The fast escaping set for quasiregular mappings

Bergweiler, Walter; Drasin, David; Fletcher, Alastair

doi:10.1007/s13324-014-0078-9

Cited by 20 publications

(37 citation statements)

References 24 publications

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“…This was first defined, for a transcendental entire function, in [11], and a detailed study of this set was given in [34]. See also [8,10], which studied the fast escaping set of a quasiregular map of R d of transcendental type. When f is a function defined on C or R d , the fast escaping set is roughly the set of points x for which |f n (x)| eventually grows faster than some iterated maximum modulus.…”

Section: The Fast Escaping Setmentioning

confidence: 99%

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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Section: The Fast Escaping Setmentioning

confidence: 99%

The dynamics of quasiregular maps of punctured space

Nicks¹,

Sixsmith²

2019

Indiana Univ. Math. J.

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“…Now, the set

A (f)

contains continua [, Theorem 1.2], and so has positive capacity. Moreover, the complement of

A (f)

contains

B O (f)

, and so also has positive capacity [, Theorem 1.4].…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…• The action of φ on S 3 1 is defined by first rotating by one half-turn, and then translating so that the upper boundary of the image of S 3 1 coincides with the right-hand lower boundary of φ(S 2 1 ). • The action of φ on S 4 1 is defined as follows, and is very similar to the action on S 2 1 . First translate S 4 1 so that its bottom left corner lies at the origin.…”

Section: Note Thatmentioning

confidence: 99%

“…• The action of φ on S 4 1 is defined as follows, and is very similar to the action on S 2 1 . First translate S 4 1 so that its bottom left corner lies at the origin. Then enlarge it by a scale factor of (s n + 2t n + 4/(s n + t n ))/2 to obtain a subset of A.…”

Section: Note Thatmentioning

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%

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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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