  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.
