Pancho, Coba, Mati, Iriar, Enzo, Mario Emilio Hasson

precious to me. Thank you also to Arrianne, who along with Afaf became an inseparable friends. We have spent so many things together, from the evenings at my house to the vacations we took and every memory by your side is something I will treasure with time. How can I forget Bastian and Josefa, with whom I have not been able to share as much as I would like, especially because time moves fast and does not stop, but I like them just the same for everything they have given me and the moments we have spent together. They are very valuable friends to me and they can always count on me.

To the people in LAMFA, Sebastian, Mariem, Clément, Gauthier, Cheryl, Jihade, Marouan, Henry, Yohan for the time we have spent together, especially at parties and the forgotten after-lunch board games before covid attacked. I would like to make

It has been a long path since I started this part of my life, but the final moments have arrived. Many things happened during this time, but above all I want to thank the people who were there with me during the whole process.

First, I want to thank my thesis director Samuel Petite for always believing in me, despite the obstacles and for all the help and constant support he has given me over these years. Thank you professor for your meticulous revisions and the long meetings where we debated about the best way to express what I wanted to say and for always paying attention to the details that I missed, it is because of your advice that I have been able to forge my path as a researcher. Without you I would not be the professional I believe I am today.

I would like to extend my sincere thanks to the reviewers of this thesis, Valérie Berthé and Reem Yassawi, for taking time in reading and the good comments received for my work. I also like to extend my thanks to the examiners Bryna Kra, María Isabel Cortez and Fabien Durand for accepting the invitation to be part of my thesis's jury.

I would also like to thank professor Alejandro Maass, because apart from all the help he gave me working in Chile and his trust, he was there for me in a very difficult moment of my life and I will always appreciate him for that. I would like to extend my sincere thanks to professor Sebastian Donoso because he was the first person who believed in me, and although we met very randomly, he is always willing to work together and support me in my projects.

To my friends, those from farther and nearer who have always been there to share good times. To Ambuli for all his years of friendship. We have been through many things together that it is hard for me to summarize it in a couple of lines, but you have always been my companion, in adventures and misadventures and that is why you will always be one of my best friends. If I look back, I can imagine that my life would not be the same if I had not met you and I am one hundred percent sure that our friendship will last no matter the time that passes and the distance that separates us. To Afaf, I'm not quite sure how it all started, but suddenly you became an indispensable person in my life. You are very important to me and despite our differences and arguments, you are always by my side and that makes our friendship so special. Thank you for the company and the good times, which are countless, and for being by my side since I arrived in Amiens. You are very

Contents

Introduction

The central objects of study of this thesis are homomorphisms between topological Z dactions T : X × Z d → X on a compact metric space X. A homomorphism is a continuous surjective map φ : (X, T, Z d ) → (Y, T, Z d ) such that for some Z d -automorphism M ∈ GL(d, Z), we have φ • T n = T M n • φ for all n ∈ Z d . This notion extends the classical dynamical one of morphism like factor, when M is the identity and conjugacy when φ is an invertible factor. Invertible homomorphisms, which will be called isomorphisms, are then conjugacies of Z d -actions, up to a GL(d, Z)-transformation.

For Z-actions, isomorphisms are nothing else than flip-conjugacies, i.e., homeomorphisms φ such that φ • T is equal to T • φ or T -1 • φ. Factors and conjugacies are referred to as endomorphisms and automorphisms, respectively when the dynamical systems are the same. While endomorphisms represent a kind of internal symmetries in the system, such as permutations, homomorphisms represent furthermore symmetries of the orbits, such as rotations and reflections. That is why they are also sometimes called extended symmetries (see for example [START_REF] Baake | A brief guide to reversing and extended symmetries of dynamical systems[END_REF][START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF][START_REF] Bustos | Extended symmetry groups of multidimensional subshifts with hierarchical structure[END_REF][START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]). From an algebraic point of view, while the automorphism group is the centralizer of the action group T in the group Homeo(X) of self-homeomorphisms in X, the isomorphism group is the normalizer of the action group T in Homeo(X), that is, the set of self-homeomorphisms φ such that φ T φ -1 = T .

The study of homomorphisms of a dynamical system (X, T, Z d ) is a classical problem. The elements in the group T generated by the action are automorphisms of the system, hence the automorphism group is always nonempty. However the existence of homomorphisms for a particular matrix M ∈ GL(d, Z) is generally an open problem. Classical questions concern their dynamical and algebraic properties in relation with the dynamical ones of (X, T, Z d ). The description for these isomorphisms of their possible subgroups, their quotients, their amenability or their action on the T -invariant measures depending on the properties of (X, T, Z d ) are natural problems. In a general context these questions are widely open. Another classical question is the determination of topological factors of a particular system. Their explicit description can be used to unravel the structure of the system. For certain aspects, they carry relevant information and enable to do concrete calculations or to study some specific structures (for example in spectral theory [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF]).

The study of isomorphisms of the particular class of minimal systems where X is a

Cantor set is motivated by algebraic reasons. Indeed, to any Z-minimal Cantor system (X, T, Z) there is an associated countable group called the topological full group and denoted by [[T ]]. The elements of this group are self-homeomorphisms of the space X that are locally power of the transformation T . Homomorphisms of the system (X, T, Z) induce group homomorphisms in [[T ]]. Conversely, the topological full group is a complete invariant of flip-conjugacy [START_REF] Giordano | Full groups of Cantor minimal systems[END_REF]. It appears such full groups [[T ]] present remarkable properties. H. Matui proved in [START_REF] Matui | Some remarks on topological full groups of Cantor minimal systems[END_REF] that its commutator subgroup is simple. Moreover, he also showed that this group is finitely generated if and only if (X, T, Z) is conjugated to a minimal subshift. Thus, these groups enable to construct finitely generated simple ones with unexpected algebraic properties. For instance any topological full group is amenable [START_REF] Juschenko | Cantor systems, piecewise translations and simple amenable groups[END_REF], giving the first examples of finitely generated simple groups in this class. Using isomorphisms of some low complexity Z-subshifts (namely linearly recurrent ones with arbitrarily long palindromic words like the substitutive Fibonacci subshift) V. Nekrashevych in [START_REF] Nekrashevych | Palindromic subshifts and simple periodic groups of intermediate growth[END_REF] succeed to construct with full groups a finitely generated simple group with intermediate growth, i.e., which is neither of polynomial nor of exponential growth. Moreover it is of Burnside type, that is, an infinite group where each element is periodic.

Another motivation for the study of isomorphisms comes from theoretical physics. The discovery in 1984 by D. Schechtman et.al. [START_REF] Shechtman | Metallic phase with long-range orientational order and no translational symmetry[END_REF] of a metal alloy structure similar to ideal crystal ones deeply influenced the study of multidimensional aperiodic structure. This alloy presented a discrete diffraction pattern, like for crystals, but had a five-fold rotational symmetry which is forbidden for ideal crystals. The term quasicrystal was then invented to describe these new classes of crystals with "forbidden" symmetry, although there is little agreement on the precise definition of a quasicrystal. Roughly speaking, a quasicrystal is a solid, which exhibits sharp bright spots (called Bragg peaks) in their X-ray diffraction pattern but has an aperiodic structure (usually manifested by the presence of a non-quasicrystallographic symmetry). The presence of Bragg peaks indicates the presence of "long-range order" in the structure (see [START_REF] Levine | Quasicrystals: a new class of ordered structures[END_REF]). This work earned D. Schechtman the Wolf Prize in Physics in 1999 and the Nobel Prize in Chemistry in 2011. One can assume that (at least approximately) a quasicrystal consists of atoms located at the vertices of an almost periodic tiling. It is possible to recover quantitative physical properties by studying a dynamical system associated with the tiling. Such systems were first introduced by D. Rudolph in [START_REF] Rudolph | Markov tilings of R n and representations of R n actions[END_REF]. Since then, a series of articles are devoted to their study (see [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF][START_REF] Frank | Substitution and tiling dynamics: introduction to self-inducing structures[END_REF][START_REF] Robinson | Symbolic dynamics and tilings of R d . In Symbolic dynamics and its applications[END_REF][START_REF] Solomyak | Dynamics of self-similar tilings[END_REF] for an extensive bibliography on this subject). The diffraction pattern is then essentially the point spectrum of the corresponding translation action [START_REF] Levine | Quasicrystals: a new class of ordered structures[END_REF]. Moreover its symmetries are reflected by the isomorphisms of its dynamical system [START_REF] Robinson | The dynamical theory of tilings and quasicrystallography[END_REF]. In 1982, A. Mackay [START_REF] Mackay | Crystallography and the Penrose pattern[END_REF] published the diffraction pattern of a tiling created by R. Penrose several years before [START_REF] Penrose | The role of aesthetics in pure and applied mathematical research[END_REF], which has very similarities with the ones discovered by D. Schechtman.

Figure 1: The quasicrystal diffraction images appearing in the original article of D. Shechtman et.al. [START_REF] Shechtman | Metallic phase with long-range orientational order and no translational symmetry[END_REF] Figure 2: The diffraction pattern of the Penrose tiling as it appears in the original article of A. Mackay [START_REF] Mackay | Crystallography and the Penrose pattern[END_REF] The Penrose tiling is then a good mathematical model of quasicrystals. It is build with only 2 tiles, up to rotations and translations. The generation of patterns is obtained by means of an algorithmic method so-called substitution. Roughly speaking, this process consists in substituting tiles by a union of tiles and applying the same rule to this new pattern. Such construction provides, at the limit, most of the simplest aperiodic tilings, in the sense that they have the lowest complexity [START_REF] Lagarias | Local complexity of Delone sets and crystallinity[END_REF].

Another interesting property of the Penrose tiling, meaningful in the crystallographic context of short range interaction, is that all the allowed tilings of the associated system are the ones verifying a finite set of local rules. This is a geometrical analogue of onedimensional subshifts of finite type. However as a difference with one-dimensional subshifts of finite type that always contains periodic points, Penrose tiling system is aperiodic. This highlights a fundamental difference between the one and two dimensional combinatorial properties which are linked with logic and computability. The seminal work of H. Wang [START_REF] Wang | Proving theorems by pattern recognition -ii[END_REF] already established relation between decidability of certain first-order logic formulas and domino problems. His student R. Berger [START_REF] Berger | The undecidability of the domino problem[END_REF] showed the undecidability of the domino problem by exhibiting a (huge) family of tiles with adjancies rules, called Wang tiles, allowing only aperiodic tilings. Later, this example was simplified by R. Robinson [START_REF] Robinson | Undecidability and nonperiodicity for tilings of the plane[END_REF] with [START_REF] Bustos | Extended symmetry groups of multidimensional subshifts with hierarchical structure[END_REF] Wang tiles (5 up to rotation and reflection). More recently by E. Jeandel and M. Rao [START_REF] Jeandel | An aperiodic set of 11 Wang tiles[END_REF] provided a similar example with only 11 Wang tiles, and showed it is the optimal bound. A powerful method to generate aperiodic Wang tiles is through the use of so-called constant-shape substitution, where the shape of the image of tiles are the same. The result of S. Mozes [START_REF] Mozes | Tilings, substitution systems and dynamical systems generated by them[END_REF] illustrates this procedure giving sufficient conditions for a (constant-shape) substitutive subshift to be a factor of a subshift of finite type. Let us also mention that both examples of R. Robinson and Jeandel-Rao contain minimal systems that are substitutive subshifts [START_REF] Gähler | Combinatorics and topology of the Robinson tiling[END_REF][START_REF] Labbé | Substitutive structure of Jeandel-Rao aperiodic tilings[END_REF]. Motivated by all these reasons, in this thesis we will focus on the study of homomorphisms between some specific multidimensional subshifts: the substitutive ones, generated by multidimensional constant-shape substitutions. Let us recall some historical results, first in the one-dimensional context. For Z-actions, isomorphisms are closely related with automorphisms. They are the same, or the isomorphism group is an index-2 group extension of the automorphism group. So most of the properties of isomorphisms can be deduced by these ones. Automorphisms of symbolic systems present rigidity properties already in the one-dimensional case. For instance, the famous Curtis-Hedlund-Lyndon theorem [START_REF] Hedlund | Endomorphisms and automorphisms of the shift dynamical system[END_REF], ensures that any factor between subshifts is a sliding block code or an cellular automata. Actually, homomorphisms are also induced by local maps, but the center changed according to the matrix of the homomorphism (Theorem 1.9). This shows both the automorphism group and the isomorphism group of a subshift are countable and discrete subgroups on the group of self-homeomorphisms of the phase space.

The automorphism group of symbolic systems was initially studied for subshifts of finite type by G. Hedlund in [START_REF] Hedlund | Endomorphisms and automorphisms of the shift dynamical system[END_REF]. This group is infinitely generated and is very large. It contains all the finite ones, free groups, the direct sum of a countable number of copies of Z, any countable collection of finite groups, etc. In particular it is not an amenable group. However, it is residually finite, so does not contain divisible groups (like Q) or the infinite symmetric group [START_REF] Boyle | The automorphism group of a shift of finite type[END_REF][START_REF] Kim | On the automorphism groups of subshifts[END_REF]. Nevertheless, there is still no general description of the automorphism group for a given subshift nor of their generators. For example, whether the automorphism groups of the two-letter full-shift and the three-letter full-shift are algebraically isomorphic is still an open problem.

Large complexity is not enough to have a large automorphism group. In [START_REF] Bulatek | Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers[END_REF][START_REF] Donoso | On automorphism groups of Toeplitz subshifts[END_REF] the authors gave a family of Toeplitz subshifts with arbitrarily large positive entropy with trivial automorphism group. Also, the size of the automorphism group imposes no restrictions on the entropy as shown in [START_REF] Donoso | On automorphism groups of Toeplitz subshifts[END_REF]. A large class of infinite finitely generated abelian groups can be realized as the automorphism group of arbitrarily large or zero entropy Toeplitz subshift.

At the opposite, low complexity of the subshift restricts algebraic properties of the automorphism group. In [START_REF] Coven | Computing automorphism groups of shifts using atypical equivalence classes[END_REF][START_REF] Cyr | The automorphism group of a shift of linear growth: beyond transitivity[END_REF][START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] it was proved the automorphism group is virtually Z for minimal subshifts with non super-linear complexity, i.e., such that lim inf n→∞ p X (n)/n < ∞ where p X (n) denote the number of words of length n. This hypothesis implies the subshift has finitely many asymptotic pairs, i.e., two different points x, y ∈ X with a common past. The strategy in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] is based under the property that automorphisms permutes asymptotic pairs and non super-linear complexity subshifts have a finite number of asymptotic pairs. For higher complexity subshifts, the growth rate of the automorphism group is bounded by the complexity of the subshift, In particular it is amenable for a large class of zero entropy subshift as proved in [START_REF] Cyr | The automorphism group of a minimal shift of stretched exponential growth[END_REF][START_REF] Cyr | The automorphism group of a shift of slow growth is amenable[END_REF]. Beyond quantitative properties, other algebraic restrictions do exist for zero-entropy subshifts. For example, in [START_REF] Cyr | Distortion and the automorphism group of a shift[END_REF] was provided the first examples of countable groups that cannot embed (as the Baumslag-Solitar groups BS(1, n)) into the automorphism group of any zero-entropy subshift. But there still some countable groups, such as the discrete Heisenberg group, that it is not known whether they can embed into the automorphism group of a one-dimensional subshift.

Let us recall some examples of subshifts having a non trivial isomorphism, i.e., that is not an automorphism (see [START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF]): The full-shift, any palindromic subshift, such as, the sturmian shifts, the period doubling shift and the Thue-Morse shift. The first one has a huge automorphism group, not even amenable, whereas the automorphism group of the second and third ones are trivial. The automorphism group of the last one is isomorphic to Z ⊕ C 2 . These examples suggest that algebraic properties of the automorphism group does not imply the existence of non trivial isomorphisms.

Few general results are known in the multidimensional context: In [START_REF] Hochman | On the automorphism groups of multidimensional shifts of finite type[END_REF] M. Hochman proved that most of the one-dimensional properties of the automorphism group are preserved for the class of subshifts of finite type with positive entropy. Nevertheless, he presented a remarkable example: A subshift of finite type with automorphism group isomorphic to Z 2 ⊕ G, where G is locally finite and factors onto a virtually simple group. This is in contrast to the one-dimensional setting, where the group has to be residually finite. Isomorphisms of the chair tiling has been studied in [START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF]. It appears its automorphism group is trivial and its isomorphism group is a semi-direct product of Z 2 by the symmetry group of the square.

The study of automorphisms is also a classical subject in the context of ergodic theory. In the measure-theoretic framework we consider (X, µ, T, Z), where (X, F, µ) is a standard probability space, T represent the action generated by a measure-preserving transformation and automorphism are defined almost everywhere and preserved the measure µ. Let us recall some important results (we refer the reader to [START_REF] Ferenczi | Systems of finite rank[END_REF] for an overview of this theme). D. Ornstein [START_REF] Ornstein | On the root problem in ergodic theory[END_REF] proved that a mixing rank one dynamical system has a trivial (measurable) automorphism group. Later, A. del Junco [START_REF] Del Junco | A simple measure-preserving transformation with trivial centralizer[END_REF] showed that the example given by Chacon [START_REF] Chacon | Weakly mixing transformations which are not strongly mixing[END_REF], also has this property. Then, J. King and J.-P. Thouvenot [START_REF] King | A canonical structure theorem for finite joining-rank maps[END_REF] proved that for mixing systems of finite rank, its measurable automorphism group is virtually Z.

In the class of substitutive subshifts, homomorphisms have presented stronger rigidity properties than the ones mentioned above. Let us make a brief description of substitutions. These are combinatorial objects which produce infinite sequences by an iteration process. Their deep understanding took several decades. We refer [START_REF] Fogg | Substitutions in dynamics, arithmetics and combinatorics[END_REF][START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] for extensive bibliographies on the earlier developments of the subject. Using the shift as the action on these infinite sequences, we obtain the substitutive subshifts, which are the simplest nontrivial zero-entropy symbolic systems. They were introduced by W.H. Gottschalk in [START_REF] Gottschalk | Substitution minimal sets[END_REF]. Their simplicity makes them appear in many different fields of mathematics, such as, combinatorics on words (see [START_REF] Berstel | The origins of combinatorics on words[END_REF]), number theory (especially in transcendental number theory [1]), numeration systems (see [START_REF] Cobham | On the base-dependence of sets of numbers recognizable by finite automata[END_REF]), diophantine approximations (see [2]), and computer science (especially automata theory [3]).

B. Host and F. Parreau in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] gave a complete description of factors between subshifts arising from certain constant-length substitutions like for instance the Thue-Morse substitution defined by 0 → 01, 1 → 10. They proved that any measurable factor induces a continuous one, and the automorphism group is isomorphic to a direct product of Z with a finite group. Moreover, any finite group can be realized as a quotient group Aut(X, S, Z)/ S for these subshifts as proved by M. Lemańczyk and M. K. Mentzen in [START_REF] Lemańczyk | On metric properties of substitutions[END_REF]. Later, I. Fagnot [START_REF] Fagnot | Sur les facteurs des mots automatiques[END_REF] proved that the problem of whether there exists a factor map between two constant-length substitutions subshifts is decidable, using the first-order logic framework of Presburger arithmetic. Some years later, F. Durand in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] showed that linearly recurrent subshifts (in particular substitutive subshifts) have finitely many subshift factors, up to conjugacy. Also in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] it was proved that topological Cantor factors of substitutive subshifts are either substitutive subshifts or odometer systems. V. Salo and I. Törmä provided in [START_REF] Salo | Block maps between primitive uniform and Pisot substitutions[END_REF] a renormalization process of the factor maps to extend the description obtained in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF]. Next, C. Müllner and R. Yassawi [START_REF] Müllner | Automorphisms of automatic shifts[END_REF] demonstrated that any aperiodic symbolic factor of a constant-length substitutive subshift is conjugated via a letter-to-letter map to a constant-length substitutive subshift. More recently, F. Durand and J. Leroy [START_REF] Durand | Decidability of the isomorphism and the factorization between minimal substitution subshifts[END_REF] showed the decidability of the existence problem of a factor map between two minimal substitutive subshifts.

Presentation of main results

In this thesis, we study homomorphisms between multidimensional substitutive subshifts generated by constant-shape substitutions. In our context, L ∈ M(d, Z) is an integer expansion matrix, i.e, L > 1 and L -1 < 1. A constant-shape substitution ζ is a map A → A F , where A is a finite alphabet and F is a fundamental domain of L(Z d ) in Z d . The set F is called the support of the substitution. Constant-shape substitutions are a multidimensional analogue of one-dimensional constant-length substitutions. Here the "length" of the substitution is represented by the expansion matrix L. For every n > 0, any interation ζ n of the substitutions can also be obtained by a constant-shape substitution, with expansion matrix L n and support F n . At the differences with the one-dimensional case, these substitutions may not be linearly recurrent (Example 3.6), and can have topological Cantor factors that are neither expansive nor equicontinuous (Example 4.3). Some known results of the one-dimensional case are still preserved for these constant-shape substitutions, such as, they are finite extensions of a specific odometer system given by the data of the substitution (Lemma 3.9) and their maximal equicontinuous factors have a similar structure to the one-dimensional case (Proposition 3.15).

In [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF] the authors studied isomorphisms, called extended symmetries, for a class of constant shape substitutions, called bijective block substitutions. A constant-shape substitution is bijective if for any index f ∈ F we have |{ζ(a) f : a ∈ A}| = |A|. Block substitutions are constant-shape substitutions with diagonal expansion matrix and a parallelepiped support. The authors proved in [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF] that the set of matrices which define isomorphisms is a finite group. Although it is not proved in the article, it can be deduced that the group of isomorphisms is virtually generated by the shift action.

We extend the study of homomorphisms, by describing the isomorphism group for general constant-shape substitutions. As a difference with [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF], our strategy is by describing the set of nondeterministic directions (Theorem 5.2). The matrices defining isomorphisms preserve the set of nondeterministic directions (Proposition 1.12). A vector v ∈ S d-1 is said to be nondeterministic for the subshift (X, S, Z d ) if there is two different points x = y ∈ X such that are equal in the half-space H v = {t ∈ R d : t, v < 0}. This is a multidimensional analogue of asymptotic pairs [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF]. Nondeterministic directions were introduced in [START_REF] Boyle | Expansive subdynamics[END_REF] with the notion of nonexpansive subspaces to study sub-actions of a given Z d -action, for d > 1. This notion appears to be meaningful in symbolic dynamics: As an example let us mention [START_REF] Cyr | Nonexpansive Z 2 -subdynamics and Nivat's conjecture[END_REF] in which these objects were used to prove a weak version of Nivat's conjecture.

We give a description of the set of nondeterministic directions for bijective constantshape substitutions.

Theorem A (Theorem 5.2). Let ζ be an aperiodic bijective primitive constant-shape substitution. The set of nondeterministic directions of its substitutive subshift is the intersection of S d-1 with a nonempty union of limits of opposite normal cones of faces of the convex hull support of ζ n , for integers n > 0.

This theorem gives topological constraints on the set of nondeterministic directions for bijective substitutions. Under geometrical conditions of the support we get stronger properties about nondeterministic directions. A bijective constant-shape substitution is polytope when the convex hull of the compact set defined as the limit of L -n (F n ) (called digit tile), with respect to the Hausdorff metric (see Section 1.7), is a polytope. In this case the set of nondeterministic directions is much more restricted: it is a finite union of closed balls (eventually degenerated). For instance, in the two-dimensional case it cannot be a Cantor set (Corollary 5.3). In contrast with the result of M. Boyle and D. Lind in [START_REF] Boyle | Expansive subdynamics[END_REF] and M. Hochman in [START_REF] Hochman | Non-expansive directions for Z 2 actions[END_REF], where they proved any compact set of S 1 can be realized as the set of nonexpansive directions of a subshift. The work in this thesis gives the first descriptions of the set of nondeterministic directions for minimal Z d -actions.

When the rank of nondeterministic directions is maximal, thanks to the former description, we get the following constraints on the homomorphisms of substitutive subshifts.

Theorem B (Proposition 5.15 and Theorem 5.17). Let ζ be an aperiodic bijective primitive polytope substitution. If the substitutive subshift (X ζ , S, Z d ) has d linearly independent nondeterministic directions, then:

1. Any homomorphism on the substitutive subshift (X ζ , S, Z d ) is invertible.

2. The group of isomorphisms is virtually generated by the shift action.

Block substitutions are particular cases of polytope substitutions and it can be easily to check it satisfies the hypothesis on the rank of the set of nondeterministic directions, so Theorem B generalizes the ones in [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]. Actually we can relax the notion of bijectivity by the one called bijective on extremities (see Chapter 5) and reducibility (see Chapter 4) and is enough to keep the same conclusion. This hypothesis of reducibility is almost optimal because we provide in Chapter 6 an example of a constant-shape substitutive subshift with an infinite set of matrices defining isomorphisms.

The hypothesis on the rank of the set of nondeterministic directions is weak since we didn't know a substitution that does not satisfy it. We provide an algorithm to check if this hypothesis is satisfied (Lemma 5.12). Furthermore, given the result in [START_REF] Guillon | Determinism in subshifts[END_REF], the hypothesis is true for a generic family of two-dimensional bijective constant-shape substitutions. In a private communication, P. Guillon [62] mentioned this result was already proved for higher dimensions, but nowhere published.

To get Theorem B, we need some control on the block maps defining the isomorphisms. For this, we follow the strategy of B. Host and F. Parreau in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF]. Moreover we also get a strong rigidity property when the matrices commute with the expansion matrix of the substitution. Recall that substitutive subshifts are uniquely ergodic, so any continuous endomorphism induces a measurable one. We provide a partial converse:

Theorem C (Theorem 4.1, simplified version). Let (X ζ , S, Z d ) be a subshift generated by an aperiodic, primitive reduced constant-shape substitution. For every measurable endomorphism φ, there exists j ∈ Z d such that S j φ is equal to a continuous endomorphism ψ, satisfying the following two properties:

1. The endomorphism ψ has a bounded radius given by the substitution.

2. There exist an integer n > 0 and p ∈ Z d such that,

S p ψζ n 1 = ζ n 2 ψ.
Theorem C implies that, under reduciblity, any measurable endomorphism induced a continuous one. In fact, we prove the set of measurable endomorphisms is countable. Theorem C is a multidimensional analogue of the one proved by B. Host and F. Parreau in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF].

Counterexamples of Theorem C are provided by substitutive subshifts that are metrically isomorphic to their maximal equicontinuous factors. This occurs when a substitution has a combinatorial condition called coincidence [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] (see Example 4.3). The set of measurable endomorphisms of odometer systems is then uncountable and any element of the odometer system represent a measurable endomorphism via addition. So, as in the original article [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF], reducibility is an optimal hypothesis for Theorem C.

We then get some dynamical consequence of Theorem C. Under the reducibility condition, the substitutive subshifts are coalescent (Proposition 4.7), i.e., any endomorphism on the substitutive subshift is invertible. This was already known for linearly recurrent subshifts, first in the one-dimensional case in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], then in higher dimensions in [START_REF] Cortez | Linearly repetitive Delone systems have a finite number of nonperiodic Delone system factors[END_REF].

The chair tiling (see [START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF]) and the half-hex substitutive subshifts, studied in Chapter 6, are examples of substitutive subshifts having a coincidence. Hence their measurable endomorphisms form an uncountable set. Nevertheless both examples are coalescent and their automorphism groups are virtually generated by the shift action. Outside the reducibility condition, we do not know whether all aperiodic substitutive subshifts satisfy these last properties.

The half-hex substitutive subshift, mentioned before, does not satisfy neither the hypothesis of Theorem C nor Theorem B. Nevertheless we are able to describe its maximal equicontinuous factor and characterize its isomorphisms. The symmetry group of a subshift is the set of matrices M ∈ GL(d, Z) defininig an isomorphism. Thanks to this example we get the following result.

Theorem D (Theorem 6.3). There exists a minimal aperiodic subshift (in fact a substitutive one) with an infinite symmetry group.

More precisely, the isomorphism group of the half-hex substitutive subshift is isomorphic to a semidirect product between Z 2 and GL(2, Z), so its symmetry group is the largest possible. Subshifts with infinite symmetry groups have been found before, as in [START_REF] Baake | Number-theoretic positive entropy shifts with small centralizer and large normalizer[END_REF], studying their relation with topological entropy. But these examples are far from being minimal.

Finally, concerning the factors of substitutive subshifts we have the following characterization.

Theorem E (Theorem 3.22). Let (Y, S, Z d ) be an aperiodic symbolic factor of a subshift generated by an aperiodic primitive constant-shape substitution ζ. Then, there exists an aperiodic primitive constant-shape substitution ζ , with the same structure of a power of ζ, generating a system (X ζ , S, Z d ) conjugated to the symbolic factor (Y, S, Z d ). This is a multidimensional analogue of a result proved by C. Müllner and R. Yassawi [START_REF] Müllner | Automorphisms of automatic shifts[END_REF] for the one-dimensional case, which is a refinement of a result proved in [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF]. This result leaves open what can be said about other topological Cantor factors of substitutive subshifts. Example 4.3 gives a substitutive subshift with a Cantor topological factor that is neither expansive nor equicontinuous. Also Example 4.3 has a symbolic factor with a non-trivial period and an infinite phase space. This is in contrast with the one-dimensional dichotomy proved in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF].

Theorem A,Theorem B, Theorem C and Theorem E are published in [START_REF] Cabezas | Homomorphisms between multidimensional constant-shape substitutions[END_REF].

Organization of this thesis

This thesis is organized as follows. The basic definitions and background to be used throughout this thesis are introduced in Chapter 1. We recall some classical notions of topological dynamical systems, ergodic theory and symbolic dynamics. We develop the relation between homomorphisms and topological factors of dynamical systems. Then, we present the example of odometer systems and Toeplitz subshifts. We obtain a characterization to define a homomorphism between odometers. We finish this chapter with a brief survey of multidimensional constant-shape substitutions, which is where we mainly study the homomorphisms.

In Chapter 2, we study the symmetry semigroup of two-dimensional constant-base odometer systems given by a matrix L. In this case, we get a description of a bifurcation phenomenon at the level of the symmetry semigroup with respect to arithmetical relations of invariants of the matrix L. The main theorem (Theorem 2.2) shows in most cases the symmetry semigroup is the centralizer of the matrix L. This will help to get a characterization of the isomorphism semigroup of aperiodic primitive constant-shape substitutions using the relation between homomorphisms and their maximal equicontinuous factors (Lemma 1.7).

The main result in Chapter 3 is the characterization about aperiodic symbolic factors of substitutive subshifts given by an aperiodic, primitive constant-shape substitution. They are conjugate to substitutive subshifts generated by aperiodic primitive constant-shape substitutions (Theorem E). Substitutive subshifts are not necessarily linearly repetitive (Example 3.6). Nevertheless, we prove a polynomial growth on the repetitivity function for constant-shape substitutions (Lemma 3.7).

Chapter 4 is devoted to prove rigidity properties about measurable factors and homomorphisms between substitutive subshifts (Theorem C). Then, we deduce that these substitutive subshifts are coalescent (Proposition 4.7) and their automorphism group is virtually generated by the shift action (Proposition 4.8).

In Chapter 5 we describe the isomorphism group for general constant-shape substitutions. We prove it is virtually generated by the shift action (Theorem B). To do this, we relate the symmetry group with different types of supports of the substitution and non-diagonal expansion matrices, via the nondeterministic directions. We characterize the nondeterministic directions through the digit tile for some weaker version of bijective substitutions (Theorem A). Moreover, these directions are computable in terms of the combinatorics of the substitution (Corollary 5.13).

Finally, in Chapter 6 we characterize the isomorphism group for two examples of constant-shape substitutions. The first one, called table substitution, satisfied the hypothesis of the results in Chapter 4 and Chapter 5. The second example does not satisfy the hypothesis for the previous results. Nevertheless, a description of its isomorphism group is provided by Theorem D.

Résumé

Cette thèse traite des homomorphismes entre des Z d -actions topologiques T : X × Z d → X sur des espaces métriques compacts X. Un homomorphisme est une surjection continue φ :

(X, T, Z d ) → (Y, T, Z d ) telle que pour un Z d -automorphisme M ∈ GL(d, Z), on ait φ•T n = T M n •φ pour tout n ∈ Z d .
Cette notion étend le concept de facteur, lorsque M est l'identité et de conjugaison lorsque φ est un facteur inversible. Les isomorphismes (homomorphismes inversibles) sont alors des conjugaisons de Z d -actions, à une transformation de GL(d, Z) près.

Pour les Z-actions, les isomorphismes sont des flip-conjugaisons, c'est-à-dire des homéomorphismes φ tels que φ • T est égal à T • φ ou T -1 • φ. Les facteurs et les conjugaisons sont appelés respectivement endomorphismes et automorphismes lorsque les systèmes dynamiques sont identiques. Alors que les endomorphismes représentent une sorte de symétrie interne du système, comme les permutations, les homomorphismes représentent en plus des symétries des orbites, comme les rotations et les réflexions. C'est pourquoi ils sont aussi parfois appelés symétries étendues (voir par exemple [START_REF] Baake | A brief guide to reversing and extended symmetries of dynamical systems[END_REF][START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF][START_REF] Bustos | Extended symmetry groups of multidimensional subshifts with hierarchical structure[END_REF][START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]). D'un point de vue algébrique, alors que le groupe d'automorphisme est le centralisateur du groupe d'action T dans le groupe Homeo(X) des homéorphismes de X, le groupe d'isomorphisme est le normalisateur du groupe d'action T dans Homeo(X), c'est-à-dire l'ensemble des homéorphismes φ tels que φ T φ -1 = T . L'étude des homomorphismes d'un système dynamique (X, T, Z d ) est un problème classique. Les éléments du groupe T engendré par l'action sont des automorphismes du système. Le groupe d'automorphisme est donc toujours non-vide. Cependant, l'existence d'homomorphismes pour une matrice particulière M ∈ GL(d, Z) est généralement un problème ouvert. Les questions classiques concernent leurs propriétés dynamiques et algébriques en relation avec la dynamique de (X, T, Z d ). La description pour ces isomorphismes de leurs sous-groupes, de leurs quotients, de leur moyennabilité ou de leur action sur les mesures invariantes de l'action T en fonction des propriétés de (X, T, Z d ) sont des problèmes naturels. Dans un contexte général, ces questions sont largement ouvertes. Une autre question classique est la détermination des facteurs topologiques d'un système particulier. Leur description explicite peut être utilisée pour décrire la structure du système. Par certains aspects, ils contiennent des informations pertinentes et permettent de faire des calculs concrets ou d'étudier certaines structures spécifiques (par exemple en théorie spectrale [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF]). L'étude des isomorphismes de la classe particulière des systèmes minimaux où X est un ensemble de Cantor est motivée par des raisons algébriques. En effet, à tout système minimal de Cantor (X, T, Z) il existe un groupe dénombrable associé appelé groupe topologique plein et noté [[T ]]. Les éléments de ce groupe sont des homéomorphismes de l'espace X qui sont localement des puissances de la transformation T . Les homomorphismes du système (X, T, Z) induisent des homomorphismes de groupe dans [[T ]]. Réciproquement, le groupe topologique plein est un invariant complet de flip-conjugaison [START_REF] Giordano | Full groups of Cantor minimal systems[END_REF]. Il apparaît que de tels groupes pleins [[T ]] présentent des propriétés remarquables. H. Matui a prouvé dans [START_REF] Matui | Some remarks on topological full groups of Cantor minimal systems[END_REF] que son sous-groupe dérivé est simple. De plus, il a également montré que ce groupe est finiment engendré si et seulement si (X, T, Z) est conjugué à un sous-shift minimal. Ainsi, ces groupes permettent de construire des groupes simples finiment engendrés avec des propriétés algébriques inattendues. Par exemple, tout groupe topologique plein est moyennable [START_REF] Juschenko | Cantor systems, piecewise translations and simple amenable groups[END_REF], ce qui donne les premiers exemples de groupes simples finiment engendrés dans cette classe. En utilisant les isomorphismes de certains sous-shifts de complexité faible (à savoir les sous-shifts linéairement récurrentes avec des mots palindromiques arbitrairement longs comme le sous-shift substitutive de Fibonacci), V. Nekrashevych dans [START_REF] Nekrashevych | Palindromic subshifts and simple periodic groups of intermediate growth[END_REF] réussit à construire avec les groupes pleins un groupe simple finiment engendré avec une croissance intermédiaire, c'est-à-dire qui n'est ni de croissance polynomiale ni de croissance exponentielle. De plus il est de type Burnside, c'est-à-dire un groupe infini où chaque élément est périodique.

Une autre motivation pour l'étude des isomorphismes provient de la physique théorique. La découverte en 1984 par D. Schechtman et.al. [START_REF] Shechtman | Metallic phase with long-range orientational order and no translational symmetry[END_REF] d'une structure d'alliage métallique similaire à celle des cristaux idéaux a profondément influencé l'étude de la structure apériodique multidimensionnelle. Cet alliage présentait un diagramme de diffraction discret, comme les cristaux, mais avait une symétrie rotationnelle d'ordre 5, ce qui est interdit pour les cristaux idéaux. Le terme quasi-cristal a alors été inventé pour décrire ces nouvelles classes de cristaux avec des symétries "interdites", s'il n'y a pas consensus sur la définition précise d'un quasi-cristal. En substance, un quasi-cristal est un solide qui présente des points brillants nets (appelés pics de Bragg) dans son diagramme de diffraction aux rayons X, mais qui a une structure apériodique généralement manifestée par la présence d'une symétrie non-quasi-cristallogique. La présence de pics de Bragg indique la présence d'un "ordre à longue portée" dans la structure (voir [START_REF] Levine | Quasicrystals: a new class of ordered structures[END_REF]). Ces travaux ont valu à D. Schechtman le prix Wolf de physique en 1999 et le prix Nobel de chimie en 2011. On peut supposer que (au moins approximativement) un quasi-cristal est constitué d'atomes situés aux sommets d'un pavage presque périodique. Il est possible de retrouver des propriétés physiques quantitatives en étudiant un système dynamique associé au pavage. De tels systèmes ont été présentés pour la première fois par D. Rudolph dans [START_REF] Rudolph | Markov tilings of R n and representations of R n actions[END_REF]. Dès lors, une série d'articles ont été consacrés à leur étude (voir [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF][START_REF] Frank | Substitution and tiling dynamics: introduction to self-inducing structures[END_REF][START_REF] Robinson | Symbolic dynamics and tilings of R d . In Symbolic dynamics and its applications[END_REF][START_REF] Solomyak | Dynamics of self-similar tilings[END_REF] pour une vaste bibliographie sur ce sujet).

