Stable windings on hyperbolic surfaces

Let Script M sign be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T¹ Script M sign of the windings of the geodesics of Script M sign around the cusps. This limit law is stable with parameter 2δ - 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of Möbius isometries associated with Script M sign. The normalization is t^-1/(2δ-1), for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves.