# Dynamics of Asymptotically Hyperbolic Manifolds

Julie Rowlett

Arxiv ID: 0809.3472•Last updated: 12/11/2020

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. A corollary of this dynamical trace formula is a dynamical
resonance-wave 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.

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