Le diagramme de diffraction est alors essentiellement le spectre ponctuel de l'action de translation correspondante [START_REF] Levine | Quasicrystals: a new class of ordered structures[END_REF]. De plus, ses symétries sont reflétées par les isomorphismes de son système dynamique [START_REF] Robinson | The dynamical theory of tilings and quasicrystallography[END_REF]. En 1982, A. Mackay [START_REF] Mackay | Crystallography and the Penrose pattern[END_REF] a publié le schéma de diffraction d'un pavage créé par R. Penrose quelques années plus tôt [START_REF] Penrose | The role of aesthetics in pure and applied mathematical research[END_REF], qui présente de grandes similitudes avec ceux découverts par D. Schechtman (voir figure 1 et figure 2).

Le pavage de Penrose est alors un bon modèle mathématique de quasi-cristaux. Il est construit avec seulement 2 tuiles, aux rotations et translations près. La génération des motifs est obtenue par une méthode algorithmique appelé substitution. Cette procédure consiste à substituer des tuiles par une union de tuiles et appliquer la même règle à ce nouveau motif. En itérant ce processus ad infinitum, cette construction fournit la plupart des pavages apériodiques les plus simples, dans le sens où ils ont la plus faible complexité possible [START_REF] Lagarias | Local complexity of Delone sets and crystallinity[END_REF].

Une autre propriété intéressante du pavage de Penrose, est que tous les pavages autorisés du système associé sont ceux qui vérifient un ensemble fini de règles locales. C'est l'analogue géométrique des sous-shifts de type fini unidimensionnels. Cependant, le système de pavage de Penrose est apériodique, à l'opposé des sous-shifts de type fini unidimensionnelles qui contiennent toujours des points périodiques. Cela met en évidence une différence fondamentale entre les propriétés combinatoires unidimensionnelles et bidimensionnelles qui sont liées à la logique et à la calculabilité. Le travail fondateur de H. Wang [START_REF] Wang | Proving theorems by pattern recognition -ii[END_REF] a déjà établi une relation entre la décidabilité de certaines formules de la logique du premier ordre et les problèmes de domino. Son étudiant R. Berger [START_REF] Berger | The undecidability of the domino problem[END_REF] a montré l'indécidabilité du problème de domino en exposant une énorme famille de tuiles avec des règles d'adjacence, appelées tuiles de Wang, ne permettant que des pavages apériodiques. Plus tard, cet exemple a été simplifié par R. Robinson [START_REF] Robinson | Undecidability and nonperiodicity for tilings of the plane[END_REF] avec 20 tuiles de Wang (5 à rotation près). Plus récemment, E. Jeandel et M. Rao [START_REF] Jeandel | An aperiodic set of 11 Wang tiles[END_REF] ont fourni un exemple similaire avec seulement 11 tuiles de Wang et ont montré que c'est la borne optimale. Une méthode puissante pour générer des tuiles de Wang apériodiques est l'utilisation de ce que l'on appelle substitution de forme constante, où la forme de l'image des tuiles est la même. Le résultat de S. Mozes [START_REF] Mozes | Tilings, substitution systems and dynamical systems generated by them[END_REF] illustre cette procédure en donnant des conditions suffisantes pour qu'un sous-shift substitutif (de forme constante) soit un facteur d'un sous-shift de type fini. Nous mentionnons également que les deux exemples de R. Robinson et Jeandel-Rao contiennent des sous-shifts substitutifs minimaux [START_REF] Gähler | Combinatorics and topology of the Robinson tiling[END_REF][START_REF] Labbé | Substitutive structure of Jeandel-Rao aperiodic tilings[END_REF].

Motivés par toutes ces raisons, nous nous concentrerons dans cette thèse sur l'étude des homomorphismes entre certains sous-shifts spécifiques: les substitutifs, générés par des substitutions multidimensionnelles de forme constante. Rappelons quelques résultats historiques dans le contexte unidimensionnel. Pour les Z-actions, les isomorphismes sont étroitement liés aux automorphismes. Le groupe d'isomorphisme est une extension de groupe d'indice au plus 2 du groupe d'automorphisme. Ainsi, la plupart des propriétés des isomorphismes peuvent être déduites de celles-ci. Les automorphismes des systèmes symboliques présentent des propriétés de rigidité déjà dans le cas unidimensionnel. Par exemple, le célèbre théorème de Curtis-Hedlund-Lyndon [START_REF] Hedlund | Endomorphisms and automorphisms of the shift dynamical system[END_REF] assure que tout facteur entre sous-shifts est un fonction de bloc glissant ou un automate cellulaire. En fait, les homomorphismes sont aussi induits par des fonctions locales, mais le centre change selon la matrice de l'homomorphisme (théorème 1.9). Cela montre que le groupe d'automorphisme et le groupe d'isomorphisme d'un sous-shift sont des sous-groupes dénombrables, discrets dans le groupe des homéomorphismes de l'espace des phases.

Le groupe d'automorphisme des systèmes symboliques a été initialement étudié pour les sous-shifts de type fini par G. Hedlund dans [START_REF] Hedlund | Endomorphisms and automorphisms of the shift dynamical system[END_REF]. Ce groupe est infiniment engendré et contient de nombreux subgroupes: n'importe quel groupe fini, les groupes libres, la somme directe d'un nombre dénombrable de copies de Z, toute collection dénombrable de groupes finis, etc. Cependant, il est résiduellement finie, et ne contient donc pas de groupes divisibles (comme Q) ou le groupe symétrique infini [START_REF] Boyle | The automorphism group of a shift of finite type[END_REF][START_REF] Kim | On the automorphism groups of subshifts[END_REF]. Néanmoins, il n'existe pas de description générale du groupe d'automorphisme ni de leurs générateurs pour un sous-shift donné. Par exemple, la question de savoir si les groupes d'automorphisme du full-shifts sur 2 et 3 lettres sont algébriquement isomorphes est un problème encore ouvert.

Une grande complexité n'est pas suffisante pour avoir un grand groupe d'automorphisme. Dans [START_REF] Bulatek | Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers[END_REF][START_REF] Donoso | On automorphism groups of Toeplitz subshifts[END_REF], les auteurs ont donné une famille de sous-shifts Toeplitz d'entropie positive arbitrairement grande presentant un groupe d'automorphisme trivial. De plus, la taille du groupe d'automorphisme n'impose aucune restriction sur l'entropie, comme prouvé dans [START_REF] Donoso | On automorphism groups of Toeplitz subshifts[END_REF]. Une grande classe de groupes abéliens infinis finiment engendrés peut être réalisée comme le groupe d'automorphisme d'un sous-shift Toeplitz d'entropie arbitrairement grande ou nulle.

À l'inverse, la complexité faible du sous-shift restreint les propriétés algébriques du groupe d'automorphisme. Il est prouvé dans [START_REF] Coven | Computing automorphism groups of shifts using atypical equivalence classes[END_REF][START_REF] Cyr | The automorphism group of a shift of linear growth: beyond transitivity[END_REF][START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] que le groupe d'automorphisme est virtuellement Z pour les sous-shifts minimaux de complexité non superlinéaire, c'està-dire tels que lim inf n→∞ p X (n)/n < ∞ où p X (n) correspond au nombre de mots de longueur n. Deux suites différents x, y ∈ X sont dites asymptotiques lorsqu'elles ont le même passé. La stratégie de [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] est basée sur la propriété que les automorphismes permutent les paires asymptotiques et les sous-shifts de complexité non super-linéaire ont un nombre fini de paires asymptotiques. Pour les sous-shifts de complexité plus élevée, le taux de croissance du groupe d'automorphisme est limité par la complexité du sous-shift. En particulier, il est moyennable pour une grande classe de sous-shifts à entropie nulle, comme prouvé dans [START_REF] Cyr | The automorphism group of a minimal shift of stretched exponential growth[END_REF][START_REF] Cyr | The automorphism group of a shift of slow growth is amenable[END_REF]. Au-delà des propriétés quantitatives, d'autres restrictions algébriques existent pour les sous-shifts à entropie nulle. Par exemple, dans [START_REF] Cyr | Distortion and the automorphism group of a shift[END_REF] a été fourni les premiers exemples de groupes dénombrables qui ne peuvent pas être des sous-groupes d'automorphisme (comme les groupes de Baumslag-Solitar BS ( L'étude des homomorphismes est également un sujet classique dans le contexte de la théorie ergodique. Dans le cadre de la théorie de la mesure, nous considérons le systèmes dynamiques (X, µ, T, Z), où (X, F, µ) est un espace de probabilité standard, T est l'action générée par une transformation préservant la mesure µ et les automorphismes sont définis presque partout et préservent la mesure µ. Rappelons quelques résultats importants (nous renvoyons le lecteur à [START_REF] Ferenczi | Systems of finite rank[END_REF] pour un aperçu de ce thème). D. Ornstein [START_REF] Ornstein | On the root problem in ergodic theory[END_REF] a prouvé qu'un système dynamique mélangeant de rang un possède un groupe d'automorphisme mesurable trivial. Plus tard, A. del Junco [START_REF] Del Junco | A simple measure-preserving transformation with trivial centralizer[END_REF] a montré que l'exemple donné par Chacon [START_REF] Chacon | Weakly mixing transformations which are not strongly mixing[END_REF], possède également cette propriété. Ensuite, J. King et J.-P. Thouvenot [START_REF] King | A canonical structure theorem for finite joining-rank maps[END_REF] ont prouvé que le groupe d'automorphisme mesurable est virtuellement Z pour les systèmes mélangeant de rang fini.

Dans la famille des sous-shifts substitutifs, les homomorphismes présentent des propriétés de rigidité plus fortes que celles mentionnées ci-dessus. Faisons une brève description des substitutions. Ce sont des objets combinatoires qui produisent des suites infinies par un processus d'itération. Leur compréhension profonde a pris plusieurs décennies. Nous renvoyons à [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF][START_REF] Fogg | Substitutions in dynamics, arithmetics and combinatorics[END_REF] pour des bibliographies complètes sur les sujet. En utilisant le shift comme action sur ces suites infinies, nous obtenons les sous-shifts substitutifs, qui sont les plus simples systèmes symboliques non triviaux d'entropie nulle. Ils ont été introduits par W.H. Gottschalk dans [START_REF] Gottschalk | Substitution minimal sets[END_REF]. Leur simplicité les fait apparaître dans de nombreux domaines des mathématiques, tels la combinatoire des mots (voir [START_REF] Berstel | The origins of combinatorics on words[END_REF]), la théorie des nombres (en particulier dans la théorie des nombres transcendants [1]), les systèmes de numération (voir [START_REF] Cobham | On the base-dependence of sets of numbers recognizable by finite automata[END_REF]), les approximations diophantiennes (voir [2]), et l'informatique (en particulier la théorie des automates [3]).

B. Host et F. Parreau dans [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] ont donné une description complète des facteurs entre les sous-shifts résultant de certaines substitutions de longueur constante, comme par exemple la substitution de Thue-Morse définie par 0 → 01, 1 → 10. Ils ont prouvé que tout facteur mesurable induit un facteur continu, et que le groupe d'automorphisme est isomorphe à un produit direct de Z avec un groupe fini. De plus, tout groupe fini peut être réalisé comme un groupe quotient Aut(X, S, Z)/ S pour ces sous-shifts comme l'ont prouvé M. Lemańczyk et M. K. Mentzen dans [START_REF] Lemańczyk | On metric properties of substitutions[END_REF]. Plus tard, I. Fagnot [START_REF] Fagnot | Sur les facteurs des mots automatiques[END_REF] a prouvé que le problème de savoir s'il existe une facteur entre deux sous-shifts substitutifs de longueur constante est décidable, en utilisant le cadre de la logique du premier ordre de l'arithmétique de Presburger. Quelques années plus tard, F. Durand dans [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] a montré que les sous-shifts linéairement récurrents (en particulier les sous-shifts substitutifs) ont un nombre fini de facteurs symboliques, à conjugaison près. De plus, dans [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], il a été prouvé que les facteurs de Cantor topologiques des sous-shifts substitutifs sont soit des sous-shifts substitutifs, soit des odomètres. V. Salo et I. Törmä ont fourni dans [START_REF] Salo | Block maps between primitive uniform and Pisot substitutions[END_REF] un processus de renormalisation des facteurs étandant la description obtenue dans [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF]. Ensuite, C. Müllner et R. Yassawi [START_REF] Müllner | Automorphisms of automatic shifts[END_REF] ont démontré que tout facteur symbolique apériodique d'un sous-shift substitutif de longueur constante est conjugué par une fonction lettre à lettre à un sous-shift substitutif de longueur constante. Plus récemment, F. Durand et J. Leroy [START_REF] Durand | Decidability of the isomorphism and the factorization between minimal substitution subshifts[END_REF] ont montré la décidabilité du problème d'existence d'une facteur entre deux sous-shifts substitutifs minimaux.

Présentation des principaux résultats

Dans cette thèse, nous étudions les homomorphismes entre les sous-shifts substitutifs multidimensionnells générées par les substitutions à forme constante. Dans notre contexte, L ∈ M(d, Z) est une matrice d'expansion entière, c'est-à-dire L > 1 et L -1 < 1. Une substitution de forme constante est une fonction ζ : A → A F , où A est un alphabet fini et F est un domaine fondamental de L(Z d ) dans Z d . L'ensemble F est appelé le support de la substitution. Les substitutions de forme constante sont les analogues multidimensionnels des substitutions de longueur constante unidimensionnel. Ici, la "longueur" de la substitution est représentée par la matrice d'expansion L. Pour chaque n > 0, toute itération ζ n des substitutions peut également être obtenue par une substitution de forme constante de une matrice d'expansion L n et de support F n . À la différence du cas unidimensionnel, ces substitutions peuvent ne pas être linéairement récurrentes (exemple 3.6) et peuvent avoir des facteurs de Cantor non expansifs ni équicontinus (exemple 4.3). Des résultats connus du cas unidimensionnel sont encore préservés pour ces substitutions de forme constante: elles sont des extensions finies d'un odomètre spécifique donné par la substitution (lemme 3.9) et leurs facteurs équicontinus maximaux ont une structure similaire au cas unidimensionnel (proposition 3.15).

Dans [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF], les auteurs ont étudié les isomorphismes, appelés symétries étendues, pour une classe de substitutions de forme constante, appelée substitutions par blocs bijectifs. Une substitution de forme constante est bijective si tout indice f ∈ F vérifie |{ζ(a) f : a ∈ A}| = |A|. Les substitutions de blocs sont des substitutions de forme constante avec une matrice d'expansion diagonale et un support parallélépipédique. Les auteurs ont prouvé dans [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF] que l'ensemble des matrices qui définissent les isomorphismes est un groupe fini. Bien que cela ne soit pas prouvé dans l'article, on peut en déduire que le groupe des isomorphismes est virtuellement engendré par l'action du shift.

Nous étendons l'étude des homomorphismes en décrivant le groupe d'isomorphisme pour les substitutions générales de forme constante. À la différence de [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF], notre stratégie consiste à décrire l'ensemble des directions non déterministes (théorème 5.2). Les matrices définissant les isomorphismes préservent l'ensemble des directions non déterministes (proposition 1.12). Un vecteur v ∈ S d-1 est dit nondéterministe pour le sous-shift (X, S, Z d ) s'il existe deux points différents x = y ∈ X qui sont égaux dans le demi-espace

H v = {t ∈ R d : t, v < 0}.
C'est un analogue multidimensionnel des paires asymptotiques [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF]. Les directions non déterministes ont été introduites dans [START_REF] Boyle | Expansive subdynamics[END_REF] avec la notion de sous-espace non expansif pour étudier les sous-actions d'une action Z d donnée, pour d > 1. Cette notion est important en dynamique symbolique. À titre d'exemple, mentionnons [START_REF] Cyr | Nonexpansive Z 2 -subdynamics and Nivat's conjecture[END_REF] où ces objets ont été utilisés pour prouver une version faible de la conjecture de Nivat.

Nous donnons une description de l'ensemble des directions non déterministes pour les substitutions bijectives de forme constante.

Théorème A (théorème 5.2). Soit ζ une substitution apériodique bijective primitive de forme constante. L'ensemble des directions non déterministes de son sous-shift substitutif est l'intersection de S d-1 avec une union non vide des limites des cônes normaux des faces de l'enveloppe convexe du support de ζ n , pour les entiers n > 0.

Ce théorème donne des contraintes topologiques sur l'ensemble des directions non déterministes pour les substitutions bijectives. Sous des conditions géométriques du support, nous obtenons des propriétés plus fortes sur les directions non déterministes. Une substitution bijective de forme constante est polytope si l'enveloppe convexe de l'ensemble limite de L -n (F n ) (appelé digit tile, voir la section 1.7) est un polytope. Dans ce cas, l'ensemble des directions non déterministes est beaucoup plus restreint: c'est une union finie de boules fermées (éventuellement dégénérées). Par exemple, dans le cas bidimensionnel, il ne peut pas être un ensemble de Cantor (corollaire 5.3). Ceci diffère du le résultat prouvé par M. Boyle et D. Lind [START_REF] Boyle | Expansive subdynamics[END_REF] et M. Hochman [START_REF] Hochman | Non-expansive directions for Z 2 actions[END_REF]. Ils assurent que tout ensemble compact de S 1 peut être réalisé comme l'ensemble des directions non expansifs d'un sousshift. Le travail de cette thèse donne les premiers description de l'ensemble des directions non déterministes pour des Z d -actions minimaux.

Lorsque le rang des directions non déterministes est maximal, grâce à la description précédente, nous obtenons les contraintes suivantes sur les homomorphismes des sous-shifts substitutifs.

Théorème B (proposition 5.15 et théorème 5.17). Soit ζ une substitution polytope primitive bijective apériodique. Si le sous-shift substitutif (X ζ , S, Z d ) a d directions non déterministes linéairement indépendantes, alors:

1. tout homomorphisme du sous-shift substitutif (X ζ , S, Z d ) est inversible.

Le groupe d'isomorphisme est virtuellement engendré par l'action du shift.

Les substitutions de blocs sont des cas particulier des substitutions polytopes et il est simple de vérifier qu'elles vérifient l'hypothèse sur le rang de l'ensemble des directions non déterministes. Ainsi le théorème B généralise les résultats de [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]. En fait, nous pouvons affaiblir la notion de bijectivité par les notions de bijectivité sur les extrémités (voir le chapitre 5) et réductibilité (voir le chapitre 4). Cela suffit pour garder la même conclusion. Cette hypothèse de réductibilité est presque optimale car nous fournissons dans l'exemple 6 un exemple de sous-shift substitutif de forme constante dont l'ensemble des matrices définissant les isomorphismes est infini.

L'hypothèse sur le rang de l'ensemble des directions non déterministes est faible puisque nous ne connaissions pas de substitution qui ne la satisfasse pas. Nous fournissons un algorithme pour vérifier si cette hypothèse est satisfaite (lemme 5.12). De plus, par le résultat dans [START_REF] Guillon | Determinism in subshifts[END_REF], cette hypothèse est vérifiée pour une famille générique de substitutions bijectives de forme constante bidimensionnelles. Dans une communication privée, P. Guillon [62] a mentionné que ce résultat étant également pour les dimensions supérieures. Malheureusement la preuve n'a jamais été publié.

Pour obtenir le théorème B, nous avons besoin d'un certain contrôle sur les fonctions de blocs définissant les isomorphismes. Pour cela, nous suivons la stratégie de B. Host et F. Parreau dans [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF]. Nous obtenons également une propriété remarquable de rigidité lorsque les matrices commuent avec la matrice d'expansion de la substitution. Nous rappelons que les sous-shifts substitutifs sont uniquement ergodiques, donc tout endomorphisme continu induit un endomorphisme mesurable. Nous fournissons une réciproque partielle.

Théorème C (théorème 4.1, version simplifiée). Soit (X ζ , S, Z d ) un sous-shift généré par une substitution apériodique primitive de forme constante. Pour tout endomorphisme mesurable φ, il existe j ∈ Z d tel que S j φ est égal à un endomorphisme continu ψ, satisfaisant les deux propriétés suivantes:

1. l'endomorphisme ψ a un rayon borné par la substitution.

Il existe des entiers

n > 0 et p ∈ Z d tels que S p ψζ n 1 = ζ n 2 ψ.
Le théorème C implique que pour une substitution réduite tout endomorphisme mesurable induit un endomorphisme continu. Ainsi l'ensemble des endomorphismes mesurés est un ensemble dénombrable. Le théorème C est un analogue multidimensionnel de celui prouvé par B. Host et F. Parreau dans [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF].

Des contre-exemples au théorème C sont fournis par des sous-shifts substitutifs qui sont métriquement isomorphes à leur facteur équicontinu maximal. Cela se produit lorsqu'une substitution possède une condition combinatoire appelée coïncidence [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] (voir l'exemple 4.3). L'ensemble des endomorphismes mesurés des odomètres est alors indénombrable et tout élément de l'odomètre représente un endomorphisme mesurable par addition. Ainsi, comme dans l'article original [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF], la réductibilité est une hypothèse optimale pour le théorème C.

Nous obtenons ensuite quelques conséquence dynamique du théorème C. Sous la condition de réductibilité, les sous-shifts substitutifs sont coalescents (proposition 4.7), c'està-dire que tout endomorphisme du sous-shift est inversible. Ceci était déjà connu pour les sous-shifts linéairement récurrents, d'abord pour le cas unidimensionnel dans [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], puis par les dimensions supérieures dans [START_REF] Cortez | Linearly repetitive Delone systems have a finite number of nonperiodic Delone system factors[END_REF].

Le pavage de la chaise (voir [START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF]) et les sous-shifts substitutifs du demi-hexagone, étudiées dans le chapitre 6, sont des exemples de sous-shifts substitutifs avec coïncidence. Par conséquent, leurs endomorphismes mesurés forment un ensemble indénombrable. Néanmoins, les deux exemples sont coalescents et leurs groupes d'automorphisme sont virtuellement engendrés par l'action de shift. Sans la condition de réductibilité, nous ne savons pas si tous les sous-shift substitutifs apériodiques satisfont ces dernières propriétés.

Le sous-shift substitutif du demi-hexagone, mentionné précédemment, ne satisfait pas les hypothèses des théorèmes C et B. Néanmoins, nous sommes capables de décrire son facteur équicontinu maximal et de caractériser ses isomorphismes. Le groupe de symétrie d'un sous-shift est l'ensemble des matrices M ∈ GL(d, Z) définissant un isomorphisme. Grâce à cet exemple, nous obtenons le résultat suivant.

Théorème D (théorème 6.3). Il existe un sous-shift apériodique minimal (en fait un sousshift substitutif) avec un groupe de symétrie infini.

Plus précisément, le groupe d'isomorphisme du sous-shift substitutif du demi-hexagone est isomorphe au produit semidirect de Z 2 avec GL(2, Z). Son groupe de symétrie est donc le plus grand possible. Des sous-shifts avec des groupes de symétrie infini ont été trouvés auparavant, comme dans [START_REF] Baake | Number-theoretic positive entropy shifts with small centralizer and large normalizer[END_REF], en étudiant leurs relation avec l'entropie topologique. Mais ces exemples sont loin d'être minimaux.

Enfin, concernant les facteurs de sous-shifts substitutifs, nous avons la caractérisation suivante.

Théorème E (théorème 3.22). Soit (Y, S, Z d ) un facteur symbolique apériodique d'un sous-shift (X ζ , S, Z d ) généré par une substitution ζ apériodique primitive de forme constante. Alors, il existe une substitution primitive apériodique de forme constante ζ , de la même structure qu'une puissance de ζ, dont le système (X ζ , S, Z d ) est conjugué au facteur symbolique (Y, S, Z d ).

Ceci est un analogue multidimensionnel d'un résultat prouvé par C. Müllner et R. Yassawi [START_REF] Müllner | Automorphisms of automatic shifts[END_REF] pour le cas unidimensionnel, qui est un raffinement d'un résultat prouvé dans [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF]. En revanche, ce résultat ne dit rien sur les autres facteurs topologiques sur un Cantor des sous-shifts substitutifs. L'exemple 4.3 donne un exemple de sous-shift substitutif avec un facteur topologique de Cantor qui n'est ni expansif ni équicontinu. De même, l'exemple 4.3 possède un facteur symbolique avec une période non triviale et une espace des phases infini. Ceci est en contraste avec la dichotomie unidimensionnelle prouvée dans [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF].

Les théorèmes A, B, C et E sont publiés dans [START_REF] Cabezas | Homomorphisms between multidimensional constant-shape substitutions[END_REF].

Organisation de cette thèse

Cette thèse est organisée comme suit. Les définitions de base et le contexte qui seront utilisés tout au long de cette thèse sont introduits dans le chapitre 1. Nous rappelons quelques notions classiques de systèmes dynamiques topologiques, de théorie ergodique et de dynamique symbolique. Nous développons la relation entre les homomorphismes et les facteurs topologiques des systèmes dynamiques. Ensuite, nous présentons l'exemple des odomètres et des sous-shifts de Toeplitz. Nous obtenons une caractérisation pour des homomorphismes entre odomètres. Nous terminons ce chapitre par un bref survol des substitutions multidimensionnelles de forme constante, dont nous étudions principalement les homomorphismes.

Dans le chapitre 2, nous étudions le semigroupe de symétrie des odomètres bidimensionnels de base constante donnée par une matrice L. Dans ce cas, nous obtenons une description d'un phénomène de bifurcation au niveau du semigroupe de symétrie par rapport aux relations arithmétiques des invariants de la matrice L. Le théorème principal (théorème 2.2) montre que dans la plupart des cas le semigroupe de symétrie est le centralisateur de la matrice L. Cela aidera à obtenir une caractérisation du semigroupe normalisateur des substitutions apériodiques primitives de forme constante en utilisant la relation entre les homomorphismes et leurs facteurs équicontinus maximaux (lemme 1.7).

Le résultat principal du chapitre 3 est la caractérisation des facteurs symboliques apériodiques des sous-shifts substitutifs donnés par une substitution apériodique primitive de forme constante. Ils sont conjugués aux sous-shifts substitutifs générés par des substitutions apériodiques primitives de forme constante (théorème E). Les sous-shifts substitutifs ne sont pas nécessairement linéairement répétitives (exemple 3.6). Néanmoins, nous donnos une croissance polynomiale sur la fonction de répétitivité pour les substitutions de forme constante (lemme 3.7).

Le chapitre 4 est consacré à la preuve de propriétés de rigidité des facteurs mesurés et les homomorphismes entre sous-shifts substitutifs (théorème C). Ensuite, nous déduisons que ces sous-shifts substitutifs sont coalescentes (proposition 4.7) et que leurs groupes d'automorphisme est virtuellement engendrés par l'action de shift (proposition 4.8).

Dans le chapitre 5, nous décrivons le normalisateur des substitutions générales de forme constante. Nous prouvons que le normalisateur est virtuellement engendré par l'action de shift (théorème B). Pour ce faire, nous relions le normalisateur à différents types de supports des substitutions, via les directions non-déterministes. Nous caractérisons les directions non-déterministes par le digit tile pour une version affaiblie de substitutions bijectives (théorème A). De plus, ces directions sont calculables en termes de combinatoire de la substitution (théorème 5.13).

Enfin, dans le chapitre 6, nous caractérisons le groupe normalisateur pour deux exemples de substitutions de forme constante. Le premier, appelé substitution de la table, satisfait les hypothèses des résultats des chapitres 4 et 5. Le pavage du demi-hexagone ne satisfait pas les hypothèses des résultats précédents. Néanmoins, une description de son groupe d'isomorphisme est fournie par le théorème D.

Chapter 1

Definitions and background

In this chapter, we fix some notations, definitions and show some general properties to be used throughout this thesis. We start with some notions on discrete, convex and fractal geometry. Then we recall some classical notions of topological dynamical systems and the ones of homomorphisms between them. In particular, we will see some relations between homomorphisms and topological factors of dynamical systems (Lemma 1.6). We also recall some basics of ergodic theory, symbolic dynamics, and nondeterministic directions (also called nonexpansive half-spaces).

In Section 1.2 we define the central objects of study of this thesis, which are homomorphisms between topological Z d -actions T : X × Z d → X on a compact metric space X. This types of morphisms have been studied in both one and higher dimensions actions. For Z-actions, the isomomorphism group (the group of invertible homomorphisms) of a topological dynamical system is either the automorphism group Aut(X, T, Z d ), or an index-2 extension of Aut(X, T, Z d ). Therefore, the study of isomorphisms for Z-actions focuses on the existence of an isomorphism, which are sometimes called reversors (see [START_REF] Baake | A brief guide to reversing and extended symmetries of dynamical systems[END_REF] for a brief guide to the study of these isomorphisms). Since GL(d, Z) is an infinite group for d > 1, the relation between automorphisms and isomorphisms becomes less clear. Isomorphisms have been studied for particular subshifts [START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF] and more recently for a class of substitutive subshifts under strong geometrical and combinatorial restrictions [START_REF] Bustos | Extended symmetry groups of multidimensional subshifts with hierarchical structure[END_REF][START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]. In this thesis we extend the study of homomorphisms by describing the isomorphism group (called the normalizer group) for general constant-shape substitutions that are defined in this chapter.

We then describe the nondeterministic directions for a subshift. This notion was introduced in [START_REF] Guillon | Determinism in subshifts[END_REF] to the study of two-dimensional subshifts. In [START_REF] Cyr | Nonexpansive Z 2 -subdynamics and Nivat's conjecture[END_REF] they were used to prove a weak version of Nivat's conjecture. This notion is motivated by the work of M. Boyle and D. Lind, about nonexpansive subspaces. When the space X is infinite such subspaces always exist [START_REF] Boyle | Expansive subdynamics[END_REF]Theorem 3.7]. In fact, they can be described only by hyperplanes [17, Theorem 3.6], hence the term of nondeterministic directions. In this thesis, nondeterministic directions are characterized for bijective substitutions (Theorem 5.2) and used to describe the isomorphisms for a big family of substitutive subshifts (Theorem 5.17). We also present the example of odometer systems and Toeplitz sequences in Section 1.6. Odometer systems are the most natural equicontinuous systems in the study of minimal Cantor systems. In fact, they are the maximal equicontinuous factor for a big family of symbolic systems, such as, some substitutions and Toeplitz sequences. Toeplitz subshifts are symbolic sytems that are the orbit closures of the regular quasi-periodic points of the subshift. We refer to [START_REF] Downarowicz | Survey of odometers and Toeplitz flows[END_REF][START_REF] Cortez | Z d Toeplitz arrays[END_REF][START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF] for the study of odometer systems for different actions. We also obtain a characterization of homomorphisms between two odometer systems (Lemma 1.14), useful to describe isomorphisms for substitutive subshifts.

We finish this chapter with a brief survey of multidimensional constant-shape substitutions. They represent the simplest nontrivial zero-entropy symbolic systems, since they are generated by finite data. By this fact, ergodic and topological properties of substitution dynamical systems have been extensively studied. They were introduced by W.H. Gottschalk in [START_REF] Gottschalk | Substitution minimal sets[END_REF] (see [START_REF] Fogg | Substitutions in dynamics, arithmetics and combinatorics[END_REF][START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] for a good bibliography on this subject). We also provide some good finite sets useful to the structure of the sequences in substitutive subshifts. As a corollary, we get a canonical form of symbolic factors of constant-shape substitutions. We will call a sequence of finite sets

Basics on discrete, convex and fractal geometry

(F n ) n>0 ⊆ Z d a F ølner sequence if for all n ∈ Z d we have that lim n→∞ |F n ∆(n + F n )| |F n | = 0. 1
For any r > 0 and F Z d we denote

F •r as the set of all elements f ∈ F such that f + (B(0, r) ∩ Z d ) ⊆ F , i.e., F •r = {f ∈ F : f + (B(0, r) ∩ Z d ) ⊆ F }.
Note that the Følner assumption implies for any r > 0

lim n→∞ |F •r n | |F n | = 1. (1.1)

Convex geometry

In the following we will give some basics on convex geometry that we will be used in the rest of the thesis. We refer to [START_REF] Auslender | Asymptotic cones and functions in optimization and variational inequalities[END_REF] for a survey of results about this field.

A set C ⊆ R d is said to be convex if for all x, y ∈ C the set [x, y] = {z ∈ R d : z = tx + (1 -t)y, t ∈ [0, 1]} is included in C.
Recall that the image of a convex set under an affine map is also a convex set, and the intersection of an arbitrary family of convex sets is also a convex set. This leads to the notion of the convex hull of a set.

If A ⊆ R d we define the convex hull of A, denoted by conv(A), as the intersection of all convex sets containing A.

A set S ⊆ R d is an affine set if for any x, y ∈ S the line {tx + (1 -t)y : t ∈ R} is contained in S. For any set A ⊆ R d we define the affine hull of A, denoted by Aff(A), as the intersection of all affine sets containing A.

A fundamental characterization of convex sets is provided by Carathéodory's theorem.

Theorem 1.1 (Carathéodory's theorem). For any A ⊆ R d , any element of conv(A) can be represented as a convex combination of no more than (d + 1) elements of A.

We now recall some basic topological concepts associated with convex sets. We say that a point x ∈ A is relative interior for A, if A contains the intersection of a ball centered at x with Aff(A), i.e., ∃r > 0, B(x, r) ∩ Aff(A) ⊆ A. The set of all relative interior points of A is called its relative interior, and is denoted by ri(A). We can also define the relative boundary ∂ ri (A) as the set difference of the closure and the relative interior, i.e.,

∂ ri (A) = cl(A) \ ri(A).
An important notion for convex sets are the supporting hyperplanes. We now recall some basic notions about cones and polyhedral sets. A nonempty set C ⊆ R d is said to be a cone if for every x ∈ C, the set C contains the positive ray R + x = {tx : t > 0} spanned by x. A translation of a cone by a non zero vector is called an affine convex cone. A cone C ⊆ R d is said to be finitely generated if it can be written as

If C ⊆ R d is a closed convex set,
C = p i=1 tu i : u i ∈ R d , t i ≥ 0, i = 1, . . . , p .
For a given nonempty set A ⊆ R d , the smallest cone containing the set A is called the positive hull (or conical hull ) of A. It is the smallest cone containing the set A, and is given by cone

A = {tx : x ∈ A, t > 0} ∪ {0}.
The positive hull is also said to be the cone generated by A.

A set P ⊆ R d is called polyhedral if it has the form P = x ∈ R d : u i , x ≤ a i , i = 1, . . . , p ,
where

u i ∈ R d , a i ∈ R.
The following is characterization of polyhedral sets.

Theorem 1.2 (Minkowski-Weyl Theorem). A cone C is polyhedral if and only if it is finitely generated.

Convex sets can be represented, but it requires the notion of extreme points and extreme rays.

A point x in a convex set C is called an extreme point, if can not be written as the convex combination of two different points in C, i.e., if

x is equal to tu + (1 -t)v for some 0 ≤ t ≤ 1, with u, v ∈ C, then u = v = x.
We denote by Ext(C) the set of the extreme points of a convex set C. Extreme points are special cases of faces of a convex set. A compact convex set is called a polytope if it has a finite number of extreme points.

A convex set F ⊆ C is called a face of C if for every x ∈ F and every y, z ∈ C such that x = ty + (1 -t)z, with 0 < t < 1, we have that y, z ∈ F . The dimension of a face F of C is the dimension of its affine hull. The 0-dimensional faces of C are exactly the extreme points of C, and the bounded 1-dimensional faces are called segments or edges. An extreme ray of a convex set C is the direction of a half-line that is a face of C. A useful result about representation of closed convex sets in R d is the following Theorem 1.3 (Krein-Milman theorem for unbounded convex sets). If a nonempty closed convex set C ⊆ R d has at least one extreme point, i.e., does not have an affine line. Then C can be written as the sum of the convex hull of its extreme points and the cone generated by its extreme rays. Some useful notion for convex sets corresponds to the normal cone. Let F be a nonempty face of a closed convex polytope C. The opposite normal cone

2 NF (C) of C at F is defined as NF (C) = {v ∈ R d : min t∈C v, t = v, p , ∀p ∈ F }.
The opposite normal fan of C is the collection of all opposite normal cones of C:

N (C) = { NF (C) : F is a proper face of C}.
The following are simple statements on the normal fan

• If F is a face of C, then dim( NF (C)) = d -dim(F ). • If F is a face of G, which is a face of C, then NG (C) is a face of NF (C).
• The set

F face of C NF (C) is equal to R d .

