Convergence groups and semi conjugacy

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 $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$, and that when $h\in $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}$($\mathbb{S}$^1)$, we have a topological conjugacy. We also construct examples, where $h$ is not continuous, for which there is no such conjugacy.