Abstract
We consider Riccati foliations ℱρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π1(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 μρ+, μρ- 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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