On the Julia set of a typical quadratic polynomial with a Siegel disk
type: Item
instance of: scholarly article
title: On the Julia set of a typical quadratic polynomial with a Siegel disk
zbMATH Open document ID: 1069.37038
author: Carsten Lunde Petersen
author: Saeed Zakeri
published in: Annals of Mathematics. Second Series
publication date: 2004-12-29T00:00:00Z
review text: The paper studies quadratic polynomials of the form \(P(z) = z^2 + rz\), where \(r=e^{2\pi i \theta}\) for some irrational number \(0<\theta < 1\). A growth condition is assumed on the integers \(a_n\) making up the continued fraction expansion of \(\theta\), namely that \(\log (a_n) = {\mathcal O}(\sqrt{n})\). The set of \(\theta\)'s satisfying this condition has full measure. The main result is that for such a \(\theta\), the Julia set of \(P\) is locally connected and has Lebesgue measure 0. It follows that there is a set of \(\theta\)'s of full measure for which the corresponding polynomials \(P\) have Siegel disks which are Jordan domains whose boundary contains the critical point of \(P\). A major part of the proof involves area and length estimates which enables one to apply a theorem of David concerning the integrability of Beltrami differentials whose dilatation is unbounded. This allows one to reduce the problem to a somewhat simpler one, which the authors solve, using an approach due to Yoccoz.
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 2123891
zbMATH Keywords: quadratic polynomials
zbMATH Keywords: growth condition
zbMATH Keywords: Julia set
zbMATH Keywords: Siegel disks
zbMATH Keywords: Jordan doamins
reviewed by: Mike Hurley
MaRDI profile type: Publication
full work available at URL: annals.2004.159.1
OpenAlex ID: W2146108974
Wikidata QID: Q120786174
DOI: 10.4007/ANNALS.2004.159.1
description: scientific article; zbMATH DE number 2123891