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Convergence groups and semiconjugacy

Published online by Cambridge University Press:  10 November 2014

DANIEL MONCLAIR*
Affiliation:
UMPA, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France email daniel.monclair@ens-lyon.fr

Abstract

We study a problem that arises from the study of Lorentz surfaces and Anosov flows. For a non-decreasing map of degree one $h:\mathbb{S}^{1}\rightarrow \mathbb{S}^{1}$, we are interested in groups of circle diffeomorphisms that act on the complement of the graph of $h$ in $\mathbb{S}^{1}\times \mathbb{S}^{1}$ by preserving a volume form. We show that such groups are semiconjugate to subgroups of $\text{PSL}(2,\mathbb{R})$ and that, when $h\in \text{Homeo}(\mathbb{S}^{1})$, we have a topological conjugacy. We also construct examples where $h$ is not continuous, for which there is no such conjugacy.

Type
Research Article
Copyright
© Cambridge University Press, 2014 

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