Essential singular points and branching of open discrete ring mappings (Q2901664)

The present article is devoted to the study of mappings with finite distortion in \({\mathbb R}^n\), \(n\geq 2\). It is proved that, if a point \(x_0\in {\mathbb R}^n\), \(n\geq3\), is an essential singularity of a discrete open ring mapping \(f:D\rightarrow \overline{{\mathbb R}^n}\) at \(x_0\), \(B_f\) is a branch set of \(f\) and \(z_0\in \overline{{\mathbb R}^n}\) is an asymptotic value of \(f\) at \(x_0,\) then \(z_0\in\overline{f (B_f\cap U)}\) for every neighborhood \(U\) of \(x_0\) under some conditions on \(Q\). In this case, every point of \(\overline {{\mathbb R}^n}\setminus f(D)\) is an asymptotic value of \(f\) at \(x_0\) and \(\overline{{\mathbb R}^n}\setminus f(D)\subset \overline{f(B_f)}\). If, in addition, \(\infty\notin f(D)\), then \(f(B_f)\) is unbounded and \(x_0\in\overline{B_f}\). The results of the work can be applied to various classes of spatial mappings with finite distortion, and, in particular, to Sobolev classes.