Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Failure of the Denjoy theorem
for quasiregular maps in dimension $n\ge 3$


Authors: Ilkka Holopainen and Seppo Rickman
Journal: Proc. Amer. Math. Soc. 124 (1996), 1783-1788
MSC (1991): Primary 30C65
MathSciNet review: 1307528
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1929 L. V. Ahlfors proved the Denjoy conjecture which states that the order of an entire holomorphic function of the plane must be at least $k$ if the map has at least $2k$ finite asymptotic values. In this paper, we prove that the Denjoy theorem has no counterpart in the classical form for quasiregular maps in dimensions $n\ge 3$. We construct a quasiregular map of $\mathbb {R}^{n},\ n\ge 3,$ with a bounded order but with infinitely many asymptotic limits. Our method also gives a new construction for a counterexample of Lindelöf's theorem for quasiregular maps of $B^{n},\ n\ge 3$.


References [Enhancements On Off] (What's this?)

  • [1] Ahlfors, L.V., Über die asymptotischen Werte der ganzen Funktionen endlichen Ordnung, Ann. Acad. Sci. Fenn. Ser. A 32;6 (1929), 1--15.
  • [2] S. Granlund, P. Lindqvist, and O. Martio, 𝐹-harmonic measure in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 233–247. MR 686642, 10.5186/aasfm.1982.0717
  • [3] James A. Jenkins, On the Denjoy conjecture, Canad. J. Math. 10 (1958), 627–631. MR 0099409
  • [4] Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644
  • [5] Seppo Rickman, Asymptotic values and angular limits of quasiregular mappings of a ball, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 185–196. MR 595190, 10.5186/aasfm.1980.0523
  • [6] Rickman, S., Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 26, Springer--Verlag, Berlin Heidelberg New York, 1993. CMP 94:01
  • [7] S. Rickman and M. Vuorinen, On the order of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 221–231. MR 686641, 10.5186/aasfm.1982.0727
  • [8] Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30C65

Retrieve articles in all journals with MSC (1991): 30C65


Additional Information

Ilkka Holopainen
Affiliation: Department of Mathematics, P.O. Box 4 (Hallituskatu 15), FIN-00014 University of Helsinki, Finland

Seppo Rickman
Affiliation: Department of Mathematics, P.O. Box 4 (Hallituskatu 15), FIN-00014 University of Helsinki, Finland
Email: ih@geom.helsinki.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03181-4
Keywords: Quasiregular maps, Denjoy theorem, Lindel\"{o}f's theorem
Received by editor(s): May 2, 1994
Received by editor(s) in revised form: November 18, 1994
Additional Notes: Supported in part by the EU HCM contract No. CHRX-CT92-0071.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1996 American Mathematical Society