Ergodicity and equidistribution in Hilbert geometry

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 rank-one properly convex projective manifolds or orbifolds, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective manifolds or orbifolds whose universal covers are strictly convex with C¹ boundary.More specifically, with respect to the Sullivan measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.