Non-removability of Sierpinski carpets

Abstract

We prove that all Sierpi\'nski carpets in the plane are non-removable for (quasi)conformal maps. More precisely, we show that for any two Sierpi\'nski carpets $S,S'\subset \hat{\mathbb{C}}$ there exists a homeomorphism $f\colon \hat{\mathbb{C}}\to \hat{\mathbb{C}}$ that is conformal in $\hat{\mathbb{C}}\setminus S$ and it maps $S$ onto $S'$. The proof is topological and it utilizes the ideas of the topological characterization of Whyburn. As a corollary, we obtain a partial answer to a question of Bishop, whether any planar continuum with empty interior and positive measure can be mapped to a set of measure zero with an exceptional homeomorphism of the plane, conformal off that set.