systems[END_REF] maps. The analysis follows the approach developed in the case of 2 sites [START_REF] Coutinho | Symbolic dynamics of two coupled lorenz maps: From uncoupled regime to synchronisation[END_REF][START_REF] Fernandez | Route to chaotic synchronisation in coupled map lattices: rigorous results[END_REF]. It primarily consider piecewise linear maps and then extends the results by continuation to CML based on C 1 -perturbations of the individual map.

Our strategy consists in analyzing the symbolic dynamics associated with the restriction of the CML to its repeller in the expanding regime. An equation for spatio-temporal symbolic sequences is established whose solutions coincide with the codes of orbits lying in the repeller. By analyzing the equation, the maximal coupling parameter up to which all sequences are solutions (and hence the CML remains conjugated to its interaction-free counterpart) is computed. Beyond that threshold (which does not depend on the chain length), many sequences are pruned and coupling-dependent estimates of both forbidden and admissible sequences are established. Inspired by the physical nature of diffusive-type interactions, the main criteria for (non-)admissibility is the size of space-time reactangles where the orbits are composed of heterogeneous configurations, viz. when their coordinates are out of sync.

The estimates imply upper and lower bounds on the topological entropy of repellers, which become sharper and sharper as the coupling is decreased back to the threshold. Not only these bounds apply to coupled periodic chains of any (even) length, but there are shown to be extensive, namely to scale linearly with the period length. In that way, we prove that while a macroscopic change of the dynamics occurs beyond the uncoupled regime, the dynamics remains "extensively" chaotic in a large part of the expanding domain.

Model and main results

We consider basic Coupled Map Lattices (CML) on periodic chains Z 2L := Z/2LZ with even number 2L of sites (L 1 is an arbitrary integer). 1 Thus the phase space is the set R Z 2L of periodic configurations x = {x s } s∈Z 2L with real components x s , endowed with uniform norm • . In this set, the dynamics is generated by the iterations of the map F ,2L defined by [START_REF] Chazottes | Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems[END_REF][START_REF] Kaneko | Theory and Applications of Coupled Map Lattices[END_REF] (

F ,2L x) s = (1 -)f (x s ) + 2 (f (x s-1 ) + f (x s+1 )), ∀s ∈ Z 2L ,
where ∈ [0, 1] is the coupling strength and the individual map f is the piecewise affine symmetric expanding map [START_REF] Guckenheimer | Nonlinear oscillators, dynamical systems and bifurcations of vector fields[END_REF][START_REF] Rand | The topological classification of lorenz attractors[END_REF][START_REF] Robinson | Dynamical Systems[END_REF], see Figure 1, left.

f (x) = ax -(a -1)H(x -1/2), ∀x ∈ R.
Here the slope a > 2 is fixed and H denotes the Heaviside function

H(x) = 0 if x < 0 1 if x 0 ∀x ∈ R.
The assumption a > 2 implies that the repeller k f of the individual dynamical system (R, f ), namely the subset of points x ∈ R whose forward orbit {f t (x)} t 0 is bounded, is a Cantor set bounded away from the discontinuity x = 1/2. As a consequence, k f is uniformly hyperbolic and hence structurally stable with respect to small C 1 -perturbations. Moreover, the system (k f , f ) is topologically conjugated to the full Bernoulli shift with two symbols and 1 Assuming even number of sites makes the analysis much simpler. However, it is likely that the existence of extensive bounds on the topological entropy does not depend on that assumption and holds for periodic chains of any number of sites. Similarly, these bounds might hold for one-dimensional lattices with other types of boundary conditions instead of periodic ones. as such, its topological entropy is equal to log 2. (For a definition and basic properties of the topological entropy in dynamical systems, see e.g. [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF][START_REF] Robinson | Dynamical Systems[END_REF].)

In brief, the individual system is the most basic example of a simultaneously chaotic and robust system. Given such simple features for the individual dynamics, we would like to get insights into the corresponding characteristics for the CML (R Z 2L , F ,2L ). In particular, we ask about the topological entropy in the repeller K ,2L (which, again, is defined as the set of points whose forward orbit remains bounded). For simplicity, we only consider the case where the CML is expanding, namely when ∈ [0, e ] where e = a-1 2a is independent of L, see equation [START_REF] Guckenheimer | Nonlinear oscillators, dynamical systems and bifurcations of vector fields[END_REF] in Appendix A. Structural stability of the individual system implies that the same property holds for the uncoupled system F 0,2L , uniformly in the lattice size [START_REF] Mackay | Dynamics of networks: Features which persist from the uncoupled limit, Stochastic and spatial structures of dynamical systems[END_REF]. As a consequence, (K ,2L , F ,2L ) and (K 0,2L , F 0,2L ) remain topologically conjugated provided that is small enough. Thanks to the piecewise affine assumption, the maximal coupling parameter up to which this property persists can be explicitly computed. When formulated in terms of the topological entropy h ,2L := h top (K ,2L , F ,2L ), the robustness of the uncoupled system with respect to small variations of coupling writes as follows.

Proposition 2.1. For every L 1, we have h ,2L = h 0,2L = (2L) log 2 iff c := a-2 2a . Notice that the maximal coupling parameter c does not depend on L and c < e . The proof of this statement relies on symbolic dynamics and is given in Section 3.2.

Since (2L) log 2 is a maximum for the CML entropy, Proposition 2.1 stipulates that this quantity must decrease beyond the uncoupled regime (i.e. h ,2L < (2L) log 2 for all L when > c ). However, it does not provide any estimate about the decay behavior when the number L of sites diverges. A more accurate analysis of non-admissible sequences beyond c shows that this decay is actually extensive. Proposition 2.2. For every > c , there exists δ > 0 such that for all L 1, we have h ,2L < (2L)(log 2 -δ).

For the proof, see Section 3.3. Of note, one expects that δ → 0 when → + c . This property will follow from the next statement.

Whereas the CML entropy is subjected to extensive decay for > c , it does not immediately drop off to zero. Uniform positive lower bounds exist for the entropy per lattice site h ,2L /(2L) provided that is sufficiently close to the threshold c . More precisely, we have the following statement whose proof is given in Section 4.

