Inversion invariant bilipschitz homogeneity
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.
scientific article; zbMATH DE number 6141584
scientific article