Satellite renormalization of quadratic polynomials

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Title: Satellite renormalization of quadratic polynomials
Authors: Cheraghi, D
Shishikura, M
Item Type: Working Paper
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).
Issue Date: 25-Sep-2015
Publisher: arXiv
Copyright Statement: © 2015 The Authors
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/M01746X/1
Keywords: math.DS
0101 Pure Mathematics
Notes: 71 pages, comments welcome
Publication Status: Published
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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