Proposition 2.3. For every 0 < δ < log 2, there exists δ > c such that for all < δ and L 1, we have h ,2L (2L)δ. This phenomenology, namely the existence of upper and lower bounds on the entropy per lattice site across the expanding coupling range, is not limited to piecewise affine CML. It extends to every CML with individual maps that are small C 1 -perturbations of the piecewise affine map f , see Figure 1, right. In order to accurately state this result, let g : R → R be a continuous and increasing map for x < 1/2 and for x 1/2 and let

a g := inf x<y< 1 2 or 1 2 x<y g(y) -g(x) y -x and g -f := sup x∈R |g(x) -f (x)|. (1)
Let also F g, ,2L denote the CML with individual map g, namely

(F g, ,2L x) s = (1 -)g(x s ) + 2 (g(x s-1 ) + g(x s+1 )), ∀s ∈ Z 2L . (2)
Theorem 2.4. For every 0 < µ < min{ c , ( ec )/2}, there exists η > 0 such that for every < e -µ and L 1, any CML F g, ,2L with individual map g satisfying g -f + |a g -a| < η has a repeller I g, ,2L on which the entropy satisfies

δ h top (I g, ,2L , F g, ,2L ) 2L δ ,
where the upper and lower bounds → δ and → δ have the following properties

δ = log 2 if < c -µ δ < log 2 if > c -µ , lim →( c -µ) + δ = log 2,
and δ = log 2 if < c + µ δ < log 2 if > c + µ
For the proof, see Section 5. A schematic illustration of the entropy bounds is given in Figure 2.

Symbolic dynamics: Basic features

Symbolic description of the CML

In order to define a symbolic dynamics for the CML, we first introduce a coding that is inspired from the natural coding of the individual map. Given a periodic configuration x ∈ R Z 2L , a periodic symbolic configuration {θ s } s∈Z 2L ∈ {0, 1} Z 2L is assigned according to the components location with respect to 1/2. Namely, we set

θ s = H(x s -1/2), ∀s ∈ Z 2L
where H still denotes the Heaviside function. Now, the code θ = {θ t } t 0 associated with x is simply the sequence of symbolic configurations (the symbolic sequence for short) associated with the iterates F t ,2L x, viz. θ t s = H((F t ,2L x) s -1/2), ∀s ∈ Z 2L , t 0. The symbolic dynamics of the CML relies on the property that configurations in the repeller can be entirely determined by their code. This is a consequence of Lemma 3.1 below. Moreover, their expression is known explicitly and depends upon the function χ ,2L defined by

χ ,2L (θ) = a -1 a +∞ k=0 a -k n∈Z 2L (k) n,2L θ k n
where the coefficients

(k)
n,2L are the entries of the inverse powers C -k ,2L of the coupling operator C ,2L (see Appendix A), namely we have

(C -k ,2L x) s = n∈Z 2L (k) n,2L x s+n , ∀s ∈ Z 2L (3)
for all k 0 and x ∈ R Z 2L . This expression follows from the fact that C ,2L is a convolution operator on R Z 2L and the inverse powers C -k ,2L are well-defined when < e and are also convolution operators [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF]. Throughout the paper, we shall extensively use various properties of the coefficients (k) n,2L . These properties are collected in Appendix A. In particular, we show in Claim A.3 that

+∞ k=0 a -k n∈Z 2L | (k) n,2L | < +∞
which implies that the function χ ,2L is well-defined. Hence, not only this function provides an expression of configurations in the repeller but it can be used to characterize those symbolic sequences that are actually the codes of such configurations. Let R sp and R ti respectively denote the space and time translations acting on symbolic sequences, i.e.

(R sp θ) t s = θ t s+1 and (R ti θ) t s = θ t+1 s , ∀s ∈ Z 2L , t 0. Lemma 3.1. A periodic configuration x ∈ R Z 2L belongs to K ,2L iff we have x s = χ ,2L • R s sp (θ), ∀s ∈ Z 2L
where θ is the code associated with x. Independently, given a symbolic sequence θ, the configuration {χ ,2L • R s sp (θ)} s∈Z 2L belongs to K ,2L iff θ solves the equation

θ t s = H χ ,2L • R s sp • R t ti (θ) -1/2 , ∀s ∈ Z 2L , t 0. ( 4
)
Notice that a version of this statement for L = 1 has been given in [START_REF] Fernandez | Route to chaotic synchronisation in coupled map lattices: rigorous results[END_REF].

Proof. We only show that if x ∈ K ,2L , then it writes {χ ,2L • R s sp (θ)} s∈Z 2L . The proofs of other properties are direct and left to the reader. The action of the mapping F ,2L can be written in an operator form as follows

F ,2L x = aC ,2L x -(a -1)C ,2L θ 0 .
When < 1/2 (which is the case when < e ), all eigenvalues of C ,2L are positive and the smallest eigenvalue is 1-2 . Hence C ,2L is invertible and we have

C -1 ,2L = (1 -2 ) -1
. By inverting the previous relation and iterating, one obtains the following expression

x = a -t C -t ,2L F t ,2L x + (a -1) t-1 k=0 a -(k+1) C -k ,2L θ k , ∀t 0.
If the configuration x ∈ K ,2L , then the definition of this set implies that the norm F t ,2L x remains bounded for all times. The assumption < e then yields lim

t→+∞ a -t C -t ,2L F t ,2L x = 0.
Hence by taking the limit t → +∞ in the expression of x above, we obtain the following expression for its components

x s = a -1 a +∞ k=0 a -k (C -k ,2L θ k ) s , ∀s ∈ Z 2L
which is the desired expression.

Every solution of the admissibility equation ( 4) is called an admissible (symbolic) sequence. Let A ,2L be the set of solutions, endowed with the product topology (of discrete topology). Lemma 3.1 can be interpreted as the existence of a conjugacy, i.e. a uniformly continuous bijection,2 between the symbolic system (A ,2L , R ti ) and the CML (K ,2L , F ,2L ). In general, this conjugacy needs not be an homeomorphismand the topological entropies of the two systems need not be equal -because A ,2L may not be compact. However, uniform continuity and monotonicity of the entropy with respect to set inclusion assert the inequalities (see e.g. Proposition 3.1.6 and its proof in [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF])

sup A⊂A ,2L , A compact, invariant h top (A, R ti ) h ,2L h top (A ,2L , R ti ), ∀ < e , (5)
where h top (A, R ti ) and h top (A ,2L , R ti ) denote the entropy of the symbolic (sub-)systems.

The uncoupled regime < c

Based on Lemma 3.1, our strategy to investigate topological properties of the CML dynamics is to determine which symbolic sequences are solutions of the admissibility equation depending on the coupling parameter. Here, we consider the simplest case where all sequences are admissible and we rely on basic properties of the coefficients (k) n,2L in χ ,2L to compute the maximal parameter up to which this property holds. Lemma 3.2. All symbolic sequences are admissible iff < c := a-2 2a . Proof. A sequence θ solves the admissibility equation ( 4) iff

χ ,2L • R s sp • R t ti (θ) < 1/2 if θ t s = 0 χ ,2L • R s sp • R t ti (θ) 1/2 if θ t s = 1 ∀s ∈ Z 2L , t 0.
Therefore, all symbolic sequences are admissible when the following inequalities hold sup

θ∈Ω 2L : θ 0 0 =0 χ ,2L (θ) < 1/2 and inf θ∈Ω 2L : θ 0 0 =1 χ ,2L (θ) 1/2.
where Ω 2L is the set of all symbolic sequences, viz.

