Hollow quasi-Fatou components of quasiregular maps
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-Fato...
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| Format: | Article |
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Cambridge University Press
2017
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| Online Access: | https://eprints.nottingham.ac.uk/35696/ |
| _version_ | 1848795140210032640 |
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| author | Nicks, Daniel A. Sixsmith, David J. |
| author_facet | Nicks, Daniel A. Sixsmith, David J. |
| author_sort | Nicks, Daniel A. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:27:21Z |
| format | Article |
| id | nottingham-35696 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:27:21Z |
| publishDate | 2017 |
| publisher | Cambridge University Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-356962020-05-04T18:48:34Z https://eprints.nottingham.ac.uk/35696/ Hollow quasi-Fatou components of quasiregular maps Nicks, Daniel A. Sixsmith, David J. 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. Cambridge University Press 2017-05-31 Article PeerReviewed 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 https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/hollow-quasi-fatou-components-of-quasiregular-maps/B8F599AA67A1CCA4FB4D5CDE988AAC16 doi:10.1017/S0305004116000840 doi:10.1017/S0305004116000840 |
| spellingShingle | Nicks, Daniel A. Sixsmith, David J. Hollow quasi-Fatou components of quasiregular maps |
| title | Hollow quasi-Fatou components of quasiregular maps |
| title_full | Hollow quasi-Fatou components of quasiregular maps |
| title_fullStr | Hollow quasi-Fatou components of quasiregular maps |
| title_full_unstemmed | Hollow quasi-Fatou components of quasiregular maps |
| title_short | Hollow quasi-Fatou components of quasiregular maps |
| title_sort | hollow quasi-fatou components of quasiregular maps |
| url | https://eprints.nottingham.ac.uk/35696/ https://eprints.nottingham.ac.uk/35696/ https://eprints.nottingham.ac.uk/35696/ |