There are numerous connections between the theory of formal languages and that of symbolic dynamics. In each, the transition from one dimension to two dimensionsis accompanied by much difficulty due in large part to the emptiness problem, which is related to the presence (or lack thereof) of periodic points and is known to be undecidable. Here, we focus on two-dimensional languages that have the property that all blocks allowed by the language can be extended to a configuration of the plane satisfying the structure of the language; for such languages the emptiness problem is not an issue. We first show that dot systems may be associated with two-dimensional languages having this property, so that we might employ these languages as varied examples. We next define a new type of finite automaton and with it, a tool for recognizing two-dimensional "strings" of data. It is then shown that these automata correctly represent the sofic shift spaces that result from the application of block maps to shifts of finite type. Thereafter, these automataare utilized to investigate properties of transitivity in the two-dimensional languages that they represent. More specifically, new definitions for different types of two-dimensional transitivity are adapted from topological dynamics and then illustrated through the use of dot systems. The appearance of periodic points in the languages represented by these automata is also explored, with a main result being that the existence of a periodic pointis guaranteed under certain conditions. Finally, issues of equivalence are introduced in the two-dimensional setting with regards to formal languages (syntactic monoids) and symbolic dynamics (the follower sets of a graph representing a sofic shift space).

Given a family of recognizable languages L1, . . . ,Lm and recognizable languages K1 ⊆ K2, the relative inclusion star height problem means to compute the minimal star height of some rational expression r over L1, . . . ,Lm satisfying K1 ⊆ L(r) ⊆ K2. We show that this problem is of elementary complexity and give a detailed analysis its complexity depending on the representation of K1 and K2 and whether L1, . . . ,Lm are singletons.

This paper deals with decision problems related to the star problem in trace monoids, which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by G. Richomme from 1994 [32, 33], we know that the star problem is decidable in trace monoids which do not contain a submonoid of the form {a,c}* x {b,d}*. Here, we consider a more general problem: Is it decidable whether for some recognizable trace language and some recognizable or finite trace language P the intersection R ∩ P* is recognizable? If P is recognizable, then we show that this problem is decidale iff the underlying trace monoid does not contain a submonoid of the form {a,c}* x b*. In the case of finite languages P, we show several decidability and undecidability results.

This paper deals with decision problems related to the star problem in trace monoids, which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by G. Richomme from 1994 [32, 33], we know that the star problem is decidable in trace monoids which do not contain a submonoid of the form {a,c}* x {b,d}*. Here, we consider a more general problem: Is it decidable whether for some recognizable trace language and some recognizable or finite trace language P the intersection R ∩ P* is recognizable? If P is recognizable, then we show that this problem is decidale iff the underlying trace monoid does not contain a submonoid of the form {a,c}* x b*. In the case of finite languages P, we show several decidability and undecidability results.

In this doctoral dissertation, we studied and solved several questions regarding positional and abstract numeration systems. Each particular problem is the focus of a chapter. The first problem concerns the study of the preservation of recognizability under multiplication by a constant in abstract numeration systems built on polynomial regular languages. We obtained several results generalizing those from P. Lecomte and M. Rigo. The second problem we considered is a decidability problem, which was already studied, most notably, by J. Honkala and A. Muchnik. For our part, we studied this problem for two new cases: the linear positional numeration systems and the abstract numeration systems. Next, we focused on the extension to the multidimensional setting of a result of A. Maes and M.~Rigo regarding S-automatic infinite words. We obtained a characterization of multidimensional S-automatic words in terms of multidimensional (non-necessarily uniform) morphisms. This result can be viewed as the analogous of O. Salon's extension of a theorem of A. Cobham. Finally, generalizing results of P. Lecomte and M. Rigo, we proposed a formalism to represent real numbers in the general framework of abstract numeration systems built on languages that are not necessarily regular. This formalism encompasses in particular the rational base numeration systems, which have been recently introduced by S. Akiyama, Ch. Frougny, and J. Sakarovitch. Finally, we ended with a list of open questions in the continuation of this work./Dans cette dissertation, nous étudions et résolvons plusieurs questions autour des systèmes de numération abstraits. Chaque problème étudié fait l'objet d'un chapitre. Le premier concerne l'étude de la conservation de la reconnaissabilité par la multiplication par une constante dans des systèmes de numération abstraits construits sur des langages réguliers polynomiaux. Nous avons obtenus plusieurs résultats intéressants généralisant ceux de P. Lecomte et M. Rigo. Le deuxième problème auquel je me suis intéressée est un problème de décidabilité déjà étudié notamment par J. Honkala et A. Muchnik et ici décliné en deux nouvelles versions : les systèmes de numération de position linéaires et les systèmes de numération abstraits. Ensuite, nous nous penchons sur l'extension au cas multidimensionnel d'un résultat d'A. Maes et de M. Rigo à propos des mots infinis S-automatiques. Nous avons obtenu une caractérisation des mots S-automatiques multidimensionnels en termes de morphismes multidimensionnels (non nécessairement uniformes). Ce résultat peut être vu comme un analogue de l'extension obtenue par O. Salon d'un théorème de A. Cobham. Finalement, nous proposons un formalisme de la représentation des nombres réels dans le cadre général des systèmes de numération abstraits basés sur des langages qui ne sont pas nécessairement réguliers. Ce formalisme englobe notamment le cas des numérations en bases rationnelles introduits récemment par S. Akiyama, Ch. Frougny et J. Sakarovitch. Nous terminons par une liste de questions ouvertes dans la continuité de ce travail.