The foliated geodesic flow on Riccati equations

We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 and hyperbolic, corresponding to the unique complete metric of curvature-1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivisation of the solutions of a linear ordinary differential equations over a finite Riemann surface of hyperbolic type $S $ , and may be described by a representation $\rho $ : $\pi_{1}(S) $ $arrow GL(n, \mathbb{C}) $. We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. That is, there are measures $\mu^{+} $ and $\mu^{-} $ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to $72^{-} $ and for positive time to $\mu^{+} $ (i.e. $\mu+ $ is the unique SRB-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow