Informal Group Action Seminar

We show that closed negatively curved locally symmetric spaces are characterized among nearby Riemannian manifolds by the Lyapunov exponents of their geodesic flow along periodic orbits. Our methods extend to locally characterize the geodesic flows of these symmetric spaces by their periodic orbit exponents even among arbitrary smooth perturbations of the dynamical system. We also introduce a new dimensional invariant for these perturbed flows thatis closely related to the 1/4-pinching rigidity theorem for complex hyperbolic manifolds.