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.
