Dynamics of Asymptotically Hyperbolic Manifolds

We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of Duistermaat-Guillemin for compact manifolds and the results of GuillopéZworski, Perry, and Guillarmou-Naud for hyperbolic manifolds with infinite volume. A corollary of this dynamical trace formula is a dynamical resonancewave trace formula for compact perturbations of convex co-compact hyperbolic manifolds which we use to prove a growth estimate for the length spectrum counting function. We next 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 which 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 co-compact hyperbolic manifolds is related to the dynamics of the geodesic flow.