# A prime orbit theorem, trace formulae, and interactions between quantum and classical mechanics on asymptotically hyperbolic manifolds

# A prime orbit theorem, trace formulae, and interactions between quantum and classical mechanics on asymptotically hyperbolic manifolds

Asymptotically hyperbolic manifolds are a natural generalization of infinite volume hyperbolic manifolds and enjoy similar features. They are of particular interest in physics because all Poincar\'e-Einstein manifolds, which arise in adS-CFT correspondence, are asymptotically hyperbolic. 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 two ``dynamical wave trace formulae.'' Based on the prime orbit theorem and the trace formulae, 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.