Convergence groups and semi conjugacy

Abstract

We study a simple problem that arises from the study of Lorentz surfaces and Anosov flows. For a non decreasing map of degree one $h:{\mathbb{S}}^1\to {\mathbb{S}}^1$, 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 semi conjugate to subgroups of $\mathrm{PSL}(2,\mathbb{R})$, and that when $h\in \mathrm{Homeo}({\mathbb{S}}^1)$, we have a topological conjugacy. We also construct examples, where $h$ is not continuous, for which there is no such conjugacy.

arXiv comment27 pages, 7 figures. arXiv admin note: text overlap with arXiv:1402.0424