Density and Equidistribution of One-Sided Horocycles of a Geometrically Finite Hyperbolic Surface

TL;DR

This paper establishes conditions for the density and equidistribution of one-sided horocycles on geometrically finite hyperbolic surfaces, extending prior results from symmetric horocycles to one-sided cases.

Contribution

It provides necessary and sufficient conditions for the density of one-sided horocycles and proves their equidistribution, broadening understanding of horocycle dynamics.

Findings

Necessary and sufficient conditions for density of one-sided horocycles.

All dense one-sided horocycles are equidistributed.

Extension of symmetric horocycle results to one-sided horocycles.

Abstract

On geometrically finite negatively curved surfaces, we give necessary and sufficient conditions for a one-sided horocycle $(h^s u)_{s\ge 0}$ to be dense in the nonwandering set of the geodesic flow. We prove that all dense one-sided orbits $(h^su)_{s\ge 0}$ are equidistributed, extending results of [Bu] and [Scha2] where symmetric horocycles $(h^su)_{s\in\R}$ were considered.