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 ...
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...
Understanding the topological structure of phase space for dynamical systems inhigher dimensions is critical for numerous applications, including transport of objects in the solar system, systems of uids, and charged particles in crossed magnetic and electric elds. Many topological techniques have been developed to study maps of two-dimensional (2D...
The Domino Problem on ℤ² asks if it is possible to tile the plane with a given set of Wang tiles; it is a classical decision problem which is known to be undecidable. The purpose of this article is to parameterize this problem to explore the frontier between decidability and undecidability. To do so we fix some horizontal constraints H on the tiles...
This paper proves the decidability of several important properties of additive cellular automata over finite abelian groups. First of all, we prove that equicontinuity and sensitivity to initial conditions are decidable for a nontrivial subclass of additive cellular automata, namely, the linear cellular automata over 𝕂ⁿ, where 𝕂 is the ring ℤ/mℤ. T...
We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Gio...
Dans la vie quotidienne, nous avons souvent besoin de représenter des nombres à l’aide de systèmes de numération. Dans ce mémoire de HDR nous envisageons de tels systèmes sous plusieurs aspects différents. Nous commençons par étudier les bases possibles pour ces systèmes et la distribution de la longueur des partitions enentiers dans l’écriture des...
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 ...
Suppose that Alice and Bob are given each an infinite string, and they want to decide whether their two strings are in a given relation. How much communication do they need? How can communication be even defined and measured for infinite strings? In this article, we propose a formalism for a notion of infinite communication complexity, prove that i...
