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arXiv · 2210.09280

Uniform a priori bounds for neutral renormalization

Dzmitry Dudko·Mikhail Lyubich
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Abstract

We prove uniform ``pseudo-Siegel'' a priori bounds for Siegel disks of bounded type that give a uniform control of oscillations of their boundaries in all scales. As a consequence, we construct the Mother Hedgehog controlling the postcritical set for any quadratic polynomial with a neutral periodic point and show that this hedgehog has a star-like structure. Pseudo-Siegel bounds imply uniform a priori bounds of the Sector Renormalization, which gives an opportunity to extend Siegel/Pacman Renormalization Theory and Near-Parabolic Renormalization Theory to all near-neutral quadratic polynomials. Various applications beyond quadratic polynomials are also underway.

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Dzmitry Dudko, Mikhail Lyubich. 2022-10-17. Uniform a priori bounds for neutral renormalization. https://arxiv.org/abs/2210.09280

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