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 Möbius maps, of dimension 2d. We prove that this invariant family identifies with a relative character variety χ(π1Σ,PSL(2, ℝ),h) where Σ is a d-holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group MCG(Σ). 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.

中文翻译：

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带断点的圆微分同胚重整化的辛结构

本文的主要目的是揭示与带断点的圆图重整化相关的辛结构。我们首先证明了𝒞r圆微分同胚d休息，r > 2，在给定的中断大小下，收敛到一个不变的分段莫比乌斯图族，维数2d. 我们证明了这个不变的族与一个相对性格多样性 χ(π1Σ,PSL(2, ℝ),H)在哪里Σ是一个d-holed torus，并且重整化运算符与映射类组的子动作标识心电图(Σ). 这个动作允许我们引入通过重整化保留的辛形式。不变辛形式与 Guruprasad 描述的辛形式有关等人. [群系统，群，和抛物线丛的模空间，杜克数学。J。 89(2) (1997) 377-412]，并回到 Goldman 的早期工作 [The symplectic nature of basic groups ofsurfaces,进阶。数学。 54(2) (1984) 200–225]。据我们所知，非线性设置中的重整化与辛动力学之间的联系尚未被揭示。