Quasiconformal maps with thin dilatations

Abstract

. We give an estimate that quantifies the fact that a normalized quasi-conformal 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 a construction of transcendental wandering domains given by Fagella, Godillon and Jarque [7].