Stable windings on hyperbolic surfaces |
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Authors: | Nathanaël Enriquez Jacques Franchi Yves Le Jan |
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Institution: | (1) Laboratoire de Probabilités de Paris 6, 4 place Jussieu, tour 56, 3ème étage, 75252 Paris Cedex 05. e-mail: enriquez@ccr.jussieu.fr, FR;(2) Faculté des Sciences de Paris 12, 61 avenue de Gaulle, 94010 Créteil Cedex. e-mail: franchi@math.u-strasbg.fr, FR;(3) Université Paris Sud, Mathématiques, Batiment 425, 91405 Orsay. e-mail: yves.lejan@math.u-psud.fr, FR |
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Abstract: | 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
1ℳ of the windings of the geodesics of ℳ 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 M?bius isometries associated with ℳ. 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.
Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000 |
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Keywords: | Mathematics Subject Classification (2000): 58F17 58G32 60J60 51M10 |
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