FAILURE OF THE DENJOY THEOREM FOR QUASI-REGULAR MAPS IN DIMENSION N-GREATER-THAN-OR-EQUAL-TO-3

In 1929 L. V. Ahlfors proved the Denjoy conjecture which states that t he order of an entire holomorphic function of the plane must be at lea st k if the map has at least 2k finite asymptotic values. In this pape r, we prove that the Denjoy theorem has no counterpart in the classica l form for quasiregular maps in dimensions n greater than or equal to 3. We construct a quasiregular map of R(n), n greater than or equal to 3, with a bounded order but with infinitely many asymptotic limits. O ur method also gives a new construction for a counterexample of Lindel of's theorem for quasiregular maps of B-n, n greater than or equal to 3.