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Quasiconformally homogeneous planar domains. (English) Zbl 1191.30008
The authors explore the ambient quasiconformal homogeneity of planar domains and their boundaries. They show that the quasiconformal homogeneity of a domain \( D\) and its boundary \( E\) implies that the pair \( (D,E)\) is in fact quasiconformally bi-homogeneous. They also give a geometric and topological characterization of the quasiconformal homogeneity of \( D\) or \( E\) under the assumption that \( E\) is a Cantor set captured by a quasicircle. A collection of examples is provided to demonstrate that certain assumptions are the weakest possible. The notion of uniform perfectness in the sense of Pommerenke has an important role in this paper.

MSC:
30C62 Quasiconformal mappings in the complex plane
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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