 
Summary: DYNAMICS OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS
J. M. ROWLETT
Abstract. We prove a dynamical wave trace formula for asymptotically hy
perbolic (n + 1) dimensional manifolds with negative (but not necessarily con
stant) sectional curvatures which equates the renormalized wave trace to the
lengths of closed geodesics. This result generalizes the classical theorem of
DuistermaatGuillemin for compact manifolds and the results of Guillop´e
Zworski, Perry, and GuillarmouNaud 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 which we use to prove a growth estimate for the length spectrum
counting function. We next define a dynamical zeta function and prove its an
alyticity 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 or
bit theorem, using our dynamical resonancewave 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.
1. Introduction