Fractal Geometry

In the following we present some definitions and some properties satisfied for some fractals sets which are defined by iterated function systems or IFS. We refer to [START_REF] Kirat | Remarks on self-affine fractals with polytope convex hulls[END_REF][START_REF] Strichartz | Geometry of self-affine tiles[END_REF][START_REF] Vince | Digit tiling of Euclidean space[END_REF] for some results that will be used throughout this thesis. Let C(R d ) denote the collection of all nonempty compact subsets of R d . The Hausdorff metric H on C(R d ) is defined as follows:

H(A, B) = inf{ε : A ⊆ B ε ∧ B ⊆ A ε }, where A ε = {t ∈ R d : t -y ≤ ε, for some y ∈ A}. We have that (C(R d ), H) is a complete metric space. A map f : R d → R d is said to be a contraction if there exists 0 < c < 1 such that f (x) -f (y) ≤ c x -y for all x, y ∈ R d . Let {f i } N
i=1 be a set of contraction maps on R d , and define the map

F : C(R d ) → C(R d ) A → N i=1 f i (A)
This map is a contraction on (C(R d ), H), so by the Banach fixed-point Theorem, there exists a unique set

T ∈ C(R d ) (called digit tile) such that T = N i=1 f i (T ).
A way to approximate this set is by iterations

T = lim n→∞ F n (T 0 ), (1.2)
where T 0 is an arbitrary compact set in R d and the limit is with respect to the Hausdorff metric.

Since the convex hull of a compact set in R d is compact, the map conv : C(R d ) → C(R d ) which gives for any set A ∈ C(R d ) its convex hull is well defined, and is well known to be continuous.

Topological dynamical systems

In this section, we will present the basic definitions and some properties of topological dynamical systems. We also define the central object of study of this thesis, which are homomorphisms between topological dynamical systems, and present some basic results about them. We finish this section with a survey on some results about the compatibility of homomorphisms between topological factors of topological dynamical systems. We mention [4] for an extensive bibliography of this area.

Basic definitions

A topological dynamical system is a triple (X, T, G), where (X, ρ) is a compact metric space, G is a group of homeomorphisms of the space X into itself, and T : X × G → X is a continuous map, satisfying T (x, e) = x, and T (T (x, g), h) = T (x, gh) for all x ∈ X, and g, h ∈ G. We will denote T g to the homeomorphism T (•, g).

If (X, ρ) is a compact metric space, we denote Homeo(X) the group of all homeomorphisms from X to itself, and if T ∈ Homeo(X), we use (X, T, Z) to denote the topological dynamical system (X, T, {T n : n ∈ Z}). Similarly, if T 1 , . . . , T d are d commuting homeomorphisms on X, we denote (X, T, Z d ) to denote the topological dynamical system (X, T, {T 1 , . . . , T d } ).

For a point x ∈ X, we define its orbit as the set

O(x, G) = {T g (x) : g ∈ G}. If A ⊆ X, we say that A is G-invariant if for all x ∈ A, O(x, G) is included in A. If (X, T, G) is a topological dynamical system, a subset K ⊆ X is called a minimal set if K is closed, nonempty, G-invariant, and has no proper closed nonempty invariant subsets, i.e., if N ⊆ K is closed and G-invariant, then N = ∅ or N = K. In this case, we say that (K, T | K , G) is a minimal system, where T | K : K × G → K corresponds to the restriction of T in K.
It is easy to see that a system is minimal if and only if it it is the closure orbit of all of its points.

An important type of topological dynamical systems are the so-called equicontinuous systems. A topological dynamical system (X, T, Z d ) is said to be equicontinuous if the set of maps {T n : n ∈ Z d } forms an equicontinuous family of homeomorphisms. The equicontinuous systems are, in some sense, the simplest dynamical systems. In fact, there exists a complete characterization of them [4].

Homomorphisms between topological dynamical systems

In the following, we define the homomorphisms between topological dynamical systems, which are the central object of study on this thesis. Homomorphisms represent internal symmetries of a given topological dynamical system, such as rotations and reflections. Invertible homormophisms, which will be called isomorphisms, are then conjugacies of Z d -actions, up to a GL(d, Z)-transformation. We refer to [START_REF] Baake | A brief guide to reversing and extended symmetries of dynamical systems[END_REF] for a brief guide to homomorphisms both for one-dimensional systems and higher-dimensional dynamical systems.

Notation and basic properties

Definition 1.4. Let (X, T, Z d ), (Y, T, Z d ) be two topological dynamical systems and M ∈ GL(d, Z). A homomorphism associated with M is a surjective continuous map φ : X → Y such that for all n ∈ Z d we have that φ • T n = T M n • φ. If φ is invertible, then φ is an isomorphism.
In the following we fix the different notations that we will used throughout this thesis:

• We denote the set of all homomorphisms associated with M between (X, T, Z d ) and

(Y, T, Z d ) by Hom M (X, Y, T, Z d ).

• The set of homomorphisms between two dynamical systems, is defined as the collection of all of homomorphisms, i.e., Hom(X, Y, T,

Z d ) = M ∈GL(d,Z)
Hom M (X, Y, T, Z d ).

• In the special case M is the identity matrix, homomorphisms are called factors and we denote Fac(X, Y, T, Z d ) the collection of all factors between (X, T, Z d ) and (Y, T, Z d ).

If a factor is invertible, then it is called a conjugacy.

• In the case (X, T, Z d ) = (Y, T, Z d ), we simply denote these sets as N M (X, T, Z d ) and N (X, T, Z d ) and we call the last one, the normalizer semigroup of (X, T, Z d ). A factor map is called an endomorphism, and a conjugacy is called an automorphism. We denote the set of all endomorphisms and automorphisms of a topological dynamical system as End(X, T, Z d ) and Aut(X, T, Z d ), respectively.

• We define the symmetry semigroup N (X, T, Z d ) of (X, T, Z d ) as the collection of all matrices M ∈ GL(d, Z) with N M (X, T, Z d ) = ∅.

• A topological dynamical system (X, T, Z d ) is said to be coalescent if every endomorphism of (X, T, Z d ) is an automorphism.

Note that the symmetry semigroup of a topological dynamical system is an invariant under conjugation.

As an example, we can define an isomorphism for the Z-action T : S 1 → S 1 given by the rotation T α (x) = x + α, α ∈ S 1 , by the map φ(x) = -x. Indeed, we have that

φ • T = T -1 • φ.
For the Z 2 -action on the torus T 2 generated by the actions

T 1 (x) = x+α, T 2 (x) = x+β, an isomorphism is given by the map ψ : T 2 → T 2 defined as ψ(x) = -x.
Isomorphisms of a dynamical system has been studied before. In [START_REF] Baake | A brief guide to reversing and extended symmetries of dynamical systems[END_REF] they are called as reversors and the normalizer group N * (X, T, Z) (generated by isomorphisms) is called the reversing symmetry group. Their study was inspired by the time-reversal symmetry of many fundamental equations in physics. In this case N * (X, T, Z) = Aut(X, T, Z) or N * (X, T, Z) is an index-2 extension of Aut(X, T, Z). Isomorphisms are always elements of even or infinite order. The existence of isomorphisms have been studied for particular subshifts. Evidence suggest that algebraic properties of the automorphism group does not affect its existence:

• The full shift (A Z , S, Z). The automorphism group is huge (not amenable).

• Any sturmian subshift, which is always palindromic [START_REF] Droubay | Palindromes and Sturmian words[END_REF]. Its automorphism group is trivial, i.e., Aut(X, S, Z) = S .

• The period doubling shift, defined by the primitive substitution 0 → 01, 1 → 00.

• The Thue-Morse shift, defined by 0 → 01, 1 → 10.

• The square-free shift, obtained as the orbit closure of the characteristic function of the square-free integers.

For higher-dimensional systems they are sometimes called extended symmetries (see for example [START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF][START_REF] Bustos | Extended symmetry groups of multidimensional subshifts with hierarchical structure[END_REF][START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]). Since GL(d, Z) is an infinite group for d > 2, the relation between N * (X, T, Z d ) and Aut(X, T, Z d ) is less clear than in the one-dimensional case.

If

φ ∈ N M 1 (X, T, Z d ), and ψ ∈ N M 2 (X, T, Z d ), then φψ is in N M 1 M 2 (X, T, Z d ), so the sets N M (X, T, Z d ) are not semigroups (except if M is the identity matrix). Now, even though the matrices M ∈ GL(d, Z) are invertible in Z d , the symmetry semigroup N (X, T, Z d
) is not necessarily a group, since the existence of a homomorphism associated with a matrix M does not necessarily imply the existence of a homomorphism associated with M -1 . Nevertheless, we get the following result when a dynamical system is coalescent.

Proposition 1.5. Let (X, T, Z d ) be a coalescent system. If N (X, T, Z d ) is a group, then any homomorphism in N (X, T, Z d ) is invertible.
Proof. Let φ, ψ be two homomorphisms onto (X, T, Z d ) associated with M , M -1 , respectively. Then φψ is a factor map onto (X, T, Z d ). Since (X, T, Z d ) is coalescent, then φψ is invertible. We conclude that φ and ψ are invertible maps.

Equicontinuous systems are examples of coalescent systems [4]. See [START_REF] Parry | Minimal skew-product homeomorphisms and coalescence[END_REF] for an example of a non-coalescent system.

When the action generated by T is free, the normalizer group N * (X, T, Z d ) (which is the group of isomorphisms of (X, T, Z d )) is the normalizer of T in Homeo(X). This is because g T g 1 = T , for g ∈ Homeo(X) is only possible if the generators of T , which are T e i for 1 ≤ i ≤ d, are conjugated into generators of T . In the particular case T is not free, then norm Homeo ( T ) contains N * (X, T, Z d ) as a subgroup, but possibly further elements (see [START_REF] Baake | The structure of reversing symmetry groups[END_REF] for some examples). Nevertheless, in this thesis we are only interested in free actions.

The groups T and Aut(X, T, Z d ) are normal subgroups of N * (X, T, Z d ), and the centers of N * (X, T, Z d ) and Aut(X, T, Z d ) are the same. In fact, we have the following short exact sequences For every topological dynamical system, there exists at least one equicontinuous factor, which is the system given by one point. Furthermore, for every topological dynamical system, there exists its maximal equicontinuous factor, i.e., a factor π eq : (X, T, Z d ) → (X eq , T eq , Z d ) such that (X eq , T eq , Z d ) is an equicontinuous system, and for every equicontinuous factor π : (X, T, Z d ) → (Y, T, Z d ), there exists a factor map φ : (X eq , T eq , Z d ) → (Y, T, Z d ) such that π = φ • π eq . In the one-dimensional case, the maximal equicontinuous factor of a minimal system is a rotation on a compact monothetic topological group [96, Theorem 2.11], which is a group G for which there exists an element g ∈ G such that the subgroup generated by g is dense. Such groups are always abelian. Also, in the particular case, where π : (X, T, Z d ) → (Y, T, Z d ) is an almost 1-to-1 factor and (Y, T, Z d ) is an equicontinuous system, then (Y, T, Z d ) is the maximal equicontinuous factor of (X, T, Z d ). For instance, odometer systems (defined in Section 1.6) are almost 1-to-1 factors of Toeplitz systems [START_REF] Cortez | Z d Toeplitz arrays[END_REF][START_REF] Downarowicz | Survey of odometers and Toeplitz flows[END_REF].

1 → T → Aut(X, T, Z d ) → Aut(X, T, Z d )/ T → 1 (1.3) 1 → Aut(X, T, Z d ) → N * (X, T, Z d ) → N * (X, T, Z d ) → 1. (1.4)

Compatibility properties of homomorphisms

We say a factor map π : (X, T, Z d ) → (Y, T, Z d ) is compatible if for any endomorphism φ ∈ End(X, T, Z d ), and every x, y ∈ X, if π(x) is equal to π(y), then π(φ(x)) is equal to π(φ(y)). With the same spirit, we say a factor π is compatible with homomorphisms if for any homomorphism φ ∈ N (X, T, Z d ), and every x, y ∈ X, if π(x) is equal to π(y), then π(φ(x)) is equal to π(φ(y)).

The compatibility property allow us to study homomorphisms of some topological dynamical system as shown in the following result.

Lemma 1.6. Let (X, T, Z d ), (Y, T, Z d ) be two minimal systems, such that π : (X, T, Z d ) → (Y, T, Z d ) is a compatible factor. Then, there is a semigroup homomor- phism π : End(X, T, Z d ) → End(Y, T, Z d ) such that 1. π(φ)(π(x)) = π(φ(x)) for all φ ∈ End(X, T, Z d ) and x ∈ X. 2. π(Aut(X, T, Z d )) ≤ Aut(Y, T, Z d ). 3. For all ψ ∈ End(Y, T, Z d ), |π -1 ({ψ})| ≤ min y∈Y |π -1 (y)|.
Moreover, if π is compatible with homomorphisms, there is an extension of

π : N (X, T, Z d ) → N (Y, T, Z d ) defined as in Item 1. for all φ ∈ N (X, T, Z d ), such that π(N M (X, T, Z d ) ≤ N M (Y, T, Z d ), for any M ∈ GL(d, Z). Furthermore, if c = min y∈Y |π -1 (y)|, then for each M ∈ GL(d, Z), the map π : N M (X, T, Z d ) → N M (Y, T, Z d ) is at most c-to-1.
Proof. Set φ ∈ End(X, T, Z d ). By definition, the map π(φ) : Y → Y given by π(φ)(π(x)) = π(φ(x)) is well defined and, by minimality of (Y, T, Z d ), it is a surjective map, so π(φ) is an endomorphism of (Y, T, Z d ). Moreover, if φ is an automorphism of (X, T,

Z d ), then π(φ) is invertible. Indeed, π(φ) • π(φ -1 ) • π = π • φ • φ -1 = π, so we conclude that π(φ) • π(φ -1 ) = id Y . Now, set ψ ∈ End(Y, T, Z d ) and suppose that min y∈Y |π -1 ({y})| = c < ∞ (if not, then
there is nothing to prove). Let x 0 ∈ X and y 0 ∈ Y be such that |π -1 ({y 0 })| = c, and y 0 = ψ(π(x 0 )). Assume there exists c + 1 endomorphisms φ 0 , . . . , φ c of (X, T, Z d ), in π({ψ}) -1 . By the pigeonhole principle,

y 0 = ψ(π(x 0 )) = π(φ 0 (x 0 )) = • • • = π(φ c (x 0 )). So, there must exists two different indices 0 ≤ i, j ≤ c such that φ i (x 0 ) = φ j (x 0 ), which, by minimality of (X, T, Z d ), implies φ i = φ j .
Finally, note that the proof for homomorphisms use similar arguments.

We will use Lemma 1.6 to describe isomorphisms of particular examples in Chapter 6.

It is known that factor maps between equicontinuous systems are compatible [4], but as we will see in the next section, they are not necessarily compatible with homomorphisms (see Remark 2.7). Nevertheless, the maximal equicontinuous factor is an example of a factor compatible with homomorphisms as proved in [10, Theorem 5 and Corollary 3]. The next result summarizes the compatibility properties of the maximal equicontinuous factor. Lemma 1.7. [10, Theorem 5 and Corollary 3] For any minimal topological dynamical system (X, T, Z d ) such that the action of the maximal equicontinuous factor T eq is free, the maximal equicontinuous factor π eq : (X, T, Z d ) → (X eq , T eq , Z d ) is compatible with homomorphisms.

In this case, there exists a semigroup homomorphism θ : (X, T, Z d ) → Homeo(X eq ), with θ(End(X, T, Z d )) = id such that π eq (φ(x)) = π(φ) + θ(φ)(π eq (x)), for all x ∈ X, and φ ∈ N (X, T, Z d ), i.e., any homomorphism φ ∈ N (X, T, Z d ) induces a unique homomorphism in X eq given by z → π(φ) + θ(φ)(z).

Moreover, if c = min z∈Xeq |φ -1 eq ({z})| is finite, then for all n ≥ c, and φ ∈ N (X, T, Z d ), we have that {z ∈ X eq : |π -1 eq ({z})| = n} = θ(φ)({z ∈ X eq : |π -1 eq ({z})| = n}) + π(φ),
In some cases the quantity min y∈Y |π -1 (y)| can be computed. We refer to [START_REF] Coven | Computing automorphism groups of shifts using atypical equivalence classes[END_REF] for the class of substitutive systems generated by constant-length substitutions. Also Lemma 3.9 is the analogue for constant-shape substitutions.

Measure-preserving systems

In the following, we present the basics on ergodic theory. We mention [START_REF] Petersen | Ergodic theory, volume 2 of Cambridge Studies in Advanced Mathematics[END_REF][START_REF] Walters | An introduction to ergodic theory[END_REF] for classical references on this theme. A measure-preserving system is a 4-tuple (X, µ, T, G), where (X, F, µ) is a probability space and G is a countable group of measurable and measure-preserving transformations acting on X (where the action is denoted by T ), i.e., ∀A ∈ F, ∀g ∈ G, µ(T g -1 A) = µ(A).

We say that (X, µ, T, G) is ergodic if for all A ∈ F we have that

(∀g ∈ G) µ(T g -1 (A)∆A) = 0 =⇒ µ(A) = 0 ∨ µ(A) = 1.
We now recall the notions of measurable homomorphisms in the measure-theoretic framework.

Let (X, µ, T, G) and (Y, ν, T, G) be measure-preserving systems and M ∈ GL(d, Z). A measurable homomorphism associated with M is a measure-preserving map φ :

X → Y where X , Y are measurable subset of X, Y respectively, µ(X ) = ν(Y ) = 1 and for any g ∈ G, T g (X ) ⊆ X , T g (Y ) ⊆ Y such that for any n ∈ Z d we have that φ • T n = T M n • φ in X .
If there is a measurable factor φ between X and Y , then X is said to be an extension of Y . If φ is a bi-measurable bijection, we say that φ is a measurable conjugacy and in this case (X, µ, T, G) and (Y, ν, T, G) are metrically isomorphic.

For Z d -actions, we always have at least one invariant probability measure for topological dynamical systems (in fact, at least one ergodic probability measure). We define M(X, T, Z d ) the set of all invariant probability measures. This set is convex and compact on the weak-* topology. We say that (X, T,

Z d ) is uniquely ergodic if |M(X, T, Z d )| = 1,
and strictly ergodic if it is minimal and uniquely ergodic.

In the special case of strictly ergodic topological dynamical systems (X, T, Z d ), (Y, T, Z d ) we denote m Hom(X, Y, T, Z d ), m Fac(X, Y, T, Z d ) the collection of all measurable homomorphisms and factors between X and Y , respectively. We recall that a map φ is in m Hom M (X, Y, T, Z d ), for a particular matrix M ∈ GL(d, Z), if φ is measure-preserving and there exists two subsets

X ⊆ X, Y ⊆ Y with µ X (X ) = 1, µ Y (Y ) = 1 such that for all n ∈ Z d , φ • S n = S M n • φ for µ X -a.e in X .

Symbolic Dynamics

In this section, we will present the basic definitions and some background about symbolic dynamics that will be used in the rest of this thesis. We refer to [START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF] for a classical reference in the one-dimensional case, and [START_REF] Ceccherini-Silberstein | Cellular automata and groups[END_REF] for actions on more abstract groups.

Let A be a finite alphabet and d ≥ 1 be an integer. We define a topology on A Z d by endowing A with the discrete topology, and considering in A Z d the product topology, which is generated by cylinders. Since A is finite, A Z d is a metrizable compact space. In this space Z d acts by translations, defined for every n ∈ Z d by:

S n (x) k = x n+k , x ∈ A Z d , k ∈ Z d . The Z d -action (A Z d , S, Z d ) is called the fullshift.
Let P ⊆ Z d be a finite subset. A pattern is an element p ∈ A P . We say that P is the support of p, and we denote P = supp(p). A pattern occurs in x ∈ A Z d , if there exists n ∈ Z d such that p = x| n+P , in this case we denote it p x, and we call this n an occurrence in x of p.

A subshift (X, S, Z d ) is given by a closed subset X ⊆ A Z d which is invariant by the Z d -action. A subshift can also be defined by its language. For P Z d we define

L P (X) = {p ∈ A P : ∃x ∈ X, p x}.
We define the language of a subshift X by

L(X) = P Z d L P (X).
Let (X, S, Z d ) be a minimal subshift and x ∈ X. We say that p ∈ Z d is a period of x if for all n ∈ Z d , x n+p = x n . We say that (X, S, Z d ) is aperiodic if there are no nontrivial periods.

Let B be other finite alphabet, and Y ⊆ B Z d be a subshift. For P Z d , we define a P -block map as a map of the form Φ : L P (X) → B. This map induce a factor φ : X → Y given by φ(x) n = Φ(x| n+P ).

The map φ is called the sliding block code induced by Φ, and P is the support of the map φ. In most of the cases we may assume the support of the sliding block codes is a ball of the form B(0, r), for r ∈ N. We define the radius (and we denote by r(φ)) as the infimum of r ∈ N such that we can define a B(0, r)-block map which induced it. The next theorem characterizes the factor maps between two subshifts. Theorem 1.8 (Curtis-Hedlund-Lyndon theorem). Let (X, S, Z d ) and (Y, S, Z d ) be two subshifts. A map φ : (X, S, Z d ) → (Y, S, Z d ) is a factor if and only if there exists a B(0, r)block map Φ :

L B(0,r) (X) → L 1 (Y ), such that φ(x) n = Φ(x| n+B(0,r) ), for all n ∈ Z d and x ∈ X.
For homomorphisms we have a similar characterization, but we need to make a slight variation of this theorem. Theorem 1.9 (Curtis-Hedlund-Lyndon theorem for homomorphisms). Let (X, S, Z d ) and (Y, S, Z d ) be two subshifts, and

M ∈ GL(d, Z). A map φ : (X, S, Z d ) → (Y, S, Z d ) is a homomorphism associated with M if and only if there exists a B(0, r)-block map Φ : L B(0,r) (X) → L 1 (Y ), such that φ(x) n = Φ(x| M -1 n+B(0,r) ), for all n ∈ Z d and x ∈ X. Proof. Let (x n ) n∈N be a convergent sequence to x ∈ X. This implies, (∀p > 0)(∃N ∈ N)(∀n ≥ N ) x n | B(0,p) = x| B(0,p) .
Set m ∈ Z d . Let p(m) ∈ N be large enough such that M -1 m + B(0, r) ⊆ B(0, p(m)). Hence, for any n ≥ N (p(m)) we have that

φ(x n ) m = Φ(x n | M -1 m+B(0,r) ) = Φ(x| M -1 m+B(0,r) ) = φ(x) m . We conclude that φ is continuous. Now, set m, n ∈ Z d . Then for all x ∈ X φ(S n x) m = Φ((S n x)| M -1 m+B(0,r) ) = Φ(x| n+M -1 m+B(0,r) ),
and

S M n φ(x) m = φ(x) M n+m = Φ(x| n+M -1 m+B(0,r) ).
We conclude that φ • S n = S M •n • φ, so φ is a homomorphism associated with M . On the other hand, let φ : (X, S, Z d ) → (Y, S, Z d ) be a homomorphism associated with M , and let r > 0 be such that x| B(0,r) = y| B(0,r) , implies φ(x) 0 = φ(y) 0 . Then, the local map Φ(x| B(0,r) ) = φ(x) 0 is well defined by the very definition of r. Finally, note that φ(x) h = S h φ(x) 0 = φ(S M -1 h x) 0 = Φ(x| M -1 h+B(0,r) ), which proves the claim. This means, for any homomorphism φ we can define a radius (also denoted by r(φ)), as the infimum of r ∈ N such that we can define a B(0, r)-block map which induced it.

An immediate consequence of Theorem 1.9 is that for any subshift (X, S, Z d ), its normalizer group N * (X, S, Z d ) is a countable and discrete subset in Homeo(X).

Nondeterministic directions of a topological dynamical system

An interesting notion in the study of higher-dimensional dynamical systems, with the objective to study sub-actions of one, is the so-called nonexpansive subspaces, introduced by M. Boyle and D. Lind in [START_REF] Boyle | Expansive subdynamics[END_REF]. When the space X is infinite such subspaces always exist [START_REF] Boyle | Expansive subdynamics[END_REF]Theorem 3.7]. In fact, nonexpansive subspaces are always subsets of nonexpansive hyperplanes [START_REF] Boyle | Expansive subdynamics[END_REF]Theorem 3.6]. A key step on the proof of this result, was one proved by S. Schwartzman. S. Schwartzman proved that there are no "one-sided expansive" homeomorphisms, except on finite spaces. Although he never published this result. A proof can be found in [START_REF] Gottschalk | Topological dynamics[END_REF].

We will only focus on hyperplanes in R d , which lead to the notion of deterministic/nondeterministic directions. Let S d-1 be the unit (d -1)-dimensional sphere. For

v ∈ S d-1 define H v = {x ∈ R d : x, v < 0} to
be the open half-space with outward unit normal v. We identify the set H d of all half-spaces in R d with the sphere S d-1 using the parametrization v ←→ H v . Definition 1.10. Let (X, S, Z d ) be a subshift and v be a unit vector of R d . Then v is deterministic for (X, T, Z d ) if for all x, y ∈ X we have that

x| Hv∩Z d = y| Hv∩Z d =⇒ x = y.
If v does not satisfies this condition, we say that v is nondeterministic for (X, T, Z d ).

In this thesis, we are only interested in the set of nondeterministic directions. In the one-dimensional case, this lead to the notion of asymptotic pairs, and in the finitary version to the notion of special words [START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF]. For a subshift (X, S, Z d ) we denote ND(X, S, Z d ) the sets of nondeterministic directions for (X, S, Z d ). In [START_REF] Boyle | Expansive subdynamics[END_REF] it was proved an analogous result of the following, that we mention in our context Theorem 1.11. [17, Lemma 3.4 and Theorem 3.7] For any subshift (X, S, Z d ), with an infinite phase space X, the set of nondeterministic directions ND(X, S, Z d ) is a non-empty compact set.

The notion of direction of determinism was introduced in [START_REF] Guillon | Determinism in subshifts[END_REF] for two-dimensional subshifts, and in [START_REF] Cyr | Nonexpansive Z 2 -subdynamics and Nivat's conjecture[END_REF] they were used to prove a weak version of Nivat's conjecture.

The following result establishes a link between nondeterministic directions and the symmetry group of a subshift, which we will use to describe the symmetry group for a big family of substitutive systems (Theorem 5.17). Proposition 1.12. Let (X, S, Z d ) be a subshift. Then, for all v ∈ ND(X, S, Z d ) and M ∈ N (X, S, Z d ), we have that

(M * ) -1 v/ (M * ) -1 v ∈ ND(X, S, Z d ). Proof. If v is in ND(X, S, Z d ), there exists x = y ∈ X with x| Hv∩Z d = y| Hv∩Z d . Set M ∈ N (X, T, Z d ) and φ ∈ N M (X, S, Z d ). Then, we have that φ(x)| (M Hv)+n = φ(y)| (M Hv)+n ,
where n is a vector of radius r(φ). We note that S n φ(x)| M Hv = S n φ(y)| M Hv , and we conclude that (M * ) -1 v/ (M * ) -1 v ∈ (X, S, Z d ).

Odometer systems

In the following we present the example of odometer systems and Toeplitz sequences. Odometer systems are the most natural equicontinuous systems in the study of minimal Cantor systems. They have been described by algebraic reasons. They present interesting recurrence properties. The return times to clopen sets contain infinite arithmetic progressions. In fact, they are the maximal equicontinuous factor for a big family of symbolic systems, such as, some substitutions and Toeplitz sequences. In this section we describe the symmetry semigroup of odometer systems (Lemma 1.14) which we then use to completele characterize it for two-dimensional constant-base odometer systems (Theorem 2.2). Note that, in [START_REF] Giordano | Z d -odometers and cohomology[END_REF] there was a first attempt of studying homomorphisms between two-dimensional odometer systems.

Toeplitz sequences have been introduced in dynamical systems by K. Jacobs and M. Keane in [START_REF] Jacobs | 0 -1-sequences of Toeplitz type[END_REF]. Toeplitz subshifts are symbolic sytems that are the orbit closures of the regular quasi-periodic points of the subshift. N. Markley and M. Paul characterize them in [START_REF] Markley | Almost automorphic symbolic minimal sets without unique ergodicity[END_REF] as the minimal almost 1-1 extensions of odometer systems. They have been used to provide a series of examples with interesting dynamical properties. We refer to [START_REF] Downarowicz | Survey of odometers and Toeplitz flows[END_REF] for the study of odometer systems and Toeplitz sequences in the one-dimensional case, [START_REF] Cortez | Z d Toeplitz arrays[END_REF] for higher dimensional actions, and [START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF] for more abstract actions given by residually finite groups. Here we will follow the same notation than in [START_REF] Cortez | Z d Toeplitz arrays[END_REF].

Let

Z 0 ≥ Z 1 ≥ . . . ≥ Z n ≥ Z n+1 ≥ .
. . be a nested sequence of finite index subgroups of Z d such that n≥0 Z n = {0}, and let α n : Z d /Z n+1 → Z d /Z n be the function induced by the inclusion map. Following the notation in [START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF], we consider the inverse limit of these groups

← - Z d (Zn) = lim ←n (Z d /Z n , α n ), i.e., ← - Z d (Zn) is the subset of the product n≥0 Z d /Z n consisting of the elements ← -g = (g n ) n≥0
such that α n (g n+1 ) = g n (mod Z n ) for all n ≥ 0. This set is a group equipped with the addition defined coordinate-wise, i.e.,

← -g + ← - h = (g n + h n ) n≥0 .
Every group Z d /Z n is endowed with the discrete topology, then

n≥0 (Z d /Z n ) is a compact
metric space, and ← -Z d (Zn) is a compact topological group whose topology is spanned by the cylinder sets

[a] n = ← -g ∈ ← - Z d (Zn) : g n = a ,
with a ∈ Z d /Z n , and n ≥ 0. Now, consider the group homomorphism κ (Zn) :

Z d → n≥0 Z d /Z n defined for n ∈ Z d by κ (Zn) (n) = [n (mod Z n )] n≥0 .
The image of

Z d by κ (Zn) is dense in ← - Z d (Zn) , so the Z d -action n( ← -g ) = κ (Zn) (n)+ ← -g , with n ∈ Z d , ← -g ∈ ← - Z d (Zn)
, is well defined and (

← - Z d (Zn) , + (Zn) , Z d
) is a minimal equicontinuous system as proved in [START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF]. We call ( ← -Z d (Zn) , + (Zn) , Z d ) an odometer system. From now on, we will denote the odometer system ( ← -

Z d (Zn) , + (Zn) , Z d ) just as ← - Z d (Zn)
, and in the constant base case (where Z n = L n (Z d ), for some matrix L ∈ M(d, Z)), we will denote it as ← -Z d (L n ) . Odometer systems have been extensively studied before (see [START_REF] Cortez | Z d Toeplitz arrays[END_REF][START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF][START_REF] Downarowicz | Survey of odometers and Toeplitz flows[END_REF]). The next result characterizes the factor odometers systems of a fixed odometer system.

Lemma 1.13. [28, Lemma 1] Let ← - Z d (Z j
n ) be two odometer systems (j = 1, 2). There exists a factor map π :

← - Z d (Z 1 n ) → ← - Z d (Z 2
n ) if and only if for every Z 2 n there exists some

Z 1 m such that Z 1 m ≤ Z 2 n .
The proof of Lemma 1.13 can be modified to provide a characterization for the matrices

M ∈ GL(d, Z) defining a homomorphism φ : ← - Z d (Z 1 n ) → ← - Z d (Z 2 n ) . Lemma 1.14. Set M ∈ GL(d, Z). There exists a homomorphism associated with M from ← - Z d (Z 1 n ) to ← - Z d (Z 2 n ) , if and only if for all n ∈ N, there exists m M (n) ∈ N such that M Z 1 m M (n) ≤ Z 2 n . (Normalizer Condition)
Proof. First we prove the necessity. Let φ :

← - Z d (Z 1 n ) → ← - Z d (Z 2
n ) be a homomorphism associated with a matrix M ∈ GL(d, Z). By continuity, for any n ≥ 0 and

g ∈ Z d /Z 2 n , there exists m ≥ 0 and f ∈ Z d /Z 1 m such that [f ] m ⊆ φ -1 ([g] n ). Set h ∈ Z 1 m . Note that for all ← - f ∈ [f ] m , we have that h( ← - f ) ∈ [f ] m , which implies φ(h( ← - f )) = M h( ← - f ) ∈ [g] n . Since φ( ← - f ) is in [a] n , then m ∈ Z d : m(φ( ← -g )) ∈ [a] n is equal to Z 2 n , which implies M h ∈ Z 2 n .
For the sufficiency, assume that M ∈ GL(d, Z) satisfies (Normalizer Condition). Since the sequences {Z i n } n>0 , i = 1, 2 are decreasing, we may assume that m(n) ≤ m(n + 1) for all n > 0. Thus, we have a homomorphism φ m(n) :

Z d /Z 1 m M (n) → Z d /Z 2 n , given by φ m(n) (m) = M m.
To finish the proof, we remark that φ :

← - Z d (Z 1 n ) → ← - Z d (Z 2 n ) defined as φ( ← -g = (g m(n) ) n>0 ) = (φ m(n) (g m(n) )) n>0 is a homomorphism associated with M .
In [START_REF] Giordano | Z d -odometers and cohomology[END_REF] there was a first attempt of studying homomorphisms between two-dimensional odometer systems. Note that for all n ≥ 0, we may assume that m M (n) is arbitrarily large. Consider, for all n ∈ N a matrix L n,i ∈ M(d, Z) such that L n,i (Z d ) = Z i n for i ∈ {1, 2}. This matrix is unique, up to a conjugation with a matrix in GL(d, Z). Then, (Normalizer Condition) is equivalent to, for all n ≥ 0, there exists m M (n) ≥ 0 such that

L -1 n,2 M L m M (n),1 is an endomorphism in Z d . Since det(L)L -1 = adj(L)
, where adj(L) is the adjugate matrix of L, then (Normalizer Condition) is equivalent to

∀n ∈ N, ∃m M (n) ∈ N, adj(L n,2 )M L m M (n),1 ≡ 0 (mod det(L n,2 )). ( NC 2)
In the constant-base case, ignoring the condition M ∈ GL(d, Z), then L satisfies (NC 2). Moreover, the set of integer matrices M ∈ M(d, Z) satisfying (NC 2) is an additive group. In particular, any polynomial in L with integer coefficients also satisfies (NC 2).

A direct corollary of Lemma 1.14 is the following.

Corollary 1.15. The following are consequences of Lemma 4.10 easily to verified.

1. If L = pM is an integer expansion matrix, with p ∈ Z and M ∈ GL(d, Z), then the Then Per(x, Z) = a∈A Per(x, Z, a) for all x ∈ A Z d . When Per(x, Z) is non-empty, we say Z is a group of periods of x. We say Z ⊆ Z d is a group generated by essential periods of x if for all finite index subgroup Z ⊆ Z d , Per(x, Z) ⊆ Per(x, Z ) implies that Z ⊆ Z. We say that x is a Z d -Toeplitz sequence if for all n ∈ Z d , there exists a finite index subgroup Z ⊆ Z d such that n ∈ Per(x, Z). The following is a characterization of Toeplitz sequences Proposition 1.16. [28, Proposition 14] The following statements concerning x ∈ A Z d are equivalent:

symmetry semigroup N ( ← - Z d (L n ) ) is GL(d, Z).
1. x is a Toeplitz sequence.

2. There exists a sequence of positive integer numbers {p n } n≥0 such that p n < p n+1 , p n divides p n+1 and -n, . . . , n d ⊆ Per(x, p n Z d ) for all n ≥ 0.

Toeplitz subshifts, which are subshifts generated by a Toeplitz sequence, are almost 1-to-1 extensions of odometer systems. In fact Proposition 1.17. [28, Proposition 20] Let x ∈ A Z d be a Toeplitz sequence. If {Z n } n≥0 is a sequence of groups generated by essential periods of x such that Z n+1 ⊆ Z n and

n≥0 Per(x, Z n ) = Z d , then ( ← - Z d Zn , + Zn , Z d
) is the maximal equicontinuous factor of the subshift generated by x.

Multidimensional constant-shape substitutions

In this section we will define the main object where we study homomorphisms in this thesis, which are multidimensional substitutive subshifts. They represent the simplest nontrivial zero entropy symbolic systems, since they are generated by finite data. By that, ergodic and topological properties of substitution dynamical systems have been extensively studied. They were introduced by W.H. Gottschalk in [START_REF] Gottschalk | Substitution minimal sets[END_REF] (see [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] for a good bibliography on this subject).

Unlike the one-dimensional case, the notion of multidimensional substitutions is not clear. In our case we extend the notion of constant-length substitutions to the multidimensional framework and we call them constant-shape substitution. To do this, we use the fact that if L ∈ M(d, Z) is an integer matrix with det(L) = 0 and F is a fundamental domain of L(Z d ) (with 0 ∈ F ), i.e., a set of representative classes of Z d /L(Z d ), then for any n ∈ Z d , there exists unique p ∈ Z d and f ∈ F such that n = L(p) + f . This is a multidimensional interpretation of the Euclidean division.

In our context, L ∈ M(d, Z) is an integer expansion matrix, i.e, L > 1 and

L -1 < 1, such that L(Z d ) ⊆ Z d .
Let F be a fundamental domain of L(Z d ) with 0 in F , and A be a finite alphabet. A multidimensional constant-shape substitution ζ is a map A → A F . The set F is called the support of the substitution. An analogue of constant-shape substitutions in the one-dimensional case are the constant-length substitutions. For any f ∈ F ζ 1 we define p f the restriction of ζ in f , i.e., for a ∈ A, we have that p f (a) = ζ(a) f . We say the substitution is bijective if for all f ∈ F ζ 1 the maps p f are bijective. A substitution is called primitive if there exists a positive integer n > 0, such that for every a, b ∈ A, b occurs in ζ n (a). Fig. 1.2 is an example of a bijective and primitive constant-shape substitution. The language of a substitution is the set of all patterns that appear in ζ n (a), for some n > 0, a ∈ A, i.e.,

L ζ = {p : p ζ n (a), for some n > 0, a ∈ A}.
Using the language we define the subshift X ζ associated with a substitution as the set of all sequences x ∈ A Z d such that every pattern occurring in x is in L ζ , and we denote (X ζ , S, Z d ) the substitutive subshift. If ζ is a primitive constant-shape substitution, the existence of periodic points is well known, i.e., there exists at least one point x 0 ∈ X ζ such that ζ p (x 0 ) = x 0 for some p > 0. In the primitive case, the subshift is preserved replacing the substitution by a power of it, i.e., X ζ n is equal to X ζ for all n > 0. Then, we may assume that the substitution possess at least one fixed point, i.e., there exists a point x ∈ X ζ such that x = ζ(x). It is well known that this subshift is uniquely ergodic (in [START_REF] Lee | Consequences of pure point diffraction spectra for multiset substitution systems[END_REF] can be found a proof for substitution tiling systems seen as substitution Delone sets for R d -actions that can be adapted for our context with the assumption of the supports (F ζ n ) n>0 being a Følner sequence). The unique ergodic measure is characterized in terms of the expansion matrix of ζ, and we denote this measure as µ ζ . For a cylinder set [p] n , where p is a pattern in L ζ , the quantity µ ζ ([p]) represents the frequency of the pattern p in any sequence in X ζ .

