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Quasiconformally homogeneous compacta in the complex plane. (English) Zbl 0960.30017
A subset \(E\neq\emptyset\) of the extended complex plane \(\widehat{\mathbb{C}}\) is said to be quasiconformally homogeneous (\(K\)-quasiconformally homogeneous) if for each pair of points \(a,b\in E\) there exists a quasiconformal mapping (\(K\)-quasiconformal mapping) \(f: \widehat{\mathbb{C}}\to \widehat{\mathbb{C}}\) such that \(f(E)= E\) and \(f(a)= b\). This paper summarizes some old (and forgotten) facts and new findings of the authors on these sets \(E\).
New results: If \(E\neq\widehat {\mathbb{C}}\) is compact and \(K\)-quasiconformally homogeneous, then \(H\)-\(\dim(E)\leq d<\infty\) where \(d= d(K)\). If \(E\) is compact and quasiconformally homogeneous, then one of the conditions holds:
(i) \(E= \widehat{\mathbb{C}}\), (ii) \(E\) is a finite set, (iii) \(E\) is a union of a finite and disjoint collection of quasicircles, (iv) \(E\) is a Cantor set, with \(H\)-\(\dim(E)< 2\). There are several examples. For instance, the authors show that for each \(K>1\) there exists a \(K\)-quasiconformally homogeneous linear Cantor set.

MSC:
30C62 Quasiconformal mappings in the complex plane
28A78 Hausdorff and packing measures
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