We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one 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 projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ 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 μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (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 behaviour of the leaves of the foliation.

Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$,having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$, and tangent to $V := F^{-1}(0)$. Consider the ${\cal O}_{V,0}$-complex obtained by contracting the germs of Kähler differential forms of $V$ at $0$\renewcommand{\theequation}{0.\arabic{equation}}\begin{equation}\Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0}\end{equation}with the vector field <formula form="inline" disc="math" id="frm14"><formtex notation="AMSTeX">$X:=\tilde{X}|_V$ on $V$:\begin{equation}0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0.\end{equation}