In the literature, constant-shape substitutions with a positive diagonal expansion matrix L = diag(l i ) i=1,...,d and support equal to the standard d-dimensional parallelepiped

F 1 = d i=1 0, l i -1 are called block substitutions.
Examples of constant-shape substitutions can be generated via constant-length substitutions as follows: Let {ζ i } d i=1 be d aperiodic one-dimensional constant-length substitutions with alphabet A i and length q i for 1 ≤ i ≤ d. We define the product substitution of {ζ i } d i=1 as the constant-shape substitution ζ with alphabet A =

d i=1 A i , expansion matrix given by L ζ = diag(l i ) and support F ζ 1 = d i=1 0, l i -1 , defined as ζ(a 1 , . . . , a d ) j = (ζ 1 (a 1 ) j 1 , . . . , ζ d (a d ) j d ).
It is straightforward to check that if {ζ i } d i=1 are primitive, then the product substitution is also primitive. The same is true for bijectivity.

Since L ζ is an expansion matrix, then

L -1 ζ is a contraction map in R d . For any g ∈ F ζ 1 define the map f g (•) = L -1 ζ (• + g).
As mentioned in Section 1.1.3, there exists a nonempty compact subset T ζ (or denoted T (L, F 1 ) when there is no substitution defined) in

R d such that L ζ (T ζ ) = g∈F ζ 1 T ζ + g.
As in [START_REF] Vince | Digit tiling of Euclidean space[END_REF] we call this set the digit tile of the substitution. Using T 0 = {0} in (1.2) we get that

T ζ = lim n→∞ n-1 i=0 L -i ζ (F ζ 1 ) = lim n→∞ L -n ζ (F ζ n ), (1.5)
where the limit is with respect to the Hausdorff metric. The expansion matrix and fundamental domains of these examples are the following:

(a) L (a) = 2 0 0 2 , F (a) 1 = 0 0 , 1 0 , 0 1 , -1 -1 . (b) L (b) = 3 0 0 3 , F (b) 1
= 0 0 , 1 1 , 2 2 , -1 0 , -2 0 , - 1
1 , 0 -1 , 0 -2 , 1 -1 .
(c)

L (c) = 3 0 0 3 , F (c) 1 = 0 0 , 1 0 , 2 0 , 0 1 , 0 2 , 2 2 , 4 4 , 2 1 , 1 2 .
(d)

L (d) = 1 -1 1 1 , F (d) 1 = 0 0 , 1 0 .
As in the one-dimensional case, the following proposition shows that for any multidimensional constant-shape substitution there exists a finite subset K Z d whose iteration under the substitution fill the whole Z d . This set determine the fixed points of a primitive constant-shape substitution.

Proposition 1.18. Let ζ be a multidimensional constant-shape substitution. Then, the set

K ζ = m>0 ((id -L m ζ ) -1 (F ζ m ) ∩ Z d ) is finite and satisfies n≥0 L n ζ (K ζ ) + F ζ n = Z d , using the notation F ζ 0 = {0}.
Proof. Set n ∈ Z d and consider the sequence (a n ) n≥0 ⊆ Z d given by a 0 = n and for any n ≥ 0, a n+1 is defined as the unique element in Z d such that there exists an element

f n ∈ F ζ 1 with a n = L ζ (a n+1 ) + f n . Note that for any n ≥ 0, a n+1 ≤ L -1 ζ • a n + L -1 ζ • F ζ 1 , which implies a n ≤ L -1 ζ n n + L -1 ζ F ζ 1 (1 -2 L -1 ζ n ) 1 -L -1 ζ , hence (a n ) n≥0 is a bounded sequence in Z d
. By the Pigeonhole Principle there exist n ≥ 0 and k > 0 such that a n = a n+k , i.e.,

a n = L k ζ (a n ) + f , for some f ∈ F ζ k . It follows the set K ζ = m>0 ((id -L m ζ ) -1 (F ζ m ) ∩ Z d
) satisfies the property. Now we prove that K ζ is finite.

Note that for any m > 0

(id -L m ζ ) -1 (F ζ m ) = m-1 i=0 (id -L m ζ ) -1 L i (F ζ 1 ) ≤ F ζ 1 m-1 i=0 (id -L m ζ ) -1 L i ζ ≤ F ζ 1 (id -L m ζ ) -1 (id -L m ζ )(id -L ζ ) -1 ≤ F ζ 1 (id -L ζ ) -1 .
We conclude that K ζ is a finite set.

Remark 1.19. The following statements can be easily verified.

(1) In the one-dimensional case for any constant-length substitution ζ, we have that

K ζ = {-1, 0}.
Moreover, for any d-dimensional block substitution, i.e., L ζ equal to the diagonal matrix diag(l 1 , . . . , l d ), and

F ζ 1 equal to the standard d-dimensional parallelepiped support d i=1 0, l i -1 , then K ζ = -1, 0 d . (2) If L ζ is such that | det(L ζ -id)| = 1, then K ζ is equal to (id -L -1 ζ ) -1 (F ζ 1 ).
(3) Since K ζ is a finite set, there exists j > 0 such that

K ζ = j m=0 ((id -L m ζ ) -1 (F ζ m ) ∩ Z d ).
When the substitution ζ is primitive, up to considering a power of ζ, we may assume that the set K ζ is of the form (id

-L ζ ) -1 (F ζ 1 ) ∩ Z d .
The argument used in the proof of Proposition 1.18 is inspired by the Euclidean Division Algorithm. A similar idea can be used to find different sets satisfying specific statements involving the supports (F ζ n ) n>0 that will be needed for some proofs. From now on for any n ∈ Z d we denote by

d n ∈ Z d and f n ∈ F ζ 1 the unique elements such that n = L ζ (d n ) + f n .
The following result will be useful in a series of results throughout this thesis.

Proposition 1.20. Set A Z d and let F Z d be such that F ζ 1 ⊆ F . Define B = {d n } n∈F +A .
Then, there exists a finite subset C of Z d satisfying the following conditions:

1. B ⊆ C. 2. C + F + A ⊆ L ζ (C) + F ζ 1 . 3. C ≤ B + L -1 ζ A + F + F ζ 1 / 1 -L -1 ζ .
Proof. We define two sequences of finite sets of

Z d , (B n ) n≥0 , (C n ) n≥0 , with B 0 = B, C 0 = B + F + A, and for any n ≥ 0, set B n+1 Z d such that B n+1 = {d n } n∈Cn , and C n+1 Z d such that C n+1 = B n+1 + F + A. Note that B n+1 ≤ L -1 ζ C n + F ζ 1 ≤ L -1 ζ B n + A + F + F ζ 1 ≤ L -1 ζ B n + L -1 ζ A + F + F ζ 1 .
Hence, for any n > 0 we have that

B n ≤ B L -1 ζ n + 1 -L -1 ζ n 1 -L -1 ζ L -1 ζ A + F + F ζ 1 . Since L -1 ζ is strictly smaller than 1, then B n ≤ B + L -1 ζ A + F + F ζ 1 / 1 -L -1 ζ . This implies there exists N ∈ N such that n≤N B n = n≤N +1 B n . We conclude the proof taking C = N n=0 B n .
Remark 1.21. The following statements will be useful on the rest of the thesis.

(1) Condition 2. implies

C + A + F ζ 1 ⊆ L ζ (C) + F ζ 1
, and a direct induction proves that for all n ≥ 0, we have that

L n ζ (C + A) + F ζ n ⊆ L n+1 ζ (C) + F ζ n+1 .
(

) Using F = F ζ 1 + F ζ 1 and A = {0}, we obtain a set C Z d such that C + F ζ 1 + F ζ 1 ⊆ L ζ (C) + F ζ 1 . Since 0 ∈ F ζ 1 , then 0 is in C, which implies F ζ 1 + F ζ 1 ⊆ L ζ (C) + F ζ 1 . 2
Assume that for some n > 0 we have that

F ζ n + F ζ n ⊆ L n ζ (C) + F ζ n . Then, we obtain that F ζ n+1 + F ζ n+1 = F ζ n + F ζ n + L n ζ (F ζ 1 ) + L n ζ (F ζ 1 ) ⊆ L n ζ (C) + F ζ n + L n ζ (F ζ 1 ) + L n ζ (F ζ 1 ) ⊆ L n+1 ζ (C) + L n ζ (F ζ 1 ) + F ζ n = L n+1 ζ (C) + F ζ n+1 ,
so, by induction we prove that for all n > 0,

F ζ n + F ζ n ⊆ L n ζ (C) + F ζ n .
The

Chapter 2

The symmetry semigroup of Z 2 -odometers

In this chapter, we will study the symmetry semigroup of some odometer systems. In [START_REF] Giordano | Z d -odometers and cohomology[END_REF] there is a first attempt to characterize this semigroup for Z 2 -odometers. We start with the d-dimensional universal odometer system. This odometer is universal in the sense that any d-dimensional odometer system is a topological factor of this one. We get that its symmetry semigroup is the largest one and equal to GL(d, Z). Then, we will restrict on two dimensional constant base odometer systems ← -Z 2 (L n ) , where L ∈ M(2, Z) is an integer expansion matrix. In this case, we get a description of a bifurcation phenomenon at the level of the symmetry semigroup with respect to arithmetical relations of invariants of the matrix. The main theorem (Theorem 2.2) shows that in most cases the symmetry semigroup is the centralizer of the matrix L. This will help to get a characterization of the normalizer semigroup of aperiodic primitive constant-shape substitutions using the relation between homomorphisms and their maximal equicontinuous factors (Lemma 1.7).

The universal odometer

Let (Γ n ) n∈N be a enumeration of all finite index subgroups of Z d . We define the ddimensional universal odometer system as follows: Set Z 0 = Γ 0 , and for any n > 1 set

Z n = Λ n-1 ∩ Γ n . Then, we define the d-dimensional universal odometer as ← - Z d (Zn)
. This odometer is universal in the sense that by Lemma 1.13 any odometer system is a topological factor of the universal odometer. For instance, the 1-dimensional universal odometer system is equal to ← -Z (n!) . With respect to its symmetry semigroup, (NC 2) implies the following result.

Proposition 2.1. The symmetry semigroup of the universal odometer is GL(d, Z).

Proof. Consider L n ∈ M(d, Z) such that L n (Z d ) = Λ n . A matrix M ∈ GL(d, Z) is in N ( ← - Z d (Λn) ) if and only if M satisfies (NC 2). Now, for any n ∈ N we can choose m(n) ∈ N large enough such that Λ m(n) ≤ det(L n )Z d , so adj(L n )M L m(n) ≡ 0 (mod det(L n )). We then conclude that N ( ← - Z d (Λn) ) = GL(d, Z).
2.2 Bifurcation phenomenon of the normalizer semigroup for constant base Z 2 -odometer systems

To describe the different cases of the normalizer semigroup of odometer systems of the form

← - Z 2 (L n )
, where L ∈ M(2, Z) is an integer expansion matrix, we need some extra notations. For any positive integer n > 0, the radical rad(n) of n is defined as the product of the distinct prime numbers dividing n, if n < 0 we define rad(n) just as rad(-n). We define the centralizer of a matrix L on GL(2, Z) Cent GL(2,Z) (L) as the subgroup of all matrices in GL(2, Z) commuting with L. Recall by Corollary 1.15 the centralizer Cent GL(2,Z) (L) is always a subgroup of the symmetry semigroup N ( ← -

Z d (L n ) )
. From now on, we will fix the following notation. An integer expansion matrix will be denoted as L = p q r s and its powers as

L n = p(n) q(n) r(n) s(n) . We will denote a matrix in GL(2, Z) as M = m 11 m 12 m 21 m 22 .
The following theorem summarizes the different cases of the symmetry group N ( ← -Z 2 (L n ) ) depending on the matrix L.

Theorem 2.2. Let L ∈ M(2, Z) be an integer expansion matrix.

• If rad(det(L)) divides trace(L), then N ( ← - Z 2 (L n ) ) is equal to GL(2, Z).
• Otherwise 1. If the spectrum of the matrix L is disjoint from the integers, then the symmetry semigroup

N ( ← - Z 2 (L n ) ) is the centralizer Cent GL(2,Z) (L). Moreover, if the spec- trum of L is disjoint from the real line, then N ( ← - Z 2 (L n )
) is a finite group. 2. When the spectrum of L contains an integer value, then the matrix coefficients of elements in

N ( ← - Z 2 (L n (Z 2 ))
) are characterized by linear relations with respect to the coefficients given by the one of the matrix L. In particular, under explicit arithmetical properties of the coefficients of L, N ( ← -

Z 2 (L n (Z 2 )) ) can be isomorphic to Z/2Z, Z 2 /2Z × 2Z, or N ( ← - Z 2 (L n (Z 2 )) )/(Cent GL(2,Z) (L)) can be virtually to Z.
Along the proof of Theorem 2.2 we will get more precise information about the symmetry semigroup, when we have more restrictions on the matrix L.

Since odometer systems are coalescent systems, then as a consequence of the proof of Theorem 2.2 and Proposition 1.5 we get the following result.

Proposition 2.3. Let L ∈ M(2, Z) be an integer expansion matrix. Then, the symme-

try semigroup of ← - Z 2 (L n ) is a group. In particular, any homomorphism in N ( ← - Z 2 (L n ) ) is invertible.
Proposition 2.3 will be proved in Section 2.5.

The following examples illustrate the different consequences of Theorem 2.2 according to the expansion matrix L. . We have that trace(L 1 ) = 5, and det(L 1 ) = 7, so by Theorem 2.2 a

matrix M ∈ GL(2, Z) is in N ( ← - Z 2 (L n 1
) ) if and only if M commutes with L 1 . Note that trace(L 1 ) 2 -4 det(L 1 ) = -3, so L 1 has complex eigenvalues (which are 5/2 ± i √ 3/2), and Cent GL(2,Z) (L 1 ) is equal to

       1 1 -1 0 , -1 -1 1 0 , 0 -1 1 1 0 -1 1 -1 , 1 0 0 1 , -1 0 0 -1        . ( 2
)

The diagonal case

We will decompose the proof of Theorem 2.2 starting with the case where the expansion matrix has integer eigenvalues. First we will study the diagonal case, i.e., q = r = 0, since we will get more precise results about the symmetry semigroup. Note that det(L) = ps and trace(L) = p + s, so the condition rad(det(L)) divides trace(L) is equivalent to rad(p) = rad(s). The following result is a particular case of Theorem 2.2 for diagonal matrices.

Proposition 2.6. Let L = diag(p, s) be an expansion 2 × 2 diagonal matrix such that rad(det(L)) does not divides trace(L). We have the following:

1. If (rad(p) does not divide s and rad(s) divides p), or (rad(p) divides s and rad(s)

does not divide p), then N ( ← - Z 2 (L n ) )/(Cent GL(2,Z) (L)) is virtually Z.
2. If rad(p) does not divide s and rad(s)

does not divide p, then N ( ← - Z 2 (L n ) ) is isomorphic to Z/2Z × Z/2Z.
Proof. Let M ∈ GL(2, Z) satisfying (NC 2). Then, for any n > 0, we get the following equations

adj(L n )M L m(n) = m 11 p m(n) s n m 12 s m(n)+n m 21 p m(n)+n m 22 s m(n) p n ≡ 0 0 0 0 (mod p n s n ), (2.1)
Assuming m(n) > 0 is large enough, we simplify the equations in the anti-diagonal, and we get

m 12 s m(n) ≡ 0 (mod p n ), (2.2)
m 21 p m(n) ≡ 0 (mod s n ) (2.3)
If rad(p) does not divide s, there exists a prime number t dividing p such that for all m > 0, and all n > 0, s m is an invertible element in Z/t n Z. This implies m 12 ≡ 0 (mod t n ) for all n > 0, hence m 12 = 0. On the other hand, if rad(p) divides s, then for any n > 0, there exists m(n) > 0 large enough such that p n divides s m(n) . In this case any m 12 ∈ Z is solution of (2.2). Using the same arguments we have the same conclusion for m 21 with respect to rad(s) and p.

Assuming rad(p) does not divide s and rad(s) does not divide p, we get m 12 = 0 and m 21 = 0, so the only matrices satisfying (NC 2) are diagonal matrices. We conclude

N ( ← - Z 2 (L n ) ) is isomorphic to Z 2 /(2Z × 2Z).
Remark 2.7. The proof of Proposition 2.6 can be easily generalized to higher dimensions in the following way: Suppose that L = diag(p 1 , . . . , p d ) is an expansion d × d diagonal matrix. If rad(p i ) does not divide p j for some 1 ≤ i, j ≤ d, then for any

M ∈ N ( ← - Z d (L n )
), we have that m i,j = 0. In particular, if for any pair of distinct indices 1

≤ i = j ≤ d, rad(p i ) does not divide p j , then by (NC 2), N ( ← - Z d (L n ) ) is isomorphic to Z d /(2Z × • • • × 2Z
), as can be deduced by the proof of Theorem 28 in [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF].

The triangular case

We will next consider the case where the expansion matrix is a triangular matrix. We will focus on the upper triangular case, i.e., q = 0 and r = 0, since the lower triangular case can be deduce by this one via conjugation with the matrix 0 1 1 0 . For all n ∈ Z we have that

L n = p n q(n) 0 s n ,
where q(n) = q(p n -s n )/(p -s) = q n-1 i=0 p i s n-1-i . As in the diagonal case, det(L) = ps and trace(L) = p + s, so the condition rad(det(L)) divides trace(L) is equivalent to rad(p) = rad(s). In this case, there is a similar result to the one obtained for the diagonal case, as it is shown in the following proposition:

Proposition 2.8. Let L ∈ M(2, Z) be an expansion upper triangular matrix such that rad(det(L)) does not divide trace(L). Then, we have the following:

1. If rad(p) does not divide s and rad(s) divides p, then a matrix M ∈ GL(2, Z) is in

N ( ← - Z 2 L n ) if and only if (p -s) 2 m 12 = m 21 q 2 + (p -s)(m 11 -m 22 )q. In particular, if (p -s)|q, then N ( ← - Z 2 L n )/(Cent GL(2,Z) (L)) is virtually Z.
2. Assume that rad(p) divides s and rad(s) does not divide p. Then

N ( ← - Z 2 (L n ) )/(Cent GL(2,Z) (L)) is virtually Z.
3. If rad(p) does not divide s and rad(s) does not divide p, we have two cases: (2.4) Suppose rad(s) does not divide p. Then, there exists a prime number t dividing s such that for all n > 0 and m > 0, p m is an invertible element in Z/t n Z. Hence, m 21 ≡ 0 (mod t n ), which implies m 21 = 0, so m 21 = 0. Now, by (2.4) we get that m 12 s m ≡ 0 (mod p n ).

• If 2q ∈ (p -s)Z then N ( ← - Z 2 (L n ) ) is isomorphic to Z/2Z × Z/2Z. • Otherwise, N ( ← - Z 2 (L n ) ) is isomorphic to Z/2Z. Proof. Let M be in N ( ← - Z 2 (L n ) ). Define the matrix M = (p -s)M -(m 11 -m 22 )L -(p • m 22 -m 11 • s) id
(

We have two cases:

• If rad(p) does not divide s, then (2.5) implies m 12 = 0. We conclude that M = 0 0 0 0 , i. It is not difficult to see that M 2 is the identity matrix. We conclude that Finally, if rad(s) divides p, then for any n > 0, and any m large enough s n divides p m and q(m). Let t be a prime number dividing p that does not divide s. Then, by (2.4) we obtain (p -s) 2 m 12 s n+m ≡ m 21 q 2 s n+m (mod t n ).

N ( ← - Z 2 (L n ) ) is isomorphic to Z/2Z × Z/2Z. If 2q / ∈ (p -s)Z, then N ( ← - Z 2 (L n ) ) is isomorphic to Z/2Z.
(2.6) Since t does not divide s, then for any n, m > 0, s n+m is an invertible element in Z/t n Z, so (2.6) is reduced to

(p -s) 2 m 12 ≡ m 22 q 2 (mod t n ), (2.7)
which implies (p -s) 2 m 12 = m 21 q 2 . Thus, we get that

(p -s) 2 m 12 = m 21 q 2 + (p -s)(m 11 -m 22 )q. (2.8)
Now, if (p -s) divides q, we write q = k(p -s) for some k ∈ Z. By (2.8), we have that

       1 -m • k -mk 2 m 1 + m • k , 1 -m • k 2k -mk 2 m m • k -1 -1 -m • k -2k -mk 2 m 1 + m • k , -1 -m • k -mk 2 m -1 + m • k : m ∈ Z        .
In this case, L is conjugate to the matrix p 0 0 s , via the matrix 1 k 0 1 , so by the diagonal case (Proposition 2.6) we conclude that N ( ← -

Z 2 (L n ) )/(Cent GL(2,Z) (L)) is virtually Z.

Proof of Theorem 2.2

Now we will prove Theorem 2.2. Since we already proved the triangular case, we assume that q = 0 and r = 0.

Proof of Theorem 2.2. Let L ∈ M(2, Z) be an integer expansion matrix. The Cayley-Hamilton theorem implies L 2 = trace(L) • L -det(L) • id R 2 . If rad(det(L)) divides trace(L), then L 2 ≡ 0 (mod rad(det(L))). Hence, for all n > 0 and any m(n) > 0 large enough, we get that L m(n) ≡ 0 (mod det(L) n ). So any matrix in GL(2, Z) satisfies (NC 2), and we conclude that N ( ← -Z 2 (L n ) ) = GL(2, Z). Suppose now that rad(det(L)) does not divide trace(L). By induction, for any n > 0 we have that L n ≡ trace(L) n-1 L (mod det(L)). Since rad(det(L)) does not divide trace(L), there exists a prime number t dividing det(L) that does not divide p or s. Without loss of generality (up to a conjugation with 0 1 1 0 ) we may assume that t does not divide s. Since s(n) ≡ trace(L) n-1 s (mod det(L)), then, for all n > 0, and m > 0, s(m) is an invertible element in Z/t n Z.

Let M = m 11 m 12 m 21 m 22 be in N ( ← - Z 2 (L n ) ). Define the matrix M = rM -m 21 L -(r • m 11 -p • m 21 ) id R 2 .
Then, the matrix M satisfies (Normalizer Condition) and has the form

M = 0 m 12 0 m 22 ,
where

m 12 = r • m 12 -q • m 21 , m 22 = r • m 22 -s • m 21 -r • m 11 + p • m 21 , with m 12 , m 21 ∈ Z. Now, for all n, m > 0 adj(L n )M L m = r(m)(m 12 s(n) -m 22 q(n)) s(m)(m 12 s(n) -m 22 q(n)) r(m)(-m 12 r(n) + m 22 p(n)) s(m)(-m 12 r(n) + m 22 p(n)) ≡ 0 0 0 0 (mod det(L) n ).
(2.9)

Since s(m) is an invertible element in Z/t n Z, (2.9) implies

m 12 s(n) -m 22 q(n) ≡ 0 (mod t n ), (2.10)
-m 12 r(n) + m 22 p(n) ≡ 0 (mod t n ), (2.11)
which is equivalent to , so E is an adj(L)-invariant Z-module of rank at most 1.

adj(L n ) m 12 m 22 ≡ 0 0 (mod t n ). ( 2
If L does not have integer eigenvalues, E must have rank 0. This implies m 12 = m 22 = 0.

Hence M = 0 0 0 0 and then rM (so M ) commutes with L.

Since m 12 = m 22 = 0, we have that

r • m 12 -q • m 21 = 0 r • m 22 -s • m 21 -r • m 11 + p • m 21 = 0.
(2.13)

• Suppose p = s. In this case, (2.13) implies m 11 = m 22 and m 21 = m 11 • r/q. Note that L has complex eigenvalues if and only if (2p) 2 -4(p 2 -qr) < 0, i.e., qr < 0. Since | det(M )| = 1, then m 2 11 -m 2 12 • r/q = ±1, so the condition qr < 0 implies there exists a finite number of points (m 11 , m 12 ) ∈ Z 2 satisfying (2.13).

• If p = s, then (2.13) implies m 12 = q(m 11 -m 22 )/(p-s) and m 21 = r(m 11 -m 22 )/(ps). Since M ∈ GL(2, Z), we get that

m 11 m 22 -(m 11 -m 22 ) 2 qr (p -s) 2 = ±1. (2.14)
In this case, there is a finite number of solutions if (p -s) 2 -4qr < 0, which is equivalent to trace(L) 2 -4 det(L) < 0. Since trace(L) 2 -4 det(L) is the discriminant of the characteristic polynomial of L, this is equivalent to L having complex eigenvalues.

Now we will prove Proposition 2.3

Proof of Proposition 2.3.

First note that if M = m 11 m 12 m 21 m 22 is in GL(2, Z), then M -1 =            m 22 -m 12 -m 21 m 11 if det(M ) = 1, -m 22 m 12 m 21 -m 11 if det(M ) = -1.
Then, we will prove that if M satisfies the arithmetical relations given by the proof of Theorem 2.2, then M -1 also satisfies them. We will do it by cases as in the proof of Theorem 2.2:

• The statement is easily to verifies when L is a diagonal matrix.

• If L = p q 0 s is an upper triangular matrix. We only need to prove it for Item 1 in Proposition 2.8. The rest of the cases are easily to verify.

We recall that a matrix M satisfies (Normalizer Condition) in this case if and only if (p -s) 2 m 12 = m 21 q 2 + (p -s)(m 11 -m 22 )q. We will see that either of the cases det(M ) = 1 or det(M ) = -1, M -1 also satisfies this arithmetical relation. Indeed

-Assume that det(M ) = 1, then (p -s) 2 (-m 12 ) = -m 21 q 2 + (p -s)(m 22 -m 11 )q (p -s) 2 m 12 = m 21 q 2 + (p -s)(m 11 -m 22 )q.
-If det(M ) -1, we have that

(p -s) 2 m 12 = m 21 q 2 + (p -s)((-m 22 ) -(-m 11 ))q (p -s) 2 m 12 = m 21 q 2 + (p -s)(m 11 -m 22 )q.
We conclude that the statement is true when L is an upper triangular case. We recall the lower triangular case is deduced by this one.

• In the general case, we need to separate it in two cases:

-

If
-q • m 21 r • m 22 -s • m 21 -r • m 11 + p • m 21
is an eigenvector of adj(L) (with associated eigenvalue having a prime divisor not dividing trace(L)). A direct computation shows that in both cases (det(M ) = 1 or det(M ) = -1), if M ∈ GL(2, Z) satisfies this property, then the inverse matrix M -1 also satisfies it.

We conclude that the statement is true for all the different cases.

Remark 2.9.

1. In the particular case gcd(trace(L), det(L)) = 1, we can simplify the proof noting that (2.10), (2.11) imply the existence of two sequences

(k 1 n ) n>0 , (k 2 n ) n>0 ⊆ Z such that det(L) n k 1 n = m 12 s(n) -m 22 q(n) and det(L) n k 2 n = -m 12 r(n) + m 22 p(n), i.e., k 1 n = m 12 s(n) det(L) n -m 22 q(n) det(L) n k 2 n = -m 12 r(n) det(L) n + m 22 p(n) det(L) n . (2.15)
Since L is an expansion matrix, then L -1 is a contraction, so we have that

lim n→∞ p(n) det(L) n = lim n→∞ q(n) det(L) n = lim n→∞ r(n) det(L) n = lim n→∞ s(n) det(L) n = 0,
this implies for all n large enough, k 1 n = k 2 n = 0, and we conclude that m 12 = m 22 = 0.

Chapter 3

The recognizability property of constant-shape substitutions

The recognizability property of substitutions is a combinatorial property that provides a form of invertibility of the morphism, to uniquely decompose points in the substitutive subshift. For Z-actions, this property has a dynamical interpretation, since it implies the presence of refining Kakutani-Rokhlin partitions which allows encoding the dynamics in an infinite graph [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF] (called Bratteli diagrams). This coding enabled, among other results, the study of topological orbital equivalence of minimal Cantor systems [START_REF] Giordano | Topological orbit equivalence and C *crossed products[END_REF], to characterize continuous and measurable eigenvalues [START_REF] Durand | Eigenvalues of minimal Cantor systems[END_REF] and to analyze their invariant measures [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF]. It was first proved for any aperiodic primitive substitution by B. Mossé in [START_REF] Mossé | Puissances de mots et reconnaissabilité des points fixes d'une substitution[END_REF], and then in [START_REF] Bezuglyi | Aperiodic substitution systems and their Bratteli diagrams[END_REF] for the non-primitive case. Also, V. Berthé, W. Steiner, J. M. Thuswaldner, and R. Yassawi [START_REF] Berthé | Recognizability for sequences of morphisms[END_REF] studied a recognizability property for different types of one-dimensional morphisms ζ : A → B Z , where A and B can be different alphabets. In the multidimensional context, B. Solomyak showed in [START_REF] Solomyak | Nonperiodicity implies unique composition for self-similar translationally finite tilings[END_REF] that aperiodic translationally finite self-affine tilings of R d satisfy a recognizability property (called unique composition property).

In this chapter, we will get a similar result as [START_REF] Solomyak | Nonperiodicity implies unique composition for self-similar translationally finite tilings[END_REF], but for aperiodic symbolic factors (Proposition 3.3). We then present several consequences on invariants of factors and homomorphisms of substitutive subshifts: There exists a finite number of orbits in X ζ which are invariant by the substitution map (Proposition 3.4). They are not necessarily linearly repetitive (Example 3.6). Nevertheless, there is a polynomial growth of the repetitivity function of substitutive subshifts (Lemma 3.7). We determine their maximal equicontinuous factors, which is an explicit odometer system (Proposition 3.15). Thanks to these last descriptions, we get that any aperiodic symbolic factor of a substitutive subshift is conjugate to a substitutive subshift via a letter-to-letter map (Theorem 3.22). This extends the result proved in [START_REF] Müllner | Automorphisms of automatic shifts[END_REF]. In the original article, a key part is a characterization of periodic sequences by their complexity function. Our proof does not require this characterization, which is known as Nivat's conjecture in the two-dimensional case (and it is known to be false for higher dimensions [START_REF] Cassaigne | Subword complexity and periodicity in two or more dimensions[END_REF]).

The recognizability property of aperiodic symbolic factors of substitutive subshifts

The substitution ζ seen as a map from X ζ to ζ(X ζ ) is continuous. Moreover, when the constant-shape substitution is aperiodic and primitive this map is actually a homeomorphism. This comes from the notion of recognizability of a substitution.

Definition 3.1. A constant-shape substitution ζ is said to be recognizable, if there exists R > 0 such that for any x, y

∈ X ζ satisfying x| B(0,R)∩Z d = y| B(0,R)∩Z d , then x is in ζ(X ζ ) if and only if y is in ζ(X ζ ).
This implies for every x ∈ X ζ there exist a unique x ∈ X ζ and a unique j

∈ F ζ 1 such that x = S j ζ(x ). With this, the set ζ(X ζ ) is a clopen subset of X ζ and {S j ζ(X ζ ) : j ∈ F ζ
1 } is a clopen partition of X ζ (a proof for the one-dimensional case can be found in [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF], which can be adapted to our case). This property is also satisfied for the iterations ζ n , for all n > 0. The recognizability property was first proved for any aperiodic primitive substitution by B. Mossé in [START_REF] Mossé | Puissances de mots et reconnaissabilité des points fixes d'une substitution[END_REF] for the one-dimensional case, and in the multidimensional case by B. Solomyak in [START_REF] Solomyak | Nonperiodicity implies unique composition for self-similar translationally finite tilings[END_REF]. In this section we will prove it for aperiodic symbolic factors of substitutive subshifts (Proposition 3.3). This property will allows us to determine its maximal equicontinuous factors. The proof of Proposition 3.3 is a multidimensional analogue of the one given by P. Kůrka in [START_REF] Kurka | Topological and symbolic dynamics, volume 11 of Cours Spécialisés[END_REF] with the use of the following repulsion property for constant-shape substitutions proved in [START_REF] Solomyak | Nonperiodicity implies unique composition for self-similar translationally finite tilings[END_REF].

Proposition 3.2 (Repulsion Property

). [107, Lemma 3.2] Let ζ be an aperiodic primitive constant-shape substitution, and x ∈ X ζ . Then, there exists N > 0 such that, for all n > 0, and every neighborhood V ⊆ R d of the origin, if a pattern p x, with

(L n ζ (V ) ∩ Z d ) + s ⊆ supp(p) for some s ∈ Z d , has two occurrences j 1 , j 2 ∈ Z d such that j 1 -j 2 ∈ L n-N ζ (V ), then j 1 is equal to j 2 .
As proved in Lemma 1.22, we may assume that an aperiodic symbolic factor of a substitutive subshift is induced by a letter-to-letter map.

Proposition 3.3 (Recognizability property of aperiodic symbolic factors of substitutive subshifts).

Let A, B be two finite alphabets, ζ be an aperiodic primitive constant-shape substitution from the alphabet A, and T : A → B be a map such that (τ (X ζ ), S, Z d ) is an aperiodic subshift. Let x ∈ X ζ be a fixed point of ζ and y = τ (x). Then, there exists R > 0 such that if y| i+B(0,R) = y| j+B(0,R) and i

∈ L ζ (Z d ), then j ∈ L ζ (Z d ).
Proof. Let N given by Proposition 3.2 and V ⊆ R d be a neighborhood of 0 such that

L N ζ ([-1, 1] d ) ⊆ V and let B = d i=1 I i be a box containing (V + [-1, 1] d )
, where I i is a finite interval of Z. Assume the contrary, then for every n > 0 there exist

i n ∈ L ζ (Z d ), j n / ∈ L ζ (Z d ) and a pattern u n ∈ L B (X ζ ) such that y| in+L n ζ (B∩Z d ) = τ (ζ n (u n )) = y| jn+L n ζ (B) . τ (ζ n (u n )) • • • • • • • • • • • • • • • • • • τ (ζ n (u n )) • • • • • • • • • • • • • • • • • • i n × j n × j n -i n Figure 3.1: Illustration of the patterns τ (ζ n (u n )).
The pattern in i n (black) comes from the fact that x = ζ(x). We will prove that the pattern in j n (blue) also comes from this fact.

Since x = ζ(x), there exist a finite subset

C (n) Z d , a pattern v n ∈ L C (n) (X ζ ) and an occurrence c n ∈ L ζ (Z d ) of ζ(v n ) with j n + (L n ζ (B) ∩ Z d ) ⊆ c n + L n ζ (C (n) ) ⊆ (F ζ n +n)∩(jn+L n ζ (B)) =∅ n∈Z d F ζ n + n. In particular, ζ n (u n ) ζ n (v n ),
as illustrated in Fig. 3.2:

τ (ζ n (v n )) τ (ζ n (u n )) • • • • • • • • • • • • • • • • • • j n × Figure 3.2: Illustration of the patterns τ (ζ n (v n )) and τ (ζ n (u n )) in j n .
By the Pigeonhole Principle, there are an infinite set

E ⊆ N, a finite set C Z d and patterns u ∈ L B (X ζ ), v ∈ L C (X ζ ) such that for all n ∈ E, C (n) is equal to C, u n = u, and v n = v. Consider D = {n ∈ C : n + -1, 1 d ⊆ C}, and let w = x| kn+D where k n ∈ Z d is such that L n ζ (k n + D) is strictly contained in j n + (L n ζ (B) ∩ Z d ) and set a n = x| (jn+L n ζ (B))\(L n ζ (kn+D)) . We have that supp(a n ) ⊆ (∂(j n + L n ζ (B)) + L n ζ ([-1, 1] d ))
as illustrated in Fig. 3.3:

τ (ζ n (v)) τ (ζ n (u)) τ (ζ n (w)) • • • • • • • • • • • • • • • τ (a n ) L n ζ (k n ) × j n ×
m-n (ζ n (w)) = ζ m (w). If ζ m-n (a n ) = a m , there is two occurrences of ζ m (w) in ζ m (u).
Since the supports of ζ m-n (a n ) and a m are in the same set, the distance between these two occurrences is smaller than max

t∈[-1,1] d L m ζ (t)
. If these occurrences are not the same, the repulsion property (Proposition 3.2) gives a contradiction, so ζ m-n (a n ) = a m as illustrated in Fig. 3.4:

τ (ζ m (v)) τ (ζ m (u)) τ (ζ m (w)) • • • • • • • • • • • • • • • τ (ζ m-n (a n )) = τ (a m ) L m-n ζ (k n ) L m-n ζ (j n ) × × Figure 3.4: Illustration of the patterns ζ m-n (a n ) in L m-n ζ (j n ). Now, since x = ζ m-n (x), y| L m-n ζ (jn+L n ζ (B)) is equal to τ (ζ m (u)). Hence, we have L m-n ζ (j n ) -L m-n ζ (k n ) = j m -L m ζ (k m ). This implies that j m ∈ L ζ (Z d
) which gives a contradiction.

