On dilatations and substantial boundary points of homeomorphisms of Jordan curves
type: Item
instance of: scholarly article
title: On dilatations and substantial boundary points of homeomorphisms of Jordan curves
zbMATH Open document ID: 0872.30011
DOI: 10.1007/BF03322160
author: Shanshuang Yang
published in: Results in Mathematics
publication date: 1997-10-20T00:00:00Z
review text: Let \(D\) be a Jordan domain in the plane and let \(h:\partial D\to\partial D\) be a sense-preserving homeomorphism. The dilatation \(K_h\) of \(h\) is the supremum of the ratio of the 2-moduli of curve families joining two disjoint nondegenerate continua on \(\partial D\) and their images under \(h\) in \(D\). Another dilatation, \(K_h^*\), is obtained as \(\inf K(f)\) where \(f\) runs over all quasiconformal extensions of \(h\) to \(D\). In general, \(K_h\leq K_h^*\) [see \textit{M. Lehtinen}, Ann. Acad. Sci. Fenn., Ser. A I 9, 133-139 (1984; Zbl 0539.30010) and \textit{J. M. Anderson} and \textit{A. Hinkkanen}, Comment. Math. Helv. 70, No. 3, 455-474 (1995; Zbl 0839.30017)]. The author shows that if \(K_h= K_h^*\), then either \(h\) is induced by an affine mapping or \(h\) has a substantial boundary point \(p\); the latter condition means that all local extensions at \(p\) have the minimum dilatation \(=K_h^*\).
reviewed by: O. Martio
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 1003997
zbMATH Keywords: quasisymmetric maps
zbMATH Keywords: dilatation
MaRDI profile type: Publication
cites work: Q5512337
cites work: Quadrilaterals and extremal quasiconformal extensions
cites work: The boundary correspondence under quasiconformal mappings
cites work: Über extremale quasikonforme Abbildungen
cites work: Symmetric Structures on a Closed Curve
cites work: Quasiextremal distance domains and integrability of derivatives of conformal mappings
cites work: Remarks on the maximal dilatation of the Beurling-Ahlfors extension
cites work: Q3744437
cites work: Q5685358
cites work: Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises
cites work: Extremal quasiconformal mappings
full work available at URL: https://doi.org/10.1007/bf03322160
OpenAlex ID: W2018373004
description: scientific article; zbMATH DE number 1003997