Periodic domains of quasiregular maps
We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an addition...
| Main Authors: | , |
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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/36602/ |
| _version_ | 1848795313122312192 |
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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 consider the iteration of quasiregular maps of transcendental type from Rd to Rd. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function.
We construct a quasiregular map of transcendental type from R3 to R3 with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from R3 to R3 which is equal to the identity map in a half-space. |
| first_indexed | 2025-11-14T19:30:06Z |
| format | Article |
| id | nottingham-36602 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:30:06Z |
| publishDate | 2017 |
| publisher | Cambridge University Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-366022020-05-04T18:37:53Z https://eprints.nottingham.ac.uk/36602/ Periodic domains of quasiregular maps Nicks, Daniel A. Sixsmith, David J. We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from R3 to R3 with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from R3 to R3 which is equal to the identity map in a half-space. Cambridge University Press 2017-03-14 Article PeerReviewed Nicks, Daniel A. and Sixsmith, David J. (2017) Periodic domains of quasiregular maps. Ergodic Theory and Dynamical Systems . ISSN 1469-4417 https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/div-classtitleperiodic-domains-of-quasiregular-mapsdiv/22A2C10D6415782810D8904515341212 doi:10.1017/etds.2016.116 doi:10.1017/etds.2016.116 |
| spellingShingle | Nicks, Daniel A. Sixsmith, David J. Periodic domains of quasiregular maps |
| title | Periodic domains of quasiregular maps |
| title_full | Periodic domains of quasiregular maps |
| title_fullStr | Periodic domains of quasiregular maps |
| title_full_unstemmed | Periodic domains of quasiregular maps |
| title_short | Periodic domains of quasiregular maps |
| title_sort | periodic domains of quasiregular maps |
| url | https://eprints.nottingham.ac.uk/36602/ https://eprints.nottingham.ac.uk/36602/ https://eprints.nottingham.ac.uk/36602/ |