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Non-removability of Sierpiński spaces. (English) Zbl 1437.30008

The authors prove that Sierpiński spaces in \(n\)-spheres \(\mathbb S^n\), \(n\geq2\), are non-removable for (quasi)conformal maps. A compact set \(K\subset\mathbb S^n\) is (quasi)conformally removable if every homeomorphism \(f:\mathbb S^n\to\mathbb S^n\) that is (quasi)conformal in \(\mathbb S^n\setminus K\) is (quasi)conformal everywhere. The authors pose the following question.
Question 1. Let \(K\subset\mathbb S^n\), \(n\geq2\), be a compact set of positive Lebesgue measure. Does there exist a homeomorphism \(f\) of \(\mathbb S^n\) that is quasiconformal on \(\mathbb S^n\setminus K\) and maps \(K\) (or a subset of \(K\) of positive measure) to a set of positive measure?
A continuum \(X\subset\mathbb S^n\), \(n\geq2\), is an \((n-1)\)-dimensional Sierpiński space if its complement \(\mathbb S^n\setminus X\) consists of countably many components \(U_k\), \(k\in\mathbb N\), satisfying the conditions: \(\mathbb S^n\setminus U_k\) is an \(n\)-cell for each \(k\in\mathbb N\), \(\overline U_k\cap\overline U_j=\emptyset\) for \(k\neq j\), \(\bigcup_{k\in\mathbb N}U_k\) is dense in \(\mathbb S^n\) and \(\text{diam}(U_k)\to0\) as \(k\to\infty\). The main result of the paper is in the next theorem.
Theorem 1.3. Let \(X\subset\mathbb S^n\), \(n\geq2\), be an \((n-1)\)-dimensional Sierpiński space. Then there exist a Sierpiński space \(Y\subset\mathbb S^n\) of positive Lebesgue measure and a homeomorphism \(f:\mathbb S^n\to\mathbb S^n\) which maps \(X\) onto \(Y\) and is conformal on \(\mathbb S^n\setminus X\).
Corollary 1.4. All \((n-1)\)-dimensional Sierpiński spaces in \(\mathbb S^n\), \(n\geq2\), are non-removable for (quasi)conformal maps.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
32D20 Removable singularities in several complex variables

Citations:

Zbl 1426.30012
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References:

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