Anna Frid

Bryna Kra

Samuel Petite

Sebastián Donoso

France, Paulina, Jenny, Bitar Álvaro Samuel

Vincent Special Thanks To Julien

Cassaigne for accepting being part of the jury, and to Sébastien Ferenczi and Van Cyr

Introduction

An idea that became unavoidable to study zero entropy symbolic dynamics is that the dynamical properties of a system induce in it a combinatorial structure. The first use of this approach was in the works of Morse and Hedlund [MH38;[START_REF] Morse | Symbolic Dynamics II. Sturmian Trajectories[END_REF], where Sturmian sequences were studied based on a structure given by what the authors called derivative sequences. As the theory developed, more examples like this one emerged. Relevant ones include substitutive and linearly recurrent subshifts [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], Toeplitz systems [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF], (natural codings of) interval exchange transformations [START_REF] Gjerde | Bratteli-Vershik models for cantor minimal systems associated to interval exchange transformations[END_REF], dendric sequences [START_REF] Gheeraert | S-adic characterization of minimal dendric shifts[END_REF] and general minimal subshifts [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF].

In this thesis, we investigate these combinatorial structures under two approaches. The first one assumes a given combinatorial structure and focuses on studying the dynamical properties of the systems supporting such a structure. We will consider finite topological rank systems, a class of systems possessing two desirable but opposite properties: It is a large class and contains most of the known zero entropy symbolic systems, yet it presents strong dynamical restrictions. Thus, the finite topological rank class provides a good framework for proving general and interesting theorems. We exploit this idea by studying automorphisms and symbolic factors of finite topological rank subshifts. Several theorems, describing rigidity properties for these objects, are obtained in Chapters 2 and 3.

The second approach consists of finding new combinatorial structures for systems of interest. We study one of the major questions in this direction -the Sadic conjecture, which asks for a structure theorem for linear-growth complexity subshifts. In the final chapter, we solve this conjecture and, furthermore, extend it to nonsuperlinear-growth complexity subshifts. An important consequence of our results is that these complexity classes gain access to the S-adic machinery. We show how this provides a unified framework and simplified proofs of several known results, including the pioneering 1996 Cassaigne's Theorem.

We will now discuss the thesis topics in more detail.

Basic terminology

Let us briefly review the modern standard for describing symbolic systems and their structures. An alphabet is a finite set A and a word is a finite concatenation

INTRODUCTION

of letters, i.e., elements of A. The full-shift on A is the set A Z endowed with the product topology of the discrete topology of A. We define the shift as the map S : A Z → A Z defined by (x n ) n∈Z → (x n+1 ) n∈Z . A symbolic system or subshift is a closed subset X of A Z such that S(X) = X. We will mostly consider minimal subshifts X, that is, such that {S n x : n ∈ Z} is dense in X for all x ∈ X.

A substitution is a map τ : A + → B + that substitutes the letters a i of a word w = a 1 . . . a ℓ by τ (a i ). Then, a sequence of substitutions τ of the form (τ n : A + n+1 → A + n ) n≥0 is called an S-adic sequence and generates a subshift X τ ⊆ A Z 0 given by requiring that x ∈ X τ if and only if, for all ℓ ≥ 0, x [-ℓ,ℓ) occurs in τ 0 τ 1 . . . τ n-1 (a) for some n ≥ 1 and a ∈ A n .

Finite topological rank systems

An ordered Bratteli diagram is an infinite directed graph B = (V, E, ≤) such that the vertex set V and the edge set E are partitioned into levels V = V 0 ∪ V 1 ∪ . . . , E = E 0 ∪ . . . so that E n are edges from V n+1 to V n , V 0 is a singleton, each V n is finite and ≤ is a partial order on E such that two edges are comparable if and only if they start at the same vertex. The order ≤ can be extended to the set X B of all infinite paths in B, and the Vershik action V B on X B is defined when B has unique minimal and maximal infinite paths with respect to ≤. We say that (X B , V B ) is a BV representation of the Cantor system (X, S) if both are conjugate. Bratteli diagrams are a tool coming from C * -algebras that, at the beginning of the 90', Herman et. al. [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF] used to study minimal Cantor systems. Their success at characterizing the strong and weak orbit equivalence for systems of this kind marked a milestone in the theory that motivated many posterior works. Some of these works focused on studying with Bratteli diagrams specific classes of systems and, as a consequence, many of the classical minimal systems have been characterized as Bratteli-Vershik systems with a specific structure. Some examples include odometers as those systems that have a BV representation with one vertex per level, substitutive subshifts as stationary BV (all levels are the same) [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], certain Toeplitz sequences as "equal row-sum" BV [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF], and (codings of) interval exchanges as BV where the diagram codifies a path in a Rauzy graph [START_REF] Gjerde | Bratteli-Vershik models for cantor minimal systems associated to interval exchange transformations[END_REF]. Now, almost all of these examples share certain coarse dynamical behavior: they are subshifts, have finitely many ergodic measures, are not strongly mixing, have zero entropy, and their BV representations have a bounded number of vertices per level, among many others. It turns out that just having a BV representation with a bounded number of vertices per level (or, from now on, having finite topological rank) implies the previous properties (see, for example, [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF], [START_REF] Downarowicz | Finite-rank Bratteli-Vershik diagrams are expansive[END_REF]). In particular, finite topological rank systems are subshifts. Hence, the finite topological rank class arises as a possible framework for studying minimal subshifts and proving general theorems.

This idea has been exploited in many works: Durand et. al., in a series of papers (being [START_REF] Durand | Eigenvalues of minimal Cantor systems[END_REF] the last one), developed techniques from the wellknown substitutive case and obtained a criteria for any BV of finite topological rank to decide if a given complex number is a continuous or measurable eigenvalue, Bezugly et. al. described in [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF] the simplex of invariant measures together with natural conditions for being uniquely ergodic, Giordano et. al. bounded the rational rank of the dimension group by the topological rank ( [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF]), among other works. It is important to remark that these works were inspired by or first proved in the substitutive case. Now, since Bratteli-Vershik whose topological rank is at least two are conjugate to a subshift [START_REF] Downarowicz | Finite-rank Bratteli-Vershik diagrams are expansive[END_REF], it is interesting to try to define them directly as a subshift. This can be done by codifying the levels of the Bratteli diagram as substitutions and then iterate them to obtain a sequence of symbols defining a subshift conjugate to the initial BV system. This procedure also makes sense for arbitrary nested sequences of substitutions (called directive sequences), independently from the Bratteli diagram and the various additional properties that its codifying substitutions have. Subshifts obtained in this way are called S-adic (substitution-adic) and may be non-minimal (see for example [START_REF] Berthé | Recognizability for sequences of morphisms[END_REF]).

Although there are some open problems about finite topological rank systems depending directly on the combinatorics of the underlying Bratteli diagrams, others are more naturally stated in the S-adic setting (e.g., when dealing with endomorphisms, it is useful to have the Curtis-Hedlund-Lyndon Theorem) and, hence, there exists an interplay between S-adic subshifts and finite topological rank systems in which theorems and techniques obtained for one of these classes can sometimes be transferred to the other. The question about which is the exact relation between these classes has been recently addressed in [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF] and, in particular, the authors proved:

Theorem 1 ([DDMP21]). A minimal subshift (X, S) has topological rank at most K if and only if it is generated by a proper, primitive and recognizable S-adic sequence of alphabet rank at most K.

In Chapters 2 and 3, we will use the S-adic formalism to study automorphisms and factors of finite topological rank systems.

Automorphisms

Let X be a subshift. The automorphism group of (X, S), Aut(X, S), is the set of homeomorphisms from X onto itself that commute with S. The study of the automorphism group of low complexity subshifts (X, S) has attracted a lot of attention in recent years. By complexity, we mean the increasing function p X : N → N which counts the number of words of length n ∈ N appearing in points of the subshift (X, S). In contrast to the case of non trivial mixing shifts of finite type or synchronized systems, where the algebraic structure of this group can be very rich [BLR88; KR90; FF96], the automorphism group of low complexity subshifts is expected to present high degrees of rigidity. The most relevant example illustrating this fact are minimal subshifts of non-superlinear complexity, where the automorphism group is virtually Z [CK15; DDMP16]. Interestingly, in [START_REF] Salo | Toeplitz subshift whose automorphism group is not finitely generated[END_REF] (and then in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] in a more general class) the author INTRODUCTION provides a Toeplitz subshift with complexity p X (n) ≤ Cn 1.757 , whose automorphism group is not finitely generated. So some richness in the algebraic structure of the automorphism groups of low complexity subshifts can arise. Other low complexity subshifts have been considered by Cyr and Kra in a series of works. In [START_REF] Cyr | The automorphism group of a shift of subquadratic growth[END_REF] they proved that for transitive subshifts, if lim inf n→+∞ p X (n)/n 2 = 0, then the quotient Aut(X, S)/⟨S⟩ is a periodic group, where ⟨S⟩ is the group spanned by the shift map; and in [START_REF] Cyr | The automorphism group of a minimal shift of stretched exponential growth[END_REF] for a large class of minimal subshifts of subexponential complexity they also proved that the automorphism group is amenable. All these classes and examples show that there is still a lot to be understood on the automorphism groups of low complexity subshifts.

In Chapter 2, we study the automorphism group of minimal S-adic subshifts of finite or bounded alphabet rank. This class of minimal subshifts is somehow the most natural class containing minimal subshifts of non-superlinear 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]. Moreover, this class contains several well studied minimal symbolic systems. Among them, substitution subshifts, linearly recurrent subshifts, symbolic codings of interval exchange transformations, dendric subshifts and some Toeplitz sequences. Thus, this class represents a useful framework for both, proving general theorems in the low complexity world and building subshifts with interesting dynamical behavior. The descriptions made in [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF] of its invariant measures and in [START_REF] Durand | Eigenvalues of minimal Cantor systems[END_REF] of its eigenvalues are examples of the former, and the well-behaved Sadic codings of high dimensional torus translations from [BST20] is an example of the latter.

The main result of Chapter 2 is the following rigidity theorem:

Theorem 2. Let (X, S) be a minimal S-adic subshift given by an everywhere growing directive sequence τ = (τ n : A + n+1 → A + n ) n≥0 . Suppose that τ is of finite alphabet rank, i.e., lim inf n→+∞ #A n < +∞. Then, Aut(X, S) is virtually Z.

A minimal S-adic subshift of finite topological rank, as stated in [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF], is defined as an S-adic subshift in which the defining directive sequence τ is proper, primitive, recognizable and with finite alphabet rank. In particular, τ is everywhere growing. Therefore, Theorem 2 includes all minimal S-adic subshifts of finite topological rank. Also, in the same paper, the authors prove that minimal subshifts of non-superlinear complexity are S-adic of finite topological rank. Thus, Theorem 2 can be seen as a generalization to a much broader class of the already mentioned results from [START_REF] Cyr | The automorphism group of a shift of linear growth: beyond transitivity[END_REF] and [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF]. Finally, by results stated in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF], Theorem 2 also applies to all level subshifts of minimal Bratteli-Vershik systems of finite topological rank and its symbolic factors.

The proof of Theorem 2 follows from a fine combinatorial analysis of asymptotic classes of S-adic subshifts of finite alphabet rank. This idea already appeared in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF], where the authors prove that the automorphism group of a minimal system is virtually Z whenever it has finitely many asymptotic classes. The following theorem summarizes this combinatorial analysis.

Theorem 3. Let W ⊆ A + be a set of nonempty words and define ⟨W⟩ := min w∈W length(w). Then, there exists B ⊆ A ⟨W⟩ with #B ≤ 122(#W) 7 such that: if x, x ′ ∈ A Z are factorizable over W, x (-∞,0) = x ′ (-∞,0) and x 0 ̸ = x ′ 0 , then x [-⟨W⟩,0) ∈ B.

Here, the important point is that, despite the fact that the length of the elements in B is ⟨W⟩, the cardinality of B depends only on #W, and not on ⟨W⟩.

Finally, we get a bound for the asymptotic classes of an S-adic subshift of finite alphabet rank. This result does not require minimality.

Theorem 4. Let (X, S) be an S-adic subshift (not necessarily minimal) given by an everywhere growing directive sequence of finite alphabet rank K. Then, (X, S) has at most 122K 7 asymptotic classes.

Factors

In the context of finite topological rank systems, a fundamental question is the following:

Question 1. Are subshift factors of finite topological rank systems of finite topological rank? Indeed, the topological rank controls various coarse dynamical properties (number of ergodic measures, rational rank of dimension group, among others) which cannot increase after a factor map, and we also know that big subclasses of the finite topological rank class are stable under symbolic factors, such as the linearly recurrent and the non-superlineal complexity classes [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF], so it is expected that this question has an affirmative answer. However, when trying to prove this using Theorem 1, we realize that the naturally inherited S-adic structure of finite alphabet rank that a symbolic factor has is never recognizable. Moreover, this last property is crucial for many of the currently known techniques to handle finite topological rank systems (even in the substitutive case it is a deep and fundamental theorem of Mossé), so it is not clear why it would be always possible to obtain this property while keeping the alphabet rank bounded or why recognizability is not connected with a dynamical property of the system. Thus, an answer to this question seems to be fundamental to the understanding of the finite topological rank class.

In Chapter 3, we obtain the optimal answer to Question 1 in a more general, non-minimal context: Theorem 5. Let (X, S) be an S-adic subshift generated by an everywhere growing and proper directive sequence of alphabet rank equal to K, and π : (X, S) → (Y, S) be an aperiodic subshift factor. Then, (Y, S) is an S-adic subshift generated by an everywhere growing, proper and recognizable directive sequence of alphabet rank at most K.

Here, a directive sequence σ = (σ n : A + n+1 → A + n ) n∈N is everywhere growing if lim n→∞ min a∈An |σ 0 . . . σ n-1 (a)| = ∞, and a system (X, S) is aperiodic if INTRODUCTION every orbit {S n x : n ∈ Z} is infinite. Theorem 5 implies that the topological rank cannot increase after a factor map (Corollary 7). Theorem 5 implies the following sufficient condition for a system to be of finite topological rank: Corollary 1. Let (X, S) be an aperiodic minimal S-adic subshift generated by an everywhere growing directive sequence of finite alphabet rank. Then, the topological rank of (X, S) is finite.

An interesting corollary of the underlying construction of the proof of Theorem 5 is the coalescence property for this kind of systems, in the following stronger form: Corollary 2. Let (X, S) be an S-adic subshift generated by an everywhere growing and proper directive sequence of alphabet rank equal to K, and (X, S) π1 → (X 1 , S) π2 → . . . π L → (X L , S) be a chain of aperiodic subshift factors. If L > log 2 K, then at least one π j is a conjugacy.

One of the results in [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] is that factor maps between aperiodic linearly recurrent subshifts are finite-to-one. In particular, they are almost k-to-1 for some finite k. For finite topological rank subshifts, we prove: Theorem 6. Let π : (X, S) → (Y, S) be a factor map between aperiodic minimal subshifts. Suppose that (X, S) has topological rank equal to K. Then π is almost k-to-1 for some k ≤ K.

We use this theorem, in Corollary 9, to prove that Cantor factors of finite topological rank subshifts are either odometers or subshifts.

In [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], the author proved that linearly recurrent subshifts have finite topological rank, and that this kind of systems have finitely many aperiodic subshifts factors up to conjugacy. Inspired by this result, we use ideas from the proof of Theorem 5 to obtain: Theorem 7. Let (X, S) be a minimal subshift of topological rank K. Then, (X, S) has at most (3K) 32K aperiodic subshift factors up to conjugacy.

Altogether, these results give a rough picture of the set of totally disconnected factors of a given finite topological rank system: they are either equicontinuous or subshifts satisfying the properties in Theorems 5, 2, 7 and 6. Now, in a topological sense, totally disconnected factors of a given system (X, S) are "maximal", so, the natural next step in the study of finite topological rank systems is asking about the connected factors. As we have seen, the finite topological rank condition is a rigidity condition. By this reason, we think that the following question has an affirmative answer: Question 2. Let (X, S) be a minimal system of finite topological rank and π : (X, S) → (Y, T ) be a factor map. Suppose that Y is connected. Is (Y, T ) an equicontinuous system?

We remark that the finite topological rank class contains all minimal subshifts of non-superlinear complexity [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF], but even for the much smaller class of linear complexity subshifts the author is not aware of results concerning Question 2.

Low complexity subshifts

Structure theorems

Theorems that describe a combinatorial structure of a given class of subshifts are usually an S-adic characterization, namely, of the form: a subshift X belongs to the class C if and only if X is generated by an S-adic sequence satisfying certain property P. The structure then appears as an infinite desubstitution process for the points of X.

In the context of structure theorems, an interesting intuition is that a subshift of low enough complexity should be very restricted, and thus hide a strong structure. Here, low complexity is a vague term referring to a slow growth of the complexity function p X (n), defined as the number of words of length n that occur in some point of X. This intuition dates back to the 70s, and matured in the 80s and 90s until it was finally established as the following more concrete question.

Question 3. Consider the class (L) of linear-growth complexity subshifts, defined by requiring that p X (n) ≤ dn for some d > 0. Is there an S-adic characterization of the class (L)?

Question 3 is known as the S-adic conjecture. The first time it was explicitly stated was in [START_REF] Ferenczi | Rank and symbolic complexity[END_REF], where the author attributes the idea to B. Host, who, in turn, attributes the idea to the whole Marseille community.

The attempts to solve this conjecture have identified two major difficulties. The first one is that, in contrast to what happens with other structure theorems, there is no clear structure induced by the complexity. For example, in the substitutive case, it was always clear that the substitution itself should produce a self-similar structure; the main obstruction was technical and referred to whether the desubstitution process was properly defined [START_REF] Mossé | Reconnaissabilité des substitutions et complexité des suites automatiques[END_REF]. Similarly, in the Sturmian and IET cases, the known structure came from the geometric counterpart (more precisely, from the Rauzy induction). The second challenge is that the condition P we are looking for in Question 3 is ill-defined. To exemplify this point, observe that a corollary of [START_REF] Cassaigne | Terminologie S-adique et propriétés[END_REF] is the following S-adic characterization of (L): a subshift is in (L) if and only if there exist τ generating it and such that X τ is in (L). This tautological answer to Question 3 does not provide information. Certain restrictions on Question 3 have been proposed to avoid this type of trivial answer, but none of them is considered satisfactory; we refer the reader to [START_REF] Durand | Do the Properties of an S-adic Representation Determine Factor Complexity?[END_REF] for a full discussion.

In Chapter 4, we completely solve the S-adic conjecture for minimal subshifts by proving the following theorem. Theorem 8. A minimal subshift X has linear-growth complexity, i.e., X satisfies lim sup n→+∞ p X (n)/n < +∞, if and only if there exist d > 0 and an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that, for every n ≥ 0, the following holds:

INTRODUCTION (P 1 ) #(root σ [0,n) (A n )) ≤ d † . (P 2 ) |σ [0,n) (a)| ≤ d • |σ [0,n) (b)
| for every a, b ∈ A n .

(P 3 ) |σ n-1 (a)| ≤ d for every a ∈ A n .

Our techniques extend to the case of nonsuperlinear complexity subshifts (NSL).

Theorem 9. A minimal subshift X has nonsuperlinear-growth complexity, i.e., X satisfies lim inf n→+∞ p X (n)/n < +∞, if and only if there exist d > 0 and an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that, for every n ≥ 0, the following holds:

(P 1 ) #(root σ [0,n) (A n )) ≤ d. (P 2 ) |σ [0,n) (a)| ≤ d • |σ [0,n) (b)| for every a, b ∈ A n .
The case of non-minimal subshifts does not pose additional intrinsic difficulties and follows from methods similar to those given here. However, we did not include it to avoid over saturating an already technical presentation.

An important consequence of our main results is that the classes (L) and (NSL) gain access to the S-adic machinery. We show in Section 4.10 how this provides a unified framework and simplified proofs of several known results on (L) and (NSL), including Cassaigne's Theorem [START_REF] Cassaigne | Special Factors of Sequences with Linear Subword Complexity[END_REF]. Further applications of our main results, which include a new proof of partial rigidity for (NSL) [START_REF] Creutz | Measure-Theoretically Mixing Subshifts of Minimal Word Complexity[END_REF] using the technique in [BKMS13, Theorem 7.2], will be presented in a future work.

We prove, in the more specialized Theorems 33 and 34, that when X is in (L) or in (NSL), then τ can be assumed to be recognizable. Observe that the conditions (P i ) in Theorems 8 and 9 are optimal in the sense that if we remove any of them then the corresponding theorem is false. Conditions (P 2 ) and (P 3 ) also occur in the positive substitutive case ‡ and in linearly recurrent subshifts, but the behavior in our theorems is very different since we do not impose positiveness.

With regard to (P 1 ) and (P 3 ), these were designed on the basis of two conditions that are present in most works that involve S-adic sequences. The first is having bounded alphabets (BA), which requires that #A n is uniformly bounded, and the second is finitariness, which asks for the set {τ n : n ≥ 0} to be finite. Note that finitariness implies both (BA) and Conditions (P 1 ) and (P 3 ), that (BA) implies (P 1 ), and that, under (P 3 ), finitariness and (BA) are equivalent. There are several papers in which a finitary S-adic sequence is looked for a subshift in (L) (see [START_REF] Leroy | An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) -p(n) ≤ 2[END_REF] and the references therein), and S-adic sequences with (BA) have shown to be closely connected with (L) and (NSL)

INTRODUCTION

Introduction (en français)

Une idée devenue incontournable pour étudier la dynamique symbolique à entropie nulle est que les propriétés dynamiques d'un système induisent en lui une structure combinatoire. Cette approche a été utilisée pour la première fois dans les travaux de Morse et Hedlund [MH38; MH40], où les suites Sturmiennes ont été étudiées sur la base d'une structure donnée par ce que les auteurs ont appelé les suites dérivées.

Au fur et à mesure que la théorie s'est développée, d'autres exemples comme le Sturmien sont apparus. Parmi les exemples pertinents, on peut citer les sousshifts substitutifs et linéairement récurrents [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], les systèmes de Toeplitz [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF], les (codages naturels des) échange d'intervalles [START_REF] Gjerde | Bratteli-Vershik models for cantor minimal systems associated to interval exchange transformations[END_REF], les suites dendriques [START_REF] Gheeraert | S-adic characterization of minimal dendric shifts[END_REF] et les sous-shifts minimaux généraux [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF].

Dans cette thèse, nous étudions ces structures combinatoires selon deux approches. La première suppose une structure combinatoire donnée et se concentre sur l'étude des propriétés dynamiques des systèmes supportant une telle structure. Nous considérerons systèmes de rang topologique fini, une classe de systèmes possédant deux propriétés souhaitables mais opposées : C'est une grande classe qui contient la plupart des systèmes symboliques à entropie nulle connus, mais qui présente de fortes restrictions dynamiques. Ainsi, la classe de rang topologique fini fournit un bon cadre pour prouver des théorèmes généraux et intéressants. Nous exploitons cette idée en étudiant les automorphismes et les facteurs symboliques des sous-shifts de rang topologique fini. Plusieurs théorèmes, décrivant des propriétés de rigidité pour ces objets, sont obtenus dans les chapitres 2 et 3.

La seconde approche consiste à trouver des nouvelles structures combinatoires pour des systèmes d'intérêt. Nous étudions l'une des questions majeures dans cette direction : la conjecture S-adique, qui demande un théorème de structure pour les sous-shifts de complexité linéaire. Dans le dernier chapitre, nous résolvons cette conjecture et, de plus, nous l'étendons aux sous-shifts de complexité non superlinéaire. Une conséquence importante de nos résultats est que ces classes de complexité gagnent l'accès à la machinerie S-adique. Nous montrons comment cela fournit un cadre unifié et des preuves simplifiées de plusieurs résultats connus, comprenant le théorème de Cassaigne de 1996.

Nous aborderons maintenant les thèmes de la thèse plus en détail.

INTRODUCTION

Terminologie de base

Passons brièvement en revue la notation moderne qui décrive des systèmes symboliques et leurs structures. Un alphabet est un ensemble fini A et un mot est une concaténation finie de lettres, i.e., éléments de A. Le full-shift sur A est l'ensemble A Z doté de la topologie produit de la topologie discrète de A. Nous définissons le shift comme l'application S : A Z → A Z définie par (x n ) n∈Z → (x n+1 ) n∈Z . Un système symbolique ou sous-shift est un sousensemble fermé X de A Z tel que S(X) = X. Nous considérerons principalement des sous-ensembles X minimaux, c'est-à-dire tels que {S n x : n ∈ Z} est dense dans X pour tout x ∈ X.

Une substitution est une application τ : A + → B + qui remplace les lettres a i d'un mot w = a 1 . . . a ℓ par τ (a i ). Une suite de substitutions τ de la forme (τ n : A n+1 → A + n ) n≥0 est appelée une séquence S-adique et génère un sous-shift

X τ ⊆ A Z 0 défini par la condition suivante : x ∈ X τ si et seulement si, pour tout ℓ ≥ 0, x [-ℓ,ℓ) apparaît dans τ 0 τ 1 . . . τ n-1 (a) pour tout n ≥ 1 et a ∈ A n .

Systèmes de rang topologique fini

Un diagramme de Bratteli ordonné est un graphe infini dirigé B = (V, E, ≤) tel que l'ensemble des sommets V et l'ensemble des arêtes E sont divisés en niveaux

V = V 0 ∪ V 1 ∪ . . . , E = E 0 ∪ . . . de sorte que E n sont les arêtes de V n+1 à V n , V 0
est un singleton, chaque V n est fini et ≤ est un ordre partiel sur E tel que deux arêtes sont comparables si et seulement si elles commencent au même sommet. L'ordre ≤ peut être étendu à l'ensemble X B de tous les chemins infinis dans B, et l'action de Vershik V B sur X B est définie lorsque B a des chemins infinis minimaux et maximaux uniques par rapport à ≤. Nous disons que (X B , V B ) est une représentation BV du système de Cantor (X, S) si les deux sont conjugués. Les diagrammes de Bratteli sont un outil issu de l'étude des algèbres C * que, au début des années 90, Herman et. al. [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF] ont utilisé pour étudier les systèmes de Cantor minimaux. Leur succès à caractériser l'équivalence orbitale forte et faible pour les systèmes de ce type a marqué une étape importante dans la théorie qui a motivé de nombreux travaux postérieurs. Certains de ces travaux se sont concentrés sur l'étude de classes spécifiques de systèmes à l'aide de diagrammes de Bratteli et, en conséquence, de nombreux systèmes minimaux classiques ont été caractérisés comme des systèmes de Bratteli-Vershik avec une structure spécifique. Parmi les exemples, on peut citer les odomètres en tant que systèmes qui ont une représentation BV avec un sommet par niveau, les sousshifts substitutifs en tant que BV stationnaire (tous les niveaux sont identiques) [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], les séquences de Toeplitz en tant que BV ayant "niveaux avec la même somme" [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF], et (codages des) échanges d'intervalles en tant que BV où le diagramme codifie un chemin dans le graphe de Rauzy [START_REF] Gjerde | Bratteli-Vershik models for cantor minimal systems associated to interval exchange transformations[END_REF]. Pratiquement tous ces exemples partagent certains comportements dynamiques grossiers : ils ont un nombre fini de mesures ergodiques, ne sont pas fortement mélangeurs, sont à entropie nulle, sont des sous-shifts, et leurs représentations BV ont un nombre borné de sommets par niveau, parmi beaucoup d'autres. Il s'avère que le simple fait d'avoir une représentation BV avec un nombre borné de sommets par niveau (ou, dorénavant, d'avoir rang topologique fini) implique les propriétés précédentes (voir, par exemple, [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF], [START_REF] Downarowicz | Finite-rank Bratteli-Vershik diagrams are expansive[END_REF]). Par conséquent, la classe de rang topologique fini apparaît comme un cadre possible pour étudier les sous-shifts minimaux et prouver des théorèmes généraux. Comme les systèmes de Bratteli-Vershik dont le rang topologique est au moins égal à deux sont conjugués à un sous-shift [START_REF] Downarowicz | Finite-rank Bratteli-Vershik diagrams are expansive[END_REF], il est intéressant d'essayer de les définir directement comme un subshift. Cela peut se faire en codifiant les niveaux du diagramme de Bratteli comme des substitutions, puis en les itérant pour obtenir une suite de symboles définissant un sous-shift conjugué au système BV initial. Cette procédure est également valable pour des suites de substitutions arbitraires (appelées "suites S-adiques"), indépendamment du diagramme de Bratteli et des diverses propriétés supplémentaires que possèdent les substitutions qui le codifient. Les sous-shifts ainsi obtenus sont appelés Sadiques (substitution-adiques) et peuvent être non minimaux (voir par exemple [START_REF] Berthé | Recognizability for sequences of morphisms[END_REF]).

Bien qu'il y a quelques problèmes ouverts sur les systèmes de rang topologique fini dépendant directement de la combinatoire des diagrammes de Bratteli sousjacents, d'autres sont plus naturellement énoncés dans le cadre S-adique (e.g., lorsqu'il s'agit d'endomorphismes, il est utile d'avoir le théorème de Curtis-Hedlund-Lyndon) et, par conséquent, il existe une interaction entre les sousshifts S-adiques et les systèmes de rang topologique fini dans laquelle les théorèmes et les techniques obtenus pour l'une de ces classes peuvent parfois être transférés à l'autre. La question de savoir quelle est la relation exacte entre ces classes a été récemment abordée dans [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF] et, en particulier, les auteurs ont prouvé : Theorem 10 ([DDMP21]). Un sous-shift minimal (X, S) a un rang topologique au plus égal à K si et seulement s'il est généré par une suite S-adique propre, primitive et reconnaissable de rang alphabétique au plus égal à K.

Dans les chapitres 2 et 3, nous utiliserons le formalisme S-adique pour étudier les automorphismes et les facteurs des systèmes de rang topologique fini.

INTRODUCTION

Automorphismes

Soit X un sous-shift. Le groupe des automorphismes de (X, S), Aut(X, S), est l'ensemble des homéomorphismes de X sur lui-même qui commuent avec S. L'étude du groupe d'automorphismes des sous-shifts de faible complexité (X, S) a attiré beaucoup d'attention ces dernières années. Par complexité, nous entendons la fonction p X : N → N qui compte le nombre de mots de longueur n ∈ N apparaissant en des points du sous-shift (X, S). Contrairement au cas des sous-shifts non triviaux de type fini ou des systèmes synchronisés, où la structure algébrique de ce groupe peut être très riche [BLR88; KR90; FF96], le groupe d'automorphisme des sous-shifts de faible complexité présentent des degrés élevés de rigidité. L'exemple le plus pertinent illustrant ce fait est celui des sous-shifts minimaux de complexité non superlinéaire, où le groupe d'automorphisme est virtuellement Z [CK15; DDMP16]. Il est intéressant de noter que dans [START_REF] Salo | Toeplitz subshift whose automorphism group is not finitely generated[END_REF] (puis dans [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] dans une classe plus générale), l'auteur fournit un sous-shift de Toeplitz de complexité p X (n) ≤ Cn 1,757 , dont le groupe d'automorphisme n'est pas finiment engendré. La structure algébrique des groupes d'automorphismes des sous-shifts de faible complexité peut donc s'avérer riche. D'autres sous-shifts de faible complexité ont été étudiés par Cyr et Kra dans une série de travaux. Dans [START_REF] Cyr | The automorphism group of a shift of subquadratic growth[END_REF], ils ont prouvé que pour les sous-shifts transitifs, si lim inf n→+∞ p X (n)/n 2 = 0, alors le quotient Aut(X, S)/⟨S⟩ est un groupe périodique, où ⟨S⟩ est le groupe engendré par le shift ; et dans [START_REF] Cyr | The automorphism group of a minimal shift of stretched exponential growth[END_REF] pour une grande classe de sous-shifts minimaux de complexité sousexponentielle, ils ont également prouvé que le groupe d'automorphisme est moyennable. Toutes ces classes et tous ces exemples montrent qu'il reste encore beaucoup à comprendre sur les groupes d'automorphisme des sous-shifts de faible complexité.

Dans le chapitre 2, nous étudions le groupe d'automorphisme des sous-shifts minimaux S-adiques de rang alphabétique fini ou borné. Cette classe de sousshifts minimaux est en quelque sorte la classe la plus naturelle contenant des sous-shifts minimaux de complexité non superlinéaire, mais elle est beaucoup plus large, comme cela a été montré dans [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]. De plus, cette classe contient plusieurs systèmes symboliques minimaux bien étudiés. Parmi eux, les sous-shifts substitutifs, les sous-shifts linéairement récurrents, les codages symboliques des échanges d'intervalles, les sous-shifts dendriques et certaines suites de Toeplitz. Ainsi, cette classe représente un cadre utile à la fois pour prouver des théorèmes généraux dans le monde de la faible complexité et pour construire des sous-shifts ayant un comportement dynamique intéressant. Les descriptions faites dans [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF] de ses mesures invariantes et dans [START_REF] Durand | Eigenvalues of minimal Cantor systems[END_REF] de ses valeurs propres sont des exemples du premier cas, et les codages S-adiques bien équilibrés des translations de tore de haute dimension de [BST20] sont un exemple du second cas.

Le résultat principal du chapitre 2 est le théorème de rigidité suivant :

Theorem 11. Soit (X, S) un sous-shift minimal donné par une suite S-adique Un sous-shift S-adique minimal de rang topologique fini, comme indiqué dans [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF], est défini comme un sous-shift engendré par une suite S-adique τ qui est propre, primitive, reconnaissable et avec un rang alphabétique fini. En particulier, τ est partout croissant. Par conséquent, le théorème 11 inclut tous les sous-shifts minimaux de rang topologique fini. De plus, dans le même article, les auteurs prouvent que les sous-shifts minimaux de complexité non superlinéaire sont de rang topologique fini. Ainsi, le théorème 11 peut être considéré comme une généralisation à une classe beaucoup plus large des résultats déjà mentionnés dans [START_REF] Cyr | The automorphism group of a shift of linear growth: beyond transitivity[END_REF] et [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF]. Finalement, par des résultats énoncés dans [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF], le théorème 11 s'applique également à tous les sous-shifts des niveaux des systèmes minimaux de Bratteli-Vershik de rang topologique fini et à ses facteurs symboliques.

τ = (τ n : A + n+1 → A + n ) n≥0 qui
La preuve du théorème 11 découle d'une analyse combinatoire fine des classes asymptotiques de sous-shifts S-adiques de rang alphabétique fini. Cette idée est déjà apparue dans [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF], où les auteurs prouvent que le groupe d'automorphisme d'un système minimal est virtuellement Z lorsqu'il possède un nombre fini de classes asymptotiques. Le théorème suivant résume cette analyse combinatoire.

Theorem 12. Soit W ⊆ A + un ensemble de mots non vides et définissons ⟨W⟩ := min w∈W length(w). Alors, il existe B ⊆ A ⟨W⟩ avec B ≤ 122(W) 7 tel que

: si x, x ′ ∈ A Z sont factorisables sur W, x (-∞,0) = x ′ (-∞,0) et x 0 ̸ = x ′ 0 , alors x [-⟨W⟩,0) ∈ B.
Ici, le point important est que, malgré le fait que la longueur des éléments de B est ⟨W⟩, la cardinalité de B ne dépend que de #W, et non de ⟨W⟩.

Enfin, nous obtenons une borne pour le nombre de classes asymptotiques d'un sous-shift S-adique de rang alphabétique fini. Ce résultat ne nécessite pas de la minimalité.

Theorem 13. Soit (X, S) un sous-shift (pas nécessairement minimal) donné par une suite S-adique partout croissante de rang alphabétique fini K. Alors, (X, S) a au plus 122K 7 classes asymptotiques.

INTRODUCTION

En effet, le rang topologique contrôle diverses propriétés dynamiques grossières (nombre de mesures ergodiques, rang rationnel du groupe de dimension, entre autres) qui ne peuvent pas augmenter après le passage à un facteur, et nous savons également que des grandes sous-classes de la classe de rang topologique fini sont stables sous facteurs symboliques, comme les classes linéairement récurrentes et de complexité non superlinéaire [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF]. Donc, on s'attend que cette question a une réponse affirmative. Cependant, en essayant de le prouver à l'aide du théorème 10, nous nous rendons compte que la structure S-adique de rang d'alphabet fini qu'un facteur symbolique naturellement hérite n'est jamais reconnaissable. De plus, cette dernière propriété est cruciale pour de nombreuses techniques actuellement connues pour traiter les systèmes de rang topologique fini (même dans le cas substitutif, il s'agit d'un théorème profond et fondamental de Mossé), de sorte qu'il n'est pas clair pourquoi il serait toujours possible d'obtenir cette propriété tout en gardant le rang alphabétique fini ou pourquoi la reconnaissabilité n'est pas liée à une propriété dynamique du système. Ainsi, une réponse à cette question semble être fondamentale pour la compréhension de la classe de rang topologique fini.

Dans le chapitre 3, nous obtenons la réponse optimale à la question 4 dans un contexte plus général et non minimal : Theorem 14. Soit (X, S) un sous-shift S-adique généré par une suite S-adique partout croissante et propre de rang alphabétique égal à K, et π : (X, S) → (Y, S) un facteur symbolique apériodique. Alors, (Y, S) est engendré par une suite Sadique partout croissante, propre, reconnaissable et de rang alphabétique au plus égal à K.

Ici, un système (X, S) est apériodique si toute orbite {S n x : n ∈ Z} est infinie. Le théorème 14 implique que le rang topologique ne peut pas augmenter après passer à un facteur symbolique (corollaire 7). Le théorème 14 implique la condition suffisante suivante pour qu'un système soit de rang topologique fini : Corollary 3. Soit (X, S) un sous-shift minimal apériodique engendré par une suite S-adique partout croissante de rang alphabétique fini. Alors, le rang topologique de (X, S) est fini.

Un corollaire intéressant de la construction sous-jacente de la preuve du théorème 14 est la propriété de coalescence pour ce type de systèmes, sous la forme plus forte suivante : Corollary 4. Soit (X, S) un sous-shift généré par une suite S-adique propre et partout croissante de rang alphabétique égal à K, et (X, S)

π1 → (X 1 , S) π2 → . . . π L → (X L , S
) une chaîne de facteurs symboliques apériodiques. Si L > log 2 K, alors au moins un π j est une conjugaison.

L'un des résultats de [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] est que les applications facteur entre les sousshifts linéairement récurrents apériodiques sont finies-à-un. En particulier, elles sont presque k-à-1 pour un certain k fini. Pour les sous-shifts de rang topologique fini, nous prouvons : Theorem 15. Soit π : (X, S) → (Y, S) un facteur entre des sous-shifts minimaux apériodiques. Supposons que (X, S) a rang topologique égal à K. Alors π est presque k-à-1 pour quelque k ≤ K.

Nous utilisons ce théorème, dans le corollaire 9, pour prouver que les facteurs de Cantor des sous-shifts de rang topologique fini sont des odomètres ou des sous-shifts.

Dans [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], l'auteur a prouvé que les sous-shifts linéairement récurrents ont rang topologique fini, et que ce type de systèmes a un nombre fini de facteurs symboliques apériodiques sauf conjugaison. Inspirés par ce résultat, nous utilisons les idées de la preuve du théorème 14 pour obtenir : Theorem 16. Soit (X, S) un sous-shift minimal de rang topologique K. Alors, (X, S) a au plus (3K) 32K facteurs symboliques apériodiques sauf conjugaison.

Dans l'ensemble, ces résultats donnent une image approximative de l'ensemble des facteurs totalement discontinu d'un système de rang topologique fini donné : il s'agit de systèmes équicontinus ou sous-shifts satisfaisant les propriétés des théorèmes 14, 4, 16 et 15. Au sens topologique, les facteurs totalement discontinu d'un système donné (X, S) sont "maximaux", de sorte que la prochaine étape naturelle dans l'étude des systèmes de rang topologique fini consiste à se demander sur les facteurs connexes. Comme nous l'avons vu, la condition de rang topologique fini est une condition de rigidité. Pour cette raison, nous pensons que la question suivante a une réponse affirmative : Question 5. Soit (X, S) un système minimal de rang topologique fini et π : (X, S) → (Y, T ) un facteur. Supposons que Y est connexe. Est-ce que (Y, T ) est un système équicontinu ?

Nous remarquons que la classe de rang topologique fini contient tous les sous-shifts minimaux de complexité non super-linéaire [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF], mais même pour la classe beaucoup plus petite de sous-shifts de complexité linéaire, l'auteur n'a pas connaissance de résultats concernant la question 5.

Sous-shifts de faible complexité

Théorèmes de structure

Les théorèmes qui décrivent une structure combinatoire d'une classe donnée de sous-shifts sont généralement une caractérisation S-adique de la forme suivante : un sous-shift X appartient à la classe C si et seulement si X est généré par une suite S-adique satisfaisant une certaine propriété P. La structure apparaît alors comme un processus de désubstitution infini pour les points de X.

Dans le contexte des théorèmes de structure, une intuition intéressante est qu'un sous-shift de complexité suffisamment faible devrait être très restreint, et donc cacher une structure forte. Ici, faible complexité est un terme vague se référant à une croissance lente de la fonction de complexité p X (n). Cette intuition remonte aux années 70, et a mûri dans les années 80 et 90 pour finalement s'établir sous la forme plus concrète de la question suivante: INTRODUCTION Question 6. Considérons la classe (L) des sous-shifts de complexité linéaire, définie en exigeant que p X (n) ≤ dn pour quelque d > 0. Existe-t-il une caractérisation S-adique de la classe (L) ?

La question 6 est connue sous le nom de conjecture S-adic. La première fois qu'elle a été explicitement énoncée, c'était dans [START_REF] Ferenczi | Rank and symbolic complexity[END_REF], où l'auteur attribue l'idée à B. Host, qui, à son tour, l'attribue à l'ensemble de la communauté marseillaise.

Les tentatives de résolution de cette conjecture ont mis en évidence deux difficultés majeures. La première est que, contrairement à ce qui se passe avec d'autres théorèmes de structure, il n'y a pas de structure claire induite par la complexité. Par exemple, dans le cas des substitutifs, il a toujours été clair que la substitution elle-même devait produire une structure autosimilaire ; le principal obstacle était d'ordre technique et concernait la question de savoir si le processus de désubstitution était correctement défini [START_REF] Mossé | Reconnaissabilité des substitutions et complexité des suites automatiques[END_REF]. De même, dans les cas Sturmien et IET, la structure connue provient de la contrepartie géométrique (plus précisément, de l'induction de Rauzy). Le deuxième défi est que la condition P que nous recherchons est mal-posé. Pour illustrer ce point, observons qu'un corollaire de [START_REF] Cassaigne | Terminologie S-adique et propriétés[END_REF] est la caractérisation S-adique de (L) suivante : un sous-shift appartient à (L) si et seulement s'il existe τ le générant et tel que X τ appartient à (L). Cette réponse tautologique à la question 6 n'apporte aucune information. Certaines restrictions à la question 6 ont été proposées pour éviter ce type de réponse triviale, mais aucune est jugée satisfaisante ; nous renvoyons le lecteur à [START_REF] Durand | Do the Properties of an S-adic Representation Determine Factor Complexity?[END_REF] pour une discussion complète.

Dans le chapitre 4, nous résolvons complètement la conjecture S-adique pour les sous-shifts minimaux en prouvant le théorème suivant:

Theorem 17. Un sous-shift minimal X a complexité linéaire, i.e., X satisfait lim sup n→+∞ p X (n)/n < +∞, si et seulement s'il existe d > 0 et une suite S-adique σ = (σ n : A n+1 → A + n ) n≥0 générant X telle que, pour tout n ≥ 0, ce qui suit est vrai :

(P 1 ) #(root σ [0,n) (A n )) ≤ d § . (P 2 ) |σ [0,n) (a)| ≤ d • |σ [0,n) (b)| pour tout a, b ∈ A n . (P 3 ) |σ n-1 (a)| ≤ d pour tout a ∈ A n .
Nos techniques s'étendent au cas des sous-shifts de complexité non superlinéaire (NSL).

Theorem 18. Un sous-shift minimal X a complexité superlinéaire, i.e., X satisfait lim inf n→+∞ p X (n)/n < +∞, § Pour un mot u, root u désigne le préfixe le plus court v de u tel que u = v k pour quelque k ; pour un ensemble de mots W, root W = {root w : w ∈ W}. si et seulement s'il existe d > 0 et une suite S-adique σ = (σ n : A n+1 → A + n ) n≥0 générant X telle que, pour tout n ≥ 0, ce qui suit est vrai :

(P 1 ) #(root σ [0,n) (A n )) ≤ d. (P 2 ) |σ [0,n) (a)| ≤ d • |σ [0,n) (b)| pour tout a, b ∈ A n .
Le cas des sous-shifts non minimaux ne pose pas de difficultés intrinsèques additionnelles et découle de méthodes similaires à celles données ici. Cependant, nous ne l'avons pas inclus pour éviter de saturer une présentation déjà technique.

Une conséquence importante de nos résultats principaux est que les classes (L) et (NSL) obtiennent l'accès à la machinerie S-adique. Nous montrons dans la section 4.10 comment cela fournit un cadre unifié et des preuves simplifiées de plusieurs résultats connus sur (L) et (NSL), y compris le théorème de Cassaigne [START_REF] Cassaigne | Special Factors of Sequences with Linear Subword Complexity[END_REF]. D'autres applications de nos résultats principaux, qui incluent une nouvelle preuve de rigidité partielle pour (NSL) [START_REF] Creutz | Measure-Theoretically Mixing Subshifts of Minimal Word Complexity[END_REF] en utilisant la technique dans [BKMS13, Théorème 7.2], seront présentées dans un travail futur.

Nous prouvons, dans les théorèmes plus spécialisés 33 et 34, que lorsque X est dans (L) ou dans (NSL), alors τ peut être supposé reconnaissable. Observons que les conditions (P i ) des théorèmes 17 et 18 sont optimales dans le sens où si nous supprimons l'une d'entre elles, le théorème correspondant est faux. Les conditions (P 2 ) et (P 3 ) apparaissent également dans le cas des substitutions positives. ¶ et dans des sous-shifts linéairement récurrents, mais le comportement dans nos théorèmes est très différent puisque nous n'imposons pas la positivité.

En ce qui concerne (P 1 ) et (P 3 ), ils ont été conçus sur la base de deux conditions qui sont présentes dans la plupart des travaux qui s'agissent des suites S-adiques. La première est la condition d'alphabets bornés (BA), qui exige que #A n soit uniformément borné, et la seconde est la finitude, qui exige que l'ensemble {τ n : n ≥ 0} soit fini. Notons que la finitude implique à la fois (BA) et les Conditions (P 1 ) et (P 3 ), que (BA) implique (P 1 ), et que, sous (P 3 ), la finitude et (BA) sont équivalentes. Il existe plusieurs articles dans lesquels une suite S-adique finitaire est recherchée pour un sous-shift dans (L) (voir [START_REF] Leroy | An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) -p(n) ≤ 2[END_REF] et les références qui y figurent), et les suites S-adiques avec (BA) se sont révélées être liées à (L) et (NSL) [START_REF] Ferenczi | Rank and symbolic complexity[END_REF][START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF]. Il est alors naturel de se demander si l'on peut remplacer, dans le théorème 17, les conditions (P 1 ) et (P 3 ) par la finitude. Nous montrons dans le théorème 35 que cela n'est pas possible. Plus précisément, nous construisons un sous-shift minimal avec une complexité linéaire telle que tout τ le générant et satisfaisant (P 1 ), (P 2 ) et (P 3 ) n'est pas finitaire (de manière équivalente, (BA) n'est pas satisfaite). Cependant, dans les théorèmes 33 et 34, nous donnons une condition suffisante pour que τ soit finitaire. Les sous-shifts satisfaisant cette condition suffisante comprennent les sous-shifts substitutifs, les codages des IET et les sous-shifts dendriques.

Basics in topological dynamics

A topological dynamical system (or just a system) is a pair (X, S), where X is a compact metric space and S : X → X is a homeomorphism of X. The orbit of x ∈ X is the set {S n x : n ∈ Z}. A point x ∈ X is periodic if its orbit is a finite set and aperiodic otherwise. A topological dynamical system is aperiodic if any point x ∈ X is aperiodic and is minimal if the orbit of every point is dense in X. We use the letter S to denote the action of a topological dynamical system independently of the base set X.

Basics in symbolic dynamics

Words and subshifts

Let A be an alphabet i.e. a finite set. Elements in A are called letters and concatenations w = a 1 . . . a ℓ of them are called words. The number ℓ is the length of w and it is denoted by |w|, the set of all words in A of length ℓ is A ℓ , and A + = ℓ≥1 A ℓ . The word w ∈ A + is |u|-periodic, with u ∈ A + , if w occurs in a word of the form uu . . . u. We define per(w) as the smallest p for which w is p-periodic. We will use notation analogous to the one introduced in this paragraph when dealing with infinite words x ∈ A N and bi-infinite words x ∈ A Z . The set A + equipped with the operation of concatenation can be viewed as the free semigroup on A. It is convenient to introduce the empty word 1, which has length 0 and is a neutral element for the concatenation. In particular, A + ∪ {1} is the free monoid in A.

Let W ⊆ A * be a set of words and u ∈ A * . We write uW = {uw : w ∈ W}, Wu = {wu : w ∈ W}, and also The shift map S : A Z → A Z is defined by S((x n ) n∈Z ) = (x n+1 ) n∈Z . For x ∈ A Z and integers i < j, we denote by x [i,j) the word x i x i+1 . . . x j . Analogous notation will be used when dealing with intervals of the form [i, ∞), (i, ∞), (-∞, i] and (-∞, i). A subshift is a topological dynamical system (X, S) where X is a closed and S-invariant subset of A Z (we consider the product topology in A Z ) and S is the shift map. Classically one identifies (X, S) with X, so one says that X itself is a subshift. When we say that a sequence in a subshift is periodic (resp. aperiodic), we implicitly mean that this sequence is periodic (resp. aperiodic) for the action of the shift. Therefore, if x ∈ A Z is periodic, then per(x) is equal to the size of the orbit of x. The language of a subshift X ⊆ A Z is the set L(X) of all words w ∈ A + that occur in some x ∈ X.

⟨W⟩ := min

Morphisms and substitutions

Let A and B be finite alphabets and τ : A + → B + be a morphism between the free semigroups that they define. Then, τ extends naturally to maps from A N to itself and from A Z to itself in the obvious way by concatenation (in the case of a twosided sequence we apply τ to positive and negative coordinates separately and we concatenate the results at coordinate zero). We say that τ is primitive if for every a ∈ A, all letters b ∈ B occur in τ (a). We observe that any map τ : A → B + can be naturally extended to a morphism (that we also denote by τ ) from A + to B + by concatenation, and we use this convention throughout the document. So, from now on, all maps between finite alphabets are considered to be morphisms between their associated free semigroups.

Factorizations and recognizability

Definition 1. Let X ⊆ A Z be a subshift and σ : A + → B + be a morphism. We say that

(k, x) ∈ Z × X is a σ-factorization of y ∈ B Z in X if y = S k σ(x). If moreover k ∈ [0, |σ(x 0 )|), then (k, x) is a centered σ-factorization in X.
The pair (X, σ) is recognizable if every point y ∈ B Z has at most one centered σ-factorization in X, and recognizable with constant r ∈ N if whenever y [-r,r] = y ′ [-r,r] and (k, x), (k ′ , x ′ ) are centered σ-factorizations of y, y ′ ∈ B Z in X, respectively, we have (k, x 0 ) = (k ′ , x ′ 0 ).

BACKGROUND IN TOPOLOGICAL AND SYMBOLIC DYNAMICS 29

The cuts of (k, x) are defined by

c σ,j (k, x) = -k + |σ(x [0,j) )| if j ≥ 0, -k -|σ(x [j,0) )| if j < 0.
We write C σ (k, x) = {c σ,j (k, x) : j ∈ Z}.

Remark 1. In the context of the previous definition:

(i) The point y ∈ B Z has a (centered) σ-factorization in X if and only if y belongs to the subshift Y := n∈Z S n σ(X). Hence, (X, σ) is recognizable if and only if every y ∈ Y has a exactly one centered σ-factorization in X.

(ii) If (k, x) is a σ-factorization of y ∈ B Z in X, then (c σ,j (k, x), S j x) is a σ- factorization of y in X for any j ∈ Z.
There is exactly one factorization in this class that is centered.

(iii) If (X, σ) is recognizable, then it is recognizable with constant r for some

r ∈ N [DDMP21].
The behavior of recognizability under composition of morphisms is given by the following lemma.

Lemma 1 ([BSTY19], Lemma 3.5). Let σ : A + → B + and τ : B + → C + be morphisms, X ⊆ A Z be a subshift and Y = k∈Z S k σ(X). Then, (X, τ σ) is recognizable if and only if (X, σ) and (Y, τ ) are recognizable.

S-adic subshifts

We recall the definition of an S-adic subshift as stated in [START_REF] Berthé | Recognizability for sequences of morphisms[END_REF]. An S-adic sequence or directive sequence σ is a sequence of morphisms having the form (σ n :

A + n+1 → A + n ) n∈N . For 0 ≤ n < N , we denote by σ [n,N ) the morphism σ n • σ n+1 • • • • • σ N -1 . We say that σ is everywhere growing if lim N →+∞ ⟨σ [0,N ) ⟩ = +∞, (1.1)
and primitive if for any n ∈ N there exists N > n such that σ [n,N ) is positive. We remark that this notion is slightly different from the usual one used in the context of substitutional dynamical systems. Observe that σ is everywhere growing if σ is primitive. Let P be a property for morphisms (e.g. proper, letter-onto, etc). We say that σ has property P if σ n has property P for every n ∈ N.