Ω 2L := θ = {θ t s } s∈Z 2L ,t 0 , : θ t s ∈ {0, 1} .
By continuity of the function θ → χ ,2L (θ) and compactness of Ω 2L , these upper and lower bounds are attained. Furthermore, the normalization

n∈Z 2L (k) n,2L = 1 ∀k 0 (see Claim A.2 in Appendix A) implies that following symmetry holds χ ,2L (1 -θ) = 1 -χ ,2L (θ)
where 1 -θ := {1 -θ t s } s∈Z 2L ,t 0 . Consequently, the previous conditions are equivalent to sup

θ∈Ω 2L : θ 0 0 =0 χ ,2L (θ) < 1/2. ( 6
)
Now, using the properties

(0) n,2L = δ n,0 mod 2L and (k)
2n,2L > 0 and

(k)
2n+1,2L < 0 (see Lemma A.1 in Appendix A again) in the expression of χ ,2L , it easily follows that sup

θ∈Ω 2L : θ 0 0 =0 χ ,2L (θ) = χ ,2L (θ 0 [(10) L ] ∞ )
where θ 0 0 = 0 and θ 0 s is arbitrary for s = 0 mod 2L. Moreover [( 10) L ] ∞ is a shortcut for θ t s = H((-1) s ), ∀s ∈ Z 2L , t 1. In particular, the maximum is attained for a sequence of 2-periodic symbolic configurations (hence it does not depend on L), viz.

χ ,2L (θ 0 [(10) L ] ∞ ) = χ ,2 ([0θ 0 1 ][10] ∞ ) = a -1 a +∞ k=1 a -k (k) 0,2 = a -1 2a 1 a -1 + 1 b -1 .
where we have used the explicit expression of (k) 0,2 from equation ( 15) in Appendix A and b := a(1 -2 ). It immediately follows that the condition [START_REF] Chazottes | Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems[END_REF]

holds iff b > 2, i.e. iff < c . Proof of Proposition 2.1. Lemma 3.2 states that A (2L) = Ω 2L when < c .
Since this set is compact, the conjugacy between the symbolic system (A ,2L , R ti ) and the CML (K ,2L , F ,2L ) is a homeomorphism. Hence, we have

h ,2L = h top (Ω 2L , R ti ) = (2L) log 2 is maximal in this domain.
The equality h c ,2L = (2L) log 2 will be shown in the proof of Proposition 2.3 below. On another hand, the end of the proof of Lemma 3.2 shows that

χ ,2L (θ 0 [(10) L ] ∞ ) > 1/2
for every sequence with θ 0 0 = 0 when > c . By continuity of the map θ → χ ,2L (θ), there exists T ,2L 1 such that for any sequence θ so that

θ 0 0 = 0 and θ t = (10) L 1 t T ,2L (7)
we have χ ,2L (θ) > 1/2; hence θ is not admissible. By a standard argument that we detail below, this implies that h top (A (2L) , R ti ) < (2L) log 2 which, together with the inequality (5), implies that h ,2L < (2L) log 2 when > c , as desired.

The remaining argument relies on the fact that the topological entropy of the symbolic subshift (A ,2L , R ti ) can be characterized as follows [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF][START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF]

h top (A ,2L , R ti ) = lim T →∞ log N T,2L T = inf T 1 log N T,2L T .
where N T,2L is the number of admissible words of length T in A ,2L . In particular, we must have

h top (A ,2L , R ti ) log N T ,2L T
for every T 1. In the present case, the non-admissibility of the sequences that satisfy relation [START_REF] Coutinho | Symbolic dynamics of two coupled lorenz maps: From uncoupled regime to synchronisation[END_REF] and of symmetric sequences imply the following inequality

N T ,2L +1,2L 2 (2L)(T ,2L +1) -2 2L
which yields

h top (A ,2L , R ti ) log N T ,2L +1,2L T ,2L + 1 < (2L)(T ,2L + 1) T ,2L + 1 log 2 = (2L) log 2.
The proof is complete. 2 For > c , it is reasonable to expect that the entropy h ,2L would monotonically decrease with . While this property remains to be proved, a similar trait can be shown for the symbolic system. This is an immediate consequence of the fact that admissibility of symbolic sequences is a monotonic property in a neighborhood of c , namely

A 1,2L ⊃ A 2 ,2L ,
when 1 < 2 (are sufficiently close to c ). The proof is identical to the one of Proposition 3.2 in [START_REF] Fernandez | Route to chaotic synchronisation in coupled map lattices: rigorous results[END_REF] and essentially relies on the following property (which in turn is a consequence of the fact that (k) L,2L is the entry with smallest modulus among the configuration

{ (k) n,2L } n∈Z 2L ) sup θ 0 ∈{0,1} Z 2L : θ 0 =(10) L (C -k ,2L θ 0 ) 0 = (C -k ,2L θ0 ) 0 , ∀k 1,
where θ0 s = H((-1) s ) for all s ∈ Z 2L , except for s = L mod 2L. The details are left to the reader.

Finite spatio-temporal word non-admissibility when > c

As shown in the end of the proof of Proposition 2.1 above, beyond the uncoupled regime, the CML entropy must be smaller than its maximal value for every even number of sites, viz. h ,2L < (2L) log 2 for all L 1. In order to obtain a non-trivial uniform upper bound for the entropy per lattice site, one needs to refine the restriction (7) on admissible sequences. One way to proceed is to demonstrate that the presence of specific (large) spatio-temporal motifs in a sequence induces non-admissibility. This is the purpose of the next statement. 1 such that for every L > L , every symbolic sequence θ ∈ Ω 2L for which there exists (s 0 , t 0 ) such that (see Figure 3)

1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 t 0 t 0 t 0 t 0 t 0 t 0 L Ε L Ε
   either θ t0 s0 = 0 and θ t0+t s0+s = H((-1) s ) or θ t0 s0 = 1 and θ t0+t s0+s = H((-1) s+1 ) ∀ -L s < L , 1 t T is not admissible.
Proof. We begin to prove that if two symbolic sequences coincide on a sufficiently large rectangular space-time window that contains the origin, then their image under χ ,2L are arbitrarily close to each other. Given L, L , T 1 with L < L, let θ, θ ∈ Ω 2L be two sequences such that

θ t s = θt s , ∀ -L s < L , 0 t T.
Since we have |θ t s -θt s | 1 for the remaining pairs (s, t), it follows from the definition of the function χ ,2L above that

|χ ,2L (θ) -χ ,2L ( θ)| m T + m L ,L,T where m T = a -1 a +∞ k=T +1 a -k n∈Z 2L | (k) n,2L | and m L ,L,T = a -1 a T k=0 a -k -L n<-L or L n<L | (k) n,2L |.
The first term m T is estimated using the property

n∈Z 2L | (k) n,2L | = (1-2 ) -k (Claim A.3 in Appendix A -recall that b = a(1 -2 )) m T = a -1 a +∞ k=T +1 a -k ( (k) 0,2 - (k) 1,2 ) = a -1 a b -T b -1 .
In particular, since b > 1 when < e , we have lim T →+∞ m T = 0. In order to control the second term m L ,L,T , we consider the entries

{ (k)
n } n∈Z of the inverse powers C -k of the coupling operator C defined on bounded configurations of the infinite lattice Z. (Just as for its periodic version, the operator C is invertible when < e and its inverse powers are all convolution operators [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF].) That is to say, given x ∈ ∞ (Z) we have

(C -k x) s = n∈Z (k) n x s+n ∀s ∈ Z.

