Lyapunov Exponent Rigidity for Geodesic Flows

Clark Butler

Published 2015-01-24Version 1

We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if the derivative action of the geodesic flow on the unstable bundle has equal extremal Lyapunov exponents with respect to every invariant measure supported on a periodic orbit then the manifold is homothetic to a real hyperbolic manifold. Under the assumption that the manifold is homotopy equivalent to the appropriate locally symmetric space, we characterize the negatively curved locally symmetric spaces by the values of their Lyapunov exponents on their invariant measures supported on periodic orbits. We also show how, for real and complex hyperbolic space, the assumptions on the Lyapunov exponents of periodic orbits can be replaced with assumptions on the Lyapunov exponents of a single invariant Gibbs measure. The proofs use new results from hyperbolic dynamics including the nonlinear invariance principle of Avila and Viana and the approximation of Lyapunov exponents of invariant measures by Lyapunov exponents associated to periodic orbits which was developed by Kalinin in his proof of the Livsic theorem for matrix cocycles.