Abstract
We prove that if a multivalued mapping \( F \) of circle to circle has the \( \eta \)-BAD property (bounded distortion of generalized angles with control function \( \eta \)) then there exist a positive integer \( N \) and a quasimöbius homeomorphism \( \varphi \) of a circle into itself such that the left inverse mapping to \( F \) is of the form \( (\varphi(z))^{N} \). Moreover, \( \varphi \) is a locally \( \omega \)-quasimöbius mapping with \( \omega \) depending only on \( \eta \) and \( N \).
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References
Väisälä J., “Quasimöbius maps,” J. Anal. Math., vol. 44, no. 1, 218–234 (1984/85).
Aseev V. V., Sychev A. V., and Tetenov A. V., “Möbius-invariant metrics and generalized angles in Ptolemaic spaces,” Sib. Math. J., vol. 46, no. 2, 189–204 (2005).
Aseev V. V., “Generalized angles in Ptolemaic Möbius structures,” Sib. Math. J., vol. 59, no. 2, 189–201 (2018).
Aseev V. V., “Generalized angles in Ptolemaic Möbius structures. II,” Sib. Math. J., vol. 59, no. 5, 768–777 (2018).
Aseev V. V., “Multivalued mappings with the quasimöbius property,” Sib. Math. J., vol. 60, no. 5, 741–756 (2019).
Aseev V. V., “On coordinate vector-functions of quasiregular mappings,” Sib. Electron. Math. Rep., vol. 15, 768–772 (2018).
Aseev V. V., “Adherence of the images of points under multivalued quasimöbius mappings,” Sib. Math. J., vol. 61, no. 3, 391–402 (2020).
Kuratowski K., Topology. Vols. 1 and 2, Academic and Polish Scientific, New York etc. (1966, 1969).
Hilton P. J. and Wylie S., Homology Theory: An Introduction to Algebraic Topology, Cambridge University, Cambridge (2009).
Lehto O. and Virtanen K. I., Quasikonforme Abbildungen, Springer, Berlin, Heidelberg, and New York (1965).
Rickman S., “Quasiconformally equivalent curves,” Duke Math. J., vol. 36, 387–400 (1969).
Vuorinen M., “Quadruple and spatial quasiconformal mappings,” Math. Z., vol. 205, 617–628 (1990).
Ibragimov Z. Sh., “Metric density and quasimöbius mappings,” Sib. Math. J., vol. 43, no. 5, 812–821 (2002).
Aseev V. V. and Kuzin D. G., “Locally quasimöbius mappings on a circle,” J. Math. Sci. (New York), vol. 211, no. 6, 724–737 (2015).
Acknowledgment
The author is grateful to the referee for thoroughly checking the text and making a series of critical remarks incorporated in the new version.
Funding
The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project 0314–2019–0007).
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Aseev, V.V. Multivalued Quasimöbius Mappings from Circle to Circle. Sib Math J 62, 14–22 (2021). https://doi.org/10.1134/S003744662101002X
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DOI: https://doi.org/10.1134/S003744662101002X
Keywords
- quasimöbius mapping
- quasisymmetric mapping
- multivalued mapping
- generalized angle
- BAD property
- local quasimöbius property