Hyperbolic entire functions with bounded Fatou components

Bergweiler, Walter; Fagella, Núria; Rempe, Lasse

doi:10.4171/cmh/371

Cited by 32 publications

(50 citation statements)

References 36 publications

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“…As with the previous lemma, the proof is basically a picture; see Figure 6. Suppose J ∈ J and let R = J × [1,2]. The exponential map sends R to the annulus identifying complex conjugate points).…”

Section: Lemma 71 (Simple Folding)mentioning

confidence: 99%

“…The exponential function maps the rectangle [1,2] × J conformally to the slit annulus {e < |z| < e 2 } \ [e, e 2 ]. The map φ is chosen to map the annulus A={e < |z| < e 2 } to the slit disk {|z| < e 2 } \ [−e, e] so that it equals the identity on {|z| = e 2 } and equals 1 2 (z + e 2 z ) on {|z| = e}.…”

Section: Lemma 71 (Simple Folding)mentioning

confidence: 99%

“…This is a piecewise linear map that sends [ρ/2, ρ] to [ρ/2, 1] and sends [ρ, 2ρ] to [1,2]. The slope on both intervals is less than 2/ρ.…”

Section: Introductionmentioning

confidence: 99%

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Models for the Eremenko–Lyubich class

Bishop

2015

J. London Math. Soc.

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Abstract. If f is in the Eremenko-Lyubich class B (transcendental entire functions with bounded singular set) then Ω = {z : |f (z)| > R} and f | Ω must satisfy certain simple topological conditions when R is sufficiently large. A model (Ω, F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko-Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.

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Section: Lemma 71 (Simple Folding)mentioning

confidence: 99%

Section: Lemma 71 (Simple Folding)mentioning

confidence: 99%

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Models for the Eremenko–Lyubich class

Bishop

2015

J. London Math. Soc.

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“…However, unlike disjoint type functions, hyperbolic functions may have more than one immediate attracting basin. An example from [BFRG15] is the map Proof. Let W ⊃ J(f ) be the domain in the definition of a hyperbolic map.…”

Section: Hyperbolic Functionsmentioning

confidence: 99%

“…We also note, in passing, a technique due to MacLane and Vinberg, which can be used to construct transcendental entire functions with a pre-assigned sequence of real critical values. This technique was used in [BFRG15] to construct hyperbolic functions with certain dynamical properties.…”

Section: Constructing Functions In Class S and Class Bmentioning

confidence: 99%

Dynamics in the Eremenko-Lyubich class

Sixsmith¹

2018

Conform. Geom. Dyn.

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The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials -in other words transcendental entire functions -is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics.In this paper we survey the dynamics of functions in the Eremenko-Lyubich class, B. Among transcendental entire functions, those in this class have properties that make their dynamics most "similar" to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of class B functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases.We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry.

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On Connected Preimages of Simply-Connected Domains Under Entire Functions

Rempe

Sixsmith

2019

Geom. Funct. Anal.

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Let f be a transcendental entire function, and let U, V ⊂ C be disjoint simply-connected domains. Must one of f −1 (U ) and f −1 (V ) be disconnected?In 1970, Baker [Bak70] implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains.It was recently observed by Julien Duval that there is a flaw in Baker's argument (which has also been used in later generalisations and extensions of Baker's result). We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, for the function f (z) = e z + z, there is a collection of infinitely many pairwise disjoint simply-connected domains, each with connected preimage. We also answer a long-standing question of Eremenko by giving an example of a transcendental meromorphic function, with infinitely many poles, which has the same property.Furthermore, we show that there exists a function f with the above properties such that additionally the set of singular values S(f ) is bounded; in other words, f belongs to the Eremenko-Lyubich class. On the other hand, if S(f ) is finite (or if certain additional hypotheses are imposed), many of the original results do hold.For the convenience of the research community, we also include a description of the error in the proof of [Bak70], and a summary of other papers that are affected.

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Hyperbolic entire functions with bounded Fatou components

Cited by 32 publications

References 36 publications

Models for the Eremenko–Lyubich class

Models for the Eremenko–Lyubich class

Dynamics in the Eremenko-Lyubich class

On Connected Preimages of Simply-Connected Domains Under Entire Functions

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