Semihyperbolic entire functions

Bergweiler, Walter; Morosawa, Shunsuke

doi:10.1088/0951-7715/15/5/316

Cited by 13 publications

(12 citation statements)

References 17 publications

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“…Many results on iterations of rational functions have been generalized to functions in class S. For example, Sullivan's well-known theorem that a rational function has no wandering domains has been extended to transcendental meromorphic functions in class S (see [10]). In this paper, we extend Qiao's result to functions in class S by proving the following: Examples of functions with locally connected Julia sets can be found in [15], [25] and [30]. In particular, in [30] and [25], examples of class S functions are constructed in such a way that their Julia sets are Sierpinski curves.…”

Section: Motivations and Resultsmentioning

confidence: 86%

Residual Julia Sets of Meromorphic Functions

Zheng²,

Choi

2006

Math. Proc. Camb. Phil. Soc.

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In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function f belongs to the class S and its Julia set is locally connected, then the residual Julia set of f is empty if and only if its Fatou set F (f) has a completely invariant component or consists of only two components. We also show that if f is a meromorphic function which is not of the form α + (z − α) −k e g(z) , where k is a natural number, α is a complex number and g is an entire function, then f has buried components provided that f has no completely invariant components and its Julia set J(f) is disconnected. Moreover, if F (f) has an infinitely connected component, then the singleton buried components are dense in J(f). This generalizes a result of Baker and Domínguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.

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Section: Motivations and Resultsmentioning

confidence: 86%

Residual Julia Sets of Meromorphic Functions

Zheng²,

Choi

2006

Math. Proc. Camb. Phil. Soc.

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“…The example is based on work by Domínguez and Fagella [16], though they did not discuss local connectedness. Under certain conditions, the Julia set is also locally connected for the class of semihyperbolic entire functions investigated by Bergweiler and Morosawa in [10].…”

Section: Now the Functionmentioning

confidence: 96%

“…Bergweiler and Morosawa's result on local connectedness is the following. Lemma 6.5 (Theorem 4 in [10]). Let f be entire.…”

Section: Now the Functionmentioning

confidence: 99%

“…Rational maps with locally connected Julia sets have been much studied, and several classes of functions are known for which, if the Julia set is connected, then it is also locally connected -see, for example, [22,Chapter 19] and [13,20,31]. Some analogous results have been obtained for transcendental entire functions [10,23], but the situation is less well understood. More details are given in Section 6.…”

Section: Introductionmentioning

confidence: 99%

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

Osborne¹

2012

Ergod. Th. Dynam. Sys.

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Abstract. We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.

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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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Semihyperbolic entire functions

Cited by 13 publications

References 17 publications

Residual Julia Sets of Meromorphic Functions

Residual Julia Sets of Meromorphic Functions

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

Hyperbolic entire functions with bounded Fatou components

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