On the iteration of quasimeromorphic mappings

Warren, Luke

doi:10.1017/s030500411800052x

Cited by 4 publications

(9 citation statements)

References 18 publications

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“…There, the analysis of the Julia set also requires different techniques based on the cardinality of the backward orbit of infinity. The example constructed in Corollary 1.2 therefore shows that such maps exist, justifying the necessity for the different techniques used in [17].…”

Section: Introductionmentioning

confidence: 84%

“…When studying the dynamical behaviour of transcendental meromorphic functions on C, different techniques are used based on whether the backward orbit of infinity is finite or infinite. Recently, the Julia set for quasimeromorphic mappings of transcendental type with at least one pole has been investigated in [17]. There, the analysis of the Julia set also requires different techniques based on the cardinality of the backward orbit of infinity.…”

Section: Introductionmentioning

confidence: 99%

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Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3

Warren

2019

Preprint

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We construct a quasiregular analogue of the function z exp(z) in dimension 3, which gives the first explicit example of a quasiregular mapping of transcendental type that has exactly one zero. We then modify the construction to create a family of such quasiregular mappings and study their dynamics. From this, we also construct the first quasimeromorphic mappings with an essential singularity at infinity where the backward orbit of infinity is non-empty and finite.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3

Warren

2019

Preprint

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“…By defining the Julia set directly using the expansion property in (1.1), it has been possible to study analogues of the Fatou-Julia theory in the new setting. Recently, the Julia set for quasimeromorphic mappings of transcendental type with at least one pole has been successfully established in [37]; here, it was shown that many of the usual properties of the Julia set analogously hold as well. These are summarised below.…”

Section: Julia Set Of Quasimeromorphic Mappingsmentioning

confidence: 95%

“…For a quasimeromorphic mapping of transcendental type with at least one pole, a strong relationship between points with finite backward orbits and capacity was established in [37].…”

Section: Capacity Of a Condensermentioning

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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“…We also want to define the Julia set for a quasimeromorphic map of transcendental type with at least one pole, f : R d → R d . This was done by Warren in [40] where he defined…”

Section: Background On Quasiregular Maps and Capacitymentioning

confidence: 99%

Permutable quasiregular maps

Tsantaris¹

2021

Math. Proc. Camb. Phil. Soc.

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Let f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$ . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$ , where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

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On the iteration of quasimeromorphic mappings

Cited by 4 publications

References 18 publications

Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3

Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3

On slow escaping and non-escaping points of quasimeromorphic mappings

Permutable quasiregular maps

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