On the dynamics of Riccati foliations with nonparabolic monodromy representations
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- by Nicolas Hussenot Desenonges
- Conform. Geom. Dyn. 23 (2019), 164-188
- DOI: https://doi.org/10.1090/ecgd/337
- Published electronically: October 1, 2019
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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.References
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Bibliographic 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