Inversion invariant bilipschitz homogeneity
type: Item
instance of: scholarly article
title: Inversion invariant bilipschitz homogeneity
zbMATH Open document ID: 1264.30046
author: David M. Freeman
published in: Michigan Mathematical Journal
publication date: 2013-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.mmj/1339011533
review text: The author considers the class of metric spaces that are (i) doubling, (ii) bilipschitz homogeneous, (iii) remain bilipschitz homogeneous after metric inversion. A metric space \(X\) is called doubling if there exists an integer \(D\) such that every ball \(B\subset X\) can be covered by at most \(D\) balls of half the size of \(B\). It is bilipschitz homogeneous if there exists a constant \(L\) such that for any \(a,b\in X\) there is an \(L\)-bilipschitz homeomorphism \(f: X\to X\) such that \(f(a)=b\). The inversion of a metric space \((X,d)\) at a point \(p\in X\) puts another metric \(d_p\) on the same set \(X\) (possibly with a point added) such that \(d_p(x,y)\) is comparable to \[ \frac{d(x,y)}{d(x,p)\,d(y,p)}. \] For subsets of a Euclidean space this amounts to the familiar Möbius inversion \[ x\mapsto p+\frac{x-p}{|x-p|^2}. \] One of the main result of the article is the following theorem. Theorem 1.1. Suppose \(X\) is a proper, connected, and \(D\)-doubling metric space. If there exists a point \(p\in X\) such that both \(X\) and the inversion of \(X\) at \(p\) are \(L\)-bilipschitz homogeneous then \(X\) is Ahlfors regular, with regularity constant depending only on \(D\) and \(L\). It should be noted that Ahlfors regularity is much stronger than the doubling condition. A space \(X\) is Ahlfors regular if there exists \(Q\) such that the \(Q\)-dimensional Hausdorff measure of any ball \(B\) is comparable to \((\mathrm{diam}\,B)^Q\), with uniform multiplicative constants. As a corollary of the main result, the author improves a result from his paper [Ill. J. Math. 54, No. 2, 753--770 (2010; Zbl 1251.30009)] by showing that a Jordan curve \(\Gamma\) in \(\mathbb R^n\) is an Ahlfors regular quasicircle if and only if there exists a point \(p\in \Gamma\) such that both \(\Gamma\) and the inversion of \(\Gamma\) at \(p\) are bilipschitz homogeneous.
Mathematics Subject Classification ID: 30L05
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 30L10
zbMATH DE Number: 6141584
zbMATH Keywords: doubling metric space
zbMATH Keywords: bilipschitz homogeneity
zbMATH Keywords: metric inversion
zbMATH Keywords: Ahlfors regularity
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
OpenAlex ID: W2082244874
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DOI: 10.1307/MMJ/1339011533
description: scientific article; zbMATH DE number 6141584