Catching sets with quasicircles. (English) Zbl 0944.30013

MacManus, Paul

Rev. Mat. Iberoam. 15, No. 2, 267-277 (1999).

The following two theorems are proved: Theorem 1. For a subset \(E\) of \(\mathbb{C}\) the following statements are equivalent: i) \(E\) has empty interior and uniform complement. ii) \(E\) is uniformly disconnected. iii) \(E\) is quasiconformally equivalent to a porous subset of \(R\). The various constants depend only on each other. Theorem 2. For a compact set \(K\) in \(\mathbb{C}\) whose interior is empty the following statements are equivalent: i) \(K\) is uniformly perfect and has uniform complement. ii) \(K\) is both uniformly perfect and uniformly disconneted. iii) \(K\) is quasiconformally equivalent to the usual Cantor middle-third set. The various constants depend only on each other.

Reviewer: A.Neagu (Iaşi)

MSC:

30C62

Quasiconformal mappings in the complex plane

Keywords:

quasiconformal map; quasicircle; nul set; Cantor middle-third set