Satellite renormalization of quadratic polynomials

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).

arXiv comment71 pages, comments welcome

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• A Proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type
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Cited by in corpus (4)

• A Proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type
• The high type quadratic Siegel disks are Jordan domains
• Dynamics of asymptotically holomorphic polynomial-like maps
• A model of the cubic connectedness locus