Convergence groups and semiconjugacy

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.