Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
DYNAMICS OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS J. M. ROWLETT
 

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
Duistermaat-Guillemin for compact manifolds and the results of Guillop´e-
Zworski, Perry, and Guillarmou-Naud 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 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 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 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.
1. Introduction

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics