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

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 $B^{n}, n\ge 3$.