The action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surface

TL;DR

This paper studies how the mapping class group acts on the space of geodesic rays in a punctured hyperbolic surface, showing that the action is 'almost everywhere' wandering, revealing dynamical properties of the surface's symmetries.

Contribution

It introduces the concept of the mapping class group's action on geodesic rays and proves that this action is 'almost everywhere' wandering, a new insight into the dynamics of hyperbolic surfaces.

Findings

The action of the mapping class group on the space of geodesic rays is 'almost everywhere' wandering.

The study provides a new perspective on the dynamical behavior of surface symmetries.

The results contribute to understanding the geometric and dynamical structure of punctured hyperbolic surfaces.

Abstract

Let $\Sigma$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $\Sigma$ emanating from the puncture. Then there is a natural action of the mapping class group of $\Sigma$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.