The coefficients (k) n

have the same sign properties as the (k) n,2L in Lemma A.1, see relation [START_REF] Kaneko | Theory and Applications of Coupled Map Lattices[END_REF] in Appendix A. Moreover we have

(k) n,2L = m∈Z (k) n+m(2L) , ∀n ∈ Z 2L .
This easily implies the following inequality

-L n<-L or L n<L | (k) n,2L | -L n<-L or L n<L | (k) n | = (C -k x L ) 0
where the infinite configuration x L is defined by (see Figure 4)

s L L 1 s x L 1 1 Figure 4. Graphic representation of the configuration x L defined in the proof of Lemma 3.3. (x L ) s = (H(s -L ) + H(-s -L -1)) (-1) s , ∀s ∈ Z.
We have x L = 1 for all L 1 and the sequence {x L } L 1 point-wise converges to 0 (i.e. the configuration defined by x s = 0 for all s ∈ Z) as L → +∞. Convolution operators commute with point-wise limits of equibounded sequences [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF]. It results that for every k 0, we have lim

L →+∞ (C -k x L ) 0 = (C -k 0) 0 = 0
which yields lim L ,L→+∞ m L ,L,T = 0 for every T 1. This completes the first part of the proof. Now given > c , let η = χ ,2 ([0θ 0 1 ][10] ∞ )-1/2 > 0 where θ 0 1 is arbitrary (see end of the proof of Lemma 3.2). According to the first part of the proof, let T be sufficiently large such that m T < η/2. Then, let L be large enough such that m L ,L,T < η/2 for every L > L . By construction, any sequence θ ∈ Ω 2L where θ 0 0 = 0 and θ t s = H((-1) s ) for |s| < L and 1 t T is such that

|χ ,2L (θ) -χ ,2 ([0θ 0 1 ][10] ∞ )| m T + m L ,L
,T < η and hence χ ,2L (θ) > 1/2. Since θ 0 0 = 0, this shows that this sequence can not be admissible. By symmetry, one proves that any sequence such that θ 0 0 = 1 and θ t s = H((-1) s+1 ) for |s| < L and 1 t T is also not admissible. Finally, for any symbolic sequence θ as in the statement, the space-time translated sequence R s0 sp • R t0 ti (θ) must be of one of the previous types. As such, θ cannot be admissible and the proof is complete.

Proof of Proposition 2.2. Here L and T refer to the integers that were defined in Lemma 3.3. We are going to prove that this Lemma implies the existence of δ 1 , δ 2 > 0 for every > c such that

h top (A ,2L , R ti ) 2L(log 2 -δ 1 ) + δ 2 , ∀L > L .
Then letting L be sufficiently large such that δ1 := δ 1 -δ 2 /(2 L) > 0 and using from Proposition 2.1 that δ2 := log 2 -max 1 L< L h ,2L 2L > 0, the Proposition easily follows with δ = min{ δ1 , δ2 }.

Recall from the proof of Proposition 2.1 that we must have h

top (A ,2L , R ti ) log N T +1,2L
T +1

where N T,2L denotes the number of admissible words of length T in A ,2L . Let now L > L . A word {θ t s } t=0,••• ,T s∈Z 2L can be admissible only if each (nonoverlapping) restriction on spatial strips of width 2L namely

{θ t s } t=0,••• ,T s=r(2L ),••• ,(r+1)(2L )-1 , ∀r = 0, • • • , L L - 1
(where • denotes the floor function) is distinct from any forbidden motif given in Lemma 3.3. It means that if θ 0 r(2L )+L = 0 then there must be -L s < L and 1 t T such that θ t r(2L )+L +s = H((-1) s ) and similarly if θ 0 r(2L )+L = 1. Notice that, given θ 0 r(2L )+L in such motifs, the other symbols for t = 0 and -L s < L , s = 0 are arbitrary. Hence there are 2 2L -1 motifs for each choice of θ 0 r(2L )+L and thus there are 2 2L forbidden motifs for each r. Consequently, there remain at most 2 2L (T +1) -2 2L choices of admissible motifs on each strip s = r(2L ), • • • , (r+1)(2L )-1. Therefore, there are at most (2 2L (T +1) -2 2L ) L L admissible sub-words on the strip s = 0, • • • , L L L -1. Assuming maximal diversity for the remaining sites s = L L L , • • • , L -1, we finally obtain the following upper bound

N T +1,2L (2 2L (T +1) -2 2L ) L L 2 (L-L L L )(T +1)
from which, using basic inequalities for the floor function (viz. x-1 < x x for all x ∈ R), it easily follows that log N T +1,2L

T + 1 2L(log 2 -δ 1 ) + δ 2 as desired, where

δ 1 = -log(1-2 -2L T ) 2L (T +1)
> 0 and δ 2 = L log 2. 2

Families of admissible subshifts with positive entropy per lattice unit

The strategy of the proof of Proposition 2.3 consists in showing the admissibility of certain symbolic subshifts that are simultaneously compact and with topological entropy equal to (2L)δ where δ > 0 approaches log 2 as → + c . Compactness implies the lower bound h ,2L (2L)δ as desired. More precisely, we introduce a one-parameter family of nested subshifts with appropriate entropy in the limit of large lattices, and we prove couplingdependent admissibility for every set in this family.

