Sur le codage du flot g\'{e}od\'{e}sique dans un arbre

Abstract

Given a tree $T$ and a group $\Ga$ of automorphisms of $T$, we study the markovian properties of the geodesic flow on the quotient by $\Ga$ of the space of geodesics of $T$. For instance, when $T$ is the Bruhat-Tits tree of a semi-simple connected algebraic group $\underline{G}$ of rank one over a non archimedian local field $\wh K$, and $\Ga$ is a (possibly non uniform) lattice in $\underline{G}(\wh K)$, we prove that the type preserving geodesic flow is Bernoulli with finite entropy. Under some mild assumptions, we prove that if the quotient geodesic flow is mixing for a probability Patterson-Sullivan-Bowen-Margulis measure, then it is loosely Bernoulli.