Periodic domains of quasiregular maps

Nicks, Daniel A.; Sixsmith, David J.

Abstract

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.

Citation

Nicks, D. A., & Sixsmith, D. J. (2018). Periodic domains of quasiregular maps. Ergodic Theory and Dynamical Systems, 38(6), 2321-2344. https://doi.org/10.1017/etds.2016.116

• Journal Article Type: Article
• Acceptance Date: Aug 26, 2016
• Online Publication Date: Mar 14, 2017
• Publication Date: 2018-09
• Deposit Date: Sep 14, 2016
• Publicly Available Date: Mar 14, 2017
• Journal: Ergodic Theory and Dynamical Systems
• Print ISSN: 0143-3857
• Electronic ISSN: 1469-4417
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 38
• Issue: 6
• Pages: 2321-2344
• DOI: https://doi.org/10.1017/etds.2016.116
• Public URL: https://nottingham-repository.worktribe.com/output/850700
• Publisher URL: https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/div-classtitleperiodic-domains-of-quasiregular-mapsdiv/22A2C10D6415782810D8904515341212
• Related Public URLs: http://arxiv.org/abs/1509.06723
• Additional Information: © Cambridge University Press, 2017