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

Abstract

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 ≥ 3. We construct a quasiregular map of R n , n ≥ 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 ≥ 3.