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

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 $(h^{s}v{)}_{s\in \mathbb{R}}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^{s}v{)}_{s\ge 0}$ 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.