On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality
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 KI (x, f) and the outer di...
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| Date: | 2011 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2011
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2699 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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 KI (x, f) and the outer dilatation
KΟ (x, f) of the mapping f at a point x satisfy certain conditions.
Denote by Bf the set of branch points of f. Then for an arbitrary neighborhood V of the point y0, a set V ∩ f(Bf ) 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. |
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