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NC\({}_ p\)-sets of finite area. (English. Russian original) Zbl 0727.30021

Sib. Math. J. 31, No. 5, 862-864 (1990); translation from Sib. Mat. Zh. 31, No. 5(183), 194-196 (1990).
In the paper, the author shows that the set \(B_ f\) of all branching points of a mapping f:G \(\to {\mathbb{R}}^ n\) with a bounded distortion, having a finite (n-1)-dimensional Hausdorff measure, is the so-called \(NC_ p\)-set [W. M. Gol’dshtein and Yu. G. Reshetnyak, An introduction to the theory of functions with generalized derivatives and quasiconformal mappings, Moscow; Science 1983] in the region G for each \(p\in (1,+\infty)\). He also shows that, for such mappings, the full pre- images of an \(NC_ n\)-set is an \(NC_ n\)-set and finds a structure formula for an \(NC_ p\)-set of a finite (n-1)-dimensional Hausdorff measure.

MSC:

30C99 Geometric function theory
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References:

[1] Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982). · Zbl 0487.30011
[2] V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mapping [in Russian], Nauka, Moscow (1983).
[3] H. Federer, Geometric Measure Theory, Springer-Verlag, New York (1969). · Zbl 0176.00801
[4] V. M. Miklyukov, ?On the removable singularities of quasiconformal mappings in the space,? Dokl. Akad. Nauk SSSR,188, No. 3, 525-527 (1969). · Zbl 0204.39401
[5] I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).
[6] J. L. Kelley, General Topology, Van Nostrand, Princeton (1955).
[7] V. V. Aseev and A. V. Sychev, ?On the sets that are removable for quasiconformal spatial mappings,? Sib. Mat. Zh.,15, No. 6, 1213-1227 (1974).
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