Skip to Main Content

Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the dynamics of Riccati foliations with nonparabolic monodromy representations
HTML articles powered by AMS MathViewer

by Nicolas Hussenot Desenonges PDF
Conform. Geom. Dyn. 23 (2019), 164-188 Request permission

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.
References
Similar Articles
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
  • Received by editor(s): July 8, 2016
  • Published electronically: October 1, 2019
  • © Copyright 2019 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 23 (2019), 164-188
  • MSC (2010): Primary 37F75, 37A50, 37C85, 37H15, 32D15
  • DOI: https://doi.org/10.1090/ecgd/337
  • MathSciNet review: 4013742