Le flot g\'eod\'esique des quotients geometriquement finis des g\'eom\'etries de Hilbert

TL;DR

This paper investigates the dynamics of geodesic flows on geometrically finite quotients of Hilbert geometries, establishing hyperbolicity under certain conditions and exploring the relationship between geometric properties and dynamical behavior.

Contribution

It proves hyperbolicity of geodesic flow under a cusp assumption and links dynamics to geometric features, extending rigidity results to non-compact quotients.

Findings

Geodesic flow is uniformly hyperbolic under cusp assumptions.

Existence of quotients with zero Lyapunov exponent without cusp assumptions.

Volume entropy equals critical exponent for finite volume quotients.

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.