For n ∈ N, we define

X (n) σ = x ∈ A Z n : ∀ℓ ∈ N, x [-ℓ,ℓ] occurs in σ [n,N ) (a) for some N > n, a ∈ A N .
This set clearly defines a subshift that we call the nth level of the S-adic subshift generated by σ. We set X σ = X (0)

σ and simply call it the S-adic subshift generated by σ. If σ is everywhere growing, then every

X (n) σ , n ∈ N, is nonempty; if σ is primitive, then X (n)
σ is minimal for every n ∈ N. There are non-everywhere growing directive sequences that generate minimal subshifts.

The relation between levels of an S-adic subshift is given by the following lemma.

Lemma 2 ([BSTY19], Lemma 4.2). Let σ = (σ n :

A + n+1 → A + n ) n∈N be a direc- tive sequence of morphisms. If 0 ≤ n < N and x ∈ X (n) σ , then there exists a (centered) σ [n,N ) -factorization in X (N ) σ . In particular, X (n) σ = k∈Z S k σ [n,N ) (X (N ) σ ).
We define the alphabet rank of a directive sequence τ as

AR(τ ) = lim inf n→+∞ #A n . A contraction of τ is a sequence τ = (τ [n k ,n k+1 ) : A + n k+1 → A + n k ) k∈N
, where 0 = n 0 < n 1 < n 2 < . . . . Observe that any contraction of τ generates the same S-adic subshift X τ . When the context is clear, we will use the same notation to refer to τ and its contractions. If τ has finite alphabet rank, then there exists a contraction τ = (τ

[n k ,n k+1 ) : A + n k+1 → A + n k ) k∈N of τ in which A n k has cardinality AR(τ ) for every k ≥ 1.
Chapter 2 Automorphisms

Introduction

Automorphism groups of low complexity subshifts have gained considerable attention in recent years. Unlike the case of mixing shifts of finite type, where the algebraic structure of this group can be very rich [BLR88; KR90; FF96], the automorphism group of low complexity subshifts has a high degree of rigidity. The most relevant example illustrating this fact is the case of minimal subshifts of nonsuperlinear-growth complexity, in which the automorphism group is virtually Z [CK15; DDMP16]. In this chapter, we study the automorphism group of minimal S-adic subshifts of finite alphabet rank. This class of subshifts contains all minimal subshifts of nonsuperlinear-growth 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 main result of this chapter is the following rigidity theorem:

Theorem 19. Let (X, T ) be a minimal S-adic subshift generated by an everywhere growing S-adic sequence τ = (τ n :

A + n+1 → A + n ) n≥0
. Suppose that τ is of finite alphabet rank, i.e. lim inf n→+∞ #A n < +∞. Then, Aut(X, T ) is virtually Z.

The proof of Theorem 19 is a consequence of a fine combinatorial analysis of asymptotic classes of S-adic subshifts of finite alphabet rank, which we summarize in the following theorem.

Theorem 20. Let W ⊆ A + be a set of nonempty words and define ⟨W⟩ := min w∈W length(w). Then, there exists B ⊆ A ⟨W⟩ with #B ≤ 122(#W) 7 such that:

if x, x ′ ∈ A Z are factorizable over W, x (-∞,0) = x ′ (-∞,0) and x 0 ̸ = x ′ 0 , then x [-⟨W⟩,0) ∈ B.
This chapter was published as a standalone article in [START_REF] Espinoza | Symbolic factors of S-adic subshifts of finite alphabet rank[END_REF].

Organization

The chapter is organized as follows. In the next section we give additional background in topological and symbolic dynamics. In Section 2.3 we introduce some special ingredients allowing to prove the main theorems: the notions of interpretation and reducibility of sets of words together with its properties and the key Proposition 1, whose technical proof is given in Section 2.5. In Section 2.4 we restate our main results and provide complete proofs.

Additional background

An automorphism of the topological dynamical system (X, S) is a homeomorphism φ : X → X such that φ • S = S • φ. We use the notation φ : (X, S) → (X, S) to indicate the automorphism. The set of all automorphisms of (X, S) is denoted by Aut(X, S) and is called the automorphism group of (X, S). It has a group structure given by the composition of functions. It is said that Aut(X, S) is virtually Z if the quotient Aut(X, S)/⟨S⟩ is finite, where ⟨S⟩ is the subgroup generated by S.

We write ≤ p and ≤ s for the relations in A * of being prefix and suffix, respectively. We also write u < p v (resp. u < s v) when u ≤ p v (resp. u ≤ s v) and u ̸ = v. When v = sut, we say that u occurs in v or that u is a subword of v. We also use these notions and notations when considering prefixes, suffixes and subwords of infinite sequences.

Notion of Interpretation

In this section we introduce the concepts of interpretation and double interpretation of a word together with its basic properties. The definitions we provide here are variants of the same notion used seldom in combinatorics of words, see for example [START_REF] Lothaire | Combinatorics on words[END_REF]. The key Proposition 1, where we provide a fundamental upper bound for the number of irreducible sets of simple double interpretations, is announced here and proved in the last section of the chapter.

For the rest of this section we fix an alphabet A and a finite set of nonempty words W ⊆ A + . If u, v, w ∈ A * are such that w = uv, then we write u = wv -1 and v = u -1 w.

Interpretations and simple double interpretations

Definition 2. Let d ∈ A + . A W-interpretation of d is a sequence of words I = d L , d M , d R , a such that: (1) d M ∈ W * and a ∈ A; (2) there exist u L , u R ∈ W such that 1 ̸ = d L ≤ s u L , d R a ≤ p u R ; (3) d = d L d M d R .
See Figure 2.1 for an illustration of this definition. Note that d M and d R can be the empty word. The extra letter a will be crucial to handle asymptotic pairs and W-interpretations later. If the context is clear, we will say interpretation instead of W-interpretation. Now we make an observation that will be useful when we want to inherit interpretations of a given word to some of its subwords. We state it as a lemma without proof.

Lemma 3. Let I = d L , d M , d R , a be a W-interpretation of d ∈ A + . Suppose that d ′ ≤ p d satisfy |d ′ | ≥ |d L |. Then, d ′ has a W-interpretation of the form I ′ = d L , d ′ M , d ′ R , a ′ such that d ′ a ′ ≤ p da.
The proofs of our main theorems are based in a procedure allowing to reduce the so called double interpretations (defined below) to a special class called simple double interpretations.

Definition 3. Let d ∈ A + . A W-double interpretation (written for short W- d.i.) of d is a tuple D = (I; I ′ ), where I = d L , d M , d R , a, I ′ = d ′ L , d ′ M , d ′ R , a ′ are W-interpretations of d such that a ̸ = a ′ . We say that D is simple if in addition (1) d ′ M d ′ R ≤ s d R , and
(2)

d ′ L ∈ W or |d ′ L | ≥ |u| for some u ∈ W having d R a as a prefix.
Again, if there is no ambiguity, we will omit W and simply say double interpretation or d.i.

Note that if D is simple, then D ′ = (I ′ ; I) is a d.i., which is not necessarily simple. Condition (1) in the previous definition says that d ′ L , the left-most word of I ′ , "touches" d R , the right-most word of I; see Figure 2.2 for an illustration of this. Condition (2) is more technical and we will comment about it at the end of the Subsection 2.3.2. The next lemma will be useful to build a simple double interpretation from a word having a double interpretation.

Lemma 4. Let D = (I = d L , d M , d R , a; I ′ = d ′ L , d ′ M , d ′ R , a ′ ) be a double inter- pretation of a word d ∈ A + . Suppose that d ′ L ∈ W and |d L | ≤ |d ′ L d ′ M |.
Then, there exists e ≤ s d with a simple double interpretation.

Proof. By considering the shortest suffix of d verifying the hypotheses of the lemma we can assume without loss of generality that this suffix is d itself. We consider three cases.

(1) d ′ L < p d L . This condition and the hypotheses of the lemma imply that

d ′ L < p d L ≤ p d ′ L d ′ M .
Therefore, d ′ M is not the empty word and we can write

d ′ M = uv, with u ∈ W and v ∈ W * . Then, e := d ′ M d ′ R < s d has the inter- pretations J = (d ′ L ) -1 d L , d M , d R ,
a (here we are using that (d ′ L ) -1 d L ̸ = 1) and

J ′ = u, v, d ′ R , a ′ . But u ∈ W and |(d ′ L ) -1 d L | ≤ |(d ′ L ) -1 d ′ L d ′ M | = |uv|
, so e is a strict suffix of d having a d.i. E := (J; J ′ ) verifying the hypotheses of the lemma, which contradicts the minimality of d. Thus, this case is incompatible with the hypotheses.

(2)

d L < p d ′ L . If D is not a simple d.i. we have d R < s d ′ M d ′ R since d ′ L ∈ W and then d L < p d ′ L ≤ p d L d M .
This implies that d M is not the empty word. Then, we can write d M = uv with u ∈ W and v ∈ W * . We have that E

= (J = d -1 L d ′ L , d ′ M , d ′ R , a ′ ; J ′ = u, v, d R , a) is a d.i. of e := d M d R < s d which, in addition, satisfies u ∈ W and |d -1 L d ′ L | ≤ |uv|.
This contradicts the minimality of d and D must be simple.

(3

) d L = d ′ L . If d M = 1 or d ′ M = 1,
′ M = u ′ v ′ , with u, u ′ ∈ W and v, v ′ ∈ W * . Let e := d M d R = d ′ M d ′ R , J = u, v, d R , a and J ′ = u ′ , v ′ , d ′ R , a ′ . Observe that when |u ′ | ≤ |u|, E = (J ′ ; J) is a d.i.
A point x ∈ A Z is factorizable over W if there exist a point y ∈ W Z and k ∈ Z such that x [k,∞) = y 0 y 1 y 2 • • • and x (-∞,k) = • • • y -3 y -2 y -1 . For example, if τ is a directive sequence, 0 ≤ n < N and x ∈ X (n) τ , from Lemma 2 we see that x is factorizable over τ [n,N ) (A N ).
The last lemma of this subsection gives the relation between asymptotic pairs that are factorizable over the set of words W and simple double interpretations over W. This lemma is crucial to reduce our combinatorial studies in next sections to the case of simple double interpretations.

Lemma 5. If x, x ′ ∈ A Z are factorizable over W, x (-∞,0) = x ′
(-∞,0) and x 0 ̸ = x ′ 0 , then there exists a word e ≤ s x (-∞,0) having a simple double interpretation over W.

Proof. Let l ≥ 2|W| and d := x [-l,0) . Then d inherits in a natural way interpretations

I = d L , d M , d R , a and I ′ = d ′ L , d ′ M , d ′ R , a
′ from the factorizations of x and x ′ respectively. Since a = x 0 ̸ = x ′ 0 = a ′ , the tuple D := (I; I ′ ) is a d.i. Moreover, by choosing adequately l we can suppose that

d ′ L ∈ W. Also, |d L | ≤ |W| ≤ l -|d ′ R | = |d ′ L d ′ M |
, so the hypotheses of Lemma 4 hold. Thus d (and of course x (-∞,0) ) has a suffix e with a simple double interpretation over W. This proves the lemma.

Reducible and irreducible simple double interpretations

In this section we introduce the notions of reducible and irreducible sets of simple double interpretations. In Proposition 1 we provide an upper bound for the size of irreducible sets of simple d.i. (the proof of this proposition is very technical and is postponed until Section 2.5). Thus, even if in some cases it is not necessary, most of the notions appearing in this section will be considered only for simple d.i. For the rest of the chapter each time we use a letter D to denote a d.i. on W, then it double interprets the word d ∈ A + and is written D

= (I D = d L , d M , d R , a D ; I ′ D = d ′ L , d ′ M , d ′ R , a ′ D ). Definition 4. Given U = (u M , u R , u ′ L , u ′ M , u ′ R , ℓ) ∈ W 5 × N, we define D U as the set of simple W-d.i. D such that: (1) either d M ∈ W * u M or d M = 1 and d L ≤ s u M ; (2) d R a D ≤ p u R and |u R | = min{|w| : d R a D ≤ p w, w ∈ W}; (3) d ′ R a ′ D ≤ p u ′ R , d ′ L ≤ s u ′ L and |u ′ L | = min{|w| : d ′ L ≤ s w, w ∈ W}; (4) d ′ M = 1 or d ′ M = v 1 • • • v n ∈ W + , v 1 = u ′ M and max 1≤j≤n |v j | = ℓ.
It is easy to see that

D := U ∈W 5 ×N D U
is the set of all simple W-d.i. of words in A + . Moreover, from ((4)) of Definition 4 we have that ℓ ∈ {|w| : w ∈ W} ∪ {0} when D U ̸ = ∅, so D is the union of no more than #W 5 (#W + 1) sets D U . (2) there exists D ∈ D ′ that reduces to some simple d.i. If D ′ is not reducible, we say that it is irreducible.

The main combinatorial result about irreducible sets of simple d.i. is the following proposition, whose proof will be carried out in Section 2.5.

Proposition 1. Let U ∈ W 5 × N. Any irreducible subset of D U has at most 61(#W) elements.
The use of condition (2) of Definition 3 appears during the proof of this proposition. This proof consists in directly showing that sets D ′ ⊆ D U with more than 61(#W) elements are reducible. For this, one finds elements in D ′ that are equivalent or can be reduced. In this process, one observes that eliminating condition (2) in the definition of simple d.i. has two opposite effects. On one hand, it should be easier to find a reduction of a given simple d.i., since more d.i. are simple; but on the other hand, without condition (2) being simple means less structure, so it is more difficult to actually find the desired reductions during the proof. Balancing this trade-off is the reason behind the technical condition (2). It is worth mentioning that this condition (2) is only used in the proof of Lemma 9.

Proof of main results

In this section we prove our main results. As we commented in the introduction, the proof of Theorem 19 is based on two general steps: first we use a proposition from [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF] relating the number of asymptotic components with the "size" of the automorphism group and secondly we develop a complete combinatorial analysis of the asymptotic classes arising in an S-adic subshift of finite alphabet rank.

Let (X, S) be a topological dynamical system. Two points x, x ′ ∈ X are (negatively) asymptotic if lim n→-∞ dist(S n x, S n x ′ ) = 0. We define the relation ∼ in X as follows: x ∼ x ′ whenever x is asymptotic to S k x ′ for some k ∈ Z. It is easy to see that ∼ is an equivalence relation. An equivalence class for ∼ that is not the orbit of a single point is called an asymptotic class, and we write Asym(X, S) for the set of asymptotic classes of (X, S). Observe that if (X, S) is a subshift, then x ∼ x ′ if and only if x (-∞,k) = x ′ (-∞,ℓ) for some k, ℓ ∈ Z. The following proposition, which is a direct consequence of Corollary 3.3 in [START_REF] Donoso | On automorphism groups of low complexity subshifts[END_REF], gives a relation between the number of asymptotic classes and the cardinality of Aut(X, S)/⟨S⟩ under conditions that any infinite minimal subshift satisfies.

Proposition 2. Let (X, S) be a topological dynamical system. Assume there exists a point x 0 ∈ X with ω(x 0 ) := n≥0 {S k x 0 : k ≥ n} = X that is asymptotic to a different point. Then, #Aut(X, S)/⟨S⟩ ≤ #Asym(X, S)!.

Now we prove our first combinatorial theorem.

Theorem 20. Let W ⊆ A + be a set of nonempty words. Then, there exists

B ⊆ A ⟨W⟩ with #B ≤ 122(#W) 7 such that: if x, x ′ ∈ A Z are factorizable over W, x (-∞,0) = x ′ (-∞,0) and x 0 ̸ = x ′ 0 , then x [-⟨W⟩,0) ∈ B.
As will be clear from the proof, the bound "122(#W) 7 " is not necessarily optimal. Here, the important point is that, despite the fact that the length of the elements in B is ⟨W⟩, the cardinality of B depends only on #W, and not on ⟨W⟩.

Proof. We start by defining the set B.

For each U = (u M , u R , u ′ L , u ′ M , u ′ R , ℓ) ∈ W 5 × N, fix D ′
U ⊆ D U an irreducible subset of maximal size (we consider the empty set as an irreducible set, so there always exists such set D ′ U ). We define

B := w ∈ A ⟨W⟩ : ∃ U ∈ W 5 × N, D ∈ D ′ U , w ≤ s d ,
where in this set d ∈ A + represents the word that is double interpreted by D. We note that this makes sense because |d| ≥ ⟨W⟩ for all simple d.i. As we observed previously, we have ℓ ∈ {|w| : w ∈ W} ∪ {0} when D U is nonempty. Thus, there are no more than #W 5 (#W + 1) choices for U such that D U is nonempty. Using this and Proposition 1 we get:

#B ≤ 61#W • #{U ∈ W 5 × N : D U ̸ = ∅}, ≤ 61#W • #W 5 (#W + 1) ≤ 122(#W) 7 .
It rests to prove the main property of the theorem. In this purpose, let x, x ′ ∈ A Z be factorizable over W with x (-∞,0) = x ′ (-∞,0) and x 0 ̸ = x ′ 0 . From Lemma 5 we can find a simple d.i.

D of d ≤ s x (-∞,0) . Let D =: D(0) ⇒ D(1) ⇒ D(2) ⇒ • • • ⇒ D(n)
be a sequence of reductions that starts with D (where, possibly, n = 0 and D has no reduction). We write, for convenience, D(j) = (I(j); I ′ (j)) and d(j) for the word that is double interpreted by D(j). Since |d(0)| > |d(1)| > . . . , any sequence like this ends after a finite number of steps. In particular, we can take (and we are taking) this sequence so that n is maximal. This implies that D(n) has no reduction.

Since D = U ∈W 5 ×N D U , we can find U ∈ W 5 × N satisfying D(n) ∈ D U . We claim that there is a word e with a simple d.i. E = (I E ;

I ′ E ) ∈ D ′ U such that D(n) is equivalent to E. Indeed, if D(n) ∈ D ′ U then, since D(n) is equivalent to itself, we can take E := D(n). If D(n) is not in D ′
U , then, from the maximality of D ′ U we see that D ′ U ∪ {D(n)} is reducible. Since D(n) has no reduction and D ′ U is irreducible, there exists E ∈ D ′ U equivalent to D(n). This proves the claim.

Then, using the definitions of reduction and equivalence of simple d.i., we have that the suffix w ∈ A ⟨W⟩ of e satisfies

w ≤ s d(n) < s d(n -1) < s • • • < s d(0) ≤ s x (-∞,0) ,
and w ∈ B since E ∈ D ′ U . This finishes the proof.

Now we have all the ingredients to compute the number of asymptotic classes in the case of S-adic subshifts of finite alphabet rank.

Theorem 4. Let (X, S) be an S-adic subshift given by an everywhere growing directive sequence of alphabet rank K. Then, (X, S) has at most 122K 7 asymptotic classes.

Proof. Set K ′ = 122K 7 . We are going to prove the following stronger result.

Claim 20.1. Let P be the set of pairs (x, y) ∈ X × X such that x (-∞,0) = y (-∞,0) and x 0 ̸ = y 0 . Then, #{x (-∞,0) : (x, y) ∈ P} ≤ K ′ .

First, we show how this claim implies the theorem. Suppose the claim is true and let C 0 , . . . , C K ′ be asymptotic classes for (X, S). For each j ∈ {0, . . . , K ′ } we choose (z j , z ′ j ) ∈ C j such that z j and z ′ j do not belong to the same orbit. Then, there exist m j , m ′ j ∈ Z such that x j := S mj z j and y j := S m ′ j z ′ j satisfy (x j ) (-∞,0) = (y j ) (-∞,0) and (x j ) 0 ̸ = (y j ) 0 , ∀j ∈ {0, . . . , K ′ }.

(2.1) Thus, (x j , y j ) ∈ P for all j ∈ {0, . . . , K ′ } and, by the claim and the Pigeonhole Principle, there exist different j, j ′ ∈ {0, . . . ,

K ′ } such that (x j ) (-∞,0) = (x j ′ ) (-∞,0)
. This implies C j = C j ′ and, thus, that (X, S) has at most K ′ asymptotic classes. Now we prove the claim. Let τ = (τ n : A + n+1 → A + n ) n≥0 be an everywhere growing directive sequence of alphabet rank K generating X. By doing a contraction, if required, we can suppose that #A n = K for every n ≥ 1.

For n ≥ 1 put W n = τ [0,n) (A n ) and let B n ⊆ A +
0 be the set given by Theorem 20 when it is applied to

W n . By hypothesis, #W n ≤ #A n = K, so #B n ≤ 122(#W n ) 7 ≤ 122K 7 = K ′ .
For j ∈ {0, . . . , K ′ } let (x j , y j ) ∈ P. We have to show that (

x j ) (-∞,0) = (x j ′ ) (-∞,0) for different j, j ′ ∈ {0, . . . , K ′ }.
Since for all n ≥ 1 and j ∈ {0, . . . , K ′ } the points x j and y j are factorizable over W n (Lemma 2), from Theorem 20 we have that (

x j ) [-⟨Wn⟩,0) ∈ B n . But #B n ≤ K ′ so by the Pigeon- hole Principle there exist j n , j ′ n ∈ {0, . . . , K ′ } with j n ̸ = j ′ n such that (x jn ) [-⟨Wn⟩,0) = (x j ′ n ) [-⟨Wn⟩,0) . (2.2)
Thus, again by the Pigeonhole Principle, we can choose 1

≤ n 1 < n 2 < . . . such that j n1 = j n2 = • • • = j ̸ = j ′ = j ′ n1 = j ′ n2 = . . . By (2.2), (x j ) [-⟨Wn i ⟩,0) = (x j ′ ) [-⟨Wn i ⟩,0) , ∀i ≥ 1. (2.3)
Since τ is everywhere growing, ⟨W n ⟩ goes to infinity when n → +∞. Thus, (2.3) implies that (x j ) (-∞,0) = (x j ′ ) (-∞,0) , as desired. This completes the proof.

We remark again that in previous result we do not assume minimality. This hypothesis is needed in the next proof (of Theorem 19) only because we bound the size of the automorphism group by the number of asymptotic classes via Proposition 2. Thus, Theorem 19 is mainly a consequence of combinatorial facts inherent to S-adic subshifts.

Theorem 19. Let (X, S) be a minimal S-adic subshift given by an everywhere growing sequence of finite alphabet rank K. Then, its automorphism group is virtually Z.

Proof. From Proposition 2 and Theorem 4 we get

#Aut(X, S)/⟨S⟩ ≤ #Asym(X, S)! ≤ 122K 7 ! < +∞.
This inequality proves that Aut(X, S) is virtually Z.

Proof of Proposition 1

In this last section we prove Proposition 1. All but one result we need (Lemma 4) are presented and proved here, so the section is almost self contained.

We fix, for the rest of this section, a finite set of words W ⊆ A + and a sequence

U = (u M , u R , u ′ L , u ′ M , u ′ R , ℓ) ∈ W 5 × N.
For D ∈ D U , we define:

d := d R (d ′ M d ′ R ) -1 = (d L d M ) -1 d ′ L .
We need a last definition: two words u, v ∈ A * are prefix dependent

(resp. suffix dependent) if u ≤ p v or v ≤ p u (resp. u ≤ s v or v ≤ s u).
In this case, u and v share a common prefix (resp. suffix) of length min(|u|, |v|). Lemma 6. Consider different elements D, E in D U . If any of the following conditions holds, then the set {D, E} is reducible:

(i) d ′ M d ′ R a D , e ′ M e ′ R a E are prefix dependent; (ii) |d R | = |e R |; (iii) | d| ≤ |ẽ| ≤ | dd ′ M | or |ẽ| ≤ | d| ≤ |ẽe ′ M |.
Proof. We will show that under conditions of the lemma one of the following relations occurs: D ∼ E, E reduces to a simple d.i. or D reduces to a simple d.i.

(i) Without loss of generality, we can suppose that

d ′ M d ′ R a D ≤ p e ′ M e ′ R a E . We distinguish two cases: (1) d ′ M d ′ R a D = e ′ M e ′ R a E . Using item ((3)) of Definition 4 we can write d = d ′ L d ′ M d ′ R ≤ s u ′ L d ′ M d ′ R . Similarly, e ≤ s u ′ L e ′ M e ′ R .
This and hypothesis (a) imply that d and e are suffix dependent. But, since D and E are simple d.i., by Remark 2 we have that |d|, |e| ≥ ⟨W⟩. We conclude that d and e share a suffix of length at least min(|d|, |e|) ≥ ⟨W⟩, which implies D ∼ E.

(2) d ′ M d ′ R a D < p e ′ M e ′ R a E (so, d ′ M d ′ R a D ≤ p e ′ M e ′ R )
. We claim that ℓ > 0 in the definition of U . Suppose that ℓ = 0. Then, d ′ M = e ′ M = 1 and we can write:

d ′ R a D ≤ p e ′ R ≤ p u ′ R .
Since by ((3)) of Definition 4 we also have

d ′ R a ′ D ≤ p u ′ R , we conclude that a D = a ′ D . This contradicts the fact that E is a d.i. Thus, ℓ > 0. Now, ℓ > 0 and ((4)) of Definition 4 imply that v D := (u ′ M ) -1 d ′ M ∈ W * and v E := (u ′ M ) -1 e ′ M ∈ W * . Let w := d ′ M d ′ R . Observe that J D = u ′ M , v D , d ′ R , a ′ D is an interpretation of w. Moreover, since u ′ M ≤ p w < p u ′ M v E e ′
R by hypothesis (b) and v E ∈ W * , we can obtain, using Lemma 3, an interpretation of w of the form

J E = u ′ M , e ′′ M , e ′′ R , a ′′ E such that wa ′′ E ≤ p u ′ M v E e ′ R . Next, we prove that F := (J D ; J E ) is a d.i. of w. Observe that v D d ′ R a D ≤ p v E e ′ R by hypothesis (b) and e ′′ M e ′′ R a ′′ E ≤ p v E e ′ R by the definition of J E . But v D d ′ R = (u ′ R ) -1 w = e ′′ M e ′′ R , so a D = a ′′ E . Hence, a ′ D ̸ = a D = a ′′ E and F is a d.i. of w.
Finally, we note that since J D and J E start with u ′ M ∈ W, we can use Lemma 4 with F to obtain a simple d.i. G of a word g such that g ≤ s w < s d. This corresponds to the fact that D reduces to G. (iii) We consider the case

| d| ≤ |ẽ| ≤ | dd ′ M |
, the other one is symmetric. We start with some simplifications. Observe that condition ((2)) in Definition 4 implies

d R a D = dd ′ M d ′ R a D ≤ p u R and e R a E = ẽe ′ M e ′ R a E ≤ p u R . (2.4) Then, if | d| = |ẽ|, we are in case (i), and if |d R | = |e R |,
we are in case (ii). Thus, we can suppose, without loss of generality, that

| d| < |ẽ|, (2.5) |d R | ̸ = |e R |. (2.6)
The idea of the proof is the following. We are going to define a word w, which is suffix of d or e, and that has a d.i. F satisfying the hypothesis of Lemma 4. This would imply that F (and then also D or E) reduces to a simple d.i., as desired.

From (2.5) and hypothesis (iii) we have that

| d| ̸ = | dd ′ M | and thus ℓ ̸ = 0. In particular, this last fact implies that v D := (u ′ M ) -1 d ′ M ∈ W * and v E := (u ′ M ) -1 e ′ M ∈ W * .
Also, from (2.4) and (2.5) we see that it makes sense to define

t := d-1 ẽ ̸ = 1. Then, J D = u ′ M , v D , d ′ R , a ′ D is an interpretation of d ′ M d ′ R and J E = t, e ′ M , e ′ R , a ′ E is an interpretation of te ′ M e ′ R
. Now, using (2.4) and (2.6) we also obtain that either

d ′ M d ′ R < p te ′ M e ′ R or te ′ M e ′ R < p d ′ M d ′ R .
We analyze these two cases separately:

(1) Assume d ′ M d ′ R < p te ′ M e ′ R . We define w = d ′ M d ′ R < s d. Note that J D is an interpretation of w. By hypothesis (iii), we have t ≤ p w < p te ′ M e ′ R
, so we can use Lemma 3 with J E to obtain an interpretation of w having the form

J ′ E = t, e ′′ M , e ′′ R , a and satisfying wa ≤ p e ′ M e ′ R . We set F = (J D , J ′ E ). Since wa ≤ p te ′ M e ′ R = d-1 e R ≤ p d-1 u R and wa D = d ′ M d ′ R a D = d-1 d R a D ≤ p d-1 u R , we have a = a D . Being a D ̸ = a ′ D as D is a d.i., we conclude that a ̸ = a ′ D and that F is a d.i. Recall that u ′ R ∈ W and observe that |t| ≤ |d ′ M |
by hypothesis (iii). Thus, F satisfies the hypothesis of Lemma 4. This implies that D is reducible.

(

2) Suppose te ′ M e ′ R < p d ′ M d ′ R .
Observe that from ((4)) of Definition 4 we know that there exist n ≥ 0 and, for j ∈ {1, . . . , n}, v j ∈ W with

|v j | ≤ ℓ, such that v D = v 1 • • • v n (we interpret v 1 • • • v n = 1 when n = 0). We define w = te ′ M e ′ R < s e and v n+1 = d ′ R . See Figure 2.4 for an illustration of the definitions so far. Since |w| ≥ |u ′ R |, we have u ′ M ≤ p w < p u ′ M v 1 • • • v n+1 by (b)
, and thus, there exists a least integer m ∈ {1, . . . , n + 1} such that w

≤ p u ′ M v 1 • • • v m . Being m minimal, we can write w = u ′ M v 1 • • • v m-1 v ′ m , with v ′ m ≤ p v m and wa ≤ p d ′ M d ′ R for some a ∈ A. Then, J ′ D := u ′ M , v 1 • • • v m-1 , v ′ m , a and J E are interpretations of w.
< p ẽ ≤ p ẽu ′ M v E ≤ p du ′ M v 1 • • • v n+1 ≤ p u R .
This and the definitions of w and t are represented in the figure.

We set F = (J ′ D , J E ) and claim that F is a d.i. Indeed, on the one hand, the definition of

J ′ D gives wa ≤ p d ′ M d ′ R ≤ p d-1 u R .
On the other hand, since

w = d-1 ẽe ′ M e ′ R = d-1 e R , we have wa E ≤ p d-1 u R by ((2)) of Definition 4. We conclude that a = a E . Then, a ̸ = a ′ E (because E is a d.i.
) and F is a d.i. Finally, we prove that F satisfies the hypothesis of Lemma 4. Since

J ′ D starts with u ′ M ∈ W, we only need to show that |t| ≤ |u ′ M v 1 • • • v m-1 |. By contradiction, we assume u ′ M v 1 • • • v m-1 < p t. This condition implies two things. First, that we can define t ′ = (u ′ M v 1 • • • v m-1 ) -1 t ̸ = 1, and then, since u ′ M v 1 • • • v m-1 v ′ m = te ′ M e ′ R , that v ′ m = t ′ e ′ M e ′ R . In particular, ℓ ≤ |e ′ M | < |v ′ m |.
The second fact is that m ≤ n. Indeed, by hypothesis (iii) we have

|u ′ M v 1 • • • v m-1 | < |t| ≤ |d ′ M | = |u ′ M v 1 • • • v n |. Hence, ℓ < |v ′ m | ≤ |v m | ≤ ℓ,
which is a contradiction. This proves that Lemma 4 can be applied with F , so F (and then also E) reduces to a simple d.i.

If u ∈ A + , then we write u ∞ := uuu • • • and ∞ u := • • • uuu. Recall that an integer k ≥ 1 is a period of w ∈ A + if w ≤ p u ∞ (equivalently, w ≤ s ∞ u) for some u ∈ A k .
The following result (also known as the Fine and Wilf Lemma) is classical.

Lemma 7 (Proposition 1.3.2, [START_REF] Lothaire | Combinatorics on words[END_REF]). If p, p ′ ≥ 1 are periods of w ∈ A + and p + p ′ ≤ |w|, then gcd(p, p ′ ) is also a period of w.

We fix an irreducible subset

D ′ ⊆ D U . For D, E ∈ D ′ , since d, ẽ ≤ p u R and d, ẽ ≤ s u ′
L , we have that d and ẽ are both prefix and suffix dependent. So it makes sense to define in D ′ :

D ≤ E iff d ≤ p ẽ.
Observe that Lemma 6 part (iii) implies that D = E if and only if d = ẽ. Therefore, ≤ is a total order. In particular, we can use the notation D < E when D ≤ E and D ̸ = E. In this case it is not difficult to prove that |ẽ| -| d| is a period of ẽ.

Let D(1) < • • • < D(s) be all the elements in D ′ (deployed in increasing order). We adopt the mnemotechnical notation:

D(j) = (d L (j), d M (j), d R (j), a(j); d ′ L (j), d ′ M (j), d ′ R (j), a ′ (j));
(2.7)

d(j) = d L (j)d M (j)d R (j), d(j) = (d L (j)d M (j)) -1 d ′ L (j). (2.8) For D, E ∈ D ′ , since d R a D , ẽ ≤ p u R , we have that d R a D ≤ p ẽ if and only if |d R | < |ẽ|. Thus, for j ∈ {1, . . . , s} we can define D ′ (j) := {D ∈ D ′ : d R a D ≤ p d(j)} = {D ∈ D ′ : |d R | < | d(j)|}
and D ′ (s + 1) := D ′ . By definition of the total order, this is a nondecreasing sequence. Moreover, D ′ (j) ⊆ {D(k) : k ∈ {1, . . . , j-1}} for all j ∈ {1, . . . , s+1}. In particular, D ′ (1) = ∅. Lemma 8. Let p ∈ {1, . . . , s + 1} be such that D ′ (p) is nonempty and let D(p ′ ) := max D ′ (p), where the maximum is taken with respect to the total order. Then, #(D ′ (p)\D ′ (p ′ )) ≤ 6.

Proof. We prove the lemma by contradiction. Suppose #(D

′ (p)\D ′ (p ′ )) ≥ 7 and let D(j 1 ) < D(j 2 ) < • • • < D(j 7 ) be seven different elements in D ′ (p)\D ′ (p ′ ).
We start by obtaining some relations. First, from part (iii) of Lemma 6 and the irreducibility of D ′ , we get

dd ′ M < p ẽ for all D, E ∈ D ′ (p) such that D < E.
(2.9) Thus,

d(j k ) ≤ p d(j k )d ′ M (j k ) < p d(j k+1 ) ≤ p d(j k+1 )d ′ M (j k+1
) for all k ∈ {1, . . . , 6}.

(2.10) In Figure 2.5 we illustrate these conditions. We set

v k = d(j k )d ′ M (j k ), k ∈ {1, . . . , 6}. By (2.10), v 1 < p • • • < p v 5 < p d(j 6 ) < p v 6 < p d(j 7 ).
Also, observe that for any

D ∈ D ′ (p)\D ′ (p ′ ) we have D ≤ D(p ′ ) and D ̸ ∈ D ′ (p ′ ), which gives d ≤ p d(p ′ ) ≤ p d R ≤ p u R .
(2.11) Equation (2.10), the first inequality of (2.11) used with d(j 7 ) and the second inequality of (2.11) used with d(j k ) imply that

v k < p d(j 7 ) ≤ p d(p ′ ) ≤ p d R (j k ) for all k ∈ {1, . . . , 6}.
(2.12)

From previous relations we can define the nonempty word w := v -1 1 d(j 7 ). Let q ≤ p w be such that |q| is the least period of w. We will prove that |q| divides |v -1 1 v k | for all k ∈ {1, . . . , 5}. On the one hand, the observation made before the proof shows that | d(j 6 ) -1 d(j 7 )| is a period of d(j 7 ), and thus also of w. On the other hand, if k ∈ {1, . . . , 6}, then from (2.12) and the definition of d we get

(v -1 1 v k ) -1 w = v -1 k d(j 7 ) ≤ p v -1 k d R (j k ) = d ′ R (j k ) ≤ p u ′ R , being the last step true due to item ((3)) of Definition 4. In particular, for k = 1 we get w ≤ p u ′ R . These inequalities imply w ≤ p (v -1 1 v k ) ∞ . Conse- quently, |v -1 1 v k | is a period of w. Since, by (2.10), v -1 k d(j 6
) is defined for all k ∈ {1, . . . , 5}, then for these values of k we can compute

|q| + |v -1 1 v k | ≤ | d(j 6 ) -1 d(j 7 )| + |v -1 1 v k | = |w| -|v -1 k d(j 6 )| ≤ |w|. Hence, Lemma 7 can be applied to get that gcd(|q|, |v -1 1 v k |) is a period of w for k ∈ {1, . . . , 5}. In particular, |q| = gcd(|q|, |v -1 1 v k |) and |q| divides |v -1 1 v k | for k ∈ {1, . . . , 5}.
Then, we have w ≤ p q ∞ and, by the claim, for k ∈ {1, . . . , 5} there exists

n k ≥ 0 satisfying v -1 1 v k = q n k . Moreover, from the definition of v k we have v k = v 1 q n k , which implies d ′ R (j k )a(j k ) = v -1 k d R (j k )a(j k ) ≤ p v -1 k u R = q -n k v -1 1 u R and d ′ R (j k )a ′ (j k ) ≤ p u ′ R . Thus, since a(j k ) ̸ = a ′ (j k ), we deduce that d ′ R (j k ) is the maximal common prefix of q -n k v -1 1 u R and u ′ R . Now, let n, n ′ ≥ 0 and r, r ′ < p q be maximal such that q n r ≤ p v -1 1 u R and q n ′ r ′ ≤ p u ′ R . We conclude that d ′ R (j k ) = q n-n k r if n -n k < n ′ and d ′ R (j k ) = q n ′ r ′ if n -n k > n ′ (2.13) for k ∈ {1, . . . , 5}.
We have all the elements to complete the proof. Since

n 2 < n 3 < n 4 < n 5 , we have n 2 < n 3 < n -n ′ or n 5 > n 4 > n -n ′ .
We are going to show that both cases give a contradiction, proving, thereby, the lemma.

First, suppose that n 2 < n 3 < n -n ′ . Then, for k ∈ {2, 3}, we have n -n k > n ′ , and thus, by (2.13),

d ′ R (j k ) = q n ′ r ′ . If ℓ = 0, d(j k ) = d ′ L (j k )d ′ R (j k ) ≤ s u ′ L q n ′ r ′ . Then, d(j 2 ) and d(j 3 ) are suffix dependent, which gives that D(j 2 ) is equivalent to D(j 3 ), contradicting the irreducibility of D ′ . If ℓ > 0, we have d R (j k ) = v 1 (v -1 1 v k )d ′ R (j k ) = v 1 q n k +n ′ r ′ . Then, using (2.10), |q n k | = |v -1 1 v k | ≥ |v -1 1 v 2 | ≥ |d ′ M (j 2 )| ≥ |u ′ M | ≥ ⟨W⟩, and hence d(j 2 ) and d(j 3 ) share a common suffix of length ⟨W⟩. This is, D(j 2 ) ∼ D(j 3 ), which is a contradiction. Finally, assume n 5 > n 4 > n -n ′ . We have, by (2.13), that d ′ R (j k ) = q n-n k r for k ∈ {4, 5}. Hence, d R (j k ) = v 1 (v -1 1 v k )d ′ R (j k ) = v 1 q n k d ′ R (j k ) = v 1 q n r.
In particular, condition (ii) of Lemma 6 holds for {D(j 4 ), D(j 5 )}, contradicting the irreducibility of D ′ . This completes the proof. Proof. Note that p ′′ < p ′ < p. Before proving the main statement of the lemma, we highlight two useful relations. First, note that

d L (p ′′ )d M (p ′′ ) d(p ′′ ) = d ′ L (p ′′ ) (2.14) as D(p ′′ ) is simple. Second, since u R and u ′ L are, by Definition 4, the short- est words in W satisfying d R (p ′′ )a(p ′′ ) ≤ p u R and d ′ L (p ′′ ) ≤ s u ′ L , respectively, we have, by condition ((2)) of the definition of simple d.i., that |d ′ L (p ′′ )| ≥ min(|u R |, |u ′ L |) ≥ | d(k)| for k ∈ {1, . . . , s}.
This and the fact that

d ′ L (p ′′ ) and d(k) are both suffix of u ′ L imply d(k) ≤ s d ′ L (p ′′ ) for k ∈ {1, . . . , s}.
(2.15)

Now we are ready to prove the main statement of the lemma. Using (2.15) and d(p

′ ) ≤ p d(p), we have (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) ≤ p d ′ L (p ′′ ). In addition, d L (p ′′ ) ≤ p d ′ L (p ′′ ) by the simplicity of D(p ′′ ). Thus, (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) and d L (p ′′ ) are prefix dependent.
In what follows, we split the proof in two cases according to which of these words is prefix of the other.

(a) (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) ≤ p d L (p ′′ ). Observe that d(s) ≤ s u ′ L and d M (p ′′ ) d(p ′′ ) ≤ s d ′ L (p ′′ ) ≤ s u ′ L , so d(s) and d M (p ′′ ) d(p ′′
) are suffix dependent. In addition, from (2.14) and (a) we get

|d M (p ′′ ) d(p ′′ )| = |d ′ L (p ′′ )| -|d L (p ′′ )| (2.16) ≤ |d ′ L (p ′′ )| -|(d ′ L (p ′′ ) d(p) -1 ) d(p ′ )| = | d(p)| -| d(p ′ )| ≤ | d(s)|. We conclude that d M (p ′′ ) d(p ′′ ) ≤ s d(s).
Thus, it makes sense to define

w ′ := d(s)(d M (p ′′ ) d(p ′′ )) -1 . Clearly, w ′ ≤ p d(s) d(p ′′ ) -1 . Let w ∈ W be a word satisfying d L (p ′′ ) ≤ s w,
w ′ ≤ s d ′ L (p ′′ )(d M (p ′′ ) d(p ′′ )) -1 = d L (p ′′ ) ≤ s w, so w and w ′ are suffix dependent. It is left to prove that |w ′ | ≥ | d(s)|-| d(p)| and |w| ≥ | d(p ′ )|.
For this, we note that in (2.16) it was shown that |d

M (p ′′ ) d(p ′′ )| ≤ | d(p)| -| d(p ′ )|. Thus, |w ′ | ≥ | d(s)| -| d(p)| + | d(p ′ )| ≥ max(| d(s)| -| d(p)|, | d(p ′ )|). We conclude that |w ′ | ≥ | d(s)| -| d(p)| and, since w ′ ≤ s w, |w| ≥ |w ′ | ≥ | d(p ′ )|.
This completes the proof in case (a).

(b) d L (p ′′ ) < p (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ).
We start by claiming that

| d(p ′′ )| + | d(p ′ )| < | d(p)|.
(2.17)

Assume that (2.17) does not hold. Let q be the shortest word satisfying d(p) ≤ s ∞ q. As we commented before Lemma 8, condition p ′ , p ′′ < p implies that d(p ′ ), as well as d(p ′′ ), are prefixes and suffixes of d(p).

So | d(p)| -| d(p ′ )| and | d(p)|-| d(p ′′ )| are periods of d(p).
Moreover, since we are assuming (2.17) is not true, we also have that

(| d(p)| -| d(p ′ )|) + (| d(p)| -| d(p ′′ )|) ≤ | d(p)|. Then, by Lemma 7, we obtain that |q| divides | d(p)| -| d(p ′ )| and | d(p)| -| d(p ′′ )|. Hence, there exists n ′ , n ′′ ∈ N such that q n ′ = d(p ′ ) -1 d(p) and q n ′′ = d(p ′′ ) -1 d(p). Now, since p ′ , p ′′ ∈ D ′ (p), we can write d ′ M (p ′ )d ′ R (p ′ )a(p ′ ) = d(p ′ ) -1 d R (p ′ )a(p ′ ) ≤ p d(p ′ ) -1 d(p) = q n ′ ≤ p q ∞ and, in a similar way, d ′ M (p ′′ )d ′ R (p ′′ )a(p ′′ ) ≤ p q ∞ . Thus, {D(p ′ ), D(p ′′ )}
is reducible by part (i) of Lemma 6, which contradicts the irreducibility of D ′ . This proves the claim.

From (2.17) and (2.14) we get

|(d ′ L (p ′′ ) d(p) -1 ) d(p ′ )| = |d ′ L (p ′′ )| -| d(p)| + | d(p ′ )| < |d ′ L (p ′′ )| -| d(p ′′ )| = | d(p ′′ ) -1 d ′ L (p ′′ )| = |d L (p ′′ )d M (p ′′ )|.
Then, since

(d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) ≤ p (d ′ L (p ′′ ) d(p) -1 ) d(p) = d ′ L (p ′′ ) = d L (p ′′ )d M (p ′′ ) d(p ′′ ), we obtain that (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) < p d L (p ′′ )d M (p ′′ ).
This and (b) can be written together as

d L (p ′′ ) < p (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) < p d L (p ′′ )d M (p ′′ ). (2.18) Since d L (p ′′ )d M (p ′′ ) d(p ′′ ) = d ′ L (p ′′ )

By (2.18), we can write d

L (p ′′ )d M (p ′′ ) = vwv ′ , where v ∈ d L (p ′′ )W * , w ∈ W, v ′ ∈ W * and v < p (d ′ L (p ′′ ) d(p) -1 ) d(p ′ ) ≤ p vw. (2.19)
The word w is the one we need in the statement of the lemma. To define w ′ , we first note that d(s)

≤ s d ′ L (p ′′ ) and v ′ d(p ′′ ) ≤ s d L (p ′′ )d M (p ′′ ) d(p ′′ ) = d ′ L (p ′′ ), so d(s) and v ′ d(p ′′ ) are suffix dependent. Moreover, using (2.19) we get |v ′ d(p ′′ )| = |d ′ L (p ′′ )| -|vw| ≤ |d ′ L (p ′′ )| -|(d ′ L (p ′′ ) d(p) -1 ) d(p ′ )| = | d(p)| -| d(p ′ )|. (2.20) Then, |v ′ d(p ′′ )| ≤ | d(p)| -| d(p ′ )| ≤ | d(s)| and v ′ d(p ′′ ) ≤ s d(s). Now it makes sense to define w ′ := d(s)(v ′ d(p ′′ )) -1 , which clearly verifies w ′ ≤ p d(s) d(p ′′ ) -1 .
It is also clear that w and w ′ are suffix dependent. Indeed, from (2.15) and (2.14) we have (2.21)

w ′ ≤ s d ′ L (p ′′ )(v ′ d(p ′′ )) -1 = vw. Now, from (2.20), |w ′ | ≥ | d(s)| -| d(p)| + | d(p ′ )| ≥ | d(s)| -| d(p)|,
First, we prove that it makes sense to define the word

w ′′ := ((d ′ L (p ′′ ) d(p) -1 ) -1 v) -1 d R (p ′ ) ∈ A + . (2.22) From (2.19) and (2.21) we get |v| ≥ |(d ′ L (p ′′ ) d(p) -1 ) d(p ′ )|-|w| > |d ′ L (p ′′ ) d(p) -1 |. But, v ≤ p d L (p ′′ )d M (p ′′ ) ≤ p d ′ L (p ′′ ) and d ′ L (p ′′ ) d(p) -1 ≤ p d ′ L (p ′′ ), so d ′ L (p ′′ ) d(p) -1 < p v and (d ′ L (p ′′ ) d(p) -1
) -1 v exists and is not the empty word. Hence, by (2.19),

(d ′ L (p ′′ ) d(p) -1 ) -1 v < p d(p ′ ) ≤ p d R (p ′ ) (2.23)
and w ′′ is well defined.

Now, we have vw

≤ p d L (p ′′ )d M (p ′′ ) ≤ p d ′ L (p ′′ ) and, using p ′ ∈ D ′ (p), that (d ′ L (p ′′ ) d(p) -1 )d R (p ′ ) ≤ p (d ′ L (p ′′ ) d(p) -1 ) d(p) = d ′ L (p ′′ ). Thus, vw and (d ′ L (p ′′ ) d(p) -1 )d R (p ′ ) are prefix dependent. Therefore, there are two cases: vw is prefix of (d ′ L (p ′′ ) d(p) -1 )d R (p ′ ) and (d ′ L (p ′′ ) d(p) -1 )d R (p ′
) is a strict prefix of vw; in each of these cases we will build a reduction for D(p ′ ), producing a contradiction.

(b.1) vw ≤ p (d ′ L (p ′′ ) d(p) -1 )d R (p ′ ). We start by building a d.i. of w ′′ . Note that w ′′ a(p ′ ) ≤ p wv ′ d(p ′′ ). (2.24) Indeed, since D(p ′ ) ∈ D ′ (p) and (d ′ L (p ′′ ) d(p) -1 ) d(p) = d ′ L (p ′′ ) = vwv ′ d(p ′′ ), we have d R (p ′ )a(p ′ ) ≤ p d(p) = (d ′ L (p ′′ ) d(p) -1 ) -1 vwv ′ d(p ′′ ), which implies (2.24). Now, since w ∈ W, v ′ ∈ W * and d(p ′ ) < p u R , the word wv ′ d(p ′ ) has an interpretation of the form J = w, v ′ , d(p ′ ), a. Moreover, using (b.1) we can get |w ′′ | = |d R (p ′ )| + |d ′ L (p ′′ ) d(p) -1 | -|v| ≥ |w|.
Hence, by (2.24), Lemma 3 can be applied with J to obtain an interpretation of w ′′ having the form I ′ = w, r, r ′ , a(p ′ ). We need another interpretation of w ′′ . Note that in the middle step of (2.23) we showed that (d

′ L (p ′′ ) d(p) -1 ) -1 v < p d(p ′ ). In particular, the word ((d ′ L (p ′′ ) d(p) -1 ) -1 v) -1 d(p ′ ) is nonempty and is a suffix of u ′ L ∈ W. Then, I := ((d ′ L (p ′′ ) d(p) -1 ) -1 v) -1 d(p ′ ), d ′ M (p ′ ), d ′ R (p ′ ), a ′ (p ′ )) is an interpretation of w ′′ (here, we used that d(p ′ )d ′ M (p ′ )d ′ R (p ′ ) = d R (p ′ )). We set D = (I, I ′ ). Since a(p ′ ) ̸ = a ′ (p ′ ), D is a d.i. of w ′′ . Now we can conclude the proof of this case. From (2.19) we have |v| ≥ |(d ′ L (p ′′ ) d(p) -1 ) d(p ′ )| -|w|, which implies |((d ′ L (p ′′ ) d(p) -1 ) -1 v) -1 d(p ′ )| ≤ |w| ≤ |wr|.
This and that w ∈ W allow us to use Lemma 4 to obtain a simple d.i. E of a word e such that e ≤ s w ′′ . Since

w ′′ < s d R (p ′ ) < s d(p ′ ), we have that D(p ′ ) reduces to E. This is the desired contradiction. (b.2) (d ′ L (p ′′ ) d(p) -1 )d R (p ′ ) < p vw.
We are going to build a simple d.i.

D = (I; I ′ ) of d R (p ′ ) < s d(p ′ ), proving, thereby, that D(p ′ ) has a reduction. Let I ′ = d(p ′ ), d ′ M (p ′ ), d ′ R (p ′ ), a ′ (p ′ ). It is clear that I ′ is an interpretation of d R (p ′ ) since d(p ′ ) ≤ s u ′ L , d ′ M (p ′ ) ∈ W * , d ′ R (p ′ )a ′ (p ′ ) ≤ p u ′ R and | d(p ′ )| > | d(p ′′ )| ≥ 0. To define I, observe that in the proof of (2.22) we showed that (d ′ L (p ′′ ) d(p) -1
) -1 v exists and is not the empty word. But, moreover, from

v ∈ d L (p ′′ )W * we see that we can write (d ′ L (p ′′ ) d(p) -1 ) -1 v = rr ′ in such a way
that r is a nonempty suffix of some word in W and r ′ ∈ W * . Since, by definition, d R (p ′ ) = rr ′ w ′′ , to prove that

I := r, r ′ , w ′′ , a(p ′ ) is an interpretation of d R (p ′ ) it is enough to show that w ′′ a(p ′ ) ≤ p w. From (b.2) we get rr ′ w ′′ = d R (p ′ ) < p rr ′ w, so w ′′ a ′ ≤ p w for some a ′ ∈ A. Then, using that vw ≤ p vwv ′ d(p ′′ ) = d ′ L (p ′′ ), we obtain d R (p ′ )a ′ ≤ p rr ′ w = (d ′ L (p ′′ ) d(p) -1 ) -1 vw ≤ p (d ′ L (p ′′ ) d(p) -1 ) -1 d ′ L (p ′′ ) = d(p) ≤ p u R . Since we also have d R (p ′ )a(p ′ ) ≤ p u R , we deduce that a ′ = a(p ′ ). Hence, w ′′ a(p ′ ) ≤ p w and I is an interpretation of d R (p ′ ). Being a(p ′ ) ̸ = a ′ (p ′ ), we conclude that D := (I; I ′ ) is a d.i. of d R (p ′ ).
Finally, we prove that D is simple. Using the middle step in (2.23) we get Remark that in the last paragraph it was the first time that in a proof we build a reduction to a simple d.i. satisfying the second condition of ((2)) in Definition 3.

rr ′ = (d ′ L (p ′′ ) -1 d(p)) -1 v < p d(p ′ ). This implies that d ′ M (p ′ )d ′ R (p ′ ) = d(p ′ ) -1 d R (p ′ ) ≤ s (rr ′ ) -1 d R (p ′ ) = w ′′ ,

Proof of Proposition 1

Proposition [. 1] Any irreducible subset of D U has at most 61(#W) elements.

