Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension
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 \(\widehat\Gamma^*\) denotes an inversion of \(\widehat\Gamma\). If \(\infty\in\widehat\Gamma\), we require that \(\infty\notin\widehat\Gamma^*\), and conversely. Theorem 1.2: Let \(\widehat\Gamma\subset\widehat{\mathbb R}^n\) be a Jordan curve such that \(\{0,\infty\}\subset\widehat\Gamma\). Then \(\widehat\Gamma\) is \((C,\alpha)\)-fractal chordarc if and only if \(\widehat\Gamma\) is of \(B\)-bounded turning and both \(\widehat\Gamma\) and \(\widehat\Gamma\cap\mathbb R^n\) are \(L\)-bilipschitz homogeneous. This equivalence is quantitative. In Section 7, Theorem 1.2 is illustrated by constructing planar curves that are bilipschitz homogeneous with respect to the Euclidean distance but not the chordal distance and vice-versa. These construction methods lead to a means of calculating the dimensions of planar bilipschitz homogeneous Jordan curves, which is described in Section 8.
scientific article; zbMATH DE number 5981651