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 $Γ$ is any group, if $ρ_1$, $ρ_2$ are representations of $Γ$ in $\mathrm{PSL}(2,\mathbb{R})$, one of them being non elementary and non discrete, and if for all $γ\inΓ$, $ρ_1(γ)$ and $ρ_2(γ)$ have the same rotation number, then $ρ_1$ and $ρ_2$ 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 $\mathrm{PSL}(2,\mathbb{R})$.

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