Quasisymmetric embeddings of slit Sierpiński carpets

Abstract

We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela [Acta Math. 181 (1998), pp. 1–61]. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpetXXcan be associated to a “pillowcase sphere”X^\widehat {X}which is a metric space homeomorphic to the sphereS2\mathbb {S}^2. We show thatXXquasisymmetrically embeds into the plane if and only ifX^\widehat {X}is quasisymmetric toS2\mathbb {S}^2if and only ifX^\widehat {X}is Ahlfors22-regular.