Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres

Abstract:

Let X be a complete locally compact metric tree and $\Gamma$ a group of isometries of X acting properly on this space. The space of bi-infinite geodesics in X constitutes a space GX on which $\Gamma$ acts properly. Let $\Omega$ be the quotient of GX by this action. The geodesic flow associated to $\Gamma$ is the flow on $\Omega$ which is the quotient of the geodesic flow on GX, defined by the time-shift on geodesics. To any $\Gamma$-conformal measure on the boundary $\partial X$ there is an associated measure m on $\Omega$ which is invariant by the geodesic flow. We prove the following results: The geodesic flow on $(\Omega ,m)$ is either conservative or dissipative. If it is conservative, then it is ergodic, If it is dissipative, then it is not ergodic unless it is measurably conjugate to the action of $\mathbb {R}$ on itself by conjugation. We prove also a dichotomy in terms of the conical limit set ${\Lambda _c} \subset \partial X$ of $\Gamma$: the flow on $(\Omega ,m)$ is conservative if and only if $\mu ({\Lambda _c}) = \mu (\partial X)$, and it is dissipative if and only if $\mu ({\Lambda _c}) = 0$. The results are analogous to results of E. Hopf and D. Sullivan in the case of Riemannian manifolds of constant negative curvature.