Proof. Let D ′ be an irreducible subset of D U . Recall that, with the notation introduced above,

D(1) < • • • < D(s) are the elements of D ′ deployed in in- creasing order, D ′ (s + 1) = D ′ and D ′ (j) = {D ∈ D ′ : d R a D ≤ p d(j)} = {D ∈ D ′ : |d R | < | d(j)|} for j ∈ {1, . . . , s}.
We define recursively a finite decreasing sequence (p i ) t+1 i=0 . We start with

p 0 = s + 1. Then, for i ≥ 0: a) if #D ′ (p i ) ≤ 1 we put p i+1 = 1 and the procedure stops; b) if #D ′ (p i ) > 1, set D(p i+1 ) = max D ′ (p i ). Observe that D ′ (p i+1 ) ⊊ D ′ (p i ). Let t ≥ 0 be the first integer for which #D ′ (p t ) ≤ 1, so that D ′ (p t+1 ) = D ′ (1) = ∅. This construction gives D ′ = t i=0 D ′ (p i )\D ′ (p i+1 ).
From Lemma 8 we get that #D ′ ≤ 6t + 1. To complete the proof we are going to show that t ≤ 8#W + 2.

We proceed by contradiction, so we suppose t > 8#W + 2. This will imply that D ′ is reducible, which contradicts our hypothesis.

Let 1 ≤ i ≤ t -1. Since p i ̸ = s + 1 and #D ′ (p i ) > 1, we can define D(p ′′ i ) = max D ′ (p i ) \ {D(p i+1
)} and use Lemma 9 with D ′ (p i ) to obtain suffix dependent words w i ∈ W and

w ′ i ∈ A * such that (i) |w i | > | d(p i+1 )|, (ii) |w ′ i | ≥ | d(s)| -| d(p i )|, (iii) w ′ i ≤ p d(s) d(p ′′ i ) -1 .
(2.25) Then, by the Pigeonhole Principle, we can find 1

≤ i 5 < • • • < i 1 ≤ t -1 such that (a) w := w i1 = • • • = w i5 and (b) i k+1 + 2 ≤ i k for any k ∈ {1, . . . , 4}.
Using (a) and (b) we are going to obtain relations (2.26) and (2.27) below.

First, we use (b) to prove that

d(s) d(p i k+1 ) -1 < p w ′ i k+1 ≤ p d(s) d(p i k ) -1 < p w ′ i k for any k ∈ {1, . . . , 4}. (2.26) Let k ∈ {1, . . . , 4}. By (b), we have i k+1 ≤ i k+1 + 1 < i k+1 + 2 ≤ t -1. Thus, D(p i k+1 +2 ) < D(p i k+1 +1 ) and D(p i k+1 +1 ), D(p i k+1 +2 ) ∈ D ′ (p i k+1 ), which implies that p ′′ i k+1 ≥ p i k+1 +2 by the definition of p ′′ i k+1 . Being p i k+1 +2 ≥ p i k by (b), we obtain p ′′ i k+1 ≥ p i k . This and (iii) of (2.25) imply w ′ i k+1 ≤ p d(s) d(p ′′ i k+1 ) -1 ≤ p d(s) d(p i k ) -1 . This proves the middle inequality of (2.26). Let k ∈ {1, . . . , 5}. Since w ′ i k ≤ p d(s) d(p ′′ i k ) -1 ≤ p d(s) by (iii) of (2.25) and d(s) d(p i k ) -1 ≤ p d(s), we have that w ′ i k and d(s) d(p i k ) -1 are prefix dependent. Moreover, |w ′ i k | > | d(s) d(p i k ) -1 | by (ii) of (2.25), so d(s) d(p i k ) -1 < p w ′ i k .
This proves the first and last inequality of (2.26), completing the proof.

Thanks to (2.26), the word ( d(s) d(p i k ) -1 ) -1 w ′ i k ′ exists for any 1 ≤ k ′ ≤ k ≤ 5. We will use this fact freely through the proof.

Next, we want to obtain from (a) that

( d(s) d(p i4 ) -1 ) -1 w ′ i k ≤ s w for k ∈ {1, . . . ,
|( d(s) d(p i4 ) -1 ) -1 w ′ i k | ≤ | d(s) d(p ′′ i k ) -1 | -| d(s) d(p i4 ) -1 | ≤ | d(p i4 )| ≤ |w|.
But, being w and (

d(s) d(p i4 ) -1 ) -1 w ′ i k suffix dependent since w and w ′ i k have the same property and ( d(s) d(p i4 ) -1 ) -1 w ′ i k ≤ s w ′ i k , we obtain that ( d(s) d(p i4 ) -1 ) -1 w ′ i k ≤ s w,
v := ( d(s) d(p i4 ) -1 ) -1 w ′ i1 . More precisely, we claim that if q ∈ A + is the shortest word satisfying v ≤ p q ∞ , then |q| divides | d(p i4 )| -| d(p i k )| for k ∈ {2, 3}. Fix k ∈ {2, 3}. First, observe that v ≤ s w and v((w ′ i2 ) -1 w ′ i1 ) -1 = ( d(s) d(p i4 ) -1 ) -1 w ′ i2 ≤ s w by (2.27). Being (w ′ i2 ) -1 w ′ i1 ̸ = 1 by (2.25), we deduce that v ≤ s ∞ ((w ′ i2 ) -1 w ′ i1 ). This implies that |q| ≤ |(w ′ i2 ) -1 w ′ i1 |. Thus, |q| + | d(p i4 ) d(p i k ) -1 | ≤ |(w ′ i2 ) -1 w ′ i1 | + | d(p i4 ) d(p i k ) -1 | (2.28) = |v| + |( d(s) d(p i k ) -1 ) -1 w ′ i2 | ≤ |v|, where ( d(s) d(p i k ) -1 ) -1 w ′ i2 exists because k ≥ 2. Second, since w ′ i1 ≤ p d(s) by (iii) of (2.25), we have that v = ( d(s) d(p i4 ) -1 ) -1 w ′ i1 ≤ p d(p i4 ) ≤ p u R and ( d(s) d(p i k ) -1 ) -1 w ′ i1 ≤ p d(p i k ) ≤ p u R . Therefore, v ≤ p u R and ( d(p i4 ) d(p i k ) -1 ) -1 v = ( d(s) d(p i k ) -1 ) -1 w ′ i1 ≤ p u R .
This and the fact that, by (2.25), (

d(p i4 ) d(p i k ) -1 ) ̸ = 1 imply that v ≤ p ( d(p i4 ) d(p i k ) -1 ) ∞ . Hence, | d(p i4 ) d(p i k ) -1 | is a period of v. ( 2
k ∈ {2, 3} that d(p i4 ) d(p i k ) -1 = q n k for some n k ≥ 1. Then, since | d(p i4 ) d(p i k ) -1 | is a period of d(p i4 ) as p i k < p i4 , we obtain d(p i4 ) ≤ p ( d(p i4 ) d(p i k ) -1 ) ∞ = q ∞ and q ≤ p q. Since, v ≤ p d(p i4 ) ≤ p
q∞ , we also have q ≤ p q. Therefore, q = q. Now we can finish the proof of the proposition. Since d(p i4 ) ≤ p q ∞ , there are n ≥ 0 and r < p q such that d(p i4 ) = q n r. Then, for k ∈ {2, 3}, we have

d(p i k ) = q -n k d(p i4 ) = q n-n k r. Being p i2 , p i3 ∈ D ′ (p i4 ), we get d′ M (p i k ) d′ R (p i k )a(p i k ) = d(p i k ) -1 dR (p i k )a(p i k ) ≤ p d(p i k ) -1 d(p i4 ) = r -1 q n k ≤ p r -1 q ∞ .
Thus, condition (i) of Lemma 6 holds, which implies that {D(p i2 ), D(p i3 )} is reducible, contradicting our hypothesis.

Chapter 3

Symbolic factors 3.1 Introduction

The class of finite topological rank subshifts have shown to be both a broad class of symbolic systems [DDMP16; DDMP21], containing many of the most studied types of subshifts, and to present high degrees of rigidity [BKMS13; BDM10; EM21]. Hence, it arises as a possible framework for studying minimal subshifts and proving general theorems.

In this direction, a fundamental question is the following:

Question 7. Is the finite topological rank class closed under symbolic factors?

Indeed, the topological rank aims to measure how complex is the system, so an affirmative answer is expected to this question. However, symbolic factors inherit a natural yet non-recognizable S-adic structure with finite alphabet rank from their extensions, and thus it is not clear if a structure that is, in addition, recognizable can always be obtained. Thus, an answer to this question seems to be fundamental to the understanding of finite topological rank systems.

In this chapter, we obtain the optimal answer to Question 7 in a more general, non-minimal context: Theorem 21. Let (X, S) be an S-adic subshift generated by an everywhere growing and proper directive sequence of alphabet rank equal to K, and π : (X, S) → (Y, S) be an aperiodic subshift factor. Then, (Y, S) is an S-adic subshift generated by an everywhere growing, proper and recognizable directive sequence of alphabet rank at most K.

Theorem 21 implies that the topological rank cannot increase after a factor map (Corollary 7).

We are also able to prove the following theorems, which give a finer description of symbolic factors.

Corollary 5. Let (X, S) be an S-adic subshift generated by an everywhere growing and proper directive sequence of alphabet rank equal to K, and (X, S)

π1 → 53 (X 1 , S) π2 → . . . π L → (X L , S
) be a chain of aperiodic subshift factors. If L > log 2 K, then at least one π j is a conjugacy.

Theorem 22. Let π : (X, S) → (Y, S) be a factor map between aperiodic minimal subshifts. Suppose that (X, S) has topological rank equal to K. Then π is almost k-to-1 for some k ≤ K.

Theorem 23. Let (X, S) be a minimal subshift of topological rank K. Then, (X, S) has at most (3K) 32K aperiodic subshift factors up to conjugacy.

This chapter was published as a standalone article in [START_REF] Espinoza | Symbolic factors of S-adic subshifts of finite alphabet rank[END_REF].

Organization

This chapter consists of 6 sections. In the first one, we give the additional needed background in topological and symbolic dynamics. Section 3.3 is devoted to prove some technical combinatorial lemmas. The main results about the topological rank of factors are stated and proved in Section 3.4. Next, in Section 3.5, we prove Theorem 22. In Section 3.6, we study the problem about the number of symbolic factors and prove Theorem 23. The last section contains a combinatorial proof of Proposition 3.

Preliminaries

The hyperspace of (X, S) is the system (2 X , S), where 2 X is the set of all closed subsets of X with the topology generated by the Hausdorff metric d H (A, B) = max(sup x∈A d(x, A), sup y∈B d(y, A)), and S the action A → S(A).

A factor between the topological dynamical systems (X, S) and (Y, T ) is a continuous function π from X onto Y such that π•S = T •π. We use the notation π : (X, S) → (Y, T ) to indicate the factor. A factor map π : (X, S) → (Y, T ) is almost K-to-1 if #π -1 (y) = K for all y in a residual subset of Y . We say that π is distal if whenever π(x) = π(x ′ ) and x ̸ = x ′ , we have inf k∈Z dist(S k x, S k x ′ ) > 0.

Given a system (X, S), the Ellis semigroup E(X, S) associated to (X, S) is defined as the closure of {x → S n x : n ∈ Z} ⊆ X X in the product topology, where the semi-group operation is given by the composition of functions. On X we may consider the E(X, S)-action given by x → ux. Then, the closure of the orbit under S of a point x ∈ X is equal to the orbit of x under E(X, S). If π : (X, S) → (Y, T ) is a factor between minimal systems, then π induces a surjective map π * : E(X, S) → E(Y, T ) which is characterized by the formula π(ux) = π * (u)π(x) for all u ∈ E(X, S) and x ∈ X.

If the context is clear, we will not distinguish between u and π * (u). When u ∈ E(2 X , S), we write u • A instead of uA, the last symbol being reserved to mean uA = {ux : x ∈ A}. We can describe more explicitly u • A as follows: it is the set of all x ∈ X for which we can find nets x λ ∈ A and m λ ∈ Z such that lim λ S m λ x λ = x and lim λ S m λ = u. Finally, we identify X with {{x} ⊆ 2 X : x ∈ X}, so that the restriction map E(2 X , S) → E(X, S) which sends u ∈ E(2 X , S) to the restriction u| X : X → X is an onto morphism of semigroups. As above, we will not distinguish between u ∈ 2 X and u| X .

Basics in symbolic dynamics

Words and subshifts

The pair (x, x)

∈ A Z × A Z is right asymptotic if there exist k ∈ Z satisfying x (k,∞) = x(k,∞) and x k ̸ = xk . If moreover k = 0, (x, x) is a centered right asymptotic.
A right asymptotic tail is an element x (0,∞) , where (x, x) is a centered right asymptotic pair. We make similar definitions for left asymptotic pairs and tails.

Morphisms and substitutions

We say that τ is positive if for every a ∈ A, all letters b ∈ B occur in τ (a), is rproper, with r ≥ 1, if there exist u, v ∈ B r such that τ (a) starts with u and ends with v for any a ∈ A, is proper when is 1-proper, and is letter-onto if for every b ∈ B there exists a ∈ A such that b occurs in a. The minimum and maximum length of τ are, respectively, the numbers ⟨τ ⟩ := ⟨τ Let X ⊆ A Z and Z ⊆ C Z be subshifts and π : (X, S) → (Z, S) a factor map. The classic Curtis-Hedlund-Lyndon Theorem asserts that π has a local code, this is, a function ψ : A 2r+1 → C, where r ∈ N, such that π(x) = (ψ(x [i-r,i+r] )) i∈Z for all x ∈ X. The integer r is called the a radius of π. The following lemma relates the local code of a factor map to proper morphisms.

Lemma 10. Let σ : A + → B + be a morphism, X ⊆ A Z and Z ⊆ C Z be subshifts, and Y = k∈Z S k σ(X). Suppose that π : (Y, S) → (Z, S) is a factor map of radius r and that σ is r-proper. Then, there exists a proper morphism τ :

A + → C + such that |τ (a)| = |σ(a)| for any a ∈ A, Z = k∈Z S k τ (X) and the following diagram commutes: X Y Z σ τ π (3.1)
Proof. Let ψ : A 2r+1 → B be a local code of radius r for π and u, v ∈ B r be such that σ(a) starts with u and ends with v for all a ∈ A. We define τ : A → C + by τ (a) = ψ(vσ(a)u). Then, since σ is r-proper, τ is proper and we have π(σ(x)) = τ (x) for all x ∈ X (this is, Diagram (3.1) commutes). In particular, k∈Z

S k τ (X) = k∈Z S k π(σ(X)) = π(Y ) = Z.

S-adic subshifts

The levels X

(n)

σ can be described in an alternative way if σ satisfies the correct hypothesis.

Lemma 11. Let σ = (σ n : A + n+1 → A + n ) n∈N be an everywhere growing and proper directive sequence. Then, for every n ∈ N,

X (n) σ = N >n k∈Z S k σ [n,N ) (A Z N ) (3.2)
Proof. Let Z be the set in the right-hand side of (3.2). Since, by Lemma 2,

X (n) σ = k∈Z S k σ [n,N ) (X (N ) σ ) for any N > n, we have that X (n) σ included in Z. Conversely, let x ∈ Z and ℓ ∈ N. We have to show that x [-ℓ,ℓ) occurs in σ [n,N ) (a) for some N > n and a ∈ A N . Let N > n be big enough so that σ [n,N ) is ℓ-proper. Then, by the definition of Z, there exists y ∈ A Z N such that x [-ℓ,ℓ) occurs in σ [n,N ) (y). Since ⟨σ [n,N ) ⟩ ≥ ℓ (as σ [n,N ) is ℓ-proper), we deduce that x [-ℓ,ℓ) occurs in σ [n,N ) (ab)
for some word ab of length 2 occurring in y. (3.3) Hence, by denoting by u and v the suffix and prefix of length

ℓ of τ [n,N ) (a) and τ [n,N ) (b), respectively, we have that x [-ℓ,ℓ) occurs in σ [n,N ) (a), in τ [n,N ) (b),
or in uv. In the first two cases, we are done. In the last case, we observe that since σ [n,N ) is ℓ-proper, the following is true: for every

M > N such that ⟨σ [N,M ) ⟩ ≥ 2, vu ⊑ σ [n,M ) (c) for any c ∈ A M . In particular, x [-ℓ,ℓ) ⊑ τ [n,M ) (c) for such M and c. We have proved that x ∈ X (n) σ .
Finite alphabet rank S-adic subshifts are eventually recognizable:

Theorem 24 ([DDMP21], Theorem 3.7). Let σ be an everywhere growing directive sequence of alphabet rank equal to K. Suppose that X σ is aperiodic. Then, at most log 2 K levels (X (n) σ , σ n ) are not recognizable. We will also need the following property.

Theorem 25 ([EM21], Theorem 3.3). Let (X, S) be an S-adic subshift generated by an everywhere growing directive sequence of alphabet rank K. Then, X has at most 144K 7 right (resp. left) asymptotic tails.

Proof. In the proof of Theorem 3.3 in [START_REF] Espinoza | On the automorphism group of minimal S-adic subshifts of finite alphabet rank[END_REF] the authors show the following: the set consisting of pairs (x, y) ∈ X × X such that x (-∞,0) = y (-∞,0) and x 0 ̸ = y 0 has at most 144K 7 elements. In our language, this is equivalent to saying that X has at most 144K 7 left asymptotic tails. Since this is valid for any S-adic subshift generated by an everywhere growing directive sequence of alphabet rank K, 144K 7 is also an upper bound for right asymptotic tails.

Combinatorics on words lemmas

In this section we present several combinatorial lemmas that will be used throughout the chapter.

Lowering the rank

Let σ : A + → B + be a morphism. Following ideas from [START_REF] Rozenberg | Handbook of Formal Languages: Volume 1. Word, Language, Grammar. Handbook of Formal Languages[END_REF], we define the rank of σ as the least cardinality of a set of words D ⊆ B + such that σ(A + ) ⊆ D + . Equivalently, the rank is the minimum cardinality of an alphabet C in a decomposition into morphisms A + q -→ C + p -→ B + such that σ = pq. In this subsection we prove Lemma 15, which states that in certain technical situation, the rank of the morphism σ under consideration is small and its decomposition σ = pq satisfies additional properties.

We start by defining some morphisms that will be used in the proofs of this subsection. If a ̸ = b ∈ A are different letters and ã is a letter not in A, then we define ϕ a,b :

A + → (A \ {b}) + , ψ a,b : A + → A + and θ a,ã : A + → (A ∪ {ã}) + by ϕ a,b (c) = c if c ̸ = b, a if c = b. ψ a,b (c) = c if c ̸ = b, ab if c = b. θ a,ã (c) = c if c ̸ = a, ãa if c = a.
Observe that these morphisms are letter-onto. Before stating the basic properties of these morphisms, we need one more set of definitions.

For a morphism σ : A + → B + , we define |σ| 1 = a∈A |σ(a)|. When u, v, w ∈ A + satisfy w = uv, we say that u is a prefix of w and that v a suffix of w. Recall that 1 stands for the empty word.

Lemma 12. Let σ : A + → B + be a morphism.

σ ′ (c) = σ(c) if c ̸ = b, t if c = b. (3.4) (iii) If σ(a) = st for some s, t ∈ B + and a ∈ A, then σ = σ ′ θ a,ã , where σ ′ : (A ∪ {ã}) + → B + is defined by σ ′ (c) =      σ(c) if c ̸ = a, ã, s if c = ã, t if c = a.
(3.5)

Proof. The lemma follows from unraveling the definitions. For instance, in case (ii), we have σ

′ (ψ a,b (a)) = σ ′ (a) = σ(a), σ ′ (ψ a,b (b)) = σ ′ (ab) = σ(a)t = σ(b), and σ ′ (ψ a,b (c)) = σ ′ (c) = σ(c) for all c ̸ = a, b, which shows that σ ′ ψ a,b = σ.
Lemma 13. Let {σ j : A + → B + j } j∈J be a set of morphisms such that for every fixed a ∈ A, ℓ a := |σ j (a)| is constant for any chosen j ∈ J, (3.6) and u, v ∈ A + , with u of length at least ℓ := a∈A ℓ a . Assume that u and v start with different letters and that σ j (u) is a prefix of σ j (v) for every j ∈ J.

Then, there exist a letter-onto morphism q : A + → C + , with #C < #A, and morphisms {p j : C + → B + j } j∈J satisfying a condition analogous to (3.6) and such that σ j = p j q.

Remark 3. If in the previous lemma we change the last hypothesis to "u and v end with different letters and σ j (u) is a suffix of σ j (v) for every j ∈ J", then the same conclusion holds. This observation will be used in the proof of Lemma 26.

Proof (of Lemma 13). By contradiction, we assume that u, v and {σ j } j∈J , are counterexamples for the lemma. Moreover, we suppose that ℓ is as small as possible.

Let us write u = au ′ and v = bv ′ , where a, b ∈ A. Since σ j (u) is a prefix of σ j (v), we have that for every j ∈ J, one of the words in {σ j (a), σ j (b)} is a prefix of the other.

(3.7)

We consider two cases. First, we suppose that ℓ a = ℓ b . In this case, (3.7) implies that σ j (a) = σ j (b) for every j ∈ J. Hence, we can use (1) of Lemma 12 to decompose each σ j as σ ′ j ϕ a,b , where σ ′ j is the restriction of σ j to (A \ {b}) + . Since ϕ a,b is letter-onto and ℓ c = |σ ′ j (c)| for every j ∈ J, c ∈ A \ {b}, the conclusion of the lemma holds, contrary to our assumptions.

It rests to consider the case in which ℓ a ̸ = ℓ b . We only do the case ℓ a < ℓ b as the other is similar. Then, by (3.7), for every j ∈ J there exists a nonempty word t j ∈ B ℓ b -ℓa j of length ℓ b -ℓ a such that σ j (b) = σ j (a)t j . Thus, we can use (2) of Lemma 12 to write, for any j ∈ J, σ j = σ ′ j ψ a,b , where σ ′ j is defined as in (3.4).

Let ũ = ψ a,b (u ′ ) and ṽ = bψ a,b (v ′ ). We want now to prove that ũ, ṽ and {σ ′ j : j ∈ J} satisfy the hypothesis of the lemma. First, we observe that for every j ∈ J,

if c ̸ = b, then |σ ′ j (c)| = ℓ c , and |σ ′ j (b)| = |t j | = ℓ b -ℓ a . (3.8)
Therefore, {σ ′ j } j∈J satisfy condition (3.6). Also, since ψ a,b (c) never starts with b, we have that ũ, ṽ start with different letters.

(3.9) Furthermore, by using the symbol ≤ p to denote the prefix relation, we can compute:

σ j (a)σ ′ j (ũ) = σ j (a)σ j (u ′ ) = σ j (u) ≤ p σ j (v) = σ ′ j (ψ a,b (v)) = σ ′ j (a)σ ′ j (ṽ).
This and the fact that σ j (a) is equal to σ ′ j (a) imply that σ ′ j (ũ) is a prefix of σ ′ j (ṽ) for every j ∈ J.

(3.10)

Finally, we note |ũ| ≥ |u| -1 ≥ c∈A ℓ c -ℓ a =: ℓ ′ . (3.11)
We conclude from equations (3.8), (3.9), (3.10) and (3.11) that ũ, ṽ and {σ ′ j : j ∈ J} satisfy the hypothesis of this lemma. Since ℓ ′ < ℓ, the minimality of ℓ implies that there exist a letter-onto morphism q ′ : A + → C + , with #C < #A, and morphisms {p j : C + → B + j } j∈J satisfying σ ′ j = p j q ′ and a property analogous to (3.6). But then q := q ′ ψ a,b is also letter-onto and the morphisms {p j } j∈J satisfy σ j = p j q and a property analogous to (3.6). Thus, the conclusion of the lemma holds for {σ j } j∈J , contrary our assumptions.

Lemma 14. Let σ : A + → B + be a morphism, u, v ∈ A + , a, b be the first letters of u, v, respectively, and σ(a) = st be a decomposition of σ(a) in which t is nonempty. Assume that σ(u) is a prefix of sσ(v), |u| ≥ |σ| 1 + |s|, and either that s = 1 and a ̸ = b or that s ̸ = 1.

Then, there exist morphisms q : A + → C + and p :

C + → B + such that #C ≤ #A, q is letter-onto, |p| 1 < |σ| 1 , and σ = pq.
Remark 4. As in Lemma 13, there are symmetric hypothesis for the previous lemma that involve suffixes instead of prefixes and which give the same conclusion. We will use this in the proof of Lemma 15.

Proof (of Lemma 14). Let us write u = au ′ and v = bv ′ . We first consider the case in which s = 1. In this situation, u and v start with different letters, so Lemma 13 can be applied (with the index set J chosen as a singleton) to obtain a decomposition A + q → C + p → B + such that q is letter-onto, #C < #A, and σ = pq. Since C has strictly fewer elements than A, we have |p| 1 < |σ| 1 . Hence, the conclusion of the lemma holds in this case.

We now assume that s ̸ = 1. In this case, t and s are nonempty, so we can use (3) of Lemma 12 to factorize σ = σ ′ θ a,ã , where ã is a letter not in A and σ ′ is defined as in (3.5). We set ũ = aθ a,ã (u ′ ) and ṽ = θ a,ã (v). Our plan is to use Lemma 13 with ũ, ṽ and σ ′ .

Observe that θ a,ã (c) never starts with a, so ũ, ṽ start with different letters.

(3.12) Also, by using, as in the previous proof, the symbol ≤ p to denote the prefix relation, we can write:

sσ ′ (ũ) = sσ ′ (a)σ ′ (θ a,ã (u ′ )) = stσ(u ′ ) = σ(u) ≤ p sσ(v) = sσ ′ (θ a,ã (v)) = sσ ′ (ṽ), which implies that σ ′ (ũ) is a prefix of σ ′ (ṽ). (3.13)
Finally, we use (3.5) to compute:

|ũ| ≥ |u| -1 ≥ |σ| 1 + |s| -1 ≥ |σ| 1 = |σ ′ | 1 . (3.14)
We conclude, by equations (3.12), (3.13) and (3.14), that Lemma 13 can be applied with ũ, ṽ and σ ′ (and J as a singleton). Thus, there exist morphisms q ′ : (A ∪ {ã}) + → C + and p : C + → B + such that #C < #(A ∪ {ã}), q ′ is letter-onto and σ ′ = pq ′ . Then, #C ≤ #A, q := q ′ θ a,ã is letter-onto and σ = pq ′ θ a,ã = pq. Moreover, since θ a,ã is not the identity function, we have

|p| 1 < |σ| 1 .
The next lemma is the main result of this subsection. To state it, we introduce additional notation. For an alphabet A, let A ++ be the set of words w ∈ A + in which all letters occur. Observe that σ :

A + → B + is letter-onto if and only if σ(A ++ ) ⊆ B ++ . Lemma 15. Let ϕ : A + → C + , τ : B + → C + be morphisms such that τ is ℓ- proper, with ℓ ≥ |ϕ| 4 1 , and ϕ(A + ) ∩ τ (B ++ ) ̸ = ∅. Then, there exist B + q -→ D + p -→ C + such that (i) #D ≤ #A, (ii) τ = pq, (
iii) q is letter-onto and proper.

Proof. By contradiction, we suppose that the lemma does not hold for ϕ and τ and, moreover, that |ϕ| 1 as small as possible.

That

ϕ(A) + ∩τ (B ++ ) is nonempty means that there exist u = u 1 • • • u n ∈ A + and w = w 1 • • • w m ∈ B ++ with ϕ(u) = τ (w). If m = 1,
then, since w ∈ B ++ , we have B = {w 1 } and the conclusion of the lemma trivially holds for D = {a ∈ C : a occurs in τ (w 1 )}, q : B + → D + , w 1 → τ (w 1 ), and p : D + → C + the inclusion map, contradicting our initial assumption. Therefore, m ≥ 2 and {1, . . . , m -1} is nonempty.

Let k ∈ {1, . . . , m -1}. We define i k as the smallest number in {1, . . . , n} for which |τ (w

1 • • • w k )| < |ϕ(u 1 • • • u i k )| holds. Since |ϕ(u 1 )| ≤ |ϕ| 1 ≤ ℓ ≤ |τ (w 1 • • • w k )|, i k is at least 2 and, thus, |ϕ(u 1 • • • u i k -1 )| ≤ |τ (w 1 • • • w k )| by minimality of i k .
Hence, there exists a decomposition ϕ(u i k ) = s k t k such that t k is nonempty and

t k ϕ(u i k +1 . . . u n ) = τ (w k+1 . . . w m ).
(3.15)

Our next objective is to use Lemma 14 to prove that s k and u k have a very particular form:

Claim 25.1. For every k ∈ {1, . . . , m -1}, s k = 1 and u 1 = u i k .
Proof. To prove this, we suppose that it is not true, this is, that there exists k ∈ {1, . . . , m -1} such that

s k ̸ = 1 or u 1 ̸ = u i k .
(3.16) Let ũ := u i k . . . u i k +|ϕ| 2 1 -1 and ṽ := u 1 . . . u |ϕ| 3 1 . We are going to check the hypothesis of Lemma 14 for ũ, ṽ and ϕ.

First, we observe that, since ϕ(u) = τ (v), we have that ϕ(ṽ) is a prefix of τ (v). Moreover, given that |ϕ(ṽ)| ≤ |ϕ| 4 1 ≤ ℓ and that τ is ℓ-proper, ϕ(ṽ) is a prefix of τ (b) for every b ∈ B. In particular, ϕ(ṽ) is a prefix of τ (w k ).

(3.17)

Second, from (3.15) and the inequalities

|t k ϕ(u i k +1 . . . u i k +|ϕ| 2 1 -1 )| ≤ |ϕ| 3 1 ≤ ℓ ≤ |τ (w k )| we deduce that t k ϕ(u i k +1 . . . u i k +|ϕ| 2 1 -1 ) is a prefix of τ (w k ). Therefore, ϕ(ũ) = s k t k ϕ(u i k +1 . . . u i k +|ϕ| 2 1 -1 ) is a prefix of s k τ (w k ). (3.18)
We conclude from (3.17), (3.18) and the inequality |ϕ(ũ

)| ≤ |ϕ| 3 1 = |ṽ| ≤ |s k ϕ(ṽ)| that ϕ(ũ) is a prefix of s k ϕ(ṽ).
This, the inequality |ũ| ≥ |ϕ| 1 +|s k | and (3.16) allow us to use Lemma 14 with ũ, ṽ and ϕ and obtain morphisms

A + q -→ Ã+ φ -→ C + such that # Ã ≤ #A, ϕ = φq and | φ| 1 < |ϕ| 1 . Then, ℓ ≥ |ϕ| 4 1 > | φ| 4
1 and φ( Ã+ )∩τ (B ++ ) contains the element φ(q(u)) = τ (w), and so τ and φ satisfy the hypothesis of this lemma. Therefore, by the minimality of |ϕ| 1 , there exists a decomposition B + q → D + p → C + of τ satisfying (i-iii) of this lemma, contrary to our assumptions. We refer the reader to Remark 4 for further details. Now we can finish the proof. First, from (3.15) and the first part of the claim we get that τ

(w k ) = ϕ(u i k-1 • • • u i k -1 ) for k ∈ {2, . . . , m -1}, τ (w 1 ) = ϕ(u 1 • • • u i1-1 ) and τ (w m ) = ϕ(u im-1 • • • u n ).
Being w ∈ B ++ , these equations imply that each τ (b), b ∈ B, can be written as a concatenation x 1 • • • x N , with x j ∈ ϕ(A). Moreover, by the second part of the claim and (3.19), we can choose this decomposition so that x 1 = u 1 and x N = u n . This defines (maybe non-unique) morphisms

B + q -→ D + 1 p1 -→ C + such that τ = p 1 q, #D 1 ≤ #{ϕ(u 1 ), . . . , ϕ(u n )} ≤ #A
and q is proper. If we define D as the set of letters d ∈ D 1 that occur in some w ∈ q(B), and p as the restriction of p 1 to D, then we obtain a decomposition B + q -→ D + p -→ C + that still satisfies the previous properties, but in which q is letter-onto. Hence, p and q met conditions (i), (ii) and (iii).

Periodicity lemmas

We will also need classic results from combinatorics on words. We follow the presentation of [RS97, Chapter 6].

Let w ∈ A * be a nonempty word. We say that p is a local period of w at the position |u| if w = uv, with u, v ̸ = 1, and there exists a word z, with |z| = p, such that one of the following conditions holds for some words u ′ and v ′ :

         (i) u = u ′ z and v = zv ′ ; (ii) z = u ′ u and v = zv ′ ; (iii) u = u ′ z and z = vv ′ ; (iv) z = u ′ u = vv ′ . ( 3

Rank of symbolic factors

In this section we prove Theorem 21. We start by introducing the concept of factor between directive sequences and, in Proposition 3, its relation with factor maps between S-adic subshifts. These ideas are the S-adic analogs of the concept of premorphism between ordered Bratteli diagrams from [START_REF] Amini | The category of Bratteli diagrams[END_REF] and their Proposition 4.6. Although Proposition 3 can be deduced from Proposition 4.6 in [START_REF] Amini | The category of Bratteli diagrams[END_REF] by passing from directive sequences to ordered Bratteli diagrams and backwards, we consider this a little bit artificial since it is possible to provide a direct combinatorial proof; this is done in the Appendix. It is interesting to note that our proof is constructive (in contrast of the existential proof in [START_REF] Amini | The category of Bratteli diagrams[END_REF]) and shows some additional features that are consequence of the combinatorics on words analysis made. Next, we prove Theorem 21. We apply these results, in Corollary 7, to answer affirmatively Question 7 and, in Theorem 5, to prove a strong coalescence property for the class of systems considered in Theorem 21. It is worth noting that this last result is only possible due the bound in Theorem 21 being optimal. We end this section by proving that Cantor factors of finite topological rank systems are either subshifts of odometers.

Rank of factors of directive sequences

The following is the S-adic analog of the notion of premorphism between ordered Bratteli diagrams in [START_REF] Amini | The category of Bratteli diagrams[END_REF].

Definition 7. Let σ = (A + n+1 → A + n ) n∈N , τ = (B + n+1 → B + n ) n∈N be directive sequences. A factor ϕ : σ → τ is a sequence of morphisms ϕ = (ϕ n ) n∈N ,
where ϕ 0 : A + 1 → B + 0 and ϕ n : A + n → B + n for n ≥ 1, such that ϕ 0 = τ 0 ϕ 1 and ϕ n σ n = τ n ϕ n+1 and for every n ≥ 1.

We say that ϕ is proper (resp. letter-onto) if ϕ n is proper (resp. letter-onto) for every n ∈ N.

Remark 5. Factors are not affected by contractions. More precisely, if

0 = n 0 < n 1 < n 2 < . . . , then ϕ ′ = (ϕ n k ) k∈N is a factor from σ ′ = (σ [n k ,n k+1 ) ) k∈N to τ ′ = (τ [n k ,n k+1 ) ) k∈N .
The next lemma will be needed at the end of this section.

Lemma 16. Let ϕ = (ϕ n ) n≥1 : σ → τ be a factor. Assume that σ and τ are everywhere growing and proper and that ϕ is letter-onto. Then,

X τ = k∈Z S k ϕ 0 (X (1) σ ) and X (n) τ = k∈Z S k ϕ n (X (n) σ ) for every n ≥ 1.
Proof. We start by proving that

X (n) τ ⊆ k∈Z S k ϕ n (X (n) σ ). Let y ∈ X (n) τ and ℓ ∈ N. There exist N > n and b ∈ B n such that y [-ℓ,ℓ] occurs in τ [n,N ) (b). In addition, since ϕ N is letter-onto, there exists a ∈ A N for which b occurs in ϕ N (a). Then, y [-ℓ,ℓ] occurs in τ [n,N ) ϕ N (b) and, consequently, also in ϕ n σ [n,N ) (b) as τ [n,N ) ϕ N = ϕ n σ [n,N )
. Hence, by taking the limit ℓ → ∞ we can find (k

′ , x) ∈ Z × X (n) σ such that y = S k ′ ϕ n (x). Therefore, y ∈ k∈Z S k ϕ n (X (n) σ ).
To prove the other inclusion, we use Lemma 11 to compute:

ϕ n (X (n) σ ) = N >n k∈Z S k ϕ n σ [n,N ) (A Z N ) = N >n k∈Z S k τ [n,N ) ϕ N (A Z N ) ⊆ N >n k∈Z S k τ [n,N ) (B Z N ) = X (n) τ .
As we mentioned before, the following proposition is consequence of the main result in [START_REF] Amini | The category of Bratteli diagrams[END_REF]. We provide a combinatorial proof in the Appendix.

Proposition 3. Let σ be a letter-onto, everywhere growing and proper directive sequence. Suppose that X σ is aperiodic. Then, there exist a contraction σ ′ = (σ ′ n ) n∈N , a letter-onto, everywhere growing, proper and recognizable τ = (τ n ) n∈N generating X σ , and a letter-onto factor ϕ :

σ ′ → τ , ϕ = (ϕ n ) n∈N , such that ϕ 0 = σ ′ 0 .
The next proposition is the main technical result of this section. To state it, it is convenient to introduce the following concept. The directive sequences σ and τ are equivalent if σ = ν ′ , τ = ν ′′ for some contractions ν ′ , ν ′′ of a directive sequence ν. Observe that equivalent directive sequences generate the same S-adic subshift.

Proposition 4. Let ϕ : σ → τ be a letter-onto factor between the everywhere growing and proper directive sequences. Then, there exist a letter-onto and proper factor ψ : σ ′ → ν, where (1) σ ′ is a contraction of σ;

(2) ν is letter-onto, everywhere growing, proper, equivalent to τ , AR(ν) ≤ AR(σ), and the first coordinate of ψ and ϕ coincide;

(3) if τ is recognizable, then ν is recognizable.

Proof. Let us write σ = (A

+ n+1 → A + n ) n∈N and τ = (B + n+1 → B + n ) n∈N .
Up to contractions, we can suppose that for every n ≥ 1, #A n = AR(σ) and that τ n is |ϕ n | 4 1 -proper (for the last property we used that τ is everywhere growing and proper).

Using that ϕ n+1 is letter-onto, we can compute:

τ n (B ++ n+1 ) ⊇ τ n (ϕ n+1 (A ++ n+1 )) = ϕ n (σ n (A ++ n+1 )) ⊆ ϕ n (A + n ),
where in the middle step we used the commutativity property of ϕ. We deduce that n such that (i) #D n+1 ≤ #A n , (ii) τ n = p n q n+1 , (iii) q n+1 is letter-onto and proper.

τ n (B ++ n+1 ) ∩ ϕ n (A + n ) ̸ = ∅
We define ν 0 := p 0 , the morphisms ν n := q n p n : D + n+1 → D + n and ψ n := q n ϕ n : A + n → D + n , n ≥ 1, and the sequences ν = (ν n ) n∈N and ψ = (ψ n ) n∈N , where ψ 0 := ϕ 0 . We are going to show that these objects satisfy the conclusion of the proposition.

We start by observing that it follows from the definitions that the diagram below commutes for all n ≥ 1: In particular, ν n ν n+1 = q n τ n p n+1 , so ⟨ν [n,n+1] ⟩ ≥ ⟨τ n ⟩. Being τ everywhere growing, this implies that ν has the same property. We also observe that (iii) implies that ν n = q n p n is letter-onto and proper. Altogether, these arguments prove that, up to contracting the first levels, ν is everywhere growing and proper. Next, we note that ν and τ are equivalent as both are contractions of (p 0 , q 1 , p 1 , q 2 , . . . ). This implies, by Lemma 1, that ν is recognizable if τ is recognizable. Further, by (i), ν has alphabet rank at most AR(σ).

A + n B + n D + n A + n+1 B + n+1 D +
It is only left to prove that ψ is a letter-onto and proper factor. By unraveling the definitions we can compute:

ψ 0 = ϕ 0 = τ 0 ϕ 1 = p 0 q 1 ϕ 1 = ν 0 ψ 1 ,
and from the diagram we have σ n ψ n = ψ n+1 τ n for all n ≥ 1. Therefore, ψ is a factor. Finally, since q n is letter-onto and proper by (iii) and ϕ was assumed to be letter-onto, ψ n = q n ϕ n is letter-onto and proper.

Rank of factors of S-adic subshifts

In this section, we will prove Theorem 21 and its consequences. We start with a technical lemma.

The next lemma will allow us to assume without loss of generality that our directive sequences are letter-onto.

Lemma 17. Let τ = (τ n : A + n+1 → A + n ) n∈N be an everywhere growing and proper directive sequence. If

Ãn = A n ∩ L(X (n)
σ ), τn is the restriction of τ n to Ãn+1 and τ = (τ 0 , τ1 , . . . ), then τ is letter-onto and

X (n) τ = X (n) τ for every n ∈ N. Conversely, if τ is letter-onto, then A n ⊆ L(X (n) τ ) for every n ∈ N.
Proof. By Lemma 2, τn is letter-onto mapping Ã+ n+1 into Ãn . Moreover, that lemma also gives that for every x ∈ X

S k τ [n,N ) ( ÃZ N ) ⊇ X (n) τ
Now, τ is everywhere growing and proper, so we can apply Lemma 11 to obtain that X

(n) τ = Z ⊇ X (n) τ . Since it is clear that X (n) τ ⊆ X (n) τ as ÃN ⊆ A N for every N , we conclude that X (n) τ = X (n) τ . If τ is letter-onto, then A n ⊆ L( k∈Z S k τ [n,N ) (A Z N )
) for every N > n, and hence, by the formula in Lemma 11, A n ⊆ L(X

(n) τ ).
Now we are ready to prove Theorem 21. We re-state it in a more precise way.

Theorem 21. Let π : (X, S) → (Y, S) be a factor map between aperiodic subshifts. Suppose that X is generated by the everywhere growing and proper directive sequence σ = (σ n :

A + n+1 → A + n ) n∈N of alphabet rank K.
Then, Y is generated by a letter-onto, everywhere growing, proper and recognizable directive sequence τ of alphabet rank at most K.

Moreover, if σ is letter-onto, then, up to contracting the sequences, there exists a proper factor ϕ : σ → τ such that π(σ 0 (x)) = ϕ 0 (x) for all x ∈ X Proof. Thanks to Lemma 17, we can assume without loss of generality that σ is letter-onto. Moreover, in this case we have:

A n ⊆ L(X (n)
σ ) for every n ∈ N.

(3.21)

Let us write σ = (σ n :

A + n+1 → A + n ) n∈N .
By contracting σ, we can further assume that σ 0 is r-proper and π has radius r. Then, Lemma 10 gives us a proper morphism τ : In particular, π(σ

A + 1 → B + ,
[0,n) (x)) = τ σ [1,n) (x) and |σ [0,n) (a)| = |τ σ [1,n) (a)| for all n ∈ N, x ∈ X (n)
σ and a ∈ A n , so (3.22) holds for any contraction of σ. We define σ = (τ, σ 1 , σ 2 , . . . ) and observe this is a letter-onto, everywhere growing and proper sequence generating Y . This and that Y is aperiodic allow us to use Proposition 3 and obtain, after a contraction, a letter-onto factor φ : σ → τ , where φ0 = σ0 = τ and τ is a letter-onto, everywhere growing, proper and recognizable directive sequence generating Y . The sequence τ has all the properties required by the theorem but having alphabet rank bounded by K. To overcome this, we use Proposition 4 with φ and do more contractions to obtain a letter-onto and proper factor ϕ : σ → τ such that ϕ 0 = φ0 = τ and τ is a letter-onto, everywhere growing, proper and recognizable directive sequence generating Y and satisfying AR(τ ) ≤ AR( σ) = AR(σ).

It is left to prove the last part of the theorem. Observe that since σ and σ differ only at their first coordinate, ϕ is also a factor from σ to τ . Corollary 6. Let (X, S) be an aperiodic minimal subshift of generated by an everywhere growing and proper directive sequence of alphabet rank K. Then, the topological rank of X is at most K.

Proof. We can use Theorem 21 to obtain an everywhere growing, proper and recognizable directive sequence τ = (τ n : B + n+1 → B + n ) n∈N generating X and having of alphabet rank at most K. Due to Lemma 17, we can assume that τ is letter-onto. In particular, B n ⊆ L(X

(n) τ ) for every n ∈ N. We claim that X (n) τ is minimal. Indeed, if Y ⊆ X (n) τ is a subshift, then τ [0,n) (Y ) is closed (as τ [0,n) : X (n) τ → X τ is continuous), so k∈Z S k τ [0,n) (Y ) = |k|≤|τ [0,n) | S k τ [0,n) (Y ) is a subshift in X τ which, by minimality, is equal to it. Thus, any point x ∈ X (n) τ has a τ [0,n) -factorization (k, z) with z ∈ Y . The recognizability property of (X (n) τ , τ [0,n) ) then implies that Y = X (n)
τ . Now, we prove that for any n ∈ N there exists N > n such that τ [n,N ) is positive. This would imply that the topological rank of X is at most K and hence would complete the proof. Let n ∈ N and R be a constant of recognizability for (X

(n) τ , τ [0,n) ). Since X (n) τ
is minimal, there exists a constant L ≥ 1 such that two consecutive occurrences of a word w ∈ L(X

(n) τ ) ∩ B 2R+1 n in a point x ∈ X (n) τ
are separated by at most L. Let N > n be big enough so that ⟨τ [0,N ) ⟩ ≥ L + 2R. Then, for all a ∈ B N ⊆ L(X We can now prove Corollary 1.

(N ) τ ) and w ∈ L(X (n) τ ) ∩ B 2R+1 n , w occurs at a position i ∈ {R, R + 1, . . . , |τ [0,N ) (a)| -R} of τ [0,N ) (a). Since R is a recognizability constant for (X (n) τ , τ [0,n) ),
Corollary 1. Let (X, S) be an aperiodic minimal subshift generated by an everywhere growing directive sequence of finite alphabet rank. Then, the topological rank of (X, S) is finite.

Proof. We are going to prove that X is generated by an everywhere growing and proper directive sequence τ of finite alphabet rank. This would imply, by Corollary 6, that the topological rank of X is finite. Let σ = (σ n : A + n+1 → A + n ) n∈N be an everywhere growing directive sequence of finite alphabet rank generating X. We contract τ in a way such that #A n ≤ K for every n ≥ 1.

We are going to inductively define subshifts X n , n ∈ N. We start with X 0 := X. We now assume that X n is defined for some n ∈ N. Then the set

X ′ n+1 = {x ∈ X (n+1) σ : σ n (x) ∈ X n } is a subshift. We define X n+1 as any minimal subshift contained in X ′ n+1 . It follows from the definition of X n+1 that k∈Z S k σ n (X n+1 ) ⊆ X n . Being X n minimal, we have k∈Z S k σ n (X n+1 ) = X n . ( 3
w |w| ∈ B + 0 , then η(w ′ ) := w 1 . . . w |w|-1 ∈ A + 0 . Observe that η : B + 0 → A + 0 is a morphism. We now define τ . Let τ n : B + n+1 → B + n be the unique morphism such that τ n ( a b ) = χ n (v n (a)u n (a)b n ) for every a b ∈ B n+1 . Observe that since v n (a)u n (a)b n ∈ L(X n ), it is indeed the case that τ n ( a b ) ∈ B + n .
We set τ = (ητ 0 , τ 1 , τ 2 , . . . ). )| ≥ 2. Therefore, ⟨τ n ⟩ ≥ 2 and τ is everywhere growing. Also, #B n ≤ #A 2 n ≤ K 2 for every n ∈ N, so the alphabet rank of τ is finite.

It remains to prove that X = X τ . By minimality, it is enough to prove that X ⊇ X τ . Observe that since τ n χ n+1 (ab) = χ n (v n (a)u n (b)b n ), the word τ n χ n+1 (ab) occurs in χ n σ n (ab). Moreover, for every w = w 1 . . . w |w| ∈ L(X (n) σ ), τ n χ n+1 (w) occurs in χ n σ n (w). Then, by using the symbol ⊑ to denote the "subword" relation, we can write for every n ∈ N and ab ∈ L(X

(n) σ ): τ [0,n) χ n (ab) ⊑ τ [0,n-1) χ n-1 σ n-1 (ab) ⊑ τ [0,n-2) χ n-2 σ [n-2,n) (ab) ⊑ • • • ⊑ χ 0 σ [0,n) (ab) Hence, ητ [0,n) ( a b ) ⊑ ηχ 0 σ [0,n) (ab) ⊑ σ [0,n) (ab). We conclude that X τ ⊆ X σ = X.
Corollary 7. Let (X, S) be a minimal subshift of topological rank K and π : (X, S) → (Y, S) a factor map, where Y is an aperiodic subshift. Then, the topological rank of Y is at most K.

Proof. By Theorem 1, (X, S) is generated by a proper and primitive directive sequence σ of alphabet rank equal to K. In particular, σ is everywhere growing and proper, so we can use Theorem 21 to obtain an everywhere growing, proper and recognizable directive sequence τ = (τ n : B + n+1 → B + n ) n≥0 generating (Y, S) and having of alphabet rank at most K. Then, the hypothesis of Corollary 6 hold for (Y, S), and thus the topological rank of (Y, S) is at most K.

The following notion will be used in the proof of the theorem below: σ = (σ n :

A + n+1 → A n ) n≥0 has exact alphabet rank at most K if #A n ≤ K for all n ≥ 1.
Corollary 5. Let (X, S) be an S-adic subshift generated by an everywhere growing and proper sequence of alphabet rank K, and π j : (X j+1 , S) → (X j , S), j = 0, . . . , L, be a chain of aperiodic symbolic factors, with X L = X. Suppose that L > log 2 (K). Then π j is a conjugacy for some j.

Proof. We start by using Theorem 21 with the identity function id : (X, S) → (X, S) to obtain a letter-onto, everywhere growing, proper and recognizable directive sequence σ L of alphabet rank at most K generating X. By doing a contraction, we can assume that σ L has exact alphabet rank at most K.

By Theorem 21 applied to π L-1 and σ L , there exists, after a contraction of σ L , a letter-onto factor ϕ L-1 : σ L → σ L-1 , where σ L-1 is letter-onto, everywhere growing, proper, recognizable, has alphabet rank at most K, generates X L-1 , and, if ϕ L-1,0 and σ L,0 are the first coordinates of ϕ L-1 and σ L , respectively, then π L-1 (σ L,0 (x)) = ϕ L-1,0 (x) for every x ∈ X

(1) σL and |σ L,0 (a)| = |ϕ L-1,0 (a)| for every letter a in the domain of σ L,0 . By contracting these sequences, we can also suppose that σ L-1 has exact alphabet rank at most K. The same procedure applies to π L-2 and σ L-1 . Thus, by continuing in this way we obtain for every j = 0, . . . , L -1 a letter-onto factor ϕ j : σ j+1 → σ j such that • σ j is letter-onto, everywhere growing, proper, recognizable, has exact alphabet rank at most K, generates X j , π j (σ j+1,0 (x)) = ϕ j,0 (x) for every x ∈ X

(1) Here, we are using the notation σ j = (σ j,n :

σ j+1 ,
A + j,n+1 → A + j,n ) n∈N , ϕ j = (ϕ j,n : A + j+1,n → A + j,n ) n∈N and X (n) j = X (n)
σj . We note that (△ 1 ) for every x ∈ X

(1) j+1 , π j (σ j+1,0 (x)) = ϕ j,0 (x) = σ j,0 ϕ j,1 (x) since ϕ j,0 = σ j,0 ϕ j,1 ;

(△ 2 ) X (1) j = k∈Z S k ϕ j,1 (X (1)
j+1 ) by Lemma 16. Hence, the following diagram commutes:

X (1) 0 • • • X (1) j X (1) j+1 • • • X (1) L X (0) 0 • • • X (0) j X (0) j+1 • • • X (0) L σ0,0 σj,0 ϕ0,1 σj+1,0 ϕj,1 σ L,0 ϕ L-1,1 π0 πj π L-1 Claim 26.1. If (X (1)
j+1 , ϕ j,1 ) is recognizable, then π j is a conjugacy.

Proof. Let us assume that (X

j+1 , ϕ j,1 ) is recognizable and let, for i = 0, 1,

x i ∈ X (1)
j+1 such that y = π j (x 0 ) = π j (x 1 ). We have to show that x 0 = x 1 . First, we use Lemma 2 to find a centered σ j+1,0 -factorization (k i , z i ) of x i in X

(1) j+1 . Then, equation △ 1 allows us to compute:

S k 0 σ j,0 ϕ j,1 (z 0 ) = S k 0 π j (σ j+1,0 (z 0 )) = π j (x 0 ) = π j (x 1 ) = S k 1 σ j,0 ϕ j,1 (z 1 ). This implies that (k i , z i ) is a σ j,0 ϕ j,1 -factorization of y in X (1)
j+1 for i = 0, 1. Moreover, these are centered factorizations as, by •, |σ j,0 ϕ j,1 (a)| = |σ j+1,0 (a)| for all a ∈ A j+1,1 . Now, being (X (1) j , σ 0,j ) and (X (1) j+1 , ϕ j,1 ) recognizable, Lemma 1 gives that (X (1) j+1 , σ j,1 ϕ j,1 ) is recognizable, and thus we have that (k 0 , z 0 ) = (k 1 , z 1 ). Therefore, x 0 = x 1 and π is a conjugacy. □ Now we can finish the proof. We assume, by contradiction, that π j is not a conjugacy for all j. Then, by the claim, (X

(1) j , ϕ 1,j ) is not recognizable for every j ∈ {0, . . . , L -1}.

