Dynamical rigidity of non discrete representations in PSL(2,R)

The aim of this note is to advertise on a result, not stated explicitly, but proved, in arXiv:0802.0512. Namely, if $\mathrm{\Gamma }$ is any group, if ${\rho }_1$, ${\rho }_2$ are representations of $\mathrm{\Gamma }$ in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$, one of them being non elementary and non discrete, and if for all $\gamma \in \mathrm{\Gamma }$, ${\rho }_1(\gamma )$ and ${\rho }_2(\gamma )$ have the same rotation number, then ${\rho }_1$ and ${\rho }_2$ are conjugate in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$. In particular, if two non discrete, non elementary representations yield semi-conjugate actions on the circle, then they are conjugate in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$.