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Stable windings on hyperbolic surfaces
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  • Published: February 2001

Stable windings on hyperbolic surfaces

  • Nathanaël Enriquez1,
  • Jacques Franchi2 &
  • Yves Le Jan3 

Probability Theory and Related Fields volume 119, pages 213–255 (2001)Cite this article

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  • 9 Citations

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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 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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Authors and Affiliations

  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

    Nathanaël Enriquez

  2. Faculté des Sciences de Paris 12, 61 avenue de Gaulle, 94010 Créteil Cedex. e-mail: franchi@math.u-strasbg.fr, , , , , , FR

    Jacques Franchi

  3. Université Paris Sud, Mathématiques, Bâtiment 425, 91405 Orsay. e-mail: yves.lejan@math.u-psud.fr, , , , , , FR

    Yves Le Jan

Authors
  1. Nathanaël Enriquez
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  2. Jacques Franchi
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  3. Yves Le Jan
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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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  • Issue Date: February 2001

  • DOI: https://doi.org/10.1007/PL00008759

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  • Mathematics Subject Classification (2000): 58F17, 58G32, 60J60, 51M10
  • Keywords or phrases: Geodesic flow – Hyperbolic geometry – Patterson-Sullivan measure – Diffusion paths
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