Satellite renormalization of quadratic polynomials
Satellite renormalization of quadratic polynomials
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发表时间：
2015-09
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通讯作者：
D. Cheraghi;Mitsuhiro Shishikura
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作者：
D. Cheraghi;Mitsuhiro Shishikura
证明了作用于无限维全纯变换空间的近抛物型重整化算子的一致双曲性。这暗示了标度定律的普适性，这是物理学家在70年代推测的，适用于二元分岔类。通过近抛物型重整化，首次成功地研究了卫星型的类多项式重整化，并引入了分析具有此类无限重整化结构的映射的精细尺度动力学特征的新技术。特别地，我们在组合学上证实了二次增长条件下的刚性猜想。本文讨论的映射类包括在小比例尺上具有退化几何的无限可重整映射（缺乏先验边界）。
We prove the uniform hyperbolicity of the near-parabolic renormalization op- erators acting on an infinite-dimensional space of holomorphic transformations. This im- plies the universality of the scaling laws, conjectured by physicists in the 70's, for a com- binatorial class of bifurcations. Through near-parabolic renormalizations the polynomial- like renormalizations of satellite type are successfully studied here for the first time, and new techniques are introduced to analyze the fine-scale dynamical features of maps with such infinite renormalization structures. In particular, we confirm the rigidity conjecture under a quadratic growth condition on the combinatorics. The class of maps addressed in the paper includes infinitely-renormalizable maps with degenerating geometries at small scales (lack of a priori bounds).