Hollow quasi-Fatou components of quasiregular maps

Abstract

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.

References in corpus (4)

• The escaping set of a quasiregular mapping
• The size and topology of quasi-Fatou components of quasiregular maps
• Hollow quasi-Fatou components of quasiregular maps
• Fixed points of composite entire and quasiregular maps

Cited by in corpus (5)

• The size and topology of quasi-Fatou components of quasiregular maps
• Hollow quasi-Fatou components of quasiregular maps
• The dynamics of quasiregular maps of punctured space
• The escaping set in transcendental dynamics
• Permutable Quasiregular Maps