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Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy

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Abstract

We consider Riccati foliations ℱρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π 1(S) → PSL(2,ℂ) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ+ and µ− for positive and negative times, respectively. These measures have a common projection to a harmonic measure μρ for the Riccati foliation. We describe μ +ρ , μ -ρ and μρ in terms of the Patterson-Sullivan construction, and we show that the measures μρ provide examples of the conformal harmonic measures introduced by M. Brunella.

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Authors and Affiliations

  1. Universidad Autónoma de Ciudad Juárez, Unidad Multidisciplinaria de la UACJ en Cuauhtémoc, Carretera Cuauhtémoc-Anáhuac Km. 61.5 Calle Ejército Nacional 5220 Col. Ejido Cuauhtémoc, C.P. 31600, Anáhuac, Municipio de Cuauhtémoc, Chih., México

    Elifalet López-González

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  1. Elifalet López-González
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Correspondence to Elifalet López-González.

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López-González, E. Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy. Bull Braz Math Soc, New Series 44, 393–412 (2013). https://doi.org/10.1007/s00574-013-0018-6

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  • DOI: https://doi.org/10.1007/s00574-013-0018-6

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