Hollow quasi-Fatou components of quasiregular maps

Nicks, Daniel A. and Sixsmith, David J. (2017) Hollow quasi-Fatou components of quasiregular maps. Mathematical Proceedings of the Cambridge Philosophical Society, 162 (3). pp. 561-574. ISSN 1469-8064

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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 Rd is called hollow if it has a bounded complementary component. We show that for each d≥2 there exists a quasiregular map of transcendental type f:Rd→Rd 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.

Item Type: Article
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1017/S0305004116000840
Related URLs:
Depositing User: Eprints, Support
Date Deposited: 04 Aug 2016 08:00
Last Modified: 01 May 2017 14:27
URI: http://eprints.nottingham.ac.uk/id/eprint/35696

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