Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension
type: Item
instance of: scholarly article
title: Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension
zbMATH Open document ID: 1251.30009
author: David M. Freeman
published in: Illinois Journal of Mathematics
publication date: 2011-11-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1318598680
review text: A homeomorphism \(f: (X,d_X)\to(Y,d_Y)\) is \(L\)-bilipschitz if \(L^{-1}d_X(x,y)\leq d_Y(f(x),f(y))\leq Ld_X(x,y)\) for all \(x,y\in X\). A metric space \(X\) is \(L\)-bilipschitz homogeneous if for any two points \(x,y\in X\), there exists an \(L\)-bilipschitz self map of \(X\) sending \(x\) to \(y\). The author utilizes bilipschitz homogeneity and Möbius maps to provide a new characterization of fractal chordarc curves. Let \(\widehat{\mathbb R}^n\) denote \(\mathbb R^n\cup\{\infty\}\) under the chordal distance \(\chi\) defined by \[ \chi(z,w):=\frac{2|z-w|}{\sqrt{1+|z|^2}\sqrt{1+|w|^2}},\quad \chi(z,\infty):=\frac{2}{\sqrt{1+|z|^2}}. \] \(\text{Möb}(\widehat{\mathbb R}^n)\) denotes the collection of Möbius maps \(M:\widehat{\mathbb R}^n\to\widehat{\mathbb R}^n\). Given a Jordan curve \(\Lambda\) in \((X,d)\), and two points \(x,y\in\Lambda\), \(\Lambda[x,y]\) denotes the smaller (with respect to the diameter) closed subarc of \(\Lambda\) joining \(x\) to \(y\). \(\Lambda\) is of \(B\)-bounded turning if for all \(x,y\in\Lambda\), \(\text{diam}(\Lambda[x,y])\leq Bd(x,y)\). A curve \(\Lambda\) is \((C,\alpha)\)-fractal chordarc if there exist constants \(C\geq1\) and \(\alpha>0\) such that, for all \(x,y\in\Lambda\), \(C^{-1}d(x,y)^{\alpha}\leq\mathcal H^{\alpha}(\Lambda[x,y])\leq Cd(x,y)^{\alpha}\), where \(\mathcal H^{\alpha}\) denotes the \(\alpha\)-dimensional Hausdorff measure. Theorem 1.1: Let \(\widehat{\Gamma}\subset\widehat{\mathbb R}^n\) be a Jordan curve of \(B\)-bounded turning. The following three statements are quantitatively equivalent (i.e. the constants for each condition depend only on the constants for the other): (i) \(\widehat\Gamma\) is a \((C,\alpha)\)-fractal chordarc curve. (ii) For every \(M\in\text{Möb}(\widehat{\mathbb R}^n)\), \(M(\widehat\Gamma)\cap\mathbb R^n\) is \(L\)-bilipschitz homogeneous. (iii) Both \(\widehat\Gamma\cap\mathbb R^n\) and \(\widehat\Gamma^*\cap\mathbb R^n\) are \(L\)-bilipschitz homogeneous, where \(\widehat\Gam…
reviewed by: Dmitri V. Prokhorov
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 51F99
zbMATH DE Number: 5981651
zbMATH Keywords: fractal chordarc curve
zbMATH Keywords: bi-Lipschitz homogeneity
zbMATH Keywords: dimension
zbMATH Keywords: \(B\)-bounded turning
MaRDI profile type: Publication
cites work: Q4341343
cites work: Bi-Lipschitz homogeneous curves in \(\mathbb{R} ^2\) are quasicircles
cites work: Bilipschitz homogeneity and inner diameter distance
cites work: Higher dimensional Ahlfors regular sets and chordarc curves in \(R^n\)
cites work: Bilipschitz homogeneous Jordan curves
cites work: Bilipschitz group actions and homogeneous Jordan curves
cites work: Trajectories of one-parameters groups of quasi-isometries
cites work: Quasicircles modulo bilipschitz maps
description: scientific article; zbMATH DE number 5981651