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Aspects of quasiconformal homogeneity. (English) Zbl 1233.30014

“Let \(\Omega\) and \(\Omega'\) be domains in \(\mathbb R^n\). A homeomorphism \(f:\Omega\to\Omega'\) is said to be \(K\)-quasiconformal if \(f\in W_{\text{loc}}^{1,n}(\Omega,\mathbb R^n)\) and \(|Df(x)|^n\leq KJ(x,f)\) almost everywhere in \(\Omega\). The map \(f\) is called quasiconformal if \(f\) is \(K\)-quasiconformal for some finite \(K\). Let \(QC(\Omega)\) be the group of all quasiconformal self homeomorphisms of a domain \(\Omega\subset \overline{\mathbb R^n}=\mathbb R^n\cup\{\infty\}\) under composition. A quasiconformal group is a subgroup \(G\) of the group \(QC(\Omega)\), and \(G\) is said to be a uniformly quasiconformal group if there exists \(K<\infty\) such that each \(f\in G\) is \(K\)-quasiconformal. Let \(E\) be a nonempty subset of a domain \(\Omega\) and let \(F\) be a subfamily of \(QC(\Omega)\), we say \(F\) acts transitively on \(E\) relative to \(\Omega\) if for each \(x,y\in E\) there exists \(f\in F\) such that \(f(x)=y\) and \(f(E)=E\).
Let \(E\) be a subset of a domain \(\Omega\) in \(\overline{\mathbb R^n}\). \(E\) is said to be quasiconformal homogeneous relative to \(\Omega\) if there exists a subfamily \(F\subset QC(\Omega)\) which acts transitively on \(E\). We say that \(E\) is \(K\)-quasiconformally homogeneous relative to \(\Omega\) if \(F\) can be taken to be a \(K\)-quasiconformal subfamily of \(QC(\Omega)\). In particular, if \(E=\Omega\), then we say \(\Omega\) is quasiconformally homogeneous, or \(K\)-quasiconformally homogeneous, respectively.
Quasiconformal groups were introduced by F. W. Gehring and B. P. Palka [“Quasiconformal homogeneous domains”, J. Anal. Math. 30, 172–199 (1976; Zbl 0349.30019)] in their study of quasiconformally homogeneous domains. An elementary argument of Gehring and Palka shows that every domain \(\Omega\subset\mathbb R^n\) is quasiconformally homogeneous. The family of \(K\)-quasiconformally homogeneous domains is much more restricted; there are a number of interesting relevant results in the plane, but little is known about the higher dimensional situation, though there are some interesting results concerning the quasiconformal homogeneity of hyperbolic manifolds and Riemann surfaces.”
In the paper under review, the authors discuss the quasiconformal homogeneity in various different settings and give an example of a compact set \(E\) which is homogeneous with respect to a quasiconformal family of maps of a domain containing \(E\), but which is not quasiconformally homogeneous with respect to \(\mathbb R^n\). They also construct Cantor sets which are \(K\)-quasiconformally homogeneous for each \(K>1\) and explicitly identify their Hausdorff dimension.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30C62 Quasiconformal mappings in the complex plane

Citations:

Zbl 0349.30019
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