Hollow quasi-Fatou components of quasiregular maps

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ${\mathbb{R}}^d$ is called hollow if it has a bounded complementary component. We show that for each $d\ge 2$ there exists a quasiregular map of transcendental type $f: $\mathbb{R}$^d \to $\mathbb{R}$^d$ with a quasi-Fatou component which is hollow. Suppose that $U$ is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if $U$ is bounded, then $U$ has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if $U$ is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if $J(f)$ has an isolated point, or if $J(f)$ is not equal to the boundary of the fast escaping set. Finally, we deduce that if $J(f)$ has a bounded component, then all components of $J(f)$ are bounded.