On the dynamics of Riccati foliations with non parabolic monodromy representations

TL;DR

This paper investigates the asymptotic behavior and analytic continuation of holonomy maps in Riccati foliations over non-compact Riemann surfaces with hyperbolic monodromy, extending previous results from parabolic cases.

Contribution

It generalizes the understanding of holonomy dynamics in Riccati foliations to cases with hyperbolic monodromy, characterizing the behavior of holonomy maps along Brownian paths.

Findings

Characterization of the asymptotic classes of holonomy maps for hyperbolic monodromy

Extension of analytic continuation results to non-parabolic monodromy cases

Holonomy germs can be analytically continued along Brownian paths when the monodromy group is sufficiently large

Abstract

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.