On mappings of a spatial domain which preserve angles and dilatations along a system of rays
In 1933, D. E. Men'shov has proven that if a univalent continuous mapping acting from a planar domain into the plane preserves angles or dilations along a system of rays at every point of the domain then this mapping is holomorphic [Math. Ann. 109, 101-159 (1933; Zbl 0007.41502)]. A number of mathematicians strived for weakening the assumptions of this Men'shov's theorem. It turns out that the claim of the theorem is true even if the mapping is neither univalent nor continuous (see, for example, the author's article [Sib. Mat. Zh. 38, No. 5, 1005-1007 (1997; Zbl 0880.30004)] for more detail). Less is known about mappings in the 3-space. In his article [Sov. Math., Dokl. 7, 1321-1323 (1966; Zbl 0181.08801)], \textit{A. P. Kopylov} announced the following theorem: If \(D\) is a domain in \(\mathbb R^3\) and \(f\: D\to\mathbb R^3\) is a homeomorphism preserving angles between rays at every point of~\(D\), then \(f\) is conformal. In the article under review, the author proves the following theorem: Let \(Q\) be an open cube in \(\mathbb R^3\). For every point \(x\) of \(Q\), there is an everywhere dense countable system of rays \(t_i(x)\) \((i=1,2,\dots)\) in \(\mathbb R^3\) and a discontinuous function \(f\: Q\to \mathbb R^3\) such that \(f\) preserves dilatations and angles along the system of rays \(t_i(x)\) \((i=1,2,\dots)\) at every point \(x\) of \(Q\).
scientific article; zbMATH DE number 1431635