Dynamique sur le rayon modulaire et fractions continues en caractéristique $p$

Abstract

Let $\wh K$ be the field of formal Laurent series in $X^{-1}$ over the finite field $k$, and let $A$ be the ring of polynomials in $X$ over $k$. One of the main results of the paper is to give a particularly nice coding of the geodesic flow on the quotient of the Bruhat-Tits tree $\TT$ of ${\rm PGL}\_2(\wh K)$ by ${\rm PGL}\_2(A)$, by using the continued fraction expansion of the endpoints of the geodesic lines in $\TT$ (the space of ends of $\TT$ identifies with $\PP\_1(\wh K)$). This allows in particular to prove in a dynamical way the invariance of the Haar measure by the Artin map.