Stable windings on hyperbolic surfaces

Nathanaël Enriquez, Jacques Franchi, Yves Le Jan

Research output: Contribution to journalArticlepeer-review


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 T1 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 languageEnglish (US)
Pages (from-to)213-255
Number of pages43
JournalProbability Theory and Related Fields
Issue number2
StatePublished - Feb 2001


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

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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