Dynamique sur le rayon modulaire et fractions continues en caract\'{e}ristique $p$

Let $\text{wh}K$ 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 $\text{TT}$ of $\mathrm{P}\mathrm{G}\mathrm{L}\mathrm{\_}2(\text{wh}K)$ by $\mathrm{P}\mathrm{G}\mathrm{L}\mathrm{\_}2(A)$, by using the continued fraction expansion of the endpoints of the geodesic lines in $\text{TT}$ (the space of ends of $\text{TT}$ identifies with $\text{PP}\mathrm{\_}1(\text{wh}K)$). This allows in particular to prove in a dynamical way the invariance of the Haar measure by the Artin map.