On the set where the iterates of an entire function are neither escaping nor bounded

Osborne, John W.; Sixsmith, David J.

doi:10.5186/aasfm.2016.4134

Cited by 35 publications

(36 citation statements)

References 45 publications

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“…The main question that we sought to address in this paper was the following. The function f (z) = e z has the property that I(f ) is connected and there is an unbounded connected set in the complement of I(f ); see [19,Example 2]. Note that in our proof of Theorem 1.1 we make strong use of the fact that the unbounded connected set Γ in I(f ) c is closed.…”

Section: Open Questionsmentioning

confidence: 99%

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Eremenko points and the structure of the escaping set

Rippon

Stallard

2019

Trans. Amer. Math. Soc.

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Much recent work on the iterates of a transcendental entire function f has been motivated by Eremenko's conjecture that all the components of the escaping set I(f ) are unbounded. Here we show that if I(f ) is disconnected, then the set I(f ) \ D has uncountably many unbounded components for any open disc D that meets the Julia set of f . For the set A R (f ), which is the 'core' of the fast escaping set, we prove the much stronger result that for some R > 0 either A R (f ) is connected and has the structure of an infinite spider's web or it has uncountably many components each of which is unbounded. There are analogous results for the intersections of these sets with the Julia set when no multiply connected wandering domains are present, but strikingly different results when they are present. In proving these, we obtain the unexpected result that multiply connected wandering domains can have complementary components with no interior, indeed uncountably many.

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Section: Open Questionsmentioning

confidence: 99%

“…There also exist entire functions for which A R (f ) and A(f ) are disconnected but I(f ) is connected, such as Fatou's function f (z) = z + 1 + e −z [12, Exemple 1]. Note that for Fatou's function I(f ) is a spider's web [10], whereas for f (z) = e z it is not [19,Example 2].…”

Section: Introductionmentioning

confidence: 99%

Eremenko points and the structure of the escaping set

Rippon

Stallard

2019

Trans. Amer. Math. Soc.

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“…Quasiconformal folding is a method of associating entire functions to certain infinite planar graphs introduced in [2], and was applied there to construct various new examples, such as a wandering domain for an entire function in the Eremenko-Lyubich class. Other applications have been given by Fagella, Godillon and Jarque [7], Lazebnik [9], Osborne and Sixsmith [11], and Rempe-Gillen [12]. We will review the basic folding construction in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Prescribing the postsingular dynamics of meromorphic functions

Bishop

Lazebnik

2019

Math. Ann.

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We show that any dynamics on any discrete planar sequence S can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of S. This work is motivated by an analogous result of DeMarco, Koch and McMullen in [5] for finite S in the rational setting. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set of f and the geometry of f , using the Folding Theorem of [2] and a classical fixpoint theorem [14]. arXiv:1807.04581v2 [math.CV]

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“…The set

B O (f)

for a transcendental entire function

f

was studied in [, ]. If

f

is transcendental, then

B U (f)

is non‐empty; indeed the Hausdorff dimension of

B U (f) \cap J (f)

is greater than zero [, Theorem 5.1]. The properties of

B U (f)

were studied in and subsequently in .…”

Section: Introductionmentioning

confidence: 99%

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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On the set where the iterates of an entire function are neither escaping nor bounded

Cited by 35 publications

References 45 publications

Eremenko points and the structure of the escaping set

Eremenko points and the structure of the escaping set

Prescribing the postsingular dynamics of meromorphic functions

The bungee set in quasiregular dynamics

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