Density of half-horocycles on geometrically infinite hyperbolic surfaces

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 $(h^sv)_{s\in {\mathbb R}}$ , 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 non-wandering set $\mathcal {E}$ of the horocyclic flow. We give also a counterexample to this result when this assumption is not satisfied.