Dynamique symbolique des systèmes 2D et des arbres infinis. (Symbolic dynamics on multidimensional systems and infinite trees)

This thesis is devoted to the study of subshifts, or symbolic dynamical systems, defined on some finitely presented monoids like Zd or the infinite binary tree. The main result concerning multidimensional subshifts establishes that any effective subshift of dimension d can be obtained by factor map and projective subaction of a subshift of finite type of dimension d+ 1. This result has many applications, and in particular we prove that multidimensional effective S-adic subshifts are sofic. On tree-shifts we prove a decomposition theorem, which implies that the conjugacy problem between two tree-shifts of finite type is decidable. We then investigate the class of sofic tree-shifts that are exactly those recognized by tree automata. We prove that any sofic tree-shift has a unique deterministic, reduced, irreducible and synchronized tree automaton that recognized it. Finally we prove that it is decidable wether a sofic tree-shift belong to the sub-class of AFT tree-shifts.