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

Abstract

For the open discrete mappings f: D \ {b} → R³ of the domain D ⊂ R³ satisfying relatively general geometric conditions in D \ {b} and having the essential singularity b ∈ R³, we prove the following statement. Let y₀ belong to R³ \ f (D \ {b}) and let the inner dilatation K_I (x, f) and the outer dilatation K_Ο (x, f) of the mapping f at a point x satisfy certain conditions. Denote by B_f the set of branch points of f. Then for an arbitrary neighborhood V of the point y₀, a set V ∩ f(B_f ) cannot be contained in the set A such that g(A) = I, where I = {t ∈ R: |t| < 1} and g : U → Rⁿ is a quasiconformal mapping of the domain U ⊂ Rⁿ such that A ⊂ U