A prime orbit theorem and interactions between quantum and classical mechanics

Asymptotically hyperbolic manifolds are a natural generalization of infinite volume hyperbolic manifolds and enjoy similar features.  In this talk, we'll recall the definition of these spaces and see some examples.  After a brief discussion of their spectral theory and dynamics, I will present a prime orbit theorem and a "dynamical wave trace formula."  Based on the prime orbit theorem and the trace formula, we will determine a relationship between the existence of pure point spectrum and the topological entropy of the geodesic flow.  We can interpret this physically as an interaction between the quantum and classical mechanics.