Excursion and return times of a geodesics to a subset of a hyperbolic Riemann surface

TL;DR

This paper analyzes the average return and excursion times of geodesics on hyperbolic Riemann surfaces, providing asymptotic rates and distribution results for geodesic behavior near subsurfaces and collars.

Contribution

It introduces new formulas for average return and excursion times of geodesics on hyperbolic surfaces, extending understanding of geodesic dynamics in these geometries.

Findings

Calculated asymptotic average return rates of geodesics to subsurfaces.

Derived distribution of excursion depths into collar neighborhoods.

Established average time geodesics spend in specified subsurfaces.

Abstract

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to a subsurface with geodesic boundary. As a consequence we get the average time a generic geodesic spends in such a subsurface. Related results are obtained for excursions into a collar neighborhood of a simple closed geodesic and the associated distribution of excursion depths.