(3.26)

Let ν = (ϕ 0,1 , ϕ 1,1 , ϕ 2,1 , . . . , ϕ L-1,1 , σ L,1 , σ L,2 , σ L,3 , . . . ).
The idea is to use Theorem 24 with ν to obtain a contradiction. To do so, we first note that, since ν and σ (L) have the same "tail",

X (m+L) ν = X (m+1) L
for all m ∈ N. Moreover, △ 2 and the previous relation imply that

X (j) ν = k∈Z S k ϕ j,1 (X (j+1) ν ) = • • • = k∈Z S k ϕ j,1 . . . ϕ L-1,1 (X (L) ν ) = k∈Z S k ϕ j,1 . . . ϕ L-1,1 (X (1) L ) = k∈Z S k ϕ j,1 . . . ϕ L-2,1 (X (1)
L-1 ) = • • • = X (1) j .
This and (3.26) imply that for every j ∈ {1, . . . , L -1}, the level (X

(j)
ν , ϕ j,1 ) of ν is not recognizable. Being ν everywhere growing as σ L has this property, we conclude that Theorem 24 can be applied and, therefore, that X

(1) 0

= X ν is periodic. But then X 0 = k∈Z S k σ 0,0 (X (1)
0 ) is periodic, contrary to our assumptions.

A system (X, S) is coalescent if every endomorphism π : (X, S) → (X, S) is an automorphism. This notion has been relevant in the context of topological dynamics; see for example [START_REF] Downarowicz | The royal couple conceals their mutual relationship: A noncoalescent toeplitz flow[END_REF].

Corollary 8. Let (X, S) be an S-adic subshift generated by an everywhere growing and proper directive sequence of finite alphabet rank. Then, (X, S) is coalescent. Remark 6. A linearly recurrent subshift of constant C is generated by a primitive and proper directive sequence of alphabet rank at most C(C + 1) 2 ([Dur00], Proposition 6). In [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], the authors proved the following Theorem 27 ([DHS99], Theorem 3). For a linearly recurrent subshift X of constant C, in any chain of factors π j : (X j , S) → (X j+1 , S), j = 0, . . . , L, with X 0 = X and L ≥ (2C(2C + 1) 2 ) 4C 3 (2C+1) 2 there is at least one π j which is a conjugacy.

Thus, Theorem 5 is not only a generalization of this result to a much larger class of systems, but also improves the previous super-exponential constant to a logarithmic one.

In Proposition 28 of [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF], the authors proved that Cantor factors of linearly recurrent systems are either subshifts or odometers. Their proof only uses that this kind of systems satisfy the strong coalescence property that we proved in Corollary 8 for finite topological rank systems. Therefore, by the same proof, we have: Corollary 9. Let π : (X, S) → (Y, T ) be a factor map between minimal systems. Assume that (X, S) has finite topological rank and that (Y, T ) is a Cantor system. Then, (Y, T ) is either a subshift or an odometer.

Proof. We sketch the proof from [DHS99] that we mentioned above.

Let (P n ) n∈N be a sequence of clopen partitions of Y such that P n+1 is finer than P n and their union generates the topology of Y . Also, let Y n be the subshift obtained by codifying the orbits of (Y, T ) by using the atoms of P n . Then, the fact that P n is a clopen partition induces a factor map π n : (Y, T ) → (Y n , S). Moreover, since P n+1 is finer than P n , there exists a factor map ξ n : (Y n+1 , S) → (Y n , S) such that ξ n π n+1 = π n . Hence, we have the following chain of factors:

(X, S) π -→ (Y, T ) πn -→ (Y n , S) ξn-1 -→ (Y n-1 , S) ξn-2 -→ . . . ξ1 -→ (Y 0 , S).
We conclude, by also using the fact that the partitions P n generate the topology of Y , that (Y, S) is conjugate to the inverse limit ←lim n→∞ (Y n ; ξ n ). Now we consider two cases. If Y n is periodic for every n ∈ N, then Y is the inverse limit of periodic system, and hence an odometer. In the other case, we have, by Corollary 5, that ξ n is a conjugacy for all big enough n ∈ N, and thus that (Y, S) is conjugate to one of the subshifts Y n .

Fibers of symbolic factors

The objective of this section is to prove Theorem 22, which states that factor maps π : (X, S) → (Y, S) between S-adic subshifts of finite topological rank are always almost k-to-1 for some k bounded by the topological rank of X. We start with some lemmas from topological dynamics.

Lemma 18 ([Aus88]

). Let π : X → Y be a continuous map between compact metric spaces. Then π -1 : Y → 2 X is continuous at every point of a residual subset of Y .

Next lemma gives a sufficient condition for a factor map π to be almost k-to-1. Recall that E(X, S) stands for the Ellis semigroup of (X, S).

Lemma 19. Let π : (X, S) → (Y, T ) be a factor map between topological dynamical systems, with (Y, T ) minimal, and K ≥ 1 an integer. Suppose that for every y ∈ Y there exists u ∈ E(2 X , S) such that #u • π -1 (y) ≤ K. Then, π is almost k-to-1 for some k ≤ K.

Proof. First, we observe that by the description of u • A in terms of nets at the end of Subsection 3.2, we have

#u • A ≤ #A, ∀u ∈ E(2 X , S), A ∈ 2 X .
(3.27) Now, by previous lemma, there exists a residual set Ỹ ⊆ Y of continuity points for π -1 . Let y, y ′ ∈ Ỹ be arbitrary. Since Y is minimal, there exists a sequence (n ℓ ) ℓ such that lim ℓ T n ℓ y = y ′ . If w ∈ E(2 X , S) is the limit of a convergent subnet of (S n ℓ ) ℓ , then wy = y ′ . By the continuity of π -1 at y ′ and (3.27), we have #π -1 (y ′ ) = #π -1 (wy) = #w • π -1 (y) ≤ #π -1 (y).

We deduce, by symmetry, that #π -1 (y ′ ) = #π -1 (y). Hence, k := π -1 (y) does not depend on the chosen y ∈ Ỹ . To end the proof, we have to show that k ≤ K. We fix y ∈ Ỹ and take, using the hypothesis, u ∈ E(2 X , S) such that #u • π -1 (y) ≤ K. As above, by minimality, there exists v ∈ E(2 X , S) such that vuy = y. Then, by the continuity of π -1 at y,

π -1 (y) = π -1 (vuy) = (vu) • π -1 (y) = v • (u • π -1 (y)).
This and (3.27

) imply that k = #π -1 (y) ≤ #u • π -1 (y) ≤ K.
Let σ : A + → B + be a morphism, (k, x) a centered σ-factorization of y ∈ B Z in A Z and ℓ ∈ Z. Note that there exists a unique j ∈ Z such that ℓ ∈ [c σ,j (k, x), c σ,j+1 (k, x)) (recall the notion of cut from Definition 9). In this context, we say that (c σ,j (k, x), x j ) is the symbol of (k, x) covering position ℓ of y.

Theorem 22. Let π : (X, S) → (Y, S) be a factor between subshifts, with (Y, S) minimal and aperiodic. Suppose that X is generated by a proper and everywhere growing directive sequence σ of alphabet rank K. Then, π is almost k-to-1 for some k ≤ K.

Proof. Let σ = (σ n : A n+1 → A n ) n≥0 be a proper and everywhere growing directive sequence of alphabet rank at most K generating X. Due the possibility of contracting σ, we can assume without loss of generality that #A n ≤ K for every n ≥ 1 and that σ 0 is r-proper, where r is the radius of π. Then, by Lemma 10, Y is generated by an everywhere growing directive sequence of the form τ = (τ, σ 1 , σ 2 , . . . ), where τ :

A + 1 → B + is such that τ (x) = π(σ 0 (x)) for every x ∈ X (1) τ = X (1)
σ . We will use the notation τ [0,n) = τ σ [1,n) . Further, for y ∈ Y and n ≥ 1, we write F n (y) to denote the set of τ

[0,n) -factorizations of y in Y (n) τ .
Before continuing, we prove the following claim.

Claim 27.1. There exist ℓ n ∈ Z and G n ⊆ Z × B n+1 with at most K elements such that if (k, x) ∈ F n (y), then the symbol of (k, x) covering position ℓ n of y is in G n .

First, since Y is aperiodic, there exists L ∈ N such that all words w ∈ L(Y ) of length ≥ L have least period greater than |τ [0,n) |.

(3.28) We assume, by contradiction, that the claim does not hold. In particular, for every ℓ ∈ [0, L) there exist

K + 1 τ [0,n) -factorizations (x, k) of y in Y (n) τ
such that their symbols covering position ℓ of y are all different. Now, since #τ [0,n) (A n+1 ) ≤ K, we can use the Pigeonhole Principle to find two of such factorizations, say (k, x) and (k ′ , x ′ ), such that if (c, a) and (c ′ , a ′ ) are their symbols covering position ℓ of y then a = a ′ and c < c ′ . Then,

y (c,c+|τ [0,n) (a)|] = τ [0,n) (a) = y (c ′ ,c ′ +|τ [0,n) (a)|] and, thus, y (c,c ′ +|τ [0,n) (a)|] is (c ′ -c)-periodic. Being ℓ ∈ (c ′ , c + |τ [0,n) (a)|), we deduce that the local period of y [0,L) at ℓ is at most c ′ -c ≤ |τ [0,n) |.
Since this true for every ℓ ∈ [0, L) and since, by Theorem 26, per(y [0,L) ) = per(y [0,L) , y [0,ℓ) ) for some ℓ ∈ [0, L), we conclude that per(y [0,L) ) ≤ |τ [0,n) |. This contradicts (3.28) and proves thereby the claim. Now we prove the theorem. It is enough to show that the hypothesis of Lemma 19 hold. Let y ∈ Y and Fn (y) ⊆ F n (y) be such that # Fn (y) = #G n and the set consisting of all the symbols of factorizations (k, x) ∈ Fn (y) covering position ℓ n of y is equal to

G n . Let z ∈ π -1 (y) and (k, x) be a σ [0,n) -factorization of z in X (n) σ . Then, S k τ [0,n) (x) = S k π(σ [0,n) (x)) = π(z) = y and (k, x) is a τ [0,n) -factorization of y in Y (n)
τ . Thus, we can find (k ′ , x ′ ) ∈ Fn (y) such that the symbols of (k, x) and (k ′ , x ′ ) covering position ℓ n of y are the same; let (m, a) be this common symbol. Since σ is proper, we have

z [m-⟨σ [0,n-1) ⟩,m+|σ [0,n) (a)|+⟨σ [0,n-1) ⟩] = z ′ [m-⟨σ [0,n-1) ⟩,m+|σ [0,n) (a)|+⟨σ [0,n-1) ⟩] ,
where

z ′ = S k ′ σ [0,n) (x ′ ) ∈ X is the point that (k ′ , x ′ ) factorizes in (X (n) σ , σ [0,n) ). Then, as ℓ n ∈ (m, m + |σ [0,n) (a)|], z (ℓn-⟨σ [0,n-1) ⟩,ℓn+⟨σ [0,n-1) ⟩] = z ′ (ℓn-⟨σ [0,n-1) ⟩,ℓn+⟨σ [0,n-1) ⟩] .
Thus, dist(S ℓn z, S ℓn P n (y)) ≤ exp(-⟨σ [0,n-1) ⟩), where P n (y) ⊆ π -1 (y) is the set of all points S k ′′ σ [0,n) (x ′′ ) ∈ X such that (k ′′ , x ′′ ) ∈ Fn (y). Since this holds for every n ≥ 1, we obtain that d H (S ℓn π -1 (y), S ℓn P n (y)) converges to zero as n goes to infinity (where, we recall, d H is the Hausdorff distance). By taking an appropriate convergent subnet u ∈ E(2 X , S) of (S ℓn ) n∈N we obtain #u • π -1 (y) ≤ sup n∈N #P n = sup n∈N #G n ≤ K. This proves that the hypothesis of Lemma 19 holds. Therefore, π is almost k-to-1 for some k ≤ K.

Number of symbolic factors

In this section we prove Theorem 23. In order to do this, we split the proof into 3 subsections. First, in Lemma 22 of subsection 3.6.1, we deal with the case of Theorem 23 in which the factor maps are distal. Next, we show in Lemma 26 from Subsection 3.6.2 that in certain technical situation -which will arise when we consider non-distal factor maps-it is possible to reduce the problem to a similar one, but where the alphabet are smaller. Then, we prove Theorem 23 in subsection 3.6.3 by a repeated application of the previous lemmas.

Distal factor maps

We start with some definitions. If (X, S) is a system, then we always give X k the diagonal action

S [k] := S × • • • × S. If π : (X, S) → (Y, T ) is a factor map and k ≥ 1, then we define R k π = {(x 1 , . . . , x k ) ∈ X k : π(x 1 ) = • • • = π(x k )}. Observe that R k π is a closed S [k] -invariant subset of X k .
Next lemma follows from classical ideas from topological dynamics. See, for example, Theorem 6 in Chapter 10 of [START_REF] Auslander | Minimal Flows and Their Extensions[END_REF].

Lemma 20. Let π : (X, S) → (Y, T ) be a distal almost k-to-1 factor between minimal systems, z = (z 1 , . . . , z k ) ∈ R k π and Z = orb S [k] (z). Then, π is k-to-1 and Z is minimal

We will also need the following lemma:

Lemma 21 ([Dur00], Lemma 21). Let π i : (X, S) → (Y i , T i ), i = 0, 1, be two factors between aperiodic minimal systems. Suppose that π 0 is finite-to-one. If x, y ∈ X are such that π 0 (x) = π 0 (y) and π 1 (x) = T p 1 π 1 (y), then p = 0.

Lemma 22. Let (X, S) be an infinite minimal subshift of topological rank K and J an index set of cardinality #J > K(144K 7 ) K . Suppose that for every j ∈ J there exists a distal symbolic factor π j : (X, S) → (Y j , S). Then, there are i ̸ = j ∈ J such that (Y i , S) is conjugate to (Y j , S).

Proof. We start by introducing the necessary objects for the proof and doing some general observations about them. First, thanks to Theorem 22, we know that π j is almost k j -to-1 for some k j ≤ K, so, by the Pigeonhole Principle, there exist J 1 ⊆ J and k ≤ K such that #J 1 ≥ #J/K > (144K 7 ) K and k j = k for every j ∈ J 1 . For j ∈ J 1 , we fix z j = (z j 1 , . . . , z j k ) ∈ R k πj with z j n ̸ = z j m for all n ̸ = m. Let Z j = orb S [k] (z j ) and ρ : X k → X be the factor map that projects onto the first coordinate. By Lemma 20, π j is k-to-1 and Z j minimal. This imply that if x = (x 1 , . . . , x k ) ∈ Z j , then {x 1 , . . . , x k } = π -1 j (π j (x n )) for all n ∈ {1, . . . , k}, (3.29)

x n ̸ = x m for all n, m ∈ {1, . . . , k}.

(3.30) Indeed, since Z j is minimal, (S [k] ) n ℓ z → x for some sequence (n ℓ ) ℓ , so,

inf n̸ =m dist(x n , x m ) ≥ inf n̸ =m,l∈Z dist(S l z n , S l z m ) > 0,
where in the last step is due the fact that π j is distal. This gives (3.30). For (3.29) we first note that {x 1 , . . . , x k } ⊆ π -1 j (π j (x n )) as x ∈ R πj , and then that the equality must hold since #π -1 j (π j (x n )) = k = #{x 1 , . . . , x k } by (3.30). The next step is to prove that asymptotic pairs in Z j are well-behaved: Claim 27.2. Let j ∈ J 1 and (x j = (x j 1 , . . . , x j k ), xj = (x j 1 , . . . , xj k )) be a right asymptotic pair in Z j , this is,

lim n→-∞ dist((S [k] ) n x j , S [k] xj ) = 0 and x j ̸ = xj .
(3.31)

Then, (x j n , xj n ) is right asymptotic for every n ∈ {1, . . . , k}.

Proof. Suppose, with the aim to obtain a contradiction, that (x j n , xj n ) is not right asymptotic for some n ∈ {1, . . . , k}. Observe that (3.31) implies that for every m ∈ {1, . . . , k}, either (x j m , xj m ) is right asymptotic or x j n = xj n .

(3.32) Therefore, x j n = xj n . Using this and that x j , xj ∈ R k πj we can compute:

π j (x j m ) = π j (x j n ) = π j (x j n ) = π j (x j l
) for all m, l ∈ {1, . . . , k}, and thus, by (3.29),

{x j 1 , . . . , x j k } = π -1 j (π j (x j n )) = π -1 j (π j (x j n )) = {x j 1 , . . . , xj k }.
The last equation, (3.30) and that x j ̸ = xj imply that there exist m ̸ = l ∈ {1, . . . , k} such that xj l = x j m . This last equality and (3.32) tell us that x j m and x j l are either asymptotic or equal. But in both cases a contradiction occurs: in the first one with the distality of π and in the second one with equation (3.30). □ Let j ∈ J 1 . Since Y j is infinite, Z j is a infinite subshift. It is a well-known fact from symbolic dynamics that this implies that there exists a right asymptotic pair (x j = (x j 1 , . . . , x j k ), xj = (x j 1 , . . . , xj k )) in Z j . We are now going to use Theorem 25 to prove the following:

Claim 27.3. There exists i, j ∈ J 1 , i ̸ = j, such that Z i = Z j .
Proof. On one hand, by the previous claim, (x j n , xj n ) ∈ X 2 is right asymptotic for every n ∈ {1, . . . , k} and j ∈ J 1 . Let p j n ∈ Z be such that (S p j n x j n , S p j n xj n ) is centered right asymptotic. On the other hand, Theorem 25 asserts that the set {x (0,∞) : (x, x) is centered right asymptotic in X} has at most 144K 7 elements. Since #J 1 > (144K 7 ) K , we conclude, by the Pigeonhole principle, that there exist i, j ∈ J 1 , i ̸ = j, such that S p i n x i n and S p j n x j n agree on (0, ∞) for every n ∈ {1, . . . , k}.

(3.33)

We are going to show that Z i = Z j . Using (3.33), we can find u ∈ E(X, S) such that uS p i n x i n = uS p j n x j n for every n. Then, by putting y i n = ux i n , y j n = ux j n and q n = p j n -p i n , we have

y i := (y i 1 , . . . , y i k ) ∈ Z i , y j := (y j 1 , . . . , y j k ) ∈ Z j and y i n = S qn y j n .
Hence, π(y i n ) = S qn π(y j n ) and Lemma 21 can be applied to deduce that q := q n has the same value for every n. We conclude that y i = S q y j ∈ S q Z j = Z j , that Z i ∩ Z j is not empty and, therefore, that Z i = Z j as these are minimal systems.

□

We can now finish the proof. Let i ̸ = j ∈ J 1 be the elements given by the previous claim, so that Z := Z i = Z j . Let y ∈ Y i and x = (x 1 , . . . , x k ) ∈ ρ -1 π -1 i (y) ∩ Z. Then, by (3.29), π -1 i (y) = {x 1 , . . . , x k } = π -1 j (π j (x 1 )), and so π j π -1 i (y) contains exactly one element, which is π j (x 1 ). We define ψ : Y i → Y j by ψ(y) = π j (x 1 ).

Observe that π -1 i : Y i → 2 X is continuous (as π i is distal, hence open) and commutes with S. Being π j a factor map, ψ is continuous and commutes with S. Therefore, ψ : (Y i , S) → (Y j , S) is a factor map. A similar construction gives a factor map ϕ : Y j → Y i which is the inverse function of ψ. We conclude that ψ is a conjugacy and, thus, that Y i and Y j are conjugate.

Non-distal factor maps

To deal with non-factor maps, we study asymptotic pairs belonging to fibers of this kind of factors. The starting point is the following lemma.

Lemma 23. Let π : (X, S) → (Y, S) be a factor between minimal subshifts. Then, either π is distal or there exists a fiber π -1 (y) containing a pair of right or left asymptotic points.

Proof. Assume that π is not distal. Then, we can find a fiber π -1 (y) and proximal points x, x ′ ∈ π -1 (y), with x ̸ = x ′ . This implies that for every k ∈ N there exist a (maybe infinite) interval

I k = (a k , b k ) ⊆ Z, with b k -a k ≥ k,
for which x and x ′ coincide on I and I k is maximal (with respect to the inclusion) with this property. Since x ̸ = x ′ , then a k > -∞ or b k < ∞. Hence, there exists an infinite set E ⊆ N such that a k > -∞ for every k ∈ E or b k < ∞ for every k ∈ E. In the first case, we have that (S b k (x, x ′ )) k∈E has a left asymptotic pair (z, z ′ ) as an accumulation point, while in the second case it is a right asymptotic pair (z, z ′ ) who is an accumulation point of (S a k (x, x ′ )) k∈E . In both cases we have that (z, z ′ ) ∈ R 2 π since (S b k (x, x ′ )) k∈E and (S a k (x, x ′ )) k∈E are contained in R 2 π and R 2 π is closed. Therefore, the fiber π -1 (π(z)) contains a pair z, z ′ of asymptotic points.

The next lemma allows us to pass from morphisms σ : X → Y to factors π : X ′ → Y in such a way that X ′ is defined on the same alphabet as X and has the "same" asymptotic pairs. We remark that its proof is simple, but tedious.

Lemma 24. Let X ⊆ A + be an aperiodic subshift, σ : A + → B + be a morphism and Y = k∈Z S k σ(X). Define the morphism i σ : A + → A + by i σ (a) = a |σ(a)| , a ∈ A, and X ′ = k∈Z S k i σ (X). Then, centered asymptotic pairs in X ′ are of the form (i σ (x), i σ (x)), where (x, x) is a centered asymptotic pair in X, and there exists a factor map π : (X ′ , S) → (Y, S) such that π(i σ (x)) = τ (x) for all x ∈ X. Proof. Our first objective is to prove that (X, i σ ) is recognizable. We start by observing that if (k, x), ( k, x) are centered i σ -factorizations of y ∈ X ′ , then x 0 = x0 . (3.34) Indeed, since the factorization are centered, we have

x 0 = i σ (x 0 ) k = y 0 = i σ (x 0 ) k = x0 .
Let Λ be the set of tuples (k, x, k, x) such that (k, x), ( k, x) are centered i σfactorizations of the same point. Moreover, for R ∈ {=, >}, let Λ R be the set of those (k, x, k, x) ∈ Λ satisfying k R k.

Claim 27.4. If (k, x, k, x) ∈ Λ = , then (0, Sx, 0, S x) ∈ Λ = , and if

(k, x, k, x) ∈ Λ > , then (|i σ (x 0 )| -k + k, x, 0, Sx) ∈ Λ > .
Proof. If (k, x, k, x) ∈ Λ = , then, since x 0 = x0 by (3.34), we can write i σ (Sx) = S k i σ (x) = S ki σ (x) = i σ (S x). Thus, (0, Sx, 0, S x) ∈ Λ = . Let now (k, x, k, x) ∈ Λ > and y := S k i σ (x) = S ki σ (x). We note that

S |iσ(x0)|-k+ ki σ (x) = S |iσ(x0)|-k y = S |iσ(x0)| i σ (x) = i σ (Sx), so (|i σ (x 0 )| -k + k, x
) and (0, Sx) are i σ -factorization of the same point. Now, since x 0 = x0 (by (3.34)) and (k, x), ( k, x) are centered, we have k, k ∈ [0, |i σ (x 0 )|). This and and the fact that

k > k imply that k -k ∈ (0, |i σ (x 0 )|). Therefore, |i σ (x 0 )|-k + k ∈ (0, |i σ (x 0 )|) and, consequently, (|i σ (x 0 )|-k + k, x, 0, Sx) ∈ Λ > . □
We prove now that (X, i σ ) is recognizable. Let (k, x, k, x) ∈ Λ. We have to show that (k, x) = ( k, x). First, we consider the case in which k = k. In this situation, the previous claim implies that (0, Sx, 0, S x) ∈ Λ = . We use again the claim, but with (0, Sx, 0, S x), to obtain that (0, S 2 x, 0, S 2 x) ∈ Λ = . By continuing in this way, we get (0, S n x, 0, S n x) ∈ Λ = for any n ≥ 0. Then, (3.34) implies that x n = xn for all n ≥ 0. A similar argument shows that x n = xn for any n ≤ 0, and so (k, x) = ( k, x). We now do the case k > k. Another application of the claim gives us (p 1 , x, 0, Sx) ∈ Λ > for some p 1 ∈ Z. As before, we iterate this procedure to obtain that (p 2 , Sx, 0, S x) ∈ Λ > , (p 3 , S x, 0, S 2 x) ∈ Λ > and so on. From these relations and (3.34) we deduce that x 0 = x0 , x0 = (Sx) 0 = x 1 , x 1 = (Sx) 0 = (S x) 0 = x1 , x1 = (S x) 0 = (S 2 x) 0 = x 2 , etc. We conclude that x n = xn = x 0 for any n ≥ 0. Then, by compacity, the periodic point • • • x 0 .x 0 x 0 • • • belongs to X, contrary to our aperiodicity hypothesis on X. Thus, the case k > k does not occurs. This proves that (X, i σ ) is recognizable.

Using the property we just proved, we can define the factor map π : X ′ → Y as follows: if x ′ ∈ X ′ , then we set π(x ′ ) = S k τ (x) ∈ Y , where (k, x) is the unique centered i σ -factorization of x ′ in X. To show that π is indeed a factor map, we first observe that since

|τ (a)| = |i σ (a)| for all a ∈ A, (3.35)
π commutes with S. Moreover, thanks to (iii) in Remark 1, π is continuous. Finally, if y ∈ Y , then by the definition of Y there exist a centered (k, x) τfactorization of y in X. Thus, by (3.35), (k, x) is a centered i σ factorization of x ′ := S k i σ (x). Therefore, π(x ′ ) = y and π is onto. Altogether, these arguments show that π is a factor map. That π(i σ (x)) = τ (x) for every x ∈ X follows directly from the definition of π.

It is left to prove the property about the asymptotic pairs. We only prove it for left asymptotic pairs since the other case is similar. We will use the following notation: if Z is a subshift, then A(Z) denotes the set of centered left asymptotic pairs. To start, we observe that (i σ (x), i σ (x ′ )) ∈ A(X ′ ) for every (x, x) ∈ A(X). Let now (z, z) ∈ A(X ′ ), and (k, x) and ( k, x) be the unique centered i σ -factorizations of z and z in X, respectively. We have to show that k = k = 0 and that (x, x) ∈ A(X). Due to (iii) in Remark 1, (X, i σ ) has a recognizability constant. This and the fact that (z, z) is centered left asymptotic imply that (k, x) and ( k, x) have a common cut in (-∞, 0], this is, that there exist p, q ≤ 0 such that

m := -k -|i σ (x [p,0) )| = -k -|i σ (x [q,0) )| ∈ (-∞, 0].
We take m as big as possible with this property. Then, x p ̸ = xq . Moreover, being z m = x p and zm = xp by the definition of i σ , we have that z m ̸ = zm and consequently, by also using that (z, z) is centered left asymptotic, that m ≥ 0. We conclude that m = 0, this is, that k

+ |i σ (x [p,0) )| = k + |i σ (x [q,0) )| = 0.
Hence, k = k = p = q = 0. Now, it is clear that x (-∞,p] = x(-∞,q] , so from the last equations we obtain that (x, x) ∈ A(X). This completes the proof.

We will also need the following lemma to slightly strengthen Proposition 25.

Lemma 25. Let X ⊆ A Z be an aperiodic subshift with L asymptotic tails. Then, (X, S) has at most 2L 2 • #A 2 centered asymptotic pairs. Proof. Let P r be the set of centered right asymptotic pairs in X and T r = {x (0,∞) : (x, x) ∈ Λ} ⊆ A N ≥1 be the set of right asymptotic tails, where N ≥1 = {1, 2, . . . }. We are going to prove that

#P r ≤ #T 2 r • #A 2 . (3.36)
Once this is done, we will have by symmetry the same relation for the centered left asymptotic pairs P l , and thus we are going to be able to conclude that the number of centered asymptotic pairs in X is at most

(#T 2 r + #T 2 l ) • #A 2 ≤ 2L 2 • #A 2 , completing the proof.
Let (x, x) ∈ P r and R x = {k ≤ 0 : x (k,∞) ∈ T r }. We claim that #R x ≤ #T r . Indeed, if this is not the case, then, by the Pigeonhole principle, we can find k ′ < k and w ∈ T r such that w = x (k,∞) = x (k ′ ,∞) . But this implies that w has period k -k ′ , and so X contains a point of period k -k ′ , contrary to the aperiodicity hypothesis. Thus, R x is finite and, since R x is nonempty as it contains x (0,∞) , k x := min R x is a well-defined non-positive integer.

Let now ϕ : P r → T 2 r × A 2 be the function defined by

ϕ(x, x) = (x (kx,∞) , x(kx,∞) , x kx , xkx )
If ϕ is injective, then (3.36) follows. Let us then prove that ϕ is injective. We argue by contradiction and assume that there exist (x, x) ̸ = (y, ỹ) such that ϕ(x, x) = ϕ(y, ỹ) = (z, z, a, ã). Without loss of generality, we may assume that x ̸ = y. Then, x (kx,∞) = z = y (ky,∞) and x kx = a = y ky . Being x ̸ = y, this implies that (x, y) is asymptotic. Furthermore, it implies that there exist p < k and q < ℓ such that (S p x, S q y) is centered right asymptotic. In particular, x (p,∞) ∈ T r and p < k x , contrary to the definition of k x . We conclude that ϕ is injective and thereby complete the proof of the lemma.

Lemma 26. Let X ⊆ A Z be a subshift of topological rank K, J be an index set and, for j ∈ J, let τ j : A + → B + j be a morphism. Suppose that for every j ∈ J (I) Y j = k∈Z S k τ j (X) is aperiodic;

(II) for every fixed a ∈ A, |τ j (a)| is equal to a constant ℓ a independent of j ∈ J.

Then, one of the following situations occur:

(1) There exist i, j ∈ J, i ̸ = j, such that (Y i , S) is conjugate to (Y j , S).

(2) There exist ϕ :

A + → A + 1 with #A 1 < #A, a set J 1 ⊆ J having at least #J/2#A 2 (144K 7 ) 2 -K(144K 7 ) K elements, and morphisms τ ′ j : C + 1 → B j , j ∈ J 1 , such that τ j = τ ′ j ϕ.
In particular, the hypothesis of this lemma hold for X 1 := k∈Z S k ϕ(X) and τ ′ j , j ∈ J 1 .

Proof. Let i : A + → A + be the morphism defined by i(a) = a ℓa , a ∈ A, and X ′ = k∈Z S k i(X). We use Lemma 24 with X and τ j to obtain a factor map π j : (X ′ , S) → (Y j , S) such that π(i(x)) = τ j (x) for every x ∈ X.

(3.37) If π j is distal for K(144K 7 ) K + 1 different values of j ∈ J, then by Lemma 22 we can find i, j such that (Y i , S) is conjugate to (Y j , S). Therefore, we can suppose that there exists J ′ ⊆ J such that #J ′ ≥ #J -K(144K 7 ) K and π j is not distal for every j ∈ J ′ .

(3.38)

From this and Lemma 23 we obtain, for every j ∈ J ′ , a centered asymptotic pair (x (j) , x(j) ) in X ′ such that π j (x (j) ) = π j (x (j) ). This and (3.37) imply that τ j (x (j) ) = π j (x (j) ) = π j (x (j) ) = τ j (x (j) ).

(3.39) Now, by Lemma 25, X has at most 2#A 2 (144K 7 ) 2 centered asymptotic pairs and thus, thanks to Lemma 24, the same bound holds for X ′ . Therefore, by the Pigeonhole principle, there exist

J 1 ⊆ J satisfying #J 1 ≥ #J ′ /2#A 2 (144K 7 ) 2 ≥ #J/2#A 2 (144K 7 ) 2 -K(144K 7
) K and a centered asymptotic pair (x, x) in X ′ such that (x, x) = (x (j) , x(j) ) for every j ∈ J 1 . We assume that (x, x) is right asymptotic as the other case is similar. Then, equation (3.39) implies that if ℓ = a∈A ℓ a , then, for every j ∈ J 1 , one of the words in {τ j (x [0,ℓ) ), τ j (x [0,ℓ) )} is a prefix of the other.

(3.40)

This, hypothesis (II) and the fact that, since (x, x) a centered asymptotic pair, x 0 ̸ = x0 allow us to use Lemma 13 with u := x [0,ℓ) , v := x[0,ℓ) , J := J 1 and w j := τ j (x [0,∞) ) [0,ℓ) and obtain morphisms ϕ : A + → A + 1 and τ ′ j : A + 1 → B + j , j ∈ J 1 , such that #A 1 < #A, τ j = τ ′ j ϕ and for every a ∈ A 1 , ℓ ′ a := |τ ′ j (c)| does not depend on the chosen j ∈ J. (3.41) Finally, we observe that X 1 and τ ′ j , j ∈ J 1 , satisfy the hypothesis of the lemma: condition (I) holds since, by the relation τ j = τ ′ j ϕ, the subshift X 1 := k∈Z S k ϕ(X) satisfies that k∈Z S k τ ′ j (X 1 ) = Y j is aperiodic; condition (II) is given by (3.41).

Proof of main result

We now prove Theorem 23. We restate it for convenience.

Theorem 23. Let (X, S) be an minimal subshift of topological rank K. Then, (X, S) has at most (3K) 32K aperiodic symbolic factors up to conjugacy.

Proof. We set R = (3K) 32K . We prove the theorem by contradiction: assume that there exist X ⊆ A Z of topological rank K and, for j ∈ {0, . . . , R}, factor maps π j : (X, S) → (Y j , S) such that (Y i , S) is not conjugate to (Y j , S) for every i ̸ = j ∈ {0, . . . , R}. We remark that X must be infinite as, otherwise, it would not have any aperiodic factor.

To start, we build S-representations for the subshifts X and Y j . Let σ = (σ n : A + n+1 → A + n ) n∈N be the primitive and proper directive sequence of alphabet rank K generating X given by Theorem 1. Let r ∈ N be such that every π j has a radius r and let B j the alphabet of Y j . By contracting σ, we can assume that σ 0 is r-proper and #A n = K for all n ≥ 1. Then, we can use Lemma 10 to find morphisms τ j : A + 1 → B + j such that π j (σ 1 (x)) = τ j (x) for all x ∈ X (1) σ and |τ j (a)| = |σ 0 (a)| for all a ∈ A 1 . (3.42) Next, we inductively define subshifts X n ⊆ C Z n and morphisms {τ n,j :

C + n → B j : j ∈ J n } such that (i) X n has topological rank at most K; (ii) Y j = k∈Z τ n,j (X n ); (iii) for every c ∈ C n , ℓ n,a := |τ n,j (c)| does not depend on the chosen j ∈ J n . First, we set X 0 = X (1)
σ , C 0 = A 1 , J 0 = J and, for j ∈ J 0 , τ 0,j = τ j , and note that by the hypothesis and (3.42), they satisfy (i), (ii) and (iii). Let now n ≥ 0 and suppose that X n ⊆ C Z n and τ n,j , j ∈ J n , has been defined in a way such that (i), (ii) and (iii) hold. If #J n /2#A 2 (144K 7 ) 2 -K(144K 7 ) K ≤ 1, then the procedure stops. Otherwise, we define step n + 1 as follows. Thanks to (i), (ii), (iii) we can use Lemma 26, and since there are no two conjugate (Y i , S), this lemma gives us a morphism ϕ :

C + n → C + n+1 , a set J n+1 ⊆ J n and morphisms {τ n+1,j : C + n+1 → B + j : j ∈ J n+1 } such that #C n+1 < #C n , #J n+1 ≥ #J n /2#C 2 n (144K 7 ) 2 -K(144K 7 ) K and τ n,j = τ n+1,j ϕ n .
Furthermore, X n+1 := k∈Z S k ϕ n (X n ) and τ n+1,j satisfy the hypothesis of that lemma, that is, conditions (ii) and (iii) above. Since (ϕ n . . . ϕ 0 σ 1 , σ 2 , σ 3 , . . . ) is a primitive and proper sequence of alphabet rank K generating X n+1 , Theorem 21 implies that condition (i) is met as well.

Since #C 0 > #C 1 > . . . , there is a last C N defined. Our next objective is to prove that N ≥ K. Observe that #C n ≤ K, so

#J n+1 ≥ #J n /2K 2 (144K 7 ) 2 -K(144K 7 ) K for any n ∈ {0, . . . , N -1}.
Using this recurrence and the inequalities #J 0 > (3K) 32K and K ≥ 2, it is routine to verify that the following bound holds for every n ∈ {0, . . . , K -1} such that the nth step is defined:

#J n /2#C 2 n (144K 7 ) 2 -K(144K 7 ) K > 1 Therefore, N ≥ K. We conclude that #C N ≤ #C 0 -K = 0, which is a contradiction.
Remark 7. In Theorem 1 of [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF], the author proved that linearly recurrent subshifts have finitely many aperiodic symbolic factors up to conjugacy. Since this kind of systems have finite topological rank (see Remark 6), Theorem 23 generalizes the theorem of [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF] to the much larger class of minimal finite topological rank subshifts.

Appendix

To prove Proposition 3, we start with some lemmas concerning how to construct recognizable pairs (Z, τ ) for a fixed subshift Y = k∈Z S k τ (Z).

Codings of subshifts

If Y ⊆ B Z is a subshift, U ⊆ Y and y ∈ Y , we denote by R U (y) the set of return times of y to U , this is, R U (y) = {k ∈ Z : S k y ∈ U }. We recall that the set C τ (k, z) in the lemma below corresponds to the cuts of (k, z) (see Definition 9 for further details).

Remark 8. If U ⊆ Y satisfies (III), then U is ρ := min per(L ℓ (Y ))-separated. Indeed, if U ∩ S k U ̸ = ∅ for some k > 0, then [v] ∩ S k [v] ̸ = ∅, where v ∈ A ℓ is such that U ⊆ [v]. Hence, v is k periodic and k ≥ ρ. Proof. Let y ∈ Y . By (I), the sets R U (y) ∩ [0, ∞), R U (y) ∩ (-∞, 0] are infinite. Thus, we can write R U (y) = {. . . k -1 (y) < k 0 (y) < k 1 (y) . . . }, with min{i ∈ Z : k i (y) > 0} = 1. Let W = {y [ki(y),ki+1(y)) : y ∈ Y, i ∈ Z} ⊆ B + . By (I), W
is finite, so we can write C := {1, . . . , #W} and choose a bijection ϕ : C → W.

Then, ϕ extends to a morphism τ :

C + → B + . As B ⊆ L(Y ), ϕ is letter-onto. We define ψ : Y → C Z by ψ(y) = (ϕ -1 (y [ki(y),ki+1(y)) )) i∈Z and set Z = ψ(Y ).
We are going to prove that τ and Z satisfy (1-4).

Claim 27.5. , we deduce that k 0 (y) = k 0 (y ′ ) and k 1 (y) = k 0 (y ′ ). Hence, ψ(y) 0 = ϕ -1 (y [k0(y),k1(y)) ) = ϕ -1 (y ′ [k0(y ′ ),k1(y ′ )) ) = ψ(y ′ ) 0 . To prove (ii) we compute:

(i) If y [-d-r,d+r] = y ′ [-d-r,d+r] , then ψ(y) 0 = ψ(y ′ ) 0 , (ii) τ (ψ(y)) = S k0(y) y, (
τ (ψ(y)) = τ (• • • ϕ -1 (y [k-1(y),k0(y)) ).ϕ -1 (y [k0(y),k1(y)) ) • • • ) = • • • y [k-1(y),k0(y)) .y [k0(y),k1(y)) • • • = S k0 y.
Finally, for (iii) we write, for k ∈ [k j (y), k j+1 (y)),

S j ψ(y) = . . . ϕ -1 (y [kj-1(y),kj (y)) ).ϕ -1 (y [kj (y),kj+1(y)) ) • • • = ψ(S k y).
□ Now we prove the desired properties of τ and Z.

(1) From (i), we see that ψ is continuous and, therefore, Z is closed. By (iii), Z is also shift-invariant and, then, a subshift. By (ii), Y = n∈Z S n τ (Z). The condition C ⊆ L(Y ) follows from the definition of W and τ .

(2) We claim that the only centered τ -interpretation in Z of a point y ∈ Y is (-k 0 (y), ψ(y)). Indeed, this pair is a τ -interpretation in Z by (ii), and it is centered because k 0 (y) ≤ 0 < k 1 (y) implies -k 0 (y) ∈ [0, k 1 (y) -k 0 (y)) = [0, |ψ(y) 0 |). Let (n, z) be another centered τ -interpretation of y in Z. By the definition of Z, there exists y ′ ∈ Y with z = ψ(y ′ ). Then, by (ii),

S n+k0(y ′ ) y ′ = S n τ (ψ(y ′ )) = S n τ (z) = y.
(3.43)

Now, on one hand, we have |τ

(z 0 )| = |τ (ψ(y ′ ) 0 )| = k 1 (y ′ ) -k 0 (y ′ ). On the other hand, that (n, ψ(y ′ )) is centered gives that n ∈ [0, |τ (z 0 )|). Therefore, n + k 0 (y ′ ) ∈ (k 0 (y ′ ), k 1 (y ′ )]
. We conclude from this, (iii) and (3.43) that ψ(y ′ ) = ψ(y). Hence, y = S n τ ψ(y ′ ) = S n τ ψ(y) = S n+k0(y) y, which implies that n = -k 0 (y) as Y is aperiodic. This proves that (-k 0 (y), ψ(y)) is the only τ -interpretation of y in Z. From this and (i) we deduce property (2).

(

) Since U is d-syndetic, |τ (ψ(y) i )| = |y [ki(y),ki+1(y)) | = k i+1 (y) -k i (y) ≤ d for y ∈ Y and i ∈ Z, so |τ | ≤ d. Similarly, we can obtain ⟨τ ⟩ ≥ ρ using that U is ρ-separated. Let u, v ∈ B ℓ satisfying U ⊆ [u.v]. Since k i , k i+1 ∈ R U (y), we have that u = y [ki(y),ki(y)+|u|) , v = y [ki+1(y)-|v|,ki+1(y)) 3
and, thus, that τ is min(ℓ, ⟨τ ⟩)-proper. In particular, it is min(ℓ, ρ)-proper.

(4) This follows directly from the definition of τ and R U (y).

Lemma 28. For j ∈ {0, 1}, let σ j : A + j → B + be a morphism and X j ⊆ A Z j be a subshift such that Y := n∈Z S n σ j (X j ) and A j ⊆ L(X j ) for every j ∈ {0, 1}. Suppose that:

(1) (X 0 , σ 0 ) is recognizable with constant ℓ,

(2) σ 1 is ℓ-proper,

(3) C σ0 (k 0 , x 0 )(y) ⊇ C σ1 (k 1 , x 1 )(y) for all y ∈ Y and σ j -factorizations (k j , x j ) of y in X j , j = 0, 1.

Then, there exist a letter-onto and proper morphism ν : A + 1 → A + 0 such that σ 1 = σ 0 ν and X 0 = k∈Z S k ν(X 1 ).

Proof. Since σ 1 is ℓ-proper, we can find u, v ∈ B ℓ such that σ 1 (a) starts with u and ends with v for every a ∈ A 1 . We define ν as follows. Let a ∈ A 1 and x ∈ X 1 such that a = x 0 . Since σ 1 is ℓ-proper, the word v.σ 1 (a)u occurs in σ 1 (x) ∈ Y at position 0. By (3), we can find w ∈ L(X 0 ) with σ 1 (x 0 ) = σ 0 (w). We set ν(a) = w. Since (X 0 , σ 0 ) is recognizable with constant ℓ and u, v have length ℓ, w uniquely determined by v.σ 1 (a)u and, therefore, ν is well defined. Moreover, the recognizability implies that the first letter of ν(a) depends only on v.u, so ν is left-proper. A symmetric argument shows that ν is right-proper and, in conclusion, that it is proper. We also note that ν is letter-onto as A 0 ⊆ L(X 0 ). It follows from the definition of ν that σ 1 = σ 0 ν. Now, let x ∈ X 1 and (k, x ′ ) be a centered σ 0 -factorization of σ 1 (x) in X 0 . By (3), k = 0 and σ 1 (x j ) = σ 0 (x ′ [kj ,kj+1) ) for some sequence ... < k -1 < k 0 < ... Hence, by the definition of ν, ν(x) = x ′ ∈ X 0 . This argument shows that X ′ 0 := n∈Z S n ν(X 1 ) ⊆ X 0 . Then, n∈Z S n σ 0 (X ′ 0 ) = n∈Z S n σ 0 ν(X 1 ) = Y , where in the last step we used that σ 0 ν = σ 1 . Since the points in Y have exactly one σ 0 -factorization, we must have X ′ 0 = X 0 . This ends the proof.

Factors of S-adic sequences

Now we are ready to prove Proposition 3. For convenience, we repeat its statement.

Proposition 5. Let σ = (σ n : A n → A n-1 ) n≥0 be a letter-onto, everywhere growing and proper directive sequence. Suppose that X σ is aperiodic. Then, there exists a contraction σ ′ = (σ n k ) k∈N and a letter-onto and proper factor ϕ : σ ′ → τ , where τ is letter-onto, everywhere growing, proper, recognizable and generates X σ .

Proof. We start by observing that from Lemma 17 we can get that

A n ⊆ L(X (n) σ ) for every n ∈ N. (3.44) Let p n = min{per(σ [0,n) (a)) : a ∈ A n }.
Since σ is everywhere growing and X σ is aperiodic, lim n→∞ p n = ∞. Hence, we can contract σ in a way such that, for every n ≥ 2,

(I n ) p n ≥ 3|σ [0,n-1) |, (II n ) σ [0,n) is 3|σ [0,n-1) |-proper, For n ≥ 2, let U n = u,v∈A 2 n [σ [0,n) (u.v)]. Observe that U n is |σ [0,n) |-syndetic, has radius 2|σ [0,n) |, is 3|σ [0,n-1)
|-proper and, by Remark 8, is p n -separated. Thus, by (I n ), U is 3|σ [0,n-1) |-separated. We can then use Lemma 27 with (X

(n) σ , σ [0,n) ) to obtain a letter-onto morphism ν n : B + n → A + 0 and a subshift Y n ⊆ B Z n such that (P 1 n ) X σ = k∈Z S k ν n (Y n ) and B n ⊆ L(Y n ), (P 2 n ) (Y n , ν n ) is recognizable with constant 3|σ [0,n) |, (P 3 n ) |ν n | ≤ |σ [0,n) |, ⟨ν n ⟩ ≥ 3|σ [0,n-1) |, and ν n is 3|σ [0,n-1) |-proper, (P 4 n ) C νn (k, y) = R Un (x) for all x ∈ X σ and ν n -factorization (k, y) of x in Y n . We write C νn (x) := C νn (k, y) if x ∈ X σ and (k, y) is the unique ν n -factorization of x in Y n . Observe that U n+1 ⊆ U n for n ≥ 2. Thus, C νn+1 (x) = R Un+1 (x) ⊆ R Un (x) = C νn (x)
for all x ∈ X σ . This, (P 2 n ) and (P 3 n+1 ) allow us to use Lemma 28 with (Y n+1 , ν n+1 ) and (Y n , ν n ) and find a letter-onto and proper morphism

τ n : B + n+1 → B + n such that ν n τ n = ν n+1 and Y n = k∈Z S k τ n (Y n+1 ). Next, we claim that C νn (x) ⊇ C σ [0,n+1) (k, z) for all x ∈ X σ and σ [0,n+1) - factorization (k, z) of x in X (n+1) σ . Indeed, if j ∈ Z, then S cσ [0,n+1) ,j (k,z) x ∈ [σ [0,n+1) (z j-1 .z j z j+1 )] ⊆ [σ [0,n) (a.bc)] ⊆ U n ,
where a is the last letter of σ n (z j-1 ) and bc the first two letters of σ n (z j z j+1 ), so c σ [0,n+1) ,j (k, z) ∈ R Un (x) = C νn (x), as desired.

Thanks to the claim, (P 2 n ), (I n+1 ) and (3.44), we can use Lemma 28 with (Y n , ν n ) and (X

(n+1) σ , σ [0,n+1) ) to obtain a proper morphism ϕ n : A + n+1 → B + n such that σ [0,n+1) = ν n ϕ n and Y n = k∈Z S k ϕ n (X (n+1) σ
). Now we can define the morphisms τ 1 := ν 2 and ϕ 1 := ν 2 ϕ 2 and the sequences:

ϕ = (ϕ n ) n≥1 , τ = (τ n ) n≥1 and σ ′ = (σ [0,2) , σ 2 , σ 3 , . . . ) n≥2 .
We are going to prove that ϕ, σ ′ , and τ are the objects that satisfy the conclusion of the Proposition.

These sequences are letter-onto as each ν n and each ϕ n is letter-onto. Next, we show that ϕ is a factor. The relation ϕ 1 = τ 1 ϕ 2 follows from the definitions. To prove the other relations, we observe that from the commutative relations for τ n and ϕ n , we have that

ν n ϕ n σ n+1 = σ [0,n+1) σ n+1 = σ [0,n+2) = ν n+1 ϕ n+1 = ν n τ n ϕ n+1 .
(3.45)

In particular,

ν n ϕ n σ n+1 (x) = ν n τ n ϕ n+1 (x) for any x ∈ X (n+2) σ . Since ϕ n σ n+1 (x) and τ n ϕ n+1 (x) are both elements of Y n and (Y n , ν n ) is recognizable, we deduce that ϕ n σ n+1 (x) = τ n ϕ n+1 (x) for any x ∈ X (n+2) σ
. Thus, one of the words in

{ϕ n σ n+1 (x 0 ), τ n ϕ n+1 (x 0 )} is a prefix of the other. Since A n+2 ⊆ L(X (n+2) σ
), we deduce that, for any a ∈ A n+2 , one of the words in {τ n ϕ n+1 (a), ν n ϕ n σ n+1 (a)} is a prefix of the other. But, by (3.45), the words ν n τ n ϕ n+1 (a) and ν n ϕ n σ n+1 (a) have the same length, so ϕ n σ n+1 (a) must be equal to τ n ϕ n+1 (a) for every n ≥ 2. This proves that ϕ n σ n+1 = τ n ϕ n+1 for every n ≥ 2 and that ϕ : σ ′ → τ is a factor.

The following commutative diagram, valid for all n ≥ 2, summarizes the construction so far:

A + n+2 A + n+1 A + 0 B + n+1 B + n σn+1 ϕn+1 σ [0,n+1) ϕn τn νn+1 νn
As shown in the diagram, we have that ν n τ n = ν n+1 for n ≥ 2. Thus,

τ 1 τ 2 • • • τ n = ν n+1 , and hence ⟨τ 1 τ 2 • • • τ n ⟩ ≥ ⟨ν n+1 ⟩ ≥ p n → n→∞ ∞.
Therefore, τ is everywhere growing. Also, by using Lemma 1 with

(Y n , ν n ) = (Y n , τ 1 τ 2 • • • τ n-1 ),
we deduce that (Y n , τ n-1 ) is recognizable for every n ≥ 2, which implies that τ is recognizable. Finally, as each τ n is proper, τ is proper.

Chapter 4

A solution to the S-adic conjecture

A interesting intuition in symbolic dynamics of zero entropy is that a subshift of low enough complexity should be very restricted, and thus hide a strong structure. This idea dates back to the 70s, and matured in the 80s and 90s until it was finally established as the following more concrete question.

Question 8. Consider the class (L) of linear-growth complexity subshifts, defined by requiring that p X (n) ≤ dn for some d > 0. Is there an S-adic characterization of the class (L)?

Question 8 is known as the S-adic conjecture.

In this chapter, we completely solve the S-adic conjecture for minimal subshifts by proving the following theorem.

Theorem 28. A minimal subshift X has linear-growth complexity, i.e., X satisfies lim sup n→+∞ p X (n)/n < +∞, if and only if there exist d > 0 and an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that, for every n ≥ 0, the following holds:

(P 1 ) #(root σ [0,n) (A n )) ≤ d † . (P 2 ) |σ [0,n) (a)| ≤ d • |σ [0,n) (b)| for every a, b ∈ A n . (P 3 ) |σ n-1 (a)| ≤ d for every a ∈ A n .
We are able to give a similar structure for nonsuperlinear complexity subshifts (NSL). using d. For controlling the periodic words, we rely on tricks from combinatorics on words. These two ideas are used to obtain, in Sections 4.6 to 4.7, two variations of (X ′ n , σ ′ n ), with increasingly better properties, and where the last one is (X n , σ n ).

Organization

The chapter has three parts. The first one consists of Sections 4.1, 4.2 and 4.3 and provide the necessary background and some lemmas for handling periodic words. Then, in Sections 4.4 to 4.8, we carry out the proofs of Theorems 28 and 29. Finally, we prove Theorem 35 and present applications of our main results in Sections 4.9 and 4.10.

