Periodic domains of quasiregular maps

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Nicks, Daniel A. and Sixsmith, David J. (2017) Periodic domains of quasiregular maps. Ergodic Theory and Dynamical Systems . ISSN 1469-4417

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Official URL: https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/div-classtitleperiodic-domains-of-quasiregular-mapsdiv/22A2C10D6415782810D8904515341212

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.

Item Type:

Article

RIS ID:

https://nottingham-repository.worktribe.com/output/850700

Additional Information:

Schools/Departments:

University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences

Identification Number:

https://doi.org/10.1017/etds.2016.116

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