Dynamic rays of bounded-type entire functions

Rottenfußer, Günter; Rückert, Johannes; Rempe, Lasse; Schleicher, Dierk

doi:10.4007/annals.2010.173.1.3

Cited by 73 publications

(179 citation statements)

References 31 publications

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“…To prove Theorem 1.9, we use similar ideas as in the proof of Theorem 1.4, together with the results of [3,38], which establish the existence of Cantor bouquets in the Julia sets of the functions under consideration. The difference between the general case and that of exponential maps is that we do not have such explicit information about the position and behaviour of rays as we did through our model F. In particular, there may not be an analogue of the characterisation of "fast" addresses in terms of a simple growth condition.…”

Section: Beyond the Exponential Familymentioning

confidence: 99%

“…Then there is again a natural notion of "escaping endpoints": these are those points z 0 ∈ I ( f ) with the following property. Let n 0 ≥ 0 be any number that is sufficiently large to ensure that the external address of f n 0 (z 0 ) is defined in the sense of [38]. If z = f n 0 (z 0 ) is an escaping point with the same external address, then | f n (z)| > | f n+n 0 (z 0 )| for all sufficiently large n.…”

Section: Beyond the Exponential Familymentioning

confidence: 99%

“…In this article, we focus primarily on the family of exponential maps. However, there is a large class of transcendental entire functions for which the existence of rays, in analogy to the exponential family, was established in [38] (and, independently, in [2] for a more restrictive class of maps). It makes sense to ask about the properties of the set of escaping endpoints in this context also.…”

Section: Conjecture 16 (Escaping Endpoints Always Explode) Let a ∈ Cmentioning

confidence: 99%

“…In this direction, we shall prove the following extension of Theorem 1.4. (Recall that an entire function has finite order if log log [38]. Then the setẼ( f ) ∪ {∞} is connected.…”

Section: Conjecture 16 (Escaping Endpoints Always Explode) Let a ∈ Cmentioning

confidence: 99%

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Escaping Endpoints Explode

Alhabib

Rempe

2016

Comput. Methods Funct. Theory

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In 1988, Mayer proved the remarkable fact that ∞ is an explosion point for the set E( f a ) of endpoints of the Julia set of f a : C → C; e z + a with a < −1; that is, the set E( f a ) is totally separated (in particular, it does not have any non-trivial connected subsets), but E( f a )∪{∞} is connected. Answering a question of Schleicher, we extend this result to the setẼ( f a ) of escaping endpoints in the sense of Schleicher and Zimmer, for any parameter a ∈ C for which the singular value a belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the setẼ( f ) ∪ {∞} is connected for any transcendental entire function f of finite order with bounded singular set. We also discuss corresponding results for all endpoints in the case of exponential maps; to do so, we establish a version of Thurston's no wandering triangles theorem for exponential maps.

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Section: Beyond the Exponential Familymentioning

confidence: 99%

Section: Beyond the Exponential Familymentioning

confidence: 99%

Section: Conjecture 16 (Escaping Endpoints Always Explode) Let a ∈ Cmentioning

confidence: 99%

“…In this direction, we shall prove the following extension of Theorem 1.4. (Recall that an entire function has finite order if log log [38]. Then the setẼ( f ) ∪ {∞} is connected.…”

Section: Conjecture 16 (Escaping Endpoints Always Explode) Let a ∈ Cmentioning

confidence: 99%

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Escaping Endpoints Explode

Alhabib

Rempe

2016

Comput. Methods Funct. Theory

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“…To prove that H(s) is a hair we follow Rottenfußer, Rückert, Rempe and Schleicher [19] and use the following lemma from [17]. …”

Section: Proof Of the Propositionsmentioning

confidence: 99%

Dynamics of a higher dimensional analog of the trigonometric functions

Bergweiler¹,

Erëmenko²

2011

Ann. Acad. Sci. Fenn. Math.

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Abstract. We introduce a quasiregular analog F of the sine and cosine function such that, for a sufficiently large constant λ, the map x → λF (x) is locally expanding. We show that the dynamics of this map define a representation of R d , d ≥ 2, as a union of simple curves γ : [0, ∞) → R d which tend to ∞ and whose interiors γ * = γ ((0, ∞)) are disjoint such that the union of all γ * has Hausdorff dimension 1.

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A landing theorem for entire functions with bounded post-singular sets

Benini

Rempe

2020

Geom. Funct. Anal.

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The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

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Dynamic rays of bounded-type entire functions

Cited by 73 publications

References 31 publications

Escaping Endpoints Explode

Escaping Endpoints Explode

Dynamics of a higher dimensional analog of the trigonometric functions

A landing theorem for entire functions with bounded post-singular sets

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