On convergence of a sequence of mappings with inverse modulus inequality to a discrete mapping

Authors

  • E. Sevost’yanov Ivan Franko Zhytomyr State University; Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Sloviansk, Donetsk region.
  • V. Targonskii Ivan Franko Zhytomyr State University

DOI:

https://doi.org/10.3842/umzh.v77i6.8952

Keywords:

mappings with finite distortion, moduli of families of paths, convergence of mappings, discreteness

Abstract

UDC 517.5

We consider mappings satisfying the Poletsky-type inverse inequality in a domain of the Euclidean space. It is shown that the uniform limit of the family of these mappings is a discrete mapping. We separately consider  domains that are locally connected on their boundaries and regular domains in the quasiconformal sense.

References

1. M. Cristea, Open discrete mappings having local $ACL^n$ inverses, Complex Var. and Elliptic Equat., 55, № 1–3, 61–90 (2010). DOI: https://doi.org/10.1080/17476930902998985

2. B. Klishchuk, R. Salimov, M. Stefanchuk, On the asymptotic behavior at infinity of one mapping class, Proc. Int. Geom. Center, 16, № 1, 50–58 (2023). DOI: https://doi.org/10.15673/tmgc.v16i1.2394

3. D. Kovtonyuk, I. Petkov, V. Ryazanov, Prime ends in theory of mappings with finite distortion in the plane, Filomat, 31, № 5, 1349–1366 (2017). DOI: https://doi.org/10.2298/FIL1705349K

4. O. Martio, S. Rickman, J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Math., 448, 1–40 (1969). DOI: https://doi.org/10.5186/aasfm.1969.448

5. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Science + Business Media, LLC, New York (2009).

6. V. Ryazanov, S. Volkov, On the boundary behavior of mappings in the class $W^{1,1}_{loc}$ on Riemann surfaces, Complex Anal. and Oper. Theory, 11, 1503–1520 (2017). DOI: https://doi.org/10.1007/s11785-016-0618-4

7. M. Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in $n$-space, Ann. Acad. Sci. Fenn. Math. Diss., 11, 1–44 (1976).

8. J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin etc. (1971). DOI: https://doi.org/10.1007/BFb0061216

9. E. Sevost'yanov, On boundary discreteness of mappings with a modulus conditions, Acta Math. Hung., 171, № 1, 67–87 (2023). DOI: https://doi.org/10.1007/s10474-023-01381-z

10. Є. О. Севостьянов, С. О. Скворцов, О. П. Довгопятий, Про негомеоморфні

відображення з оберненою нерівністю Полецького, Укр. мат. вісн., 17, № 3, 414–436 (2020); English translation: J. Math. Sci., 252, № 4, 541–557 (2021).

11. E. A. Sevost’yanov, S. A. Skvortsov, On the convergence of mappings in metric spaces with direct and inverse modulus conditions, Ukr. Math. J., 70, № 7, 1097–1114 (2018). DOI: https://doi.org/10.1007/s11253-018-1554-4

12. К. Куратовский, Топология, т. 2, Мир, Москва (1969).

13. J. Herron, P. Koskela, Quasiextremal distance domains and conformal mappings onto circle domains, Complex Var. and Theor. Appl., 15, 167–179 (1990). DOI: https://doi.org/10.1080/17476939008814448

14. O. Martio, U. Srebro, Automorphic quasimeromorphic mappings in ${R}^n$, Acta Math., 135, 221–247 (1975). DOI: https://doi.org/10.1007/BF02392020

15. E. A. Sevost'yanov, Mappings with direct and inverse Poletsky inequalities, Developments in Mathematics (DEVM, vol. 78), Springer Nature Switzerland AG, Cham (2023).

16. R. Näkki, Prime ends and quasiconformal mappings, J. Anal. Math., 35, 13–40 (1979). DOI: https://doi.org/10.1007/BF02791061

17. D. P. Ilyutko, E. A. Sevost'yanov, On prime ends on Riemannian manifolds, J. Math. Sci., 241, № 1, 47–63 (2019). DOI: https://doi.org/10.1007/s10958-019-04406-7

18. D. A. Kovtonyuk, V. I. Ryazanov, On the theory of prime ends for space mappings, Ukr. Math. J., 67, № 4, 528–-541 (2015). DOI: https://doi.org/10.1007/s11253-015-1098-9

19. E. A. Sevost'yanov, On discrete boundary extension of mappings in terms of prime ends, Укр. мат. вісн., 21, № 4, 231–254 (2024; English translation: J. Math. Sci., 284, 365–382 (2024). DOI: https://doi.org/10.1007/s10958-024-07356-x

20. C. J. Titus, G. S. Young, The extension of interiority with some applications, Trans. Amer. Math. Soc., 103, 329–340 (1962). DOI: https://doi.org/10.1090/S0002-9947-1962-0137103-6

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Published

01.08.2025

Issue

Vol. 77 No. 6 (2025)

Section

Research articles

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Copyright (c) 2025 E. Sevost’yanov, V. Targonskii

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How to Cite

Sevost’yanov, E., and V. Targonskii. “On Convergence of a Sequence of Mappings With Inverse Modulus Inequality to a Discrete Mapping”. Ukrains’kyi Matematychnyi Zhurnal, vol. 77, no. 6, Aug. 2025, pp. 426–440, https://doi.org/10.3842/umzh.v77i6.8952.