Jordan and Julia
type: Item
instance of: scholarly article
title: Jordan and Julia
zbMATH Open document ID: 0879.30015
published in: Mathematische Annalen
publication date: 1997-07-06T00:00:00Z
review text: The author considers the problem: which is the Julia set \(J(f)\) of a rational function \(f\) a quasiconformal Jordan arc or curve? In the case when \(J(f)\) is a Jordan arc (for which necessary and sufficient conditions are known) it is shown that \(J(f)\) is always a quasiconformal arc. If the Julia set is a Jordan curve then the complementary Fatou set \(F(f)\) consists of two components \(U\) and \(V\). In the case when \(U\) and \(V\) are attracting then D. Sullivan has proved that \(J\) is a quasicircle. The author shows that the only other case which gives rise to a quasicircle is when \(U\) and \(V\) form the Leau flower of the same parabolic fixed point. The proofs depend on a lemma about fusing critical points in a parabolic basis using quasiconformal surgery.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 1031547
zbMATH Keywords: Julia set
zbMATH Keywords: quasiconformal arc
zbMATH Keywords: quasicircle
Wikidata QID: Q122556396
author: Norbert Steinmetz
reviewed by: I. N. Baker
MaRDI profile type: Publication
full work available at URL: s002080050049
OpenAlex ID: W2070862469
DOI: 10.1007/S002080050049
description: scientific article; zbMATH DE number 1031547