The subshift definition relies on the following terminology. The symbolic configurations 0 2L and 1 2L are called homogeneous configurations. Any other configuration is said to be heterogeneous. Now, given a configuration {θ s } := {θ s } s∈Z 2L , let the function σ : {0, 1} Z 2L → {0, 1} be defined by

σ({θ s }) = 0 if {θ s } is homogeneous 1 if {θ s } is heterogeneous
The idea behind the definition of the subshifts is to restrict repeated occurrences of heterogeneous configurations. Recall that symbolic sequences are denoted by θ = {θ t } t 0 where each θ t is a symbolic configuration. Given

ν ∈ [0, 1], let Ω ν,2L = θ ∈ Ω 2L : t2-1 t=t1 σ(θ t ) ν(t 2 -t 1 ) , ∀0 t 1 < t 2
where • is the ceiling function. Notice that Ω 1,2L contains all symbolic sequences, Ω 0,2L only consists of homogeneous sequences and Ω ν1,2L Ω ν2,2L when ν 1 < ν 2 . Moreover, we shall prove below that the sequence {h top (Ω n n+1 ,2L , R ti )} n 0 (regarded as a sequence of sequences indexed by L) uniformly converges to {(2L) log 2} L 1 . Together with the following statement, this property suffices to ensure that Proposition 2.3 holds. Note that the proof of Lemma 4.2 proves that each sequence {h top (Ω n n+1 ,2L , R ti )} n 0 is strictly increasing. We believe that the function ν → h top (Ω ν,2L , R ti ) is actually strictly increasing. Lemma 4.1. (i) For every > c , there exists ν ∈ [0, 1) such that for every ν < ν we have

Ω ν,2L ⊂ A ,2L , ∀L 1.
Moreover, for every ν > ν and L 1, some of the sequences in Ω ν,2L are not admissible. (ii) The map → ν is a decreasing Devil's staircase with limit lim → + c ν = 1.

Proof. Since each set Ω ν,2L is compact and is invariant under the action of θ → 1 -θ, the reasoning at the beginning of the proof of Lemma 3.2 can be repeated here to conclude that Ω ν,2L ⊂ A ,2L iff sup

θ∈Ω ν,2L : θ 0 0 =0 χ ,2L (θ) < 1/2 (8)
The strategy to determine the optimal ν is to obtain a manageable expression for this supremum. Recall that the definition of χ ,2L , namely

χ ,2L (θ) = a -1 a +∞ k=0 a -k (C -k ,2L θ k ) 0 involves inverse powers C -k ,2L
of the coupling operator whose entries

(k) n,2L
(see equation ( 3)) have properties listed in Lemma A.1. In particular, the sign property implies that (C -k ,2L θ k ) 0 (k 1) is maximum for θ k = ( 10)

L (i.e. θ k s = H((-1) s ) for all s ∈ Z 2L ). Moreover the normalization n∈Z 2L (k)
n,2L = 1 (Claim A.2) indicates that, among the two homogeneous configurations, (C -k ,2L θ k ) 0 reaches its maximum for θ k = 1 2L . Therefore, given any sequence θ ∈ Ω ν,2L such that θ 0 0 = 0, the sequence θ defined by θt =    θ 0 if t = 0 (10) L if t 1 and θ t is heterogeneous 1 2L if t 1 and θ t is homogeneous also belongs to Ω ν,2L (because σ( θt ) = σ(θ t ) for all t 0) and satisfies χ ,2L (θ) χ ,2L ( θ). Accordingly, the supremum in ( 8) is attained in the subset where all sequences are composed (for t 1) of configurations that are either (10) L or 1 2L and in particular 2-periodic, viz. we have sup

θ∈Ω ν,2L : θ 0 0 =0 χ ,2L (θ) = sup θ∈Ω ν,2L : θ 0 0 =0 and θ t ∈{(10) L ,1 2L }, ∀t 1 χ ,2L (θ) = sup θ∈Ων,2 : θ 0 0 =0 and θ t ∈{10,11}, ∀t 1 χ ,2 (θ)
Finally, we proved in [START_REF] Coutinho | Symbolic dynamics of two coupled lorenz maps: From uncoupled regime to synchronisation[END_REF] the existence of a map → ν such that sup θ∈Ων,2 : θ 0 0 =0 and θ t ∈{10,11}, ∀t 1

χ ,2 (θ) < 1/2 iff ν < ν
We also showed that this map is a decreasing Devil's staircase with limit lim → + c ν = 1. This completes the proof of the Lemma. We now study properties of the subshift entropy h top (Ω ν,2L , R ti ). This quantity is an increasing function of ν that can be characterized by an algebraic equation (see equation ( 14) in [START_REF] Coutinho | Symbolic dynamics of two coupled lorenz maps: From uncoupled regime to synchronisation[END_REF] for the equation of h top (Ω ν,2 , R ti )).

For our purpose here, it is enough (and simpler) to consider the entropy h n,2L := h top (Ω n n+1 ,2L , R ti ) of the sets Ω n n+1 ,2L that consist of all symbolic sequences for which the length of any heterogeneous word is at most n. 3 Lemma 4.2. For every n 1, there exists δ n > 0 with limit lim n→∞ δ n = log 2 such that we have h n,2L (2L)δ n for all L 1.

Proof. h n,2L is the exponential rate of increase of the number N t of admissible words of length t in Ω n n+1 ,2L [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF][START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF]. Words in this set conclude with either an homogeneous configuration or an heterogeneous word of length at most n. Given t 1 (not smaller than n), let N 0 t be the number of admissible words of length t with homogeneous suffix and for k = 1, • • • , n, let N 1 k t be the number of admissible words of length t with heterogeneous suffix of length (exactly) k. Obviously we have N t = N 0 t + n k=1 N 1 k t and in order to obtain the growth rate of N t , we are going to establish an induction relation for N 0 t . Firstly, any homogeneous configuration can follow any admissible word to form another admissible word. Since there are two homogeneous configurations, this implies that N 0 t+1 = 2N t . Moreover, it is easy to see that we have

N 1 k+1 t+1 = α 2L N 1 k t for k = 1, • • • , n -1 where α 2L = 2 2L -2 is the number of heterogeneous configurations. Similarly, we have N 1 1 t+1 = α 2L N 0 t .
Based on these relations, we get the desired induction

N 0 t+1 = 2 N 0 t + n k=1 N 1 k t = 2 n k=0 α k 2L N 0 t-k
Since all coefficients in the right hand side are positive, the companion matrix associated with this induction is non-negative and irreducible. By the Perron-Frobenius Theorem, it results the quantity N 0 t has the following behavior

lim t→+∞ N 0 t+1 N 0 t = λ n,2L
where λ n,2L is the largest positive solution of the equation

λ n+1 -2 n k=0 α k 2L λ n-k = 0 (9)
(the only one for which all components of the corresponding eigenvector are positive). The previous limit implies that

h n,2L := lim t→+∞ log(N t ) 1/t = lim t→+∞ log(N 0 t ) 1/t = log λ n,2L .
In order to complete the proof, we use the equation above to show that for every 0 < δ < log 2, there exists n δ 1 such that for all n n δ , we have

h n,2L (2L)δ ∀L 1.
We begin to prove that lim n→+∞ λ n,2L = 2 2L for every L 1. The equation ( 9) is equivalent to p n,2L (λ) = 1 where the function p n,2L is defined by

p n,2L (λ) = 2 n k=0 α k 2L λ -(k+1) .
This function has the following properties

