Abstract
In this chapter, we study mappings acting between domains of two factor spaces by certain groups of Möbius automorphisms of the unit ball that act discontinuously and do not have fixed points. Note that the study of quotient spaces is of the utmost importance, for example, for mappings of Riemannian surfaces. In particular, by the Poincaré uniformization theorem, each such surface is homeomorphic to its quotient space; Riemannian surfaces of hyperbolic type are homeomorphic to the quotient space with respect to some group of linear fractional mappings of the unit disk onto itself, acting discontinuously and having no fixed points. For mappings acting between domains of two factor spaces, we have established estimates for the distortion of the modulus of families of paths, which are similar to the well-known Poletsky and Väisälä inequalities. As applications, we have obtained several important results on the local and boundary behavior of mappings. The chapter also contains useful and rather complex examples of such mappings.
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References
Apanasov, B.N.: Conformal Geometry of Discrete Groups and Manifolds. Walter de Gruyter, Berlin/New York (2000)
Cristea, M.: Quasiregularity in metric spaces. Rev. Roumaine Math. Pures Appl. 51(3), 291–310 (2006)
Cristea, M.: Direct products of quasiregular mappings on metric spaces. Rev. Roumaine Math. Pures Appl. 53(4), 285–296 (2008)
Cristea, M.: Local homeomorphisms having local ACLn inverses. Complex Var. Elliptic Equations 53(1), 77–99 (2008)
Cristea, M.: Open discrete mappings having local ACLn inverses. Complex Var. Elliptic Equations 55(1–3), 61–90 (2010)
Federer H.: Geometric Measure Theory. Springer, Berlin etc. (1969)
Guo, C.-Y.: Mappings of finite distortion between metric measure spaces. Conform. Geom. Dyn. 19, 95–121 (2015)
Guo, C.-Y., Williams, M.: The branch set of a quasiregular mapping between metric manifolds. C. R. Math. Acad. Sci. Paris 354(2), 155–159 (2016)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer Science+Business Media, New York (2001)
Ignat’ev, A., Ryazanov, V.: Finite mean oscillation in the mapping theory. Ukr. Mat. Visn. 2(3), 395–417 (2005) (in Russian); translation in Ukr. Math. Bull. 2(3), 403–424 (2005)
Ilyutko, D.P., Sevost’yanov, E.A.: Boundary behaviour of open discrete mappings on Riemannian manifolds. Mat. Sb. 209(5), 3–53 (2018) (in Russian); translation in Sb. Math. 209(5), 605–651 (2018)
Kuratowski, K.: Topology, vol. 1. Academic Press, New York/London (1968)
Maly, J., Martio O.: Lusin’s condition N and mappings of the class \(W_{\mathrm {loc}}^{1,n}\). J. Reine Angew. Math. 458, 19–36 (1995)
Martio, O., Rickman, S., Väisälä, J.: Definitions for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I. Math. 448, 1–40 (1969)
Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Mappings with finite length distortion. J. d’Anal. Math. 93, 215–236 (2004)
Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Moduli in Modern Mapping Theory. Springer Science + Business Media, LLC, New York (2009)
Martio, O., Srebro, U.: Automorphic quasimeromorphic mappings in \(\mathbb {R}^n.\) Acta Math. 135, 221–247 (1975)
Martio, O., Srebro, U.: Periodic quasimeromorphic mappings. J. Analyse Math. 28, 20–40 (1975)
Poleckii, E.A.: The modulus method for nonhomeomorphic quasiconformal mappings. Mat. Sb. 83(125), 2(10), 261–272 (1970) (in Russian); translation in Math. USSR-Sb. 12(2), 260–270 (1970)
Reimann, H.M., Rychener, T.: Funktionen Beschränkter Mittlerer Oscillation. Lecture Notes in Mathematics, vol. 487. Springer (1975)
Rickman, S.: Quasiregular Mappings. Springer, Berlin etc. (1993)
Ryazanov, V., Volkov, S.: On the Boundary Behavior of Mappings in the Class \(W^{1,1}_{loc}\) on Riemann Surfaces. Complex Anal. Oper. Theory 11, 1503–1520 (2017)
Ryazanov, V., Volkov, S.: Prime ends in the Sobolev mapping theory on Riemann surfaces. Mat. Stud. 48, 24–36 (2017)
Saks S.: Theory of the Integral. Dover Publications Inc., New York (1964)
Salimov, R.R., Sevost’yanov, E.A.: The Poletskii and Väisälä inequalities for the mappings with (p, q)-distortion. Complex Var. Elliptic Equations 59(2), 217–231 (2014)
Sevost’yanov, E.A.: Local and boundary behavior of maps in metric spaces. Algebra i analiz 28(6), 118–146 (2016) (in Russian); translation in St. Petersburg Math. J. 28(6), 807–824 (2017)
Sevost’yanov, E.A., Markysh, A.A.: On Sokhotski–Casorati–Weierstrass theorem on metric spaces. Complex Var. Elliptic Equations 64(12), 1973–1993 (2019)
Soultanis, E., Williams, M.: Marshall Distortion of quasiconformal maps in terms of the quasihyperbolic metric. J. Math. Anal. Appl. 402(2), 626–634 (2013)
Väisälä, J.: Two new characterizations for quasiconformality. Ann. Acad. Sci. Fenn. Ser. A 1 Math. 362, 1–12 (1965)
Väisälä, J.: Discrete open mappings on manifolds. Ann. Acad. Sci. Fenn. Ser. A 1 Math. 392, 1–10 (1966)
Väisälä, J.: Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin etc., (1971)
Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319. Springer, Berlin etc. (1988)
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Sevost’yanov, E. (2023). On the Local and Boundary Behavior of Mappings of Factor Spaces. In: Mappings with Direct and Inverse Poletsky Inequalities. Developments in Mathematics, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-031-45418-9_18
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