## Abstract

Let Script M sign 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} Script M sign of the windings of the geodesics of Script M sign 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 Script M sign. 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.

Original language | English (US) |
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Pages (from-to) | 213-255 |

Number of pages | 43 |

Journal | Probability Theory and Related Fields |

Volume | 119 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2001 |

## Keywords

- Diffusion paths
- Geodesic flow
- Hyperbolic geometry
- Patterson-Sullivan measure

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty