Characterizing Symmetric Spaces by their Lyapunov Spectra

This thesis uses methods from hyperbolic dynamics, Riemannian geometry, and analysis on metric spaces to obtain new rigidity results for negatively curved Riemannian manifolds.,We prove that closed, negatively curved locally symmetric spaces are locally characterized up to isometry by the Lyapunov spectra of the periodic orbits of their geodesic flows. This is done by constructing a new invariant measure for the geodesic flow that we refer to as the horizontal measure. We show that the Lyapunov spectrum of the horizontal measure alone suffices to locally characterize these locally symmetric spaces up to isometry. ,Our methods extend to give rigidity theorems for smooth flows obtained as perturbations of the geodesic flows of these locally symmetric spaces. The techniques developed in this paper are focused on the symmetric spaces of nonconstant negative curvature and extend many methods used to prove rigidity theorems for uniformly quasiconformal Anosov diffeomorphisms and flows.