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Hollow quasi-Fatou components of quasiregular maps

Published online by Cambridge University Press:  23 September 2016

DANIEL A. NICKS  and
DAVID J. SIXSMITH
DANIEL A. NICKS
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD. e-mails: Dan.Nicks@nottingham.ac.uk; David.Sixsmith@open.ac.uk
DAVID J. SIXSMITH
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD. e-mails: Dan.Nicks@nottingham.ac.uk; David.Sixsmith@open.ac.uk
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 3 , May 2017 , pp. 561 - 574
Copyright
Copyright © Cambridge Philosophical Society 2016 

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