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The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property. (English. Russian original) Zbl 1239.30006

Sib. Math. J. 53, No. 1, 29-46 (2012); translation from Sib. Mat. Zh. 53, No. 1, 38-58 (2012).
Summary: The Möbius midpoint condition, introduced by K. P. Goldberg in [Mich. Math. J. 21, 49–62 (1974; Zbl 0286.30020)] as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by the author and D. G. Kuzin in [Sib. Math. J. 39, No. 6, 1057–1066 (1998); translation from Sib. Mat. Zh. 39, No. 6, 1225–1235 (1998; Zbl 0958.30009)] in the same role for the topological embeddings of the line into \(\mathbb R^{ n }\), yields no information on the quasiconformality or quasisymmetry of a topological embedding of \(\mathbb R^{ k }\) into \(\mathbb R^{ n }\) for \(1 < k \leq n\). In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” \(\text{MMC}(f) \leq H < 1\). We prove that if this condition is fulfilled then every homeomorphism of domains in \(\overline{\mathbb R^{n}}\) is \(K(H)\)-quasiconformal, while a topological embedding of the sphere \(\overline{\mathbb R^{k}}\) into \(\overline {\mathbb R^{n}}\) (for \(1 \leq k \leq n\)) is \(\omega _{H}\)-quasimöbius. The quasiconformality coefficient of \(K(H)\) and the distortion function \(\omega _{H}\) depend only on \(H\) and are expressed by explicit formulas showing that \(K(H) \rightarrow 1\) and \(\omega _{H} \rightarrow \text{id}\) as \(H \rightarrow 1/2\). Since\(\text{MMC}(f) = 1/2\) is equivalent to the Möbius property of \(f\), the resulting formulas yield the closeness of the mapping to a Möbius mapping for \(H\) near 1/2.

MSC:

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