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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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