MacManus, Paul; Näkki, Raimo; Palka, Bruce Quasiconformally homogeneous compacta in the complex plane. (English) Zbl 0960.30017 Mich. Math. J. 45, No. 2, 227-241 (1998). 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. Reviewer: O.Martio (Helsinki) Cited in 8 Documents MSC: 30C62 Quasiconformal mappings in the complex plane 28A78 Hausdorff and packing measures Keywords:quasiconformally homogeneous sets PDFBibTeX XMLCite \textit{P. MacManus} et al., Mich. Math. J. 45, No. 2, 227--241 (1998; Zbl 0960.30017) Full Text: DOI