On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D{ b } ? mathbbR³ f:Dbackslash left{ b right} to {mathbb{R}^3} of a domain D ì mathbbR³ D subset {mathbb{R}^3} satisfying relatively general geometric conditions in D {b} and having an essential singularity at a point b ? mathbbR³ b in {mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(mathbbR³)] f( D{ b } ) overline {{mathbb{R}^3}} backslash fleft( {Dbackslash left{ b right}} right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? mathbbR:| t | < 1 } I = left{ {t in mathbb{R}:left| t right| < 1} right} and g:U ? mathbbRⁿ g:U to {mathbb{R}^n} is a quasiconformal mapping of a domain U ì mathbbRⁿ U subset {mathbb{R}^n} such that A ⊂ U.