Preliminaries

The word w ∈ A + is |u|-periodic, with u ∈ A + , if w occurs in u n for some n ≥ 1. We denote by per(w) the least p for which w is p-periodic.

In order to describe more precisely the periodicity properties of w, we use the notion of root, which will play a key role throughout the chapter.

Definition 8. The minimal root, or just root for short, of w ∈ A * is the shortest prefix u of w for which w = u k for some k ≥ 1, and it is denoted by root w.

We remark that per(w) is an integer but that root w is a word, and that per(w) is in general different from | root w|.

Let X ⊆ A Z be a subshift and v ∈ A + . We will use the notation v ∞ = vvv • • • ∈ A N and v Z = . . . vv.vv • • • ∈ A Z . We denote by Pow X (v) the set of words v k , where k ≥ 1, for which there exist u, w ∈ A + \ {v} of length |v| such that uv k w ∈ L(X). The power complexity of X is the number pow-com(X) = sup v∈A + #Pow X (v). Remark that pow-com(X) may be infinite. Examples with finite power complexity include linearly recurrent subshifts and subshifts in which the extension graph of every long enough word is acyclic (in particular, Sturmian subshifts and codings of minimal interval exchange transformations).

Morphisms and codings

We say that τ is positive if for every a ∈ A, all letters b ∈ B occur in τ (a), that τ is proper if there exist letters a, b ∈ B such that τ (c) starts with a and ends with b for any c ∈ A, and that τ is injective on letters if for all a, b ∈ B, τ (a) = τ (b) implies a = b.

Factorizations and recognizability

We now introduce factorizations, the recognizability property and the associated notation.

Definition 9. Let Y ⊆ B Z be a subshift and τ : B + → A + be a morphism. We say that (k,

y) ∈ Z × Y is a τ -factorization of x ∈ A Z in Y if x = S k τ (y) and 0 ≤ k < |τ (y 0 )|.
The pair (Y, τ ) is recognizable if every point x ∈ A Z has at most one τfactorization in Y . We say that (Y, τ ) is d-recognizable, with d ≥ 1, if whenever (k, y) and ( k, ỹ) are τ -factorizations of x, x ∈ A Z in Y , respectively, and

x [-d,d) = x[-d,d) , we have that k = k and y 0 = ỹ0 .
The cut function c : Z → Z of the τ -factorization (k, y) of x in Y is defined by

c j = -k + |τ (y [0,j) )| if j ≥ 0, -k -|τ (y [j,0) )| if j < 0. (4.1)
When (Y, τ ) is recognizable, we write (c, y) = F (Y,τ ) (x) and (c 0 , y 0 ) = F 0 (Y,τ ) (x). Remark 9. In the context of the previous definition:

(1) If (Y, τ ) is recognizable, then a compacity argument shows that it is drecognizable for some d ≥ 1.

(2) Suppose that (Y, τ ) is recognizable. Let x ∈ X, (c, y) = F (Y,τ ) (x) and i ∈ Z. Then, there exists a unique j ∈ Z such that i ∈ [c j , c j+1 ). Note that the last condition is equivalent to F 0 (Y,τ ) (S i x) = (c j -i, y j ). Lemma 29. Let σ : C → B + and τ : B → C + be morphisms and Z ⊆ C Z be a subshift. We set Y = k∈Z S k σ(Z) and X = k∈Z S k τ (Y ). Suppose that (Z, τ σ) is recognizable. Let

|σ(z) [0,m) | ≤ ℓ < |σ(z) [0,m] |. (4.2)
Therefore, as (ℓ, z) is a τ σ-factorization of x, we can write

S ℓ-|τ (σ(z) [0,m) )| τ (S m σ(z)) = S ℓ τ σ(z) = x.
This and (4.2) ensure that (ℓ

-|τ (σ(z) [0,m) )|, S m σ(z)) is a τ -factorization of x in Y . We conclude, using that (Y, τ ) is recognizable by Lemma 1, that ℓ - |τ (σ(z) [0,m) )| = k and S m σ(z) = y.

Codings of a subshift

We fix a subshift X ⊆ A Z . A coding of X is a pair (Y, τ ), where Y ⊆ B Z is a subshift and τ : B + → A + a morphism satisfying X = k∈Z S k τ (Y ). We present in Proposition 6 a general method for building recognizable codings of a subshift X. This idea occurs commonly in the literature under several different names and with different degrees of generality. Our Proposition 6 is inspired by the coding based on return words from [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF].

Let U ⊆ X be a clopen (i.e., open and closed) set. We say that U is Thus, there exists m ∈ [n, 2n) satisfying p X (m + 1) -p X (m) ≤ 2d.

Lemma 31. Let X ⊆ A Z be a subshift and d ≥ 1 be such that p X (n) ≤ dn for infinitely many n ≥ 1. Then, there are infinitely many m such that p X (m) ≤ 3dm and p X (m + 1) -p X (m) ≤ 2d.

Proof. Let n ≥ 1 be arbitrary. The hypothesis permits to find k ≥ 2n such that p X (k) ≤ dk. We now observe that

1 ⌈k/2⌉ k m=⌊k/2⌋ p X (m + 1) -p X (m) ≤ 1 ⌈k/2⌉ p X (k) ≤ 2d.
Therefore, there exists m such that ⌊k/2⌋ ≤ m ≤ k and p X (m+1)-p X (m) ≤ 2d.

The first condition ensures that m ≥ n and p X (m) ≤ p X (k) ≤ dk ≤ 3dm.

Some combinatorial lemmas

In order to prove our main results, we will need to extensively deal with strongly periodic words. The objective of this section is to give the necessary tools for doing so.

A basic result on periodicity of words is the Fine and Wilf Theorem, which we state below.

Theorem 31. Let u, v, w ∈ A + and suppose that w is a prefix of u ∞ and v ∞ . If |w| ≥ |u| + |v| -1, then there exists t ∈ A + such that u and v are powers of t.

A proof of Theorem 31 can be found in [RS97, Chapter 6, Theorem 6.1].

Lemma 32. Let u be a word such that |u| ≥ 2| root u|. Then, | root u| = per(u).

Proof. Note that u is a prefix of (root u) Z and thus that per(u) ≤ | root u|. It is left to prove the other inequality.

Let t be the prefix of u of length per(u). Then u is a prefix of both t ∞ and (root u) ∞ . We deduce, as |u| ≥ 2 per(u) ≥ |t| + | root u|, that the hypothesis of Theorem 31 is complied. Hence, t and root u are powers of a common word r. In particular, u is a power of r, so we must have that root u = r. Therefore,

| root u| = |r| ≤ |t| = per(u).
Remark 11. The previous lemma ensures that if u is a word and k ≥ 1, then root u k = root u. In particular, if u and v are powers of a common word, then they have the same root. These basic relations will be freely used throughout the chapter.

The next proposition will allow us to synchronize occurrences of strongly periodic words.

Proposition 7. Let t, s ∈ A + .

(1) Suppose that ℓ ≥ |s| + |t| -1 and i, j ∈ Z are such that t Z

[i,i+ℓ) = s Z [j,j+ℓ)
Then, S i t = S j s.

(2) An integer i satisfies S i t Z = t Z if and only if i = 0 (mod | root t|).

Proof. We first prove Item (1). Let

t 0 = t Z [i,i+|t|) , s 0 = s Z [j,j+|s|) and w = t Z [i,i+ℓ) = s Z [j,j+ℓ) . Then, w is a prefix of both t ∞ 0 and s ∞ 0 . Since |w| = ℓ ≥ |s| + |t| -1 = |s 0 | + |t 0 | -1,
we can use Theorem 31 to deduce that s 0 and t 0 are powers of a common word r. We then have

S i s Z = s Z 0 = r Z = t Z 0 = S j t Z . We now prove Item (2). It is clear that if i = 0 (mod | root t|) then S i t Z = t Z .
Let us suppose that S i t Z = t Z . We argue by contradiction and assume that i ̸ = 0 (mod | root t|). We write root t = ss ′ , where |s| = i (mod | root t|). Then, (s ′ s) Z = S i t Z = t Z = (ss ′ ) Z , so Theorem 31 implies that s ′ s and ss ′ are powers of a common word r. In particular, root(s ′ s) = root(ss ′ ) = root r. This implies that

| root(s ′ s)| = | root(ss ′ )| = | root root t| = | root t| = |ss ′ | = |s ′ s|, so root(s ′ s) = s ′ s.
Hence, s ′ s = ss ′ . Now, since i ̸ = 0 (mod | root t|), s and s ′ are not the empty word. This and the condition s ′ s = ss ′ imply that s ′ s is a prefix of s ∞ and of s ′ ∞ . We can then use Theorem 31 to deduce that s and s ′ are powers of a common word r ′ . Therefore, as root = ss ′ , ss ′ = root t = root s = root s ′ . This is possible only if s = 1 or s ′ = 1. Consequently, |s| ∈ {0, | root t|} and i = |s| = 0 (mod | root t|), contradicting our assumptions.

The rest of the section is devoted to prove Propositions 8 and 9. These results describe situations in which information about the global period of a word can be retrieved from small subwords of it. We remark that Propositions 8 and 9 can be obtained as a direct consequence of the Critical Factorization Theorem, a fundamental result in combinatorics on words; here we give proofs that depend only on Theorem 31 in order to maintain our presentation as self-contained as possible.

Lemma 33. Let u, v, w, s and t be words in A.

(1) Suppose that uv occurs in t ∞ and that vw occurs in s ∞ . If, |v| ≥ |t|+|s|-1, then uvw occurs both in t ∞ and in s ∞ .

(2) Suppose that uv is a prefix of t ∞ and that vw is a suffix of ∞ t. If |v| ≥ 2|t|, then uvw is a power of root t.

(3) If |v| ≥ per(uv) + per(vw), then per(uvw) = per(uv) = per(vw).

Proof. Assume that the hypothesis of Item (1) holds. Then,

uv = t Z [i,i+|uv|) and vw = s Z [j,j+|vw|) for some i, j ∈ Z. Hence, t Z [i+|u|,i+|uv|) = s Z [j,j+|v|)
. This and the inequality |v| ≥ |t| + |s| -1 allows us to use Item (1) in Proposition 7 to get that S i+|u| t Z = S j s Z . We conclude that

uvw = t Z [i,i+|uv|) s Z [j+|v|,j+|vw|) = t Z [i,i+|uv|) t Z [i+|u|+|v|,i+|u|+|vw|) = t Z [i,i+|uvw|) ,
and that uvw occurs in t ∞ . Similarly, uvw occurs in s ∞ . We now assume that the hypothesis of Item (2) holds. Let 3) give that uvw = (t Z 0 ) [0,|uvw|) is a power of t 0 = root t. We finally prove Item (3). Clearly, per(uv) ≤ per(uvw) and per(vw) ≤ per(uvw). Let t 0 be the prefix of uv of length per(uv) and s 0 be the prefix of vw of length per(vw). Then, uv occurs in t 0 and vw occurs in s 0 . This and the inequality |v| ≥ |u| + |v| ≥ |t 0 | + |s 0 | allow us to use Item (1) of this lemma to deduce that uvw occurs in t Z 0 and s Z 0 . We deduce that per(uvw) ≤ |t 0 | = per(uv) and per(uvw) ≤ |s 0 | = per(vw). Therefore, per(uvw) = per(uv) = per(vw).

t 0 = root t Then, uv = (t Z 0 ) [0,|uv|) and vw = (t Z 0 ) [-|vw|,0) . This implies that (t Z 0 ) [|u|,|uv|) = (t Z 0 ) [-|vw|,-|w|) ,
Proposition 8. Let V ⊆ A + and u ∈ A + be such that |u| ≥ 2|V|. Suppose that for any subword v of u with length |v| = 2|V| there exists w v ∈ V such that v occurs in w Z v . Then, for any such word v, u occurs in w Z v . In particular, per(u) ≤ |V|.

Proof. The case |u| = 2|V| follows directly from the hypothesis. Suppose the lemma is true for words u ′ of length 2|V| ≤ |u ′ | < |u|. Let v be a subword of u with length |v| = 2|V|. We have to prove that w occurs in w Z v . Let us write u = au ′ = u ′′ b for certain letters a, b and words u ′ , u ′′ . There is no loss of generality in assuming that v occurs in u ′ . Since |u| > 2|V|, we can take a subword v ′ of u ′′ with length |v ′ | = 2|V|. Then, the inductive hypothesis can be used to deduce that u ′ occurs in w Z v and that u ′′ occurs in w Z v ′ . Now, u ′ and u ′′ have a common subword of length |u| -

2 ≥ 2|W| -1 ≥ |w v | + |w v ′ | -1.
Therefore, Item (1) of Lemma 33 can be applied and we deduce that w occurs in w Z v . This proves the inductive step and thereby the proposition. Proposition 9. Let u be a word.

(1) If t is a word occurring in u and |t| ≥ 2 per(u), then per(t) = per(u).

(2) Let k ≥ 1. If |u| ≥ 2k and per(u) > k, then there exists t occurring in u with |t| = 2k and per(t) > k.

Proof. We start with Item (1). Note that per(t) ≤ per(u), so we only have to prove the other inequality. Let s (resp. s ′ ) be the prefix of t of length per(t) (resp. per(u)). Then, t occurs in s Z and s ′ Z . Being |t| ≥ 2 per(u) ≥ |s| + |s ′ |, we can use Item (1) in Proposition 7 to deduce that s Z = S ℓ s ′ Z for some ℓ ∈ Z. This implies, as u occurs in s ′ Z , that u occurs in s Z . In particular, per(u) ≤ |s| = per(t).

Next, we prove Item (2) by contradiction. Assume that k ≥ 1 and u are such that |u| ≥ 2k and per(u) > k, but that for all word t occurring in u of length |t| = 2k we have that per(t) ≤ k. We define, for all such t, s t as the prefix of t of length per(t), and note that t occurs in s Z t and that |s t | ≤ k. Therefore, the set V consisting of the words s t and the word u comply with the hypothesis of Proposition 8. We conclude that per(u) ≤ |V| ≤ k.

The classic coding based on special words

The notion of right-special word is an important concept for studying lineargrowth complexity subshifts. In this section, we present basic results on rightspecial words and the coding associated to them. Most of these ideas are common to many works on the S-adic conjecture and related problems. One of the new ingredients of our work is Proposition 12.

Definition 11. Let X be a subshift. A word w ∈ L(X) is called right-special if there exist two different letters a and b such that wa, wb ∈ L(X). We denote by RS n (X) the set of all right special words of X having length n.

Remark 12. We can also define left-special words, which together with rightspecial words form the set of special words of X. In our work, we will only use right-special words.

The next proposition summarizes the facts about RS n (X) and its relation to the complexity of X that are important for us. A return word to a clopen set U is an element w ∈ A + such that there exists x ∈ X satisfying

x [0,|w|) = w, S k x ∈ U if k ∈ {0, |w|} and S k x ̸ ∈ U if k ∈ (0, |w|). A clopen set U is k-recurrent, for d ≥ 0, if for all x ∈ X there is i ∈ [0, d) such that S i x ∈ U .
Proposition 10. Let X ⊆ A Z be an aperiodic subshift and U the clopen set

U = {x ∈ X : x [0,n) ∈ RS n (X)}.
(1) We have the following bounds on the number of right-special words:

1 #A (p X (n + 1) -p X (n)) ≤ # RS n (X) ≤ p X (n + 1) -p X (n). (2) The set U is (p X (n) + n)-recurrent in X.
(3) The number of return words to U is at most

#A • # RS n (X).
Proof. A proof of Items (1), ( 2) and (3) can be found, with a different notation, in [START_REF] Leroy | A Combinatorial Proof of S-adicity for Sequences with Linear Complexity[END_REF].

We can combine Propositions 10 and 6 to obtain the following proposition.

Proposition 11. Suppose that X ⊆ A Z is an aperiodic subshift and let d be the maximum of ⌈p X (n)/n⌉, p X (n+1)-p X (n) and #A. Let (Z ⊆ C Z , τ : C + → A + ) be the coding obtained from Proposition 6 with U = {x ∈ X :

x [0,n) ∈ RS n (X)}.
Then:

(1) #C ≤ d 3 .

(2) |τ (a)| ≤ (d + 1)n for all a ∈ C.

(

) (Z, τ ) is (d + 2)n-recognizable. (4) If x ∈ X, (c, z) = F (Z,τ ) (x) and i ∈ Z, then i = c j for some j ∈ Z if and only if x [i,i+n) ∈ RS(X). 3
Proposition 11 is the starting point of other works on the S-adic conjecture; see for example [START_REF] Ferenczi | Rank and symbolic complexity[END_REF][START_REF] Leroy | Some improvements of the S-adic conjecture[END_REF].

Proposition 12. Let (Z, τ ) be the coding in Proposition 11. Let x ∈ X and (c, z) = F (Z,τ ) (x) and suppose that i, j ∈ Z satisfy i + d < j and ℓ :

= max{|τ (z k )| : k ∈ [i, j)} ≤ n/6d. Then, per(x [ci-n/3,c j-d ) ) ≤ dℓ.
Proof. We start by noticing that, since x [cm-n,cm) ∈ RS n (X) for all m ∈ Z and since # RS n (X) ≤ d, we can use the Pigeonhole principle to obtain, for each

k ∈ [i, j -d), integers p k , q k ∈ [k, k + d) such that p k < q k and x [cp k -n,cp k ) = x [cq k -n,cq k ) . These conditions imply that per(x [cp k -n,cq k ) ) ≤ c q k -c p k ≤ dℓ. Therefore, as c p k -n ≤ c k + dℓ -n ≤ c k -2n/3 and c q k ≥ c k+1 , x [c k -2n/3,c k+1 ) for all k ∈ [i, j -d). (4.4)
We will use (4.4) to prove the lemma by contradiction. Assume that per(

x [ci-n/3,c j-d ) ) > dℓ. Then, by Item (2) in Lemma 9, there exists m ∈ [c i -n/3 + 2dℓ, c j-d ) such that per(x [m-2dℓ,m) ) > dℓ. Now, the condition m ∈ [c i -n/3 + 2dℓ, c j-d ) allows us to find k ∈ [i, j -d) such that m ∈ [c k -n/3, c k+1 ). Hence, as 2dℓ ≤ n/3, x [m-2dℓ,m) occurs in x [c k -2n/3,c k+1 ) , which yields per(x [c k -n/3,c k+1 ) ) ≥ per(x [m-2dℓ,m) ) > ε.
This contradicts (4.4) and completes the proof.

The first coding

In this section, we begin the proof of the main results: Theorems 33 and 34. We start by constructing the codings described in Proposition 13. Then, in Sections 4.5, 4.6, and 4.7, we will modify these codings to obtain new versions of them, each with better properties than the previous one. We will show in Subsection 4.7.2 that the final codings can be connected with morphisms, and we will use this fact in Section 4.8 to complete the proof of the main results.

Proposition 13. Let X be a minimal infinite subshift, n ≥ 1 and let d be the maximum of ⌈p X (n)/n⌉, p X (n + 1) -p X (n), #A and 10 4 . Then, there exist a coding (Z ⊆ C Z , τ : C → A + ) of X and ε ∈ [n/d 2d 3 +4 , n/d) satisfying the following conditions:

(1) C has at most d 3 elements.

(2) |τ (a)| ≤ 3dn for all a ∈ C.

(3) (Z, τ ) is 3dn-recognizable.

(4) The periodicity properties in Proposition 14 are satisfied.

Proposition 14. Consider the coding described in Proposition 13. Let z ∈ Z, x = τ (z) and (c, z) = F (Z,τ ) (x). We define Q p (z) as the set of integers

j ∈ Z such that | root τ (z j )| ≤ ε and x [cj -99ε,cj+1+99ε) = (root τ (z j )) Z [-99ε,|τ (zj )|+99ε) .
(1) 0 ̸ ∈ Q p (z) and |τ (z 0 )| > 401ε implies that per(x [c0+97ε,c1-97ε) ) > ε.

(2) Suppose that 0

̸ ∈ Q p (z) and |τ (z 0 )| ≤ 401ε. If -1 ∈ Q p (z) or 1 ∈ Q p (x), then per(x [c0+97ε,c1-97ε) ) > ε.
(

) If k > d and |τ (z j )| ≤ 401ε for all j ∈ [0, k), then [0, k) ⊆ Q p (z). 3
(4) Let z ′ ∈ Z and assume that 0

∈ Q p (z), 0 ∈ Q p (z ′ ) and that root τ (z 0 ) is conjugate to root τ (z ′ 0 ). Then root τ (z 0 ) = root τ (z ′ 0 )
. We fix, for the rest of the section, the following notation. Let X ⊆ A Z be a minimal infinite subshift, n ≥ 0 and d be the maximum of p X (n)/n, p X (n + 1) -p X (n), #A and 10 4 . We denote by (Y ⊆ B Z , σ : B → A + ) the coding given by Proposition 11 when it is used with X and n. We now define a set W ε ⊆ A + that will be important for controlling the periodicity properties in Proposition 13. We start by introducing classic notions related to periodicity of words. Recall that two words u, v ∈ A + are conjugate if ur = rv for some r ∈ A * . The relation u ∼ R v iff u and v are conjugate is an equivalence relation, and a ∼ R -equivalence class is called a rotation class. A word u ∈ A + is primitive if u = root u ‡ . We fix a set W ε ⊆ A + consisting of one element of the rotation class of each primitive word w ∈ A + such that |w| ≤ ε.

Construction of the first coding

Lemma 35. Let t ∈ A + be such that per(t) ≤ ε and |t| ≥ 198ε + per(t). Then, for some s ∈ W ε , s Z [-99ε,99ε) occurs in t. ‡ We recall the reader that root u is the shortest prefix v of u such that u = v k for some k ≥ 1

Proof. Let u be the prefix of t of length per(t). Note that u is primitive as otherwise per(t) ≤ | root u| < |u| = per(t), which is a contradiction. The primitiveness of u and the inequality |u| = per(t) ≤ ε imply that there exist s ∈ W ε and a suffix u ′ of s such that |s| = |u| and u ′ s is a prefix of uu. Being per(t) = |u| = |s|, we then have that

t is a prefix of u ′ s ∞ . (4.6) We set k = 99ε-|u ′ | |s| . Observe that, since |s| ≤ ε. |u ′ s k | = |u ′ | + k|s| ≤ per(t) + 99ε ε ε = 99ε + per(t).
Hence, |u ′ s k | + 99ε ≤ |t|. From this and Equation (4.6) we deduce that if v is the prefix of s ∞ of length 99ε, then u ′ s k v is a prefix of t. Now, we have the bound

|u ′ s k | = |u ′ | + 99ε -|u ′ | |s| |s| ≥ 99ε. Hence, s Z [-99ε,99ε) is a suffix of u ′ s k v. We conclude that s Z [-99ε,99ε) occurs in t.
Lemma 36. Let w be a word of length n. Then, there exists a decomposition w = vuu ′ v ′ satisfying one of the following sets of conditions. does not occur in uu ′ for all s ∈ W ε .

Proof. Since |w| ≥ 2 • 500ε, there is a decomposition w = v 0 tv ′ 0 , where

|v 0 | = ⌊n/2 -500ε⌋, |v ′ 0 | ≥ n/2 -500ε and |t| = 2 • 500ε.
There are two cases:

(i) s Z [-99ε,99ε) occurs in t for some s ∈ W ε . (ii) s Z [-99ε,99ε) does not occur in t for all s ∈ W ε . Suppose first that case (i) occurs. It is then possible to write t = v 1 uu ′ v ′ 1 , where uu ′ = s Z [-99ε,99ε) and |u| = |u ′ | = 99ε. We set v = v 0 v 1 and v ′ = v ′ 1 v ′ 0 and note that w = vuu ′ v ′ satisfies Condition (a).
We now assume that (ii) holds. Being the length of t equal to 2•500ε, we can write w 0 = uu ′ , where |u| = |u ′ | = 500ε. Then, the decomposition w = v 0 uuv ′ 0 satisfies Condition (b).

We now can define (Z, τ ).

Definition 12. For w ∈ RS n (X), we use Lemma 36 to fix a decomposition w = v w u w u ′ w v ′ w satisfying one of the following conditions:

(P a ) |u w | = |u ′ w | = 99ε, |v w |, |v ′ w | ≥ n/2 -500ε, and u w u ′ w = s Z [-99ε,99ε) for some s ∈ W ε . (P b ) |u w | = |u ′ w | = 500ε, |v w | = ⌊n/2-500ε⌋, |v ′ w | ≥ n/2-500ε, and s Z [-99ε,99ε)
does not occur in u w u ′ w for all s ∈ W ε . Moreover, we choose this decomposition so that |v w u w | is as small as possible. We define (Z ⊆ C Z , τ : C → A + ) as the coding of X obtained from Proposition 6 and the clopen set

U = {x ∈ X : ∃w ∈ RS n (X), x [-|vwuw|,|u ′ w v ′ w |) = w}.

Basic properties of the first coding

Lemma 37. Let x ∈ X and i, j ∈ Z with i < j. Suppose that

x [i-|vwuw|,i+|u ′ w v ′ w |) = w and x [j-|v w u w |,j+|u ′ w v ′ w |) = w for some w, w ∈ RS n (X). Then, i + |u ′ w v ′ w | < j + |u ′ wv ′ w|.
Proof. We assume, with the aim of obtaining a contradiction, that i

+ |u ′ w v ′ w | ≥ j + |u ′ wv ′ w|.
First, we consider the case i

+ |u ′ w v ′ w | = j + |u ′ wv ′ w|. Then, w = x [i+|u ′ w v ′ w |-n,i+|u ′ w v ′ w |) = x [j+|u ′ w v ′ w |-n,j+|u ′ w v ′ w |) = w. Hence, u ′ w v ′ w = u ′ wv ′
w, and therefore

i = (i + |u ′ w v ′ w |) -|u ′ w v ′ w | = (j + |u ′ wv ′ w|) -|u ′ wv ′ w| = j.
This contradicts that i < j.

Next, we assume that

i + |u ′ w v ′ w | > j + |u ′ wv ′ w|.
(4.7)

Note that this is equivalent to i -|v w u w | > j -|v wu w|. This fact will be freely used through the proof.

We consider the following two cases:

(i) i + |u ′ w | < j + |u ′ w|. (ii) i + |u ′ w | ≥ j + |u ′ w|.
Suppose first that case (i) occurs. We are going to define a decomposition w = vuu ′ v ′ as the one in Definition 12 and such that |vu w | < |v wu w|. This would contradict the minimality of |v wu w|.

We start by noting that, thanks to (4.7), if we set

v = x [j-|v w u w |,i-|uw|) and v ′ = x [i+|u ′ w |,j+|u ′ w v ′ w |) , then w = vu w u ′ w v ′ .
Note that from (4.7) we have that

|v| = |x [j-|v w u w |,i-|vwuw|) | + |v w | ≥ n/2 -500ε.
Also, (i) implies that

|v ′ | = |x [i+|u ′ w |,j+|u ′ w v ′ w |) | ≥ |x [j+|u ′ w |,j+|u ′ w v ′ w |) | = |v ′ w| ≥ n/2 -500ε.
We conclude, as

w = v w u w u ′ w v ′ w satisfies

Next, we assume that

i + |u ′ w | ≥ j + |u ′ w|.
Then, as i < j, we have that

[j, j + |u ′ w|) ⊊ [i, i + |u ′ w |).
This implies two things. First, since |u w| = |u ′ w| and

|u w | = |u ′ w |, that [j -|u w|, j + |u ′ w|) ⊊ [i -|u w |, i + |u ′ w |). (4.8) Second, that |u ′ w| < |u ′ w |. Being |u ′ w |, |u ′ w| ∈ {99ε, 500ε},
w = v w u w u ′ w v ′ w . In particular, we can find s ∈ V ε such that u wu ′ w = s Z [-99ε,99ε) . This implies, by (4.8), that s Z [-99ε,99ε) = u wu ′ w occurs in u w u ′ w . But then Condition (P b ) cannot hold for w = v w u w u ′ w v ′ w , contradicting our assumptions. It is convenient to introduce some notation. Let x ∈ X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). For j ∈ Z, we define w j (x) = x [cj -n,cj ) ∈ RS n (X), v j (x) = v wj (x) , u j (x) = u wj (x) , u ′ j (x) = u ′ wj (x) and v ′ j (x) = v ′ wj (x) . Then, x [cj -n,cj ) = w j (x) = v j (x)u j (x)u ′ j (x)v ′ j (x). Observe that if j ∈ Z then x [fj -|vwuw|,fj +|u ′ w v ′ w |) = w for some w ∈ RS n (X), so there exists i ∈ Z such that f j + |u ′ i (x)v ′ i (x)| = c i .
We define ϕ x (j) as the smallest integer such that

f j + |u ′ ϕx(j) (x)v ′ ϕx(j) (x)| = c ϕx(j) . (4.10)
Then, by Lemma 37, ϕ x (i) < ϕ x (j) for all x ∈ X and i < j. (4.11)

Lemma 38. Let x ∈ X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). If i ∈ Z and k ∈ [ϕ x (i), ϕ x (i + 1)), then f i + |u ′ k (x)v ′ k (x)| = c k . Proof. Observe that, since x [c k -n,c k ) = w k (x), there exists j ∈ Z such that f j = c k -|u ′ k (x)v ′ k (x)|.
We are going to prove that j = i. First, we note that, since k ∈ [ϕ x (i), ϕ x (i + 1)) and

c k = f j + |u ′ k (x)v ′ k (x)|, f i + |u ′ ϕx(i) (x)v ′ ϕx(i) (x)| = c ϕx(i) ≤ f j + |u ′ k (x)v ′ k (x)| < c ϕx(i+1) = f i+1 + |u ′ ϕx(i+1) (x)v ′ ϕx(i+1) (x)|. (4.12)
This implies, by Lemma (37), that i ≤ j ≤ i + 1. Now, if j = i + 1, then Equation (4.12) ensures that

f i+1 + |u ′ k (x)v ′ k (x)| is strictly smaller than f i+1 + |u ′ ϕx(i+1) (x)v ′ ϕx(i+1) (x)
|, which contradicts the minimality of ϕ x (i + 1). We conclude that j = i. Lemma 39. The set C has at most d 3 elements.

Proof. Let x ∈ X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). We drop the dependency on x in ϕ x and just write ϕ. The lemma follows from the following claim.

(•) For j ∈ Z, let ζ(j) = (w ϕ(j+1)-1 , x c ϕ(j+1)-1 ) ∈ RS n (X) × A. Then, ζ(i) = ζ(j) implies that x [fi,fi+1) = x [fj ,fj+1) .
Indeed, being Z minimal (as X is minimal and (Z, τ

) is recognizable), (•) implies that #τ (C) = #{x [fj ,fj+1) : j ∈ Z} ≤ # RS n (X) • #A ≤ d 3 ,
where we used that # RS n (X) ≤ #A • (p X (n + 1) -p X (n)) by Item (1) in Proposition 10. This implies, as τ is injective on letters by Proposition 6, that

#C = #τ (C) ≤ d 3 . Let us prove the claim. Suppose that i, j ∈ Z satisfy ζ(i) = ζ(j) = (w, a
). We start with some observations. First, the condition

ζ(i) = ζ(j) = (w, a) implies that (i) w = x [c ϕ(j+1)-1 -n,c ϕ(j+1)-1 ) = x [c ϕ(i+1)-1 -n,c ϕ(i+1)-1 ) ; and (ii) a = x c ϕ(j+1)-1 = x c ϕ(i+1)-1 .
Also, Equation (4.11) ensures that ϕ(i) < ϕ(i + 1) and ϕ(j) < ϕ(j + 1), so ϕ(i) ≤ ϕ(i + 1) -1 < ϕ(i + 1) and ϕ(j) ≤ ϕ(j + 1) -1 < ϕ(j + 1). (4.13)

We now prove the claim (•). The definition of c ϕ(j+1)-1 and c ϕ(j+1) guarantees that the words

x [k-n,k) , k ∈ (c ϕ(j+1)-1 , c ϕ(j+1) ), are not right-special. Thus, x [c ϕ(j+1)-1 ,c ϕ(j+1) ) is determined by x [c ϕ(j+1)-1 -n,c ϕ(j+1)-1 ) and x c ϕ(j+1)-1 . A similar observation holds for x [c ϕ(i+1)-1 ,c ϕ(i+1) )
. Combining these two things with (i) and (ii) yields that

x [c ϕ(j+1)-1 ,c ϕ(j+1) ) = x [c ϕ(i+1)-1 ,c ϕ(i+1) ) . (4.14)
Then, by (i),

w ϕ(j+1) (x) = x [c ϕ(j+1) -n,c ϕ(j+1) ) = x [c ϕ(i+1) -n,c ϕ(i+1) ) = w ϕ(i+1) (x).
Let us write w = w ϕ(j+1) (x) = w ϕ(i+1) (x). With this notation, we have, by (4.10), that

x [fj+1,c ϕ(j+1) ) = x [fi+1,c ϕ(i+1) ) = u ′ wv ′ w.
(4.15)

Now, Equation (4.13) allows us to use Lemma 38 with ϕ(i+1)-1 and ϕ(j+1)-1; we deduce, as w = w ϕ(j+1)-1 = w ϕ(i+1)-1 , that

f j + |u ′ w v ′ w | = c ϕ(j+1)-1 and f i + |u ′ w v ′ w | = c ϕ(i+1)-1 .
In particular,

x [fj ,c ϕ(j+1)-1 ) = x [fi,c ϕ(i+1)-1 ) = u ′ w v ′ w .
This and Equation (4.14) then give that

x [fj ,c ϕ(j+1) ) = x [fi,c ϕ(i+1) ) .
We conclude using (4.15) that x [fj ,fj+1) = x [fi,fi+1) . This completes the proof of the claim and thereby the proof of the lemma.

Lemma 40. Let x ∈ X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). Then: (1) |σ(y k )| ≤ |τ (z 0 )| + 2 • 401ε for any k ∈ [ϕ x (0), ϕ x (1)). (2) |τ (z 0 )| ≤ |σ(y ϕ(1)-1 )| + 2 • 401ε.
Proof. We write, for simplicity, ϕ = ϕ x . Let k ∈ [ϕ(0), ϕ(1)). Then, by Lemma 38,

f 0 + |u ′ k (x)v ′ k (x)| = c k . Hence, τ (z 0 ) • u ′ ϕ(1) (x)v ′ ϕ(1) (x) = x [f0,f1) • x [f1,c ϕ(1) ) = x [f0,c k ) • x [c k ,c ϕ(1) ) = u ′ k (x)v ′ k (x) • σ(y [k,ϕ (1)) ).
In particular, (4.17)

|τ (z 0 )| -|σ(y [k,ϕ(1)) )| = |u ′ ϕ(1) (x)v ′ ϕ(1) (x)| -|u ′ k (x)v ′ k (x)| . (4.16)
Item (1) of this lemma follows. Moreover, since

y ϕ(1)-1 = y [ϕ(1)-1,ϕ(1)) , Item (2)
is also a consequence of (4.17).

Proof of Propositions 13 and 14

We now prove Proposition 13.

Proof of Proposition 13. Item (1) follows directly from Lemma 39. Let us prove Items (2) and (3). We define

U ′ = {x ∈ X : x [0,n) ∈ RS n (X)} and U = {x ∈ X : ∃w ∈ RS n (X), x [-|uwvw|,|u ′ w v ′ w |) = w}.
First, we recall that (Z, τ ) is defined as the coding of X obtained from U as in Proposition 6. Observe that, since |v w u w | ≤ |w| = n and |u ′ w v ′ w | ≤ |w| = n for all w ∈ RS n (X), U has radius n. Also, Item (2) in Proposition 10 ensures that U ′ is (d + 1)n-syndetic, and thus that U is (d + 3)n-syndetic. Therefore, Proposition 6 ensures that |τ (a)| ≤ (d + 3)n for all a ∈ C and that (Z, τ ) is (d + 4)n-recognizable. Since d ≥ #A ≥ 2, Items (2) and (3) follow.

The rest of the section is devoted to prove Proposition 14. Lemma 41. Let x ∈ X and (c, z) = F (Z,τ ) (x). We use the notation w = w ϕx(0) and w = w ϕx(1) . Then, the following are equivalent: Moreover, if any of the previous condition holds, then root τ (z 0 ) ∈ W ε .

(1) | root τ (z 0 )| ≤ ε and x [c0-99ε,c1+99ε) = (root τ (z 0 )) Z [-99ε,|τ (
Proof. We assume that Item (1) holds. Let s = root τ (z 0 ) and note that Item (1) ensures that

|s| ≤ ε and x [c0-99ε,c1+99ε) = s Z [-99ε,|τ (z0)|+99ε) . (4.18)
This allows us to use Lemma 35 with x [c0-99ε,c0+|s|+99ε) and find

t ∈ W ε such that t Z [-99ε,99ε) occurs in x [c0-99ε,c0+|s|+99ε) . Since |s| ≤ ε, we have in par- ticular that t Z [-99ε,99ε) occurs in x [c0-500ε,c0+9ε
) . This is incompatible with the decomposition w = v w u w u ′ w v ′ w satisfying Condition (P b ) in Definition 12; therefore, w = v w u w u ′ w v ′ w satisfies Condition (P a ). A similar argument shows that w = v wu wu ′ wv ′ w also satisfies Condition (P a ). Finally, it follows from (4.18) that per

(x [c0+97ε,c1-97ε) ) ≤ |s| ≤ ε.
We assume that Item (2) holds. Then, by Definition 12, there exist s, s ∈ W ε such that

s Z [-99ε,99ε) = u w u ′ w = x [c0-99ε,c0+99ε) and sZ [-99ε,99ε) = u wu ′ w = x [c1-99ε,c1+99ε) . (4.19) We claim that per(x [c0-99ε,c1+99ε) ) ≤ ε. (4.20)
Assume, with the objective of obtaining a contradiction, that (4.20) is not satisfied. Then, Item (2) in Proposition

9 gives i ∈ [c 0 -98ε, c 1 + 98ε) such that per(x [i-ε,i+ε) ) > ε. We consider three cases. If i ∈ [c 0 -98ε, c 0 + 98ε), then x [i-ε,i+ε) occurs in u w u ′ w .
Thus, by (4.19) and since s ∈ W ε implies that |s| ≤ ε, per(x [i-ε,i+ε) ) ≤ |s| ≤ ε. This contradicts our assumptions. In the case i ∈ [c 1 -98ε, c 1 + 98ε), a similar argument gives a contradiction. Finally, if i ∈ [c 0 + 98ε, c 1 -98ε), then x [i-ε,i+ε) occurs in x [c0+97ε,c1-97ε) and thus, by the hypothesis, per(x [i-ε,i+ε) ) ≤ ε. This proves (4.20).

Our next objective is to use the claim for proving that [-99ε,99ε) . This and the fact that |s| ≤ ε (as s ∈ W ε ) allow us to use Item (1) in Proposition 7 and deduce that s Z = S |τ (z0)| s Z . Item (2) of Proposition 7 then gives that |τ (z 0 )| = 0 (mod |s|). We conclude that x [c0,c1) is a power of s and that root τ (z 0 ) = root s = s. Item (1) of this lemma is a consequence of the last relation and (4.22). This also shows that if Item (2) of the lemma holds, then root τ (z 0 ) ∈ W ε .

|s| = |s| = per(x [c0-99ε,c1+99ε) ). ( 4
Lemma 42. Let x ∈ X, (f, z) = F (Z,τ ) (x) and i, j ∈ Z with j > i + d. Suppose that |τ (z k )| ≤ 401ε for all k ∈ [i, j). Then: (1) root τ (z k ) = root τ (z i ) for all k ∈ [i, j) and | root τ (z i )| ≤ ε. (2) x [fi-99ε,fj +99ε) = (root τ (z i )) Z [-99ε,|τ (z [i,j) )|+99ε) .
Proof. Let (c, y) = F (Y,σ) (x). We will use Lemma 12 with y and [ϕ x (i), ϕ x (j)) to prove the following:

per(x [fi-500ε,fj +500ε
c ϕx(k) -f k = |u ′ ϕx(k) v ′ ϕx(k) | ∈ [n/2 -401ε, n/2 + 401ε)
Hence, as (4.25) ensures that c ϕx(j)-d ≥ c ϕ(j) -ε and since ε ≤ n/10 4 , we have that x [fi-500ε,fj +500ε) occurs in x [c ϕx (i) -n/3,c ϕx (j)-d ) . Therefore, (4.24) holds.

Next, we use (4.24) to prove the following:

∀k ∈ [i, j], the decomposition w ϕx(k) = v ϕx(k) u ϕx(k) u ′ ϕx(k) v ′ ϕx(k)
(k) = v ϕx(k) u ϕx(k) u ′ ϕx(k) v ′ ϕx(k) , then s Z [-99ε,99ε) occurs in u ϕx(k) u ′ ϕx(k) = x [c k -500ε,c k +500ε) , contradicting (P b ). Therefore, w ϕx(k) = v ϕx(k) u ϕx(k) u ′ ϕx(k) v ′ ϕx(k)
satisfies (P a ) and (4.26) is proved.

We now prove the properties in the statement of the lemma. Let k ∈ [i, j). Then, Equations (4.24) and (4.26) imply that Item (1) in Lemma 41 is satisfied. Hence, for all k ∈ [i, j),

| root τ (z k )| ≤ ε and x [c k -99ε,c k+1 +99ε) = (root τ (z k )) Z [-99ε,|τ (z k )|+99ε) . (4.27)
In particular, we have for every k ∈ [i, j -1) that

(root τ (z k )) Z [0,99ε) = x [c k ,c k +99ε) = x [c k+1 ,c k+1 +99ε) = (root τ (z k+1 )) Z [0,99ε) .
This and the inequalities | root τ (z k )| ≤ ε and | root τ (z k+1 )| ≤ ε allow us to use Theorem 31 to deduce that root τ (z k ) and root τ (z k+1 ) are powers of a common word, and thus that root τ (z k ) = root τ (z k+1 ). And inductive argument then yields Item (1) of this lemma, and therefore, by (4.27), that Item (2) holds as well.

We have all the necessary elements to prove Proposition 14.

Proof of Proposition 14. We prove Item (1) by contradiction, Suppose that 0 ̸ ∈ Q p (z), |τ (z 0 )| > 401ε and that per(x [c0+97ε,c1-97ε) ) is at most ε. Let us write w = w ϕx(0) and w = w ϕx(1) . Then, the condition 0 ̸ ∈ Q p (z) ensures that Item (1) in Lemma 41 does not hold. Hence, Item (2) does not hold either. This implies, as per(x [c0+97ε,c1-97ε) ) at most ε, that one of the decompositions w = v w u w u ′ w v ′ w or w = v wu wu ′ wv ′ w satisfies (P b ) in Definition 12. We assume, without loss of generality, that w = v w u w u ′ w v ′ w satisfies (P b ). Then, for any s ∈ W ε , s Z [-99ε,99ε) does not occur in u w u ′ w = x [c0-401ε,c0+401ε) . In particular, s Z [-99ε,99ε) does not occur in x [c0+97ε,c0+304ε) . Being this valid for all s ∈ W ε and since x [c0+97ε,c0+304ε) has length at least 2ε, we deduce from Lemma 35 that per(x

[c0+97ε,c0+304ε) ) > ε. But c 1 -c 0 = |τ (z 0 )| > 401ε so x [c0+97ε,c0+304ε) occurs in x [c0+97ε,c1-97ε
) and thus per(x [c0+97ε,c1-97ε) ) > ε. This contradicts our assumptions and thereby proves Item (1).

We continue with Item (2). The proof is by contradiction. We assume that the hypothesis of Item (2) holds and that per(x [c0+97ε,c1-97ε) ) ≤ ε. Let us further assume, without losing generality, that 1 ∈ Q p (z). We will use the notation w = w ϕx(0) and w = w ϕx(1) . Then, the condition 1 ∈ Q p (z) is equivalent to Item (1) of Lemma 41 being satisfied by Sz; hence, Item (2) of that lemma holds with Sz. In particular, w = v wu wu ′ wv ′ w satisfies (P a ) in Definition 12, that is,

x [c1-99ε,c1+99ε) = u wu ′ w = s Z [-99ε,99ε) for some s ∈ W ε .
Now, the condition 0 ̸ ∈ Q p (z) implies, by Lemma 41, that Item (2) of that lemma is not satisfied by z. This implies, since w = v wu wu ′ wv ′ w satisfies (P a ) and since we assumed that per(x [c0+97ε,c1-97ε) ) ≤ ε, that w = v w u w u ′ w v ′ w satisfies (P b ). Therefore,

x [c1-99ε,c1+99ε) = s Z [-99ε,99ε) does not occur in x [c0-500ε,c0+500ε) . (4.28) But, since c 1 -c 0 = |τ (z 0 )| ≤ 401ε, we have that [c 1 -99ε, c 1 + 99ε) is contained in [c 0 -500ε, c 1 +500ε
), and thus that x [c1-99ε,c1+99ε) occurs in x [c0-500ε,c1+500ε) . This contradicts (4.28), finishing the proof of Item (2).

Next, we consider Item (3). Assume that k > d and that |τ (z j )| ≤ 401ε for all j ∈ [0, k). Then, we can use Lemma 42 to deduce that (1) root τ (z j ) = root τ (z 0 ) for all j ∈ [0, k) and | root τ (z 0 )| ≤ ε;

(2) x [f0-99ε,f k +99ε) = (root τ (z 0 )) Z [-99ε,|τ (z [0,k) )|+99ε) .
In particular,

x [cj -99ε,cj+1+99ε) = (root τ (z j )) Z [-99ε,|τ (zj )|+99ε) and | root τ (z j )| ≤ ε for all j ∈ [0, k). We conclude that [0, k) ⊆ Q p (z).
Finally, we prove Item (4). Let z ′ ∈ Z and assume that 0

∈ Q p (z) ∩ Q p (z ′ ) and that root τ (z 0 ) is conjugate to root τ (z ′ 0 ). The condition 0 ∈ Q p (z) ∩ Q p (z ′
) permits to use Lemma 41 to deduce that root τ (z 0 ) and root τ (z ′ 0 ) belong to W ε . Since root τ (z 0 ) conjugate to root τ (z ′ 0 ), the definition of W ε ensures that root τ (z 0 ) = root τ (z ′ 0 ).

The second coding

We continue the proof of the main theorems. The main result of this section is Proposition 15, which describes a modification of the coding in Proposition 4.4. The principal new element in Proposition 15 is a period dichotomy for the words τ (a). This property is shared by the codings constructed in Sections 4.6 and 4.7, so we introduce it as a definition.

Definition 13. Let (Z ⊆ C Z , τ : C → A + ) be a recognizable coding of the subshift X ⊆ A Z , C ap ∪ C p be a partition of C, and ε ≥ 1. We say that (Z, τ ) has dichotomous periods w.r.t. (C ap , C p ) and ε if for x ∈ X and (c, z) = F (Z,τ ) (x) the following holds:

(1) z 0 ∈ C ap implies that per(x [c0+ε,c1-ε) ) > ε.

(2)

z 0 ∈ C p implies that | root τ (z 0 )| ≤ ε and that x [c0-ε,c1+ε) is equal to (root τ (z 0 )) Z [-ε,|τ (z0)|+ε) .
(3) If a ∈ C p and root τ (z 0 ) is conjugate to root τ (a), then root τ (z 0 ) = root τ (a).

Proposition 15. Let X be a minimal infinite subshift, n ≥ 0 and let d be the maximum of ⌈p X (n)/n⌉, p X (n + 1) -p X (n), #A and 10 4 . There exist a recognizable coding (Z ⊆ C Z , τ : C → A + ) of X, a partition C = C ap ∪ C p , and ε ∈ [n/d 2d 3 +4 , n/d) such that:

(1) #C ap ≤ 2d 3d+6 , #C p ≤ 2d 3d+9 pow-com(X) and # root τ (C) ≤ 3d 3d+9 .

(2) |τ (a)| ≤ 10d 2 n for a ∈ C ap and |τ (a)| ≥ 80ε for a ∈ C.

(3) (Z, τ ) satisfies the recognizability property in Proposition 16.

(4) (Z, τ ) has dichotomous periods w.r.t. (C ap , C p ) and 8ε.

(5) The set C p satisfies the following: if z ∈ Z, then z 0 and z 1 does not simultaneously belong to C p .

Proposition 16. Consider the coding described in Proposition 15. Let x, x ∈ X be such that per(x

[-ε,ε) ) > ε and x [-7d 2 n,7d 2 n) = x[-7d 2 n,7d 2 n) . Then, F 0 (Z,τ ) (x) is equal to F 0 (Z,τ ) (x).
The strategy for proving Proposition 15 is as follows. We consider the coding (Y, σ) given by Proposition 13 and, for a point y ∈ Y , we glue together letters y i to form words y I , where I is an interval, in such a way that y I corresponds either to a maximal periodic part of σ(y) or to an aperiodic part of σ(y) of controlled length. This will produce a new coding where the letters are in correspondence with the words y I and that satisfies all the properties in Proposition 15 except for the lower bound in Item (2) for the letters associated to periodic parts y I . We solve this by slightly moving the edges of the words σ(y I ).

We start, in Subsection 4.5.1, by defining stable intervals, which correspond to the intervals I described in the last paragraph. The definition of the coding of Proposition 15 is given in Subsection 4.5.2, together with the proof of its basic properties. In the final subsection, we prove Propositions 15 and 16.

We fix the following notation for the rest of the section. Let X ⊆ A Z be a minimal infinite subshift, n ≥ 0 and let d be the maximum of ⌈p X (n)/n⌉, p X (n + 1) -p X (n), #A and 10 4 . Then, Proposition 13 applied to X and n gives a recognizable coding (Y ⊆ B Z , σ : B → A + ) of X and an integer ε ∈ [n/d 2d 3 +4 , n/d).

Stable intervals

Let y ∈ Y , x = σ(y) and (c, y) = F (Y,σ) (x). We define

Q short (y) = {i ∈ Z : |σ(y i )| ≤ 401ε} and Q long (y) = Z \ Q long . Let Q p (y) be the set of integers i ∈ Z such that | root σ(y i )| ≤ ε and x [ci-99ε,ci+1+99ε) = (root σ(y i ) Z ) [-99ε,|τ (yi)|+99ε) . (4.29)
We set Q ap (y) = Z \ Q p (y). Remark that the definition of Q p (y) coincides with the one in Proposition 14. Definition 14. A stable interval for y is a finite interval I = [i, j) ⊆ Z satisfying one of the following conditions.

(1) I ⊆ Q p .

(2)

I ⊆ Q ap , #(I ∩ Q long ) ≤ 1, and if i ∈ Q short then i -1 ∈ Q p .
The interval I is of periodic type if it satisfies Item (1) of this definition, and of aperiodic type if it satisfies Item (2). We say that I is a maximal stable interval if for all stable interval I ′ ⊇ I we have that I ′ = I.

Remark 13. We stress on the fact that the previous definition does not depend just on y, but also on σ and ε.

Lemma 43. Let I = [i, j) be a stable interval set for y of periodic type. Then:

(1) root σ(y [i,j) ) = root σ(y k ) for all k ∈ [i, j) and x [ci-99ε,cj +99ε) is equal to

(root σ(y i )) Z [-99ε,|σ(y [i,j) )|+99ε) .
(2) If I is maximal,

I ′ = [i ′ , j ′
) is a stable interval and either i ′ = j or j ′ = i, then I ′ is of aperiodic type.

(

) If y ′ ∈ Y , 0 ∈ Q p (y ′ ) and root σ(y i ) is conjugate to root σ(y ′ 0 ), then root σ(y i ) = root σ(y ′ 0 ). 3
Proof. We first prove Item (1). Let s k = root σ(y k ). Being I of periodic type, we have by Definition 14 that

|s k | ≤ ε and x [c k -99ε,c k+1 +99ε) = (s Z k ) [-99ε,|σ(y k )|+99ε) for all k ∈ [i, j). (4.30)
Then, for any k ∈ [i, j -1),

(s Z k ) [0,99ε) = x [c k ,c k +99ε) = x [c k+1 ,c k+1 +99ε) = (s Z k+1 ) [0,99ε) .
Combining this with the inequalities |s k |, |s k+1 | ≤ ε and Theorem 31 produces a word t such that s k and s k+1 are powers of t. Hence, as s k and s k+1 are roots of a word, s k = s k+1 = t for any k ∈ [i, j -1). Item (1) of the lemma follows from this and (4.30). For Item (2), we note that if I ′ is of periodic type then I ∪ I ′ is an interval contained in Q p (y), and so I ∪ I ′ is a stable interval for y. This would contradict the maximality of I; therefore, I ′ is of aperiodic type.

Let us now assume that the hypothesis of Item (3) holds. Then, since i ∈ I ⊆ Q p (y), 0 ∈ Q p (y ′ ) and root σ(y i ) is conjugate to root σ(y ′ 0 ), the points S i y and y ′ comply with the hypothesis of Item (4) of Proposition 14. We conclude tat root σ(y i ) = root σ(y ′ 0 ). Lemma 44. Let y ∈ Y and I = [i, j) be a stable interval for y of aperiodic type. Then,

(1) per(x [ci+97ε,cj -97ε) ) > ε;

(2) I has length at most 2d + 1;

(3) 195ε ≤ |σ(y I )| ≤ 9d 2 n.

