Multivalued quasim\"{o}bius property and bounded turning

Abstract

The class of multivalued mappings with bounded angular distortion (BAD) property in metric spaces can be considered as a multivalued analog for quasim\"{o}bius mappings. We study the connections between quasimeromorphic self-mappings of $X= \bar{R}^n$ and multivalued mappings $F: X\to 2^X$ with BAD property. The main result of the paper concerns the multivalued mappings $F: D\to 2^{\bar{\bf C}}$ with BAD property of a domain $D\subset \bar{\bf C}$. If the image $F(x)$ of each point $x\in D$ is either a point or a continuum with bounded turning then $F$ is proved to be a single-valued quasim\"{o}bius mapping. The crucial point in the proof of this result is the local connectedness of the set $F(X)$ for the multivalued continuous mapping $F: X\to 2^Y$ with BAD property. We obtain sufficient conditions providing $F(X)$ to have local connectedness or bounded turning property in the most general case.