Growth Rate of Periodic orbits for Geodesic Flows on Surfaces with Regions of Positive Curvature

Open Access

Author:

Weaver, Bryce A.

Graduate Program:

Mathematics

Degree:

Doctor of Philosophy

Document Type:

Dissertation

Date of Defense:

May 07, 2008

Committee Members:

Anatole Katok, Committee Chair/Co-Chair Yakov B Pesin, Committee Member Omri Sarig, Committee Member David Gerard Abler, Committee Member

Keywords:

non-uniformly hyperbolic growth of periodic orbits dynamical systems geodesic flows positive curvature Margulis measure

Abstract:

We construct a Margulis measure on the unit tangent bundle of compact surfaces that have regions of positive curvature. This measure is then used to obtain a lower bound C$grave{e}$saro estimate on $ds P_epsilon(t)$, which is the number of periodic orbits of period $t pm epsilon$ for the geodesic flow. This is the first step toward obtaining precise asymptotics and suggests that $ds P_epsilon(t)$ grows like $ds frac{e^{kt}}{t}$, for some constant $k$.