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 Möbius 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.
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Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000
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Enriquez, N., Franchi, J. & Le Jan, Y. Stable windings on hyperbolic surfaces. Probab Theory Relat Fields 119, 213–255 (2001). https://doi.org/10.1007/PL00008759
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DOI: https://doi.org/10.1007/PL00008759
- Mathematics Subject Classification (2000): 58F17, 58G32, 60J60, 51M10
- Keywords or phrases: Geodesic flow – Hyperbolic geometry – Patterson-Sullivan measure – Diffusion paths