Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds

Abstract

We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-C^1 geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson-Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen-Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen-Margulis measure.