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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
DOI: https://doi.org/10.1090/S0002-9939-96-03181-4
MathSciNet review: 1307528
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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?)

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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: https://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

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