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.