Le flot géodésique des quotients geometriquement finis des géométries de Hilbert

Abstract

We study the geodesic flow of geometrically finite quotients $\Omega /\Gamma$ of Hilbert geometries, in particular its recurrence properties. We prove that, under a geometrical assumption on the cusps, the geodesic flow is uniformly hyperbolic. Without this assumption, we provide an example of a quotient whose geodesic flow has a zero Lyapunov exponent. We make the link between the dynamics of the geodesic flow and some properties of the convex set $\Omega$ and the group $\Gamma$. As a consequence, we get various rigidity results which extend previous results of Benoist and Guichard for compact quotients. Finally, we study the link between volume entropy and critical exponent; for example, we show that they coincide provided the quotient has finite volume.

45 pages, 11 figures. To appear in Pacific Journal of Mathematics