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QUASICONFORMALLY BI-HOMOGENEOUS COMPACTA IN THE COMPLEX PLANE

Published online by Cambridge University Press:  01 January 1999

PAUL MACMANUS
Affiliation:
Department of Mathematics, The King’s Buildings, Mayfield Road, University of Edinburgh, Edinburgh EH9 3JZ E-mail: [email protected]
RAIMO NÄKKI
Affiliation:
Department of Mathematics, University of Jyväskylä, P.O. Box 35, SF-40351 Jyväskylä, Finland. E-mail: [email protected]
BRUCE PALKA
Affiliation:
Department of Mathematics University of Texas at Austin Austin TX 78712 U.S.A. E-mail: [email protected]
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Abstract

This article addresses the question of 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 $\widehat{\mathbb C}$. 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.

Type
Research Article
Copyright
© 1999 The London Mathematical Society

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