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Quasiconformally bi-homogeneous compacta in the complex plane. (English) Zbl 0974.30012
This article addresses the question: which non-empty, compact, proper subsets \(E\) of the extended complex plane \(\widehat {\mathbb{C}}\) have the feature that, for some \(K\) in \([1,\infty)\), the family of \(K\)-quasiconformal self-mappings of \(\widehat {\mathbb{C}}\) which leave \(E\) invariant acts transitively on the set \(E\times E^c\), where \(E^c\) is the complement of \(E\) in \(\mathbb{C}\)? The main result in the paper asserts that the class of sets with this property comprises all one- and two-point subsets of \(\widehat {\mathbb{C}}\), all quasicircles in \(\widehat {\mathbb{C}}\) and all images of the Cantor ternary set under quasiconformal self-mappings of \(\widehat {\mathbb{C}}\). It is shown that the third category includes the limit set of any non-cyclic, finitely generated Schottky group.
Reviewer: B.Palka (Austin)

MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30C62 Quasiconformal mappings in the complex plane
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