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Conformal Geometry and Dynamics

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On the dynamics of Riccati foliations with nonparabolic monodromy representations

Author: Nicolas Hussenot Desenonges
Journal: Conform. Geom. Dyn. 23 (2019), 164-188
MSC (2010): Primary 37F75, 37A50, 37C85, 37H15, 32D15
Published electronically: October 1, 2019
MathSciNet review: 4013742
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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.

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Nicolas Hussenot Desenonges
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, Ilha do Fundao, 68530, CEP 21941-970, Rio de Janeiro, RJ, Brasil
Address at time of publication: INSA de Rennes, 20 avenue des buttes de Coesmes, 35700 Rennes, France

Keywords: Conformal dynamics, foliation, Brownian motion, Lyapunov exponents
Received by editor(s): July 8, 2016
Published electronically: October 1, 2019
Article copyright: © Copyright 2019 American Mathematical Society