• λ → p n,2L (λ) is strictly decreasing, • p n,2L (1) > 1, • p n+1,2L (λ) > p n,2L (λ) • p n,2L (α 2L + 2) < p ∞,2L (α 2L + 2) = 1 which immediately imply that every sequence {λ n,2L } n 1 is strictly increas- ing in [1, α 2L + 2]. Moreover, a direct computation shows that p n,2L (α 2L ) = 2 n+1 α 2L which yields λ 2 2L-1 -2,2L = α 2L . Accordingly, the series p ∞,2L (λ) is uniformly convergent for λ 2 2L-1 -1,2L λ α 2L + 2.
This function is strictly increasing and it easily follows that lim n→+∞ λ n,2L = α 2L + 2 = 2 2L as desired. Uniform control on the asymptotic behavior of λ n,2L will be granted by the following relation which is a consequence of ( 9)

λ n+1 n,2L = 2 n k=0 α k 2L λ n-k n,2L > α n 2L
Accordingly, we have

h n,2L > n n + 1 log α 2L > n n + 1 log(2 2L-1 ) > (2L) 1 - 1 n + 1 - 1 2L log 2
Now, given 0 < δ < log 2, let L δ > 1 be such that 1 -1 L δ log 2 δ. Using that lim n→+∞ λ n,2L = 2 2L , let ñδ be sufficiently large such that

h ñδ ,2L (2L)δ, ∀L ∈ {1, • • • , 2L δ -1}.
The inequality above and the definition of L δ imply that

h 2L δ -1,2L > (2L)δ, ∀L 2L δ .
Letting n δ := max{ñ δ , 2L δ -1}, the monotonicity of the sequences {h n,2L } n 1 finally imply that for every n n δ , we have h n,2L (2L)δ for all L 1.

Proof of Proposition 2.3. This is a direct consequence of the statements in this section. By Lemma 4.1, for every n 1, there exists n > c such that all the Ω n n+1 ,2L (L 1) are admissible for every < n . It results from compactness of these subshifts, from the previous Lemma and from the left inequality in [START_REF] Bunimovich | Space-time chaos in coupled map lattices[END_REF], that h ,2L (2L)δ n for all L when < n and the Proposition follows from the limit lim n→∞ δ n = log 2. In addition, the inequality n > c for all n 1 implies that for = c we have

h c ,2L (2L) sup n 1 δ n = (2L) log 2
as claimed in Proposition 2.1. 2

Coupled map lattices with piecewise increasing individual maps

In this section, the results obtained for piecewise affine CML are extended by continuation to CML F g, ,2L with piecewise increasing individual map g, see equation ( 2) in Section 2. The strategy consists in showing that the estimates on admissible sequences persist for CML based on small C 1 perturbations of f .

To that goal, we preliminary assert the existence of a symbolic dynamics for piecewise expanding CML F g, ,2L . Recall the quantity a g defined in equation [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF]. Given any real map g as defined before Theorem 2.4 and with a g > 1, let g := ag-1 2ag < 1/2. Similarly to as in the piecewise linear case, the code θ associated with any point x ∈ R Z 2L is the symbolic sequence defined by θ t s = H((F t g, ,2L x) s -1/2), ∀s ∈ Z 2L , t 0. As before, the existence of the symbolic dynamics is due to the fact that every point in the repeller I g, ,2L of F g, ,2L is entirely determined by its code, which has to be an admissible sequence.

Lemma 5.1. For every < g and L 1, there exists a function χ g, ,2L : Ω 2L → R such that x ∈ I g, ,2L iff its coordinates are given by x s = χ g, ,2L • R s sp (θ), ∀s ∈ Z 2L where the code θ solves the admissibility equation

θ t s = H χ g, ,2L • R s sp • R t ti (θ) -1/2 , ∀s ∈ Z 2L , t 0.
Proof. Following a standard argument, the function χ g, ,2L is constructed by using the iterated function system associated with the pre-images of F g, ,2L . Let g 0 be the linear extension of the left branch of g to the whole R, i.e.

g 0 (x) = g(x) if x < 1/2 a g (x -1/2) + g(1/2 -0) if x 1/2
and similarly, let g 1 linear extension of the right branch of g. The maps g 0 and g 1 are invertible and when < g , the coupling operator C ,2L is also invertible. Hence, given any symbolic configuration {θ s } := {θ s } s∈Z 2L , the map F {θs},g, ,2L defined in R Z 2L by (F {θs},g, ,2L x) s = (1 -)g θs (x s ) + 2 g θs-1 (x s-1 ) + g θs+1 (x s+1 ) , ∀s ∈ Z 2L is invertible and its inverse F -1 {θs},g, ,2L given by

(F -1 {θs},g, ,2L x) s = g -1 θs ((C -1 ,2L x) s ), ∀s ∈ Z 2L ,
is well-defined in R Z 2L . We claim that the desired function χ g, ,2L is given by

χ g, ,2L (θ) = lim t→+∞ (F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L • • • • • F -1 θ t ,g, ,2L x) 0 .
We first need to show that the limit exists and does not depend on x. Monotonicity and the definition of a g imply the existence of a constant c g > 0 such that |g -1 i (x)| a -1 g |x| + c g , ∀x ∈ R, i = 0, 1 from where we get the following bound

F -1 {θs},g, ,2L x (a g (1 -2 )) -1 x + c g , ∀x ∈ R Z 2L , {θ s } ∈ {0, 1} Z 2L .
It follows that any sequence

{F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L • • • • • F -1 θ t ,g, ,2L x} t 0 is bounded in R Z 2L .
In addition, the definition of a g and C -1 ,2L = (1 -2 ) -1 yield the following inequality

F -1 {θs},g, ,2L x -F -1 {θs},g, ,2L y (a g (1 -2 )) -1 x -y (10)
for all x, y ∈ R Z 2L and {θ s } ∈ {0, 1} Z 2L . Since (a g (1-2 )) -1 < 1 when < g , one easily deduces that every

{F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L • • • • • F -1 θ t ,g, ,2L
x} t 0 is also a Cauchy sequence. Therefore it has a limit in R Z 2L and by the previous inequality again, this limit does not depend on x. Now, continuity of the maps g 0 and g 1 implies

F θ 0 ,g, ,2L lim t→+∞ F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L • • • • • F -1 θ t ,g, ,2L x = lim t→+∞ F -1 θ 1 ,g, ,2L • • • • • F -1 θ t ,g, ,2L
x.

Since F g, ,2L x = F {θs},g, ,2L x when x satisfies θ s = H(x s -1/2) for all s ∈ Z 2L , we conclude that the configuration {R s sp • χ g, ,2L (θ)} s∈Z 2L = {χ g, ,2L • R s sp (θ)} s∈Z 2L belongs to I g, ,2L when θ is admissible. Finally, proving that every configuration in the repeller writes {χ g, ,2L • R s sp (θ)} s∈Z 2L is easy and is left to the reader.

