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

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