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 >= 0) under the horocyclic flow. More precisely, given a full orbit (h(s) v)(s is an element of R), we prove that under a weak assumption on the vector v, both half-orbits (h(s) v)(s >= 0) and (h(s) v)(s <= 0), are simultaneously dense or not in the non-wandering set epsilon of the horocyclic flow. We give also a counterexample to this result when this assumption is not satisfied.