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Swiss cheese, dendrites, and quasiconformal homogeneity. (English) Zbl 1307.30048

Summary: Suppose that \(D\) is a simply connected domain in the extended complex plane \(\widehat{\mathbb C}\) with the following homogeneity property: for each pair of points \(a\) and \(b\) in \(D\) there exists a \(K\)-quasiconformal self-mapping \(f\) of \(\widehat{\mathbb C}\) such that \(f(D)=D\) and \(f(a)=b\). This paper classifies the simply connected plane domains \(D\) with locally connected boundaries that exhibit this property for some \(K\). Any such domain \(D\) falls into one of five (non-empty) categories, each specified by the character of the boundary \(\partial D\) of \(D\), namely \(\partial D\) is the empty set, a singleton, a quasicircle, a dendrite, or a Swiss cheese.

MSC:

30C62 Quasiconformal mappings in the complex plane
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