Ergodicity and equidistribution in strictly convex Hilbert geometry

Abstract

In this paper we show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank-one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures equipped with their Hilbert metric. More specifically, such structures admit a finite Sullivan measure; with respect to this measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.