Geodesic excursions into an embedded disc on a hyperbolic Riemann surface

TL;DR

This paper analyzes how often and how long a typical geodesic on a hyperbolic surface visits a specific embedded disc, including cases with cone points, providing asymptotic averages.

Contribution

It provides the first detailed asymptotic analysis of geodesic excursions into embedded discs on hyperbolic surfaces, including cone point cases.

Findings

Calculated asymptotic average return rate of geodesics

Determined average time geodesics spend in the disc

Included analysis for discs centered at cone points

Abstract

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.