  The Fatou-Julia iteration theory of rational and transcendental entire
functions has recently been extended to quasiregular maps in more than two real
dimensions. Our goal in this paper is similar; we extend the iteration theory
of analytic self-maps of the punctured plane to quasiregular self-maps of
punctured space.
  We define the Julia set as the set of points for which the complement of the
forward orbit of any neighbourhood of the point is a finite set. We show that
the Julia set is non-empty, and shares many properties with the classical Julia
set of an analytic function. These properties are stronger than those known to
hold for the Julia set of a general quasiregular map of space.
  We define the quasi-Fatou set as the complement of the Julia set, and
generalise a result of Baker concerning the topological properties of the
components of this set. A key tool in the proof of these results is a version
of the fast escaping set. We generalise various results of Marti-Pete
concerning this set, for example showing that the Julia set is equal to the
boundary of the fast escaping set.
