The symplectic structure of renormalisation of circle diffeomorphisms with breaks

2019/07/16 by Ghazouani, Selim, Khanin, Konstantin
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1907.07021

Abstract

In this article we prove that iterated renormalisations of 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 relative character variety χ(π₁ Σ, PSL(2,ℝ), h) where Σ is a d-holed torus, and that the renormalisation operator identifies with a sub-action of the mapping class group MCG(Σ). 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.

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