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Slow escaping points of quasiregular mappings

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Abstract

This article concerns the iteration of quasiregular mappings on \(\mathbb {R}^d\) and entire functions on \(\mathbb {C}\). 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 \(f:\mathbb {R}^d\rightarrow \mathbb {R}^d\) 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 \(f^n\) 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 \(|f^n(x)|\) is asymptotic to the iterated maximum modulus \(M^{n}(R,f)\).

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Acknowledgments

The author thanks Phil Rippon and Gwyneth Stallard for helpful discussions about their paper [22] and the possibility of extending slow escape results to the quasiregular setting.

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Correspondence to Daniel A. Nicks.

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The author was supported by Engineering and Physical Sciences Research Council Grant EP/L019841/1.

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Nicks, D.A. Slow escaping points of quasiregular mappings. Math. Z. 284, 1053–1071 (2016). https://doi.org/10.1007/s00209-016-1687-9

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  • DOI: https://doi.org/10.1007/s00209-016-1687-9

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