Conformally natural extension of homeomorphisms of the circle (Q1820892)

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.