Dynamics of asymptotically hyperbolic manifolds

Abstract

We prove a dynamical wave trace formula for asymptotically hyperbolic (n + 1)-dimensional manifolds with negative (but not necessarily constant) sectional curvatures; the formula equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of Duistermaat and Guillemin for compact manifolds and the results of Guillopé and Zworski, Perry, and Guillarmou and Naud for hyperbolic manifolds with infinite volume. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex cocompact hyperbolic manifolds. We define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a prime orbit theorem for the geodesic flow. This is the first such result for manifolds that have neither constant curvature nor finite volume. As a corollary to the prime orbit theorem, using our dynamical resonance-wave trace formula, we show that the existence of pure point spectrum for the Laplacian on negatively curved compact perturbations of convex cocompact hyperbolic manifolds is related to the dynamics of the geodesic flow.