Non-removability of Sierpinski carpets

TL;DR

This paper proves that all Sierpiński carpets in the plane are non-removable for (quasi)conformal maps, showing they can be mapped onto each other via homeomorphisms conformal outside the carpets, using topological methods.

Contribution

It establishes the non-removability of all planar Sierpiński carpets for (quasi)conformal maps, extending understanding of their conformal invariance properties.

Findings

Existence of homeomorphisms conformal outside carpets mapping one Sierpiński carpet to another

Topological proof based on Whyburn's characterization

Partial answer to Bishop's question on measure-zero sets

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.