Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 58F17 Geodesic and horocycle flows
- 58G32 Diffusion processes and stochastic analysis on manifolds
- 60J60 Diffusion processes, See also {58G32}
- 51M10 Hyperbolic and elliptic geometries (general) and generalizations

**Résumé:** Let $\;\M\; $ 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 $\; T^1\M\; $ of the windings of the geodesics of
$\;\M\; $ around the cusps. This limit law is stable with parameter $\, 2\d -1\,
$, where $\,\d\, $ is the Hausdorff
dimension of the limit set of the subgroup $\,\G\, $ of M\" obius isometries associated with $\;\M\; $. The
normalization is $\, t^{-1\over 2\d -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.

**Mots Clés:** *geodesic flow ; hyperbolic geometry ; Patterson-Sullivan measure ; diffusion paths*

**Date:** 1999-09-30

**Prépublication numéro:** *PMA-534*