Math. Proc. Cambridge Philos. Soc. 162 (2017), no. 3, 561–574. Also available on the arXiv. This is a joint work with Dan Nicks. We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in is called hollow if it has a bounded complementary component. We show that for each
there exists a quasiregular map of transcendental type
with a quasi-Fatou component which is hollow.
Suppose that is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if
is bounded, then
has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if
is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if
has an isolated point, or if
is not equal to the boundary of the fast escaping set. Finally, we deduce that if
has a bounded component, then all components of
are bounded.