Bi-Lipschitz homogeneous curves in \(\mathbb{R} ^2\) are quasicircles (Q2719039)
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scientific article; zbMATH DE number 1597930
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Bi-Lipschitz homogeneous curves in \(\mathbb{R} ^2\) are quasicircles |
scientific article; zbMATH DE number 1597930 |
Statements
14 May 2001
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bi-Lipschitz mappings
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homogeneous continua
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quasicircles
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bounded turning
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quasihomogeneous embeddings
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chord-arc
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quasiconformal mappings
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Hausdorff dimension
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0.86867845
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0.8628944
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0.86156535
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0.8601502
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0.85594356
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0.85238993
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Bi-Lipschitz homogeneous curves in \(\mathbb{R} ^2\) are quasicircles (English)
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The topic of the paper are the closed Jordan curves \(\Gamma\) in Euclidean spaces which are bi-Lipschitz homogeneous in the sense that there is a constant \(K\) and for all \(x,y\in\Gamma\) a \(K\)-bi-Lipschitz homeomorphism \(f\colon\Gamma\to\Gamma\) with \(f(x)=y\). It is shown that in \({\mathbb{R}}^2\) they are of bounded turning and thus quasicircles and that this is not true in \({\mathbb{R}}^3\). Combined with results of \textit{D. A. Herron} and \textit{V. Mayer} [Ill. J. Math. 43, No. 4, 770-792 (1999; Zbl 0935.30017)] several characterizations of them are given in the planar case.
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