Slow escaping points of quasiregular mappings

Nicks, Daniel A.

doi:10.1007/s00209-016-1687-9

Cited by 4 publications

(5 citation statements)

References 30 publications

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“…The final result in this subsection concerns continuous functions, and is almost identical to [24,Lemma 2.6]; see also [33, Lemma 1]. The proof is omitted.…”

Section: Topological Prerequisitesmentioning

confidence: 61%

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The dynamics of quasiregular maps of punctured space

Nicks¹,

Sixsmith²

2019

Indiana Univ. Math. J.

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The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space.We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space.We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Martí-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.

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“…The final result in this subsection concerns continuous functions, and is almost identical to [24,Lemma 2.6]; see also [33, Lemma 1]. The proof is omitted.…”

Section: Topological Prerequisitesmentioning

confidence: 61%

“…The next proposition is analogous to [24,Lemma 5.1]. Here if U ⊂ R d and r > 0, then we define rU := {rx : x ∈ U }.…”

Section: 5mentioning

confidence: 96%

The dynamics of quasiregular maps of punctured space

Nicks¹,

Sixsmith²

2019

Indiana Univ. Math. J.

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“…We need the following, which is taken from [, Lemma 3.3 and Lemma 3.4]. Here a quasiregular map

f : {double-struckR}^{d} \to {double-struckR}^{d}

of transcendental type has the pits effect if there exists

n \in N

such that, for all

c > 1

and

ε > 0

, there exists

r_{0}

such that if

r > r_{0}

, then the set

{x \in {double-struckR}^{d} : r ⩽ | x | ⩽ c r, | f false(x false) | ⩽ 1}

can be covered by

n

balls of radius

ε r

.…”

Section: Proof Of Theorems Andmentioning

confidence: 99%

“…Note that [, Lemma 3.4] states

f false(A (s_{n}, t) false) \supset A false(s_{n}, 2 t false)

in place of . Our stronger statement is easily derived from the proof of [, Lemma 3.4].…”

Section: Proof Of Theorems Andmentioning

confidence: 99%

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The bungee set in quasiregular dynamics

Nicks

Sixsmith

2018

Bull. London Math. Soc.

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In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non‐empty bungee set; this behaviour is impossible for an analytic homeomorphism.

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On slow escaping and non-escaping points of quasimeromorphic mappings

Warren¹

2020

Ergod. Th. Dynam. Sys.

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We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

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Slow escaping points of quasiregular mappings

Cited by 4 publications

References 30 publications

The dynamics of quasiregular maps of punctured space

The dynamics of quasiregular maps of punctured space

The bungee set in quasiregular dynamics

On slow escaping and non-escaping points of quasimeromorphic mappings

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