Triangulation extensions of self-homeomorphisms of the real line
type: Item
instance of: scholarly article
title: Triangulation extensions of self-homeomorphisms of the real line
zbMATH Open document ID: 1366.30017
DOI: 10.1215/21562261-3664950
author: Yi Qi
author: Yumin Zhong
published in: Kyoto Journal of Mathematics
publication date: 2017-04-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kjm/1489201228
review text: Sense-preserving self-homeomorphism of the real axis can be extended to a self-homeomorphism of the upper half-plane. One such extension was constructed by Hubbard with triangulation. For quasisymmetric homeomorphism of the real line, the extension problem has been developed for a long time, which is very interesting and rich in content. It is well known that the Beurling-Ahlfors extension, the Douady-Earle extension and the Carleson box extension are three important quasiconformal self-homeomorphisms of the upper half-plane. So it is natural to ask how about the triangulation extension. This paper concerns about the following two problems: one is the question whether the triangulation extension of a quasisymmetric homeomorphism is quasiconformal, the other one is to finf when the triangulation extension of homeomorphism is a David mapping. In the construction of the triangulation extension, the points \(f(k+\frac{2j}{2^i})\) play an important role, where \(k\in\mathbb{Z}\), \(i\in\mathbb{N}\), and \(j=0,1,2,\dots,2^{i-1}-2\), and \(f:\mathbb{R}\rightarrow\mathbb{R}\) is a homeomorphism. Lines from these points with slope \(1\) and \(-1\) form a triangulation of the upper half-plane. An extension mapping is defined to be piecewise linear and isometric on vertical line onto these triangles. So the maximal dilatation of the triangulation extension is bounded by the scalewise distortion \(\rho_f(t)\) of \(f(x)\), which is Lemma 2 in the paper and plays a key role in the proof of the main results.
reviewed by: Chong Wu
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 37F30
zbMATH DE Number: 6705665
zbMATH Keywords: quasisymmetric mapping
zbMATH Keywords: quasiconformal extension
zbMATH Keywords: triangulation extension
MaRDI profile type: Publication
OpenAlex ID: W2588400453
description: scientific article; zbMATH DE number 6705665