Nonuniform hyperbolicity in Hilbert geometries
Abstract: 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. ... read moreIn 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
Thesis (Ph.D.)--Tufts University, 2016.
Submitted to the Dept. of Mathematics.
Advisor: Boris Hasselblatt.
Committee: Zbigniew Nitecki, Genevieve Walsh, and Amie Wilkinson.
Keyword: Mathematics.read less