Invariant orbits of substitutive subshifts

As mentioned in the last proof, we assume that primitive constant-shape substitutions admit at least one fixed point for the map ζ : X ζ → X ζ . The orbits of these fixed points lead to the notion of ζ-invariant orbits. An orbit O(x, Z d ) is called ζ-invariant if there exists j ∈ Z d such that ζ(x) = S j x, i.e., the orbit is invariant under the action of ζ in X ζ . Since for every n ∈ Z d we have ζ • S n = S L ζ n • ζ, the definition is independent of the choice of the point in the Z d -orbit of x. The orbit of a fixed point of the substitution is an example of an invariant orbit. In the following, we will prove that for aperiodic primitive constant-shape substitutions there exist finitely many ζ-invariant orbits. This property will be used to prove other properties about some constant-shape substitutions such as coalescence (Proposition 4.7) and the automorphism group of some substitutive subshifts is virtually generated by the shift action (Proposition 4.8). Proof. Let x ∈ X ζ be such that ζ(x) = S jx x, for some j x ∈ Z d . For any m ∈ Z d , we have . Inductively, we obtain for every n ≥ 0

ζ(S m x) = S L ζ m ζ(x) = S L ζ m+jx x = S (L ζ -id)m+jx S m x,
and thus j x -j S m x ∈ (L ζ -id)Z d . Let H Z d be a fundamental domain of (L ζ -id)(Z d ) in Z d with 0 ∈ H. We may assume that x ∈ X ζ is in a ζ-invariant orbit with j x ∈ H. Set K ζ Z d be
x| n k=0 L k ζ j +L n+1 ζ (D)+F ζ n+1 = y| n k=0 L k ζ j +L n+1 ζ (D)+F ζ n+1
.

Let E 0 be equal to D and for all n > 0, define

E n = n-1 k=0 L k ζ j + L n ζ (D) + F ζ n .
We will prove that n≥0 E n = Z d . This implies x = y, which is a contradiction. To do this, we will prove that for every n ≥ 0 that Note that

L n ζ (K ζ ) + F ζ n ⊆ E
L n ζ (K) + F ζ n ⊆ E n+1 if and only if L n ζ (K -j) + n-1 k=0 L k ζ (F ζ 1 -j) ⊆ L n+1 ζ (D) + F ζ n+1 .
Claim 1. For every n ≥ 0 we have

n k=0 L k ζ (F 1 -j) ⊆ L n+1 ζ (C) + F ζ n+1 .
Proof of Claim. For n = 0, note that

F 1 -j is included in L ζ (C) + F ζ 1 by Proposition 1.20. Assume that for some n ≥ 0 n k=0 L k ζ (F 1 -j) ⊆ L n+1 ζ (C) + F ζ n+1 . We have n+1 k=0 L k ζ (F 1 -j) = n k=0 L k ζ (F 1 -j) + L n+1 ζ (F 1 -j) ⊆ L n+1 ζ (C) + F ζ n+1 + L n+1 ζ (F 1 -j) ⊆ L n+1 ζ (C + F ζ 1 -j) + F ζ n+1 ⊆ L n+2 ζ (C) + L n+1 ζ (F ζ 1 ) + F ζ n+1 (by Proposition 1.20) = L n+2 ζ (C) + F ζ n+2
We conclude that for every n ≥ 0,

n k=0 L k ζ (F 1 -j) ⊆ L n+1 ζ (C) + F ζ n+1 By Claim 1 L n ζ (K -j) + n-1 k=0 L k ζ (F 1 -j) is included in L n ζ (K -j) + L n ζ (C) + F ζ n , and
L n ζ (K + C -j) is a subset of L n+1 ζ (D) + L n ζ (F ζ 1 ), so we have L n ζ (K -j) + L n ζ (C) + F ζ n ⊆ L n+1 ζ (D) + L n ζ (F ζ 1 ) + F ζ n = L n+1 ζ (D) + F ζ n+1
and we conclude the proof.

Remark 3.5. Let ζ be an aperiodic primitive substitution with an expansion matrix

L ζ such that | det(L ζ -id R d )| = 1. This implies (L ζ -id R d )(Z d ) = Z d . Let x ∈ X ζ be a point in a ζ-invariant orbit, i.e., there exists j ∈ Z d such that ζ(x) = S j x and set m ∈ Z d such that (L ζ -id R d )(m) = -j. Then ζ(S m x) = S L ζ m+j ζ(x) = S m+(L ζ -id R d )(m)+j x = S m x.
Hence S m x is a fixed point of ζ. We conclude that the only ζ-invariant orbits in this case are the ones given by the fixed points of the substitution.

The repetitivity function of substitutive subshifts

Let ζ be an aperiodic primitive constant-shape substitution, and assume that x ∈ X ζ is a fixed point of the substitution. The minimality property implies the substitutive subshift is repetitive, i.e., for every pattern p x there is a radius R > 0 such that for every n ∈ Z d , the ball B(n, R) contains an occurrence of p in x. The repetitivity function is the map M X ζ : R + → R + defined for R > 0 as the smallest radius such that every ball B(n, M X ζ (R)) contains an occurrence of every pattern with diam(supp(p)) ≤ 2R. We say the substitution is linearly recurrent or linearly repetitive if the repetitivity function has a linear growth, i.e., there exists C > 0 such that M X ζ (R) ≤ C • R. The notion of linearly recurrent was introduced by F. Durand in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], to study the relations between the substitutive dynamical systems and the stationary dimension groups. For the multidimensional context, the same notion appears in [START_REF] Lagarias | Local complexity of Delone sets and crystallinity[END_REF] for Delone sets of R d . In fact, according to a result in [START_REF] Lagarias | Local complexity of Delone sets and crystallinity[END_REF]Theorem 2.3], the linear growth of the repetitivity function is the slowest possible for an aperiodic Delone set. F. Durand proved one-dimensional substitutive subshift are linearly recurrent [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], but in the multidimensional case this is no longer true, as we can see in Example 3.6.

Example 3.6 (A nonlinearly repetitive constant-shape substitution). Consider the substitution σ 1 , given by L σ 1 = 2 0 0 3 , and F σ 1 1 = 0, 1 × 0, 2 defined by

σ 1 : b c a c c b a → c b b → c b c → a c a b b c c b.
For p > 1, we consider the pattern

w p = σ p 1 (a)| 0,2 p-1 ×{0} ∈ L 0,2 p-1 ×{0} (X ζ ). A direct induction enable to prove that if w p σ p 1 α β γ δ
, for α, β, γ, δ ∈ {a, b, c, ε} (where ε denote the empty pattern), then one of the letters must be a. Moreover, w p only appears in the lower left corner of the pattern σ p 1 (a). These properties imply that there is only one occurrence of w p in σ p 1 (w p ), which is in the lower corner of σ p 1 (w p ) as seen in Fig. 3.5:

σ p 1 (w p ) • • • • • • • • • • • • • • • • • • w p 3 p 4 p σ p 1 (a) σ p 1 (b) Figure 3.5: Decomposition of σ p 1 (w p ).
Then, there is a ball of radius 3 p /2 in the support of σ p 1 (w p ) with no occurrences of w p . Since this is true for any p, this implies that this substitution is not linearly recurrent.

However, the repetitivity function has at most polynomial growth, with exponent depending only on the expansion map of the substitution. Lemma 3.7. Let ζ be an aperiodic primitive constant-shape substitution.

Then M X ζ (R) is O   R - log( L ζ ) log ( L -1 ζ )   . Proof. Using A = K ζ + K ζ , and F = F ζ 1 + F ζ 1 in Proposition 1.20
, where K ζ is given by Proposition 1.18, we obtain a subset C Z d such that for all n > 0,

L n ζ (K ζ + K ζ ) + F ζ n + F ζ n ⊆ L n ζ (C) + F ζ n .
The recognizability property implies for every pattern p ∈ L(X ζ ) such that its support is contained in

L n ζ (K ζ ) + F ζ n , there exists a pattern w ∈ L C (X ζ ) with p ζ n (w). Set T = M X ζ (diam(C))
. By definition, any ball of radius T contains an occurrence of every pattern in

L L n ζ (K ζ )+F ζ n (X ζ ). Set R > 0. Fix n > 0 such that L n-1 ζ (K ζ ) + F ζ n-1 ⊆ B(0, R) ⊆ L n ζ (K ζ ) + F ζ n . By definition of n, we get that inf t =1 L n-1 ζ (t) ≤ 2R, hence 1/ L -1 ζ n-1 ≤ 2R. Thus M X ζ (R) ≤ L n ζ T ≤ L ζ n T ≤ C(ζ)R -(log( L ζ ))/(log( L -1 ζ )) ,
for some constant C(ζ) independent of R.

Remark 3.8. The following statements can be easily verified.

(1) In the case of a symmetric expansion map for the substitution, a bound for

M X ζ (R) is
given by the eigenvalues of the expansion matrix: (2) In the case of a self-similar tiling (where the expansion map satisfies L ζ (t) = λ t , for some λ > 0), the norm matrix satisfies

M X ζ (R) = O R (log(|λ 1 |))/(log(|λ d |)) ,
L ζ = ( L -1 ζ
) -1 = λ, so the repetitivity function is O(R), i.e., has a linear growth. Hence self-similar substitutions are linearly recurrent, as it was proved in [START_REF] Solomyak | Nonperiodicity implies unique composition for self-similar translationally finite tilings[END_REF].

(3) The sufficiency of the previous case is not true, there exist constant-shape substitutions that are not self-similar, but are linearly recurrent.

Substitutive subshifts as extension of d-dimensional odometers

In this section, we will describe substitutive subshifts as symbolic extensions of odometer systems, which are given by the data of the substitution. Actually, the maximal equicontinuous factor of a substitutive subshift is an explicit odometer system (Proposition 3.15). This was first made by F. Dekking in [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF] for the one-dimensional case, where he introduced the notion of height to describe the return times of a letter which are coprime with the K2: Suppose that ←g does not satisfy K1. Proceeding as in the previous case, we assure the existence of two indices j 1 = j 2 such that y n

j 1 | K ζ = y n j 2 | K ζ for any n in an infinite subset E ⊆ N. This will imply that x j 1 | -gn+L n ζ (K ζ )+Fn = x j 2 | -gn+L n ζ (K ζ )+F ζ n for any n ∈ E. Now, note that for any n ∈ N, g n+1 -g n ∈ L n (F ζ 1 )
, and a direct induction prove that -

g n + L n (K ζ ) + F ζ n is included in -g n+1 + L n+1 (K ζ ) + F ζ n+1 . Since K ζ ⊆ K ζ ,
we conclude that x j 1 = x j 2 which will be a contradiction.

Remark 3.10. Note that (1) The set of points ← -g ∈ ← - Z d (L n ζ ) satisfying K1 is a G δ -set. Indeed, for any M > 0 define U M = ← -g ∈ ← - Z d (L n ζ ) : -M, M d ⊆ -g n + F ζ n , for some n > 0 . Note that U M is an open subset of ← - Z d (L n ζ ) and M >0
U M are exactly the points satisfying K1. Now, for any n in Z d and M > 0 we have that U M is included in U M +|n| , hence the set of points on the odometer satisfying K1 is invariant by the Z d -action.

(2) We will prove that the existence of a point

← -g ∈ ← - Z d (L n ζ (Z d )) satisfying K1 is equivalent to the existence of N ∈ N and f ∈ F ζ N such that B(0, R L ζ ) ∩ Z d ⊆ F ζ N -f , with R L ζ = max L ζ 2 , L -1 ζ 1 -L -1 ζ 1 + L ζ 2 .
Which is always true by the Følner assumption. We prove the nontrivial implication. We will show by induction that for every r ≥ R L ζ there exists n ∈ N and f r ∈ F ζ n such that B(0, r)∩Z d ⊆ F ζ n -f r . By hypothesis, this is true for r = R L ζ . Now, assume that for some r ≥ R L ζ there exists n ∈ N and

f r ∈ F ζ n such that B(0, r) ∩ Z d ⊆ f ζ n -f r . Then, L ζ (B(0, r) ∩ Z d ) is included in L ζ (F ζ n ) -L ζ (f r ). This implies L ζ (B(0, r) ∩ Z d ) + B 0, L ζ 2 ∩ Z d ⊆ F ζ n+N +1 -L ζ (f r ) -f . Now, we prove that B(0, r + 1) ∩ Z d ⊆ L ζ (B(0, r) ∩ Z d ) + B (0, L ζ /2) ∩ Z d . Set n ∈ B(0, r + 1) ∩ Z d . Since L ζ (Z d ) is L ζ /2-relatively dense, we can write n = m + L ζ (p), with m ∈ B (0, L ζ /2) ∩ Z d and p ∈ Z d . We have that p ≤ L -1 ζ L ζ (p) = L -1 ζ n -m ≤ L -1 ζ r + 1 + L ζ 2 .
The last expression is smaller than r whenever r

≥ (L -1 ζ /(1-L -1 ζ ))•(1 + L ζ /2), which implies p ∈ B(0, r) ∩ Z d . Since L ζ (f r ) + f is in F ζ n+N +1
, we conclude that there exist m ∈ N and f ∈ F ζ m such that B(0, r + 1) ∩ Z d ⊆ F ζ m -f . Finally, using classical compactness arguments we conclude the existence of a point ←g ∈ ← -

Z d (L n ζ (Z d )) such that n>0 F ζ n -g n = Z d .
(3) Let ζ be an aperiodic bijective primitive constant-shape substitution. The map p 0 is a permutation of A and we consider a power of p 0 such that p n 0 is equal to the identity. Since ζ is primitive, we replace it by ζ n and with this we may assume that ζ possess at least |A| fixed points. Now, let ←g satisfying K1. For any a ∈ A, consider a fixed point of ζ, denoted as x a , with x a (0) = a and define x ←g a = lim nm→∞ S gn m x a for some convergent subsequence. Since the sets {S gn [ζ n (a)]} a∈A are disjoint for any

a = b ∈ A, x ← -g a is different from x ← -g b . Finally, noticing that π(x ← -g a ) = ← -g , we have that π -1 ({ ← -g }) ≥ |A|.
By Lemma 3.9 we conclude that π -1 ({ ←g }) = |A| for any ←g satisfying K1, and then the factor map π :

(X ζ , S, Z d ) → ( ← - Z d (L n ζ ) , + (L n ζ ) , Z d ) for aperiodic bijective constant-shape substitutions is almost |A|-to-1.
In general, this d-dimensional odometer is not the maximal equicontinuous factor of aperiodic constant-shape substitutions.

In some particular cases we can explicitly compute the cardinality of the fibers given by the topological factor in Lemma 3.9, such as in the two examples studied in [START_REF] Robinson | On the table and the chair[END_REF].

The following result shows other examples where we can compute the cardinality of the fibers.

Lemma 3.11. Let L ∈ M(d, Z) be an integer expansion matrix with det(L) ≥ 3 and F 1 be a fundamental domain of L(Z d ) in Z d . Let A be a finite alphabet with cardinality |A| = |F 1 | -1 and τ : F 1 \ {0} → A be a bijection. We define a substitution σ L as the following:

∀a ∈ A, σ(a) f = a f = 0, τ (f ) f = 0.
Under the hypothesis that the sequence of supports of the iterations σ n L is a Følner sequence, σ L is an aperiodic primitive constant-shape substitution and the factor map π :

(X σ L , S, Z d ) → ( ← - Z d (L n ) , + (L n ) , Z d ) is almost 1-to-1. Moreover, we have that |π -1 ({ ← -g })| = |A| ← -g ∈ O( ← - 0 , Z d ), 1 ← -g / ∈ O( ← - 0 , Z d ).
Proof. Since τ is a bijection, by definition σ L is a primitive substitution. Now, we prove that σ L is aperiodic. To prove this we prove σ L is recognizable. We follow similar ideas of the proof of the recognizability property in [START_REF] Kurka | Topological and symbolic dynamics, volume 11 of Cours Spécialisés[END_REF].

Let x be a fixed point of σ L and n, m ∈ Z d such that n ∈ L(Z d ) and x| n+F 1 = x| m+F 1 . We prove that m ∈ L(Z d ). Indeed, since x is a fixed point of the substitution and n ∈ L(Z d ) we have that for all f ∈ F 1 \ {0},

x m+f = x n+f = τ (f ). Set g ∈ F \ {0} such that m + g / ∈ L(Z d ) (such g ∈ F 1 \ {0} exists since |F 1 | ≥ 3
). We write m + g = L(p) + h, with h ∈ F 1 . Note that h = 0. Hence

x m+g = x n+g = τ (g), but x is a fixed point of σ L so x m+g = σ L (x p ) h = τ (h),
i.e., τ (g) = τ (h). By bijectivity of τ we have that g = h, so m = L(p). We conclude that σ L is a recognizable substitution, i.e., σ L is aperiodic. Now we study the fibers π -1 ({ ←g }). Assume that ←g = ← -0 . By the proof of Lemma 3.9 we need to compute σ n L (w) for patterns w ∈ L Kσ L (X σ L ) and n > 0. Let w 1 , w 2 be two patterns in L Kσ L (X σ L ). Note that, by definition of σ L , the cardinality of the coordinates where the patterns σ n L (w 1 ), σ n L (w 2 ) differ is constant on n > 0 and is at most

|K σ L |. Indeed, if W = {k ∈ K σ L : w 1 (k) = w 2 (k)}, then for any k ∈ W , σ n L (w 1 ) L n (k) = σ n L (w 2 ) L n (k) and for any a ∈ (L n (K σ L ) + F σ L n ) \ L n (W ) we have that σ n L (w 1 ) a = σ n L (w 2 ) a .
Since the distance between these differences increase (exponentially on n > 0) and for any n > 0 and a ∈ A the patterns

σ n L (a)| F σ L n and σ n+1 L (a)| F σ L n
are the same, we get two cases:

• If w 1 (0) = w 2 (0), the patterns σ n L (w 1 ), σ n L (w 2 ) converge to two points x 1 , x 2 ∈ X σ L with x 1 (0) = x 2 (0) and for any n ∈ Z d , x 1 (n) = x 2 (n).
• At the opposite, if w 1 (0) = w 2 (0), the patterns σ n L (w 1 ), σ n L (w 2 ) converge to the same point x ∈ X σ L . This implies |π -1 ({ ← -0 })| ≤ |A| and since for any letter a ∈ A we have a fixed point x a such that x a (0) = a, then |π

-1 ({ ← - 0 })| ≥ |A|. We conclude that |π -1 ({ ← - 0 })| = |A|.
Since this property is invariant under translation, we conclude that for any

← -g ∈ O( ← - 0 , Z d ) that |π -1 ({ ← -g })| is equal to |A|.
For the other case, consider

← -g = (g n ) n>0 / ∈ O( ← - 0 , Z d ). This implies for every n > 0 exists m > n such that g m = 0. Set w 1 , w 2 ∈ L Kσ L (X σ L ) and let W = {k ∈ K σ L : w 1 (k) =
w 2 (k)} be their set of differences. By definition of σ L , for any n > 0 the coordinates where σ n L (w 1 ) and σ L (w 2 ) differ is L n (W ). Then, for any M > 0 we can find n > 0 such that

g n + -M, M d ⊆ L n (K σ L )+F σ L n and g n + -M, M d ∩L n (Z d ) = ∅. Hence σ n L (w 1 )| gn+ -M,M d = σ n L (w 2 )| gn+ -M,M d .
Moreover, a direct computation shows that for any n > 0, the patterns

σ n L (w 1 )| gn+ -M,M d , σ n+1 L (w 1 )| g n+1 + -M,M d
are the same. This implies, taking M > 0 arbitrarily large, for any w ∈ L Kσ L (X σ L ), the patterns σ n L (w 1 )| gn+ -M,M d are the same and then converge to a unique point x ∈ X σ L such that π(x) = ←g . We conclude that |π -1 ( ←g )| = 1.

We will use this result to a study the fibers of a particular case in Chapter 6.

The maximal equicontinuous factor of a substitutive subshift

In this section, we will describe the maximal equicontinuous factor of substitutive subshifts (see [START_REF] Frank | Multidimensional constant-length substitution sequences[END_REF] for the case with diagonal expansion matrices and [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF] for the one-dimensional case).

A subgroup L ≤ Z d is called a lattice if it is isomorphic to Z d , i.e., it has finite index. For a lattice L of Z d , we define the dual lattice of L, as the subgroup

L * = {x ∈ R d : x, n ∈ Z, ∀n ∈ L}.
We have that (Z d ) * = Z d , and for any R ∈ M(d, Z) the set R(Z d ) is a lattice of Z d with dual lattice equal to (R * ) -1 (Z d ), where R * stands for the algebraic adjoint of R. Let L 1 , L 2 be two lattices of Z d . We denote by L 1 ∨ L 2 the smallest lattice that contains L 1 and L 2 , i.e., if a lattice L containing L 1 and L 2 , then must contains

L 1 ∨ L 2 .
Fix x ∈ X ζ . We define the set of return times as

R(X ζ ) = {j ∈ Z d : ∃k ∈ Z d , x k+j = x k }.
By minimality, this set is well-defined independently of x ∈ X ζ , and it is syndetic, i.e., there exists a finite subset

A Z d such that R(X ζ )+A = Z d . We define L(R(X ζ )) as the smallest lattice containing R(X ζ ). The height lattice H(X ζ ) of a constant-shape substitution ζ is the smallest lattice containing L(R(X ζ )) such that H(X ζ ) ∩ L ζ (Z d ) ≤ L ζ (H(X ζ )). Notice that the last property is equivalent to H(X ζ ) ∩ L n ζ (Z d ) ≤ L n ζ (H(X ζ )), for any n > 0. The height lattice is trivial whenever H(X ζ ) = Z d .
In the following, we will give a description for the height lattice. For k ∈ Z d , we define

R k (X ζ ) = {j ∈ Z d : x j+k = x k }. Let L(R k (X ζ )) be the smallest lattice containing R k (X ζ ) and H k (X ζ ) be the smallest lattice containing L(R k (X ζ )) such that H k (X ζ ) ∩ L ζ (Z d ) ≤ L ζ (H k (X ζ ))
. We adapt the proof for the one-dimensional case [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF][START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] and obtain the following result. Lemma 3.12. Let ζ be an aperiodic primitive constant-shape substitution. Then, for any

k 1 , k 2 ∈ Z d the sets H k 1 (X ζ ), H k 2 (X ζ ) are the same. In particular, for any k ∈ Z d , H k (X ζ ) is equal to H(X ζ ). Proof. Let x ∈ X ζ be a fixed point of the substitution, k 1 , k 2 ∈ Z d and N be large enough such that x k 2 ζ N (x k 1 ). Set m such that x k 2 = ζ N (x k 1 ) m . Since x = ζ N (x) for any j ∈ R k 1 (X ζ ), L N ζ (k 1 + j) + m ∈ R k 2 (X ζ ). Hence L N ζ (j) is in L(R k 2 (X ζ )) and therefore in H k 2 (X ζ ). By definition of H k 2 (X ζ ) and invertibility of L N ζ , we conclude that j ∈ H k 2 (X ζ ). Since it is the smallest lattice satisfying the property, H k 1 (X ζ ) is a subgroup of H k 2 (X ζ ),
and by reciprocity we have that these sets are the same. We conclude the second equality by observing that H(

X ζ ) = k∈Z d H k (X ζ ).
As in the one-dimensional case, to study the maximal equicontinuous factor of substitutive subshifts, we study theirs eigenvalues. A vector x ∈ R d is said to be an eigenvalue for the topological dynamical system (X ζ , S, Z d ) (the measure-preserving system (X ζ , µ, S, Z d )) if there exists a continuous function f :

X ζ → C (f ∈ L 2 (X ζ , µ ζ )) such that for every n ∈ Z d , f • S n = e 2πi x,n f in X ζ (f • S n = e 2πi x,n f in X ζ , µ ζ -a.e. in X ζ ).
In [START_REF] Solomyak | Eigenfunctions for substitution tiling systems[END_REF] it was proved the following result, generalizing the characterization of eigenvalues for the one-dimensional case [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF]. 1. The vector x is a continuous eigenvalue for the topological dynamical system (X ζ , S, R d ).

2. The vector x is a measurable eigenvalue for the measure-preserving system

(X ζ , µ ζ , S, R d ).
3. The vector x satisfies the following condition:

lim n→∞ e 2πi L n ζ j,α = 1, ∀ j ∈ R(X ζ ).
4. The vector x satisfies the following condition:

lim n→∞ e 2πi L n ζ j,α = 1, ∀ j ∈ H(X ζ ).
Remark 3.14. We have that (1) Condition 4. in Theorem 3.13 is not proved in [START_REF] Solomyak | Eigenfunctions for substitution tiling systems[END_REF] but it can be easily checked noticing the set of points satisfying (L * ζ ) N α, j ∈ Z is a lattice.

(2) The same results is satisfied for the topological dynamical system (X ζ , S, Z d ) with the same arguments given in [START_REF] Solomyak | Eigenfunctions for substitution tiling systems[END_REF].

This implies the set of continuous (and measurable) eigenvalues of (X ζ , S, Z d ) corresponds to the set

n≥0 (L * ζ ) -n (H * (X ζ )). In particular the set of eigenvalues E(X ζ , S, Z d ) of (X ζ , S, Z d ) is a subset of Q d . A direct consequence of Theorem 3.
13 is a description of the maximal equicontinuous factor of (X ζ , S, Z d ).

Proposition 3.15. Let ζ be an aperiodic primitive constant-shape substitution. The maximal equicontinuous factor of the substitutive subshift of (X ζ , S, Z d ) is the odometer system

( ← - Z d (L n ζ (H(X ζ ))) , + L n ζ (H(X ζ )) , Z d ) = lim ←n (Z d /L n ζ (H(X ζ )), α n ), + (L n ζ (H(X ζ ))) , Z d .
For completeness, we provide another description for the maximal equicontinuous factor in a general setting. Let (X, T, Z d ) be a topological dynamical system, where X is a Cantor set, and Γ ≤ Z d a subgroup with finite index. We say (X, T, Z d ) admits a Γ-minimal partition if there exists a closed partition

g∈Z d /Γ X g = X such that for all g ∈ Z d /Γ the set of return times RT (X g ) = {n ∈ Z d : T -n (X g ) ∩ X g = ∅}
is equal to Γ and the topological dynamical system (X g , Γ) is a minimal system. Note that, since its a finite partition, the sets X g are clopen. The minimality of the induced actions implies there is at most one Γ-minimal partition up to a permutation of the sets {X g } g∈Z d /Γ . A Γ-minimal partition is associated with an equicontinuous factor of (X, T, Z d ). To see this we enumerate X g such that for all g ∈ Z d /Γ and n ∈ Z d we have that T n X g = X g+n mod Γ . Then, the map π : (X, T,

Z d ) → (Z d /Γ, + Γ , Z d ), such that π(x) = g if and only if x ∈ X g , is a factor map onto (Z d /Γ, + Γ , Z d )
, where Z d acts by quotient translations onto Z d /Γ. The following proposition shows the connection between Γ-minimal partitions and eigenvalues of a topological dynamical system. Proposition 3.16. Let (X, T, Z d ) be a minimal topological dynamical system and Γ ≤ Z d be a finite index subgroup. The system (X, T, Z d ) admits a Γ-minimal partition, if and only

if Γ * ⊆ E(X, T, Z d ). Proof. Let R ∈ M(d, Z) be such that Γ = R(Z d ). Let {X g } g∈R([0,1) d )∩Z d be a Γ-minimal partition satisfying X g = T g (X 0 ) for all g ∈ M ([0, 1) d )∩Z d . Let x be in (R * ) -1 (Z d ). Define f as the map f = g∈M ([0,1) d )∩Z d e 2πi x,g 1 Xg . Since the sets {X g } g∈R([0,1) d )∩Z d are clopen, the map f is continuous. Let x ∈ X 0 , m ∈ Z d and m 1 ∈ Z d , m 2 ∈ R([0, 1) d ) ∩ Z d be such that m = R(m 1 ) + m 2 . Note that T m x is in X m 2 and since x is in (R * ) -1 (Z d ), we have that e 2πi α,m = e 2πi α,R(m 1 )+m 2 = e 2πi( α,R(m 1 ) + α,m 2 ) = e 2πi α,m 2 .
We find a continuous map such that f (T m x) = e 2πi x,m f (x) for all x ∈ X and all m ∈ Z d . We conclude that x ∈ E(X, T, Z d ).

On the other hand, let x be in X. For j ∈ {1, . . . , d} we denote x j = (R * ) -1 (e i ). Since Γ * ⊆ E(X, T, Z d ) there exists a map f j : X → C such that f j (T m x) = e 2πi x j ,m f j (x) for all m ∈ Z d . Since the eigenspaces are one-dimensional we choose f j such that f j (x) = 1. By the previous formula, the values of e 2πi x j ,m only depend on

m ∈ R([0, 1) d ) ∩ Z d . Now, for any j ∈ {1, . . . , d} and m ∈ R([0, 1) d )∩Z d we denote X j m = f -1 j ({e 2πi α j ,m }). For each j ∈ {1, . . . , d}, the set X is equal to m∈R([0,1) d )∩Z d X j m . We define X m = j∈{1,...,d} X j m . We will prove that {X m } m∈R([0,1) d )∩Z d is a Γ-minimal partition. Note that X m is Γ-invariant. First, assume that there exist n 1 , n 2 ∈ R([0, 1) d )∩Z d such that n 1 = n 2 and X n 1 ∩X n 2 = ∅.
Using the Z d -action on X, this is equivalent to the existence of m ∈ R([0, 1) d ) ∩ Z d with m = 0 and X m ∩ X 0 = ∅. This means that for all j ∈ {1, . . . , d}, e 2πi x j ,m is equal to 1, i.e., x j , m ∈ Z. Since Γ * = {x 1 , . . . , x d } , we have that m ∈ (Γ * ) * = Γ which is a contradiction, so all of these sets are disjoint.

By minimality, we have that

X = m∈R([0,1) d )∩Z d O(T m x, Γ). Since O(T m x, Γ) is included in X m , we conclude that {X m } m∈R([0,1) d )∩Z d is a clopen partition of X and O(T m x, Γ) = X m , so the action of Γ on X m is minimal. A direct computation shows that the set of return times of each X m is Γ. Hence {X m } m∈R([0,1) d )∩Z d is a Γ-minimal partition.
Remark 3.17. The following statements can be easily verified.

(1) In the case of an aperiodic primitive constant-shape substitution ζ, the recognizability property implies for all n > 0, the sets

{S j ζ n (X ζ )} j∈F ζ n are a L n ζ (Z d )-minimal partition.
(2) By Theorem 3.13 and Proposition 3.16 for any aperiodic primitive constant-shape substitution there exists a H(X ζ )-minimal partition.

Aperiodic symbolic factors of substitutive subshifts are conjugate to substitutive subshifts

In the following we will prove Theorem E. This is a multidimensional analogue of a result proved by C. Müllner and R. Yassawi [START_REF] Müllner | Automorphisms of automatic shifts[END_REF], which is a refinement of a result proved in [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF] for constant-length substitutions. We follow the same strategy of [START_REF] Müllner | Automorphisms of automatic shifts[END_REF], with a slight difference. By Proposition 3.3 aperiodic symbolic factors of substitutive subshifts are recognizable. In the original article, this is not mentioned. They proved an odometer system (the one defined in Section 3.4) is an equicontinuous system of aperiodic symbolic factors of substitutive subshifts, following the ideas developed by T. Kamae [START_REF] Kamae | A topological invariant of substitution minimal sets[END_REF] about minimal partitions. To get this, they used a characterization of periodic sequences and the complexity function, known as Morse-Hedlund theorem [START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF]. Until now, this characterization is not known to be true in the two-dimensional case, and it is called as the Nivat's conjecture. However, in higher dimensions is known to be false [START_REF] Cassaigne | Subword complexity and periodicity in two or more dimensions[END_REF]. Then, using the fact that we can always assume that the factor between a substitutive subshift and an aperiodic symbolic system is induced by a letter-to-letter map (Lemma 1.22), we define an equivalence relation on the alphabet, calling two letters equivalent if and only if they have the same image via the local map given by the factor map. We consider the set of equivalence classes as a new alphabet, and we define a new constant-shape substitution on it. We finally prove that the substitutive subshift generated by this new substitution is conjugate to the aperiodic symbolic factor (Theorem 3.22). Now, we proceed to the proof. First, we have the following consequence of Proposition 3.3.

Remark 3.18. It is straightforward to check that Proposition 3.3 implies if x, x ∈ X ζ are such that τ (x) = τ (x ), then π(x) is equal to π(x ), where π : (X ζ , S, Z d ) → ( ← - Z d (L n ζ ) , + (L n ζ ) , Z d ) is the factor map.
To prove the result we will introduce some notions as in [START_REF] Müllner | Automorphisms of automatic shifts[END_REF].

Let ζ be an aperiodic primitive constant-shape substitution. We consider a labeled directed graph G ζ with vertex set E = A 2 and there exists an edge (a, b)

to (c, d) with label f ∈ F ζ 1 if ζ(a) f = c, ζ(b) f = d. Note that the diagonal ∆ A = {(a, a) : a ∈ A} is a stable set, i.e., E(∆ A ) = ∆ A . Let P = (a 0 , b 0 )(a 1 , b 1 )(a 2 , b
2 ) be a path, by definition, there is an edge from (a 0 , b 0 ) to (a 1 , b 1 ) with a label f 0 and an edge (a

1 , b 1 ) to (a 2 , b 2 ) with label f 1 , i.e, ζ(a 0 ) f 0 = a 1 , ζ(b 0 ) f 0 = b 1 , and ζ(a 1 ) f 1 = a 2 , ζ(b 1 ) f 1 = b 2 . Then, we have that ζ 2 (a 0 ) L ζ f 0 +f 1 = a 2 , ζ 2 (b 0 ) L ζ f 0 +f 1 = b 2 .
This means, the paths indicate the simultaneous positions of the letters in the iterates of the substitution. Remark 3.20. The following statements can be easily verified

(1) As for the case of periodic points for the substitution, we can replace the substitution ζ for an appropriate power, i.e, ζ n(ζ) , so we may assume that the substitution is pairaperiodic.

(2) If the substitution ζ is bijective, every (a, b) ∈ A 2 \ ∆ A is an asymptotic disjoint pair.

( With these definitions we are ready to prove the next result which is the multidimensional analogue to Theorem 22 in [START_REF] Müllner | Automorphisms of automatic shifts[END_REF]. We define an equivalence relation a

∼ b in A, such that a ∼ b if a, b are indistinguishable. By definition, the substitution ζ ([a]) f = [ζ(a) f ], f ∈ F ζ 1 .
in A/ ∼ and the map T : A/∼ → B, given by T ([a]) = τ (a) are well defined. These maps satisfy the following property: Every pair in A/ ∼ is distinguishable.

(

It is straightforward to check that primitivity of ζ implies primitivity of ζ .

Assume now that ([a], [b]

) is a periodic pair, i.e, there exists a cycle (c,d) repeating the labels of the path P 1 with a period k. By the Pigeonhole Principle, there exist two subpaths P 4 = (e 0 , f 0 ) . . . (e l 1 k , f l 1 k ), P 5 = (g 0 , h 0 ) . . . (g l 1 k, h l 2 k ) of P 3 , having the same labels of the edges as P 1 repeating with period k, such that e 0 = e l 1 k ,

P 1 = ([a 0 ], [b 0 ]) . . . ([a k ], [b k ]) in G ζ with ([a 0 ], [b 0 ]) = ([a k ], [b k ]) = ([a], [b]). We can consider a path P 2 = (c 0 , d 0 ) . . . (c k , d k ) in G ζ with [c i ] = [a i ] and [d i ] = [b i ] for 0 ≤ i ≤ k
h 0 = h l 2 k and [e 0 ] = [a 0 ], [h 0 ] = [b 0 ]. Now con- sider the cycle in G ζ (u 0 , v 0 ) . . . (u l 1 l 2 k 2 , v l 1 l 2 k 2 ) where u l 1 kj . . . u l 1 k(j+1) = e 0 , . . . , e l 1 k and v l 2 km . . . v l 2 k(m+1) = h 0 . . . h l 2 k for all 0 ≤ j < l 2 k, 0 ≤ m < l 1 k. Since ζ is pair- aperiodic, there exists f ∈ F ζ 1 such that ζ(e 0 ) f = e 0 , ζ(h 0 ) f = h 0 . We then conclude that ζ ([a]) f = [a], ζ ([b]) f = [b], i.e., ζ is pair-aperiodic.
On the other hand, for all n > 0, we have that τ

(ζ n (a)) = τ (ζ ([a]
)), hence Y has the same language as τ (X ζ ), so they are equal, since subshifts are uniquely determined by their language.

Finally, we prove that τ :

(X ζ , S, Z d ) → (Y, S, Z d ) is a conjugacy. Let x, x ∈ X ζ , with τ (x) = τ (x ). By the recognizability property of X ζ , we can write x = S f 1 ζ 2|A| 2 (x), x = S f 2 ζ 2|A| 2 (x ). By Remark 3.18 we have that f 1 = f 2 .
For every n ∈ Z d , (x n , x n ) ∈ A 2 are not asymptotic disjoint pair. Assume the contrary, i.e., there exists n ∈ Z d such that (x n , x n ) is an asymptotic disjoint pair. Then we can find a periodic pair ([a], [b]) and a path

P = ([a 0 ], [b 0 ]) . . . ([a k ], [b k ]) in G ζ with ([a 0 ], [b 0 ]) = (x n , x n ) and ([a k ], [b k ]) = ([a], [b]). with k ≤ |A| 2 . Since ζ is pair-aperiodic, we have that there exists f ∈ F ζ 2|A| 2 such that (ζ ) 2|A| 2 (x n ) f = [a], (ζ ) 2|A| 2 (x n ) f = [b]. Since τ (ζ 2|A| 2 (x n )) = ζ 2|A| 2 (τ (x n )) = ζ 2|A| 2 (τ (x n )) = τ (ζ 2|A| 2 (x n )), we have that τ (a) = τ (b), then ([a], [b]) are indistinguishable, which contradicts 3.1.
Thus we have that (x n , x n ) is not an asymptotic disjoint pair for any n ∈ Z d . By Remark 3.20 we have that

ζ 2|A| 2 (x n ) = ζ 2|A| 2 (x n ), i.e., x = x .
For the one-dimensional case, F. Durand proved [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] that Cantor topological factors of substitutive subshifts are either substitutive subshifts (when the action is expansive) or odometer systems (when the action is equicontinuous). This dichotomy is no longer true in the multidimensional context. Example 4.3 shows an example of substitutive subshift with a Cantor topological factor that is neither expansive nor equicontinuous. Also the example in Example 4.3 has a symbolic factor (in fact substitutive subshift) which has a non-trivial period, and the phase space is still infinite.