Proof. We prove Item (1) by considering two cases. Suppose that i ∈ Q long (y). Then, i ∈ Q long (y) ∩ Q ap (y) and we can use Item (1) of Proposition 14 to obtain that per(x [ci+97ε,ci+1-97ε) ) > ε. Assume now that i ∈ Q short (y). Then, Definition 14 ensures that i -1 ∈ Q p (y). Hence, Item (2) of Proposition 14 applies and so per(x [ci+97ε,ci+1-97ε) ) > ε. In particular, Item (1) holds.

We prove Item (2) by contradiction. Assume that #I > 2d + 1. Then, it follows from Definition 14 that there exists I ′ ⊆ I such that #I ′ > d and I ′ ⊆ Q short . These conditions allow us to use Item (3) in Proposition 14 and deduce that I ′ ⊆ Q p (y), contradicting the fact that I is of aperiodic type.

Finally, we consider Item (3). Item (2) of this lemma and Item (2) produce that |σ(y I )| ≤ (2d + 1) • 3dn, from which the upper bound in Item (3) follows.

To prove the lower bound, we consider two cases. If there exists k ∈ I ∩Q long (y), then |σ(y I )| ≥ |σ(y k )| ≥ 401ε. Assume now that I ∩ Q long (y) is empty. Then, i ∈ Q short (y), and so Definition 14 indicates that i -1 ∈ Q p (y). This allows us to use Item (2) of Proposition 14 to obtain that per(x

[ci+97ε,ci+1-97ε) ) > ε. In particular, |x [ci+97ε,ci+1-97ε) | > ε; hence, |σ(y I )| ≥ |σ(y i )| = |x [ci+97ε,ci+1-97ε) | + 2 • 97ε > 195ε.
Lemma 45. There exists a constant C depending only on X such that for any y ∈ Y and stable interval I for Y , we have that #I ≤ C. In particular, any stable interval is contained in a maximal stable interval.

Proof. Let C 0 be the length of the longest word w that occurs in some x ∈ X such that per(w) ≤ ε. We remark that C 0 is finite as X is assumed to be minimal and infinite. Let C = max{C 0 , 2d + 1}. We claim that for any y ∈ Y , any stable interval I for y has length at most C. Indeed, if I is of aperiodic type, then Item (3) of Lemma 44 implies that #I ≤ 2d + 1 ≤ C, and if I is of periodic type, then Item (1) of Lemma 43 ensures that per(σ(y I )) ≤ ε, and thus that #I ≤ |σ(y

I )| ≤ C 0 ≤ C.
Lemma 46. Let y ∈ Y . Then, the set of all maximal stable intervals for y is a partition of Z.

Proof. We first prove that any k ∈ Z is contained in a stable interval. This would imply that any k is contained in a maximal stable interval by Lemma 45.

Let k ∈ Z be arbitrary. We consider two cases. If k ∈ Q p (y) or k ∈ Q ap (y) ∩ Q long (y), then {k} is stable interval and we are finished. Suppose now that k ∈ Q ap (y) ∩ Q short (y). Let i < k be the biggest integer such that

i ̸ ∈ Q ap (y) ∩ Q short (y). Note that [i + 1, k] ⊆ Q ap (y) ∩ Q short (y). Hence, if i ∈ Q ap (y) ∩ Q long (y), then [i, k]
is stable interval of aperiodic type, and if i ∈ Q p , then [i + 1, k] is stable interval of aperiodic type. These are the only cases as i ̸ ∈ Q ap (y) ∩ Q short (y), and so we conclude that i belongs to a stable interval.

Next, we prove that for any maximal stable intervals I, I ′ , either I = I ′ or I ∩ I ′ = ∅. The lemma follows from this and the fact that any k is contained in a maximal stable interval.

Let I = [i, j) and I ′ = [i ′ , j ′ ) be maximal stable intervals with nonempty intersection. There is no loss of generality in assuming that i ≤ i ′ < j ≤ j ′ . Note that if i = i ′ or j = j ′ , then I ∪ I ′ ∈ {I, I ′ }, so I = I ′ = I ∪ I ′ by maximality. Hence, we may assume that i < i ′ < j < j ′ . Remark that this implies that j -1 ∈ I ∩ I ′ .

In order to continue, we consider three cases.

(1) If j -1 ∈ Q p (y), then, as j -1 ∈ I ∩ I ′ , Definition 14 implies that I and I ′ are of periodic type. It then follows from Definition 14 that I ∪ I ′ is stable interval of periodic type, and so I = I ′ = I ∪ I ′ by maximality.

(

) If j -1 ∈ Q ap (y) ∩ Q long . Then, since j -1 ∈ I ∩ I ′ , Definition 14 ensures that [i, j -1) ∪ [j + 1, j ′ ) ⊆ Q ap (y) ∩ Q short (y). 2
Hence, [i, j ′ ) = I ∪ I ′ is a stable interval of aperiodic type, which implies that I = I ′ = I ∪ I ′ by maximality.

(3) If j -1 ∈ Q ap (y) ∩ Q short (y). Then, as j -1 ∈ I ∩ I ′ , Definition 14 guarantees that I and I ′ are of aperiodic type. In particular, as i ′ -1 ∈ I, i ′ -1 ∈ Q ap (y) and therefore, by Definition 14, i ′ ∈ Q long (y). We conclude, using Definition 14, that

[i, i ′ ) ⊆ Q ap (y) ∩ Q short (y), i ′ ∈ Q ap (y) ∩ Q long (y) and that [i ′ + 1, j ′ ) ⊆ Q ap (y) ∩ Q short (y). Hence, [i, j ′ ) = I ∪ I ′ is

Construction of the second coding

The coding (Z, τ ) is obtained by modifying the cut function c in F (Y,σ) (x) of the points x ∈ X. We give the construction of the modified cut function as the proof of the following lemma, and we define (Z, τ ) right after.

Lemma 47. Let x ∈ X and set (c, y) = F (Y,σ) (x). There exist unique increasing sequences of integers satisfying (k x (j)) j∈Z and (r x (j)) j∈Z satisfying the following conditions.

(1) {[k x (j), k x (j + 1)) : j ∈ Z} is the set of all maximal stable intervals of y.

(2) For any j ∈ Z,

(1) if [k x (j), k x (j + 1)) is of aperiodic type, then r x (j) = c kx(j) .

(2) if [k x (j), k x (j +1)) is of periodic type, then r x (j) = c kx(j) -|s ℓ |, where s = root σ(y [kx(j),kx(j+1)) ) and ℓ = ⌈80ε/|s|⌉

(3) 0 belongs to [r x (0), r x (1)).

Moreover, in this case, r x (j + 1) ≥ r x (j) + 80ε for all j ∈ Z.

Moreover, Item (2) in Lemma 43 guarantees that [k(j+1), k(j+2)) is of aperiodic type. Hence, by Item (2) in Lemma 47, f j = c k(j) -|s ℓ | and f j+1 = c k(j+1) , where ℓ = ⌈80ε/|s|⌉. We can then compute, thanks to (4.34) and (4.33),

x [fj -8ε,fj+1+8ε) = s Z [-|s ℓ |-8ε,c k(j+1) -c k(j+1) +8ε) = s Z [-8ε,|s ℓ |+c k(j+1) -c k(j+1) +8ε) = s Z [-8ε,fj+1-fj +8ε) .
Note that the last computation also shows that τ

(z j ) = x [fj ,fj+1) is equal to s Z [-|s ℓ |,c k(j+1) -c k(j+1) ) = s Z [-|s ℓ |,0) σ(y [k(j),k(j+1))
). Hence, root τ (z j ) = root s = s and we have proved (1.c).

Observe that if (1.c) holds, then per(τ (z j )) ≤ |s| ≤ ε and z j ∈ C p by the definition of C p .

We now assume (2.a). Then, Equation (4.33) implies that per(x [fj ,fj+1) ) ≥ per(x [c k(j) ,c k(j+1) -81ε) ) > ε. Hence, by Lemma 43, [k(j), k(j + 1)) is not of periodic type, that is, [k(j), k(j + 1)) is of aperiodic type.

Let us suppose that (2.b) holds. In this case, Lemma 44 and (4.33) allows us to compute

per(x [fj +8ε,fj+1-8ε) ) ≥ per(x [c k(j) +97ε,c k(j+1) -97ε) ) > ε. Finally, if (2.c) is satisfied, then per(τ (z j )) ≥ per(x [fj +8ε,fj+1-8ε) ) > ε.
Lemma 50. Suppose that z -1 z 0 z 1 ∈ C ap C p C ap . Then, there exists a decomposition τ (z -1 z 0 z 1 ) = us m u ′ such that: (2) s ∈ root σ(B).

(3) s is not a suffix of u and is not a prefix of u ′ .

Proof. Let us denote k(j) = k x (j). We define s = root τ (z 0 ). Then, as z 0 ∈ C p , Lemma 49 ensures that |s| ≤ ε, x [f0-8ε,f1+8ε) = s Z [-8ε,f1-f0+8ε) , s = root σ(y [k(0),k(1)) ) and that [k(0), k(1)) is of periodic type in y. Thus, by Lemma 43, s = root σ(y k(0) ) ∈ σ(B). In particular, s satisfies Item (2) of this lemma. Now, we can find an interval

I = [i, j) containing [f 0 -8ε, f 1 + 8ε) such that x I = s Z [i-f0,j-f0
) and that no other interval strictly containing I satisfies the same properties. We observe that i ≥ f -1 +8ε as, otherwise, per(x [f-1+8ε,f0-8ε) ) ≤ ε, contradicting the fact that, since z -1 ∈ C ap , Item (2.c) Lemma 49 holds. Similarly, j ≤ f 2 -8ε. From these two things and the fact that I contains [f 0 -8ε, f 1 + 8ε) we obtain that

f -1 + 8ε ≤ i ≤ f 0 -8ε and f 1 + 8ε ≤ j ≤ f 2 -8ε. (4.35)
This allows us to write

x [f-1,f1) = us m v, where |u| ∈ [i -f -1 , i -f -1 + |s|), m ≥ 0 and |v| ∈ [f 2 -j, f 2 -j + |s|).
We have from (4.35) that |u| ≥ i-f -1 ≥ 8ε and |v| ≥ f 2 -j ≥ 8ε. Moreover,

as |s| ≤ ε, |u| ≤ f 0 -f -1 -7ε = |τ (a)| -7ε and |v| ≤ f 2 -f 1 -7ε = |τ (a ′ )| -7ε.
This proves that Item (1) of the lemma holds. Item (3) follows from the fact that |s| ≤ ε, (4.35) and the maximality of I.

Proof of Propositions 15 and 16

We are ready to prove the main results of this section.

Proof of Proposition 16. Let x, x ∈ X be such that per(x

[-ε,ε) ) > ε and x [-7d 2 n,7d 2 n) = x[-7d 2 n,7d 2 n) . We define (c, y) = F (Y,σ) (x), (c, ỹ) = F (Y,σ) (x), (f, z) = F (Z,τ ) (x)
and ( f , z) = F (Z,τ ) (x). Let k(j) = k x (j) and k(j) = k x(j) be the sequences from Lemma 47. With this notation, we have to prove that f 0 = f0 and z 0 = z0 .

Note that per(x [f0-8ε,f1+8ε) ) ≥ per(x [-ε,ε) ) > ε. Thus, by Lemma 49, [k(0), k(1)) is of aperiodic type in y.

We claim that (4.37)

[c k(0)-1 -3dn, c k(1)+1 + 3dn) is contained in [-7d 2 n, 7d 2 n) (4.
We now observe that (4.36) and the hypothesis guarantees that

x [c k -99ε,c k+1 +99ε) = x[c k -99ε,c k+1 +99ε) for every k ∈ [k(0) -1, k(1) + 1).
(4.38) Thus, for any such k, k ∈ Q p (y) if and only if Q p (ỹ). It is then not difficult to verify, using the definition of stable interval, that if i ∈ {0, 1} then (I) k(i) = k(i);

(II) the type of [k(i), k(i + 1)) in y and the type of [ k(i), k(i + 1)) in ỹ are equal.

Then, (I) and (4.38) imply that

x [k(0)-99ε,k(1)+99ε) = x[ k(0)-99ε, k(1)+99ε) . (4.39)
We claim that f 0 = f0 and f 1 = f1 . (4.40)

Let i ∈ {0, 1}. We consider two cases. First, we assume that [k(i), k(i + 1)) is of aperiodic type in y. Then, then by (II), [ k(i), k(i + 1)) is of aperiodic type in ỹ. Hence, by Item (2) in Lemma 47, f i = c k(i) and fi = ck (i) . This gives f i = fi by (I) and (4.37).

Next, we assume that [k(i), k(i+1)) is of periodic type in y. Then, [ k(i), k(i+ 1)) is of periodic type in ỹ by (II). Hence, by Lemma 43, s = root σ(y ) . In this situation, (I) and (4.38) ensures that

[k(i),k(i+1)) ) and s = root σ(ỹ [ k(i), k(i+1)) ) satisfy |s|, |s| ≤ ε, x [c k(i) ,c k(i) +2ε) is equal to s Z [0,2ε) and x[ck (i) ,ck (i) +2ε) is equal to sZ [0,2ε
s Z [0,2ε) = x [c k(i) ,c k(i) +2ε) = x[ck (i) ,ck (i) +2ε) = sZ [0,2ε
) . Since |s|, |s| ≤ ε, this allows us the use of Theorem 31 and deduce that s = s. Putting this and the fact that [k(i), k(i + 1)) and [ k(i), k(i + 1)) are of periodic type in Item (2) of Lemma 47 produces

f i = c k(i) -|s ℓ | and fi = ck (i) -|s ℓ |,
where ℓ = ⌈80ε/|s|⌉. Therefore, as c k(i) = ck (i) by (I) and (4.37), f i = fi . This completes the proof of (4.40).

Finally, we show that z 0 = z0 . Item (2) in Lemma 47 gives that |f i -c k(i) | ≤ 81ε. Hence, by (4.39) and (4.40), x [f0,f1) = x[f0,f1) = x[ f0, f1) . We conclude that τ (z 0 ) = τ (z 0 ), and therefore that z 0 = z0 as τ is injective on letters by Proposition 6.

We end this section with the proof of Proposition 15.

Proof of Proposition 15. Let x ∈ X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). Let k(j) = k x (j) be the sequence from Lemma 47.

We start with Item (ii). Let a ∈ C ap . By minimality, there exists j ∈ Z such that z j = a. We compute as follows:

|τ (z j )| = |f j+1 -f j | ≤ |f j+1 -c k(j+1) | + (k(j + 1) -k(j))|σ| + |f j -c k(j) |.
(4.46) implies that #C p ≤ #C 2 ap • #(∪ s∈root σ(B) Pow X (s)). Hence, #C p ≤ #C 2 ap • #B • pow-com(X).
Since #C ap ≤ 2d 3d+6 and since #B ≤ d 3 by Item (2) in Proposition 13, it follows that C p ≤ 2d 3d+9 pow-com(X).

Item (3) is a direct consequence of Proposition 16. Let us prove Item (4). Lemma 49 ensures that (Z, τ ) satisfies Items ((1)) and ((2)) of Definition 13. Let now x ∈ X, (c, z) = F (Z,τ ) (x) and a ∈ C p be such that root τ (z 0 ) is conjugate to root τ (a). We note that, by Lemma 49,

| root τ (z 0 )| = | root τ (a)| ≤ ε. Hence, per(x [c0+8ε,c1-8ε) ) ≤ | root τ (z 0 )| ≤ ε.
This implies, by Lemma 49, that z 0 ∈ C ap . We can then use (4.45) to get that root τ (z 0 ) = root σ(y 0 ) and root τ (a) = root σ(y ′ 0 ) for certain y, y ′ ∈ Y such that 0 ∈ Q p (y) ∩ Q p (y ′ ). We remark that, since root τ (z 0 ) is conjugate to root τ (a), the words root σ(y 0 ) and root σ(y ′ 0 ) are conjugate. Therefore, y and y ′ satisfy the hypothesis of Item (3) of Lemma 43. We conclude that root τ (z 0 ) = root σ(y 0 ) = root σ(y ′ 0 ) root τ (a). It is left to prove Item (5). Let j ∈ Z. We have, from Lemma 43, that [k(j), k(j + 1)) or [k(j + 1), k(j + 2)) is of aperiodic type. Hence, by Lemma 49, z j or z j+1 belongs to C ap .

The third coding

We continue refining the codings. The main addition to this version is that the words τ (a) have controlled lengths. The properties of the new coding are summarized in Proposition 17.

Proposition 17. Let X ⊆ A Z be a minimal infinite subshift, n ≥ 1 and let d be the maximum of ⌈p X (n)/n⌉, p X (n + 1) -p X (n), #A and 10 4 . There exist a recognizable coding

(Z ⊆ C Z , τ : C → A + ) of X, a partition C ap ∪ C sp ∪ C wp of C and ε ∈ [n/d 2d 3 +4 , n/d) such that: (1) #C ap ≤ 2d 3d+6 , #C sp ≤ 3d 3d+6 , #C wp ≤ 2d 3d+9 pow-com(X) 4 and # root τ (C) ≤ 5d 3d+6 .
(2) 20ε ≤ |τ (a)| ≤ 10d 2 n for all a ∈ C.

(3) (Z, τ ) satisfies the recognizability property described in Proposition 18.

( Proposition 18. Consider the coding described in Proposition 17. For any x, x ∈ X, we have that:

(

1) If per(x [-ε,ε) ) > ε and x [-7d 2 n,7d 2 n) = x[-7d 2 n,7d 2 n) , then F 0 (Z,τ ) (x) is equal to F (Z,τ ) (x). (2) If k ≥ 0, x [-50d 2 n,k+50d 2 n) = x[-50d 2 n,k+50d 2 n) and F 0 (Z,τ ) (x) is equal to F (Z,τ ) (x), then F 0 (Z,τ ) (S k x) is equal to F (Z,τ ) (S k x
). Proposition 19. The coding of Proposition 17 satisfies the following.

(1) If z ∈ Z and i < j are integers such that z k ∈ C \ C ap for all k ∈ [i, j + 1), then z k = z i ∈ C sp for all k ∈ [i, j) and root τ (z k ) = root τ (z i ) for all k ∈ [i, j + 1).

(2) If a ∈ C sp , then τ (a) = (root τ (a)) 2 r , where r is the unique integer for which 2 r | root τ (a)| belongs to [20ε, 40ε).

We now introduce the notation that will be used in this section. Let X ⊆ A Z be a minimal infinite subshift, n ≥ 1 and let d be the maximum of ⌈p X (n)/n⌉, p X (n + 1) -p X (n), #A and 10 4 . Then, Proposition 15 applied to X and n gives a recognizable coding (Y ⊆ C Z Y , σ : C → A + ) of X, a partition C Y = C p ∪C ap , and an integer ε ∈ [n/d 2d 3 +4 , n/d) satisfying the properties described in Proposition 15.

The strategy to prove the main proposition of this section is the following. The coding (Z, τ ) will be obtained from (Y, σ) by splitting the words in σ(C p ) into subwords of carefully chosen lengths. This will maintain most of the properties of (Y, σ) at the same time that we gain control on the lengths of all the words τ (a). A delicate part involves defining the splittings of the words in σ(C p ) in such a way that (Z, τ ) has the recognizability properties in Proposition 18.

Construction of the third coding

For s ∈ root σ(C p ), we define ζ(s) as the unique power of two such that ζ(s) • |s| lies in [20ε, 40ε). Note that, by Item (4) of Proposition 15, we have that ζ(s) ≥ 1. Then, for a ∈ C p , we can define p a and q a as the unique integers satisfying Note that σ(a) = ψ sp (a) pa-1 ψ wp (a) for all a ∈ C p .

We also choose bijections

ϕ sp : C sp → ψ sp (C p ) and ϕ wp : C wp → ψ wp (C p ),
where C sp , C wp and C ap are pairwise disjoint. Then, we define for a ∈ C Y ,

η(a) = a if a ∈ C ap ϕ -1 sp (ψ sp (a)) pa-1 ϕ -1 wp (ψ wp (a)) if a ∈ C wp
(a) = τ (b) pa-1 τ (c) if a ∈ C p , b = ϕ -1 sp (ψ sp (a)) and c = ϕ -1
wp (ψ wp (c)). In other words, τ is obtained from σ by slicing the words σ(a).

Proof of Propositions 17, 18 and 19

Proof of Proposition 17. We start with Item (1). Item (1) in Proposition 15 gives the bound #C ap ≤ 2d 3d+6 . Also, it follows from the definitions and Item (1) in Proposition 15 that Let us now assume that z 0 ∈ C Z \ C ap . This condition and Equation (4.49) imply that y 0 ∈ C p . Hence, we can use Item (4) in Proposition 15 to obtain that per(x [c0-8ε,c1+8ε) ) ≤ ε. We conclude, since (4.49) guarantees that [f

#C sp = #ψ sp (C p ) = # root σ(C p ) ≤ 3d 3d+6 and #C wp = #ψ wp (C p ) ≤ #C p ≤ 2d 3d+9 • pow-com(X).
0 , f 1 ) is contained in [c 0 , c 1 ), that per(x [f0-8ε,f1+8ε) ) ≤ per(x [c0-8ε,c1+8ε) ) ≤ ε.
It rests to prove that (Z, τ ) satisfies Item (3) of Definition 13. Let a ∈ C Z \ C ap be such that root τ (z 0 ) is conjugate to root τ (a). We have from (4.52) that z 0 occurs in η(y 0 ), so, by (4.49), root τ (z 0 ) = root σ(y 0 ). Similarly, a occurs in η(b) and root τ (a) = root σ(b) for some b ∈ C Y . The first condition and (4.49) imply, as a ∈ C Z \ C ap , that b ∈ C p . Now, the hypothesis ensures that root σ(y 0 ) is conjugate to root σ(b). Therefore, as (Y, σ) has dichotomous periods w.r.t. (C ap , C p ), we can use Item (3) of Definition 13 to obtain that root σ(y 0 ) = root σ(b). We conclude that root τ (z 0 ) = root τ (a), completing the proof of Item (3).

Finally, for Items (4) and (5), we present the proofs of Propositions 18 and 19 hereafter.

Lemma 51. Let C block be the set of words a ℓ b, where a ∈ C sp , b ∈ C wp , ℓ ≥ 1 and root τ (a) = root τ (b). Then, any z ∈ Z can be written as z = . . . w -1 w 0 w 1 . . . , where w j ∈ C block or w j ∈ C ap and w j w j+1 ̸ ∈ C 2 block .

We are going to prove that

if x [-50d 2 n,50d 2 n) = x[-50d 2 n,50d 2 n) , then F 0 (Z,τ ) (S f1 x) = F 0 (Z,τ ) (S f1 x). (4.58)
The lemma then follows from an inductive argument on k that uses Equations (4.57) and (4.58).

Let us assume that x [-50d 2 n,50d 2 n) = x[-50d 2 n,50d 2 n) . We consider two cases. First, we assume that z 1 or z1 belongs to C ap . There is no loss of generality in assuming that z 1 is the one belonging to C ap . Observe that the hypothesis and that |τ | ≤ 10d 2 n ensure that x [f1-7d 2 n,f1+7d 2 n) = x[f1-7d 2 n,f1+7d 2 n) . This allows us to use (4.54) and deduce that F 0 (Z,τ ) (S f1 x) = F 0 (Z,τ ) (S f1 x). We now consider the case in which z 1 and z1 belong to C Z \C ap . Observe that, since f 1 = f1 , we have that F 0 (Z,τ ) (S f1 x) = (0, z 1 ) and F 0 (Z,τ ) (S f1 x) = (0, z1 ). Thus, it is enough to show that z 1 = z1 .

We assume without loss of generality that |τ (z

1 )| ≥ |τ (z 1 )|. Let ℓ be the integer satisfying fℓ ≤ f 2 < fℓ+1 . Remark that ℓ ≥ 2 as f2 = f1 + |τ (z 1 )| ≤ f 1 + |τ (z 1 )| = f 2 . Being f1 = f 1 and fℓ ≤ f 2 , the hypothesis x [-50d 2 n,50d 2 n) = x[-50d 2 n,50d 2 n) and the bound |τ | ≤ 7d 2 n ensure that τ (z [1,ℓ) ) is a prefix of τ (z 1 ). Hence, since we assumed that z 1 ∈ C Z \ C sp , Lemma 52 yields that zi ∈ C Z \ C ap and root τ (z i ) = root τ (z 1 ) for all i ∈ [1, ℓ).
(4.59)

We set s = root τ (z 1 ). It then follows from 4.59 and 4.50 that for any i ∈ [1, ℓ)

z 1 = s ζ(s) if z 1 ∈ C sp s ζ(s)+qz 1 if z 1 ∈ C wp zi = s ζ(s) if zi ∈ C sp s ζ(s)+qz i if zi ∈ C wp (4.60)
We can use this to prove that |τ (z 1 )| = |τ (z 1 )| implies that z 1 = z1 . Indeed, in the case z 1 ∈ C sp , it follows from (4.60) and the fact that q a > 0 for all a ∈ C wp that τ (z 1 ) = τ (z 1 ) = s ζ(s) , and thus that z1 = z 1 = ψ -1 sp (s ζ(s) ). Similarly, if z 1 ∈ C wp , then (4.48) and the equation root τ (z 1 ) = root τ (z 1 ) ensure that q z1 = q z1 , and thus from (4.60) we get that τ (z 1 ) = τ (z 1 ) = s ζ(s)+qz 1 . In particular, z1 = z 1 = ψ -1 wp (s ζ(s)+qz 1 ). It is left to consider the case |τ (z 1 )| > |τ (z 1 )|, so let us assume that this condition is satisfied. Then, by (4.60) and the fact that q a > 0 for all a ∈ C wp , z 1 ∈ C wp and z1 ∈ C sp .

(4.61)

In this situation, Item (1) in Proposition 19 ensures that z 2 ∈ C ap . Now, observe that f 3 ≤ 3|τ | ≤ 30d 2 n and f 2 ≥ 0; so the hypothesis x

[-50d 2 n,50d 2 n) = x[-50d 2 n,50d 2 n) gives that x [f2-7d 2 n,f3+7d 2 n) = x[f2-7d 2 n,f3+7d 2
n) . Hence, we can use (4.54) to obtain that F 0 (Z,τ ) (S f2 x) is equal to F 0 (Z,τ ) (S f2 x). More precisely, since fℓ ≤ f 2 < fℓ+1 , we can write

( fℓ -f 2 , zℓ ) = F 0 (Z,τ ) (S f2 x) = F 0 (Z,τ ) (S f2 x) = (0, z 2 ).

Basic properties of the fourth coding

We present here the basic properties of (Z, τ ).

Lemma 53. The pair (Z, τ ) is a recognizable coding of X.

Proof. It follows from the definitions that ψ commutes with the shift and that it is continuous; hence Z = ψ(Y ) is a subshift. Also, by (4.64), we have that θψ(y) = y for any y ∈ Y , so Y = θψ(Y ) = θ(Z). Therefore, (Z, θ) is a coding of Y . It is easy to see from the definition of Z that (Z, θ) is recognizable. Hence, as (Y, σ) is recognizable as well, Lemma 1 tells us that (Z, σθ) is recognizable.

Being τ = σθ, we conclude that (Z, τ ) is recognizable.

Thanks to the last lemma, F (Z,τ ) (x) and F 0 (Z,τ ) (x) are defined for every x ∈ X.

Lemma 54. Let x ∈ X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). Then, c k = f k and ψ 0 (S k y) = z k for all k ∈ Z.
Proof. On one hand, (4.64) implies that τ ψ(y) = σθψ(y) = σ(y), and thus that S -c0 τ (ψ(y)) is equal to S -c0 σ(y) = x. On the other hand, σ(y 0 ) is equal to τ (ψ 0 (y)) = τ (ψ(y) 0 ); so, as (-c 0 , y) is a σ-factorization, [0, |σ(y 0 )|) = [0, |τ (ψ(y) 0 )|) contains -c 0 . From these two things, we conclude that (-c 0 , ψ(y)) is a τ -factorization of x. Then, since (Z, τ ) is recognizable by Lemma 53, (-c 0 , ψ(y)) and (-f 0 , z) are the same τ -factorization, that is, c 0 = f 0 and ψ(y) = z. We use this to compute, for j ≥ 0,

f j = -f 0 + |τ (z [0,j) )| = -c 0 + |τ (ψ(y) [0,j) )| = -c 0 + |σ(y [0,j) )| = c j ,
where in the last step we used that τ (ψ(y)) = σ(y). A similar computation shows that f j = c j for j < 0 as well.

The last lemma has the following important consequence. For any x ∈ X, the cut functions of its σ-factorization in Y and of its τ -factorization in Z are the same. Therefore, we can simply write (c, y) = F (Y,σ) (x) and (c, z) = F (Z,τ ) (x). This will be tacitly used in this subsection.

Lemma 55. Let x, x ∈ X, (c, z) = F (Z,τ ) (X) and (c, z) = F (Z,τ ) (x). If z 0 = z0 , then x [c L(z) ,c L(z)+1 ) = x[c L(z) ,c L(z)+1 ) and x [cj ,cj+1) = x[cj,cj+1) for j ∈ [-1, 1]. Proof. Let (c, y) = F (Y,σ) (x) and (c, ỹ) = F (Y,σ) (x)
. Then, by Lemma 54 and the hypothesis Proposition 20. The following conditions hold:

y L(y) y -1 y 0 y 1 = ψ 0 (y) = z 0 = z0 = ψ 0 (ỹ) = ỹL(ỹ) ỹ-1 ỹ0 ỹ1 . Hence, x [c L(z) ,c L(z)+1 ) = σ(y L(y) ) = σ(ỹ L(ỹ) ) = x[c L(z) ,c L(z)+1 ) . Similarly, x [c-1,c2) = σ(y -1 y 0 y 1 ) is equal to x[c-1,c2) = σ(ỹ -1 ỹ0 ỹ1 ). Let C ap = θ -1 (B ap ), C wp = θ -1 (B wp ) and C sp = θ -1 (B sp ). ( 4
(1) 20ε ≤ |τ (a)| ≤ 10d 2 n for all a ∈ C.

(

) #τ (C ap ) ≤ 2d 3d+6 , #(root τ (C)) ≤ 5d 3d+6 and #C ≤ 7 4 d 12d+36 pow-com(X) 4 . 2
( (1) If i < j are integers such that z

k ∈ C\C ap for all k ∈ [i, j), then root τ (z k ) = root τ (z i ) if k ∈ [i, j), z k ∈ C sp if k ∈ [i, j -
1), and z k = z i+1 for all i ∈ [i + 1, j -1).

(2) If z 0 ∈ C sp , then τ (a) = (root τ (a)) 2 r , where r is the unique integer for which 2 r | root τ (a)| belongs to [20ε, 40ε).

Proof. Suppose that i < j satisfy z k ∈ C \ C ap for all k ∈ [i, j + 1). Let us denote F (Y,σ) (τ (z)) by (c, y). Then, by Lemma 54,

y k = θ(z k ) ∈ B \ B ap for all k ∈ [i, j + 1). In this context, Item (1) of Proposition 19 ensures that root σ(y k ) = root σ(y i ) if k ∈ [i, j + 1) and y k = y i ∈ B sp if k ∈ [i, j) We deduce, as τ (z k ) = σ(y k ), that root τ (z k ) = root τ (z i ) for all k ∈ [i, j + 1). Also, for any k ∈ [i, j), we have that z k ∈ θ -1 (y k ) ∈ B sp . Now, since y k ∈ C \ C ap if [i, j + 1),
we have that y L(S k y) = y L(S i y) for all k ∈ [i, j + 1). Combining this with the fact that

y k = y i if k ∈ [i, j) yields ψ 0 (S k y) = y L(S k y) y k-1 y k y k+1 = y L(S i y) y i y i y i for all k ∈ [i + 1, j -1).
We conclude, using Lemma 54, that z k = z i+1 for k ∈ [i + 1, j -1).

Lemmas 56, 58 and 57 will use the following notation:

E = 50d 2 n.
Observe that |σ| ≤ E, |τ | ≤ E and that E is bigger than or equal to the constants 7d 2 and 50d 2 n appearing in Proposition 18.

Lemma 56. Let x, x ∈ X be such that x [-3E,3E) = x[-3E,3E) and per(x [-ε,ε) ) > ε. Let (c, z) = F (Z,τ ) (x) and (c, z) = F (Z,τ ) (x). Then, τ (z 0 ) = τ (z 0 ) and F 0 (Z,τ ) (S c1 x) = F 0 (Z,τ ) (S c1 x).
In particular, c 0 = c0 and c 1 = c1 .

Proof. We use the notation (c, y) = F (Y,σ) (x) and (c, ỹ) = F (Y,σ) (x). Observe that the hypothesis x [-3E,3E) = x[-3E,3E) allows us to use Item (3) in Proposition 17 to obtain that

F 0 (Y,σ) (S i x) = F 0 (Y,σ) (S i x) for all i ∈ [0, 2E]. (4.66)
In particular, c 0 = c0 and y 0 = ỹ0 . Thus, by Lemma 54, τ (z 0 ) = σ(y 0 ) = σ(ỹ 0 ) = τ (z 0 ). Also, for j ∈ {1, 2} we have that 0 ≤ c j ≤ c 2 ≤ 2E, so (4.66) can be applied to deduce that F 0 (Y,σ) (S cj x) = F 0 (Y,σ) (S cj x). Then, y [0,3) = ỹ[0,3) and c j = cj for all j ∈ [0, 3). (4.67)

Before continuing, we prove that y 0 , ỹ0 ∈ C ap . (4.68)

We note that if y 0 ∈ C \ C ap , then Item (3) in Proposition 20 gives that per(x [c0-8ε,c1+9ε) ) ≤ ε, which is impossible since we assumed that per(x [-ε,ε) ) > ε. Thus, y 0 ∈ C ap . Similarly, ỹ0 ∈ C ap as x[-ε,ε) = x [-ε,ε) .

We can now finish the proof. The condition c 1 = c1 follows from (4.67). Also, we have from Lemma 54 that z 1 = ψ 0 (Sy) = L(Sy)y 0 y 1 y 2 and z1 = ψ 0 (Sy) = L(S ỹ)ỹ 0 ỹ1 ỹ2 . Now, Equation (4.68) guarantees that L(Sy) = y 0 and L(S ỹ) = ỹ0 . Therefore, by Equation (4.67), z 1 = z1 . We conclude, using that c 1 = c1 , that

F 0 (Z,τ ) (S c1 x) = (0, z 1 ) = (0, z1 ) = F 0 (Z,τ ) (S c1 x). (4.69) Lemma 57. Let x, x ∈ X, (c, z) = F (Z,τ ) (x) and (c, z) = F (Z,τ ) (x). Sup- pose that x [-4E,4E) = x[-4E,4E) and z 0 ∈ C ap . Then, τ (z 0 ) = τ (z 0 ) and F 0 (Z,τ ) (S c1 x) = F 0 (Z,τ ) (S c1 x).
In particular, c 0 = c0 and c 1 = c1 .

Proof. The condition z 0 ∈ C ap implies, by Item (3) in Proposition 20, that per(x [c0,c1) ) > ε. Thus, by Item (2) in Lemma 9, there is

i ∈ [c 0 , c 1 ) such that per(x [i-ε,i+ε) ) > ε. Now, since |τ | ≤ E, the hypothesis ensures that (S i x) [i-ε,i+ε) is equal to (S i x) [i-ε,i+ε)
. Therefore, we can use Lemma 56 and conclude that τ (z 0 ) = τ (z 0 ) and F 0 (Z,τ ) (S c1 x) = F 0 (Z,τ ) (S c1 x).

Proof. The hypothesis implies that c ′ 0 = c′ 0 and z ′ 0 = z′ 0 . Combining this with Lemma 55 yields

x [c ′ L ′ (z ′ ) ,c ′ L ′ (z ′ )+1 ) = x[c ′ L ′ (z ′ ) ,c L ′ (z ′ )+1 ) , x [c ′ j ,c ′ j+1 ) = x[c ′ j ,c ′ j+1 ) and c ′ j = c′ j for j ∈ [-1, 1]. (4.87)
This and the lower bound in Item (1) of Proposition 20 ensure that

x [c ′ 0 -8ε ′ ,c ′ 1 +8ε ′ ) = x[c ′ 0 -8ε ′ ,c ′ 1 +8ε ′ ) . (4.88)
Next, we show that the following facts hold.

(i) (S c k x) [-8ε ′ ,8ε ′ ) = (S c l x) [-8ε ′ ,8ε ′ ) for all L ′ (y ′ ) < k, l ≤ 0. (ii) (S ck x) [-8ε ′ ,8ε ′ ) = (S cl x) [-8ε ′ ,8ε ′ ) for all L ′ (ỹ ′ ) < k, l ≤ 0. (iii) x [c ′ L ′ (y ′ ) ,c ′ L ′ (y ′ )+1 +8ε ′ ) = x[c ′ L ′ ( ỹ′ ) ,c ′ L ′ ( ỹ′ )+1 +8ε ′ ) . We start with Item (i). If L ′ (y ′ ) = -1, then (i) is vacuously true. We assume that L ′ (y ′ ) < -1. Let k be such that L ′ (y ′ ) < k < 0. The definition of L ′ ensures that z ′ k ∈ C ′ \ C ′ ap . So, by Item (3) in Proposition 20, s := root τ ′ (z ′ k ) satisfies x [c k -8ε ′ ,c k+1 +8ε ′ ) = s Z [-8ε ′ ,|τ ′ (z ′ k )|+8ε ′ ) . In particular, as |τ ′ (z ′ 0 )| = 0 (mod |s|), x [c k -8ε ′ ,c k +8ε ′ ) = s Z [-8ε ′ ,8ε ′ ) = (S |τ ′ (z ′ k )| s Z ) [-8ε ′ ,8ε ′ ) = x [c k+1 -8ε ′ ,c k+1 +8ε ′ ) .
Being this valid for all k ∈ [L ′ (z ′ ) + 1, 0), an inductive argument gives (i). Fact (ii) follows analogously. For (iii), we use (4.88), (i) and (ii) to deduce that

x [c ′ L ′ (z ′ )+1 ,c ′ L ′ (z ′ )+1 +8ε ′ ) = x [c0-8ε ′ ,c0+8ε ′ ) = x[c0-8ε ′ ,c0+8ε ′ ) = x[c ′ L ′ (z ′ )+1 ,c ′ L ′ (z ′ )+1 +8ε ′ )
. Fact (iii) follows from this and the first equality in (4.87). Now, since z ′ L ′ (z ′ ) ∈ C ′ ap and since (iii) holds, the hypothesis of Lemma 62 are satisfied. Therefore,

F 0 (Z,τ ) (S c ′ L ′ (z ′ )+1 +i x) = F 0 (Z,τ ) (S c′ L ′ (z ′ )+1 +i x) for all i ∈ [-7ε ′ , 7ε ′ ). (4.89)
In order to continue, we need to consider two cases. We first assume that L ′ (z ′ ) = -1. Then, z ′ -1 ∈ C ap , so (4.87) and Item (3) in Proposition 20 give

per(x [c ′ -1 +8ε ′ ,c ′ 0 -8ε ′ ) ) = per(x [c ′ -1 +8ε ′ ,c ′ 0 -8ε ′ ) ) > ε ′ .
This implies, by Item (3) in Proposition 20, that z-1 ∈ C ap . Hence, L ′ (z) = -1 and then (4.86) follows from (4.89). Next, we assume that L ′ (z ′ ) ≤ -2. In this case, we first prove the following.

(a) F 0 (Z,τ ) (S c ′ L ′ (y ′ )+1 +i x) = F 0 (Z,τ ) (S c ′ -1 +i x) for all i ∈ [-5ε ′ , 5ε ′ ). (b) F 0 (Z,τ ) (S c′ L ′ ( ỹ′ )+1 +i x) = F 0 (Z,τ ) (S c′ -1 +i x) for all i ∈ [-5ε ′ , 5ε ′ ).
We only prove (a) as (b) follows from an analogous argument. If L ′ (z ′ ) = -2, then (a) is trivially true. Assume then that L ′ (z ′ ) ≤ -3. The definition of L ′ ensures that z ′ j ∈ C ′ \ C ap for all L ′ (z ′ ) + 1 ≤ j ≤ -1. Thus, by Item (1) in Proposition 21, z ′ j ∈ C sp for all L ′ (z ′ )+1 ≤ j ≤ -2. This allows us to inductively apply Lemma 61 and deduce that, for any i ∈ [-5ε ′ , 5ε ′ ),

F 0 (Z,τ ) (S c ′ L ′ (z ′ )+1 +i x) = F 0 (Z,τ ) (S c ′ L ′ (z ′ )+2 +i x) = • • • = F 0 (Z,τ ) (S c ′ -1 +i x).
This shows (a). Now, combining Equation (4.89), (a) and (b) produces

F 0 (Z,τ ) (S c ′ -1 +i x) = F 0 (Z,τ ) (S c′ -1 +i x) for all i ∈ [-5ε ′ , 5ε ′ ). (4.90)
We are going to derive (4.86) from this and (4.87). Let i ∈ [c ′ 0 -4ε ′ , c ′ 2 -ε ′ ) be arbitrary. We note that (4.90) gives, in particular, that

x [c-1-5ε ′ ,c-1+5ε ′ ) = x[c-1-5ε ′ ,c-1+5ε ′ ) .
From this, (4.87) we get that

x [c-1-5ε ′ ,c2) = x[c-1-5ε ′ ,c2) .
(4.91)

In view of Equations (4.90) and (4.91) and of 3E ≤ ε ′ , the hypothesis of Lemma 58 holds; hence,

F 0 (Z,τ ) (S c ′ -1 +j x) = F 0 (Z,τ ) (S c′ -1 +j x) for all j ∈ [-4ε ′ , c ′ 2 -c ′ -1 -ε ′ ).
We set j = i -c ′ -1 and note that j ∈ [-4ε ′ , c ′ 2 -c ′ -1 -ε ′ ). Therefore, the last equation can be used to obtain that

F 0 (Z,τ ) (S i x) = F 0 (Z,τ ) (S c ′ -1 +j x) = F 0 (Z,τ ) (S c′ -1 +j x) = F 0 (Z,τ ) (S i+c ′ -1 -c ′ -1 x).
Being c′ -1 = c ′ -1 by (4.87), we deduce that (4.86) holds.

The connecting morphism

In this subsection, we build a morphism γ that connects (Z ′ , τ ′ ) with (Z, τ ). We start by introducing the auxiliary map r : Z ′ → Z and proving some properties for it. The crucial Proposition 23 will allow us to define the connecting morphism γ. We finish the section with Propositions 24, 25 and 26, which will be crucial for proving (P 1 ) in Theorems 33 and 34. For z ′ ∈ Z ′ and (c, z) = F (Z,τ ) (τ ′ (z ′ )), let Now, we know from the definition of r and the condition per(τ ′ (z ′ 0 )) = per(τ ′ (z ′ 0 )) > ε that z r(z ′ ) , zr(z ′ ) ∈ C ap . Hence, by Equations (4.102) and (4.103), we can use Lemma 57 and deduce the following: If i and j are the integers satisfying c r(z ′ ) ∈ [c i , ci+1 ) and cr(z ′ ) ∈ [c j , c j+1 ), then c r(z ′ ) = ci and cr(z ′ ) = c j . Therefore, by the definition of r, that cr(z ′ ) ≤ ci = c r(z ′ ) and c r(z ′ ) ≤ c j = cr(z ′ ) . We conclude that c r(z ′ ) = cr(z ′ ) . This and Equations (4.102) and (4.103) allow us to use Lemma 64, yielding ψ(S r(z ′ ) z) = ψ(S r(z ′ ) z).

r(z ′ ) = 0 if per(τ ′ (z ′ 0 )) ≤ ε min{i ≥ 0 : z i ∈ C ap } if per(τ ′ (z ′ 0 )) > ε
We now assume that ρ(z ′ -1 ) = ρ(z ′ -1 ) and that per(τ ′ (z ′ 0 )) > ε ′ . Then, z ′ 0 ∈ C ′ ap , so, since ρ(z ′ 0 ) = ρ(z ′ 0 ), we have that τ ′ (z ′ 0 ) = τ ′ (z ′ 0 ). Combining this with the equation ρ(z ′ -1 ) = ρ(z ′ -1 ) and Item (3) of Proposition 20 produces

τ ′ (z ′ ) [-8ε ′ ,|τ ′ (z ′ 0 )|+8ε ′ ) = τ ′ (z ′ ) [-8ε ′ ,|τ ′ (z ′ 0 )|+8ε ′ ) . ( 4
(S c r(z ′ ) τ ′ (z ′ )) [-ε ′ ,ε ′ ) = (S c r(z ′ ) τ ′ (z ′ )) [-ε ′ ,ε ′ )
and (S cr(z

′ ) τ ′ (z ′ )) [-ε ′ ,ε ′ ) = (S cr(z ′ ) τ ′ (z ′ )) [-ε ′ ,ε ′ ) . (4.106)
Since z r(z ′ ) , zr(z ′ ) ∈ C ap by (4.92), we can use Lemma 57 to deduce the following: If i and j are the integers satisfying c r(z ′ ) ∈ [c i , ci+1 ) and cr(z) ∈ [c j , c j+1 ), then c r(z ′ ) = ci and cr(z) = c j . We can then argue as in the first case to conclude that c r(z ′ ) = cr(z ′ ) and ψ(S r(z ′ ) z) = ψ(S r(z ′ ) z).

Proposition 24. Suppose that z ′ 0 , z′ 0 ∈ C ′ ap and ψ(z ′ ) = ψ(z ′ ). Then:

(1) c r(z ′ ) = cr(z ′ ) and c r(z ′ )+|γ(z ′ 0 )| = cr(z ′ )+|γ(z ′ 0 )| . (2) ψ(S r(z ′ ) z) = ψ(S r(z ′ ) z) and ψ(S r(z ′ )+|γ(z ′ 0 )| z) = ψ(S r(z ′ )+|γ(z ′ 0 )| z).

Proof. We have, from the condition ψ(z ′ ) = ψ(z ′ ), that ρ(z ′ -1 ) = ρ(z ′ -1 ) and ρ(z ′ 0 ) = ρ(z ′ 0 ). Also, since z ′ 0 ∈ C ′ ap , we have that per(τ ′ (z ′ 0 )) > ε. Hence, we can use Lemma 65 to deduce that c r(z ′ ) = cr(z ′ ) and ψ(S r(z ′ ) z) = ψ(S r(z ′ ) z).

(4.107)

Now, from Lemma 63 we have that c r(z ′ ) ∈ [-ε ′ , |τ ′ (z ′ 0 )| -8ε ′ ). Also, the hypothesis and Item (1) in Proposition 20 give

τ ′ (z ′ ) [-8ε ′ ,|τ ′ (z ′ 0 )|+8ε ′ ) = τ ′ (z ′ ) [-8ε ′ ,|τ ′ (z ′ 0 )|+8ε ′ ) (4.108)
These two things, together with the fact that z r(z ′ ) ∈ C ap , allow us to use Lemma 57 and deduce that F 0 (Z,τ ) (S c r(z ′ )+1 τ ′ (z ′ )) = F 0 (Z,τ ) (S c r(z ′ )+1 τ ′ (z ′ )). (4.109)

Main Theorems

We now complete the proofs of Theorems 33 and 34. The part of the proof in which we have to obtain a complexity restriction from the S-adic structures can be done without difficulties with Lemma 76. For the other part, we first present in Theorem 32 sufficient condition under which an S-adic structure as the ones in Theorems 33 and 34 can be obtained. Then, we check that linear-growth and nonsuperlinear-growth complexity subshifts satisfy these conditions using Lemmas 30 and 31.

A set of sufficient conditions

This subsection is devoted to prove the following theorem.

Theorem 32. Let X ⊆ A Z be an infinite minimal subshift. Let (ℓ n ) n≥0 be an increasing sequence of positive integers and d ≥ max{10 4 , #A}. Suppose that for every n ≥ 0 p X (ℓ n ) ≤ d • ℓ n , p X (ℓ n + 1) -p X (ℓ n ) ≤ d, and ℓ n+1 ℓ n ≥ 10 4 d 2d 3 +6 . (4.116)

Then, there exists a recognizable S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that for all n ≥ 1: (P 3 ) |σ n-1 (a)| ≤ 40d 2d 3 +8 • ℓn ℓn-1 for every a ∈ A n . The proof is presented as a series of lemmas. We fix an infinite minimal subshift X ⊆ A Z , an increasing sequence (ℓ n ) n≥0 of positive integers and d ≥ max{10 4 , #A} such that (4.116) holds for every n ≥ 0.

(P 1 ) #(root σ [0,n) (A n )) ≤ 35d
We start by defining σ. Let (Z n ⊆ C Z n , τ n : C n → A + ) be the coding constructed in Subsection 4.7.1 using ℓ n , and let ε n ∈ [ℓ n /d 2d 3 +4 , ℓ n /d) and C n = C n,ap ∪ C n,wp ∪ C n,sp be the constant and the partition that appear in this construction. In this context, Proposition 20 states the following: Also, the definition of C n in (4.63) guarantees that (iii) For all a ∈ C n there is z ∈ Z n such that z 0 = a.

(i) (# root τ n (C n )) ≤ 5d 3d+6 , #C n ≤ 7 4 d 12d+36 pow-com(X)
Moreover, (4.116) implies that 500d 2 ℓ n ≤ ℓ n+1 /d 2d 3 +4 , so the results from Subsection 4.7.2 can be used with (Z n+1 , ℓ n+1 ) and (Z n , τ n ). In particular, Now, thanks to Condition (iii), we can use Lemma 66, so any x ∈ X σ is an adherent point of a sequence x n ∈ ∪ k∈Z S k σ [0,n) (Z n ) = X. Therefore, X σ ⊆ X.

We conclude that X σ = X by the minimality of X.

It rests to prove that (Z n , σ [0,n) ) is recognizable. Let (k, z) and ( k, z) be two σ [0,n) -factorizations in Z n of x ∈ X. Then, Equation (4.121) implies that S k+q0,n(z) τ n (z) = S k σ [0,n) (z) = S kσ [0,n) (z) = S k+q0,n(z) τ n (z).

In particular, S ℓ τ n (z) = τ n (z) where ℓ = k+q 0,n (z)-k-q 0,n (z). Without loss of generality, we assume that ℓ ≥ 0. We can find i ≥ 0 such that (ℓ-|τ n (z [0,i) )|, S i z) is a τ n -factorization of S ℓ τ n (z) in Z n . Then, as (0, z) is a τ n -factorization of S ℓ τ n (z) in Z n , we deduce from the recognizability property of (Z n , τ n ) that ℓ = k + q 0,n (z) -k -q 0,n (z) = |τ n (z [0,i) )| and S i z = z (4.122)

Using this and the fact that (k, z) and ( k, z) are σ [0,n) -factorizations of x, we can write

σ [0,n) (z) = S k-k σ [0,n) (z) = S k-k σ [0,n) (S i z) = S q0,n(z)-q0,n(z)-|τn(z [0,i) )|+|σ [0,n) (z [0,i) )| σ [0,n) (z).
Being Z n aperiodic (as X is aperiodic), we get that q 0,n (z)-q 0,n (z)+|σ We deduce that i = 0, and then, from (4.122), that z = z and k = k.

Before continuing, we give some bounds for q 0,n .

Lemma 68. Let n ≥ 1 and z ′ ∈ Z n . Then,

-2ε n ≤ q 0,n (z ′ ) ≤ |τ n (z ′ 0 )| -7ε n . (4.123)
Moreover, if z ′ 0 ∈ C n \ C n,ap , then -2ε n ≤ q 0,n (z ′ ) ≤ 2ε n . (4.124)

Proof. Lemma (63) gives the bound -ε j ≤ c r(z) ≤ |τ j+1 (z 0 )| -8ε j for all 0 ≤ j < n and z ∈ Z j+1 . Thus, from (4.120), -ε n ≤ q n-1,n (z ′ ) ≤ |τ n (z ′ 0 )| -8ε n and q j+1,n (z ′ ) -|τ j+1 | ≤ q j,n (z ′ ) ≤ q j+1,n (z ′ ) + |τ j+1 |.. We obtain that -ε n - Putting this in (4.125) yields (4.123). Moreover, if z ′ 0 ∈ C n \ C n,ap , then Lemma 63 gives that q n-1,n (z ′ ) ∈ [-ε n , ε n ). So, the previous argument shows, in this case, that q 0,n (z ′ ) ∈ [-2ε n , 2ε n ).

Lemma 69. For every n ≥ 1 and z ′ ∈ Z n ,

5 d d 3 +4 ℓ n ≤ |σ [0,n) (z ′ 0 )| ≤ 20d 2 ℓ n . (4.126)
In particular, σ satisfies Items (P 2 ) and (P 3 ) of Theorem 32.

Proof. We first show that (4.126) implies that σ satisfies Items (P 1 ) and (P 2 ) of Theorem 32. Observe that, by Condition (iii), (4.126) gives, for every n ≥ 1 and a, b ∈ C n , that

|σ [0,n) (a)| ≤ 20d 2 ℓ n ≤ 4d d 3 +6 • |σ [0,n) (b)|. (4.127)
Thus, (P 2 ) is satisfied. For Item (P 3 ), we note that, for any pair of morphisms ξ and ξ ′ such that ξξ ′ is defined, we have that |ξξ ′ | ≥ ⟨ξ⟩|ξ ′ |. Therefore,

|σ [0,n+1) | ≥ ⟨σ [0,n) ⟩|σ n |.
Then, by Item 4.127,

|σ n | ≤ |σ [0,n+1) | ⟨σ [0,n) ⟩ ≤ 10d 2 ℓ n+1 1/4d 2d 3 +6 ℓ n = 40d 2d 3 +8 ℓ n+1 ℓ n .
We now prove (4.126). Let n ≥ 0 and z ′ ∈ Z n be arbitrary. On one hand, from (4.120) we have that

|σ [0,n) (z ′ 0 )| = |τ n (z ′ 0 )| -q 0,
n (z ′ ) + q 0,n (Sz ′ ) for any z ′ ∈ Z n .

