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

### Abstract

For the open discrete mappings*f*:

*D*\ {

*b*} →

**R**

^{3}of the domain

*D*⊂

**R**

^{3}satisfying relatively general geometric conditions in

*D*\ {

*b*} and having the essential singularity

*b*∈

**R**

^{3}, we prove the following statement. Let

*y*

_{0}belong to

**R**

^{3}\

*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*the set of branch points of

_{f }*f*. Then for an arbitrary neighborhood

*V*of the point

*y*

_{0}, a set

*V*∩

*f*(

*B*) cannot be contained in the set

_{f }*A*such that

*g*(

*A*) =

*I*, where

*I*= {

*t*∈

**R**: |

*t*| < 1} and

*g*:

*U*→

**R**

^{n}is a quasiconformal mapping of the domain

*U*⊂

**R**

^{n}such that

*A*⊂

*U*.

Published

25.01.2011

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 63, no. 1, Jan. 2011, pp. 69-79, https://umj.imath.kiev.ua/index.php/umj/article/view/2699.

Issue

Section

Research articles