Variability sets and Hamilton sequences (Q816621)

The aim of this paper are the variability sets and Hamilton sequences for extremal quasiconformal mappings of the unit disk \(\Delta\). The variability set \(V[h,z_o] \) of a given quasisymmetric boundary correspondence \(h: \partial \Delta \to \partial \Delta\) and \(z_o \in \Delta \) is the set of all \(f(z_o)\) if \(f\) is extremal for the boundary values \(h\). The variability set were introduced by \textit{K. Strebel} [J. Anal. Math. 30, 464-480 (1976; Zbl 0334.30013)]. It is proven for example that the hyperbolic diameter of \(V[h,z_o] \) is less than a constant depending only on the maximal dilatation and that \(V[h,z_o] \) is continuous in \(\Delta\) in the Hausdorff topology. The remainder of the paper deals with sufficient conditions for the existence of Hamilton sequences. A comprehensive treatment of this problems is given by \textit{E. Reich} [Handbook of complex analysis: geometric function theory. Vol 1, 75--136 (2002; Zbl 1079.30021)].