Hence, by (4.123) and Condition (ii),

|σ [0,n) (z ′ 0 )| = |τ n (z ′ 0 )| -q 0,n (z ′ ) + q 0,n (Sz ′ ) ≤ |τ n (z ′ 0 )| + |τ n (z ′ 1 )| -5ε n ≤ 20d 2 ℓ n .
Similarly,

|σ [0,n) (z ′ 0 )| = |τ n (z ′ 0 )| -q 0,n (z ′ ) + q 0,n (Sz ′ ) ≥ 5ε n ≥ 5 d d 3 +4 ℓ n .
Theorem 33. Let k ≥ 1 be arbitrary and let n ≥ 1 be the biggest integer such that ⟨σ [0,n) ⟩ ≤ k. Then, by the maximality of n and Items (P 2 ) and (P 3 ) of Theorem 33, we can compute

k < ⟨σ [0,n+1) ⟩ ≤ |σ [0,n) | • |σ n | ≤ d 2 ⟨σ [0,n) ⟩ ≤ d 2 k.
Combining this with (4.139) yields p X (k) ≤ d 5 k. This proves that X has lineargrowth complexity.

Bounded alphabet structures

Theorems 33 and 34 provide S-adic structures for linear-growth complexity subshifts and nonsuperlinear-growth complexity subshifts. These are not the only representations known for these classes: for instance, in [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF] it is proved that if X has nonsuperlinear-growth complexity, then X is generated by a recognizable, proper and primitive S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 such that #A n is uniformly bounded. The last condition is known as the bounded alphabet property and it is considered natural in the low complexity setting; see [START_REF] Ferenczi | Rank and symbolic complexity[END_REF][START_REF] Durand | Do the Properties of an S-adic Representation Determine Factor Complexity?[END_REF][START_REF] Espinoza | Symbolic factors of S-adic subshifts of finite alphabet rank[END_REF]. Note that the representations given by Theorems 33 and 34 do not necessarily satisfy this property. In fact, our construction gives a bonded alphabet S-adic sequence if and only if the subshift has finite power complexity. Thus, it is natural to ask whether it is possible to modify Theorems 33 and 34 so that they give bounded alphabet S-adic sequences. In this section, we show that such a strengthening is not possible for Theorem 33. More precisely, we prove the following: Theorem 35. There exists a minimal subshift X such that:

(1) X has linear-growth complexity.

(2) If σ = (σ n : A n+1 → A + n ) n≥0 is an S-adic sequence generating X and satisfying Items (1), (2) and (3) of Theorem 33, then sup n≥1 #A n = +∞.

We were not able to obtain an analogous result for Theorem 34, so we leave this as an open question.

It is interesting to compare Theorem 35 with the main result of [START_REF] Leroy | An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) -p(n) ≤ 2[END_REF], which describes bounded alphabet S-adic representations of minimal subshifts whose complexity function satisfies p X (n + 1) -p X (n) ≤ 2 for all n ≥ 1. We are led to ask the following.

Question 9. How small can sup n≥1 p X (n + 1) -p X (n) be made in Theorem 35?

We now turn into proving Theorem 35. We start with some technical lemmas.

Let n, n 0 , d, ℓ ≥ 1. We define P (n, n 0 , ℓ) as the set of integer sequences (p 1 , . . . , p ℓ ) such that p j n 0 ∈ [8 j n, 2 • 8 j n). Let K(n, d, ℓ) be the set of integer sequences (k 1 , . . . , k ℓ ) ∈ P (n, n 0 , ℓ) for which there exists E ⊆ [d -1 n, dn), with Being t a suffix of τ n (a), this implies that a p n ℓn is a suffix of t. Moreover, since t is a prefix of τ n (b), a p n ℓn occurs in τ n (b). But (4.142) guarantees that a p n ℓn occurs in τ n (b) only as a suffix, so we must have that t = τ n (b). Therefore, τ n (a) = τ n (b) = t.

We now prove that τ satisfies the properties of the lemma. The morphisms τ n are positive, so X is minimal. It follows from (4.142) that 0 p n 1 1 and 0 p n 0 0 belong to L(X (n) τ ) for all n ≥ 0, so τ [0,n) (0)1 and τ [0,n) (0)0 are elements of L(X). This shows that X has infinitely many right-special words, and thus that X is infinite. To prove that X has linear-growth complexity, we will show that p X (k) ≤ 1024k for all k ≥ 1. Let k ≥ 1 be arbitrary. We take n ≥ 0 such that |τ where in the last step we used Items (2) and (3) of Proposition 28. We now prove that S satisfies the desired properties. Let us start by showing that L(X) ⊆ S 2 . Let u ∈ L(X) and let n ≥ 1 be the biggest integer such that |u| ≥ ⟨τ [0,n) ⟩. Then, |u| < ⟨τ [0,n+1) ⟩ and, thus, as τ generates X, there exists a, b ∈ A n+1 such that u occurs in τ [0,n+1) (ab). Hence, by (4.160), u occurs in some w ∈ W n+1 , which implies, by (c), that u ∈ V 2 n+1,w ⊆ S 2 . It remains to prove that p S is uniformly bounded. Let S n = ∪ w∈Wn V n,w . We claim the following:

(i) #p Sn (k) ≤ 2 12 d 4 for all n ≥ 0 and k ≥ 0.

(ii) for any k ≥ 0, there are at most log 2 (d) + 7 integers n such that S n ∩ A k is not empty.

Observe that (i) and (ii) allow us to write

Topological rank

The topological rank of a minimal subshift X is the least element k ∈ [1, +∞] such that there exists a recognizable S-adic sequence τ = (τ n : A n+1 → A + n ) n≥0 satisfying, for every n ≥ 1, that #A n ≤ k and that τ n is positive and proper. The class of finite topological rank subshifts satisfies several rigidity properties, and many tools have been developed to handle it; a non-exhaustive list includes [BKMS13; DFM19; Esp22; EM21; DM08; BSTY19; HPS92].

It was proved in [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF] that a minimal subshift of nonsuperlinear-growth complexity has finite topological rank, and thus that the aforementioned rigidity properties hold for this class. We present in this subsection a new proof of this fact based in Theorem 34.

Theorem 39 ([DDMP21], Theorem 5.5). Let X be a minimal subshift of nonsuperlinear-growth complexity. Then, X has finite topological rank.

Proof. The case in which X is finite is trivial, and so we may assume that X is infinite. Then, Theorem 34 gives d and a recognizable S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that Items (1) and (2) of Theorem 34 hold. In particular, Chapter 5

X ⊆ k∈Z S k (root σ [0,n) (A n )) Z for

Perspectives and future work

In this final chapter, we will present some open questions and comments that have emerged from the thesis work. These ideas correspond to a future research plan.

More on symbolic factors

The theorems in Chapter 3 provide a fine description of the symbolic factors for a general class of subshifts: those having finite topological rank. It is then natural to search for applications within the finite topological rank class. We now describe two ideas for doing this.

Symbolic factors of eventually dendric shifts

The class of minimal dendric subshifts was introduced in [Ber+14] (under the name of tree shifts) and are a generalization of Arnoux-Rauzy subshifts and (the natural coding of) interval exchanges. This class presents interesting rigidity properties, such as that any set of return words is a basis of the free group of a fixed cardinality or that the complexity function is an affine function [START_REF] Berthé | Acyclic, connected and tree sets[END_REF]. Moreover, the closely related class of eventually dendric subshifts was independently discovered in [START_REF] Damron | The number of ergodic measures for transitive subshifts under the regular bispecial condition[END_REF] while generalizing a theorem on the number of ergodic measures of interval exchanges. Due to this, dendric and eventually dendric shifts have gained attention, and, in particular, the question about their symbolic factors has become relevant.

Problem 1. Describe the symbolic factors of (eventually) dendric shifts.

There are examples of dendric subshifts with non-dendric symbolic factors. However, all known such factors are eventually dendric. This has led to the following conjecture. 165 Conjecture 1. Are all symbolic factors of a given eventually dendric shift eventually dendric?

Interestingly, a finite topological rank structure for minimal eventually dendric subshift was recently obtained [START_REF] Gheeraert | S-adic characterization of minimal dendric shifts[END_REF]. Therefore, the methods developed in Chapter 3 can be applied to this case and, by doing so, we may be able to shed some light on the conjecture.

Symbolic factors of interval exchanges

The following is an old question regarding interval exchange transformations: Question 10. Let F be the set of interval exchange transformations that do not have non-trivial measure-theoretic factors. Is F generic?

Observe that an affirmative answer to this question has, as a particular case, the Avila-Forni Theorem, so it is probably a difficult problem. We consider instead a topological version of it.

Question 11. Let F top be the set of interval exchange transformations whose natural coding does not have non-trivial symbolic factors. Is F top generic?

In a work in progress with Vincent Delecroix, we have outlined a strategy, using the ideas of Chapter 3, for giving an affirmative answer to Question 11. This would represent progress towards Question 10.

More on the S-adic conjecture

Our work on the S-adic conjecture opened at least two new directions of research, which we now discuss.

Applications of the structure theorems

The S-adic characterization obtained in Chapter 4 permit the use of the Sadic machinery to study linear-and nonsuperlinear-growth complexity subshifts. Some cases in which this idea produces interesting results were presented in Section 4.10 of Chapter 4. We plan on continuing investigating in this direction. In particular, it seems that the absence of the strong mixing property and the partial rigidity (with respect to an ergodic measure) may be better understood using the methods in [START_REF] Bezuglyi | Finite rank Bratteli diagrams: structure of invariant measures[END_REF]. More generally, any of the currently known techniques for handling S-adic sequences can now be applied to linear-and nonsuperlinear-growth complexity subshifts, see [HPS92; BKMS13; DFM19; Ber+21]. In some cases, non-proper variations of those techniques must be developed first.

Finite alphabet rank structures

Let (L) and (NSL) be the classes of linear-and nonsuperlinear-growth complexity subshifts, respectively. We showed in Theorem 35 that the structure provided we obtained for (L) must have, in some cases, infinite alphabet rank † . Now, most of the techniques for handling S-adic sequences are designed for finite alphabet rank sequences. Although some of them can be adapted to our case, the following question seems natural: Question 12. Let C be (L) or (NSL). Is there a finite alphabet rank S-adic characterization of C?

This question is sometimes called the strong S-adic conjecture. Observe that this question is ill-defined in the same sense as the S-adic conjecture is.

In the direction of Question 12, a close inspection of the proof of Theorem 35 shows that, in some cases, the sets Pow X (w) encode certain long-range information that seems to be incompatible with finite alphabet rank S-adic sequences. Therefore, we suspect that Question 12 has a negative answer. † The alphabet rank of τ = (τn :

A + n+1 → A + n ) n≥0 is lim inf n→+∞ #An.
Contribution to the study of zero entropy symbolic dynamics: automorphisms, factors and structure

This thesis focuses on the study of minimal subshifts via S-adic sequences. First, we investigate automorphisms and factors of minimal subshifts generated by Sadic sequences with alphabets of bounded cardinality. As a result, we prove that these subshifts have virtually Z automorphism groups, finitely many infinite symbolic factors (up to conjugacy), and we give a fine description of symbolic factor maps. In the second part, we consider the S-adic conjecture, an old problem asking for a structure theorem for linear-growth complexity subshifts. We completely solve this problem by proving an S-adic characterization of this class of subshifts. Our methods extend to nonsuperlinear-growth subshifts. We show how this provides a unified framework and simplified proofs of several known results, including Cassaigne's Theorem.

Contribution à l'étude des systèmes symboliques d'entropie nulle : automorphisms, facteurs et structure

Cette thèse porte sur l'étude des systèmes symboliques minimaux via des séquences S-adiques. Dans la première partie, nous étudions les automorphismes et les facteurs des systèmes minimaux générés par des séquences S-adiques avec des alphabets de cardinalité bornée. Comme résultat, nous prouvons que les systèmes de cette classe ont des groupes d'automorphismes virtuellement Z, un nombre fini de facteurs symboliques infinis (jusqu'à la conjugaison), et une description fine des facteurs symboliques. Dans la seconde partie, nous considérons la conjecture S-adique, un vieux problème demandant un théorème de structure pour les systèmes symboliques de complexité à croissance linéaire. Nous résolvons complètement ce problème en prouvant une caractérisation S-adique de cette classe de systèmes. Les méthodes s'étendent aux systèmes à croissance non superlinéaire. Nous montrons comment cela fournit un cadre unifié et des preuves simplifiées de plusieurs résultats connus, y compris le théorème de Cassaigne de 1996.

  est partout croissant, i.e., lim n→+∞ min a∈An |τ 0 τ 1 . . . τ n-1 (a)| = +∞. Supposons que τ soit de rang alphabétique fini, i.e., lim inf n→+∞ A n < +∞. Alors, Aut(X, S) est virtuellement Z.

¶

  Une substitution σ : A → B + est positive si pour tout a ∈ A et b ∈ B, b apparaît dans σ(a) in topological and symbolic dynamics All the intervals we will consider consist of integer numbers, i.e., [a, b] = {k ∈ Z : a ≤ k ≤ b} with a, b ∈ Z. For us, the set of natural numbers starts with zero, i.e., N = {0, 1, . . . }.

  The minimum and maximum length of τ are the numbers ⟨τ ⟩ := ⟨τ (A)⟩ = min a∈A |τ (a)| and |τ | := |τ (A)| = max a∈A |τ (a)|, respectively.

Figure 2 . 1 :

 21 Figure 2.1: Diagram of the W-interpretation I = d L , d M , d R , a of d in Definition 2.

Remark 2 .

 2 From condition ((2)) in previous definition we have that |d ′ L |, |d| ≥ ⟨W⟩, whenever D is a simple W-d.i.

Figure 2 . 2 :

 22 Figure 2.2: Diagram of a d.i. of d satisfying (1) in Definition 3. Here, d R a ≤ p u R and d ′ R a ′ ≤ p u ′ R , where u R , u ′ R are the words given in condition (2) of Definition 2.

  it follows directly from definition that D = (I, I ′ ) or D ′ = (I ′ , I) are simple d.i. respectively. So we assume d M ̸ = 1 and d ′ M ̸ = 1. Therefore, we can write d M = uv and d

  of e satisfying u ∈ W and |u ′ | ≤ |uv|, and when |u| ≤ |u ′ |, E = (J; J ′ ) is a d.i. of e satisfying u ′ ∈ W and |u| ≤ |u ′ v ′ |. In both cases we get a contradiction with the minimality of d. Then, in this case either D or D ′ is a simple d.i. of d.

(

  ii) Assume |d R | = |e R |. Since, by ((2)) of Definition 4, we have that d R and e R are prefix of u R , hypothesis (ii) implies that d R = e R . In addition, from ((1)) of Definition 4 we see that d L d M and e L e M either share the suffix u M ∈ W or are suffix dependent. We conclude that d = d L d M d R and e = e L e M e R share a suffix of length at least ⟨W⟩. This is, D ∼ E.

Figure 2

 2 Figure 2.4: Diagram of the construction in Case (b) of the proof of Lemma 6. Observe that conditions (b) and (2.4) say that d< p ẽ ≤ p ẽu ′ M v E ≤ p du ′ M v 1 • • • v n+1 ≤ p u R .This and the definitions of w and t are represented in the figure.

Figure 2 . 5 :

 25 Figure 2.5: Diagram of conditions in equation (2.10). Observe that, since dd ′ M d ′ R = d R ≤ p u R for any D ∈ D ′ by ((2)) of Definition 4, all the words in the figure occur inside u R .

Lemma 9 .

 9 Let p ∈ {1, . . . , s} be such that #D ′ (p) ≥ 2 and let D(p ′ ) = max D ′ (p), D(p ′′ ) = max D ′ (p) \ {D(p ′ )}. Then, there exist w ∈ W and w ′ ≤ p d(s) d(p ′′ ) -1 such that w and w ′ are suffix dependent, |w| ≥ | d(p ′ )| and |w ′ | > | d(s)| -| d(p)|.

  by (2.14), we can represent the right-hand side of equation (2.18) as in Figure2.6.

Figure 2 . 6 :

 26 Figure 2.6: Diagram of the right-hand side of equation (2.18).

  proving the desired condition on the length of w ′ . It only rests to prove that |w| ≥ | d(p ′ )|. We argue by contradiction. Assume that |w| < | d(p ′ )|.

  which is the first condition in Definition 3. Since w ′′ a(p ′ ) ≤ p w and, by (2.21), | d(p ′ )| ≥ |w|, the second condition also holds. Hence, D is simple and D(p ′ ) reduces to it.

  (A)⟩ = min a∈A |τ (a)| and |τ | := |τ (A)| = max a∈A |τ (a)|.

  (i) If σ(a) = σ(b) for some a ̸ = b ∈ A, then σ = σ ′ ϕ a,b , where σ ′ : (A \ {b}) + → B + is the restriction of σ to (A \ {b}) + . (ii) If σ(a) is a prefix of σ(b) and σ(b) = σ(a)t for some nonempty t ∈ B + , then σ = σ ′ ψ a,b , where σ ′ : A + → B + is defined by

□

  An argument similar to the one used in the proof of the previous claim gives us that u n = u i k -1 for every k ∈ {1, . . . , m -1}. (3.19)

  .20) Further, the local period of w at the position |u|, in symbols per(w, u), is defined as the smallest local period of w at the position u. It follows directly from (3.20) that per(w, u) ≤ per(w).

Figure 3 . 1 :

 31 Figure 3.1: The illustration of a local period.

  for every n ∈ N. This and the fact that τ n is a |ϕ n | 4 1 -proper morphism allow us to use Lemma 15 to find morphisms B +

  and N > n, there exists a τ [n,N ) -factorization (k ′ , x ′ ) of x in X (N )τ . This together with the inclusion X

  and |σ 0 (a)| = |ϕ 0 (a)| for all a ∈ A 1

  where B is the alphabet of Y , such that π(σ 0 (x)) = τ (x) for all x ∈ X (1) σ and |σ 0 (a)| = |τ (a)| for every a ∈ A 1 . (3.22)

  Further, by equation (3.22) and the fact that ϕ 0 = τ , we have π(σ 0 (x)) = τ (x) = ϕ 0 (x) and |σ 0 (a)| = |ϕ 0 (a)| for every x ∈ X (1) σ and a ∈ A 1 .

  we deduce that for all a ∈ B N and b ∈ B n , b occurs in τ [n,N ) (a). Thus, τ [n,N ) is positive.

  .23) Let Ãn = A n ∩ L(X n ). Equation (3.23) and the fact that σ is everywhere growing allow us to assume without loss of generality that, after a contraction of σ, the following holds for every n ∈ N: if a ∈ Ãn+1 and w ∈ L(X n ) has length 3, then w occurs twice in σ n (a). (3.24) Let us fix a word w n = a n b n c b ∈ L(X n ) of length 3. Then, by (3.24), we can decompose σ n (a) = u n (a)v n (a) in a way such that u n (a) ends with a n , v n (a) starts with b n c n and |v n (a)| ≥ 2. (3.25) To define τ , we need to introduce additional notation first. Let B n be the alphabet consisting of tuples a b such that ab ∈ L(X n ). Also, if w = w 1 . . . w |w| ∈ L(X n ) has length |w| ≥ 2, then χ n (w) := w1 w2 w2 w3 . . . w |w|-1 w |w| ∈ B + n , and if w ′ = w1 w2 . . . w |w|-1

  It follows from (3.25) that for every n ∈ N and a b ∈ B n+1 , τ n ( a b ) starts with bn cn and ends with an bn . Thus, τ is proper. Moreover, since |v n (a)| ≥ 2, we have |v n (a)u n (a)b n | ≥ 3 and thus |τ n ( a b

  and |σ j+1,0 (a)| = |ϕ j,0 (a)| for every a ∈ A j+1,1 .

Lemma 27 .

 27 Let Y ⊆ B Z be an aperiodic subshift, with B ⊆ L(Y ). Suppose thatU ⊆ Y is (I) d-syndetic: for every y ∈ Y there exists k ∈ [0, d -1] with S k y ∈ U , (II) of radius r: U is a union of sets of the form [u.v], with u, v ∈ A r , (III) ℓ-proper: U ⊆ [u.v] for some u, v ∈ A ℓ , (IV) ρ-separated: U, SU, . . . , S ρ-1 U are disjoint.Then, there exist a letter-onto morphism τ :C + → B + and a subshift Z ⊆ C Z such that (1) Y = n∈Z S n τ (Z) and C ⊆ L(Y ), (2) (Z, τ ) is recognizable with constant r + d,(3) |τ | ≤ d, ⟨τ ⟩ ≥ ρ and τ is min(ρ, ℓ)-proper, (4) C τ (k, z) = R U (y) for all y ∈ Y and τ -factorization (k, z) of y in Z.

  iii) S j ψ(y) = ψ(S k y) for j ∈ Z and k ∈ [k j (y), k j+1 (y)). Proof. Let y, y ′ ∈ Y such that y [-d-r,d+r] = y ′ [-d-r,d+r] . By (I), we have k i+1 (y) -k i (y) ≤ d for all i ∈ Z and, thus, |k 0 (y)|, |k 1 (y)| ≤ d. Since U has radius r and y [-d-r,d+r] = y ′ [-d-r,d+r]

  x ∈ X, (k, Y ) be a τ -factorization of x in Y and (ℓ, z) be a τ σ-factorization of x in Z. Then, there exists m ∈ [0, |σ(z 0 )|) such that y = S m σ(z) and k = |σ(z [-m,0) )| + ℓ. Proof. Being (ℓ, z) a τ σ-factorization of x, we have that ℓ ∈ [0, |τ (σ(z 0 ))|). Hence, there exists m ∈ [0, |σ(z 0 )|) such that

  and then, since |v| ≥ 2|t| ≥ 2|t 0 |, Item (1) in Proposition 7 ensures that S |u| t Z 0 = S -|vw| t Z 0 and uvw = (t Z 0 ) [0,|uvw|) . (4.3) Now, from the first equation in (4.3) and Item (2) in Proposition 7 we get that |u| = -|vw| (mod |t 0 |), that is, |uvw| = 0 (mod |t 0 |). This and the second equation in (4.

Lemma 34 .

 34 Let W be a finite set of words. Then, there exists ε ∈ [|W|/d 2#W+4 , |W|/d) such that for all w ∈ W, either |w| > 10 4 ε or |w| ≤ ε/d. Proof. Let d 0 = 10 4 d and, for ℓ ∈ [1, #W + 1], W ℓ = {w ∈ W : |W|/d ℓ+1 0 < |w| ≤ |W|/d ℓ 0 }. The Pigeonhole principle ensures that W ℓ is empty for some ℓ ∈ [1, #W + 1]. We set ε = ⌊d|W|/d ℓ+1 0 ⌋ and note that for any w ∈ W, either w ∈ ∪ ℓ ′ <ℓ W ℓ ′ and |w| > 10 4 ε, or w ∈ ∪ ℓ ′ >ℓ W ℓ ′ and |w| ≤ ε/d. Also, since ℓ ∈ [1, #W + 1], we have that ε ∈ [|W|/d 2#W+4 , |W|/d). We use Lemma 34 with the set σ(B) to obtain ε ∈ [n/d 2#W+4 , n/d) such that for all a ∈ B, either |σ(a)| > 10 4 ε or |σ(a)| ≤ ε/d. (4.5) Note that ε ∈ [n/d 2d 3 +4 , n/d) as d 3 ≥ #σ(B) by Item (1) in Proposition 11.

  (a) |u| = |u ′ | = 99ε, |v|, |v ′ | ≥ n/2 -500ε, and uu ′ = s Z [-99ε,99ε) for some s ∈ W ε . (b) |u| = |u ′ | = 500ε, |v| = ⌊n/2 -500ε⌋, |v ′ | ≥ n/2 -500ε, and s Z [-99ε,99ε)

  Condition (P a ) or (P b ) in Definition 12, that w = vu w u ′ w v ′ satisfies (P a ) or (P b ). Moreover, since |vu w | = |x [j-|u w v w |,i) | and |v wu w| = |x [j-|v w u w |,j) |, we have that |v wu w| = |vu w | + j -i > |vu w |. Thus, w = vu w u ′ w v ′ satisfies (P a ) or (P b ), and |vu w | is strictly smaller than |v wu w|. This contradicts the minimality of |v wu w|.

  the last relation is possible only if |u ′ w| = 99ε and |u ′ w | = 500ε. (4.9) Therefore, Condition (P a ) holds for w = v wu wu ′ wv ′ w and Condition (P b ) holds for

  Now, Conditions (P a ) and (P b ) in Definition 12 ensure that for any w ∈ RS n (X) the inequalities n/2 -401ε ≤ |u ′ w v ′ w | ≤ n/2 + 401ε hold. Putting this in (4.16) produces |τ (z 0 )| -|σ(y [k,ϕ(1)) )| ≤ 2 • 401ε for all k ∈ [ϕ(0), ϕ(1)).

  z0)|+99ε) . (2) The decompositions w = v w u w u ′ w v ′ w and w = v wu wu ′ wv ′ w satisfy Condition (P a ) in Definition 12 and per(x [c0+97ε,c1-97ε) ) ≤ ε.

  ) ) is at most ε. (4.24) Let us check the hypothesis of Lemma 12. Let k ∈ [ϕ x (i), ϕ x (j)) be arbitrary. There exists ℓ ∈ [i, j) such that k ∈ [ϕ x (ℓ), ϕ x (ℓ + 1)). Putting the hypothesis |τ (z k )| ≤ 401ε in the inequality of Lemma 40 produces the bound |σ(y k )| ≤ |τ (z ℓ )| + 2 • 401ε ≤ 10 4 ε. Hence, by (4.5), |σ(y k )| < ε/d for all k ∈ [ϕ x (i), ϕ x (j)). (4.25) Since ε ≤ n/10 4 , Equation (4.25) and the inequalities ϕ x (j) ≥ (j -i) + ϕ x (i) > d + ϕ x (i) allow us to use Lemma 12 and deduce that per(x [c ϕx (i) -n/3,c ϕx (j)-d ) ) ≤ ε. Now, observe that, for any k ∈ Z,

  a stable interval of aperiodic type and I = I ′ = I ∪ I ′ by maximality.

( 1 )

 1 ε ≤ |u| ≤ |τ (z -1 )| -2ε and ε ≤ |u ′ | ≤ |τ (z 1 )| -2ε.

  36)Note that, by Items ((2)) and ((3)) in Lemma 47,c k(0) ≤ f 0 + 81ε ≤ 81ε and c k(1) ≥ f 1 ≥ 0. Hence, c k(1)+1 ≤ c k(0) + (k(1) -k(0) + 1)|σ| ≤ 81ε + (k(1)k(0) + 1)|σ| and c k(0)-1 ≥ c k(1) -(k(1) -k(0) + 1)|σ| ≥ -(k(1) -k(0) + 1)|σ|.Since, by Lemma 44, [k(0), k(1)) has at most 2d + 1 elements, and since |σ| ≤ 3dn by Item (2) in Proposition 13, we obtain that c k(1)+1 + 3dn ≤ 7d 2 n and c k(0)-1 -3dn ≥ -7d 2 n. This shows (4.36).Thanks to (4.36), we can use the fact that (Y, σ) is 3dn-recognizable (Item (3) of Proposition 13) to deduce that c k = ck and y k = ỹk for all k ∈ [k(0) -1, k(1) + 1).

  (4.41) On one hand, we have by Item (2) in Lemma 47 that |f j+1 -c k(j+1) | and |f j -c k(j) | are at most 81ε. On the other hand, since z j ∈ C ap , Lemma 49 ensures that [k(j), k(j + 1)) is of aperiodic type in y. Hence, by Lemma 44, #[k(j), k(j + 1)) ≤ 2d + 1. Putting these two things in (4.41) yields |τ (z j )| ≤ 2 • 81ε + (2d + 1)|σ| ≤ 10d 2 n.Let now a ∈ C and j ∈ Z be such that z j = a. Then, by Lemma 47,|τ (a)| = f j+1 -f j ≥ 80ε.Next, we consider Item (i) and the inequality #C ap ≤ 2d 3d+6 . Observe that Lemmas 43 and 44 ensure that|σ(y [k(j),k(j+1)) )| ≥ pε for all j ∈ Z. (4.42)This allows us to define u j as the prefix of σ(y[k(j),k(j+1)) ) of length 2ε. We claim that if [k(j), k(j + 1)) is of periodic type, then root σ(y [k(j),k(j+1))) is the prefix of u j of length per(u j ). (4.43) We claim that if aba ′ , a ba ′ ∈ U and π(aba ′ ) = π(a ba ′ ), then b = b. (4.46) Let τ (aba ′ ) = us m u ′ and τ (a ba ′ ) = ũs m ũ′ be the decompositions from the definition of π. With this notation, the hypothesis π(aba ′ ) = π(a ba ′ ) is equivalent to s m = s m. Then, as s = root s and s = root s, s = s and m = m. We now prove that u = ũ. First, we assume without loss of generality that |ũ| ≤ |u|. Then, Lemma 50 ensures that τ (a) is prefix of both us m and ũs m and that |ũ| ≤ |u| ≤ |τ (a)| -2ε. (4.47) This implies that s Z [0,|τ (a)|-|u|) = (S |u|-|ũ| s Z ) [0,|τ (a)|-|u|) . Combining this with the bound |τ (a)| -|u| ≥ 2ε ≥ 2|s| given by (4.47) allow us to use Item (1) in Proposition 7 and conclude that s Z = S |u|-|ũ| s Z . Then, by Item (2) of the same proposition, |u| = |ũ| (mod |s|). From this and (4.47) we deduce that u = ũs ℓ for some ℓ ≥ 0. But since, by Item (3) in Lemma 50, s is not a suffix of u, we must have that ℓ = 0. Therefore, u = ũ. We can show, in a similar fashion, that u ′ = ũ′ . This allows us to conclude that τ (aba ′ ) = τ (a ba ′ ) = us m u ′ , and thus that τ (b) = τ ( b). Being τ injective on letters by Proposition 6, b = b ′ and the claim is proved. Condition

  ) (Z, τ ) has dichotomous periods w.r.t. (C ap , C sp ∪ C wp ) and 8ε. (5) The set C sp satisfies the property described in Proposition 19. Remark 14. Assume the notation of Proposition 17. Then, a ∈ C \ C ap implies that | root τ (a)| = per(τ (a)). Indeed, Item (4) ensures that | root τ (a)| ≤ ε, and Item (2) that |τ (a)| ≥ 2ε, therefore, by Lemma 32, | root τ (a)| = per(τ (a)).

  p a • ζ(root σ(a)) + q a = |σ(a)| | root σ(a)| and 0 < q a ≤ ζ(root σ(a)). (4.48) It is important to remark that Item (2) in Proposition 15 ensures that p a ≥ 2. For a ∈ C p , let ψ sp (a) = (root σ(a)) ζ(root σ(a)) and ψ wp (a) = (root σ(a)) ζ(root σ(a))+qa .

  C Z = C ap ∪ C sp ∪ C wp and, for a ∈ C Z , we set τ if a ∈ C ap ϕ sp (a) if a ∈ C sp ϕ wp (a) if a ∈ C wp (4.50) It then follows that σ = τ η. (4.51) Finally, we set Z = k∈Z S k η(Y ). Let us comment on the definition of τ . Equation (4.51) says that σ(a) = τ (a) if a ∈ C ap and that σ

Now, using 4 .

 4 50 yields root τ (C Z \ C ap ) = root σ(C p ). Therefore, # root τ (C Z ) ≤ #C ap + # root σ(C p ), which gives # root τ (C Z ) ≤ 4d 3d+6 if we use the bounds #C ap ≤ 2d 3d+6 and # root σ(C Y ) ≤ 3d 3d+6 , where the last bound is given by Item (1) in Proposition 15. We now consider Item (2). Let a ∈ C Z . If a ∈ C ap , then by Item (2) in Proposition 15 we have that 20ε ≤ |τ (a)| = |σ(a)| ≤ 10d 2 n. If a ∈ C Z \ C ap and s = root τ (a), then (4.50) implies that |τ (a)| = ζ(s)|s| if a ∈ C sp (ζ(s) + q a )|s| if a ∈ C wp Since 20ε ≤ ζ(s)|s| < 40ε and 0 < q a ≤ |ζ(s)|, we obtain that 20ε ≤ |τ (a)| ≤ 80ε ≤ 10d 2 n. Next, we prove Item (3). Let x ∈ X, (c, y) = F (Y,σ) and (f, z) = F (Z,τ ) (x). We use Lemma 29 to obtain m ∈ [0, |η(y 0 )|) such that η(y 0 ) = z [-m,-m+|η(z0)|) and c 0 = f 0 -|τ (z [-m,0) )|. (4.52) We first assume that z 0 ∈ C ap . Then, Equation (4.52) implies that z 0 occurs in η(y 0 ), and thus, since z 0 ∈ C ap , Equation (4.49) ensures that η(y 0 ) = y 0 ∈ C ap . Hence, m = 0, z 0 = y 0 , c 0 = f 0 and c 1 = c 0 + |σ(y 0 )| = f 0 + |τ (z 0 )| = f 1 . Using this and Item (4) of Proposition 15 with x and (c, y) produces per(x [f0+8ε,f1-8ε) ) = per(x [c0+8ε,c1-8ε) ) > ε.

  .65) Note that, since B ap ∪ B wp ∪ B sp is a partition of B, the sets C ap , C wp and C sp form a partition of C = θ -1 (B).

  ) (Z, τ ) has dichotomous periods w.r.t. (C ap , C sp ∪ C wp ) and 8ε. Proof. We start with the proof of Items ((2)) and ((1)). We have, from the equation τ = σθ and (4.65), that τ (C ap ) = σ(B ap ) and τ (C) = B. Thus, by Item (1) in Proposition 17, #τ (C ap ) = #σ(B ap ) ≤ 2d 3d+6 and #(root τ (C)) = #(root σ(B)) ≤ 5d 3d+6 . Also, from the definition of C we get that #C ≤ #B 4 . Putting the bounds from Item (1) of Proposition 17 in this inequality gives that #C ≤ 7 4 d 12d+36 pow-com(X) 4 . Item (1) follows from the equation τ (C) = B and Item (2) of Proposition 17.We now prove Item (3). Let x ∈ X be arbitrary and define (c, z) = F (Z,τ ) (x) and (c, y) = F (Y,σ) (x). Equation (4.65) ensures that y 0 ∈ B ap if and only if z 0 ∈ C ap . We also note that, by Lemma 54, root τ (z 0 ) = root σ(y 0 ). Therefore, Item (3) of this proposition follows Item (4) in Proposition 17. Remark 15. As was similarly observed in Remark 14, a consequence of Items ((3)) and ((1)) in Proposition 20 is that, for all a ∈ C \ C ap , | root τ (a)| = per(τ (a)). Proposition 21. Let z ∈ Z.

(4. 92 )

 92 Lemma 63. Let z ′ ∈ Z ′ and (c, z) = F (Z,τ ) (τ ′ (z ′ )).

( 1 )

 1 If z ′ 0 ∈ C ′ ap , then c r(z ′ ) ∈ [-ε ′ , |τ ′ (z ′ 0 )| -8ε ′ ). (2) If ε < | root τ ′ (z ′ 0 )| ≤ ε ′ and i is the integer satisfying |τ ′ (z ′ 0 )| ∈ [c i , c i+1 ), then c r(z ′ ) ∈ [-ε ′ , ε ′ ) and c i+r(Sz ′ ) ∈ [|τ ′ (z ′ 0 )| -ε ′ , |τ ′ (z ′ 0 )| + ε ′ ).

  .104) Now, by Lemma 63,c r(z ′ ) , cr(z ′ ) ∈ [-ε ′ , |τ ′ (z ′ 0 )| -8ε ′ ). (4.105) Equations (4.104) and (4.105) imply that

  12d+24 and #A n ≤ 7 4 d 12d+36 • pow-com(X) 4 . (P 2 ) |σ [0,n) (a)| ≤ 4d d 3 +6 • |σ [0,n) (b)| for every a, b ∈ A n .

  4 and #τ (C n,ap ) ≤ 2d 3d+6 . (ii) 2ε n ≤ |τ n (a)| ≤ 10d 2 ℓ n for all a ∈ C n .

  [0,n) (z [0,i) )| is equal to |τ n (z [0,i) )|. Putting this in (4.122) produces |σ [0,n) (z [0,i) )| = k -k. Since k ∈ [0, |σ [0,n) (z 0 )|) and k ∈ [0, |σ [0,n) (z 0 )|), we obtain that |σ [0,n) (z [0,i) )| ≤ k < |σ [0,n) (z 0 )|.

  |τ j+1 | ≤ q 0,n (z ′ ) ≤ |τ n (z ′ 0 )| -8ε n + |τ j | ≤ 10d 2 ℓ j and 500d 2d 3 +6 •ℓ j ≤ ℓ j+1 , we have the bound d n-2-j |τ j | ≤ ℓ n-1 for every j ∈ [0, n -1). Therefore, 2-j ℓ n-1 ≤ 2dℓ n-1 ≤ ε n .

  Proof. Let A = {0, 1}. For a ∈ A, n ≥ 0 and j ∈ [1, ℓ n ], we use the notation w n,j (a) = τ [0,n) (a) p n j and W n,j (a) = w n,1 (a)w n,1 (ā) . . . w n,2 (a)w n,2 (ā) . . . w n,j (a)w n,j (ā). (4.143)Remark that W n,ℓn (a) = τ [0,n+1) (a). We start by proving the following properties of the morphisms τ n .(i) |w n,j (0)| = |w n,j (1)| and |W n,j (0)| = |W n,j (1)| for all n ≥ 0 and j ∈ [1, ℓ n ]. (ii) 8 j k n |τ [0,n) | ≤ |w n,j (a)| ≤ 2 • 8 j k n |τ [0,n) | and 2 • 8 j k n |τ [0,n) | ≤ |W n,j (a)| < 8 j+1 k n |τ [0,n) |. (iii) If n ≥ 0, a, b ∈ A and t is a word such that |t| ≥ |τ n |/2, t is a prefix of τ n (a) and t is a suffix of τ n (b), then τ n (a) = τ n (b) = t.Item (i) directly follows from (4.142). In Item (ii), the first inequality is a consequence of the equality |w n,j (a)| = p n j |τ [0,n) | and that, by the hypothesis,p n j ∈ [8 j k n , 2 • 8 j k n ). We can use this to compute |W n,j (a)| ≥ 2|w n,j (a)| ≥ 2 • 8 j k n |τ [0,n) | and |W n,j (a)| = j i=1 |w n,j (a)w n,j (ā)| ≤ 2 j i=1 2 • 8 i k n |τ [0,n) | < 8 j+1 k n |τ [0,n) |,which shows the second inequality in (ii). Finally, we prove Item (iii). We note that (4.143) implies that |τ [0,n+1) | = |W n,ℓn (0)| ≥ 2|w n,ℓn (0)|. Hence, as (i) ensures that |τ [0,n) (0)| = |τ [0,n) (1)|, |τ n | = |τ [0,n+1) |/|τ [0,n) | ≥ 2|w n,ℓn (a)|/|τ [0,n) | = 2|a p n ℓn |, which allows us to bound |t| ≥ |τ n |/2 ≥ |a p n ℓn |.

  [0,n) | ≤ k < |τ [0,n+1) |.We consider three cases. Assume first that k < |w n,1 (0)|. Then, from (4.142) we have that any w ∈ L(X) ∩ A k occurs in a word of the formw n,1 (a)w n,1 (b) for some a, b ∈ A. This implies, since |w| ≥ |τ [0,n) (a)| and w n,1 (a) = τ [0,n) (a) p n 1 for any a ∈ A, that p X (k) ≤ #A 2 • k = 4k. Let us now assume that |w n,1 (0)| ≤ k < |W n,ℓn-1 (0)|. Let j ∈ [1, ℓ n -1] be the least integer satisfying k < |W n,j (0)|. Then, by (ii), k ≤ |w i,n (a)| = |τ [0,n) (0) p ni | for all i ∈ [j + 1, ℓ n ] and a ∈ A.

p

  S (k) ≤ n:Sn∩A k ̸ =∅ p Sn (k) ≤ (log 2 d + 7) • 2 12 d 4 ,which would show that p S is uniformly bounded and would complete the proof.Let us first prove (i). The definition of S n ensures that p Sn (k) ≤ #W n • max{#V n,w ∩ A k : w ∈ W n }. Hence, by (4.159) and (4.161), p Sn (k) ≤ d 2 • 2 12 d 2 .Next, we prove (ii) by contradiction. Assume that there are more than log 2 d + 7 integers n such that S n ∩ A k ̸ = ∅. Then, we can find n and m such thatS n ∩ A k ̸ = ∅, S m ∩ A k ̸ = ∅and m > n + log 2 d + 7. We have, on one hand, that the definition of S m ensures that k ≥ min w∈Wn ⟨V n,w ⟩. Hence, by (b) and Item (3) in Proposition 28, k ≥ ⟨τ [0,m-1) ⟩/2 6 ≥ 2 m-n-7 ⟨τ [0,n) ⟩. (4.162) On the other hand, the definition of S n guarantees that k ≤ max w∈Wn |V n,w |. Combining this with (4.161) and Item (2) in Proposition 28 produces k ≤ 2|τ [0,n) | ≤ 2d⟨τ [0,n) ⟩. (4.163) Equations (4.162) and (4.163) are incompatible as m -n -7 > log 2 d. This contradiction proves (ii) and completes the proof of the theorem.

  Cette idée a été exploitée dans de nombreux travaux : Durand et. al., dans une série d'articles (le dernier étant [DFM19]), ont développé des techniques à partir du cas substitutif et ont obtenu un critère pour tout BV de rang topologique fini pour décider si un nombre complexe donné est une valeur propre continue ou mesurable, Bezugly et. al. ont décrit dans [BKMS13] le simplexe des mesures invariantes ainsi que les conditions naturelles pour être uniquement ergodique, Giordano et. al. ont borné le rang rationnel du groupe de dimension par le rang topologique ([HPS92]), parmi d'autres travaux. Il est important de noter que ces travaux ont été inspirés ou prouvés pour la première fois dans le cas substitutif.

there are two different and equivalent elements in D ′ , or

	2.3: Diagram illustrating restrictions in Definition 4 for a simple d.i. in
	the case d M , d ′ M ̸ = 1.
	Definition 5. Let D, E be simple d.i. on W. We say that,
	(1) D is equivalent to E, and we write D ∼ E, if d and e have a common
	suffix of length at least ⟨W⟩ (this makes sense by Remark 2).
	(2) D reduces to E, and we write D ⇒ E, if e < s d.
	Observe that, when D and E are simple d.i. on W with D ⇒ E, then, by
	Remark 2, D ∼ E.
	Definition 6. A subset D ′ ⊆ D of simple d.i. is reducible if
	(1)

  as desired.

	Now we use relations (2.26) and (2.27) to obtain restrictions on the smallest
	period of

  satisfies (P a ) in Definition 12. (4.26) Let k ∈ [i, j]. We note that (4.24) implies that per(x [c k -500ε,c k +500ε) ) ≤ ε. Thus, by Lemma 35, there exists s ∈ W ε such that s Z [-99ε,99ε) occurs in x [c k -500ε,c k +500ε) . This implies that if (P b ) in Definition 12 holds for the decomposition w ϕx

  all n ≥ 1. (4.164) Now, since |σ [0,n) | goes to +∞ as n → +∞ and d|σ [0,n) | ≤ ⟨σ [0,n) ⟩ by Item (2) in Theorem 34, we have that ⟨σ [0,n) ⟩ diverges to +∞ as n → +∞. Hence, since X is aperiodic, lim n→+∞ ⟨root σ [0,n) (A n )⟩ = +∞. This and (4.164) allow us to use [Esp22, Corollary 1.4] or [DDMP21, Theorem 4.3] to conclude that X has finite topological rank.

† For a word u, root u denotes the shortest prefix v of u such that u = v k for some k; for a set of words W, root W = {root w : w ∈ W}.‡ A substitution σ : A → B + is positive if for all a ∈ A and b ∈ B, b occurs in σ(a)

† For a word u, root u denotes the shortest prefix v of u such that u = v k for some k; for a set of words W, root W = {root w : w ∈ W}.

Remerciements

support during this process. A very special thanks to Josefa; without your love and patience, this journey would not have been possible. This thesis was supported by Doctoral Felowship CONICYT-PFCHA/Doctorado Nacional/2020-21202229 3

CHAPTER 4. A SOLUTION TO THE S-ADIC CONJECTURE Theorem 29. A minimal subshift X has nonsuperlinear-growth complexity, i.e., X satisfies lim inf n→+∞ p X (n)/n < +∞, if and only if there exist d > 0 and an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that, for every n ≥ 0, the following holds:

We show in Section 4.10 how these theorems provide a unified framework and simplified proofs of several known results on (L) and (NSL), including Cassaigne's Theorem [START_REF] Cassaigne | Special Factors of Sequences with Linear Subword Complexity[END_REF]. We also prove, in Theorem 35, that Condition (P 1 ) in Theorems 28 and 29 cannot be improved to a uniform bound on the cardinalities of the alphabets.

Strategy of the proof

The hard part of the proofs of Theorems 28 and 29 is constructing an S-adic sequence satisfying properties (P i ) from the complexity hypothesis. We detail here the strategy for doing so in the case of Theorem 28; the proof of Theorem 29 is similar.

It is convenient to introduce the following terminology: a coding of a subshift X ⊆ A Z is a pair (Z, σ), where Z ⊆ C Z is a subshift and σ : C → A + a substitution such that X = k∈Z S k σ(Z). It is a standard fact that if τ is an S-adic sequence then there are subshifts X Let X be a linear-growth complexity subshift and d = sup n≥1 p X (n)/n. The typical method for building an S-adic sequence for a subshift X is an inductive process: First, X 0 := X; then, a coding (X i+1 , σ i+1 ) of X i is defined. In this way, σ := (σ n ) n≥0 is an S-adic sequence that, under mild conditions, generates X. We, instead, take a more direct approach, similar to that in [START_REF] Donoso | Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity[END_REF]Theorem 4.3] and [Esp22, Corollary 1.4], but with additional technical details. We consider an increasing sequence of positive integers (ℓ n ) n≥0 with adequate growth and build codings (X n ⊆ C Z n , σ n : C n → A + ) of X ⊆ A Z satisfying (P 1 ),

for all letters a and with d ′ depending on d, and such that certain technical properties hold. These technical properties allow us to define connecting substitutions τ n : C n+1 → C + n in such a way that σ n τ n (x) is, up to a shift, equal to σ n+1 (x), for all x ∈ X n+1 . Then, we can prove that τ = (σ 0 , τ 0 , τ 1 , τ 2 , . . . ) generates X and satisfies all the properties in Theorem 28.

The main idea for constructing the codings (X n , σ n ) is that, thanks to a modification of the technique from [START_REF] Ferenczi | Rank and symbolic complexity[END_REF] Proposition 5], we can build a coding (X ′ n , σ ′ n ) of X (which is described in Proposition 15) in such a way that the words σ ′ (a) are either strongly aperiodic or strongly periodic. The aperiodic words greatly contribute to the complexity, so we can efficiently control them (1) ℓ-syndetic if for all x ∈ X there exists k ∈ [0, ℓ) such that S k x ∈ U ;

(2) of radius r if U is an union of sets of the form {x ∈ X : x [-|u|,|v|) = uv}, where u, v ∈ A r .

Remark that, in a minimal X, any nonempty clopen set U is ℓ-syndetic and of radius r for some ℓ and r.

Proposition 6. Let U ⊆ X be a nonempty clopen set. There exists a recognizable coding (Y ⊆ B Z , σ : B → A + ) of X, with σ injective on letters, such that if x ∈ X, (c, y) = F (Y,σ) (x) and i ∈ Z, then S i x ∈ U if and only if i = c j for some j ∈ Z.

If U is ℓ-syndetic and of radius r, then (Y, σ) additionally satisfies that:

(1) |σ(a)| ≤ ℓ for all a ∈ B.

(2) (Y, σ) is (ℓ + r)-recognizable.

The complexity function

The complexity function p X : Z ≥1 → Z ≥1 of a subshift X is defined by p X (n) = #L(X) ∩ A n . Equivalently, p X (n) counts the number of words of length n that occur in at least one x ∈ X.

Definition 10. We say that X has

(1) linear-growth complexity if there exists d > 0 such that p X (n) ≤ dn for all n ≥ 1;

(2) nonsuperlinear-growth complexity if there exists d > 0 such that p X (n) ≤ dn for infinitely many n ≥ 1.

Remark 10. When X is infinite, then a classic theorem of Morse and Hedlund [START_REF] Morse | Symbolic Dynamics[END_REF] ensures that p X (n) ≥ n + 1 for all n ≥ 0. Thus, an infinite subshift of linear-growth complexity satisfies n ≤ p X (n) ≤ dn, and so p X grows linearly.

The following theorem is classic.

Theorem 30 [START_REF] Cassaigne | Special Factors of Sequences with Linear Subword Complexity[END_REF]). Let X be a transitive linear-growth complexity subshift. Then, p X (n + 1) -p X (n) is uniformly bounded.

For the proof of Theorems 33 and 34 in Section 4.8, we will need only the following two weaker versions of Cassaigne's Theorem.

Lemma 30. Let X be a subshift and d ≥ 1 be such that p X (n) ≤ dn for all n ≥ 1. Then, for every n ≥ 1 there exists m ∈ [n, 2n) such that p X (m + 1)p X (m) ≤ 2d.

Proof. Let n ≥ 1. We observe that the average of p X (m + 1) -p X (m) for m ∈ [n, 2n) can be bounded as follows by using that p X (2n) ≤ 2dn:

Proof. Lemma 46 ensures that the set of all maximal stable intervals of y can be described as {[k(j), k(j + 1)) : j ∈ Z} for some increasing sequence of integers (k j ) j∈Z . The sequence (k j ) j∈Z is unique up to an index shift. We define r(j) as follows:

(i) if [k(j), k(j + 1)) is of aperiodic type, then r(j) = c k(j) .

(ii) if [k(j), k(j + 1)) is of periodic type, then r(j) = c k(j) -|s ℓ |, where s = root σ(y [k(j),k(j+1)) ) and ℓ = ⌈80ε/|s|⌉

It is important to remark that, in case (ii), Lemma 43 ensures that |s| ≤ ε.

We claim that r(j + 1) ≥ r(j) + 80ε for all j ∈ Z.

(4.31)

First, we note that the definition of r(j) and r(j + 1) guarantees that k(j) -81ε < r(j) ≤ k(j) and k(j + 1) -81ε < r(j + 1) ≤ k(j + 1). (4.32)

We now consider two cases. If [k(j), k(j + 1)) is of aperiodic type, then, by Lemma 44, |σ(y [k(j),k(j+1)) )| is at least 195ε. Combining this with (4.32) yields

Assume now that [k(j), k(j + 1)) is of periodic type. Then, [k(j + 1), k(j + 2)) is of aperiodic type by Lemma 43. In particular, r(j + 1) = c k(j+1) by (i). Also, since [k(j), k(j +1)) is of periodic type, (ii) ensures that r(j) ≤ k(j)-80ε. These two things imply that |x [r(j),r(j+1)) | ≥ |x [c k(j) -80ε,c k(j+1) ) | ≥ 80ε, completing the proof of the claim.

Equation (4.31) implies that (r(j)) j∈Z is increasing. Thus, there exists a unique ℓ ∈ Z such that 0 ∈ [r(ℓ), r(ℓ + 1)). We define k x (j) = k(j + ℓ) and r x (j) = r(j + ℓ). Then, (k x (j)) j∈Z and (r x (j)) j∈Z satisfy Items (1), (2) and (3) of the lemma. Moreover, being (r(j)) j∈Z increasing, it is clear ℓ (and then also (k x (j)) j∈Z and (r x (j)) j∈Z ) is unique.

We now define (Z, τ ). It follows from the recognizability property of (Y, σ) and Lemma 45 that the map x → r x (0) is continuous. In particular, U = {x ∈ X : r x (0) = 0} is clopen (and nonempty). We define (Z ⊆ C Z , τ : C → A + ) as the recognizable coding of X obtained from U as in Proposition 6.

Basic properties of the second coding

We fix, for the rest of the section, the following notation. Let x denote an element of X, (c, y) = F (Y,σ) (x) and (f, z) = F (Z,τ ) (x). We also define (k x (j)) j∈Z and (r x (j)) j∈Z as the sequences given by Lemma 47.

Lemma 48. We have that r x (j) = f j for all j ∈ Z.

Proof. Note that {k ∈ Z : S k x ∈ U } is equal to {r x (j) : j ∈ Z}. Then, by Item (1) in Proposition 6, there exists a bijective map g : Z → Z such that r j (x) = f g(j) for all j ∈ Z. Now, since Lemma 47 states that r x (j) < r x (j + 1) for all j ∈ Z, the map g is increasing. As it is also bijective, we conclude that there exists ℓ ∈ Z satisfying g(j) = j + ℓ for all j ∈ Z. Finally, by Item (3) in Lemma 47 and the definition of f , we have that 0

). Hence, g -1 (0) = 0, ℓ = 0 and the lemma follows.

