Hollow quasi-Fatou components of quasiregular maps

TL;DR

This paper explores the existence and properties of hollow quasi-Fatou components in quasiregular maps of transcendental type across various dimensions, revealing their topological characteristics and relation to Julia sets.

Contribution

It introduces the concept of hollow quasi-Fatou components in quasiregular maps and analyzes their properties, including boundedness, invariance, and boundary behavior, extending complex dynamics concepts to higher dimensions.

Findings

Existence of hollow quasi-Fatou components in all dimensions $d \,\geq\, 2$.

Bounded hollow components share properties with multiply connected Fatou components.

Unbounded hollow components are completely invariant with no unbounded boundary components.

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 $d \geq 2$ 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.