  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.
