Geodesic flow of nonstrictly convex Hilbert geometries

In this paper we describe the topological behavior of the geodesic flow for a class of closed 3-manifolds realized as quotients of nonstrictly convex Hilbert geometries. The structure of these 3-manifolds is described explicitly by Benoist; they are Finsler with isometrically embedded flats, but hyperbolic away from flats. We prove the geodesic flow of the quotient is topologically mixing and satisfies a nonuniform Anosov Closing Lemma, with applications to entropy and orbit counting. We also prove entropy-expansivity for the geodesic flow of any compact quotient of a Hilbert geometry, which implies existence of a measure of maximal entropy.