Precise asymptotics for periodic orbits of the geodesic flow in nonpositive curvature

Open Access
Author:
Gunesch, Roland
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
April 30, 2002
Committee Members:
  • John Metzner, Committee Member
  • Professor Dr Anatole B Katok, Committee Chair
  • Professor Dr Mark Levi, Committee Member
  • Professor Dr Serge Tabachnikov, Committee Member
Keywords:
  • dynamical systems
  • closed geodesics
  • Riemannian geometry
  • geodesic flows
  • measure of maximal entropy
  • nonpositive curvature
Abstract:
This thesis establishes the most precise asymptotic formula known for the number of homotopy classes of periodic orbits for the geodesic flow and proves it for any compact manifold of nonpositive curvature with geometric rank one. This extends a celebrated result of G.A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper.