The dynamics of quasiregular maps of punctured space
The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps...
| Main Authors: | , |
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| Format: | Article |
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Indiana University Mathematics Journal
2019
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| Online Access: | https://eprints.nottingham.ac.uk/42305/ |
| _version_ | 1848796457859022848 |
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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 | The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set. |
| first_indexed | 2025-11-14T19:48:18Z |
| format | Article |
| id | nottingham-42305 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:48:18Z |
| publishDate | 2019 |
| publisher | Indiana University Mathematics Journal |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-423052020-05-04T18:37:04Z https://eprints.nottingham.ac.uk/42305/ The dynamics of quasiregular maps of punctured space Nicks, Daniel A. Sixsmith, David J. The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set. Indiana University Mathematics Journal 2019 Article PeerReviewed Nicks, Daniel A. and Sixsmith, David J. (2019) The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal, 68 (1). pp. 323-352. ISSN 0022-2518 doi:10.1512/iumj.2019.68.7556 doi:10.1512/iumj.2019.68.7556 |
| spellingShingle | Nicks, Daniel A. Sixsmith, David J. The dynamics of quasiregular maps of punctured space |
| title | The dynamics of quasiregular maps of punctured space |
| title_full | The dynamics of quasiregular maps of punctured space |
| title_fullStr | The dynamics of quasiregular maps of punctured space |
| title_full_unstemmed | The dynamics of quasiregular maps of punctured space |
| title_short | The dynamics of quasiregular maps of punctured space |
| title_sort | dynamics of quasiregular maps of punctured space |
| url | https://eprints.nottingham.ac.uk/42305/ https://eprints.nottingham.ac.uk/42305/ |