Slow escaping points of quasiregular mappings
This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for trans...
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| Format: | Article |
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Springer
2016
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| Online Access: | https://eprints.nottingham.ac.uk/35154/ |
| _version_ | 1848795015300513792 |
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| author | Nicks, Daniel A. |
| author_facet | Nicks, Daniel A. |
| author_sort | Nicks, Daniel A. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f). |
| first_indexed | 2025-11-14T19:25:22Z |
| format | Article |
| id | nottingham-35154 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:25:22Z |
| publishDate | 2016 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-351542020-05-04T17:52:36Z https://eprints.nottingham.ac.uk/35154/ Slow escaping points of quasiregular mappings Nicks, Daniel A. This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f). Springer 2016-05-09 Article PeerReviewed Nicks, Daniel A. (2016) Slow escaping points of quasiregular mappings. Mathematische Zeitschrift . ISSN 1432-1823 http://link.springer.com/article/10.1007%2Fs00209-016-1687-9 doi:10.1007/s00209-016-1687-9 doi:10.1007/s00209-016-1687-9 |
| spellingShingle | Nicks, Daniel A. Slow escaping points of quasiregular mappings |
| title | Slow escaping points of quasiregular mappings |
| title_full | Slow escaping points of quasiregular mappings |
| title_fullStr | Slow escaping points of quasiregular mappings |
| title_full_unstemmed | Slow escaping points of quasiregular mappings |
| title_short | Slow escaping points of quasiregular mappings |
| title_sort | slow escaping points of quasiregular mappings |
| url | https://eprints.nottingham.ac.uk/35154/ https://eprints.nottingham.ac.uk/35154/ https://eprints.nottingham.ac.uk/35154/ |