Lyapunov exponents in Hilbert geometry

We study the behaviour of a Hilbert geometry when going to infinity along a geodesic line. We prove that all the information is contained in the shape of the boundary at the endpoint of this geodesic line and have to introduce a regularity property of convex functions to make this link precise. The point of view is a dynamical one and the main interest of this article is in Lyapunov exponents of the geodesic flow.Comment: 34 pages, comments are welcom

Archival data projects

pp. 301-31

Restructuring a mentoring program

pp. 75-8

Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert

To appear in Annales Mathématiques Blaise Pascal.International audienceWe prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set \O and any point x \in \O, any discrete group generated by a finite number of automorphisms of \O, which displace $x$ at a distance less than $\varepsilon_n$, is virtually nilpotent

Finitude géométrique en géométrie de Hilbert

To appear in Annales de l'Institut Fourier. 62 pages, 25 figures, with an appendix with Constantin Vernicos.We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with $\mathcal{C}^1$ boundary. In Gromov-hyperbolic spaces, geometrical finiteness is defined by a property of the group action on the boundary of the space. We show by constructing an explicit counter-example that this definition has to be strenghtened in order to get equivalent characterizations in terms of the geometry of the quotient orbifold, similar to those obtained by Brian Bowditch in hyperbolic geometry

Le flot géodésique des quotients géométriquement finis des géométries de Hilbert

45 pages. To appear in Pacific Journal of Mathematics.International audienceWe study the geodesic flow of geometrically finite quotients of Hilbert geometries.On étudie le flot géodésique des quotients géométriquement finis de géométries de Hilbert