For ring homeomorphisms f : ℝn → ℝn , we establish the order of growth at infinity.
Similar content being viewed by others
References
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York (2009).
O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York (1973).
C. J. Bishop, Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Math. Sci., 22, 1397–1420 (2003).
Yu. F. Strugov, “Compactness of classes of mappings quasiconformal in the mean,” Dokl. Akad. Nauk SSSR, 234, No. 4, 859–861 (1978).
V. M. Miklyukov, Conformal Mapping of an Irregular Surface and Its Applications [in Russian], Volgograd University, Volgograd (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn., Ser. A1, Math., 30, 49–69 (2005).
F. Ignat’ev and V. Ryazanov, “Finite mean oscillation in the mapping theory,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).
A. Ignat’ev and V. Ryazanov, “On the theory of boundary behavior of space mappings,” Ukr. Mat. Vestn., 3, No. 2, 199–211 (2006).
V. Ryazanov and R. Salimov, “Weakly flat spaces and boundaries in mapping theory,” Ukr. Mat. Vestn., 4, No. 2, 199–234 (2007).
V. I. Ryazanov and E. A. Sevost’yanov, “Equicontinuous classes of ring Q-homeomorphisms,” Sib. Mat. Zh., 48, No. 6, 1361–1376 (2007).
R. Salimov, “Absolute continuity on lines and differentiability of one generalization of quasiconformal mappings,” Izv. Ros. Akad. Nauk, Ser. Mat., 75, No. 5, 141–148 (2008).
R. Salimov and E. Sevost’yanov, “ACL and differentiability almost everywhere of ring homeomorphisms,” Tr. Inst. Prikl. Mat. Mekh. Nats. Akad. Nauk Ukr., 16, 171–178 (2008).
V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equations,” J. d’Anal. Math., 96, 117–150 (2005).
V. Ryazanov, U. Srebro, and E. Yakubov, “The Beltrami equation and ring homeomorphisms,” Ukr. Math. Bull., 4, No. 1, 79–115 (2007).
V. Gutlyanski, O. Martio, T. Sugava, and M. Vuorinen, “On the degenerate Beltrami equation,” Trans. Amer. Math. Soc., 357, No. 3, 875–900 (2005).
O. Martio, S. Rickman, and J. Väisälä, “Definitions for quasiregular mappings,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 448 (1969).
V. I. Kruglikov, “Capacities of condensers and space mappings quasiconformal in the mean,” Mat. Sb., 130, No. 2, 185–206 (1986).
F. W. Gehring, “Quasiconformal mappings,” in: Complex Analysis and Its Applications, Vol. 2, International Atomic Energy Agency, Vienna (1976), pp. 213–268.
J. Hesse, “A p-extremal length and p-capacity equality,” Ark. Mat., 13, 131–144 (1975).
V. A. Shlyk, “On the equality between p-capacity and p-modulus,” Sib. Mat. Zh., 34, No. 6, 216–221 (1993).
S. Saks, Theory of the Integral, Państwowe Wydawnictwo Naukowe, Warsaw (1937).
O. Martio, S. Rickman, and J. Väisälä, “Distortion and singularities of quasiregular mappings,” Ann. Acad. Sci. Fenn., Ser. A, Math., 30, 465 (1970).
E. Sevost’yanov, “Liouville, Picard, and Sokhotskii theorems for ring mappings,” Ukr. Mat. Visn., 5, No. 3, 366–381 (2008).
E. A. Sevost’yanov, “Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for Q-mappings,” Ukr. Mat. Zh., 61, No. 1, 116–126 (2009).
E. Sevost’yanov, “On the theory of removal of singularities of mappings with unbounded characteristic of quasiconformality,” Izv. Ros. Akad. Nauk, Ser. Mat., 74, No. 1, 159–174 (2010).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 6, pp. 829 – 836, June, 2010.
Rights and permissions
About this article
Cite this article
Salimov, R.R., Smolovaya, E.S. On the order of growth of ring Q-homeomorphisms at infinity. Ukr Math J 62, 961–969 (2010). https://doi.org/10.1007/s11253-010-0403-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0403-x