Convergence groups and semiconjugacy

We study a problem that arises from the study of Lorentz surfaces and Anosov flows. For a non-decreasing map of degree one h:S^1->S^1, we are interested in groups of circle diffeomorphisms that act on the complement of the graph of h in S1^×S^1 by preserving a volume form. We show that such groups are semiconjugate to subgroups of PSL(2,R) and that, when h∈Homeo(S^1), we have a topological conjugacy. We also construct examples where h is not continuous, for which there is no such conjugacy.