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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Stable windings on hyperbolic surfaces


Code(s) de Classification MSC:

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