A new proof of a conjecture of Yoccoz
type: Item
instance of: scholarly article
title: A new proof of a conjecture of Yoccoz
zbMATH Open document ID: 1223.37061
author: Xavier Buff
author: Arnaud Chéritat
publication date: 2011-05-26T00:00:00Z
full work available at URL: https://eudml.org/doc/219751
review text: The authors give a new proof of a conjecture of Yoccoz, which provides an upper bound for the logarithm of the conformal radius of the Siegel disk of a map \(Q_\theta(z) = e^{2i\pi\theta} z + z^2\), in terms of Yoccoz's Brjuno function \(Y(\theta)\). Arithmetical properties of the rotation number \(\theta\) of a map of type \(f(z) = e^{2i\pi \theta} z + O(z^2)\) are intimately linked to the size of the Siegel disk of \(f\) at \(0\) (where the Siegel disk is the maximal domain containing 0 on which \(f\) is conjugated to a rotation). The authors' interesting proof follows Yoccoz's initial methods. Their result is then extended to other new families of polynomial maps like \(z\mapsto z^d+c\) when \(d>2\).
reviewed by: Eugen Mihailescu
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37A45
zbMATH DE Number: 5899953
zbMATH Keywords: Siegel disks
zbMATH Keywords: quadratic polynomials
zbMATH Keywords: harmonic and subharbonic functions
zbMATH Keywords: conformal radius
zbMATH Keywords: holomorphic motions
MaRDI profile type: Publication
Wikidata QID: Q122971784
OpenAlex ID: W2272423056
cites work: Holomorphic families of injections
cites work: Virtually repelling fixed points
cites work: Upper bound for the size of quadratic Siegel disks
cites work: The Brjuno function continuously estimates the size of quadratic Siegel disks
cites work: A parabolic Pommerenke-Levin-Yoccoz inequality
cites work: On the dynamics of polynomial-like mappings
cites work: Q4938279
cites work: Q3518705
cites work: On the dynamics of rational maps
cites work: Q4052427
cites work: Q4328333
cites work: Linéarisation des perturbations holomorphes des rotations et applications
cites work: On the quasiconformal surgery of rational functions
cites work: Holomorphic Motions and Polynomial Hulls
cites work: Extending holomorphic motions
DOI: 10.5802/AIF.2603
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5899953