On the dynamics of Riccati foliations with nonparabolic monodromy representations

In this paper, we study the dynamics of Riccati foliations over noncompact 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 complex curves. When the monodromy representation is parabolic (i.e., the monodromy around any puncture is a parabolic element in $PSL_2(\mathbb{C})$), these two questions have been solved, respectively, in [Comm. Math. Phys. 340 (2015), pp. 433-469] and [Ergodic Theory Dynam. Systems 37 (2017), pp. 1887-1914]. Here, we study the more general case where at least one puncture has hyperbolic monodromy. We characterise the lower-upper, upper-upper, and upper-lower classes of the map $Hol_t(\omega )$ for almost every Brownian path $\omega$. We prove that the main result of [Ergodic Theory Dynam. Systems 37 (2017), pp. 1887-1914] still holds in this case: when the monodromy group of the foliation is ``big enough'', the holonomy germs can be analytically continued along a generic Brownian path.