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 $\Gamma$ is any group, if $\rho_1$, $\rho_2$ are representations of $\Gamma$ in $\mathrm{PSL}(2,\mathbb{R})$, one of them being non elementary and non discrete, and if for all $\gamma\in\Gamma$, $\rho_1(\gamma)$ and $\rho_2(\gamma)$ have the same rotation number, then $\rho_1$ and $\rho_2$ are conjugate in $\mathrm{PSL}(2,\mathbb{R})$. In particular, if two non discrete, non elementary representations yield semi-conjugate actions on the circle, then they are conjugate in $\mathrm{PSL}(2,\mathbb{R})$.