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A simply connected, homogeneous domain that is not a quasidisk. (English) Zbl 1071.30018

Let \(K \geq 1\) and let \({\mathcal F}(K)\) be the class of all \(K\)-quasiconformal maps of \({\overline {\mathbb C}}\, .\) A set \(S \subset {\overline {\mathbb C}}\) is said to be homogeneous with respect to \({\mathcal F}(K)\) if for each pair \(z_1, z_2 \in S\) there exists a mapping \( f \in {\mathcal F}(K)\) with \(f(S) = S\) and \(f(z_1)=z_2 \,.\) The author constructs a simply connected domain \(D\,,\) homogeneous with respect to \({\mathcal F}(K)\,\) for some \(K>1\,,\) that is not a quasidisk.

MSC:

30C62 Quasiconformal mappings in the complex plane
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