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Institut de Mathématiques de Marseille, UMR 7373
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The dynamics of non-strictly convex Hilbert geometries

11h00 - 12h30
horaire FRUMAM

Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03

Any open, properly convex domain $\Omega$ in $\R\P^n$ admits a Hilbert metric, compatible with a Finsler norm. Of particular interest are compact quotients of $\Omega$ by discrete subgroups of $\PGL(n+1,\R)$. The dynamical, topological, algebraic, and regularity properties of such quotients with the line flow have been exhaustively studied by Benoist and others in the case that $\Omega$ is strictly convex and consequently $\delta$-hyperbolic. My agenda is to expand previous studies to the convex but not strictly convex case for $\Omega$. In this talk, I will introduce a
class of higher dimensional examples, constructed by Benoist, which exhibit some hyperbolicity properties. I will explicitly describe the topological and ergodic dynamical properties of the line flow on the projective triangle, which plays a focal role as an obstruction to hyperbolicity in the non-strictly convex examples of Benoist. I will assure you that we expect this obstruction to be surmountable.

Sarah Bray