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arXiv · 1509.07843

Satellite renormalization of quadratic polynomials

Davoud Cheraghi·Mitsuhiro Shishikura
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Abstract

We prove the uniform hyperbolicity of the near-parabolic renormalization operators acting on an infinite-dimensional space of holomorphic transformations. This implies the universality of the scaling laws, conjectured by physicists in the 70's, for a combinatorial 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).

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Davoud Cheraghi, Mitsuhiro Shishikura. 2015-09-25. Satellite renormalization of quadratic polynomials. https://arxiv.org/abs/1509.07843

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