Le flot géodésique des quotients géométriquement finis des géométries de hilbert

Abstract

We study the geodesic flow of geometrically finite quotients Ω/Γ of Hilbert geometries, in particular its recurrence properties. We prove that, under a geometric 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 Ω and the group Γ. 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.