Densit\'e de demi-horocycles sur une surface hyperbolique g\'eom\'etriquement infinie

TL;DR

This paper investigates the density properties of half-orbits under the horocyclic flow on hyperbolic surfaces, establishing conditions for their simultaneous density or non-density in the nonwandering set.

Contribution

It proves that under a weak assumption on the initial vector, both half-orbits are either both dense or both not dense, and provides a counter-example when the assumption fails.

Findings

Both half-orbits are simultaneously dense or not in the nonwandering set under certain conditions.

Counter-example shows the necessity of the weak assumption for the main result.

Results contribute to understanding orbit distribution in hyperbolic geometry.

Abstract

On the unit tangent bundle of a hyperbolic surface, we study the density of positive orbits $(h^s v)_{s\ge 0}$ under the horocyclic flow. More precisely, given a full orbit $(h^sv)_{s\in \R}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ are simultaneously dense or not in the nonwandering set $\mathcal{E}$ of the horocyclic flow. We give also a counter-example to this result when this assumption is not satisfied.