The distortion of tetrads under quasimeromorphic mappings of Riemann sphere

Abstract

On the Riemann sphere, we consider the ptolemaic characteristic of a four of non-empty pairwise non-intersecting compact subsets (generalized tetrad, or generalized angle). We obtain an estimate for distortion of this characteristic under the inverse to a K-quasimeromorphic mapping of the Riemann sphere which takes each of its values at no more then N different points. The distortion function in this estimate depends only on K and N. In the case K=1, it is an essentially new property of complex rational functions.

Keywords

• generalized tetrad
• generalized angle
• ptolemaic characteristic
• value of generalized angle
• quasimeromorphic mapping
• rational function
• quasiconformal mapping

Kaynakça

1. [1] V.V. Aseev, Generalized angles in Ptolemaic Möbius structures, Siberian Math. J. 59 (2018) 189-201.
2. [2] V.V. Aseev, Multivalued quasimöbius mappings on Riemann sphere, Siberian Math. J. 64 (2023) (in print).
3. [3] G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translations of Math. Monographs 26, American Math. Society, Providence - Rhode Island 02904 (1969).
4. [4] H.P. Künzi, Quasikonforme Abbildungen, Spinger-Verlag, Berlin-Göttingen-Heidelberg (1960).
5. [5] O. Lehto and K.I. Virtanen, Quasikonforme Abbildungen, Springer-Verlag, Berlin-Heidelberg-New York (1965).
6. [6] O. Martio, S. Rickman, and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 448 (1969) 1-40.
7. [7] O. Martio, S. Rickman, and J. Väisälä, Distortion and singulaities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 465 (1970) 1-13.
8. [8] Yu.G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Math. Monographs 73, American Math. Society, Providence - Rhode Island (1989).

9. [9] S. Rickman, Characterization of quasiconformal arcs, Ann. Acad. Sci. Fenn. Ser. A I Math. 395 (1966) 1-30.
10. [10] M. Vuorinen, Quadruples and spatial quasiconformal mappings, Math. Z. 205 (1990) 617-628.