On the dynamics of Riccati foliations with non parabolic monodromy representations

In this paper, we study the dynamics of Riccati foliations over non-compact finite volume Riemann surfaces. More precisely, we are interested in two closely related questions: the asymptotic behaviour of the holonomy map Hol t ($\omega$) defined for every time t over a generic Brownian path $\omega$ in the base; and the analytic continuation of holonomy germs of the foliation along Brownian paths in transversal lines. When the monodromy representation is parabolic (i.e. the monodromy around any puncture is a parabolic element in P SL 2 (C)), these questions have already been solved in [DD2] and [Hus]. Here, we study the more general case where some puncture have hyperbolic monodromy. We characterise the lower-upper, upper-upper and upper-lower classes of the map Hol t ($\omega$) for almost every Brownian path $\omega$. And we prove that the main result of [Hus] still holds in this case: when the monodromy group is "big enough" , any holonomy germ of the foliations between two lines can be analytically continued along a generic Brownian path.