  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.
