Le flot géodésique des quotients geometriquement finis des géométries de Hilbert

Details Le flot géodésique des quotients geometriquement finis des géométries de Hilbert Le flot géodésique des quotients geometriquement finis des géométries de Hilbert

2012-04-23

arxiv.org/abs/1204.5005v1

Mathematics

Dynamical Systems

45 pages, 11 figures

Scientific paper

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.

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