Counting geodesic loops on surfaces of genus at least 2 without conjugate points

TL;DR

This paper extends classical results on counting closed geodesic loops to more general surfaces without conjugate points, providing asymptotic estimates and sector theorems that generalize known theorems for negatively curved surfaces.

Contribution

It introduces new asymptotic estimates and sector theorems for surfaces of genus at least 2 without conjugate points, broadening the scope beyond strictly negative curvature.

Findings

Asymptotic estimates for closed geodesic loops on such surfaces

Generalized sector theorems for surfaces without conjugate points

Extension of classical counting results to broader surface classes

Abstract

In this paper we prove asymptotic estimates for closed geodesic loops on compact surfaces with no conjugate points. These generalize the classical counting results of Huber and Margulis and sector theorems for surfaces of strictly negative curvature. We will also prove more general sector theorems, generalizing results of Nicholls and Sharp for special case of surfaces of strictly negative curvature.