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
DOI:
https://doi.org/10.1090/ecgd/337
Published electronically:
October 1, 2019
MathSciNet review:
4013742
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Abstract | References | Similar Articles | Additional Information
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 defined for every time
over a generic Brownian path
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
), 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
for almost every Brownian path
. 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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Additional Information
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
Email:
nicolashussenot@hotmail.fr
DOI:
https://doi.org/10.1090/ecgd/337
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