Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Stable windings on hyperbolic surfaces

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