On the dynamics of Riccati foliations with nonparabolic monodromy representations

Abstract:

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.