Spiders’ webs and locally connected Julia sets of transcendental entire functions

Osborne, John W.

doi:10.1017/s0143385712000259

Cited by 16 publications

(12 citation statements)

References 29 publications

(68 reference statements)

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“…It is known that the escaping, fast escaping, and even Julia sets of many transcendental entire functions are spiders' webs [14]. We note that the spiders' webs that arise in complex dynamics are extremely elaborate [12,14].…”

Section: Introductionmentioning

confidence: 84%

Cantor bouquets in spiders’ webs

Dourekas

2020

Conform. Geom. Dyn.

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For many transcendental entire functions, the escaping set has the structure of a Cantor bouquet, consisting of uncountably many disjoint curves. Rippon and Stallard showed that there are many functions for which the escaping set has a new connected structure known as an infinite spider's web. We investigate a connection between these two topological structures for a certain class of sums of exponentials.Theorem 1.2. Let f ∈ F . Then J(f ) is a spider's web that contains a Cantor bouquet. Additionally, the curves minus the endpoints lie in A(f ).Remark. The result holds for functions of the form λf , where f ∈ F and λ > 0. The reasoning is similar but for simplicity we have given the proof for the case λ = 1. Additionally, when p is even, f is an even function, and the result holds for negative λ as well. PreliminariesThis section contains some of the preliminary results we will use, many taken from [17] with slight modifications in order to make their purpose clearer for the requirements of this paper. We also prove a lemma that follows as a corollary from these results.

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Section: Introductionmentioning

confidence: 84%

Cantor bouquets in spiders’ webs

Dourekas

2020

Conform. Geom. Dyn.

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“…Local connectivity of Julia sets of transcendental entire functions has also been studied by a number of authors (see e.g. [Mor99,BM02,Osb13]). This problem is closely connected to the boundedness of Fatou components, mentioned above.…”

Section: Local Connectivity Of Julia Setsmentioning

confidence: 99%

Hyperbolic entire functions with bounded Fatou components

Bergweiler¹,

Fagella²,

Rempe³

2015

Comment. Math. Helv.

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We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values.

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“…For examples, the cases of hyperbolic [Mor99], [BFR15], semi-hyperbolic [BM02], strongly geometrically finite [ARS20] and strongly postcritically separated [Par21]. See also [BD00], [Mih12] and [Osb13] for related results. Our second aim in this paper is to study the local connectivity of the Julia sets of some transcendental entire functions containing Siegel disks.…”

Section: Introductionmentioning

confidence: 99%

Local connectivity of Julia sets for rational maps with Siegel disks

Wang¹,

Yang²,

Zhang³

et al. 2021

Preprint

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We prove that the Julia sets of a number of rational maps and transcendental entire functions with bounded type Siegel disks are locally connected. This is based on establishing an expanding property of a long iteration of a class of quasi-Blaschke models near the unit circle. Contents 1. Introduction 1 2. Quasi-Blaschke models and half hyperbolic neighborhoods 6 3. Admissible sequences and contraction regions 14 4. Proof of the Key Lemma 21 5. Proofs of Theorems A and B 25 References 31

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Spiders’ webs and locally connected Julia sets of transcendental entire functions

Cited by 16 publications

References 29 publications

Cantor bouquets in spiders’ webs

Cantor bouquets in spiders’ webs

Hyperbolic entire functions with bounded Fatou components

Local connectivity of Julia sets for rational maps with Siegel disks

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