Failure of the Denjoy theorem
for quasiregular maps in dimension
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 | 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 if the map has at least
finite asymptotic values. In this paper, we prove that the Denjoy theorem has no counterpart in the classical form for quasiregular maps in dimensions
. We construct a quasiregular map of
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
.
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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