Intersection Points of Closed Geodesics on Hyperbolic Surfaces of Finite Area

TL;DR

This paper proves that the intersection points of short closed geodesics on finite-area hyperbolic surfaces become evenly distributed across the surface as their length increases indefinitely.

Contribution

It establishes the equidistribution of intersection points of closed geodesics with bounded length on hyperbolic surfaces of finite area.

Findings

Intersection points are equidistributed as geodesic length T approaches infinity.

The distribution becomes uniform across the surface.

Results apply to all complete hyperbolic surfaces of finite area.

Abstract

Let X be a complete hyperbolic surface of finite area. We establish that the intersection points of closed geodesics with length <T are equidistributed on X as T goes to infinity.