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On the unit tangent bundle of a hyperbolic surface, we study the density ofpositive 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 onthe vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ aresimultaneously dense or not in the nonwandering set $\mathcal{E}$ of thehorocyclic flow. We give also a counter-example to this result when thisassumption is not satisfied.

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