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Hollow quasi-Fatou components of quasiregular maps

Nicks, Daniel A.; Sixsmith, David J.

Authors

David J. Sixsmith



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.

Journal Article Type Article
Publication Date May 31, 2017
Journal Mathematical Proceedings of the Cambridge Philosophical Society
Print ISSN 0305-0041
Electronic ISSN 1469-8064
Publisher Cambridge University Press (CUP)
Peer Reviewed Peer Reviewed
Volume 162
Issue 3
APA6 Citation Nicks, D. A., & Sixsmith, D. J. (2017). Hollow quasi-Fatou components of quasiregular maps. Mathematical Proceedings, 162(3), https://doi.org/10.1017/S0305004116000840
DOI https://doi.org/10.1017/S0305004116000840
Publisher URL https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/hollow-quasi-fatou-components-of-quasiregular-maps/B8F599AA67A1CCA4FB4D5CDE988AAC16
Related Public URLs http://arxiv.org/abs/1505.08114
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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