## Catching sets with quasicircles.(English)Zbl 0944.30013

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
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