Chapter 4

Measurable morphisms between substitutive subshifts

In this chapter, we study different types of homomorphisms between substitutive subshifts. Note that since these subshifts are uniquely ergodic, any topological endomorphism is also a measurable endomorphism preserving the ergodic measure. First, we will extend a result of B. Host and F. Parreau [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] to the multidimensional case (Theorem 4.1), showing some rigidity properties: Any measurable factor between two substitutive subshifts given by two aperiodic constant-shape substitutions with some combinatorial property induces a continuous one. We follows the same strategy for the proof. It is based on the property of these substitutive subshifts being self-induced systems, i.e., there exists non-empty clopen proper subsets (in our case ζ n (X ζ ), for all n > 0) such that the induced system

(ζ n (X ζ ), S ζ n (X ζ ) , Z d ) is conjugate to the system (X ζ , S, Z d ). Here, the action S ζ n (X ζ ) is given by S L n ζ (m) : m ∈ Z d . This implies, any measurable endo- morphism φ of (X ζ , S, Z d ) is associated with an induced measurable endomorphism φ n of (ζ n (X ζ ), S ζ n (X ζ ) , Z d ).
We prove that these induced measurable endomorphisms are stationary, i.e., there exists n = m > 0 such that φ n = φ m . Then, we prove that we can approximate these induced measurable endomorphisms by endomorphisms of radius

F ζ 1 1 + L -1 ζ 2 + 1/(1 -L -1 ζ
) . The finiteness of sliding block codes of a specific radius will let us conclude the theorem (Theorem 4.1).

The result of B. Host and F. Parreau was then extended by V. Salo and Törmä in [START_REF] Salo | Block maps between primitive uniform and Pisot substitutions[END_REF], for topological factors between constant-length substitutions and Pisot substitutions whose associated incidence matrices have the same dominant eigenvalue. As in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF], there exists a bound R such that any factor map is the composition of a sliding block code of radius R with a power of the shift map. They used a renormalization process, where they could reduce the radius of a given factor map. Then, F. Durand and J. Leroy extended this result for any pair of aperiodic substitutions [START_REF] Durand | Decidability of the isomorphism and the factorization between minimal substitution subshifts[END_REF]. This was one of the key steps to prove the decidability of the isomorphism problem between substitutive subshifts.

Our result shows the decidability of the isomorphism problem between multidimensional substitutive subshifts, when both constant-shape substitutions has the same expansion matrix and same support, and satisfy a combinatorial condition (called reducibility). This condition is always satisfied for bijective substitutions. Nevertheless, this result is far from representing the complete picture about the decidability of the isomorphism problem between multidimensional substitutive subshifts.

Then, we will deduce restrictions on endomorphisms and homomorphisms, using the finiteness of the number of invariant orbits (Proposition 3.4). Every substitutive subshift given by an aperiodic constant-shape substitution satisfying the combinatorial property is coalescent (Proposition 4.7). This was already proved in the one-dimensional case in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF]. Also we proved the automorphism group of substitutive subshift is virtually generated by the shift action (Proposition 4.8). It was already known in the one-dimensional context by [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] and [START_REF] Lemańczyk | On metric properties of substitutions[END_REF]. In the multidimensional framework was proved under more restrictive geometrical and combinatorial properties [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]. We also give some conditions to get that the automorphism group of a substitutive subshift is isomorphic to a direct product of Z d with a finite group (Corollary 4.9). Finally, we extend Theorem 4.1 to homomorphisms associated with matrices commuting with a power of the expansion matrix of the substitution (Theorem 4.12). This leads to the same rigidity properties about these homomorphisms (Proposition 4.16) and for a restricted normalizer group (Proposition 4.16). Notice that in the next chapter we will give sufficient conditions to ensure the former result is a complete characterization of the normalizer group.

Measurable factors implies continuous ones for substitutive subshifts

In this section, we will extend the result proved in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] about measurable factors between substitutive subshifts in the multidimensional context. As in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF], we will use the notion of reducibility of a substitution. Let ζ be a constant-shape substitution. For any pair of letters a, b ∈ A, and n > 0, we consider the sequence

d n (ζ n (a), ζ n (b)) = f ∈ F ζ n : ζ n (a) f = ζ n (b) f |F ζ n | .
This sequence is decreasing for all of the pairs a, b ∈ A. We say the constant-shape substitution is reduced if min

n∈N a =b∈A d n (ζ n (a), ζ n (b)) > 0.
For instance, every bijective constantshape substitution is reduced.

As mentioned in Section 1.7, the substitutive subshift (X ζ , S, Z d ) is uniquely ergodic. Denote µ ζ its unique ergodic invariant measure. Using the recognizability property, as in the one-dimensional case [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF], this unique ergodic measure satisfies

∀U ∈ F X ζ , µ ζ (U ) = 1 |F ζ n | X ζ f ∈ F ζ n : S f ζ n (x) ∈ U dµ ζ (x),
where F X ζ corresponds to the Borel sets of X ζ .

The following theorem is a multidimensional analogue of Theorem 1.3 in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF]:

Theorem 4.1. Let (X ζ 1 , S, Z d ), (X ζ 2 , S, Z d
) be two substitutive subshift from two aperiodic primitive constant-shape substitutions ζ 1 , ζ 2 from finite alphabets A and B, with the same expansion matrix L and same support

F 1 . If (X ζ 2 , S, Z d ) is reduced, then for every measurable factor φ : (X ζ 1 , µ ζ 1 , S, Z d ) → (X ζ 2 , µ ζ 2 , S, Z d ), there exists j ∈ Z d such that S j φ is equal µ ζ 1 -a.e. to a continuous factor ψ : (X ζ 1 , S, Z d ) → (X ζ 2 , S, Z d ),
satisfying the following two properties:

1. ψ is a sliding block code of radius

F ζ 1 1 + L -1 ζ 2 + 1/(1 -L -1 ζ ) .
2. There exist an integer n > 0 and p ∈ F ζ n such that,

S p ψζ n 1 = ζ n 2 ψ.
In Chapter 6 we present an example where Theorem 4.1 can be applied, describing the automorphisms of it.

Remark 4.2. The following statements can be easily verified.

(1) If L ζ is a diagonal matrix, then L -1 ζ ≤ 1/2, so ψ is a sliding block code of radius 3 F ζ 1 .
(2) Since the set of sliding block codes of radius (3) As also mentioned in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF], we may assume that p / ∈ (L n ζ -id)(Z d ), because it is equivalent to find a factor map commuting with the substitution map, i.e.,

F ζ 1 1 + L -1 ζ 2 + 1/(1 -L -1 ζ ) between X ζ 1 and X ζ 2 is
ψζ n = ζ n ψ, with p ∈ F ζ 1 .
We follows the same strategy of [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF]. Substitutive subshifts are self-induced systems, i.e., there exists non-empty clopen proper subsets (in our case ζ n (X ζ ), for all n > 0) such that the induced system

(ζ n (X ζ ), S ζ n (X ζ ) , Z d ) is conjugate to the system (X ζ , S, Z d ). We recall, the action S ζ n (X ζ ) is given by S L n ζ (m) : m ∈ Z d .
This implies, any measurable endomorphism φ of (X ζ , S, Z d ) is associated with an induced measurable endomorphism

φ n of (ζ n (X ζ ), S ζ n (X ζ ) , Z d ).
We prove that these induced measurable endomorphisms are stationary, i.e., there exists n = m > 0 such that φ n = φ m . Then, we prove that we can approximate these induced measurable endomorphisms by endomorphisms of radius

F ζ 1 1 + L -1 ζ 2 + 1/(1 -L -1 ζ
) . The finiteness of sliding block codes of a specific radius will let us conclude the theorem (Theorem 4. )

) f = [ζ(a) f ] for f ∈ F ζ 1 .
This substitution is reduced. We have a natural letter-to-letter factor map φ : (X ζ , S, Z d ) → (X ζ , S, Z d ), and is called the reduced substitution of ζ.

In the one-dimensional case if (X ζ , S, Z) does not have purely discrete spectrum, it can be proved using the results in [START_REF] Dekking | The spectrum of dynamical systems arising from substitutions of constant length[END_REF] that (X ζ , S, Z) is aperiodic. In the multidimensional case this is not true in general, as we can see in Example 4.3.

Example 4.3 (An aperiodic constant-shape substitution, with a periodic reduced substitution). Consider the substitution σ 2 with L σ 2 = 2 0 0 2 and F σ 2 1 = 0, 1 2 , given by σ 2 :

0 → 1 3 0 2 , 1 → 0 2 0 2 , 2 → 3 1 2 0 , 3 → 2 0 2 0 .
This substitution corresponds to the product substitution between the Thue-Morse substitution (σ 3 : 0 → 01, 1 → 10) and the doubling sequence substitution (σ 4 : a → ab, b → aa). The substitution does not have purely discrete spectrum, since (X T M × ← -

Z (2 n Z) , S ×+ (2 n Z) , Z 2
) is a factor of (X σ 2 , S, Z 2 ), where (X T M , S, Z) corresponds to the onedimensional Thue-Morse substitutive subshift. The reduced substitution for σ 2 is defined with the same expansion matrix and support, given by: σ2 :

a → a b a b , b → b a b a ,
where every element in {0} × Z is a nontrivial period of σ2 .

However, as proved in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] for the one-dimensional case, if the reduced substitution system is aperiodic, then (X ζ , µ ζ , S, Z d ) is metrically isomorphic to (X ζ , µ ζ , S, Z d ).

X ζ , S, Z d ) is a metric isomorphism. Proof. Let π : (X ζ , S, Z d ) → ( ← - Z d (L n ζ ) , + (L n ζ ) , Z d
) defined in Section 3.4.1. By Proposition 3.3, for every n > 0, π n (x) is equal to π n ( φ(x)). In particular, if x, y ∈ X ζ satisfies φ(x) = φ(y), then π n (x), π n (y) are equal for any n > 0.

Set U = {x ∈ X ζ : ∃y ∈ X ζ , φ(x) = φ(y), x 0 = y 0 }. It is enough to prove that U is a null-set. Let n > 0, f ∈ F ζ n and x ∈ X ζ be such that S f ζ n (x) ∈ U .
Then, there exists y ∈ X ζ with φ(y) = φ(S f ζ n (x)) and y 0 = ζ n (x 0 ) f . Then π n (y) = π n (S f ζ n (x)) and is equal to f . Moreover, there exists z ∈ X ζ with y = S f ζ n (z), so φ(x) is equal to φ(z). This implies (ζ n z 0 ) j , (ζ n x 0 ) j are equivalent for all j ∈ F ζ n . Note that (ζ n z 0 ) f = y 0 , so is different from (ζ n x 0 ) f . We define the set

G n = a,b∈A f ∈ F ζ n : [(ζ n a) f ] = [(ζ n b) f ], (ζ n a) f = (ζ n b) f .
We deduce from the previous paragraph that

µ ζ (U ) = 1 |F ζ n | f ∈ F ζ n : S f ζ n (x) ∈ U dµ(x) ≤ |G n | |F ζ n | . For any a, b ∈ A we denote D a,b n = (c, d) ∈ A 2 : ∃f ∈ F ζ n , (ζ n a) f = c, (ζ n b) f = d, [c] = [d] and E a,b n = (c, d) ∈ A 2 : ∃f ∈ F ζ n , (ζ n a) f = c, (ζ n b) f = d, [c] = [d] .
Set ε > 0 and let j > 0 be large enough such that for any a, b ∈ A we have that = 1

d j (ζ j (a), ζ j (b)) ≤ lim k→∞ d k (ζ k (a), ζ k (b)) + ε.
|F ζ n |   (c,d)∈D a,b n d j (ζ j (c), ζ j (d)) + (c,d)∈E a,b n d j (ζ j (c), ζ j (d))   ≤ 1 |F ζ n |   (c,d)∈D a,b n ( lim k→∞ d k (ζ k (c), ζ k (d)) + ε) + (c,d)∈E a,b n ε   ≤ ε(|D a,b n | + |E a,b n |) |F ζ n | + 1 |F ζ n | (c,d)∈D a,b n lim k→∞ d k (ζ k (c), ζ k (d)).
Since this is for every ε > 0 and lim When n → ∞, the right expression goes to zero, and we conclude µ ζ (U ) = 0.

It is known that if the substitutive subshift has a coincidence, then it is metrically isomorphic to its maximal equicontinuous factor [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF]. A constant-length substitution has a coincidence, if there exists n > 0 and an index j on the support of ζ n such that for all pair of letters a, b ∈ A, ζ n (a) j = ζ n (b) j . The doubling sequence substitutive subshift (see Example 4.3) is an example of a substitution having a coincidence. If a substitution has a coincidence, then cannot be reduced. In fact its reduced substitution is trivial, i.e., has an alphabet of cardinality 1. The set of measurable endomorphisms of odometer systems is not discrete. In fact it is uncountable and any element of the odometer system represent a measurable endomorphism via addition. So, as in the original article [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF], reducibility is an optimal hypothesis for Theorem C. Now, to prove Theorem 4.1, we assume that ζ 2 is an aperiodic primitive reduced constant-shape substitution. We denote by η = min

(φ) in F ζ 1 n . The set S pn(φ) φζ n 1 (X ζ 1 ) is included in ζ n 2 (X ζ 2 ) up to a µ ζ 2 -null set. Since ζ n 1 is a homeomorphism from X ζ 1 to ζ n 1 (X ζ 1 ), for µ ζ 1 -almost all x ∈ X ζ 1 there exists a unique point y ∈ X ζ 2 such that S pn(φ) φζ n 1 (x) = ζ n 2 (y), which we denote φ n (x). So, for every φ ∈ m Fac(X ζ 1 , X ζ 2 , S, Z d ) we consider a sequence (p n (φ)) n≥0 and a sequence of maps (φ n ) n ∈ m Fac(X ζ 1 , X ζ 2 , S, Z d ) such that p n (φ) ∈ F ζ 1 n , S pn(φ) φζ n 1 (x) = ζ n 2 (φ n (x)).
It is straightforward to check that the sequence satisfies the recurrence

p n+1 (φ) = p n (φ) + L n ζ 1 p 1 (φ n ) (mod L n+1 ζ 1 (Z d ))
. We also have the recurrence

(φ n ) 1 = φ n+1 . Now, for φ, ψ ∈ m Fac(X ζ 1 , X ζ 2 , S, Z d ), we denote d(φ, ψ) = µ ζ 1 ({x ∈ X ζ 1 : (φx) 0 = (ψx) 0 }).
We also denote for any r > 0 the quantity C(r) = |B(0, r) ∩ Z d |.

Applications of rigidity results on endomorphisms of substitutive subshifts

In this section, we prove two consequences of Theorem 4.1 on endomorphisms of substitutive subshifts. First, we prove that substitutive subshifts given by aperiodic primitive reduced constant-shape substitutions are coalescent. This was first proved in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] for one-dimensional linearly recurrent subshifts (in particular aperiodic subshifts). Linearly recurrent substitutive subshifts (such as the self-similar ones) are also coalescent as a consequence of a result in [START_REF] Cortez | Linearly repetitive Delone systems have a finite number of nonperiodic Delone system factors[END_REF].

Then, we deduce the automorphism group of a substitutive subshift of an aperiodic primitive reduced constant-shape substitution is virtually generated by the shift action. As a corollary, we get a condition for the automorphism group to be isomorphic to a direct product of Z d (given by the shift action) and a finite group. As we will see in the next chapter, this condition is always true for aperiodic bijective primitive constant-shape substitutions.

From now on, during this section we will always assume, up to consider a power of the substitution, that any factor map ψ ∈ End(X ζ , S, Z d ) satisfying Property 2. in Theorem 4.1 is true for n = 1.

Coalescence of substitutive subshifts

In [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] it was proved that one-dimensional linearly recurrent subshifts (in particular aperiodic substitutions) are coalescent. In the multidimensional context, linearly recurrent substitutive subshifts (such as the self-similar ones) are also coalescent as a consequence of a result in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF]. Here we will use Theorem 4.1 and the finiteness on the number of invariant orbits under the action of ζ (Proposition 3.4), to obtain that substitutive subshifts are also coalescent, for aperiodic primitive reduced constant-shape substitutions. Proof. Set φ ∈ End(X ζ , S, Z d ). Theorem 4.1 ensures there exists j ∈ Z d such that S j φ is equal to a sliding block code ψ of fixed radius satisfying S p ψζ = ζψ, for some p ∈ F ζ 1 . Let x ∈ X ζ be in a ζ-invariant orbit, i.e., there exists j ∈ Z d such that ζ(x) = S j x. Note that S p ψζ(x) = S p+j ψx = ζψx, so, if the orbit of x is in a ζ-invariant orbit, then ψx satisfies the same property. By Proposition 3.4, there exist finitely many ζ-invariant orbits, hence for n large enough, we can find x ∈ X ζ with x and ψ n (x) being in the same orbit, i.e., there exists m ∈ Z d such that S m ψ n (x) = x. By minimality of X ζ , ψ n = S -m , hence ψ is invertible, which implies φ is invertible.

m Hom M (X ζ 1 , X ζ 2 , S, Z d ) being non empty, whenever ζ 1 , ζ 2 are two aperiodic primitive constant-shape substitutions with the same expansion matrix and support. This condition is similar to the one for odometer systems (Lemma 1.14), although the proof is very different. In this case we use a relation between measurable eigenvalues and homomorphisms (which we will prove in the proof of Lemma 4.10). Then we conclude by the fact that measurable eigenvalues and continuous eigenvalues are the same for substitutive subshifts (Theorem 3.13). In some cases this condition implies the matrix M commutes with the expansion matrix L (as shown for constant-base odometer systems in Theorem 2.2).

Then, we prove an analogue of Theorem 4.1 (Theorem 4.12) establishing that measurable homomorphisms induced continuous ones for homomorphisms associated with matrices in the centralizer of some power of the expansion matrix. Finally, we give an explicit bound on the norm of these matrices for the quotient group of a restricted normalizer semigroup with respect to the shift action (Proposition 4.16).

Lemma 4.10. Let ζ 1 , ζ 2 be two aperiodic primitive constant-shape substitutions having the same expansion matrix L and same support

F . If M ∈ GL(d, Z) is such that m Hom M (X ζ 1 , X ζ 2 , S, Z d ) = ∅, then for all n > 0 there exists m(n) > 0 such that M L m(n) (H(X ζ 1 )) ≤ L n (H(X ζ 2 )).
(Normalizer Condition for substitutions)

Proof. Let φ be in m Hom M (X ζ 1 , X ζ 2 , S, Z d ), and x ∈ E(X ζ 2 , µ ζ 2 , S, Z d ). We will prove that M * x ∈ E(X ζ 1 , µ ζ 1 , S, Z d ). Indeed, let f ∈ L 2 (X ζ 2 , µ ζ 2 ) be such that for all m ∈ Z d , f • S m = e 2πi x,m • f , µ ζ 2 -a.e. in X ζ 2 .
Then, we have that

(f • φ) • S m = (f • S M m ) • φ = e 2πi x,M m • f • φ = e 2πi M * x,m • f • φ, µ ζ 1 -a.e. in X ζ 1 .
By Theorem 3.13 and Proposition 3.16, for any n > 0, the system (Z d /M -1 L n (H(X ζ 2 )), +, Z d ) is a finite factor of the odometer sys-

tem ( ← - Z d (L n (H(X ζ 1 ))) , + (L n (H(X ζ 1 ))) , Z d ),
which implies the odometer system

( ← - Z d (M -1 L n (H(X ζ 2 ))) , + (M -1 L n (H(X ζ 1 ))) , Z d ) is a factor of ( ← - Z d (L n (H(X ζ 1 ))) , + (L n (H(X ζ 1 )
)) , Z d ). By Lemma 1.13, we conclude that for any n > 0, there exists m(n) > 0 such that

L m(n) (H(X ζ 1 )) ≤ M -1 L n (H(X ζ 2 )).
A consequence of Proposition 4.7 is that a homomorphism is associated with a matrix with finite order, then is an isomorphism. Proof. Since M has finite order, there exists n > 0 such that M n = id R d . This implies φ n ∈ End(X ζ , S, Z d ). By Proposition 4.7, φ n is invertible, which implies φ is invertible. Now, we will prove an analogue of Theorem 4.1 for homomorphisms associated with matrices commuting with a power of the expansion matrix L. As mentioned before a priori this does not cover all the homomorphisms between substitutive subshifts. Theorem 4.12. Let (X ζ 1 , S, Z d ), (X ζ 2 , S, Z d ) be two substitutive subshifts from two aperiodic primitive constant-shape substitutions ζ 1 , ζ 2 from finite alphabets A and B, with the same support F 1 and expansion matrix L. Let M ∈ GL(d, Z) be a matrix commuting with a power of L, i.e., there exists n > 0 such that M L n = L n M . If (X ζ 2 , S, Z d ) is reduced, then for every measurable homomorphism associated with M , φ :

(X ζ 1 , µ ζ 1 , S, Z d ) → (X ζ 2 , µ ζ 2 , S, Z d ), there exists j ∈ Z d such that S j φ is equal µ ζ 1 -a.e. to a homomorphism associated with M ψ ∈ N M (X ζ 1 , X ζ 2 , S, Z d ),
satisfying the following two properties:

1. ψ is given by a block map of radius

F ζ 1 L -1 ζ 1 (1 + M ) 2 -L -1 ζ / 1 -L -1 ζ .
2. There exist an integer n > 0 and q ∈ F ζ n such that S q ψζ n 1 = ζ n 2 ψ.

In Chapter 6 we provide an example where the hypothesis are not satisfied, so we cannot apply Theorem 4.12. Nevertheless, we are able to describe its normalizer semigroup. For any m in Z d , we have that

S q ψ(ζ n 1 (S m x)) = ζ n 2 (ψ(S m x)), and S q ψ(ζ n 1 (S m x)) = S q+M L n ζ 1 m ψ(ζ n 1 (x)), ζ n 2 (ψ(S m x)) = S L n ζ 1 M m ζ n 2 (ψ(x)), it follows, M L n ζ 1 m = L n ζ 1 M m, i.
e., M and L n ζ 1 commute, hence this hypothesis is optimal to obtain property (2). Note that if L is an integer multiple of the identity, then any matrix M ∈ GL(d, Z) commutes with L.

The proof of Theorem 4.12 follows the same strategy as the one of Theorem 4.1, except some small modifications. Since the substitution ζ 1 is primitive, we can replace the substitution by some power ζ n 1 , so we may assume that M commutes with the expansion matrix of ζ 1 . We will replace the term p n (φ) by the map π n (x) -M -1 π n (φx) (mod L n ζ (Z d )), with π n (x) and M -1 π n (φx) being the representative classes in F ζ n . The commutation assumption implies, for any n > 0 the map M defines a bijection in Z d /L n (Z d ), also denoted by M , i.e., n = m (mod L n (Z d )), if and only if M n = M m (mod L n (Z d )). With this, the map p n (φ) is invariant under the shift action. Since (X

ζ 1 , µ ζ 1 , S, Z d ) is ergodic, the map p n (φ) ∈ F ζ n is a constant map µ ζ 1 -a.e. in X ζ 1 and the set S M pn(φ) φζ n 1 (X ζ 1 ) is included, up to a µ ζ 2 -null set, in ζ n 2 (X ζ 2 ). We can define the map φ n for µ ζ 1 -a.e. in X ζ 1 as the unique point y ∈ X ζ 2 such that S M pn(φ) φζ n 1 x = ζ n 2 y, where M p n (φ) is the representative element in F ζ n . It is straightforward to check that φ n • S n = S M n • φ n for all n ∈ Z d , so φ n is in mN M (X ζ 1 , X ζ 2 , S, Z d ).
The sequences p n (φ) and (φ n ) satisfies the same recurrences given in Section 4.1:

p n+1 (φ) = p n (φ) + L n ζ p 1 (φ n ), (φ n ) 1 = φ n+1 .
As in Theorem 4.1 we need the following adaptations of Lemma 4.5 and Lemma 4.6 for homomorphisms. The proof are the same, so we omit them.

L -1 ζ 1 (1 + M ) 2 -L -1 ζ / 1 -L -1 ζ such that d(φ n , ψ n ) → 0.
To finish the proof of Theorem 4.12, we proceed exactly as in the proof of Theorem 4.1

Proof of Theorem 4.12. For the fixed alphabets A and B, there exists a finite number of homomorphisms associated with M of radius

F ζ 1 L -1 ζ 1 (1 + M ) 2 -L -1 ζ / 1 -L -1 ζ
. By Lemma 4.15, there exist two different integers m, k ≥ 0 such that d(φ m , φ m+k ) < η/C(R), so by Lemma 4.14, we have that φ m = φ m+k , µ ζ 1 -a.e..

Let n ≥ m be a multiple of k. We have that (

φ n ) k = φ n+k = (φ m+k ) n-m = (φ m ) n-m = φ n , µ ζ 1 -a.e.
This implies for all r ∈ N, φ n is equal to (φ n ) rk , µ ζ 1 -a.e. Since the number of sliding block codes of radius

F ζ 1 L -1 ζ 1 (1 + M ) 2 -L -1 ζ / 1 -L -1
ζ is finite, by Lemma 4.15 we have that φ n is equal to a homomorphism associated with M of radius

F ζ 1 L -1 ζ 1 (1 + M ) 2 -L -1 ζ / 1 -L -1 ζ , µ ζ 1 -a.e. in X ζ 1 .
Note that φ n is equal to φ 2n , µ ζ 1 -a.e. We denote ψ = φ n and p = p n (ψ). By definition of p, we have that

S M p ψζ n 1 = ζ n 2 ψ. Set j = M (p n (φ) -p), then S j φζ n 1 = S M (pn(φ)-p) φζ n 1 = S -M p ζ n 2 ψ = ψζ n 1 , µ ζ 1 -a.e,
this implies that S j φ and ψ coincides in ζ n 1 (X ζ 1 ) µ ζ 1 -almost everywhere, and by ergodicity in the whole set X ζ 1 µ ζ 1 -almost everywhere.

In the case ζ 1 = ζ 2 , we can consider a restricted normalizer group as all the invertible homomorphisms associated with matrices commuting with a power of

L ζ 1 N C(X ζ 1 , S, Z d ) = M ∈GL(d,Z) M L n ζ 1 =L n ζ 1 M, for some n (N M (X, T, Z d ) ∩ Homeo(X)),
This set is a group under composition and S , Aut(X

ζ 1 , S, Z d ) are normal subgroups of N C(X ζ 1 , S, Z d ).
Following the same proof as Proposition 4.8, we have the map φ → (j φ , ψ φ ) is unique, so we obtain the same result with respect to this restricted normalizer group. A substitution with expansion matrix equal to L 1 in Example 2.4 is an example where Proposition 4.16 gives a complete characterization for the normalizer group.

Chapter 5

Precisions on bijective constant-shape substitutions Bijective substitutions are of great interest because of their mixed dynamic spectrum. They are never extensions almost 1-to-1 of its maximal equicontinuous factor. Bijective substitutions were studied before in [START_REF] Frank | Multidimensional constant-length substitution sequences[END_REF] for block substitutions, where it was proved that the subshift generated by a bijective constant-shape substitution, with a diagonal expansion matrix is measurable-theoretic isomorphic to a skew product of one-dimensional odometers. Also, in [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF] it was studied the normalizer group of bijective block substitutions. We extend the study by describing the normalizer group for general constant-shape substitutions. To do this, we relate the symmetry group with different types of supports of the substitution and non-diagonal expansion matrices.

First, we prove that the automorphism group of substitutive subshifts of bijective substitutions are direct products of the shift action and finite groups, given by a permutation of letters (Proposition 5.1). This is a well known result for one-dimensional and block substitutions ( [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF][START_REF] Lemańczyk | On metric properties of substitutions[END_REF][START_REF] Bustos | Extended symmetry groups of multidimensional subshifts with hierarchical structure[END_REF][START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]). In the rest of the chapter, under some geometrical conditions (called polytope substitutions), we prove that the normalizer group is virtually generated by the shift action. We then describe the symmetry group of these substitutive subshifts (Proposition 5.15 and Theorem 5.17). The strategy is to use the fact that nondeterministic directions, viewed as asymptotic pairs in the one-dimensional context in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF], are preserved by the symmetry group (Proposition 1.12). We determine the nondeterministic directions of these substitutive subshifts thanks to the supporting hyperplanes to conv(F ζ n ) (Theorem 5.2). As we see in Section 5.3, the polytope assumption imply strong constraints on such supporting hyperplanes. Conversely, we provides a checkable combinatorial condition to ensure a half-space to be nonexpansive for (X ζ , S, Z d ). (Corollary 5.13). This is the first characterization of nondeterministic directions for a family of minimal systems.

= n-1 i=0 L i (p). Set m ∈ Z d such that (p n + (F n ) •C ) ∩ (L n (m) + F n ) = ∅. Since S pn ψζ n 1 = ζ n 2 ψ
and ζ 2 is bijective, we have that x 0 determines ψ(x) m , which implies S -m ψ is a factor map of radius 0 (or a letter-to-letter factor map). Set φ = S -m ψ, we have that

S pn+m-L n (m) φζ n 1 = ζ n 2 φ,
and by bijectivity the coordinate x 0 determined two coordinates of ψ, unless for any n ∈ N large enough p n + m is in L n ζ (Z d ), i.e., for any n large enough there exists r n ∈ Z d such that p n + m = L n (r n ). Note that p + r n = L(r n+1 ), which implies

r n+1 ≤ L -1 ( r n + p ), so (r n ) is a bounded sequence. Hence there exist n > 0 and N > 1 such that r n+N = r n , which implies p N -1 ∈ (L N -id)(Z d ) which is not possible by Remark 4.2. If p n + m / ∈ L n ζ (Z d ), then x 0 determines two coordinates n 1 , n 2 of ψ(x)
and then the coordinates 0 and n 2 -n 1 of ψ 1 = S -n 1 ψ, since ψ 1 is also induced via a 0-block map. Note that the map Ψ 1 : A → B inducing ψ 1 is bijective, if not two fixed points x, y with Ψ 1 (x 0 ) = Ψ 2 (y 0 ) and x 0 = y 0 generate two points with the same image, which is a contradiction. It follows that x 0 determines x n 2 -n 1 , and then x k(n 2 -n 1 ) for all k ∈ Z, so x has a nontrivial period, which is a contradiction. Finally, we conclude by Corollary 4.9.

Nondeterministic directions of substitutive subshifts of bijective on extremities constant-shape substitutions

In this section we give a characterization of the nondeterministic directions (defined in Section 1.5) of a substitutive subshift (X ζ , S, Z d ) in the case ζ is a bijective on extremities constant-shape substitution. A starting remark is that for each n > 0, the set of directions S d-1 is stratified by the opposite normal fan N (conv(F ζ n )) (see Section 1.1.2). Our description of the nondeterministic directions is given in terms of union of these fans.

We say a constant-shape substitution ζ is bijective on extremities if the restriction

p f of ζ in f is bijective for all f ∈ Ext(conv(F ζ 1 )). Since Ext(conv(A + B)) ⊆ Ext(conv(A)) + Ext(conv(B))
), a substitution is bijective on extremities if and only if for any n > 0 and

f ∈ Ext(conv(F ζ n ) the restriction p f of ζ n in f is bijective. Since (F ζ n ) n>0 is a Følner sequence, there exists n > 0 such that conv(F ζ n
) is a nondegenerate polytope, so up to considering a power of ζ, we may assume that conv(F ζ 1 ) is a nondegenerate polytope. Using the recognizability property of substitutions and some basic results in convex geometry we prove the following result.

Theorem 5.2. Let ζ be an aperiodic primitive constant-shape substitution which is bijective on extremities. Then, the set of nondeterministic directions ND(X ζ , S, Z d ) of the substitutive subshift is the intersection of S d-1 with a nonempty union of limits of opposite normal cones of the form NGn (conv(F

ζ n )), with G n a face of conv(F ζ n ), for some integer n > 0.
This theorem gives topological constraints on the set of nondeterministic directions. Actually, we will see when L ζ = λ id R d , that the convex hull of any digit tile is a polytope, i.e. it has a finite number of extreme points (Theorem 5.6). In this case by Theorem 5.2, the set of nondeterministic directions ND(X ζ , S, Z d ) is a finite union of closed balls (eventually degenerated). More explicitly, in the two-dimensional case we have the following corollary, showing in particular, it cannot be a Cantor set. This in contrast with the result proved by M. Boyle and D. Lind in [START_REF] Boyle | Expansive subdynamics[END_REF], where they proved any compact set with cardinality at least 2 can be realized as the set of nondeterministic directions of some two-dimensional subshift.

Corollary 5.3. In the two dimensional case, under the hypothesis of Theorem 5.2, either the set ND(X ζ , S, Z 2 ) has nonempty interior, either it has at most 2 accumulation points.

Proof. Assume that the set of nondeterministic directions ND(X ζ , S, Z 2 ) has empty interior. By Theorem 5.2, the elements of ND(X ζ , S, Z 2 ) are limits of normal vectors to edges of conv(F ζ n ) for some n > 0. In [START_REF] Strichartz | Geometry of self-affine tiles[END_REF] it was proved such vectors are normalized vectors of the form (L * ζ ) -k u k , with u k ∈ S 1 a normal vector to an edge of conv(F ζ 1 ) for some k > 0. Hence, their accumulation points are accumulation points of orbits of the projective action L * ζ on the circle S 1 . A standard analysis of this action provides the cardinality of in the boundary of conv(F ζ n -π n (x 1 ) + L n ζ (z n (t))). Since this last set is a translated of conv(F ζ n -π n (x 1 )) and are both subsets of conv(D), a basic geometrical argument ensures for any n > 0, h n (t) and π n (x 1 ) are in the same face of conv(F ζ n ). The same arguments imply that h n (t 1 ), h n (t 2 ) are in the same face for any t 1 , t 2 ∈ F 1 ∩ D. Furthermore, h n (t) and π n (x 1 ) are different for large enough n ∈ E. Indeed, assume the converse, taking R > t , we have that

x 1 | t = ζ n (w i ) L n ζ (k 3 )+πn(x 1 ) , for some k 3 ∈ K ζ for infinitely many n ∈ E. Since t = L n ζ (k 1 -k 3 ) ,
(conv(D)) = n∈E NHn (conv(F n ) ζ ).
Suppose now that conv(D) is not closed. Set F 1 be a face of conv(D) of codimension 1 containing 0 and w ∈ NF 1 (conv(D)). We will find a sequence of faces

H n of conv(F ζ n ) such that NF 1 (conv(D)) = n>0 NHn (conv(F ζ n )).
f m i = h(m, i, n) -π n (x 1 ) + L n ζ (z(m, i, n)), with h(m, i, n) ∈ F ζ n and z(m, i, n) ∈ Z d . Since w, h(m, i, n) -π n (x 1 ) ≥ 0 and w, L n ζ (z(m, i, n)) ≥ 0, we have that for all n > 0 w, h(m, i, n) -π n (x 1 ) ----→ m→∞ 0 ∧ w, L n ζ (z(m, i, n)) ----→ m→∞ 0.
Since F ζ n is finite, we conclude that for all n ∈ E, there exists m(n) such that for all m ≥ m(n), w, h(m, i, n) = w, π n (x 1 ) = 0.

(5.

2)

The same argument as the former one when conv(D) is closed, gives h(m, i, n) = π n (x 1 ) and if i = j, h(m, i, n) = h(m, j, n) for all n large enough. Now, for any n > 0, we define H n as the face of conv(F ζ n ) of smallest dimension containing π n (x 1 ) and h(m, i, n

) : t ∈ F 1 , 0 ≤ i ≤ d, m ≥ m(n) with lim inf m→∞ t m i > 0 . In particu- lar, (5.2) shows that NF 1 (conv(D)) ⊆ n>0 NHn (conv(F ζ n )). We claim n>0 NHn (conv(F ζ n )) = NF 1 (conv(D))
. First, note that taking subsequences if its necessary, we get that for all n ∈ E the following limits

lim m→∞ d i=1 t m i h(m, i, n) = h n (t) ∧ lim m→∞ d i=1 t m i z(m, i, n) = z n (t).
Hence h n (t) ∈ H n for all n ∈ E and z n (t) ∈ conv(K ζ -K ζ ) for all n ∈ E large enough. A geometric argument shows that there exists t ∈ F 1 and ε > 0 small enough such that for all t ∈ F 1 , we have that z n (t ) = z n (t), so H n is a face of codimension 1 for all n ∈ E large enough. We then conclude that Then, to determine the nondeterministic directions for (X ζ , S, Z d ) we will study the supporting hyperplanes to conv(F ζ n ). To do this we will focus on the convex hull of the digit tile of the substitution. In general, this convex hull is not a polytope, i.e., can have at least a countable number of extreme points, even if the expansion matrix is diagonal, as we see in Example 5.4: A direct computation shows that for any n > 0, the extreme points of conv(F n ) is the set {(0, 0), (0, 3 n -1), ( 2

n -1, 3 n -1)} ∪ {(2 n -2 k , 3 k -3 n ) : 0 ≤ k ≤ n -1}, which implies Ext(conv(T (L, F 1 ))) = {(0, 0), (0, 1), (1, 1), (1, -1)} ∪ {(1 -2 -k , -1 + 3 -k ) : k ≥ 0}.

