The symplectic structure for renormalization of circle diffeomorphisms with breaks

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of Cr circle diffeomorphisms with d breaks, r > 2, with given size of breaks, converge to an invariant family of piecewise Mobius maps, of dimension 2d. We prove that this invariant family identifies with a relative character variety chi(pi(1)Sigma, PSL(2, R), h) where Sigma is a d-holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group MCG(Sigma). This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89(2) (1997) 377-412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math. 54(2) (1984) 200-225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet.