On dilatations and substantial boundary points of homeomorphisms of Jordan curves
Let \(D\) be a Jordan domain in the plane and let \(h:\partial D\to\partial D\) be a sense-preserving homeomorphism. The dilatation \(K_h\) of \(h\) is the supremum of the ratio of the 2-moduli of curve families joining two disjoint nondegenerate continua on \(\partial D\) and their images under \(h\) in \(D\). Another dilatation, \(K_h^*\), is obtained as \(\inf K(f)\) where \(f\) runs over all quasiconformal extensions of \(h\) to \(D\). In general, \(K_h\leq K_h^*\) [see \textit{M. Lehtinen}, Ann. Acad. Sci. Fenn., Ser. A I 9, 133-139 (1984; Zbl 0539.30010) and \textit{J. M. Anderson} and \textit{A. Hinkkanen}, Comment. Math. Helv. 70, No. 3, 455-474 (1995; Zbl 0839.30017)]. The author shows that if \(K_h= K_h^*\), then either \(h\) is induced by an affine mapping or \(h\) has a substantial boundary point \(p\); the latter condition means that all local extensions at \(p\) have the minimum dilatation \(=K_h^*\).
scientific article; zbMATH DE number 1003997