2 With the existence of symbolic dynamics given, our second ingredient is the continuity of the component map g → χ g, ,2L (θ), uniformly in the set of symbolic sequences. Lemma 5.2. Let g and g be two individual maps with a g , a g > 1 and such that g -g η. Then, for every < min{ g , g }, L 1 and θ ∈ Ω 2L , we have

|χ g, ,2L (θ) -χ g, ,2L (θ)| η a -(1 -2 ) -1 ,
where a = max{a g , a g }.

Proof: As in the previous proof, consider the linear extensions g 0 , g 1 , g0 and g1 over R of the branches of the maps g and g. Using the definition of a g and the property g -g < η, one easily shows that g i (x + η/a g ) > gi (x) and g i (x -η/a g ) < gi (x), ∀x ∈ R, i = 0, 1 which implies the following constraint on the inverse maps

g -1 i -g-1 i η/a g for i = 0, 1.
By exchanging the roles of g i and gi , it follows that the inverse CML introduced in the previous proof satisfy the following inequality

F -1 {θs},g, ,2L x -F -1 {θs},g, ,2L x η/a, ∀x ∈ R Z 2L .
Using the inequality (10), we subsequently obtain

F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L x -F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L x F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L x -F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L x + F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L x -F -1 θ 0 ,g, ,2L • F -1 θ 1 ,g, ,2L x η/a + (a g (1 -2 )) -1 F -1 θ 1 ,g, ,2L x -F -1 θ 1 ,g, ,2L
x From thereon, an induction and a similar reasoning with the roles of g and g being exchanged directly lead to the desired conclusion.

Proof of Theorem 2.4. We prove the lower bound and the upper bound separately. For the lower bound, we first show the following statement: For every 0 < µ < c , there exists η > 0 such that for every CML F g, ,2L with individual map g satisfying g -f + |a g -a| < η, we have for every

< c -µ h top (I g, ,2L , F g, ,2L ) = (2L) log 2, ∀L 1. (11)
In order to prove this fact, we begin to notice that the quantity

χ ,2 ([0θ 0 1 ][10] ∞ ) = sup θ∈Ω 2L : θ 0 0 =0 and θ t ∈{10,11}, ∀t 1 χ ,2 (θ)
introduced in the proof of Lemma 3.2 above is a strictly increasing function of which coincides with 1/2 for = c . Let then

ζ := 1/2 -χ c -µ,2 ([0θ 0 1 ][10] ∞ ) > 0. Let now η 1 > 0 be sufficiently small such that we have a g (1 -2( c -µ)) > 1 whenever a g > a -η 1 (η 1 exists because a(1 -2( c -µ)) > 2). Let η = min{η 1 , ζ(a -(1 -2( c -µ)) -1 )} > 0.
Let g be an individual map such that g -f + |a g -a| < η. Notice that the constraint g -f < +∞ implies that a g a; hence a = a in this case. The condition |a g -a| < η implies c -µ < g . Lemma 5.1 then guarantees the existence of the symbolic dynamics of F g, ,2L for every < c -µ. Moreover, the condition g -f < η and Lemma 5.2 imply that

|χ g, ,2L (θ) -χ f, ,2L (θ)| ζ a -(1 -2( c -µ)) -1 a -(1 -2 ) -1 ζ, ∀θ ∈ Ω 2L , < c -µ.
The constant ζ has been chosen such that for all L 1 sup

θ∈Ω 2L : θ 0 0 =0 χ f, ,2L (θ) < 1/2 -ζ, ∀ < c -µ.
Therefore, we have sup

θ∈Ω 2L : θ 0 0 =0 χ g, ,2L (θ) < 1/2, ∀ < c -µ,
from where the relation [START_REF] Gielis | Coupled map lattices with phase transitions[END_REF] immediately follows.

In order to obtain a lower bound for > c -µ, we observe that the quantities sup

θ∈Ω n n+1 ,2 : θ 0 0 =0 and θ t ∈{10,11}, ∀t 1 χ ,2 (θ)
(see end of the proof of Lemma 4.1) are also strictly increasing functions of [START_REF] Coutinho | Symbolic dynamics of two coupled lorenz maps: From uncoupled regime to synchronisation[END_REF]. Accordingly, a reasoning similar to the previous one concludes that for every µ < e , there exists (another) η > 0 such that for every CML F g, ,2L with individual map g satisfying g -f + |a g -a| < η, we have for every n 0

h top (I g, ,2L , F g, ,2L ) h top (Ω n n+1 ,2L , R ti ) (2L)δ n , ∀L 1, < n -µ,
where the δ n were introduced in Lemma 4.2 and the n were defined in the proof of Proposition 2.3 as the maximal coupling strength for which Ω n n+1 ,2L ⊂ A ,2L . The sequence { n } n 1 is strictly decreasing with 0 = e and lim n→+∞ n = c . It follows that the lower bound of the Theorem holds with

δ := log 2 if < c -µ δ n if > c -µ where n := max{n 1 : < n -µ}
Moreover, the fact that lim n→+∞ n = c and lim n→+∞ δ n = log 2 imply that lim →( c -µ)+ δ = log 2.

For the upper bound, the reasoning also follows the same lines. We first need an additional restriction on µ in a way that c + µ < e -µ, i.e. we take µ < min{ c , ( ec )/2}. Then letting now

ζ := χ c +µ,2 ([0θ 0 1 ][10] ∞ ) -1/2 >
0, similarly to as above, we set η > 0 (smaller than as before if necessary) such that g -f + |a g -a| < η implies g > e -µ and sup

θ∈Ω 2L , L 1 |χ g, ,2L (θ) -χ f, ,2L (θ)| ζ/2, ∀ < e -µ.
As in the proof of Proposition 2.1, this implies that h top (I g, ,2L , F g, ,2L ) < (2L) log 2 for all L 1 when > c + µ. Just as in the piecewise linear case, this does not suffice to get extensive decay. To that goal, the proof of Lemma 3.3 can be repeated mutatis mutandis to show that for every < e -µ there exists L , T 1 such that any sequence θ ∈ Ω 2L where θ 0 0 = 0 and θ t s = H((-1) s ) for |s| < L and 1 t T is so that |χ f, ,2L (θ) -χ f, ,2 ([0θ 0 1 ][10] ∞ )| ζ/2 and hence χ g, ,2L (θ) > 1/2 which we have show to imply h top (I g, ,2L , F g, ,2L ) (2L)(log 2 -δ) provided that L is sufficiently large, say L L. This concludes the proof of the Theorem. 2

Appendix A. Properties of the coupling operator C ,2L and its inverse

The CML mapping F ,2L can be regarded as the composition C ,2L and the uncoupled mapping F 0,2L given by (F 0,2L x) s = f (x s ) ∀s ∈ Z 2L .

