The symplectic structure of renormalisation of circle diffeomorphisms with breaks

In this article we prove that iterated renormalisations of $\mathcal{C}^r$ circle diffeomorphisms with $d$ breaks, $r>2$, with given size of breaks, converge to an invariant family of piecewise Moebius maps, of dimension $2d$. We prove that this invariant family identifies with a \textit{relative character variety} $\chi(\pi_1 \Sigma, \mathrm{PSL}(2,\mathbb{R}), \mathbf{h})$ where $\Sigma$ is a $d$-holed torus, and that the renormalisation operator identifies with a sub-action of the mapping class group $\mathrm{MCG}(\Sigma)$. This action is known to preserves a symplectic form, thanks to the work of Guruprasad-Huebschmann-Jeffrey-Weinstein. Its pull-back through the aforementioned identification provides a symplectic form invariant by renormalisation.