Quasiconformal maps with thin dilatations

Abstract

We give an estimate that quantifies the fact that a normalized quasiconformal map whose dilatation is non-zero only on a set of small area approximates the identity uniformly on the whole plane. The precise statement is motivated by applications of the author’s quasiconformal folding method for constructing entire functions; in particular an application to constructing transcendental wandering domains given by Fagella, Godillon, and Jarque [7].