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On Belinskii conformality in countable sets of points

Authors: Vladimir I. Ryazanov and Matti K. Vuorinen
Journal: Proc. Amer. Math. Soc. 129 (2001), 3049-3056
MSC (1991): Primary 30C62; Secondary 30G15
Published electronically: April 9, 2001
MathSciNet review: 1840111
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Abstract | References | Similar Articles | Additional Information


The local behavior of plane quasiconformal mappings is investigated. In particular, generalizing the well-known Reich-Walczak problem, we study the possibility for a quasiconformal mapping to be conformal in the sense of Belinskii at a prescribed point or in a prescribed set of points when the modulus of the complex dilatation is a fixed measurable function. The notion of the Belinskii conformality is related to the conception of asymptotical rotations by Brakalova and Jenkins.

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  • [A] AHLFORS L., Lectures on Quasiconformal Mappings, Princeton, NJ, Van Nostrand, 1966. MR 36:336
  • [B1] BELINSKII P.P., General properties of quasiconformal mappings, Novosibirsk, Nauka, 1974 (Russian). MR 53:11054
  • [B2] BELINSKII P.P., Behavior of a quasiconformal mapping at an isolated singular point, Uchenye Zapiski Lvovski Gosudarstvennyi Universitet. Ser. Mekhaniko-Matematichna 29 (1954), 58-70 (Russian). MR 18:740c
  • [Be] BERS L., On a theorem Mori and the definition of quasiconformality, Trans. Amer. Math. Soc. 84 (1957), 78-84. MR 18:646d
  • [Bo] BOJARSKI B. Generalized solutions of PDE system of the first order and elliptic type with discontinuous coefficients, Mat. Sb. 43 (85) (1957), 451-503 (Russian). MR 21:5058
  • [BJ1] BRAKALOVA M. AND J.A. JENKINS, On the local behavior of certain homeomorphisms, Kodai Math. J. 17 (1994), 201-213. MR 95h:30019
  • [BJ2] BRAKALOVA M. AND J.A. JENKINS, On the local behavior of certain homeomorphisms. II, Notes of Sci. Sem. of POMI 237 (1997), 11-20. MR 2000e:30034
  • [G] GEHRING F.W., $L^p$-integrability of the partial derivatives of quasiconformal mappings, Acta Math. 130 (1973), 265-277. MR 53:5861
  • [GR] GUTLYANSKII V.YA. AND RYAZANOV V.I., On the theory of local behavior of quasiconformal mappings, Izv. AN Rossii, Math. Ser., 59, no. 3 (1995), 31-58 (Russian). MR 96h:30025
  • [L] LEHTO O., On the differentiability of quasiconformal mappings with prescribed complex dilatation, Ann. Acad. Sci. Fenn. Ser. A I 275 (1960), 1-28. MR 23:A3260
  • [LV] LEHTO O., VIRTANEN K., Quasikonforme Abbildungen, Berlin etc., Springer-Verlag, 1965. MR 32:5872
  • [RW] REICH E. AND WALCZAK H., On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc. 117 (1965), 338-351. MR 31:345
  • [R1] RYAZANOV V.I., Criterion of differentiability by Belinskii and its consequences, Ukrain. Mat. Zh. 44 (1992), 289-294 (Russian); translation in Ukrainian Math. J. 44 (1992), no. 2, 254-258. MR 93g:30027
  • [R2] RYAZANOV V.I., Solution of the Reich-Walczak problem on conformality by Belinskii-Lavrent'ev, Ukrain. Mat. Zh. 44 (1992), 1406-1411 (Russian); translation in Ukrainian Math. J. 44 (1992), no. 10, 1292-1298. MR 94c:30025
  • [R3] RYAZANOV V.I., Convergence of characteristics of quasiconformal mappings, Ukrain. Mat. Zh. 38 (1986), 200-204, 269 (Russian). MR 87h:30046
  • [S] SAKS S., Theory of the integral, New York, Dover Publ. Inc., 1964. MR 29:4850
  • [Sh] SHABAT B.V., On generalized solutions of a PDE system, Mat. Sb. 17(59) (1945), 193-209 (Russian).
  • [T] TEICHMÜLLER O., Untersuchungen über konforme and quasikonforme Abbildungen, Deutsche Math. 3 (1938), 621-678.
  • [W] WITTICH H., Zum Beweis eines Satzes über quasikonforme Abbildungen, Math. Z. 51 (1948), 275-288. MR 10:241e

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Additional Information

Vladimir I. Ryazanov
Affiliation: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 340114, Donetsk, Ukraine
Address at time of publication: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN–00014 University of Helsinki, Finland

Matti K. Vuorinen
Affiliation: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN–00014 University of Helsinki, Finland

Keywords: Quasiconformal mappings, local behavior, conformality, asymptotical rotations
Received by editor(s): October 26, 1999
Received by editor(s) in revised form: March 9, 2000
Published electronically: April 9, 2001
Dedicated: Dedicated to Professor P.P. Belinskii
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2001 American Mathematical Society

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