The polytope case

In this section, we will focus in the case when the convex hull of the digit tile is a polytope. We will present some known results about the convex hull of the digit tile that we will use in the rest of this thesis. Definition 5.5. We say a substitution ζ is a polytope substitution if it is bijective on extremities, and the convex hull of the digit tile

T ζ = T (L ζ , F ζ 1 ) is a polytope.
From now on, we will only consider polytope substitutions. As we will see, this geometrical hypothesis implies several algebraic restrictions on the expansion matrix L ζ (Proposition 5.10) and some dynamical consequences for (X ζ , S, Z d ) (Theorem 5.17).

We recall here some results characterizing the polytope case in terms of the extreme points of conv(F ζ n ) [START_REF] Kirat | Remarks on self-affine fractals with polytope convex hulls[END_REF], and the inward unit normal vectors of the (d-1)-dimensional faces of conv(T ζ ) [START_REF] Strichartz | Geometry of self-affine tiles[END_REF]. A big family for the polytope case is when a power of L is an integer multiple of the identity, because for any fundamental domain F of L, the convex hull of the digit tile generated by L and F is a polytope. In particular, all the convex hull of the digit tiles of the examples in Fig. 1.4 are polytopes. Although it is not the only case where Theorem 5.6 can be applied. (2) As an example where the statement (3) in Theorem 5.6 to be applied is not necessarily satisfied in n = 1, consider L = -2 0 0 -2 and F 1 = {(0, 0), (1, 0), (0, 1), (-1, -1)}. We have that

F 2 = {(-1, -3), (0, -2), ( 1
, -2), (-3, -1), (-1, -1), (0, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (-2, 1), (0, 1), (1, 1), (2, 2), (3, 2), (2, 3)}, so conv(F 2 ) has 3 extreme points, while conv(F 1 ) has 6 extreme points as shown in In [START_REF] Kirat | Remarks on self-affine fractals with polytope convex hulls[END_REF] it was proved the following result about the extreme points of conv(F m ) for any m > n, where n is such that

| Ext(conv(F n ))| = | Ext(conv(F n+1 ))| and conv(T (L, F 1 )). Proposition 5.9. [77, Theorem 4.8] If | Ext(conv(F n ))| = | Ext(conv(F n+1 ))|, then all the extreme points of conv(T (L, F 1 )) are of the form j>0 L -(n+1)j n i=0 L i (f i ) , with n i=0 L i (f i ) being an extreme point of conv(F n+1 ). This implies conv(T (L, F 1 )) is equal to (L m -id) -1 conv(F m ) for all m > n.
Now, assume that we are under the condition | Ext(conv(F 1 ))| = | Ext(conv(T (L, F 1 ))| and for all n > 0, conv(T (L, F 1 )) = (L n -id) -1 conv(F n ). Let u be an inward unit normal vector of a (d-1)-dimensional face of conv(T (L, F 1 )). For each n > 0, ((L n -id) * ) -1 u is an inward normal vector of a (d -1)-dimensional face of F n . By Theorem 5.6 (1), there exists k > 0 such that ((L -id) * ) -1 u are eigenvectors of (L * ) k . Hence by commutation, u is an eigenvector of (L * ) k . Since conv(T (L, F 1 )) is a polytope, we can take n > 0 large enough such that any of the inward unit normal vectors of conv(F 1 ) is an eigenvector of the same power (L * ) n . Hence, by the same arguments, up to considering a power of L, we may assume that any of the inward unit normal vectors of the (d -1)-dimensional faces of conv(F 1 ) are eigenvector of L * . This is equivalent to the hyperplane ∂H[u] = {t ∈ R d : t, u = 0} (the vector space of an affine hull of a face of conv(F 1 )) generated by u being preserved by L, i.e., L∂H[u] = ∂H[u]. This implies the normal fan N (conv(F n )) is the same for all n > 0 and it is equal to the one of conv(T (L, F 1 )).

Since for some n > 0, conv(F n ) is nondegenerate (by the Følner condition), it has d linearly independent inward normal vectors (that has integer coordinates with no common divisor) which are eigenvectors of L * . The polytope condition implies then the following algebraic restrictions on the expansion matrix L. The proof is left to the reader. Moreover, if u 1 , u 2 , u 3 are linearly dependent inward unit normal vectors of (d -1)dimensional faces of conv(T (L, F 1 )), then L restricted to the vector space generated by these vectors acts as a integer multiple of the identity.

In particular, in the two-dimensional case if the digit tile has 3 or at least 5 edges, then it follows that the expansion matrix is an integer multiple of the identity.

Finally, up to taking an appropriate power of a substitution, we may assume the following hypothesis (PC 1) The expansion matrix L is diagonalizable with positive eigenvalues. (PC 4) The set K given by Proposition 1.18 is equal to (id -L) -1 (F 1 ) ∩ Z d , i.e., for any k ∈ K, there exists f ∈ F 1 such that k = L ζ (k) + f .

Dynamical properties of substitutive subshifts of polytope substitutions

In this section, we prove the normalizer group of substitutive subshifts given by polytope substitutions is virtually generated by the shift action. We then describe the symmetry group of these substitutive subshifts (Proposition 5.15 and Theorem 5.17). To do this, we determine the nondeterministic directions of these substitutive subshifts thanks to the supporting hyperplanes to conv(F ζ n ) (Theorem 5.2). Conversely, we provides a checkable combinatorial condition to ensure a vector to be nondeterministic for (X ζ , S, Z d ). (Corollary 5.13).

As we saw in the previous section, under the hypothesis (PC 1), (PC 2), (PC 3), and (PC 4), the normal fan is the same for all the supports of ζ n (for any n > 0 large enough) and the digit tile, so we have that the following interpretation of Theorem 5.2 in the polytope case. Hence, in the two-dimensional case the former corollary implies strong restrictions on the set of nondeterministic directions. For instance, the number of its connected components is bounded by the number of edges of conv(T ζ ). Now, as shown in the proof of Theorem 5.2, to establish which opposite normal vectors of conv(F ζ 1 ) appeared as nondeterministic directions for (X ζ , S, Z d ) we will study the convex sets conv(L n ζ (k) + F ζ n ) generated by the points k in K ζ , which depend on the combinatorics of the substitution. We say a subset W ⊆ K ζ is a set of differences if there exist two patterns w 1 , w

2 ∈ L K ζ (X ζ ) such that w 1 (k) is equal to w 2 (k) if and only if k is in K ζ \ W .
The next lemma gives a sufficient condition to ensure a vector v to be a nondeterministic for (X ζ , S, Z d ), seen as the converse of Theorem 5.2 in the polytope case. As in Lemma 3.9, consider a set C Z d such that for all n > 0,

C +F ζ n +F ζ n ⊆ L n ζ (C)+F ζ n and K ζ = K ζ +C. Lemma 5.12. Let W ⊆ K ζ be a set of differences, k ∈ W , n > 0, a point f ∈ ∂ conv(L n ζ (k) + F ζ n ) and v ∈ S d-1 be such that f + ∂H v is supporting to conv(L n ζ (k) + F ζ n ) at f . Suppose that f satisfies the following conditions: H1. f + K ζ ⊆ L n ζ (K ζ ) + F ζ n , H2. (f + K ζ ) ∩ (f + H v ) ⊆ L n ζ (K ζ \ W ) + F ζ n .
Then v is nondeterministic for (X ζ , S, Z d ).

Proof. Let w 1 , w 2 be two patterns such that w

1 (k ) = w 2 (k ) if and only if k ∈ K ζ \W . Note that Condition H1 is equivalent to for all m > 0, L m ζ (f ) + L m ζ (K ζ ) + F ζ m ⊆ L n+m ζ (K ζ ) + F ζ n+m . Since K ζ = K ζ + C, Remark 1.21 (2) implies for all m > 0, L m ζ (f ) + F ζ m + (L m ζ (K ζ ) + F ζ m ) ⊆ L n+m ζ (K ζ ) + F ζ n+m . (5.3) If f is an extreme point of conv(L n ζ (k) + F ζ n ), there exists g ∈ Ext(conv(F ζ 1 )) such that f = L n ζ (k) + n-1 i=0 L i ζ (g). If f is in the relative interior of a k-dimensional face of conv(L n ζ (k) + F ζ n ), (1 ≤ k ≤ d -1), we consider g ∈ Ext(conv(F ζ 1 )) such that L n ζ (k) + n-1 i=0 L i ζ (g) and f are in the same k-dimensional face of conv(L n ζ (k) + F ζ n ) as shown in Fig. 5.4. f L n ζ (k) + n-1 i=0 L i ζ (g) H v Figure 5.4: The hyperplane ∂H v supporting to conv(L n ζ (k) + F ζ n ) at f and L n ζ (k) + n-1 i=0 L i ζ (g). Now, Condition H2 is equivalent to for all m > 0 L m ζ ((f + K ζ ) ∩ (f + H v )) + F m ⊆ L n+m ζ (K ζ \ W ) + F ζ n+m . (5.4)
We will prove that for all m > 0

(L m ζ (f )+ m-1 i=0 L i ζ (g)+L m ζ (K ζ )+F ζ m )∩(L m ζ (f )+ m-1 i=0 L i ζ (g)+H v ) ⊆ (L m ζ (f )+K ζ )∩(f +H v )+F m .
(5.5) We have that v, g = min matrix M in the symmetry group of (X ζ , S, Z d ) permutes those hyperplanes defined by these (d -1)-dimensional faces of conv(T ζ ). By condition (PC 3), the normal vectors of these hyperplanes are invariant by a power of the expansion matrix L * ζ . Hence M * permutes n eigenspaces {Qv 1 , . . . , Qv n } of some power of L * ζ . Moreover, we can assume that these vectors have integer coordinates not having common divisors except ±1. Note that each vector is unique up to a sign and does not depend on M .

So, (M n! ) * leaves invariant these eigenspaces, i.e., (M n! ) * v i = α i v i . Since the vectors v i have integer coordinates with no common divisor, we have that α i ∈ Z, for all 1 ≤ i ≤ n. The same being true for (M -1 ) * , we get that each α i is invertible in Z, which implies |α i | = 1 so M 2n! is equal to the identity matrix, and then M has a finite order. By Proposition 4.11 any homomorphism of (X ζ , S, Z d ) is invertible. Furthermore, for any v i , we have that M * v i = λ i v j , for some 1 ≤ i, j ≤ n and λ i ∈ Q. Since M ∈ GL(d, Z) and the coordinates of v j does not have common divisors, we have that λ i ∈ Z. Moreover, note that (M 2n! ) * v i = λ i • λ i 1 • • • λ i 2n!-1 v i = v i for some λ i 1 . . . λ i 2n!-1 ∈ Z, hence |λ i | = 1 for all 1 ≤ i ≤ n. Up to a change of indices, we can assume that {v 1 , . . . , v d } is a R d -basis, so M * has the form M * = P Q M P -1 , where Q M is the matrix with columns equal to the coordinates of M v i (which for all 1 ≤ i ≤ d is equal to some v j or -v j ) in the basis {v 1 , . . . , v d }. We then conclude that the symmetry group N (X ζ , S, Z d ) is finite and by Minkowski's theorem we have that N (X ζ , S, Z d ) ≤ GL(d, Z/3Z).

Finally, note that M ≤ P • Q M • P -1 , where P , P -1 and sup M ∈ N (X ζ ,S,Z d ) Q M < ∞ only depend on the convex hull of the digit tile T ζ . Remark 5. [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF]. In particular, it follows from the proof that if n = d each matrix Q M is a permutation matrix, so N (X ζ , S, Z d ) is conjugate to a subgroup of the hyperoctaedral group W d . These recover results in [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF] for block substitutions. By the realization result [START_REF] Bustos | Admissible reversing and extended symmetries for bijective substitutions[END_REF]Theorem 35] these results obtained are optimal.

The following theorem summarizes all the properties satisfied for aperiodic primitive reduced polytope substitutions. Until now, we didn't find an aperiodic d-dimensional primitive constant-shape substitution with less than d linearly independent nondeterministic directions. In the case of aperiodic primitive block substitutions, it can be easily proved that this hypothesis is true. Moreover, given the result in [START_REF] Guillon | Determinism in subshifts[END_REF] for the two-dimensional case, by Theorem 5.2, the hypothesis is true for all of the cases where conv(T ζ ) does not have two parallel edges. In a private communication, P. Guillon [62] mentioned this result is already proved for higher dimensions, but nowhere published. This implies, we only have to deal in the case conv(T ζ ) has two parallel (d -1)-dimensional faces.

Theorem 5.17 deal only with polytope substitutions, and therefore leaves open how to characterize the normalizer semigroup for nonpolytope substitutions, i.e., where the convex hull of the digit tile is nonpolytope. In [START_REF] Strichartz | Geometry of self-affine tiles[END_REF], the authors gives a description of the convex hull of the digit tile on the nonpolytope case. This description may be useful to obtain a characterization for the normalizer semigroup and symmetry semigroup. Nevertheless, until now there are no good descriptions of the convex hull of the digit tile for higher dimensions.

The work in this thesis corresponds to the first examples of realization results about the set of nondeterministic directions for minimal actions. In [START_REF] Boyle | Expansive subdynamics[END_REF] it was proved that for any compact set of S 1 that is not a singleton containing one line with irrational slope can be realized as the set of nonexpansive directions of a Z 2 -action, and the singleton case was after proved by M. Hochmann in [START_REF] Hochman | Non-expansive directions for Z 2 actions[END_REF]. If aperiodic bijective on extremities primitive constantshape substitutions have d linearly independent nondeterministic directions, then we cannot obtain a unique nondeterministic direction with these substitutions, so we will need to use other types of substitutions, or other type of subshifts (such as Toeplitz sequences) to obtain other realization results with minimal subshifts.

In Chapter 6 we describe the normalizer group for two examples. The first one is a bijective polytope substitution, so we may apply the results proved in Chapter 4 and Chapter 5. The second example is a non-reduced constant-shape substitution, such that its reduced substitution is trivial, i.e., the subshift generated is conjugate to the one-point system. This implies the techniques developed in Chapter 4 and Chapter 5 does not apply. Nevertheless, we are able to characterize its maximal equicontinuous factor and its normalizer group.

Chapter 6

Some examples of constant-shape substitutions

As we have seen the normalizer group is very restrictive (Theorem 5.17) under strong combinatorial and geometric conditions on the constant-shape substitutions.

In this chapter we characterize the normalizer group for two examples of constant-shape substitutions. The first one is called the table substitution, which is a discretization of the table tiling. Its maximal equicontinuous factor was determined in [START_REF] Robinson | On the table and the chair[END_REF]. Here, we prove that the normalizer group of the table tiling is isomorphic to a direct product of the shift action and the dihedral group D 4 , given by the symmetries of the square (Proposition 6.1), using the techniques developed in Chapter 4 and Chapter 5.

The second example is called the half-hex substitution, which is a discretization of the so-called half-hex tiling. This is a non-reduced constant-shape substitution, such that its reduced substitution is trivial, i.e., the subshift generated is conjugate to the onepoint system. Then, one cannot apply the rigidity results developed in Chapter 4 and Chapter 5 cannot be applied. Nevertheless, we characterize its maximal equicontinuous factor. Furthermore, we prove it is a coalescent system, and its symmetry semigroup is a group. By Proposition 1.5 we have that the normalizer semigroup is actually a group. Moreover, we prove that its normalizer group is a semi-direct product of the shift action and GL(2, Z) (Theorem 6.3). This is a first example of a minimal aperiodic subshift with an infinite symmetry group.

The table substitution

The table tiling is a well known example of a rep-tile tiling (see [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF]Section 4.9] for more properties about this tiling), i.e., it is a polygon that can be tiled by a finite number of smaller, congruent copies of itself. The tile-substitution of the table tiling is shown in Fig. 6.1, 3 → 1. This permutation is given by reflecting the rectangles defining the table tiling by the x-axis. We now define the map ψ : X t → ψ(X t ) as ψ(x) n = Ψ(x M -1 2 n ) for all x ∈ X t , n ∈ Z d . A direct computation shows that S (-1,0) φ permutes the fixed points of ζ 2 t . Hence, by minimality of (X t , S, Z d ), we have that ψ(X t ) = X t , so ψ is a homomorphism associated with the matrix M 2 of (X t , S, Z d ). We also have that S (0,3) ψζ 2 t = ζ 2 t ψ. So ψ is the homomorphism associated with M 2 given by Theorem 4.12. Since D 4 = M 1 , M 2 and the homomorphisms are induced by letter-to-letter maps, the composition of arbitrary maps in {φ, ψ} is a homomorphism induced by a letter-to-letter map. This implies M 1 , M 2 ∼ = φ, ψ . To conclude, this last statement let us define a right split on the exact sequence (1.4), i.e., a group homomorphism from D 4 to N (X t , S, Z 2 ). By this, we conclude that N (X t , S, Z 2 ) = S {φ, ψ} ∼ = Z 2 D 4 .

The half-hex substitution

Another well known inflation rule is the half-hex inflation shown in Fig. 6.3 (see [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF]Section 6.4] for more properties about this tiling). In tiling vocabulary, it is a edge-to-edge inflation, which means that each inflated tile is precisely dissected into copies of the tiles so the vertices of any tile intersect only the vertices of the adjacent tiles. This inflation defines an aperiodic tiling of the plane as proved in [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF]. The following shows a pattern of the half-hex tiling: Since the largest edge of any half-hex can only meet the largest edge of the adjacents half-hexes, two half-hexes always join to form a regular hexagon, through the largest edge. With this procedure, the half-hex tiling can be decomposed in three types of hexagons which are distinguished by a single diagonal line (see [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF]). Using this full hexagons, we can define a pseudo inflation (using the vocabulary on [START_REF] Baake | Encyclopedia of Mathematics and its Applications[END_REF]), which is conjugated to the half-hex tiling as the following. From this pseudo inflation, we consider an inflation with only the four shaded hexagons in Fig. 6.6. On this tiling there is an invariant lattice (by translation of hexagons) Λ = Since { ←g ∈ ← -Z 2 (2 n Z×2 n Z) : |π -1 hh ({ ←g })| = 3} is the orbit O( ← -0 , Z 2 ), we have that πhh (φ) is in O( ← -0 , Z 2 ). Moreover, for any n ∈ Z 2 , πhh (S n ) is equal to [n (mod2 n Z × 2 n Z)] which is in O( ← -0 , Z 2 ). By injectivity of πhh we conclude that End(X hh , S, Z 2 ) = S . In particular any endomorphism of (X hh , S, Z 2 ) is an automorphism. Hence (X hh , S, Z 2 ) is a coalescent system. Now, for homomorphisms. Let τ : Z 2 /2Z × 2Z \ {(0, 0)} → A given by above equations. It is important to note that τ is a bijection. Proof of the Claim. Since the sign does not change the parity of the coordinates (m, n), the statement is true for k = 1 and k = -1. We only need to prove the claim for k > 0. Suppose that for any 1 ≤ k < k the statement is true. We separate the proof in two cases:

1. If k is even, we write k = 2j.

Then x(km, kn) = x((2j)m, (2j)n) = ζ hh (x)((2jm), 2jn) = ζ hh (x(jm, jn)) (0,0) = x(jm, jn), where the last equality is by the definition of the substitution. We conclude that x(km, kn) = x(m, n). Question.

1. Is there a recognizability property satisfied by periodic infinite symbolic factors of a constant-shape substitution? B. Solomyak [START_REF] Solomyak | Nonperiodicity implies unique composition for self-similar translationally finite tilings[END_REF] already proved this result for primitive constant-shape substitutions. An idea would be to see if we can extend this proof for symbolic factors of substitutive subshifts. The answer to this question will help to understand multidimensional substitutive subshifts and their topological Cantor factors.

2. Do non-primitive constant-shape substitutions producing an aperiodic subshift satisfy a recognizability property? In the one-dimensional case this property is true [START_REF] Bezuglyi | Aperiodic substitution systems and their Bratteli diagrams[END_REF].

3. In the non-minimal case, are the constant-shape substitutions recognizable for aperiodic points as proved in [START_REF] Berthé | Recognizability for sequences of morphisms[END_REF] for the one-dimensional case?

To a Cobham theorem for constant-shape substitutions

An open question of this thesis is how different two substitutions producing the same subshift but with different expanding matrix and/or different fundamental domains can be. For example, if a substitution is defined with a diagonal matrix, but with a triangle support, is there any square substitution (diagonal matrix and the standard square support) producing the same shift?

In [START_REF] Fagnot | Sur les facteurs des mots automatiques[END_REF] it was proved that if ζ 1 , ζ 2 are two aperiodic primitive constant-length substitutions, and (X ζ 2 , S, Z) is a symbolic factor of (X ζ 1 , S, Z), then their lengths have a common power (greater than 1). This is still true for multidimensional substitutions with expansion matrix equal to a multiple of the identity and the standard d-dimensional cubic support [START_REF] Durand | Cobham-Semenov theorem and N d -subshifts[END_REF], but there is no generalized version for all constant-shape substitutions. Note that by Theorem 3.22, if ζ 1 , ζ 2 are two aperiodic primitive constant-shape substitutions such that (X ζ 2 , S, Z d ) is a symbolic factor of (X ζ 1 , S, Z d ), then there is another constant-shape substitution ζ 3 with the same expansion matrix and support of ζ 1 such that (X ζ 2 , S, Z d ) and (X ζ 3 , S, Z d ) are conjugate. However, this does not imply combinatorial conditions for ζ 2 . See [START_REF] Durand | Decidability of the isomorphism and the factorization between minimal substitution subshifts[END_REF]Section 8.4] for an example of a non constant-length substitution which is conjugate to a constant-length one.

Moreover, it is known that constant-length substitutions are strongly related to automatic sequences. Cobham showed [START_REF] Cobham | Uniform tag sequences[END_REF] that automatic sequences are exactly letter-to-letter projection of fixed points of constant-length substitutions. Automatic sequences are generated by finite automatas, which are one of the most basic models of computation and they have a large number of interesting connections with number theory, such as transcendence theory in positive characteristic (see for example [3]), and expansion in integer bases [1].

Question.

1. Is there a version of Cobham's theorem for constant-shape substitutions, regarding the shape of their supports? Since in the one-dimensional case, substitutions are defined using only intervals as supports of them, there is no known version considering also the geometry of the supports of substitutions.

2. Are the letter-to-letter codings of all constant-shape substitutions multidimensional automatic sequences? In particular, are all constant-shape substitutions generated by a DFA?

Topological factors of constant-shape substitutions

In the one-dimensional case, substitutions (and its topological factors on Cantor sets) are included in a broader class of systems called finite rank systems. In [START_REF] Downarowicz | Finite-rank Bratteli-Vershik diagrams are expansive[END_REF] it was proved that finite rank systems are either expansive or equicontinuous. This classification result is no longer true in the multidimensional framework (Example 4.3 is an example of a constantshape substitution with a Cantor factor which neither expansive neither equicontinuous).

In [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF] it was proved that expansive Cantor factors of substitutions are conjugate to substitutions, and equicontinuous Cantor factors are conjugate to an odometer. Also, it is shown in [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF] that aperiodic substitutions have a finite number of aperiodic Cantor factors up to conjugacy. In this thesis, it was proved that aperiodic symbolic factors of aperiodic primitive constant-shape substitutions are conjugate to aperiodic primitive constant-shape substitutions Theorem 3.22, extending the result proved in [START_REF] Müllner | Automorphisms of automatic shifts[END_REF].

It can be proved using Theorem 4.1 and Theorem 3.22 that if we only consider reduced substitutions, for a substitutive subshift which is conjugate to an aperiodic primitive reduced substitution, there is a finite number of aperiodic symbolic factors, which are conjugate to a substitutive subshift given by an aperiodic primitive reduced substitution. But this does not cover all of the cases. Moreover, it is known the result is true for linearly repetitive primitive constant-shape substitutions as proved in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF].

Question.

1. Substitutive subshifts have a finite number of aperiodic symbolic factors?

2. Does there is a classification theorem for topological factors of constant-shape substitutions, as the one proved in [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF] for one-dimensional substitutions?

3. Are all substitutive subshifts coalescent?

4. Is the automorphism group of any substitutive subshift virtually generated by the shift action?

5. Is there a substitutive subshift with a nontrivial topological factor with a connected phase space? Until now, even in the one-dimensional case this question is open.

Decidability problems on constant-shape substitutions

The decidability of a problem corresponds to the existence of an algorithm to give a positive (or negative) answer to the problem. Since substitutions are defined by finite objects, it is natural to ask about the decidability of some properties about them.

On the study of homomorphisms between Z d -topological dynamical systems

We study in this thesis homomorphisms between Z d -symbolic dynamical systems generated by constant-shape substitutions. This notion extends the classical dynamical one of morphism like factor and conjugacy. Isomorphisms are conjugacies of Z d -actions, up to GL(d, Z)-transformations. We show the class of substitutive subshifts is stable under aperiodic symbolic factors. We prove any measurable factor induces a continuous one. We also get strong restrictions on the homomorphisms of a generic family of substitutive subshifts: they are invertible, the normalizer group is virtually generated by the shift action and its quotient by the automorphisms is limited by the digit tile of the substitution. We prove this by describing their set of nondeterministic directions. Finally, we show the optimality of the hypotheses by exhibiting an example of a minimal subshift with an infinite symmetry group.

Keywords: Symbolic dynamics, substitutive subshift, homomorphism, automorphism group, normalizer group, symmetry group, digit tile, nonexpansive half-space.

Sur l'étude des homomorphismes entre Z d -systèmes dynamiques topologiques

Nous étudions les homomorphismes entre des Z d -systèmes symboliques engendré par des substitutions de forme constante. Cette notion étend les concepts de facteur et conjugaison. Les isomorphismes sont alors des conjugaisons de Z d -actions, à une transformation de GL(d, Z) près. Nous montrons que la classe de sous-shifts substitutifs est stable par les facteur symbolique aperiodique. Nous prouvons que tout facteur mesuré induit un facteur continu. Nous obtenons également des restrictions fortes des homomorphismes d'une famille générique des sous-shifts substitutifs: ils son inversibles, le normalisateur est virtuellement engendré par l'action du shift et son quotient par des automorphismes est limité par le digit tile de la substitution. Nous prouvons ceci en décrivant leurs ensembles des directions non déterministes. Finalement, nous prouvons que nous hypothèses sont optimales en donnant un example d'un sous-shift substitutif avec un groupe de symétrie infini.

Mots-clés: dynamique symbolique, sous-shift substitutif, homomorphisme, groupe de automomorphisme, normalisateur, groupe de symétrie, digit tile, demi-espace nonexpansif.

Introduction Résumé 1

 1 Definitions and background 1.1 Basics on discrete, convex and fractal geometry . . . . . . . . . . . . . . . . 1.1.1 Discrete geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Convex geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Topological dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Homomorphisms between topological dynamical systems . . . . . . . 1.3 Measure-preserving systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Nondeterministic directions of a topological dynamical system . . . . . . . . 1.6 Odometer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Multidimensional constant-shape substitutions . . . . . . . . . . . . . . . . 2 The symmetry semigroup of Z 2 -odometers 2.1 The universal odometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bifurcation phenomenon of the normalizer semigroup for constant base Z 2odometer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The diagonal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The triangular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The recognizability property of constant-shape substitutions 3.1 The recognizability property of aperiodic symbolic factors of substitutive subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Invariant orbits of substitutive subshifts . . . . . . . . . . . . . . . . . . . . 3.3 The repetitivity function of substitutive subshifts . . . . . . . . . . . . . . .

Figure 3 :

 3 Figure 3: The five basic Robinson tiles (up to rotation and reflection).

1. 1 . 1

 11 Discrete geometryIf F ⊆ Z d is a finite set, it will be denoted by F Z d , and we use the notation F = max n∈F n , where • is the standard Euclidean norm of R d . The standard Cartesian product in R d will be denoted by •, • . If L ∈ M(d, R) is a matrix, we denote L = max x∈R\{0} L(x) / x as the matrix norm of L. We denote GL(d, Z) as the set of d × d matrices M with integer coefficientes such that | det(M )| = 1.

  and x ∈ C is a point in the relative boundary of C, An affine hyperplane ∂H[a; c] = {y ∈ R d : a, y = c}, for some a ∈ R d \ {0} and c ∈ R is supporting to C at x, if x ∈ ∂H[a : c] and inf y∈C a, y < a, x = c = sup y∈C a, y .

Fig. 1 .

 1 Fig. 1.1 illustrate the opposite normal cones of a triangle.

Figure 1 . 1 :

 11 Figure 1.1: Example of the opposite normal cones of a triangle, and the stratification of the circle S 1 given by them.

  If π : (X, T, Z d ) → (Y, T, Z d ) is a factor map between minimal systems, and there exists y ∈ Y such that |π -1 ({y})| = 1, then this property is satisfied in a G δ dense subset Y 0 ⊆ Y . In this case, we say that π is almost 1-to-1. If |π -1 ({y})| = K for all y in a G δ dense subset of Y , then we say that π is almost K-to-1. If |π -1 ({y})| ≤ K < ∞ for all y ∈ Y , we say that π is finite-to-1.

2 . 1 )(P L n 1 P

 211 If L 1 ∈ M(d, Z) is an integer expansion matrix and M ∈ N ( ), then for any P ∈ GL(d, Z), P M P -1 is in the symmetry semigroup N ( ← -Z d -1 ) ). 3. If M ∈ GL(d, Z) commutes with some power of the expansion matrix L, then M is in the symmetry semigroup N ( ← -Z d (L n ) ) Now we define Toeplitz sequences. Let A be a finite alphabet and Z ⊆ Z d be a finite index subgroup of Z d . For x ∈ A Z d and a ∈ A we define Per(x, Z, a) = {n ∈ Z d : x(n + m) = a, for all m ∈ Z}.

Fig. 1 .Figure 1 . 2 :Figure 1 . 3 :

 11213 Figure 1.2: An example of a constant-shape substitution over a four-letter alphabet.

Fig. 1 .Figure 1 . 4 :

 114 Figure 1.4: Approximation of some digit tiles: (a) Gasket, (b) Rocket, (c) Shooter, (d) Twin Dragon. The names of these tiles comes from[START_REF] Vince | Digit tiling of Euclidean space[END_REF]

  next result shows that for any aperiodic symbolic factor (Y, S, Z d ) of (X ζ , S, Z d ) we can change ζ for an appropriate substitution ζ with the same expansion matrix and fundamental domain such that (X ζ , S, Z d ) and (X ζ , S, Z d ) are conjugate, and there exists a factor π : (X ζ , S, Z d ) → (Y, S, Z d ) induced by a 0-block map. Lemma 1.22. Let ζ be an aperiodic primitive constant-shape substitution and φ : (X ζ , S, Z d ) → (Y, S, Z d ) be an aperiodic symbolic factor of (X ζ , S, Z d ). Then, there exists a substitution ζ having the same support and expansion matrix such that (X ζ , S, Z d ) and (X ζ , S, Z d ) are conjugate and a factor map π : (X ζ , S, Z d ) → (Y, S, Z d ) induced by a 0-block map. Proof. Suppose that φ : (X ζ , S, Z d ) → (Y, S, Z d ) is a factor map via a B(0, r)-block map. Set A = B(0, r), by Proposition 1.20 there exists a set C Z d such that B(0, r)+F ζ 1 +C ⊆ L ζ (C) + F ζ 1 . Set D = L ζ (C) + F ζ 1 . We will define a substitution ζ (D) considering the set L D (X ζ ) as the alphabet with the same expansion matrix and support of ζ in the following way: If p ∈ L D (X ζ ), then for any j ∈ F ζ 1 we set ζ (D) (p) j = ζ(p)| j+D . It is straightforward to check that x ∈ X ζ is a fixed point of the substitution ζ, if and only if y ∈ L D (X ζ ) Z d such that y n = x| n+D for all n ∈ Z d is a fixed point of the substitution ζ (D) . With this, we can define the following sliding block codes ψ 1 : (X ζ , S, Z d ) → (X ζ (D) , S, Z d ) given by the D-block map Ψ 1 (p) = p, and ψ 2 : (X ζ (D) , S, Z d ) → (X ζ , S, Z d ) given by the 0-block map Ψ 2 (p) = p 0 . These maps commute with the shift action and define a conjugacy between X ζ and X ζ (D) . We then, define a factor map φ (D) : (X ζ (D) , S, Z d ) → (Y, S, Z d ) given by a 0-block map equal to ψ 2 φ.

Example 2 . 4 (

 24 Different results for Theorem 2.2). (1) Consider the matrix L 1 = 2 -1 1 3

R 2 .where m 12 =

 212 Then, M satisfies (Normalizer Condition). Note that M has the form (p -s)m 12 -(m 11 -m 22 )q and m 21 = (p -s)m 21 , with m 12 , m 21 ∈ Z. Now, for all n, m > 0, (NC 2) implies -m 21 p m q(n) m 12 s n+m -m 21 q(m)q(n) m 21 p n+m m 21 pq(m) p n s n ).

m 22 .= m 11 (m 11 -m 22

 22111122 e., (p -s)M = (m 11 -m 22 )L + (p • m 22 -m 11 • s) id R 2 . Since m 21 = 0, then M has the form M = m 11 m 12 (m 11 , m 22 ) 0 where m 12 (m 11 , m 11 ) satisfies (p -s)m 12 (m 11 , m 11 ) = (m 11 -m 22 )q. -Note that m 11 = m 22 if and only if m 12 = 0. -If m 11 = m 22 , then m 11 -m 22 ∈ {-2, 2}, so (p -s)m 12 = ±2q. Since M has integer coefficients, this necessarily implies 2q ∈ (p -s)Z. If this condition is satisfied, then M has the form M

m 12 =

 12 m 21 k 2 + k(m 11 -m 22 ). Since | det(M )| = 1, and det(M ) = (m 11 +m 21 •k)(m 22 -m 21 •k), we get that |m 11 +m 21 •k| = 1 and |m 22 -m 21 • k| = 1. We then can parameterize the matrices in N ( ← -Z 2 (L n ) ) as follows:

Figure 3 . 3 :

 33 Figure 3.3: Illustration of the patterns ζ n (w) and a n in j n .

Proposition 3 . 4 .

 34 Let ζ be an aperiodic primitive constant-shape substitution. Then, there exist finitely many ζ-invariant orbits in the substitutive subshift X ζ . The bound is explicit and depends only on d, |A|, L -1 ζ , F ζ 1 and det(L ζ -id).

1 = 1 =

 11 from Proposition 1.18. Using -H as the set A and F = F ζ 1 in Proposition 1.20 we obtain a set C Z d such that L n ζ (-H + C) + F ζ n ⊆ L n+1 ζ (C) + F ζ n+1 for all n > 0. Define D Z d be such that D = C + K ζ -H. Suppose that there are more than |A| |D| • |H| ζ-invariant orbits. By the Pigeonhole Principle, there exist j ∈ H and two different points x = y ∈ X ζ such that x| D = y| D and ζ(x) = S j x, ζ(y) = S j y. Note that ζ(x| D ) = ζ(x)| L ζ (D)+F ζ x| j+L ζ (D)+F ζ 1 Hence, we have x| j+L ζ (D)+F ζ y| j+L ζ (D)+F ζ 1

  n+1 and we conclude by Proposition 1.18.

where |λ 1 |

 1 , |λ d | are the maximum and minimum of the absolute values of the eigenvalues of L ζ , respectively.

Theorem 3 .

 3 13. [108, Theorem 3.13] Let ζ be an aperiodic primitive tiling substitution with expansion matrix L ζ , which has a fixed point. Then the following are equivalent for x ∈ R d :

Definition 3 . 19 . 2 .

 3192 Let ζ be an aperiodic primitive constant-shape substitution. 1. We say that a pair (a, b) ∈ A 2 \ ∆ A is a periodic pair if there is a cycle in G ζ which starts and ends in (a, b). We define n(a, b) = min{|P | : P is a cycle in (a, b)} and we denote n(ζ) = lcm{n(a, b) : (a, b) is a periodic pair}, we call the substitution pair-aperiodic if n(ζ) = 1. We call a pair (a, b) ∈ A \ ∆ A an asymptotic disjoint pair if for any k > 0, there exists a path P = (a 0 , b 0 ) . . . (a k , b k ) in G ζ of length k with (a 0 , b 0 ) = (a, b) and (a k , b k ) / ∈ ∆ A .

)

  Assume that ζ is pair-aperiodic. If (a, b) ∈ A 2 is not an asymptotic disjoint pair, let k be the minimum length of a path from (a, b) such that any path of length k has an end in ∆ A . If k > |A| 2 , there exists a cycle as a subpath in P , i.e., one of the vertex (c, d) is a periodic pair. Since ζ is pair-aperiodic, there exists f ∈ F ζ 1 such that ζ(c) f = c = d = ζ(d) f . So we can create a path of length arbitrarily large from (a, b) that not reach ∆ A , which is a contradiction of not being asymptotic disjoint pair. We then have that k ≤ |A| 2 , which implies ζ |A| 2 (a) = ζ |A| 2 (b). Definition 3.21. Let A, B be two finite alphabets, ζ be an aperiodic primitive constantshape substitution with alphabet A and τ : A → B. We say that a, b ∈ A with a = b are indistinguishable (by (ζ, τ )) if for all n ≥ 0 we have that τ (ζ n (a)) = τ (ζ n (b)).

Theorem 3 . 22 .

