Quasiconformally bi-homogeneous compacta in the complex plane. (English) Zbl 0974.30012

MacManus, Paul; Näkki, Raimo; Palka, Bruce

Proc. Lond. Math. Soc., III. Ser. 78, No. 1, 215-240 (1999).

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