  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.
