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The aim of this note is to advertise on a result, not stated explicitly, butproved, in arXiv:0802.0512. Namely, if $\Gamma$ is any group, if $\rho_1$,$\rho_2$ are representations of $\Gamma$ in $\mathrm{PSL}(2,\mathbb{R})$, oneof them being non elementary and non discrete, and if for all$\gamma\in\Gamma$, $\rho_1(\gamma)$ and $\rho_2(\gamma)$ have the same rotationnumber, then $\rho_1$ and $\rho_2$ are conjugate in$\mathrm{PSL}(2,\mathbb{R})$. In particular, if two non discrete, nonelementary representations yield semi-conjugate actions on the circle, thenthey are conjugate in $\mathrm{PSL}(2,\mathbb{R})$.

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