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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
##### Keywords:
quasiconformal mapping; Schottky Group
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