Characterizing symmetric spaces by their Lyapunov spectra

Clark Butler

Published 2017-09-23Version 1

We prove that the geodesic flow of a closed negatively curved locally symmetric space is characterized among nearby smooth flows by the structure of its Lyapunov spectrum with respect to volume. We deduce that these locally symmetric spaces are locally characterized up to isometry by their Lyapunov spectra. We prove Lyapunov exponent pinching rigidity results for perturbations of the geodesic flows of locally symmetric spaces of nonconstant negative curvature under a conjectural lower bound on certain dimensional quantities associated to the flow. The techniques developed in this paper are focused on the locally symmetric spaces of variable negative curvature and extend many methods used to prove rigidity theorems for uniformly quasiconformal Anosov diffeomorphisms and flows.