 322 Let (Y, S, Z d ) be an aperiodic symbolic factor (with alphabet B) of a substitutive subshift (X ζ , S, Z d ), with ζ being an aperiodic primitive constant-shape substitution with alphabet A. Then, there exists an aperiodic primitive constant-shape substitution ζ with alphabet C having the same expansion matrix and support of a power of ζ and a conjugacy τ : (X ζ , S, Z d ) → (Y, S, Z d ) via a 0-block map. Proof. By Lemma 1.22 we can assume that the factor map τ : (X ζ , S, Z d ) → (Y, S, Z d ) is induced via a 0-block map and by Remark 3.20 we can assume that ζ is a pair-aperiodic substitution.

  having the same label of edges as P 1 . Now, repeating this process we get a path P 3 in G ζ of length (max{|[a]|, |[b]|} + 1)k from

  finite, we may consider an appropriate iteration of ζ 1 and ζ 2 such that any factor ψ satisfying 2. in Theorem 4.1 satisfies S p ψζ 1 = ζ 2 ψ.

  1). If a substitution is not reduced, we consider an equivalence relation calling two letters a, b equivalent when d n (ζ n (a), ζ n (b)) → 0. If two letters a, b ∈ A are equivalent, then (ζ(a)) f ∼ (ζ(b)) f for all f ∈ F ζ 1 . We define a substitution ζ on A/ ∼ given by ( ζ([a]

Proposition 4 . 4 .

 44 Let ζ be an aperiodic primitive constant-shape substitution. If ζ is aperiodic, the natural factor between (X ζ , S, Z d ) and (

Fix

  a, b ∈ A. Note that d n+j (ζ n+j (a), ζ n+j (b)) = 1 |F ζ n+j | (c,d)∈D a,b n |F ζ j |d j (ζ j (c), ζ j (d)) + (c,d)∈E a,b n |F ζ j |d j (ζ j (c), ζ j (d))

k→∞ d k

 k (ζ k (c), ζ k (d)) ≤ 1 we have that d n+j (ζ n+j (a), ζ n+j (b)) ≤ |D a,b n |/|F ζ n | and this is true for every j large enough, so lim k→∞ d k (ζ k (a), ζ k (b)) ≤ |D ad n (ζ n (a), ζ n (b)) -lim k→∞ d k (ζ k (a), ζ k (b)) .

n∈N a =b∈B d n (ζ n 2

 2 (a), ζ n 2 (b)) and R the radius from the recognizability property of X ζ 2 . Recall that the recognizability property implies the substitutions maps are injective. Let φ be in m Fac(X ζ 1 , X ζ 2 , S, Z d ). The map π n (x) -π n (φx) (mod L n (Z d ))is invariant under the Z d -action, so is constant µ ζ 1 -a.e. We denote this constant by p n

Proposition 4 . 7 .

 47 Let ζ be an aperiodic primitive reduced constant-shape substitution. Then (X ζ , S, Z d ) is coalescent.

Proposition 4 . 11 .

 411 Let ζ be an aperiodic primitive reduced constant-shape substitution. If M ∈ GL(d, Z) has finite order, then any homomorphism φ ∈ N M (X ζ , S, Z d ) is invertible.

Remark 4 . 13 .

 413 Let ψ ∈ Hom M (X ζ 1 , X ζ 2 , S, Z d ) satisfying property (2) of Theorem 4.12.

Lemma 4 . 14 .

 414 If φ, ψ ∈ mN M (X ζ 1 , X ζ 2 , S, Z d ) are such that d(φ, ψ) is smaller than η/C(R), then φ, ψ are equal µ ζ 1 -a.e in X ζ 1 . Lemma 4.15. Let φ ∈ mN M (X ζ 1 , X ζ 2 , S, Z d ).Then there exists a sequence (ψ n ) of homomorphisms associated with M of radius F ζ 1

Proposition 4 . 16 .

 416 Let (X ζ , S, Z d ) be a subshift from a reduced aperiodic primitive constant-shape substitution ζ from a finite alphabet. If the set of matrices M ∈ N (X ζ , S, Z d ) commuting with a power of the expansion matrix L ζ is finite, then the quotient group N C(X ζ , S, Z d )/ S is finite. A bound for |N C(X ζ , S, Z d )/ S | is given by an explicit formula depending only on d, |A|, L -1 ζ , F ζ 1 , and sup M ∈Cent GL(d,Z) (L ζ ) M . Proof. Let ψ ∈ N C(X ζ , S, Z d ), satisfying Property 2. of Theorem 4.12. Following the proof of Proposition 4.7, ψ acts as a permutation of the ζ-invariant orbits. Since the set of matrices M ∈ N (X ζ , S, Z d ) commuting with a power of L ζ is finite, there exists n > 0 such that ψ n is an automorphism of X ζ . By Proposition 4.8, we have that ψ n has finite order, which implies ψ has finite order.

n∈E

  NHn (conv(F ζ n )) = NF 1 (conv(D)).Thus, the extremal rays of NF 0 (conv(D)) are equal to sets of the form n>0 NHn (conv(F ζ n )), with H n being faces eventually of codimension 1 of conv(F ζ n ) containing π n (x 1 ).

Example 5 . 4 (F 1 =Figure 5 . 1 :

 54151 Figure 5.1: The fundamental domain and an approximation of the digit tile of Example 5.4.

Theorem 5 . 6 .

 56 Let T be the digit tile for an expansion matrix L ∈ M d (R) and a fundamental domain F 1 ⊆ R d . The following statements are equivalent:1. The convex hull of the digit tile T (L, F 1 ) is a polytope.2. [109, Theorem 4.2]The inward unit normal vectors of the (d -1)-dimensional faces of conv(F 1 ) are eigenvectors of (L * ) k for some k.

3. [ 77 ,Remark 5 . 7 .

 7757 Theorem 2.2] The cardinality of Ext(conv(F n )) and Ext(conv(F n+1 )) are the same for some n > 0. In such a case, for any m > n, | Ext(conv(F m ))| = | Ext(conv(F n ))|, and then | Ext(conv(T (L, F 1 ))| = | Ext(conv(F n ))|. In the case L = λ id R d , with λ > 1, a direct computation shows that the statements (2) and (3) of Theorem 5.6 are satisfied without taking any power of L.

Example 5 . 8 .Figure 5 . 2 :

 5852 Figure 5.2: The fundamental domain and an approximation of the digit tile of a non self-similar matrix.

Fig. 5 2 Figure 5 . 3 :

 5253 Figure 5.3: The sets F 1 and F 2 .

Proposition 5 . 10 .

 510 If | Ext(conv(F ζ 1 )| = | Ext(conv(T (L, F 1 )))|, then the eigenvalues of L are integer numbers.

(PC 2 )

 2 The convex set conv(F 1 ) is nondegenerate and| Ext(conv(F 1 ))| = | Ext(conv(T (L, F 1 ))|.(PC 3) Any inward unit normal vector of a (d -1)-dimensional face of conv(F 1 ) is an eigenvector of L * .

Corollary 5 . 11 (

 511 Nondeterministic directions in the polytope case). Let ζ be an aperiodic primitive polytope substitution. The set of nondeterministic directions ND(X ζ , S, Z d ) is the intersection of S d-1 with a nonempty union of opposite normal cones of the form NG (conv(T ζ )), where G is a face of conv(T ζ ).

  ) + L m ζ (K ζ ) + F ζ m ) ∩ (L m ζ (f ) + m-1 i=0 L i ζ (g) + H v ), there exist k 1 ∈ K ζ , j ∈ F ζ m and h ∈ H v such that n = L m ζ (f ) + exist c ∈ K ζ and l ∈ F ζ m such that m-1 i=0 L i ζ (g) + L m ζ (k 1 ) + j = L m ζ (c) + l.

Theorem 5 . 17 .

 517 Let ζ be an aperiodic reduced primitive polytope substitution. Then 1. The system (X ζ , S, Z d ) is coalescent, and any homomorphism inN (X ζ , S, Z d ) is invertible.If there are d linearly independent vectors that are nondeterministic directions for (X ζ , S, Z d ), we have that 2. The normalizer group is virtually generated by the shift action.

3 .

 3 The symmetry group N (X ζ , S, Z d ) acts as a permutation group in the set{N G (conv(T ζ )) : N G (conv(T ζ )) ⊆ ND(X ζ , S, Z d )}. In particular, if ND(X ζ , S, Z d ) = S d-1, then the symmetry group N (X ζ , S, Z d ) is isomorphic to a subgroup of the automorphism group of the normal fan of conv(T ζ ).Proof. The statement 1. is true by Proposition 4.7 and Proposition 4.11. Now, by the third isomorphism theorem we have thatN (X ζ , S, Z d ) /Aut(X ζ , S, Z d ) ∼ = N (X ζ , S, Z d ) / S /( Aut(X ζ , S, Z d ) / S ).Then, by Proposition 5.15 gives the quotient group N (X ζ , S, Z d )/ Aut(X ζ , S, Z d ) is finite and Proposition 4.8 implies Aut(X ζ , S, Z d )/ S is also finite, so we conclude that N (X ζ , S, Z d )/ S is a finite group. Finally, statement 3. is true by Proposition 5.15.

Figure 6 . 3 :

 63 Figure 6.3: Tile-substitution of the half-hex tiling.

Figure 6 . 4 :

 64 Figure 6.4: A pattern of the half-hex tiling.

Figure 6 . 5 :

 65 Figure 6.5: The three tiles as a new alphabet for the half-hex tiling.

Figure 6 . 6 :

 66 Figure 6.6: New tile-substitution conjugate to the half-hex tiling, with a discrete 2dimensional subaction in R 2 .

Claim 2 .

 2 For any (m, n) ∈ Z 2 \ 2Z 2 , the letter in x(m, n) (hence y(m, n) and z(m, n)) only depend on the parity of the coordinates.Proof. Let (m, n) ∈ Z 2 \ 2Z 2 . We can write (m, n) = 2(a, b) + (f 1 , f 2 ) with (a, b) ∈ Z 2 and (f 1 , f 2 ) ∈ F hh 1 \ {(0, 0)}. Since ←x is a fixed point of ζ hh we have that ←x (m, n) = ζ hh (x(a, b)) (f 1 ,f 2 ) ,which only depend on (f 1 , f 2 ) by the very definition of ζ hh .In particular we get that(m, n) ≡ (1, 0) (mod 2) =⇒ x(m, n) = y(m, n) = z(m, n) = 2 (m, n) ≡ (0, 1) (mod 2) =⇒ x(m, n) = y(m, n) = z(m, n) = 0 (m, n) ≡ (1, 1) (mod 2) =⇒ x(m, n) = y(m, n) = z(m, n) = 1.

Claim 3 .

 3 Let (m, n) ∈ Z 2 with gcd(m, n) = 1. Then for any k ∈ Z we have that x(km, kn) = x(m, n) (and the same for y, z).

2 .

 2 If k is odd, we write k = 2j + 1. Using the Euclidean division we have thatkm = (2j + 1)m = 2a + g 1 , m = 2a + f 1 kn = (2j + 1)n = 2b + g 2 , n = 2b + f 2 , with a, b, a, b ∈ Z 2 and (f 1 , f 2 ), (g 1 , g 2 ) ∈ F hh 1 .Since this decomposition is unique, then a = jm + a, b = jn + b, f 1 = g 1 and f 2 = g 2 . By the definition of the substitution we conclude that x(km, kn)= ζ hh (x(2a + f 1 , 2b + f 2 )) = ζ hh (x(2a, 2b)) (f 1 ,f 2 ) = x(m, n).

  Z 2 ⊕ G, où G est localement fini et se factorise sur un groupe virtuellement simple. Ceci diffère du cas unidimensionnel où le groupe doit être résiduellement fini. Les isomorphismes du pavage de la chaise ont été étudié dans[START_REF] Baake | Reversing and extended symmetries of shift spaces[END_REF] où il a été démontré que le groupe d'automorphisme est trivial et le groupe d'isomorphisme est un produit semi-direct de Z 2 pour le groupe de symétries du carré.

	dire qui n'est pas un automorphisme (voir [10]): le full-shift, tout sous-shift palindromique,
	comme les shifts sturmien, le period-doubling sous-shift et le sous-shift de Thue-Morse. Le
	premier a un énorme groupe d'automorphisme, même pas moyennable, alors que les groupes
	d'automorphisme du deuxième et du troisième sont triviaux. Le groupe d'automorphisme
	du dernier est isomorphe à Z ⊕ C 2 . Ces exemples suggèrent que les propriétés algébriques
	du groupe d'automorphisme n'impliquent pas l'existence d'isomorphismes non triviaux.
	Il y a peu de résultat generaux existent dans le contexte multidimensionnel. Dans
	[67] M. Hochman a prouvé que la plupart des propriétés unidimensionnelles des groupes
	d'automorphisme sont préservées pour la classe des sous-shifts de type fini d'entropie pos-
	itive. Cependant, il a présenté un exemple remarquable d'un sous-shift de type fini dont le
	groupe d'automorphisme est isomorphe à
	1, n)) d'un sous-shift d'entropie nulle. Mais il
	existe encore quelques groupes dénombrables, comme le groupe discret de Heisenberg, dont
	on ne sait pas s'il peut être un sous-groupe du groupe d'automorphisme d'un sous-shift
	unidimensionnel.
	Rappelons quelques exemples de sous-shifts ayant un isomorphisme non trivial, c'est-à-

  ) if and only if |m 11 m 22 -m 12 m 21 | = 1 and m 12 = 4m 21 + 2m 22 -2m 11 .

	Remark 2.5. Theorem 2.2 implies the factor map between equicontinuous systems is not
	necessarily compatible with homomorphisms. Consider X as the 2-dimensional universal
	odometer, and Y =	← -Z 2 (L n 1 ) , where L 1 =	2 -1 1 3	, so Y is an equicontinuous factor of
	X. Now, by Proposition 2.1, we can define a homomorphism associated with the matrix
	2 1 1 1	in X, but by Theorem 2.2 and Lemma 1.6, this is not possible in Y .
	Consider the matrix L 2 = hence by Theorem 2.2 the matrices in N ( 2 -1 . In this case trace(L 2 ) = 7 and det(L 2 ) = 11, 1 5 ← -Z 2 (L n 2 ) ) are the ones commuting with L 2 . Note that L 2 has real eigenvalues (which are equal to 7/2± √ 5/2) and Cent GL(2,Z) (L 2 )
	is an infinite group containing	2 -1 -1 1	.
	(3) Set L 3 =	0 6 -3 9	.	This matrix has integer eigenvalues, which are 3
	and 6. 3 ) This set of matrices is conjugate, via the matrix We will get that a matrix M = m 11 m 12 m 21 m 22 is in N ( ← -Z 2 (L n 1 -1 -1 2 , to the set
	of matrices N ( ← -Z 2 (L n	m 11 m 12 0 m 22		: |m 11 m 22 | = 1, m 12 ∈ Z . It is easy to see that

•

  If rad(p) divides s, then any m 12 ∈ Z satisfies (2.5). Thus, any matrix M = m 11 m 12 0 m 22 with |m 11 m 22 | = 1 satisfies (NC 2).

  5.1 The automorphism group of substitutive subshifts from bijective constant-shape substitutions Since bijective substitutions are reduced, Proposition 4.8 implies the automorphism group of (X ζ , S, Z d ) is virtually Z d . In fact, we have a more rigid result in the bijective case as shown in the following proposition. Proposition 5.1. Let ζ 1 , ζ 2 be two aperiodic bijective primitive constant-shape substitutions with the same expansion matrix L and support F 1 . Then, any factor ψ : X ζ 1 → X ζ 2 satisfying Property 2. in Theorem 4.1 has radius 0. In particular, the automorphism group Aut(X ζ , S, Z d ) of a substitutive subshift from an aperiodic bijective primitive constant-shape substitution is isomorphic to the direct product of Z d , generated by the shift action, with a finite group given by a permutation of letters. Proof. Let ψ ∈ Fac(X ζ 1 , X ζ 2 , S, Z d ) satisfying Property 2. in Theorem 4.1, i.e., there exists p ∈ F 1 such that S p ψζ 1 = ζ 2 ψ. Suppose that p = 0. Let n > 0 be large enough such that the set F •C n = {f ∈ F n : f +C ⊆ F n } is nonempty, where C is the set defined in Lemma 4.6. Then for any x ∈ X ζ 1 , the coordinate x 0 determines the pattern ζ n 1 (x)| Fn , hence the pattern (S pn ψζ n 1 )| pn+(Fn) •C is also completely determined by x 0 , where p n

  for infinitely many n ∈ E, we have that necessarilyk 1 = k 3 , which is a contradiction. Consider the face H n of conv(F ζ n ) of smallest dimension generated by {h n (t)} t∈F 1 .Notice that NF 1 (conv(D)) ⊆ By construction of x 1 , x 2 , the set z n is bounded (for all n large enough it belong to K ζ -K ζ ), so there exists t ∈ F 1 and ε > 0 small enough such for all t ∈ B(t, ε) ∩ F 1 we have that z n (t ) = z n (t) for all n ∈ E large enough. Hence H n is a face of codimension 1 and an argument of dimensions let us conclude that NF 1

	NHn (conv(F ζ n )). We will prove that NF 1 (conv(D)) =
	n>0
	NHn (conv(F ζ n )).
	n>0

In the literature, especially Group Theory, is common to also ask that the union of the sequence of sets (Fn)n>0 is equal to Z d for a sequence to be Følner, but we will not use it in this thesis.

The word opposite comes from the fact that the usual normal cone is related to the outward normal vectors of convex sets, and in this thesis we will use the inward normal vectors.

) )/ Cent GL(2,Z) (L 3 ) is virtually Z.

Acknowledgments

length of the substitution. Then, N. P. Frank in [START_REF] Frank | Multidimensional constant-length substitution sequences[END_REF] followed the same ideas to describe the maximal equicontinuous factor of block substitutions. Here, we follow the same ideas in [START_REF] Frank | Multidimensional constant-length substitution sequences[END_REF], using the notion of height lattice. It is a lattice of Z d containing the set of return times of a letter, satisfying a specific property with the expansion matrix of the constant-shape substitution. Using a characterization of the eigenvalues of a substitutive subshift proved in [START_REF] Solomyak | Eigenfunctions for substitution tiling systems[END_REF] we characterized the maximal equicontinuous factor of substitutive subshift.

Substitutive subshifts as a finite-to-1 extension of a d-dimensional

odometer by the recognizability property

The recognizability property establishes a factor map from the substitutive subshift to a d-dimensional odometer as follows. For every n > 0, let π n : X ζ → F ζ n be the map satisfying x ∈ S πn(x) ζ n (X ζ ). This map is well defined ( In particular, the factor map π : (X ζ , S,

) is finite-to-1. Proof. We separate the proof in these two cases. K1: Assume that |π -1 ({ ←g })| > |A|. Let x 0 , . . . , x |A| be in π -1 ({ ←g }). For all n > 0 and all j ∈ {0, . . . , |A|}, there exist y n j ∈ X ζ such that x j = S gn ζ n (y n j ). By the Pigeonhole Principle, there exist j 1 = j 2 ∈ {0, . . . , |A|} and an infinite set E ⊆ N with y n j 1 0 = y n j 2 0

. This implies for any n ∈ E, x

To complete the proof, note that for any n > 0, we have that g n+1 -g n is in

Proof. For all n ≥ 0, we denote U n = {x ∈ X ζ 1 : (φ n x) 0 = (ψ n x) 0 }. We will prove by induction on n ≥ 0 that p n (φ) = p n (ψ) and

. By hypothesis this is true for n = 0. Now, suppose that p n (φ) = p n (ψ) and

By the recognizability property, this map vanishes on the set {x ∈ X ζ 1 : (φ n x)| B(0,R) = (ψ n x)| B(0,R) } which has a positive measure by hypothesis, and we conclude that p n+1 (φ) = p n+1 (ψ). Now, let x be in U n+1 . Then, there exist at least η|F

, so we have that

Finally, the set {x ∈ X : φ(x) = ψ(x)} is the decreasing intersection of these sets, so has a positive measure. By ergodicity, φ, ψ are equal

Then there exists a sequence (ψ n ) of sliding block codes of radius

Proof. Set ε > 0. By Lusin's theorem, there exist an integer > 0 and a continuous map f : X ζ 1 → B such that f (x) only depends on x| B(0, ) and the measure of the set

So we have that

and by Remark 1.21 we have that

n | elements in J(x) ∩ J(y), so (φ n x) 0 is equal to (φ n y) 0 by definition of η. Hence, for every x in W , (φ n x) 0 only depends on x| C . Finally, to prove Theorem 4.1 we use similar arguments given in [START_REF] Host | Homomorphismes entre systèmes dynamiques définis par substitutions[END_REF] that we describe for completeness.

Proof of Theorem 4.1. For the fixed alphabets A and B, there exist a finite number of sliding block codes of radius

) . By Lemma 4.6, there exist two different integers m, k ≥ 0 such that d(φ m , φ m+k ) < η/C(R), so by Lemma 4.5, we have that φ m = φ m+k , µ ζ 1 -a.e..

Let n ≥ m be a multiple of k.

We have that (

This implies for all r ∈ N,

Since the number of sliding block codes of radius

Note that φ n is equal to φ 2n , µ ζ 1 -a.e. We denote ψ = φ n and p = p n (ψ). By definition of p, we have that

this implies that S j φ and ψ coincides in ζ n 1 (X ζ 1 ) µ ζ 1 -almost everywhere, and by ergodicity in the whole set X ζ 1 µ ζ 1 -almost everywhere.

The automorphism group of substitutive subshifts.

Since the set of sliding block codes of radius

between X ζ to itself is finite, we have the following result as a direct corollary of Theorem 4.1. In the special case, where any automorphism of (X ζ , S, Z d ) satisfies Property 2. of Theorem 4.1 with p = 0, we have a more rigid result. Proof. Note that an automorphism φ commutes with the substitution map if and only if φ is equal to φ n , for all n > 0. Now, by Lemma 4.6, the Property 2. implies Property 1. of Theorem 4.1. So, the group of automorphisms commuting with the substitution map is finite. To conclude we just need to observe that the pair (j φ , ψ φ ) in Theorem 4.1 associated with any automorphism φ is unique.

which implies (id -L n ζ )(j 2 -j 1 ) = 0, so j 2 = j 1 and then ψ 1 = ψ 2 .

We will show in Proposition 5.1 that for bijective substitutions, the automorphism group is isomorphic to a direct product of the shift map and a finite group, generated by permutation of letters.

Rigidity properties for homomorphisms between substitutive subshifts and applications

This section is devoted to homomorphisms between substitutive subshifts. We recall that for M ∈ GL(d, Z), the map φ : (X, T, Z d ) → (Y, T, Z d ) between two topological dynamical systems is said to be a homomorphism associated with M if for all m ∈ Z d we have that φ • S m = S M m • φ. We can also define measurable homomorphisms in the measuretheoretic setting. First, we establishes a necessary condition for the matrices M with the accumulation points is at most 2, when a power of one of the L ζ -eigenvalues is a real number. Otherwise, the projective orbits of L * ζ are dense in the circle. Since ND(X ζ , S, Z 2 ) is closed, it is the whole circle, which is a contradiction.

Proof of Theorem 5.2. Let v be a nondeterministic direction for (X ζ , S, Z d ), and

2. If conv(D) does not have extreme points, then contains a line. In this case the hyperplane ∂H v must be parallel to this line contained in conv(D) and using the same argument, we can assume that x 1 (0) = x 2 (0).

So we can assume that 0 is in a face F 0 of smallest dimension of conv(D) and then v ∈ NF 0 (conv(D)). In fact any element in NF 0 (conv(D)) ∩ S d-1 is a nondeterministic direction for (X ζ , S, Z d ).

Now, for any k > 0 consider R (k) > 0 be the recognizability radius for ζ k given by Proposition 3.3 and R = 4R (k) . Since x 1 and x 2 coincide in an arbitrarily large ball, they have the same image under the maximal equicontinuous factor, hence π n (x 1 ) = π n (x 2 ) ∈ F ζ n for any n > 0. By Lemma 3.9, there exist n > 0 and two words w

). Letting k to infinity we have that for all n > 0, the origin is in the boundary of conv(

We will show now the converse. We separate the proof in two cases. Suppose that conv(D) is closed. Let F 1 be a face of conv(D) containing 0 of codimension 1. We have that

Since h ∈ H v , then v, h ≤ 0, and by (5.6) we have that v, gl (i) ≤ 0, for all 1 ≤ i ≤ m -1. Since λ > 0, we conclude that v, c ≤ 0, which implies c ∈ H v .

By (5.3) and Proposition 1.18, the iterations of the substitution on the patterns w 1 , w 2 leads to two points

, for i ∈ {1, 2}. Finally, (5.4) implies x 1 | Hv = x 2 | Hv , and we conclude that v is nondeterministic for (X ζ , S, Z d ).

As described in Theorem 5.2, depending on the faces where the point f satisfying Condition H1. belongs in conv(L n ζ (k)+F ζ n ), we may have more nondeterministic directions, obtaining the following corollary:

In the following we present two examples with different behaviors given by Lemma 5.12 Example 5.14 (Different behaviors for the nondeterministic directions). (1) Consider the 2D-Thue Morse substitution with

In this case K ζ = -1, 0 2 , and we have that

The sets of differences for the 2D-Thue Morse substitution are {{(0, -1), (0, 0)}, {(-1, 0), (0, 0)}, {(-1, 0), (0, -1)}, {(-1, -1), (0, 0)}, {(-1, -1), (-1, 0)}}.

By Lemma 5.12 it can be proved 1 0 , -1 0 , 0 1 , 0 1 are the only nondeterministic directions for (X ζ T M , S, Z 2 ).

(2) Consider the substitution of the table tiling [START_REF] Robinson | On the table and the chair[END_REF], with L ζt = 2 0 0 2 , F ζt 1 = 0, 1 2 , given by ζ t :

The set K ζt is equal to -1, 0 2 and the sets of differences is equal to 2 K ζ t \ {∅, K ζt , {(-1, -1), (0, 0)}, {(0, -1), (-1, 0)}}. By Lemma 5.12, it can be proved the set of nondeterministic directions for (X ζ , S, Z 2 ) is the whole circle S 1 . Now, we proceed to determine the normalizer semigroup N (X ζ , S, Z d ) of substitutive subshifts of polytope substitutions. Set M ∈ N (X ζ , S, Z d ). By Proposition 1.12 if v is a nondeterministic direction for (X ζ , S, Z d ), then M * v/ M * v is also a nondeterministic direction for (X ζ , S, Z d ). Moreover, Theorem 5.2 ensures the matrix M acts on the opposite normal cones of conv(T ζ ) that appeared as nondeterministic directions for (X ζ , S, Z d ). In particular, the matrix M * permutes the hyperplanes defined by the (d - The following is a pattern of the table tiling: Figure 6.2: A pattern of the table tiling, using red and green rectangles.

The table tiling forms an aperiodic tiling [START_REF] Robinson | On the table and the chair[END_REF]. Also in [START_REF] Robinson | On the table and the chair[END_REF], it was proved that the Z 2subaction of the table tiling is conjugate to the following constant-shape substitution (called table substitution) ζ t with expansion matrix L t = 2 0 0 2 and support F ζt 1 = 0, 1 2 :

This is an aperiodic bijective primitive polytope constant-shape substitution. By Remark 3.10 (3), the factor map π t : (X t , S,

) is almost 4-to-1. In fact, by [START_REF] Robinson | On the table and the chair[END_REF] we have that |π

The table substitution has 24 patterns with support K ζt = -1, 0 2 , which generate the 24 fixed points under the square of the substitution:

In Example 5.14 it was mentioned that the set of nondeterministic directions for (X t , S, Z 2 ) is the whole circle S 1 . Since the table substitution is a polytope substitution and has 2 linearly independent nondeterministic directions we can apply Theorem 5.17. This implies, its symmetry group N (X t , S, Z 2 ) is isomorphic to a subgroup of 0 -1 1 0 , 1 0 0 -1 which is isomorphic to D 4 . More precisely, we have the following result:

Proposition 6.1. The normalizer group of the table substitution is isomorphic to Z 2 D 4 .

In particular Aut(X ζt , S, Z 2 ) is the group generated by the shift action.

Proof. First, we will prove that Aut(X t , S, Z 2 ) = S . Indeed, the table substitution is bijective. By Proposition 5.1, any automorphism φ can be written as φ = S n τ , where n ∈ Z 2 and τ is defined by a permutation map T on the alphabet {0, 1, 2, 3}. In particular, τ acts as a permutation on the set of fixed points of ζ 2 t . Now, note that 2 0 2 0 is the only pattern of the form i j i j , i, j ∈ {0, 1, 2, 3}. This implies T (0) = 0 and T (2) = 2. In the same way, 3 3 1 1 is the only pattern of the form i i j j , i, j ∈ {0, 1, 2, 3}, which implies T (1) = 1 and T (3) = 3, i.e., T = id {0,1,2,3} . By minimality of (X t , S, Z 2 ), we have that τ is the identity on X t . We conclude that Aut(X t , S, Z 2 ) = S . Now, for homomorphisms. Set M 1 = 0 -1 1 0 . Consider the permutation Φ : 0 → 3,

This permutation comes by the rotation of the rectangles defining the table tiling. We define the map φ : X t → φ(X t ), given by φ(x) n = Φ(x M -1 1 n ) for all x ∈ X t , n ∈ Z d . We will prove that φ is a homomorphism on X t . A direct computation shows that the map S (-1,0) φ permutes the fixed points of ζ 2 t . By minimality of (X t , S, Z 2 ) we conclude that S (-1,0) φ(X t ) = X t . Then φ is a homomorphism associated with M 1 onto (X t , S, Z 2 ). Note that, also by minimality, S (-1,0) φζ 2 t = ζ 2 t (S (-1,0) φ), which implies S (3,0) φζ 2 t = ζ 2 t φ. So φ is the homomorphism given by Theorem 4.12.

) , generated by the center of these hexagons, using the vectors u and v in Fig. 6.6. We can define a constant-shape substitution with expansion matrix L hh = 2 • id R 2 and F hh 1 = {(0, 0), (1, 0), (0, 1), (1, -1)}

where, for convenience, we identify the hexagons in Fig. 6.5 with the letters {0, 1, 2}. The associated substitutive subshift is conjugated to the Λ-subaction of the half-hex tiling. The set K hh (defined in Section 1.7) is equal to {(0, 0), (-1, 0), (0, -1), (-1, 1)}. Since this substitution has coincidences (the definition is given in Section 4.1) in all except one coordinate, it has exactly three fixed points x, y, z such that x(0, 0) = 0, y(0, 0) = 1, z(0, 0) = 2 and for all (m, n)

In fact, we can characterize the maximal equicontinuous factor of the half-hex substitutive subshift.

Proposition 6.2. The half-hex substitutive subshift (X hh , S, Z 2 ) is a Toeplitz subshift. Moreover, its maximal equicontinuous factor is the odometer system

Proof. This is a particular case of Lemma 3.11. Since the factor map π hh : (X hh , S,

) is the maximal equicontinuous factor of the half-hex substitutive subshift (X hh , S, Z 2 ) and (X hh , S, Z 2 ) is a Toeplitz subshift.

In fact, by Lemma 3.11, we have that

). We could show, using the rotational symmetries of the substitution generated by the matrix 0 -1 1 1 that Z/6Z can embed in N (X hh , S, Z 2 ). But, as shown in Theorem 6.3 these are not the only isomorphisms this tiling presents.

Theorem 6.3. The normalizer semigroup of the half-hex substitutive subshift is a group and it is isomorphic to Z 2 GL(2, Z). Moreover, Aut(X hh , S, Z 2 ) is equal to the shift group S .

Proof. First we will prove that End(X hh , S, Z 2 ) = S . In fact, since the factor map

) is injective (Lemma 1.6). Now, by Lemma 1.7, for any endomorphism φ, we have that

In fact, since M commutes with 2 • id R 2 , the matrix M maps the 2Z 2 cosets onto 2Z 2 cosets. Moreover, any matrix M ∈ GL(2, Z) induces a bijection in Z/2Z × Z/2Z via one of the following matrices in GL(2, Z/2Z):

Each of these matrices define a permutation τ M 1 on Z 2 /2Z × 2Z \ {(0, 0)}. Identifying (1, -1) in F hh 1 with (1, 1) (mod 2), for any matrix M ∈ GL(2, Z), we have a unique permutation τ M 2 on the alphabet A = {0, 1, 2} given by τ

), for any x ∈ X hh and n = (m, n) ∈ Z 2 . We prove that φ M is a homomorphism on (X hh , S, Z 2 ) associated with M . Indeed, by Claim 2, since τ is a bijection we have that for any (m,

Then, by Claim 3 we get that the image of a fixed point of ζ hh via φ M is a fixed point of ζ hh . Now, by minimality of (X hh , S, Z 2 ), we conclude that φ M (X hh ) = X hh for any M ∈ GL(2, Z). So φ M is a homomorphism onto (X hh , S, Z 2 ) associated with M . Note that for any (m, n) ∈ Z 2 we have that φ M (S (m,n) ) = S M (m,n) φ M . If φ M is the identity in X hh , this would imply that S (m,n) = S M (m,n) for any (m, n) ∈ Z 2 , i.e., M = id. This implies φ M is a nontrivial isomorphism.

Hence, any matrix M ∈ GL(2, Z) has a homomorphism φ M induced by a letter-to-letter local map. Since (X hh , S, Z 2 ) is a coalescent system, by Proposition 1.5 the normalizer semigroup N (X hh , S, Z 2 ) is a group. Furthermore, for any

). This defines a group homomorphism from GL(2, Z) to N (X hh , S, Z 2 ), so using the exact sequence (1.4) we conclude that N (X hh , S, Z 2 ) ∼ = Z 2 GL(2, Z).

Perspectives

We present here some perspectives that remain open after the results of this thesis.

On non-deterministic directions of multidimensional subshifts

Beyond polytope substitutions

Until now, we didn't find an aperiodic d-dimensional primitive constant-shape substitution with less than d linearly independent nondeterministic directions. In the case of aperiodic primitive block substitutions, it can be easily proved that this hypothesis is true. Moreover, given the result in [START_REF] Guillon | Determinism in subshifts[END_REF] for the two-dimensional case, by Theorem 5.2, the hypothesis is true for all of the cases where conv(T ζ ) does not have two parallel edges. In a private communication, P. Guillon [62] mentioned this result is already proved for higher dimensions, but nowhere published. This implies, we only have to deal in the case conv(T ζ ) has two parallel (d -1)-dimensional faces.

Another open problem is the study of the normalizer semigroup for nonpolytope substitutions, i.e., where the convex hull of the digit tile is nonpolytope. Using the description of the convex hull of the digit tile given in [START_REF] Strichartz | Geometry of self-affine tiles[END_REF] on the two-dimensional case it may be possible to obtain similar results for the normalizer semigroup and symmetry semigroup. Nevertheless, until now there are no good descriptions of the convex hull of the digit tile for higher dimensions.

Motivated by the classification of full-shifts, in [START_REF] Hartman | The stabilized automorphism group of a subshift[END_REF] was introduced the notion of stabilized automorphism group of a topological dynamical system, which is the group of selfhomeomorphisms commuting with a power of T . The authors could distinguished, up to isomorphism, various stabilized automorphism groups of non-trivial mixing shift of finite type. Moreover, in the class of fullshifts, they proved if the stabilized automorphism group of the fullshift on n and m letters are isomorphic, then n and m have the same number of distinct prime divisors. In [START_REF] Schmieding | Local P entropy and stabilized automorphism groups of subshifts[END_REF], S. Schmieding studied the relation between topological entropy and the stabilized automorphism group. Moreover, he introduced a certain kind of entropy (in fact a whole family of entropies) for groups which he called local P entropy.

For Z d -actions we can define the stabilized automorphism group of a topological dynamical system as the following. Let M ∈ M(d, Z) be an invertible integer matrix with

Realization results of nondeterministic directions for minimal actions

The work in this thesis corresponds to the first examples of realization results about the set of nondeterministic directions for minimal actions. In [START_REF] Boyle | Expansive subdynamics[END_REF] it was proved that for any compact set of S 1 that is not a singleton containing one line with irrational slope can be realized as the set of nonexpansive directions of a Z 2 -action, and the singleton case was after proved by M. Hochmann in [START_REF] Hochman | Non-expansive directions for Z 2 actions[END_REF]. If aperiodic bijective on extremities primitive constantshape substitutions have d linearly independent nondeterministic directions, then we cannot obtain a unique nondeterministic direction with these substitutions, so we will need to use other types of substitutions, or other type of subshifts (such as Toeplitz sequences) to obtain other realization results with minimal subshifts.

Question.

1. Is there an aperiodic primitive constant-shape substitution with a unique nondeterministic direction? 2. Is there an aperiodic primitive constant-shape substitution such that its set of nondeterministic directions is homeomorphic to a Cantor set?

Recognizability property of constant-shape substitutions

As mentioned in Section 3.1, the recognizability property is a combinatorial one, useful to prove other properties satisfying the constant-shape substitutions. In this thesis we proved aperiodic symbolic factors of aperiodic primitive constant-shape substitutions are recognizable. This left open the following questions:

On minimal multidimensional S-adic subshifts

All the previous problems can be expressed for multidimensional minimal S-adic subshifts.

In the one-dimensional case, this class of minimal subshifts is one of the most natural containing minimal subshifts of sublinear complexity, but it is much broader as was shown in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF][START_REF] Donoso | Interplay between finite topological rank minimal cantor systems, s-adic subshifts and their complexity[END_REF]. The class contains several well studied systems, such as substitutive subshifts, symbolic codings of interval exchange transformations, dendric subshifts, and some Toeplitz sequences.

In the multidimensional setting, we can consider the case where the morphisms σ n : A n+1 → A F n 1 n are given by a constant-shape morphism. In this context the following are open questions.

Question.

1. Is there a recognizability property for minimal multidimensional constantshape S-adic systems? 2. Is there an analogue definition of finite rank systems in the multidimensional framework?

3. What rigidity properties do homomorphisms satisfy in the context of minimal multidimensional constant-shape S-adic systems? Moreover, if the morphisms are bijective, it is possible to extend Theorem 5.17 in this context?