Quasiconformally homogeneous compacta in the complex plane
type: Item
instance of: scholarly article
title: Quasiconformally homogeneous compacta in the complex plane
zbMATH Open document ID: 0960.30017
author: Paul MacManus
author: Raimo Näkki
author: Bruce Palka
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: A subset \(E\neq\emptyset\) of the extended complex plane \(\widehat{\mathbb{C}}\) is said to be quasiconformally homogeneous (\(K\)-quasiconformally homogeneous) if for each pair of points \(a,b\in E\) there exists a quasiconformal mapping (\(K\)-quasiconformal mapping) \(f: \widehat{\mathbb{C}}\to \widehat{\mathbb{C}}\) such that \(f(E)= E\) and \(f(a)= b\). This paper summarizes some old (and forgotten) facts and new findings of the authors on these sets \(E\). New results: If \(E\neq\widehat {\mathbb{C}}\) is compact and \(K\)-quasiconformally homogeneous, then \(H\)-\(\dim(E)\leq d<\infty\) where \(d= d(K)\). If \(E\) is compact and quasiconformally homogeneous, then one of the conditions holds: (i) \(E= \widehat{\mathbb{C}}\), (ii) \(E\) is a finite set, (iii) \(E\) is a union of a finite and disjoint collection of quasicircles, (iv) \(E\) is a Cantor set, with \(H\)-\(\dim(E)< 2\). There are several examples. For instance, the authors show that for each \(K>1\) there exists a \(K\)-quasiconformally homogeneous linear Cantor set.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1548268
zbMATH Keywords: quasiconformally homogeneous sets
reviewed by: O. Martio
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132180
OpenAlex ID: W2054801921
DOI: 10.1307/MMJ/1030132180
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description: scientific article; zbMATH DE number 1548268