The last lemma allows us to drop the notation r x (j) and use only f j . In particular, Items (2) and (3) of Lemma 47 hold with f j .

We define a partition C a ∪ C ap of C as follows:

Lemma 49. Let j ∈ Z. The following are equivalent:

(1.a)

The following are also equivalent:

(2.a) z j ∈ C ap .

(2.b) [k x (j), k x (j + 1)) is of aperiodic type for y.

(2.c) per(x [fj +8ε,fj+1-8ε) ) > ε.

Proof. We start with a general observation. Let us write k(j) = k x (j). Then, Item (2) in Lemma 47 ensures that

Hence,

We now prove the lemma. Let us assume that (1.a) holds. Then, (4.33) implies that per(x

) . In particular, since

Being |s| ≤ ε, we obtain that s 2 is a prefix of u j and that per(u j ) ≤ |s|. This permits to use Item (1) in Proposition 9 to obtain that per(s 2 ) = per(u j ). Moreover, |s| = per(s 2 ) by Lemma 32; therefore, |s| = per(s 2 ) = per(u j ). Since s Z [0,2ε) = u j , this shows that s is the prefix of u j of length per(u j ), completing the proof of the claim.

We now use (4.43) to prove the following:

) and wheter z j+1 belongs to C ap . (4.44) Suppose that z j ∈ C ap . We consider two cases. If z j+1 ∈ C ap , then Lemma 49 ensures that [k(j), k(j + 1)) and [k(j + 1), k(j + 2)) are of aperiodic type in y, and so, by Item (2) in Lemma 47, that τ

) and ℓ = ⌈80ε/|s|⌉. Now, (4.43) says that s is determined by σ(y [k(j),k(j+1)) ), and the definition of ℓ depends only on s. Therefore, τ (z

). The proof of (4.44) is complete.

Finally, we bound C ap . Condition (4.44) implies that #τ (C ap ) is at most 2 times the number of words of the form σ(y [k(j),k(j+1)) ), where j ∈ Z is such that z j ∈ C ap . Note that if z j ∈ C ap then Lemma 49 gives that [k(j), k(j + 1)) is of aperiodic type, and thus, by Lemma 44, we have that the length of [k(j), k(j+1)) is at most 2d + 1. Hence, there are at most #B d+2 words σ(y [k(j),k(j+1)) ) such that z j ∈ C ap . We conclude that #τ (C ap ) ≤ 2 • #B d+2 , and therefore that #C ap ≤ 2d 3d+6 by Item (2) in Proposition 13 and the fact that τ is injective on letters Next, we prove that # root τ (C) ≤ 3d 3d+6 . Since # root τ (C ap ) ≤ #C ap ≤ 2d 3d+6 by what we just proved and since # root σ(B) ≤ #B ≤ d 3 by Item (2) in Proposition 13, it is enough to show that

Let a ∈ C p and j ∈ Z be such that z j = a. Thanks to Lemma 49, we have that

Hence, by Lemma 43, root τ (z j ) = root σ(y k(j) ). This proves (4.45) and thereby that # root τ (C) ≤ 3d 3d+6 . We now prove that #C p ≤ 2d 3d+9 pow-com(X) using Lemma 50. Let U = {z j-1 z j z j+1 : j ∈ Z, z j ∈ C p }. We define the map π : U → ∪ s∈root σ(B) Pow X (s) as follows. For aba ′ ∈ U, Lemma 50 gives a decomposition τ (aba ′ ) = us m u ′ . We set π(aba ′ ) = s m . Observe that Item (3) in Lemma 50 ensures that s m ∈ Pow X (s), and, by Item (2) of the same lemma, s ∈ root σ(B).

Proof. Let z ∈ Z and (c, y) = F (Y,σ) (τ (z)). We set w j = η(y j ). The definition of η in (4.49) ensures that w j ∈ C block ∪ C ap . Moreover, Item (5) in Propositions 15 says, in this context, that w j w j+1 ̸ ∈ C 2 block for all j ∈ Z. Finally, by Lemma 29 we have that z = S ℓ η(y) for some ℓ ∈ Z, and thus that z = . . . w -1 w 0 w 1 . . . We can now present the proof of Proposition 19.

Proof of Proposition 19. Item (2) directly follows from the definition of τ in (4.50). Let us prove Item (1). Let z ∈ Z be such that z [i,j+1) ⊆ (C Z \ C ap ) + . We write z = . . . w -1 w 0 w 1 . . . as in Lemma 51 and let C block be the set defined in Lemma 51. Then, the hypothesis

Proof. First, we note that, by Item (4) in Proposition 17, per(τ

Let s = root τ (z 0 ). In order to continue, we claim that if a ∈ C Z \ C sp and τ (a) is a prefix of s ∞ , then root τ (a) = s. We now prove that root τ (z i ) = s for i ∈ [0, ℓ) by induction on i. If i = 0, then we have from the hypothesis that τ (z 0 ) is a prefix of τ (z 0 ) and that z0 ∈ C Z \C ap . Thus, root τ (z 0 ) = s by the claim. Let us assume now that 0 < i < ℓ and that root τ (z j ) = s for j ∈ [0, i). Then, τ (z [0,i) ) is a power of s. Being τ (z [0,i] ) a prefix of τ (z 0 ) ∞ = s ∞ , we deduce that τ (z i ) is a prefix of s ∞ . This allows us to use the claim and obtain that root τ (z i ) = s. This finishes the inductive step and the proof of the lemma.

Finally, we prove Proposition 18

Proof of Proposition 18. We fix the following notation for this proof. Let

We start by proving Item (1). Assume that per(x [-ε,ε) ) > ε and that

We have to show that f 0 = f0 and z 0 = z0 . We claim that (i) c 0 = c0 and y 0 = ỹ0 ;

(ii) y 0 ∈ C ap and ỹ0 ∈ C ap ;

(iii) z 0 = y 0 , f 0 = c 0 , z0 = ỹ0 and f0 = c0 . Item (i) follows from the fact that the current hypothesis allows us to use Item (3) of Proposition 15 to get that F 0 (Y,σ) (x) = F 0 (Y,σ) (x), which is equivalent to (i). For Item (ii), we note that if

To prove Item (iii), we first note that Lemma 29 gives an integer m ∈

(4.53)

In particular, y 0 occurs in η(z 0 ). Since Item (ii) ensures that y 0 ∈ C ap , it follows from the definition of η in (4.49) that η(y 0 ) = y 0 = z 0 . Putting this in (4.53) gives that m = 0 and c 0 = f 0 . A similar argument shows that z0 = ỹ0 and f0 = c0 as well. This completes the proof of the claim. Items (i) and (iii) of the claim imply that (f 0 , z 0 ) = ( f0 , z0 ), proving Item (1) of the proposition.

Before proving Item (2), we claim that

(4.54) To prove (4.54), we start by using Item (4) in Proposition 17 to obtain that per(x [f0+8ε,f1-8ε) ) > ε. Thus, by Item (2) in Proposition 9, there exists j ∈ [f 0 + 8ε, f 1 -8ε) satisfying per(x [j-ε,j+ε) ) > ε.

(4.55)

Therefore, by the hypothesis

Combining this with (4.55) allows us to use Item (1) of this proposition and deduce that

Observe that the condition j ∈ [f 0 +8ε, f 1 -8ε) implies that F 0 (Z,τ ) (S j x) = (f 0j, z 0 ). Let i be the integer satisfying fi ≤ j < fi+1 and note that F 0 (Z,τ ) (S j x) = ( fi -j, zi ). Then, by (4.56), f 0 = fi and z 0 = zi . In particular, fi = f 0 ≤ 0 < f 1 = fi+1 , so i = 0. We conclude that F 0 (Z,τ ) (x) = (f 0 , z 0 ) = ( f0 , z0 ) = F 0 (Z,τ ) (x). We now prove Item (2). Assume that F 0 (Z,τ ) (x) = F 0 (Z,τ ) (x). The is equivalent to z 0 = z0 and f 0 = f0 , so f 1 = f1 as well. Hence,

Hence, the equation F 0 (Z,τ ) (S f2 x) = F 0 (Z,τ ) (S f2 x) is equivalent to fℓ = f 2 and zℓ = z 2 .

(4.62)

In particular, τ (z [1,ℓ) ) = τ (z 1 ). Now, since z 1 ∈ C wp , we have by Item (1) in Proposition 19 that zℓ = z 2 ∈ C ap . Therefore, Item (1) in Proposition 19 guarantees that zi ∈ C sp for all i ∈ [0, ℓ -1) and zℓ-1 ∈ C sp . Combining this with (4.60) and the equality

Since q z1 ≤ ζ(s) and q zℓ-1 > 0, we conclude that ℓ ≤ 1. But then τ (z 1 ) = τ (z [1,ℓ) ) is the empty word, which contradicts the definition of τ . Therefore, the case |τ (z 1 )| > |τ (z 1 )| does not occur and the proof is complete.

The fourth coding

In this section, we give the final versions of the codings needed in the proof of the main theorems. The new element of these codings is that it is possible to connect them using morphisms.

The section has two parts. In the first one, we construct the new codings, using Proposition 17 and a modified higher block construction, and present their basic properties. Then, in the second one, we show how we can connect two of these codings using the morphism described in Subsection 4.7.2.

Construction of the fourth coding

Let X ⊆ A Z be an infinite minimal subshift, n ≥ 0 and let d be the maximum of ⌈p X (n)/n⌉, p X (n + 1) -p X (n), #A and 10 4 . We use Proposition 17 with X and n to obtain a recognizable coding (Y ⊆ B Z , σ : B → A + ) of X, a partition B = B ap ∪ B sp ∪ B wp and an integer ε ∈ [n/d 2d 3 +4 , n/d) satisfying Items (1) to (5) of Proposition 17.

We start with the following observation. Since (Y, σ) is a recognizable coding of a minimal subshift, Y is minimal; thus, for all y ∈ Y there exists k < 0 such that y k ∈ B ap . This observation allows us to define the map L(y) = max{k < 0 : y k ∈ B ap } that returns the index of the first-to-the-left symbol in B ap .

Let ψ 0 : Y → B 4 be the map y → y L(y) y -1 y 0 y 1 and ψ(y) = (ψ 0 (S j y)) j∈Z . We treat ψ(y) as a sequence over the alphabet B 4 and define

Let θ(aa -1 a 0 a 1 ) = a 0 for aa -1 a 0 a 1 ∈ C and τ = σθ. Remark that θψ(y) = y for any y ∈ Y . (4.64)

We abuse a bit of the notation and define L(z) = max{k < 0 :

CHAPTER 4. A SOLUTION TO THE S-ADIC CONJECTURE Lemma 58. Let x, x ∈ X and k ≥ 0 be an integer. Suppose that

Proof. We only prove that

) , as then an inductive argument on i gives the lemma.

Let us write (c, y) = F (Y,σ) (x) and (c, ỹ) = F (Y,σ) (x). Then, by Lemma 54, z = ψ(y) and z = ψ(ỹ) satisfy (c, z) = F (Z,τ ) (x) and (c, z) = F (Z,τ ) (x). Hence, the hypothesis F 0 (Z,τ ) (x) = F 0 (Z,τ ) (x) is equivalent to z 0 = z0 and c 0 = c0 . This implies two things:

We deduce that, for any i ∈ [0, c 1 ),

In particular, if

) and the proof is complete.

We now assume that c 1 = 1 (so c1 = 1 as well by (ii)). In this case, F 0 (Z,τ ) (Sx) = (0, z 1 ) and F 0 (Z,τ ) (S x) = (0, z1 ); thus, it is enough to prove that z 1 = z1 .

We observe that, since z 1 = ψ 0 (Sy) and z1 = ψ 0 (S ỹ),

(1) z 1 = y 0 y 0 y 1 y 2 if y 0 ∈ B ap and z 1 = y L(y) y 0 y 1 y 2 if y 0 ̸ ∈ B ap , and

(2) z1 = ỹ0 ỹ0 ỹ1 ỹ2 if ỹ0 ∈ B ap and z1 = ỹL(ỹ) ỹ0 ỹ1 ỹ2 if ỹ0 ̸ ∈ B ap .

From these relations and (i) we deduce that

where we used that c 2 = c2 by (ii). It follows that y 2 = ỹ2 and thus that

Connecting two levels

In this subsection, we consider two of the codings constructed in Subsection 4.7.1 and prove several lemmas that relate them. We start by fixing the necessary notation.

Let X be a minimal infinite subshift, n, n ′ ≥ 1 be integers and let d be the maximum of ⌈p and10 4 . Let E = 50d 2 n and E ′ = 50d 2 n ′ . We will assume throughout the subsection that

We consider the recognizable codings (Z ⊆ C Z , τ :

, n/d) be the constants defined in Subsection 4.7.1, and let us denote by

The crucial relation between (Z ′ , τ ′ ) and (Z, τ ) is the following inequality, which is a consequence of (4.70):

(4.71)

Preliminary lemmas

We fix, for the rest of Subsection 4.7.2, a point x ∈ X and the notation (c, z) =

Lemma 59. Suppose that per(τ

Proof. The condition per(τ

, and so per(τ (z j )) ≤ ε, which implies, by Item (3) in Proposition 20, that z j ∈ C \ C ap . We can then use Item (1) in Proposition 21 to get that

so we in particular have that

(4.74)

We are now going to prove that

The lemma would follow from this and (4.74).

Note that τ

. Also, Item (1) in Proposition 20 ensures that τ ′ (z ′ 0 ) and τ (z 0 ) have length at least 2ε. Then, as

) and (4.73), we can use Item (1) of Proposition 9 to deduce that

Our plan is to derive the lemma using Lemma 56 with S i x and S i x. First, we note that

Equations (4.75) and (4.77) allow us to use Lemma 56 and deduce that

. Furthermore, the last equation and (4.76) are the hypothesis of Lemma 58; hence,

Proof. We consider two cases. First, we assume that per(τ ′ (z ′ 0 )) ≤ ε. This allows us to use Lemma 59 and obtain that,

) be arbitrary and denote by k and ℓ the integers satisfying

We note that the definition of k and ℓ ensures that

If we use the equality c ℓ = c k + (ℓ -k)|τ (z 0 )|, which is a consequence of (4.79), and that c ′

This and (i) yield

Then, the hypothesis of Lemma 60 is satisfied for S c ′ 0 +i x and S c ′ 1 +i x, and thus we obtain that

This and (4.82) allow us to use Lemma 56 and deduce that

The last equation and (4.83) imply that the hypothesis of Lemma 58 holds; therefore, for every j

We will derive (4.81) from this. Let i ∈ [-7ε ′ , 7ε ′ ) be arbitrary. Then, (4.82) and the inequality

Now, we observe that

. Therefore, the lemma follows from (4.85).

Proposition 22. Let x, x ∈ X and suppose that

We start with Item (1). Being r(z ′ ) nonnegative by the definition of r, we have that c r(z ′ ) ≥ c 0 . Hence, c r(z ′ ) ≥ -|τ | ≥ -ε ′ . To prove the other inequality, we note that the condition z ′ 0 ∈ C ′ ap implies, by Item (3) in Proposition 20, that per(τ 3) in Proposition 20. Also, since c j+1 ≥ k ≥ 0, we have that j ≥ 0. We conclude, by the minimality condition in the definition of r, that r(z ′ ) ≤ j. Therefore, c r(z

Observe that, by (4.93) and Item (3) in Proposition 20, z j0 and z j1 belong to

| (where i is the element defined in the statement of the lemma), we have that j 0 ≥ 0 and j 1 ≥ i. We conclude, from the definition of r, that r(z ′ ) ≤ j 0 and i + r(Sz

This completes the proof of Item (2). For Item (3), we note that the condition |s| ≤ ε implies that per(τ

We define i and j as the integers satisfying |τ

Proof. We start with some observations that will be used throughout the proof. Since z ′ 0 = z′ 0 , Lemma 55 gives that Note that Lemma (63) ensures that

Hence, from (4.95) we get that

, we get, from (4.92), that r(z ′ ) ≤ ℓ. In particular, cr(z ′ ) ≤ cℓ = c r(z ′ ) . A symmetric argument shows that c r(z ′ ) ≤ cr(z ′ ) , which allows us to conclude that cr(z) = c r(z ′ ) . Then, it follows from (4.98) that cr(z) = cℓ . Therefore, r(z) = ℓ, and thus z r(z ′ ) = zℓ = zr(z) by (4.98). This proves (4.96).

Observe that (4.96) implies that

This, (4.94) and (4.97) permit to use Lemma 58 and obtain that

Then, since c r(z ′ ) = cr(z ′ ) , we have, for any

To continue, we consider two cases. Assume that per(τ ′ (z ′ 1 )) ≤ ε. We note that, since τ ′ (z ′ 1 ) = τ ′ (z ′ 1 ), per(τ ′ (z ′ 1 )) ≤ ε. Hence, by (4.92), r(Sz ′ ) = r(S z′ ) = 0. Now, the definition of i and j and (4.100) imply that c i = cj and z i = zj . Therefore, c i+r(Sz ′ ) = cj+r(Sz ′ ) and z i+r(Sz ′ ) = zj+r(Sz ′ ) . This completes the proof in this case.

Let us now assume that per(τ ′ (z ′ 1 )) > ε. We are going to argue as in the proof of (4.96). Being τ ′ (z ′ 1 ) = τ ′ (z ′ 1 ), we have that per(τ ′ (z ′ 1 )) > ε. Hence, by the definition of r, z i+r(Sz ′ ) and zj+r(Sz

by Lemma 63, it follows from (4.99) that

The last two things imply, by the definition of r(S z′ ), that cj+r(Sz ′ ) ≤ ck = c i+r(Sz ′ ) . Similarly, c i+r(Sz ′ ) ≤ cj+r(Sz ′ ) . We conclude that k = j + r(S z′ ), cj+r(Sz ′ ) = c i+r(Sz ′ ) and that zj+r(Sz ′ ) = z i+r(Sz ′ ) .

Definition 15. The last proposition allows us to define γ :

We call γ the connecting morphism from (Z ′ , τ ′ ) to (Z, τ ).

). This relation will be freely used throughout this subsection.

The rest of this section is devoted to prove the main properties of γ. We first introduce some notation. Let ρ(a

Let ρ(a) and ψ(z) be defined analogously for a ∈ C and z ∈ Z.

We fix, for the rest of the section, points z ′ , z′ ∈ Z ′ and the notation (c, z) =

Proof. The hypothesis implies that (S c0 x) [-E,E) = (S c0 x) [-E,E) . Then, as z 0 ∈ C ap , we can use Lemma 57 to deduce that c 0 = c0 and τ (z 0 ) = τ (z 0 ). It is left to show that ρ(z -1 ) = ρ(z -1 ) and ρ(z 1 ) = ρ(z 1 ). We will only prove the first equality as the other follows from a similar argument.

There are three cases. Assume first that z -1 ∈ C ap . Then, the hypothesis ensures that (S c0-1 x) [-E,E) = (S c0-1 x) [-E,E) . Since z -1 ∈ C ap , this permits using Lemma 57 with S c0-1 x and S c0-1 x to deduce that τ (z -1 ) = τ (z -1 ). The case z-1 ∈ C ap is analogous.

Let us now assume that z -1 , z-1 ∈ C \ C ap . We define s = root τ (z -1 ) and s = root τ (z -1 ). We have to prove that s = s. Observe that, by Item

. Then, by Theorem 31, s and s are power of a common word, which implies that s = s.

Proof. We first prove the lemma in the case per(τ

and Items ((1)) and ((3)) in Proposition 20 guarantee that per(τ

The first thing and Lemma 63 give

The second thing and the hypothesis ρ(z

In particular, c r(z ′ )+1 = cr(z ′ )+1 .

To continue, we have to consider two cases. We first assume that z ′ 1 ∈ C ′ ap . Then, since ψ(z ′ ) = ψ(z ′ ), we can use Lemma 65 to obtain that c r(z

This and the condition ρ(z n 0 ) = ρ(z n 0 ) permit to use Lemma 65 to obtain that c r(z ′ ) = cr(z ′ ) . Then, since z ′ 0 ∈ C ′ \ C ′ ap , from Lemma 63 we have that

Now, the hypothesis allows us to use Lemma 60 and deduce that

We use again Lemma 60 to obtain that

for all i ∈ [0, k) and j ∈ [0, ℓ). Then:

(1) There is t such that |t| = |s| and t = root τ

(

Proof. We note that, since |s| ≤ ε, Lemma 63 implies that

Hence, by Lemma 59, every i ∈ Z such that

Similarly, for all j ∈ Z such that cj

We will use these relations to prove the following:

we deduce that t and t are conjugate. Therefore, by Item (3) in Proposition 20, t = t. Putting this in (4.115) and then using Item (2) of Lemma 7 yields c r(z ′ ) = cr(z ′ ) (mod |s|). This completes the proof of (4.114).

Let α be the integer satisfying |τ ′ (z ′ [0,k) )| ∈ [c α , c α+1 ). We have, by (4.112), that z i ∈ C \ C ap for all i ∈ [r(z ′ ), α). Also, by the definition of r, we have that z i ∈ C \ C ap for all i ∈ [α, α + r(S k z ′ )). Hence, by ((1)) in Proposition 21, root τ (z i ) = root τ (z r(z ′ ) ) = t for every i ∈ [r(z ′ ), α + r(S k z ′ )). In particular,

We can prove in a similar way that cr(z ′ )+|γ(z ′ [0,ℓ) )| = cr(z ′ ) (mod |s|) and that root τ (z j ) = t for every j ∈ [r(z ′ ), r(z ′ ) + |γ(z ′ [0,ℓ) )|). Being t equal to t, we obtain Item (1). Moreover, since c r(z ′ ) = cr(z ′ ) (mod |s), we also have Item (2).

It is left to prove Item (3). We note that, since t = t, Equations (4.112) and (4.113) imply that ψ(S r(z ′ ) z) = ψ(S r(z ′ ) z) = (t, t, t). Let us now assume that ρ(z ′ k ) = ρ(z ′ ℓ ). There are two cases. First, we assume that

CHAPTER 4. A SOLUTION TO THE S-ADIC CONJECTURE (iv) Propositions 23, 24, 25 and 26 can be used with (Z n+1 , ℓ n+1 ) and (Z n , τ n ).

We define the map r n as follows. If z ′ ∈ Z n+1 and (c, z) = F (Zn,τn) (τ n+1 (z ′ )), then

Note that this is analogous to the definition of r in (4.92). Therefore, Proposition 23 ensures that the connecting morphism σ n : C n+1 → C + n described in Definition 15 is well-defined. The morphism σ n satisfies the following:

We set σ 0 = τ 0 and σ = (σ n ) n≥0 .

Next, we describe σ [0,n) (z ′ 0 ) in terms of τ n (z ′ 0 ) and the auxiliary functions q j,n that we now define. For z ′ ∈ Z n , we set q n,n (z ′ ) = 0 and then inductively define, for 0 ≤ j < n,

where (c, z) = F (Zj+1,τj+1) (S qj+1,n(z ′ ) τ n (z ′ )). An inductive use of (4.119) yields the formula

In particular,

) for all n ≥ 1 and z ′ ∈ Z n . (4.121)

We now prove that σ satisfies all the conditions in Theorem 32.

Lemma 66. Let τ = (τ n : A n+1 → A + n ) n≥0 be an S-adic sequence. Suppose there are subshifts Z n ⊆ A Z n satisfying A n ⊆ L(Z n ). Then, for every x ∈ X τ there are sequences (n ℓ ) ℓ≥0 and x ℓ ∈ ∪ k∈Z S k τ [0,n ℓ ) (Z n ℓ ) such that x is the limit of (x ℓ ) ℓ≥0 .

Proof. Let x ∈ X τ . Then, for all ℓ ≥ 0 there exist n ℓ ≥ 0 and a ℓ ∈ A n ℓ for which x [-ℓ,ℓ) occurs in τ [0,n ℓ ) (a ℓ ). The hypothesis permits to find

. Then, x is the limit of (x ℓ ) ℓ≥0 . The lemma follows.

Lemma 67. The S-adic sequence σ is recognizable and generates X.

Proof. First, we show that σ generates X. Note that (4.121) ensures that

We now introduce some notation. Let ρ(a) = a if a belongs to C n,ap for some n ≥ 1 and let ρ(a) = root τ n (a) if n ≥ 1 and a ∈ C n \ C n,ap . We set ψ(z) = (ρ(z -1 ), ρ(z 0 ), ρ(z 1 )) for n ≥ 1 and z ∈ Z n . Note that these definitions are consistent with the ones in Subsection 4.7.2.

The proof of the following lemma will be postponed until the end of the subsection.

Lemma 71. Item (P 1 ) of Theorem 32 is satisfied by σ.

Proof. The inequality #C n ≤ 7 4 d 12d+36 pow-com(X) 4 in Item (P 1 ) follows from Condition (i). To prove the other inequality, we note that Lemma 70 implies that

Combining this with the bounds given by Condition (i) yields

It only rests to prove Lemma 70. We start by fixing some notation. Let z n , zn ∈ Z n be such that ψ(z n ) = ψ(z n ). We set s = root τ n (z n 0 ) = root τ n (z n 0 ). For j ∈ [0, n), we inductively define z j = σ j (z j+1 ) and zj = σ j (z j+1 ). Let (c j , y j ) = F (Zj ,τj ) (τ j+1 (z j+1 )) and (c j , ỹj ) = F (Zj ,τj ) (τ j+1 (z j+1 ))

With the notation introduced, we have, for every j ∈ [0, n), that S rj (z j+1 ) y j = z j (4.128)

and that

We can also write, thanks to (4.119),

for every j ∈ [0, n). Similar relations hold for zn .

The next three lemmas are the core of the proof of Lemma 70.

Lemma 72. Suppose that ψ(z n ) = ψ(z n ) and that ε n < |s|. Then, for every j ∈ [0, n], the following holds:

(a) q j,n (z n ) = q j,n (z n ) and q j,n (Sz n ) = q j,n (S zn ).

We prove the claim by induction on j. The case j = n is a direct consequence of the hypothesis. Assume that j ∈ [0, n) and that the claim is true for j + 1. The inductive hypothesis gives that z j+1 (2) ψ(z j ) = ψ(S rj (z j+1 ) y j ) = ψ(S rj (ỹ j+1 ) zj ) = ψ(z j ).

Putting the first equation and Item (a) of the induction hypothesis in the definition of q j,n yields

). The rest of the inductive step follows from similar arguments.

Lemma 73. Suppose that ρ(z n 0 ) = ρ(z n 0 ) and ε n-1 < |s| ≤ ε n . Then, for every j ∈ [0, n), the following holds:

(a) q j,n (z n ) = q j,n (z n ) = q j,n (Sz n ) = q j,n (S zn ).

Proof. We first assume that j = n -1. Let us write r = r n-1 , c j = c n-1 j , z = z n-1 , y = y n-1 , etc. Since ε n-1 < |s| ≤ ε n and ρ(z n 0 ) = ρ(z n 0 ), we can use Lemma 25 with z n 0 and zn 0 to deduce the following:

) implies, by (4.128), that Item (b) of the claim holds for j = n -1. Also, since |s| > ε n-1 , the definition of r ensures that z 0 = y r(z n ) ∈ C n-1,ap , so Item (c) of the claim holds. For Item (a), we note that, since q n,n ≡ 0, the definition of q n-1,n ensures that

Therefore, Item (a) of the claim follows from Item (a'). We now assume that j ∈ [0, n -1) and that the claim holds for j + 1. Item (c') of the induction hypothesis ensures that

Then, by the definition of r j and (4.128), z j 0 = y j rj (z j+1 ) ∈ C j,ap and zj 0 ∈ C j,ap . Equation (4.131) also allows us to use Lemma 24 with z j+1 and zj+1 and deduce the following:

(b') ψ(S rj (z j+1 ) y j ) = ψ(S rj (z j+1 ) ỹj ).

Equation (4.128) ensures that Item (b') is equivalent to ψ(z j ) = ψ(z j ). Now, putting Item (a') and Item (a) of the induction hypothesis in the definition of q j,n yields

Similar arguments, which rely on using Lemma 24 with

zj ) and q j,n (Sz n ) = q j,n (S zn ).

To complete the proof, it is enough to show that ψ(z j ) = ψ(S |σ [j,n) (z n )| z j ) and q j,n (z n ) = q j,n (Sz n ). We observe that Item (b) of the inductive hypothesis guarantees that ψ(

ap , we can use Lemma 24 with z j+1 and S |σ [j+1,n) (z n 0 )| z j+1 to obtain the following:

Item (b") implies, by (4.128), that ψ(z j ) = ψ(S |σ [j,n) (z n 0 )| z j ). Also, using the definition of q j+1,n and Item (a") we can write

This and Item (a) of the induction hypothesis gives that q j,n (Sz n ) = q j,n (z n ).

Lemma 74. Suppose that ρ(z n 0 ) = ρ(z n 0 ) and that |s| ≤ ε n-1 . Let j 0 ∈ [0, n) be the least element satisfying |s| ≤ ε j0 . Then, for every j ∈ [j 0 , n], the following holds:

(a) There is s j such that |s j | = |s| and

Proof. The case j = n follows directly from the hypothesis. Assume that j ∈ [j 0 , n) and that the claim holds for j + 1. We observe that, by Items (a) and (c) of the induction hypothesis, z j+1

comply with the hypothesis of Lemma 26. Therefore, by (4.128), Items (a) and (c) hold for j. Moreover, we have that

(4.132) Now, Item (b) of the induction hypothesis gives that q j+1,n (z n ) = q j+1,n (z n ) = q j+1,n (Sz n ) = q j+1,n (S zn ) (mod |s|). Hence, by the definition of q j,n ,

We note that, since Item (a) holds for j, we have that |τ (z j

Hence, by the definition of q j,n ,

Thus, by (4.132) and Item (b) of the induction hypothesis, q j,n (Sz n ) = q j,n (z n ). Similarly, q j,n (S zn ) = q j,n (z n ). We conclude that Item (b) holds for j.

The last ingredient for the proof of Lemma 70 is the following lemma.

Lemma 75. Suppose that ψ(z n ) = ψ(z n ).

(1) If z n 0 ∈ C n,ap , q 0,n (z n ) = q 0,n (z n ) and q 0,n (Sz n ) = q 0,n (S zn ), then

Assume that the hypothesis of Item (1) holds. We also assume, without loss of generality, that |τ n (z n 1 )| ≤ |τ n (z n 1 )|. We start by noticing that, since z n 0 ∈ C n,ap and ψ(z n ) = ψ(z n ), we have that τ n (z n 0 ) is equal to τ n (z n 0 ). Furthermore, by Condition (ii) we have that

(4.133)

Now, from Lemma 68 and the hypothesis we get that

Next, we assume that the hypothesis of Item (2) holds. The condition |s| ≤ ε n enables us to use (4.124) from Lemma 68, so

Now, by the hypothesis, there is k ∈ Z such that k = q 0,n (z n ) = q 0,n (Sz n ) (mod |s|). We deduce from (4.134) that

Being ψ(z n ) equal to ψ(z n ), we have that s = root τ n (z n 0 ). Hence, we can give similar arguments to prove that root σ ) , where k = q 0,n (z n ) = q 0,n (S zn ) (mod |s|). We conclude, as the hypothesis ensures that

. We have all the necessary elements to prove Lemma 70.

. We split the proof into two cases. Let us first assume that |s| > ε n . Then, we can use Lemma 72 and deduce that q 0,n (z n ) = q 0,n (z n ) and q 0,n (Sz n ) = q 0,n (S zn ). Thus, by Lemma 75,

. Next, we assume that |s| ≤ ε n . Let j ∈ [0, n] be the least element satisfying |s| ≤ ε j . We claim that the following is true:

If j = n, then the claim is equivalent to the hypothesis ψ(z ′ ) = ψ(z ′ ). We assume that j < n. Then, |s| ≤ ε n-1 , which permits to use Lemma 74 and conclude that Items (b) and (c) of the claim hold. Moreover, Lemma 74 also states that there is t such that |t| = |s| and t = root

In particular, Item (a) holds. This completes the proof of the claim.

Next, we now prove that

If j = 0, then (4.135) follows from the claim. Let us assume that j > 0. Then, ε j-1 < |s| ≤ ε j . This and Items (a) and (c) of the claim allow us to use Lemma CHAPTER 4. A SOLUTION TO THE S-ADIC CONJECTURE 73 twice, first with z j and zj , and then with

We get q 0,j (z j ) = q 0,j (z j ). and q 0,j (S

Moreover, since root τ j (z j ) is equal to root τ j (S |σ [j,n) (z n )|-1 z j ) and has length |s| ≤ ε j , we can use Lemma 73 with z j and S |σ [j,n) (z n )|-1 z j to obtain that q 0,j (z j ) = q 0,j (S |σ [j,n) (z n )| z j ). Therefore,

Now, from the definition of q 0,n we have that q 0,n (z n ) = q j,n (z n ) + q 0,j (z j ) and q 0,n (z n ) = q j,n (z n ) + q 0,j (z j ). Putting (4.136) and Item (b) of the claim in this relation produces q 0,n (z n ) = q 0,n (z n ) (mod |s|). The rest of the equalities in (4.135) follow from (4.136) and Item (b) of the claim in the same way. The proof of (4.135) is complete.

We recall that we assumed that ψ(z n ) = ψ(z n ) and |s| ≤ ε n . These two things and (4.135) permit to use Lemma 75 and conclude that root

Proof of the main theorems

Lemma 76. Let X be a subshift and W a set of words such that

Proof. The hypothesis implies that any w of length ⟨W⟩ occurring in some x ∈ X occurs in a word of the form uv, where u, v ∈ W. In particular, w occurs in (root u) |W| (root v) |W| . There are at most |W| • #(root W) 2 words satisfying this condition, so p X (⟨W⟩) ≤ |W| • #(root W) 2 .

Theorem 33. A minimal subshift X has linear-growth complexity i.e. lim sup n→+∞ p X (n)/n < +∞, if and only if there exist d ≥ 1 and an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 such that for every n ≥ 0:

If X is infinite and has linear-growth complexity, then σ can be chosen to be recognizable and satisfying #A n ≤ d • pow-com(X) 4 for all n ≥ 0.

Theorem 34. A minimal subshift X has nonsuperlinear-growth complexity i.e. lim inf n→+∞ p X (n)/n < +∞, if and only if there exists d ≥ 1 and an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 such that for every n ≥ 0

If X is infinite and has nonsuperlinear-growth complexity, then σ can be chosen to be recognizable and satisfying #A n ≤ d • pow-com(X) 4 for all n ≥ 0.

We prove Theorems 33 and 34 simultaneously.

Proof of Theorems 33 and 34. Let d = lim inf k→+∞ p X (k)/k and d ′ = sup k≥0 p X (k)/k. We first assume that p X has nonsuperlinear-or linear-growth and show that there exists an S-adic sequence as the ones in Theorems 33 and 34, respectively.

If p X has nonsuperlinear-growth, then d is finite and so, using Lemma 31, we obtain a sequence (ℓ n ) n≥0 such that for all n ≥ 0

(4.137)

If p X has linear growth, then d ′ is finite and using Lemma 30 we get a sequence (ℓ n ) n≥0 that satisfies (4.137) and

We use Theorem 32 with the sequence (ℓ n ) n≥0 . This produces a recognizable S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 generating X such that for every n ≥ 1:

3 ) |σ n-1 (a)| ≤ dℓ n /ℓ n-1 for every a ∈ A n . In particular, the conclusion of Theorem 34 holds. Moreover, if p X has linear growth, then Equation (4.138) holds, so we also have the bound |σ n-1 (a)| ≤ dd ′ for every n ≥ 1 and a ∈ A n . Therefore, in this case, σ satisfies the conclusion of Theorem 33.

We now assume that there exists an S-adic sequence σ = (σ n : A n+1 → A + n ) n≥0 satisfying the conclusion of Theorem 34 or the one of Theorem 33. Note that since σ generates X, we have that

Items (P 1 ) and (P 2 ) of Theorems 34 and 33 then imply that

This proves that X has nonsuperlinear-growth complexity. It rests to prove that X has linear-growth complexity when the conclusion of Theorem 33 holds. We assume that σ satisfies Items (P 1 ), (P 2 ) and (P 3 ) of at most d elements, such that every k j can be written as e∈E α e e, where α e ∈ Z ≥0 .

Lemma 77. Suppose that n > n 3 0 , n > ℓ 18ℓ 2 and ℓ > 3d > 8. Then, P (n, n 0 , ℓ)\ K(n, d, ℓ) is nonempty.

Proof. We will show that #P (n, n 0 , ℓ) > K(n, d, ℓ), which implies that P (n, n 0 , ℓ)\ K(n, d, ℓ) is nonempty. We first estimate #P (n, n 0 , ℓ). Note that there are at least (2•8 j n-8 j n)/n 0 ways of choosing p j in a sequence (p 1 , . . . , p ℓ ) ∈ P (n, n 0 , ℓ). Thus,

where we used that ℓ > 3d > 8. Now, since we assumed that n > n 3 0 , we have that (n/n 0 ) ℓ > n 2ℓ/3 . Hence, as the hypothesis ensures that ℓ > 3d and n > ℓ 18ℓ 2 , (n/n 0 ) ℓ > n d n ℓ/3 > n d ℓ 6ℓ 2 .

This and Equations (4.140) and (4.141) imply that P (n, n 0 , ℓ) \ K(n, d, ℓ) is nonempty.

Lemma 78. Let (ℓ n ) n≥0 be a sequence of positive integers. We consider, for each n ≥ 0, k n ≥ 1 and a sequence (p n 1 , . . . , p n ℓn ) such that p n j ∈ [8 j k n , 2 • 8 j k n ). For a ∈ {0, 1} and ā = 1 -a, we define

and τ = (τ n ) n≥0 . Then:

(1) X τ is infinite, minimal and with linear-growth complexity.

(

(3) (X

(4) 10 p n j 1 ∈ L(X

Using this, (4.142) and the definition of w n,i (a) we deduce that any w ∈ L(X) ∩ A k occurs in a word having either the form W n,j (a)τ [0,n) (b) p n j+1 or the form

, where a, b ∈ A. Therefore,

Putting that |W n,j (0

We conclude that p X (k) ≤ 1024k for every k ≥ 1 and that X has lineargrowth complexity.

Items (2) and (4) of the lemma follow from (4.142). Thus, it only left to prove Item (3). We note that, since -k) is both a suffix of τ n (ỹ -1 ) and a prefix of τ n (y 0 ). Hence, by (iii),

We can now prove Theorem 35. . Then, we can use Lemma 77 to find (p 0 1 , p 0 2 , . . . , p 0 ℓ0 ) ∈ P (M 0 , m 0 , ℓ 0 ) \ K(M 0 , d 0 , ℓ 0 ).

We define τ 0 using (p ) as in (4.142). We set τ = (τ n ) n≥0 . Then, Items (1) to (4) in Lemma 78 hold. In particular, X τ is minimal and has linear-growth complexity. We prove that X τ satisfies the conclusion of the theorem by contradiction. Suppose that there exist d and σ = (σ n : A n+1 → A + n ) n≥0 satisfying Items (1), ( 2) and (3) of Theorem 33 and #A n ≤ d for all n ≥ 1.

We claim that there exists n, n ′ ≥ 0 such that

We take n ≥ 0 big enough so that d n ≥ 12d 4 + d and m n ≥ 12d 2 . Let n ′ ≥ 0 be the integer satisfying

Then, by Items (2) and (3) in Theorem 33, 1 6

Also, since we chose M n and n so that M n > m 3 n and m n ≥ 12d 2 , we have that m n ≤ 1 12d 2 M n . This completes the proof of the claim. Let w a = τ [0,n) (a) for a ∈ {0, 1}. Then, by Item (4) in Lemma 78, 10 p n j 1 ∈ L(X (n) τ ) for every j ∈ [1, p n ℓn ]. Being X τ generated by σ, there exist u j ∈ A + such that w 1 w p n j 0 w 1 occurs in σ [0,n ′ ) (u j ). Moreover, we can take u j so that the following condition holds: If a j is the first letter of u j and b j is the last letter of u j , then there exists a prefix s j of σ [0,n ′ ) (a j ) and a suffix t j of σ [0,n ′ ) (b j ) such that s j w 1 w

We define a j a ′ j a ′′ j as the first three letters of u j and b ′′ j b ′ j b j as the last three letters of u j .

We claim that

, where q i ≥ 1 and r i is a prefix of w 0 different from w 0 . Similarly, σ [0,n) (a j a ′ j ) = s j w 1 w qj 0 r j for some q j ≥ 2 and prefix r j of w 0 different from w 0 . Then, as the hypothesis implies that σ [0,n) (a i a ′ i a ′′ i ) = σ [0,n) (a j a ′ j a ′′ j ), Item (4) in Lemma 78 can be used to obtain r i = r j . We obtain that

Therefore, if q i ̸ = q j then w 0 = w 1 , which contradicts the fact that X is infinite. We conclude that q i = q j , and thus that s i = s j . A similar argument shows that t i = t j , and the claim follows.

Thanks to the claim, we have that the set

, where in the last step we used (4.145). Therefore, as

Applications

We present in this section new and simpler proofs, based on Theorems 33 and 34, of known results about linear-growth and nonsuperlinear-growth complexity subshifts.

Cassaigne's Theorem

A classic result on linear-growth complexity subshift is Cassaigne's Theorem [START_REF] Cassaigne | Special Factors of Sequences with Linear Subword Complexity[END_REF], which states that, for any transitive subshift X in this complexity class, p X (n + 1) -p X (n) is uniformly bounded. We show in this subsection how to use Theorem 33 to give a different proof of this result, in the case in which X is minimal.

We start with a lemma containing the technical core of our approach.

Lemma 79. Let x, y ∈ A Z , p 1 , . . . , p n ∈ Z be a collection of different integers and ℓ 1 , . . . , ℓ n ≥ 1. Suppose that:

(1) x [pj ,pj +ℓj ) = y [0,ℓj ) for all j ∈ [1, n].

(2) |p j -p i | ≤ 1 2 ℓ k for all i, j, k ∈ [1, n]. Then, there exists w ∈ A + such that, for all i, j ∈ [1, n], x [pi,pj ) is a power of w and x [pi,pj +min(ℓi,ℓj )) is a prefix of w ∞ .

Proof. Being the p j different, there is no loss of generality in assuming that p 1 < p 2 < • • • < p n . We define, for i, j ∈ [1, n] with i < j, w i,j = root x [pi,pj ) and ℓ i,j = min(ℓ i , ℓ j ). Then, Item (1) in the statement of the lemma ensures that x [pi,pi+ℓi,j ) = x [pj ,pj +ℓi,j ) , and thus that x [pi,pj +ℓi,j ) is a prefix of w ∞ i,j . In particular, as p i < p j and w i,j = root x [pi,jj ) , we have that for all i, j ∈ [1, n] with i < j,

x [pi,pi+ℓi,j ] and x [pj ,pj +ℓi,j ) are prefixes of w ∞ i,j . (4.151)

Therefore, it is enough to find w such that w = w i,j for all i < j. First, we show that

Observe that, in this situation, we have from (4.151

Now, by Item (2) in the statement of the lemma and the definition of w i,k and w j,k , w i,k and w j,k have length at most m/2. This and (4.153) permits to use Theorem 31 and obtain that w i,k and w j,k are powers of a common word. This implies, since w i,k and w j,k are defined as roots, that w i,k = w j,k . We now note that if i, j ∈ [1, n] and i < j, then (4.152) ensures that w 1,j = w i,j . Hence, as x [p1,pj ) = x [p1,pi) x [pi,pj ) , w 1,i = w 1,j = w i,j . Being i, j arbitrary, this implies that w 1,2 = w 1,j = w i,j . Therefore, the lemma follows from defining w := w 1,2 .

The proposition below uses Lemma 79 to give a bound for p X (n + 1) -p X (n) in a very general context. Proposition 27. Let W ⊆ A + and X ⊆ k∈Z S k W Z . Then, for any ℓ < ⟨W⟩,

Proof. We prove the proposition by contradiction. Suppose that ℓ < ⟨W⟩ and that p X (ℓ + 1) -p X (ℓ) ≥ 256#A • #(root W) 2 |W| 2 /ℓ 2 . Then, by Proposition 10, we can find at least 256#(root W) 2 |W| 2 /ℓ 2 right-special words {u i : i ∈ I} of length ℓ in X. Let u i a i,0 and u i a i,1 be two different right extensions for u i in X. We are going to prove that a i,0 = a i,1 for some i ∈ I, contradicting the fact that u i a i,0 and u i a i,1 are different.

In particular, each u i a i,j occurs in some x i,j = . . .

We use (b) and the Pigeonhole principle to obtain a set I ′ ⊆ I and x 0 , x 1 ∈ X ′ such that #I ′ ≥ 256|W| 2 /ℓ 2 and x j = x i,j for all i ∈ I ′ and j ∈ {0, 1}. We use again the Pigeonhole principle to find I ′′ ⊆ I ′ satisfying #I ′′ ≥ #I ′ /(8|W|/ℓ) 2 ≥ 4 and

and γ p ̸ = γ q . Item (i) follows from (4.155). For Item (ii), we first note that the definition of γ i ensures that m i := ℓ + β i,1 -β = ℓ + β i,0 -γ i , and that (i) gives 3 4 ℓ ≤ m i ≤ ℓ. Thus, u i has a suffix u ′ i of length m i ≥ 3 4 ℓ such that, by (4.154), satisfies Item (ii). It is left to prove (iii). Assume that p, q ∈ I ′′ and γ p + |u ′ p | = γ q + |u ′ q |. Then, β p,0 = γ p + |u ′ p | -ℓ = γ q + |u ′ q | -ℓ = β q,0 and hence u p = u q , which implies that p = q. Let us now suppose that γ p = γ q . Note that if |u ′ p | = |u ′ q | then γ p + |u ′ p | = γ q + |u ′ q |, and so p = q by what we just proved. Thus, there is no loss of generality in assuming that |u ′ p | < |u ′ q |. Then, (ii) allows us to write u

. Therefore, u ′ p a p,0 = u ′ p a p,1 , which contradicts the definition of a p,0 and a p,1 . This shows that the case |u ′ p | < |u ′ q | does not occur, so |u ′ p | = |u ′ q | and p = q. This completes the proof of the claim. Thanks to the claim, we have that (

Moreover, all the γ i are different by (iii). Therefore, we can use Lemma 79 and deduce that there exists w ∈ A + such that for any p, q ∈ I ′′ , ) is a prefix of w ∞ . We conclude that u ′ r a r,1 is a prefix of w ∞ . But then (4.157) implies that u ′ r a r,1 = u ′ r a r,0 , contradicting our assumptions. Theorem 36 [START_REF] Cassaigne | Special Factors of Sequences with Linear Subword Complexity[END_REF]). Let X be a minimal linear-growth complexity subshift. Then, p X (ℓ + 1) -p X (ℓ) is uniformly bounded.

Proof. Let ℓ ≥ 1 be arbitrary. The theorem is trivial if X is finite, so we assume that X is infinite. Then, we can use Theorem 33 to obtain an S-adic sequence σ = (σ n : A + n+1 → A + n ) n≥1 generating X and d ≥ 1 such that the conditions (1), (2) and (3) of Theorem 33 hold.

Let n ≥ 1 be the least integer such that ⟨σ [0,n) ⟩ > ℓ. Then, X is a subset of k∈Z S k σ [0,n) (A Z n ), so Proposition 27 and the conditions in Theorem 33 give the bounds:

Now, the minimality of n ensures that ⟨σ [0,n-1) ⟩ ≤ ℓ, so by Items (2) and (3) in Theorem 33,

Therefore, p X (ℓ + 1) -p X (ℓ) ≤ 256#A 0 • d 6 and p X (ℓ + 1) -p X (ℓ) is uniformly bounded.

A theorem of Cassaigne, Frid, Puzynina and Zamboni

Let S ⊆ A * be a set of words. We use the notation S 2 = {uv : u, v ∈ S} and p S (n) = #(S ∩ A n ). The following result was proven in [START_REF] Cassaigne | A characterization of words of linear complexity[END_REF].

Theorem 37. Let x ∈ A N be an infinite sequence. The following conditions are equivalent:

(1) x has linear-word complexity.

(2) There exists S ⊆ A * such that S 2 ⊇ L(x) and sup n≥1 p S (n) < +∞.

In this subsection, we give a different proof of Theorem 37 for the case of minimal subshifts. We start by proving the following corollary of Theorem 33.

Proposition 28. Let X be an infinite minimal subshift of linear-growth complexity. There exists d ≥ 1 such that for any d ′ ≥ 2 we can find τ = (τ n : A n+1 → A + n ) n≥0 generating X such that:

(1) #(root τ [0,n) (A n )) ≤ d.

( Proof. Let τ ′ = (τ n : A n+1 → A + n ) n≥0 and d be the elements given by Theorem 33 when it is applied with X, and let d ′ ≥ 1 be arbitrary. We will construct τ by carefully contracting τ ′ .

Let n 0 = 0 and inductively define n k+1 as the smallest integer such that n k+1 > n k and ⟨τ [n k ,n k+1 ) ⟩ ≥ 2. We observe that, since ⟨τ [n k ,n k+1 -1) ⟩ = 1 by the minimality of n k+1 , we have that ⟨τ Lemma 80. Let w ∈ A + and ℓ ≤ |w|. There exists a set of words V such that:

(1) #V ∩ A n ≤ 2 5 |w|/ℓ for all n ≥ 1.

(2) ⟨V⟩ ≥ ℓ, |V| ≤ |w|.

(3) For any u occurring in w of length |u| ≥ 2 6 ℓ we have that u ∈ V 2 .

Proof. For i ≥ 0 and j ∈ [0, 7], let w = u i,j (1)u i,j (2) . . . u i,j (2 i ) be the (unique) decomposition of w into 2 i words u i,j (k) ∈ A * such that |u i,j (1) . . . u i,j (k)| = ⌊(8k + j)|w|/2 i+3 ⌋ for all k ∈ [1, 2 i ]. We define V i as the set of words that are a prefix or a suffix of length at least ℓ of some u i,j (k). Set V := ∪ 0≤i<log 2 (|w|/ℓ) V i . It follows from the definition of V that ⟨V⟩ ≥ ℓ and that |V| ≤ |w|, so Item (2) holds. For Item (1), we note that if n ≥ 1, then each u i,j (k) has at most one prefix of length n and at most one suffix of length n. Hence, #V i ∩A n is bounded by above by 2•8•2 i = 2 i+4 . Therefore, #V ∩A n ≤ 0≤i<log 2 |w|/ℓ 2 i+4 ≤ 2 5 |w|/ℓ.

We now prove Item (3). Let u be a word of length |u| ≥ 2 6 ℓ that occurs in w. Let us write w = tus, where t, s ∈ A * , and take i ≥ 0 such that |w|/2 i+1 < |u| ≤ |w|/2 i . Remark that i < log 2 (|w|/ℓ) as |u| ≥ 2 6 ℓ. We also consider the unique pair (k, j) ∈ [1, 2 i ] × [0, 7] such that (8k + j)|w|/2 i+3 ≤ |t| + |w|/2 < (8k + j + 1)|w|/2 i+3 .

Then, we can write u = u ′ u ′′ in such a way that |tu ′ | = ⌊(8k + j)|w|/2 i+3 ⌋. It is not difficult to check that u ′ is a suffix of u i,j (k) and that u ′′ is a prefix of u i,j (k + 1). Moreover, |u ′ |, |u ′′ | ≥ |w|/2 i ≥ ℓ, so u ′ , u ′′ ∈ V i and u ∈ V 2 .

Theorem 38. Let X ⊆ A Z be an minimal subshift. The following conditions are equivalent:

(1) X has linear-word complexity.

(2) There exists S ⊆ A * such that S 2 ⊇ L(X) and sup n≥1 p S (n) < +∞.

Proof. We first suppose that X satisfies Condition (2) and define d = sup n≥1 p S (n). Then L(X) ∩ A n ⊆ {ss ′ : s, s ′ ∈ S, |s| = n -|s ′ |}. Hence, p X (n) ≤ Let us now suppose that X has linear-growth complexity. The case in which X is finite is trivial; hence, we assume that X is infinite. Using Proposition 28 with d ′ = 2, we can find a constant d and an S-adic sequence τ = (τ n : A n+1 → A + n ) n≥0 generating X such that Items (1), (2) and (3) in Proposition 28 hold. We define S as follows. Let n ≥ 1. For u ∈ root τ [0,n) (A n ), we take p n,u ≥ 1 such that |τ [0,n) | ≤ |u pn,u | < 2|τ [0,n) |. We define W n = {u pn,u v pn,v : u, v ∈ root τ [0,n) (A n )}. For each w ∈ W n , we use Lemma 80 with ℓ = ⟨τ [0,n-1) ⟩/2 6 to obtain a set V n,w satisfying the following: (c) if u occurs in w and |u| ≥ ⟨τ [0,n-1) ⟩, then w ∈ V 2 n,w . We set S = ∪ n≥1 ∪ w∈Wn V n,w .

Before continuing, we make some observations about the definitions. It follows from the definition of W n and Item (1) in Proposition 28 that

(4.159)

We also have that