A.1. Eigenvalues and the condition for expanding CML

The coupling operator is linear and commutes with the spatial translations R sp on the chain Z 2L defined by (R sp x) s = x s-1 for all s ∈ Z 2L . As such, it can be diagonalized in the Fourier basis and a direct calculation shows that its eigenvalues are given by ĉ( , k 2L ) for k ∈ Z 2L where [2] ĉ( , ω) = 1 -(1 -cos 2πω), ∀ω ∈ R/Z.

(Notice that the eigenvalues for k = 0, L mod 2L have multiplicity 2 and the two other ones have simple multiplicity.) In particular, all eigenvalues are nonnegative when ∈ [0, 1/2] and the smallest eigenvalue is ĉ( , 1/2) = (1 -2 ).

It follows that the eigenvalues of the (constant) derivative of F ,2L outside discontinuities are given by aĉ( , k 2L ). In particular, the smallest eigenvalue is aĉ( , 1/2) = a(1 -2 ); hence for ∈ [0, 1] the CML is expanding iff a(1 -2 ) > 1 i.e. iff 0 < e := a -1 2a .

A.2. Coefficients ,2L exists when < 1/2 and is a convolution operator [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF]. Hence, the coefficients as desired.

Figure 1 .

 1 Figure 1. Left: Graph of the individual map f for a = 3. Right: Graph of a small C 1 -perturbation of f .

Figure 2 .

 2 Figure 2. Schematic illustrations of upper and lower bounds (black step functions) on the CML entropy (red curves) per lattice site. Propositions 2.1, 2.2 and 2.3 and Theorem 2.4 attest that these bounds are independent of the lattice period 2L. Left: Piecewise affine CML F ,2L . Right: CML F g, ,2L with piecewise increasing individual map g.

Figure 3 .

 3 Figure 3. Graphic representation of the first forbidden word in Lemma 3.3. The other word is obtained by exchanging the 0's and the 1's.

  are the entries of the inverse powers C -k ,2L of the coupling operator, see relation (3) above. The inverse C -1

  can be regarded as being generated by the following induction

  ,2L , ∀n ∈ Z 2L , k 0 where (0) n,2L = δ n,0 mod 2L are the entries of the identity in R Z 2L (δ n,m being the Kronecker symbol). Together with the same properties for the configuration { (1) n,2L } n∈Z 2L , this induction relation implies the following properties.

Lemma A. 1 .

 1 Every configuration { (k) n,2L } n∈Z 2L (k 1) has the properties | for all n ∈ {0, • • • , L -1}. Proof. We first check that if {a n } n∈Z 2L and {b n } n∈Z 2L are two configurations with the properties as in the statement, then the 'convoluted' configuration {(a * b) n } n∈Z 2L defined by (a * b) n := m∈Z 2L a m b n-m also satisfies the same properties. Indeed, the equality (a * b) -n = (a * b) n follows from changing m → -m in the previous sum. Moreover, the property (-1) n (a * b) n > 0 directly follows from the relation(-1) n (a * b) n = m∈Z 2L (-1) m a m (-1) n-m b n-m . Now, in order to show the inequality |(a * b) n+1 | < |(a * b) n, | for all n ∈ {0, • • • , L -1},given that the signs of (a * b) n are alternating, it suffices to check that we have(-1) n ((a * b) n + (a * b) n+1 ) > 0, ∀n ∈ {0, • • • , L -1}.To that we goal, we shall need the following inequality(-1) n+m (a n-m + a n+m+1 ) > 0, ∀n, m ∈ {0, • • • , L -1}(13)that holds for every sequence satisfying the properties of the statement. Indeed, when n, m ∈ {0, • • • , L -1}, we have-L + 1 n -m L -1 and 1 n + m + 1 2L -1and we consider separately the cases n + m + 1 L and L + 1 n + m + 1.In the first case, we certainly have |n -m| n + m + 1 and the inequality|a n | > |a n+1 | for all n ∈ {0, • • • , L -1} implies |a n-m | = |a |n-m| | > |a n+m+1 |. If L + 1 n + m + 1 2L -1, we have 1 2L -n -m -1 L -1 and |n -m| 2L -n -m -2 which implies |a n-m | = |a |n-m| | > |a 2L-n-m-1 | = |a n+m+1 |.In both cases, it results that (-1) n+m (a n-m + a n+m+1 ) = |a n-m | -|a n+m+1 | > 0.

  • F 0,2L of the coupling operator C ,2L defined in R Z 2L by (C ,2L x) s = (1 -)x s + 2 (x s-1 + x s+1 ) ∀s ∈ Z 2L

This work was funded by EU Marie Curie fellowship PIOF-GA-2009-235741.

We use the term conjugacy when the correspondence is a uniformly continuous bijection and topological conjugacy when the map is a homeomorphism.

Acknowledgment. BF acknowledges stimulating discussions with Lai-Sang Young and is grateful to Courant Institute for hospitality.

Now, using [START_REF] Jiang | Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations[END_REF] we have

from where the desired inequality immediately follows.

It remains to show that the original configuration {

n,2L } n∈Z 2L possesses the desired properties. To that goal we shall use the following explicit expression of the entries [START_REF] Afraimovich | Topological properties of linearly coupled expanding map lattices[END_REF] n of the inverse C -1 of the coupling operator acting in ∞ (Z)

Now, the expression of

n,2L = m∈Z (1) n+2mL

from which the properties

-n,2L =

n,2L and (-1) n (1) n,2L > 0 easily follow. In addition, together with [START_REF] Jost | Spectral properties and synchronization in coupled map lattices[END_REF], this expression provides an explicit formula for the coefficients, namely

where e 0 = {1} s∈Z 2L is the eigenvector associated with the eigenvalue ĉ( , 0) = 1.

Claim A.3. For all k 0, we have

n,2L | converges for every < e . Proof. According to Lemma A.1, we have

where e L = {(-1) s } s∈Z 2L is the eigenvector associated with the eigenvalue ĉ( , 1/2) (i.e. k = L in the expression ĉ( , k 2L ) above). The desired expression then immediately follows from the relation ĉ( , 1/2) = 1 -2 . Furthermore, the series

n,2L | converges when a(1 -2 ) > 1 i.e. exactly when < e .

Finally the expression of the (k) n,2 is recalled from [START_REF] Fernandez | Route to chaotic synchronisation in coupled map lattices: rigorous results[END_REF] (k)

The entries

of the inverse powers C -k of the coupling operator acting in ∞ (Z) (see proof of Lemma 3.3) have similar properties to the ones in Lemma A.1. In particular, using the definition (14) of the