Abstract
Suppose that a multi-valued mapping \( F:D\to 2^{\overline{}} \) of a domain \( D \) in the sphere \( \overline{} \) with disjoint images of distinct points boundedly distorts the Ptolemaic characteristic of generalized tetrads (quadruples of disjoint compact sets). Suppose that the image \( F(x) \) of each \( x\in D \) has at most \( N \) components, each of which is a continuum of bounded turning. Then \( F \), up to the values at some isolated branch points, is the inverse of a mapping with bounded distortion in the sense of Reshetnyak. In particular, if \( D=\overline{} \) then the left inverse to \( F \) is the composition of a quasiconformal automorphism of \( \overline{} \) and a rational function.
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Acknowledgment
The author is grateful to the referee for the favorable opinion and suggestions that were implemented in the final version of this article.
Funding
The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0005).
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 3, pp. 450–464. https://doi.org/10.33048/smzh.2023.64.302
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Aseev, V.V. The Multi-Valued Quasimöbius Mappings on the Riemann Sphere. Sib Math J 64, 514–524 (2023). https://doi.org/10.1134/S0037446623030023
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DOI: https://doi.org/10.1134/S0037446623030023
Keywords
- quasiconformal mapping
- mapping with bounded distortion
- quasimeromorphic mapping
- Ptolemaic characteristic tetrad
- continuum of bounded turning
- multi-valued mappings of BAD class