Steinhaus-Kosinski problem in complex space

Zelinsky Yuriy Borisovych

Zelinsky Yuriy Borisovych ¹

¹ Department of Complex Analysis and Potential Theory, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, 01004 , Ukraine

Keywords: Steinhaus-Kosinski problem, complex space

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Abstract

We consider a generalization of some problem of Steinhaus to higher-dimensional complex spaces. We show that for any continuous multi-valued acyclic map $Ф : \mathbb{B}^{2n} → \mathbb{C}$ on a ball $\mathbb{B}^{2n}, n≥2,$ whose restiction to the boundary of the ball is a Hopf fibration, there is a point $y ∈ \mathbb{C}$ whose preimage $Ф^{-1}(y) = \{x ∈ \mathbb{B}^{2n} : y ∈ Ф(x)\}$ contains at least three points. If the map $Ф$ is zero- dimensional then the set $\{x ∈ \mathbb{B}^{2n} : |Ф^{-1}(x)| ≥ 3\}$ has non-empty interior in $\mathbb{B}^3$.

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Cite

ACS Style: Zelinsky, Y.B. Steinhaus-Kosinski problem in complex space. Bukovinian Mathematical Journal. 2016, 3
AMA Style: Zelinsky YB. Steinhaus-Kosinski problem in complex space. Bukovinian Mathematical Journal. 2016; 3(3-4).
Chicago/Turabian Style: Yuriy Borisovych Zelinsky. 2016. "Steinhaus-Kosinski problem in complex space". Bukovinian Mathematical Journal. 3 no. 3-4.

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