The aim of the poster is to showcase the interplay between group theory, algebraic topology and combinatorics on words. A result that allows to display this is the return theorem by Berté et al. in 2015. The poster will contain an introduction to fundamental groups of graphs, dendric words as well as a new result concerning return groups of eventua...
Since 2015, dendric shifts (a generalisation of Sturmian words) have been widely studied. One of the results concerning these shift spaces is the return theorem. It describes the groups generated by the return words of a dendric shift. The proof uses the fundamental group of the Rauzy graph of the shift space. Later, eventually dendric shifts were ...
La géométrie fractale, qui entretient des liens étroits avec la théorie ergodique et la théorie des probabilités, est un sujet très dynamique à l’échelle internationale. Un grand intérêt se porte naturellement sur les ensembles et mesures auto-similaires ou auto-affines. Le cadre aléatoire a été beaucoup étudié s’agissant des ensembles autosimilair...
Given a finite set of symbols and a list of rules specifying which of them can appear next to each other, one can build a -- possibly empty -- set of infinite lines of symbols in two directions obeying these rules, named configurations. A set of configurations is called a one-dimensional subshift, and it is the mathematical object at the core of sy...
We study in this thesis homomorphisms between Z^d-symbolic dynamical systems generated by constant-shape substitutions. This notion extends the classical dynamical one of morphism like factor and conjugacy. Isomorphisms are conjugacies of Z^{d}-actions, up to GL(d,Z)-transformations. We show the class of substitutive subshifts is stable under aperi...
Nous étudions dans cette thèse la notion de quasipériodicité,introduite par Apostolico et Ehrenfeucht au début des années 1990,puis étendue aux mots infinis par Solomon Marcus au début des années2000. Un mot (fini ou infini) w est quasipériodique s'il peut êtrecouvert par des occurrences, éventuellement chevauchantes, d'un autremot, fini, appelé sa...
Shift spaces are sets of colorings of a group which avoid a set of forbidden patterns and are endowed with a shift action. These spaces appear naturally as discrete versions of dynamical systems: they are obtained by partitioning the phase space and mapping each element into the sequence of partitions visited by its orbit.Severa! breakthroughs in t...
This thesis is devoted to the study of different problems in ergodic theory and topological dynamics related to og cube structures fg. It consists of six chapters. In the General Presentation we review some general results in ergodic theory and topological dynamics associated in some way to cubes structures which motivates this thesis. We start by ...
Since early 90s, timed automata and timed languages are extensively used for modelling and verification of real-time systems, and thoroughly explored from a theoretical standpoint. Recently Asarin and Degorre introduced the notions of volume and entropy of timed languages to quantify the size of these languages and the information content of their ...
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