The analogue of Picard's theorem for quasiregular mappings in dimension three (Q1821902)

A map \(f: R^ n\to R^ n\) is K-quasiregular (qr) if f is \(ACL^ n\) and \(| f'(x)| \leq KJ(x,f)\) a.e. For \(n=2\), \(f=\phi \circ h\) where \(\phi\) is analytic and \(h: R^ 2\to R^ 2\) is a K-quasiregular homeomorphism, i.e. a quasiconformal mapping. Hence in the plane the value distribution of quasiregular maps is essentially the same as that of analytic functions. Contrary to this it is shown that for each positive integer p there exists a nonconstant quasiregular mapping \(f: R^ 3\to R^ 3\) which omits p points in \(R^ 3\). The example for \(p=1\) was given by \textit{V. A. Zorich} [Mat. Sb., Nov. Ser. 74(116), 417-433 (1967; Zbl 0181.087)]. The general construction makes use of a special two dimensional deformation theory and rather complicated turbular neighborhoods. The author [J. Anal. Math. 37, 100-117 (1980; Zbl 0451.30012)] has previously shown that for each \(K\geq 1\) and \(n\geq 3\) there is \(q=q(n,K)\) such that each K-qr map \(f: R^ n\to R^ n\setminus \{a_ 1,...,a_ q\}\) is constant. Thus this theorem is best possible for \(n=3\).