Dynamics of Asymptotically Hyperbolic Manifolds

TL;DR

This paper establishes a dynamical wave trace formula for asymptotically hyperbolic manifolds, linking wave traces to geodesic lengths, and proves a prime orbit theorem for these complex geometries, revealing new spectral-dynamical relationships.

Contribution

It introduces a novel dynamical trace formula and prime orbit theorem for asymptotically hyperbolic manifolds with variable negative curvature, extending spectral and dynamical analysis beyond constant curvature cases.

Findings

Derived a wave trace formula relating wave traces to geodesic lengths.

Proved a prime orbit theorem for the geodesic flow on these manifolds.

Established a connection between Laplacian spectrum and geodesic dynamics.

Abstract

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.