  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 linear ordinary differential equations
over a finite Riemann surface of hyperbolic type S, and may be described by a
representation r:pi_1(S) -> GL(n,C). We give conditions under which the
foliated geodesic flow has a generic repellor-attractor statistical dynamics.
That is, there are measures m- and m+ 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 m- and for positive
time to m+ (i.e. m+ is the unique SRB-measure and its basin has total Lebesgue
measure). These measures are ergodic with respect to the foliated geodesic
flow. These measures are also invariant under a foliated horocycle flow and
they project to a harmonic measure for the Riccati foliation, which plays the
role of an attractor for the statistical behavior of the leaves of the
foliation.
