This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold.

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold.

Cette thèse est composée de deux parties : l’une traite des propriétés combinatoires de mots infinis et l’autre des problèmes de colorations des graphes.La première partie du manuscrit concerne les structures régulières dans les mots apériodiques infinis, à savoir les sous-séquences arithmétiques et les premiers retours complets.Nous étudions la fonction qui donne la longueur maximale d’une sous-séquence arithmétique monochromatique (une progression arithmétique) en fonction de la différence commune d pour une famille de mots morphiques uniformes, qui inclut le mot de Thue-Morse. Nous obtenons la limite supérieure explicite du taux de croissance de la fonction et des emplacements des progressions arithmétiques de longueurs maximales et de différences d. Pour étudier des sous-séquences arithmétiques périodiques dans des mots infinis, nous définissons la notion d'indice arithmétique et obtenons des bornes supérieures et inférieures sur le taux de croissance de la fonction donnant l’indice arithmétique dans la même famille de mots.Dans la même veine, une autre question concerne l’étude de deux nouvelles fonctions de complexité de mots infinis basées sur les notions de mots ouverts et fermés. Nous dérivons des formules explicites pour les fonctions de complexité ouverte et fermée pour un mot d'Arnoux-Rauzy sur un alphabet de cardinalité finie.La seconde partie de la thèse traite des colorations parfaites (des partitions équitables) de graphes infinis de degré borné. Nous étudions les graphes de Caley de groupes additifs infinis avec un ensemble de générateurs fixé. Nous considérons le cas où l'ensemble des générateurs est composé d'entiers de l'intervalle [-n, n], et le cas où les générateurs sont des entiers impairs de [-2n-1, 2n+1], où n est un entier positif. Pour les deux familles de graphes, nous obtenons une caractérisation complète des colorations parfaites à deux couleurs / The content of the thesis is comprised of two parts: one deals with combinatorial properties of infinite words and the other with graph coloring problems.The first main part of the manuscript concerns regular structures in infinite aperiodic words, such as arithmetic subsequences and complete first returns.We study the function that outputs the maximal length of a monochromatic arithmetic subsequence (an arithmetic progression) as a function of the common difference d for a family of uniform morphic words, which includes the Thue-Morse word. We obtain the explicit upper bound on the rate of growth of the function and locations of arithmetic progressions of maximal lengths and difference d. To study periodic arithmetic subsequences in infinite words we define the notion of an arithmetic index and obtain upper and lower bounds on the rate of growth of the function of arithmetic index in the same family of words.Another topic in this direction involves the study of two new complexity functions of infinite words based on the notions of open and closed words. We derive explicit formulae for the open and closed complexity functions for an Arnoux-Rauzy word over an alphabet of finite cardinality.The second main part of the thesis deals with perfect colorings (a.k.a. equitable partitions) of infinite graphs of bounded degree. We study Caley graphs of infinite additive groups with a prescribed set of generators. We consider the case when the set of generators is composed of integers from the interval [-n,n], and the case when the generators are odd integers from [-2n-1,2n+1], where n is a positive integer. For both families of graphs, we obtain a complete characterization of perfect 2-colorings