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Conformally natural extension of homeomorphisms of the circle. (English) Zbl 0615.30005
The purpose of the paper is to extend any homeomorphism \(\phi\) of \(S^ 1\) to a homeomorphism \(\Phi =E(\phi)\) of \(\bar D,\) \(D=\{z\in {\mathbb{C}}:| z| <1\}\) in a conformally natural way. To do this the authors assign at given \(\phi\) the measure \(\phi_*(\eta_ z)\) on \(S^ 1\) to each \(z\in D\). Then they define the conformal barycenter \(w\in D\) of this measure and set \(w=\Phi (z)\). Each of these steps is done in a conformally natural way. The last step is to show that \(\Phi\) is a homeomorphism.
Reviewer: V.Z.Enolskij

MSC:
30C20 Conformal mappings of special domains
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