On local injectivity and asymptotic linearity of quasiregular mappings

V. Gutlyanskiĭ; O. Martio; V. Ryazanov; M. Vuorinen

Studia Mathematica (1998)

  • Volume: 128, Issue: 3, page 243-271
  • ISSN: 0039-3223

Abstract

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It is shown that the approximate continuity of the dilatation matrix of a quasiregular mapping f at x 0 implies the local injectivity and the asymptotic linearity of f at x 0 . Sufficient conditions for l o g | f ( x ) - f ( x 0 ) | to behave asymptotically as l o g | x - x 0 | are given. Some global injectivity results are derived.

How to cite

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Gutlyanskiĭ, V., et al. "On local injectivity and asymptotic linearity of quasiregular mappings." Studia Mathematica 128.3 (1998): 243-271. <http://eudml.org/doc/216485>.

@article{Gutlyanskiĭ1998,
abstract = {It is shown that the approximate continuity of the dilatation matrix of a quasiregular mapping f at $x_0$ implies the local injectivity and the asymptotic linearity of f at $x_0$. Sufficient conditions for $log|f(x) - f(x_0)|$ to behave asymptotically as $log|x - x_0|$ are given. Some global injectivity results are derived.},
author = {Gutlyanskiĭ, V., Martio, O., Ryazanov, V., Vuorinen, M.},
journal = {Studia Mathematica},
keywords = {quasiregular mappings},
language = {eng},
number = {3},
pages = {243-271},
title = {On local injectivity and asymptotic linearity of quasiregular mappings},
url = {http://eudml.org/doc/216485},
volume = {128},
year = {1998},
}

TY - JOUR
AU - Gutlyanskiĭ, V.
AU - Martio, O.
AU - Ryazanov, V.
AU - Vuorinen, M.
TI - On local injectivity and asymptotic linearity of quasiregular mappings
JO - Studia Mathematica
PY - 1998
VL - 128
IS - 3
SP - 243
EP - 271
AB - It is shown that the approximate continuity of the dilatation matrix of a quasiregular mapping f at $x_0$ implies the local injectivity and the asymptotic linearity of f at $x_0$. Sufficient conditions for $log|f(x) - f(x_0)|$ to behave asymptotically as $log|x - x_0|$ are given. Some global injectivity results are derived.
LA - eng
KW - quasiregular mappings
UR - http://eudml.org/doc/216485
ER -

References

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  1. [A 1 ] L. Ahlfors, On a class of quasiconformal mappings, Österreich. Akad. Wiss. Math. Naturwiss. Kl. S. B. II 185 (1976), 5-10. Zbl0351.30019
  2. [A 2 ] L. Ahlfors, Quasiconformal deformations and mappings in R n , J. Analyse Math. 30 (1976), 74-97. Zbl0338.30017
  3. [B] P. P. Belinskiĭ, General Properties of Quasiconformal Mappings, Nauka, Novosibirsk, 1974 (in Russian). Zbl0281.30018
  4. [Bell] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1970. 
  5. [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in n , Ann. Acad. Sci. Fenn. Ser. A I 8 (1983), 257-324. Zbl0548.30016
  6. [BIK] B. Bojarski, T. Iwaniec and R. Kopiecki, Riemannian manifolds with non-smooth metric tensors and QC-maps, in: Monge-Ampère Equations and Related Topics (Firenze, 1980), Ist. Naz. Alta Mat., Roma, 1982, 123-167. 
  7. [D] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. 
  8. [Fer] J. Ferrand, Regularity of conformal mappings of Riemannian manifolds, in: Lecture Notes in Math. 743, Springer, 1979, 191-203. 
  9. [Fi] G. M. Fichtengolz, Differential- und Integralrechnung, Band 1, Deutscher Verlag Wiss., Berlin, 1964. 
  10. [FG] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon Press, Oxford, 1995. Zbl0852.47030
  11. [Gol] V. M. Gol'dshteĭn, On behavior of mappings with bounded distortion under distortion coefficient close to 1, Sibirsk. Mat. Zh. 12 (1971), 1250-1258 (in Russian). Zbl0231.30031
  12. [GMRV 1 ] V. Gutlyanskiĭ, O, Martio, V. Ryazanov and M. Vuorinen, On asymptotical behavior of quasiconformal mappings in the space, to appear. 
  13. [GMRV 2 ] V. Gutlyanskiĭ, O, Martio, V. Ryazanov and M. Vuorinen, Convergence theorems for quasiregular mappings in the space, Forum Math., to appear. 
  14. [H] P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed., Van Nostrand Reinhold, New York, 1958. Zbl0107.01404
  15. [Iw] T. Iwaniec, Regularity theorems for solutions of partial differential equations related to quasiregular mappings in several variables, Dissertationes Math. 198 (1982). 
  16. [LV] O. Lehto and K. Virtanen, Quasikonforme Abbildungen, Springer, Berlin, 1965. 
  17. [MRV 1 ] O. Martio, S. Rickman and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 448 (1969), 1-40. Zbl0189.09204
  18. [MRV 2 ] O. Martio, S. Rickman and J. Väisälä, Distortion and singularities of quasiregular mappings, ibid. 465 (1970), 1-13. 
  19. [MRV 3 ] O. Martio, S. Rickman and J. Väisälä, Topological and metric properties of quasiregular mappings, ibid. 488 (1971), 1-31. 
  20. [MSA] O. Martio and J. Sarvas, Injectivity theorems in plane and space, ibid. 4 (1978/1979), 383-401. Zbl0406.30013
  21. [MSR] O. Martio and U. Srebro, Universal radius of injectivity for locally quasiconformal mappings, Israel J. Math. 29 (1978), 17-23. Zbl0383.30009
  22. [O] A. Ostrowski, Solution of Equations and Systems of Equations, 2nd ed., Academic Press, New York, 1966. Zbl0222.65070
  23. [RR] T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin, 1955. Zbl0067.03506
  24. [Re 1 ] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monographs 73, Amer. Math. Soc., 1989. 
  25. [Re 2 ] Yu. G. Reshetnyak, Stability Theorems in Geometry and Analysis, Nauka, Novosibirsk, 1982 (in Russian). 
  26. [RS] S. Rickman and U. Srebro, Remarks on the local index of quasiregular mappings, J. Analyse Math. 46 (1986), 246-250. Zbl0603.30025
  27. [S] S. Saks, Theory of the Integral, Dover Publ., New York, 1964. Zbl1196.28001
  28. [Sar] J. Sarvas, Coefficient of injectivity for quasiregular mappings, Duke Math. J. 43 (1976), 147-158. Zbl0357.30016
  29. [V 1 ] J. Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I 392 (1966), 1-10. 
  30. [V 2 ] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math. 229, Springer, Berlin, 1971. Zbl0221.30031
  31. [Vu] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 1319, Springer, Berlin, 1988. 
  32. [Z] V. A. Zorich, M. A. Lavrent'ev's theorem on quasiconformal space mappings, Mat. Sb. 74 (1967), 417-433 (in Russian). 

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