Swiss Cheese, Dendrites, and Quasiconformal Homogeneity

Suppose that is a simply connected domain in the extended complex plane with the following homogeneity property: for each pair of points and in there exists a -quasiconformal self-mapping of such that and . This paper classifies the simply connected plane domains with locally connected boundaries that exhibit this property for some . Any such domain falls into one of five (non-empty) categories, each specified by the character of the boundary of , namely is the empty set, a singleton, a quasicircle, a dendrite, or a Swiss cheese.