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

We consider Riccati foliations ℱ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π( 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 μ for the Riccati foliation. We describe μ, μ and μ in terms of the Patterson-Sullivan construction, and we show that the measures μ provide examples of the conformal harmonic measures introduced by M. Brunella. [ABSTRACT FROM AUTHOR]