Approximation of space mappings with bounded distortion by similarities (Q1097365)

The author investigates quasiregular mappings in the Möbius space \({\bar {\mathbb{R}}}^ n={\mathbb{R}}^ n\cup \{\infty \}\) with maximal dilatation K close to 1 and proves e.g. the following theorem. Theorem. Let U be a domain in \({\bar {\mathbb{R}}}^ n\) of class U(\(\alpha\),\(\beta)\) with \(\infty \in U\). Then there are constants \(\epsilon_ 0>0\) and \(c<\infty\) depending only on \(\alpha\),\(\beta\),n such that given a \(K_ f\)-quasiregular mapping f:U\(\to {\bar {\mathbb{R}}}^ n\) with \[ K_ f=1+\epsilon \leq 1+\epsilon_ 0 \] and \(f(\infty)=\infty\) and a ball \(B(x_ 0,r)\), there exists a similarity T such that \[ | Tf(x)-x| \leq cr\epsilon \text{ for all } x\in \bar B^ n(x_ 0,r)\cap U. \] Here U(\(\alpha\),\(\beta)\) denotes the class of all (\(\alpha\),\(\beta)\)-uniform domains.