Non-removability of Sierpinski carpets

Abstract

We prove that all Sierpiński carpets in the plane are non-removable for (quasi)conformal maps. More precisely, we show that for any two Sierpiński carpets $S,S^{\prime }\subset \hat{\mathbb{C}}$ there exists a homeomorphism $f:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ that is conformal in $\hat{\mathbb{C}}\setminus S$ and it maps $S$ onto $S^{\prime }$. 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.

7 pages; corrected typos, added details in the last section

References in corpus (1)

• Non-removability of the Sierpinski Gasket

Cited by in corpus (4)

• Non-removability of the Sierpinski Gasket
• CNED sets: countably negligible for extremal distances
• On the inverse absolute continuity of quasiconformal mappings on hypersurfaces
• Non-removability of Sierpinski spaces