| This thesis is a comprehensive case study of the topological dynamics, asymptotic geometry, and ergodic theory for the geodesic flow of a class of 3-manifolds which have a non-Riemannian and nonuniformly hyperbolic geometric structure. These 3-manifolds arise as Hilbert geometries, and they were discovered by Benoist. The geometric structure forces irregularity of the geodesic flow. In particular, there are four major features of the geometry and dynamics which place this dynamical system outside the scope of any existing theory to date. First, the 3-manifolds are non-Riemannian and the geodesic flow is nonuniformly hyperbolic. Geodesic flows in each of those contexts have been studied independently but not simultaneously. Moreover, the manifolds are not CAT(0), and the geodesic flow is not differentiable. In this thesis we are able to extend the long developed framework of smooth ergodic theory to this class of geodesic flows far from the classical setting of Riemannian negative curvature. The main result is ergodicity and mixing of the Bowen--Margulis measure, which is a measure of maximal entropy for the geodesic flow. We conjecture uniqueness of the Bowen--Margulis measure and propose natural extensions of this work to equilibrium states and construction of a natural volume measure. |
