Density and Equidistribution of One-Sided Horocycles of a Geometrically Finite Hyperbolic Surface

On geometrically finite negatively curved surfaces, we give necessary and sufficient conditions for a one-sided horocycle $(h^{s}u{)}_{s\ge 0}$ to be dense in the nonwandering set of the geodesic flow. We prove that all dense one-sided orbits $(h^{s}u{)}_{s\ge 0}$ are equidistributed, extending results of [Bu] and [Scha2] where symmetric horocycles $(h^{s}u{)}_{s\in \mathbb{R}}$ were considered.