Escaping Endpoints Explode

Alhabib, Nada; Rempe, Lasse

doi:10.1007/s40315-016-0169-8

Cited by 23 publications

(45 citation statements)

References 38 publications

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“…Here

J_{prefixr} false(f_{a} false)

is the radial Julia set , a set of particular importance. The results of naturally suggest the question whether

\infty

is an explosion point for

J_{prefixr} false(f_{a} false)

also. It is known [, Section 2] that

J_{prefixr} false(f_{a} false)

has Hausdorff dimension strictly greater than one, which is compatible with this possibility.…”

Section: Introductionmentioning

confidence: 96%

“…Hence, infinity is an explosion point for

E false(f_{a} false) \cup false{\infty false}, a < - 1

. Following the terminology used in , we will also simply say that infinity is an explosion point for

E (f_{a})

.…”

Section: Introductionmentioning

confidence: 99%

“…Alhabib and the second author recently proved [, Theorem 1.3] that Mayer's result holds also for the smaller set of escaping endpoints of

J (f_{a})

; that is, for the set

\tilde{E} false(f_{a} false) : = E false(f_{a} false) \cap I false(f_{a} false)

of endpoints that belong to the escaping set

I false(f_{a} false) = false{z \in double-struckC 0pt: f_{a}^{n} (z) \to \infty as n \to \infty false} .

Still assuming that

a \in (- \infty, - 1)

, the complementary set of non‐escaping endpoints of

f_{a}

satisfies the following identities:

E false(f_{a} false) ∖ \tilde{E} false(f_{a} false) = J false(f_{a} false) ∖ I false(f_{a} false) = J_{prefixr} false(f_{a} false);

see Corollary and Proposition . Here

J_{prefixr} false(f_{a} false)

is the radial Julia set , a set of particular importance.…”

Section: Introductionmentioning

confidence: 99%

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Non-escaping endpoints do not explode

Evdoridou

Rempe

2018

Bull. London Math. Soc.

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The family of exponential maps fa(z) = e z + a is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(fa). When a ∈ (−∞, −1), and more generally when a belongs to the Fatou set F (fa), it is known that J(fa) can be written as a union of hairs and endpoints of these hairs. In 1990, Mayer proved for a ∈ (−∞, −1) that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where a ∈ F (fa), and showed that it holds even for the smaller set of all escaping endpoints.We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a 'spider's web'; in particular we give a new topological characterisation of spiders' webs that may be of independent interest. We also show how our results can be applied to Fatou's function,Still assuming that a ∈ (−∞, −1), the complementary set of non-escaping endpoints of f a satisfies the following identities:see Corollary 2.2 and Proposition 2.4. Here J r (f a ) is the radial Julia set, a set of particular importance. The results of [2] naturally suggest the question whether ∞ is an explosion point for J r (f a ) also. It is known [36, Section 2] that J r (f a ) has Hausdorff dimension strictly greater than one, which is compatible with this possibility. Nonetheless, we prove here that the sets of escaping and non-escaping endpoints are topologically very different from each other.Theorem 1.2 (Non-escaping endpoints do not explode). Let a ∈ (−∞, −1). Then the set J r (f a ) ∪ {∞} is totally separated.

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“…Here

J_{prefixr} false(f_{a} false)

is the radial Julia set , a set of particular importance. The results of naturally suggest the question whether

\infty

is an explosion point for

J_{prefixr} false(f_{a} false)

also. It is known [, Section 2] that

J_{prefixr} false(f_{a} false)

has Hausdorff dimension strictly greater than one, which is compatible with this possibility.…”

Section: Introductionmentioning

confidence: 96%

“…Hence, infinity is an explosion point for

E false(f_{a} false) \cup false{\infty false}, a < - 1

. Following the terminology used in , we will also simply say that infinity is an explosion point for

E (f_{a})

.…”

Section: Introductionmentioning

confidence: 99%

“…Alhabib and the second author recently proved [, Theorem 1.3] that Mayer's result holds also for the smaller set of escaping endpoints of

J (f_{a})

; that is, for the set

\tilde{E} false(f_{a} false) : = E false(f_{a} false) \cap I false(f_{a} false)

of endpoints that belong to the escaping set

I false(f_{a} false) = false{z \in double-struckC 0pt: f_{a}^{n} (z) \to \infty as n \to \infty false} .

Still assuming that

a \in (- \infty, - 1)

, the complementary set of non‐escaping endpoints of

f_{a}

satisfies the following identities:

E false(f_{a} false) ∖ \tilde{E} false(f_{a} false) = J false(f_{a} false) ∖ I false(f_{a} false) = J_{prefixr} false(f_{a} false);

see Corollary and Proposition . Here

J_{prefixr} false(f_{a} false)

is the radial Julia set , a set of particular importance.…”

Section: Introductionmentioning

confidence: 99%

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Non-escaping endpoints do not explode

Evdoridou

Rempe

2018

Bull. London Math. Soc.

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“…Here we say that X ⊂ C (or C) is totally separated if for any two points a, b ∈ X there exists a relatively open and closed set U ⊂ X such that a ∈ U and b / ∈ U . As observed in [1], it is now known that Mayer's result holds for all Cantor bouquets.…”

Section: Introductionmentioning

confidence: 71%

“…However, we can obtain a result similar to Theorem 1.1 if we consider instead the set of endpoints which both escape and meander; this is defined by E Q (f ) := E(f ) \ A(f ). Once again the first conclusion is [1,Remark 7.2], and is included here for emphasis. Escaping endpoints as we define them are called escaping endpoints in the strong sense in [1,Remark 7.2].…”

Section: Introductionmentioning

confidence: 99%

The Topology of the Set of Non-Escaping Endpoints

Evdoridou

Sixsmith

2019

International Mathematics Research Notices

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There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties of this set of finite endpoints. It was recently shown that, for certain functions in the exponential family, there is a strong dichotomy between the topological properties of the set of endpoints which escape and those of the set of endpoints which do not escape. In this paper, we show that this result holds for large families of functions in the Eremenko-Lyubich class. We also show that this dichotomy holds for a family of functions, outside that class, which includes the much-studied Fatou function defined by f (z) := z + 1 + e −z . Finally, we show how our results can be used to demonstrate that various sets are spiders' webs, generalising results such as those in [8].

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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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Escaping Endpoints Explode

Cited by 23 publications

References 38 publications

Non-escaping endpoints do not explode

Non-escaping endpoints do not explode

The Topology of the Set of Non-Escaping Endpoints

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

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