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arXiv · 1204.5005

Le flot géodésique des quotients geometriquement finis des géométries de Hilbert

Mickaël Crampon·Ludovic Marquis
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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 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 $Ω$ 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.

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BibTeXRIS

Mickaël Crampon, Ludovic Marquis. 2013-02-21. Le flot géodésique des quotients geometriquement finis des géométries de Hilbert. https://arxiv.org/abs/1204.5005

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