Geodesic
On the Zero-Dimensionality of the Limit of the Sequence of Generalized Quasiconformal Mappings
Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 586-596.

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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 $f\mspace{2mu}\colon D\to {\mathbb R}^n$ of a domain $D\subset{\mathbb R}^n$, $n\ge 2$, 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.
Keywords: discrete mapping, bounded distortion mapping (quasiregular mapping). 
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E. A. Sevost'yanov. On the Zero-Dimensionality of the Limit of the Sequence of Generalized Quasiconformal Mappings. Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 586-596. http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a10/

[1] E. S. Afanaseva, V. I. Ryazanov, R. R. Salimov, “Ob otobrazheniyakh v klassakh Orlicha–Soboleva na rimanovykh mnogoobraziyakh”, Ukr. matem. vestnik, 8:3 (2011), 319–342

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

[3] M. Cristea, “Open discrete mappings having local $ACL^n$ inverses”, Complex Var. Elliptic Equ., 55:1-3 (2010), 61–90 | DOI | MR | Zbl

[4] V. Ya. Gutlyanskii, V. I. Ryazanov, U. Srebro, E. Yakubov, The Beltrami Equation. A Geometric Approach, Dev. Math., 26, Springer, New York, 2012 | MR | Zbl

[5] D. V. Isangulova, “Klass otobrazhenii s ogranichennym udelnym kolebaniem i integriruemost otobrazhenii s ogranichennym iskazheniem na gruppakh Karno”, Sib. matem. zhurn., 48:2 (2007), 313–334 | MR | Zbl

[6] T. Iwaniec, G. Martin, Geometric Function Theory and Non-Linear Analysis, The Clarendon Press, Oxford, 2001 | MR

[7] I. Markina, “Singularities of quasiregular mappings on Carnot groups”, Sci. Ser. A Math. Sci. (N.S.), 11 (2005), 69–81 | MR | Zbl

[8] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009 | MR | Zbl

[9] Yu. G. Reshetnyak, Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982 | MR | Zbl

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

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

[12] S. Rickman, Quasiregular Mappings, Ergeb. Math. Grenzgeb. (3), 26, Springer-Verlag, Berlin, 1993 | MR | Zbl

[13] V. I. Ryazanov, R. R. Salimov, “Slabo ploskie prostranstva i granitsy v teorii otobrazhenii”, Ukr. matem. vestnik, 4:2 (2007), 199–234

[14] E. S. Smolovaya, “Granichnoe povedenie koltsevykh $Q$-gomeomorfizmov v metricheskikh prostranstvakh”, Ukr. matem. zhurn., 62:5 (2010), 682–689 | MR | Zbl

[15] J. Väisälä, Lectures on $n$-Dimensional Quasiconformal Mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971 | DOI | MR | Zbl

[16] E. A. Sevostyanov, “Ob otkrytosti i diskretnosti otobrazhenii s neogranichennoi kharakteristikoi kvazikonformnosti”, Ukr. matem. zhurn., 63:8 (2011), 1128–1134 | MR | Zbl

[17] W. Hurewicz, H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, 1948 | MR | Zbl

[18] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001 | MR | Zbl

[19] V. G. Mazya, Prostranstva S. L. Soboleva, Izd-vo Leningradsk. un-ta, L., 1985 | MR | Zbl

[20] P. Hajlasz, “Sobolev spaces on metric-measure spaces”, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003, 173–218 | DOI | MR | Zbl

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

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

[23] C. J. Titus, G. S. Young, “The extension of interiority with some applications”, Trans. Amer. Math. Soc., 103 (1962), 329–340 | DOI | MR | Zbl

[24] S. Saks, Teoriya integrala, IL, M., 1949 | MR | Zbl

[25] V. I. Ryazanov, R. R. Salimov, E. A. Sevost'yanov, “On convergence analysis of space homeomorphisms”, Siberian Adv. Math., 23:4 (2013), 263–293 | DOI | MR | Zbl

[26] E. Sevost'yanov, On the Lightness of the Limit of Sequence of Mappings Satisfying Some Modular Inequality, 2015, arXiv: 1510.06280v1