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

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- by Ilkka Holopainen and Seppo Rickman PDF
- Proc. Amer. Math. Soc.
**124**(1996), 1783-1788 Request permission

## 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

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## Additional Information

**Ilkka Holopainen**- Affiliation: Department of Mathematics, P.O. Box 4 (Hallituskatu 15), FIN-00014 University of Helsinki, Finland
- MR Author ID: 290418
**Seppo Rickman**- Affiliation: Department of Mathematics, P.O. Box 4 (Hallituskatu 15), FIN-00014 University of Helsinki, Finland
- Email: ih@geom.helsinki.fi
- 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
- © Copyright 1996 American Mathematical Society
- 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