The escaping set of a quasiregular mapping

Bergweiler, Walter; Fletcher, Alastair; Langley, J. K.; Meyer, Janis

doi:10.1090/s0002-9939-08-09609-3

Cited by 25 publications

(43 citation statements)

References 17 publications

(55 reference statements)

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“…Note that [16,Lemma 3.4] states f (A(s n , t)) ⊃ A(s n , 2t) in place of (5). Our stronger statement is easily derived from the proof of [16,Lemma 3.4].…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 87%

“…Proof of Theorem We first use a technique from [, Section 6], (see also [, Section 4]), to define a quasiregular map

g : C \to C

of transcendental type which is equal to the identity in the upper half‐plane

double-struckH

. In particular we choose

δ > 0

small, and then set

g false(z false) : = \{\begin{matrix} left z, & left for prefixIm z ⩾ 0, \\ left z - δ (Im z) prefixexp (- z^{2}), & left for prefixIm z \in false[- 1, 0 false), \\ left z + δ prefixexp false(- z^{2} false), & left otherwise . \end{matrix}

It can be shown by a calculation that if

δ

is sufficiently small, then

g

is quasiregular.…”

Section: Examplesmentioning

confidence: 99%

“…The definitions of

I (f), B O (f)

and

B U (f)

can be modified in an obvious way to apply to quasiregular maps of

R^{d}

. In the quasiregular setting, the escaping set has been studied in , and the bounded orbit set in . Our goal in this paper is to study

B U (f)

in the case that

f

is quasiregular and of transcendental type.…”

Section: Introductionmentioning

confidence: 99%

“…Proof of Theorem 3. We first use a technique from [5,Section 6], (see also [15,Section 4]), to define a quasiregular map g : C → C of transcendental type which is equal to the identity in the upper half-plane H.…”

mentioning

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.

show abstract

“…Note that [16,Lemma 3.4] states f (A(s n , t)) ⊃ A(s n , 2t) in place of (5). Our stronger statement is easily derived from the proof of [16,Lemma 3.4].…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 87%

“…Proof of Theorem We first use a technique from [, Section 6], (see also [, Section 4]), to define a quasiregular map

g : C \to C

of transcendental type which is equal to the identity in the upper half‐plane

double-struckH

. In particular we choose

δ > 0

small, and then set

g false(z false) : = \{\begin{matrix} left z, & left for prefixIm z ⩾ 0, \\ left z - δ (Im z) prefixexp (- z^{2}), & left for prefixIm z \in false[- 1, 0 false), \\ left z + δ prefixexp false(- z^{2} false), & left otherwise . \end{matrix}

It can be shown by a calculation that if

δ

is sufficiently small, then

g

is quasiregular.…”

Section: Examplesmentioning

confidence: 99%

“…The definitions of

I (f), B O (f)

and

B U (f)

can be modified in an obvious way to apply to quasiregular maps of

R^{d}

. In the quasiregular setting, the escaping set has been studied in , and the bounded orbit set in . Our goal in this paper is to study

B U (f)

in the case that

f

is quasiregular and of transcendental type.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The bungee set in quasiregular dynamics

Nicks

Sixsmith

2018

Bull. London Math. Soc.

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show abstract

“…Here for x ∈R d = R d ∪ {∞}, we use the notation O − f (x) to denote the backward orbit of x, while we use O + f (x) to denote the forward orbit of x. Using similar techniques to those from [17] and [5], it has been possible to extend the slow escape result to the case of quasimeromorphic mappings of transcendental type with at least one pole. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

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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Iteration of Quasiregular Mappings

Bergweiler

2010

Comput. Methods Funct. Theory

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The escaping set of a quasiregular mapping

Cited by 25 publications

References 17 publications

The bungee set in quasiregular dynamics

The bungee set in quasiregular dynamics

On slow escaping and non-escaping points of quasimeromorphic mappings

Iteration of Quasiregular Mappings

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