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Periodic domains of quasiregular maps

Published online by Cambridge University Press:  14 March 2017

DANIEL A. NICKS Open the ORCID record for DANIEL A. NICKS [Opens in a new window]  and
DAVID J. SIXSMITH
DANIEL A. NICKS
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email [email protected], [email protected]
DAVID J. SIXSMITH
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email [email protected], [email protected]
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Abstract

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. 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 the 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 $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ 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 bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2321 - 2344
Copyright
© Cambridge University Press, 2017 

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