Skip to main content
SpringerLink
Log in

On the zero-dimensionality of the limit of the sequence of generalized quasiconformal mappings

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The paper is devoted to the study of the properties of a class of space mappings that is more general than that of bounded distortion mappings (aka quasiregular mappings). It is shown that the locally uniform limit of a sequence of mappings f: D → ℝn of a domain D ⊂ ℝn, n ≥ 2, satisfying one inequality for the p-modulus of families of curves is zero-dimensional. This statement generalizes a well-known theorem on the openness and discreteness of the uniform limit of a sequence of bounded distortion mappings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. S. Afanas’eva, V. I. Ryazanov, and R. R. Salimov, “On the mappings in Orlicz–Sobolev classes on Riemannian manifolds,” Ukr. Mat. Vestnik 8 (3), 319–342 (2011).

    Google Scholar 

  2. C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Math. Sci. 22, 1397–1420 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Cristea, “Open discrete mappings having local ACLn inverses,” Complex Var. Elliptic Equ. 55 (1-3), 61–90 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  4. V. Ya. Gutlyanskii, V. I. Ryazanov, U. Srebro, and E. Yakubov, Beltrami Equation. A Geometric Approach, in Dev. Math. (Springer, New York, 2012), Vol.26.

    Google Scholar 

  5. D. V. Isangulova, “The class of mappings with bounded specific oscillation, and integrability of mappings with bounded distortion on Carnot groups,” Sibirsk. Mat. Zh. 48 (2), 313–334 (2007) [Sib. Math. J. 48 (2), 249–267 (2007)].

    MATH  MathSciNet  Google Scholar 

  6. T. Iwaniec and G. Martin, Geometric Function Theory and Non-Linear Analysis (The Clarendon Press, Oxford, 2001).

    MATH  Google Scholar 

  7. I. Markina, “Singularities of quasiregular mappings on Cardoes not groups,” Sci. Ser. A Math. Sci. (N. S.) 11, 69–81 (2005).

    MATH  MathSciNet  Google Scholar 

  8. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory (Springer, New York, 2009).

    MATH  Google Scholar 

  9. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion (Nauka, Novosibirsk, 1982; Amer. Math. Soc., Providence, RI, 1989).

    MATH  Google Scholar 

  10. O. Martio, S. Rickman, and J. Väisälä, “Definitions for quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A I 448, 1–40 (1969).

    MATH  MathSciNet  Google Scholar 

  11. O. Martio, S. Rickman, and J. Väisälä, “Distortion and singularities of quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A I 465, 1–13 (1970).

    MATH  MathSciNet  Google Scholar 

  12. S. Rickman, Quasiregular Mappings, in Ergeb. Math. Grenzgeb. (3) (Springer-Verlag, Berlin, 1993), Vol.26.

    Google Scholar 

  13. V. I. Ryazanov and R. R. Salimov, “Weakly planar spaces and boundaries in the theory of mappings,” Ukr. Mat. Vestnik 4 (2), 199–234 (2007).

    MathSciNet  Google Scholar 

  14. E. S. Smolovaya, “Boundary behavior of ring Q-homeomorphisms in metric spaces,” Ukrain. Mat. Zh. 62 (5), 682–689 (2010) [UkrainianMath. J. 62 (5), 785–793 (2010)].

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1971), Vol.229.

  16. E. A. Sevost’yanov, “On the openness and discreteness of mappings with unbounded characteristic of quasiconformality,” Ukrain. Mat. Zh. 63 (8), 1128–1134 (2011) [Ukrainian Math. J. 63 (8), 1298–1305 (2012)].

    MathSciNet  Google Scholar 

  17. W. Hurewicz and H. Wallman, Dimension Theory (Princeton Univ. Press, Princeton, 1948).

    MATH  Google Scholar 

  18. J. Heinonen, Lectures on Analysis on Metric Spaces (Springer, New York, 2001).

    Book  MATH  Google Scholar 

  19. V. G. Maz’ya, Sobolev Spaces (Izd. Leningrad Univ., Leningrad, 1985) [in Russian].

    MATH  Google Scholar 

  20. P. Hajlasz, “Sobolev spaces on metric-measure spaces,” in Contemp. Math., Vol. 338: Heat Kernels and Analysis on Manifolds, Graphs and Metric Spaces (Amer. Math. Soc., Providence, RI, 2003), pp. 173–218.

    Google Scholar 

  21. T. Adamowicz and N. Shanmugalingam, “Non-conformal Loewner type estimates for modulus of curve families,” Ann. Acad. Sci. Fenn. Math. 35 (2), 609–626 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Näkki, “Boundary behavior of quasiconformal mappings in n-space,” Ann. Acad. Sci. Fenn. Ser. A I 484, 1–50 (1970).

    MATH  MathSciNet  Google Scholar 

  23. C. J. Titus and G. S. Young, “The extension interiority with some applications,” Trans. Amer. Math. Soc. 103, 329–340 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Saks, Theory of the Integral (USA, 1939; Inostr. Lit., Moscow, 1949).

    Google Scholar 

  25. V. I. Ryazanov, R. R. Salimov and E. A. Sevost’yanov, “On convergence analysis of space homeomorphisms,” Siberian Adv. Math. 23 (4), 263–293 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Sevost’yanov, On the Lightness of the Limit of Sequence of Mappings Satisfying Some Modular Inequality, arXiv:1510.06280v1 (2015).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ivan Franko Zhytomir State University, Zhytomir, Ukraine

    E. A. Sevost’yanov

Authors
  1. E. A. Sevost’yanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. A. Sevost’yanov.

Additional information

Original Russian Text © E. A. Sevost’yanov, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 4, pp. 586–596.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sevost’yanov, E.A. On the zero-dimensionality of the limit of the sequence of generalized quasiconformal mappings. Math Notes 102, 547–555 (2017). https://doi.org/10.1134/S0001434617090279

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434617090279

Keywords

Search

Navigation