A transcendental Julia set of dimension 1

Bishop, Christopher J.

doi:10.1007/s00222-017-0770-0

Cited by 21 publications

(81 citation statements)

References 46 publications

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“…We have the following corollary of Theorem 7.1 and Lemma 6.5 (c), which was also given in Baumgartner's PhD thesis [3, Other examples with finite inner connectivity were given in [15] and in [8].…”

Section: Complementary Components Of Multiply Connected Wandering Dommentioning

confidence: 88%

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: Complementary Components Of Multiply Connected Wandering Dommentioning

confidence: 88%

Eremenko points and the structure of the escaping set

Rippon

Stallard

2019

Trans. Amer. Math. Soc.

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“…We begin with a result about pre-images of annuli bounded by Jordan curves. Related results appear in [15,Lemma 11.1] and [37,Lemma 5].…”

Section: Example 3 (Three Eventually Isometric Wandering Domains)mentioning

confidence: 99%

“…Apart from wandering domains arising from lifting constructions, as far as we are aware these are the first examples of simply connected wandering domains for which it is possible to obtain information concerning the boundary. Examples of multiply connected wandering domains for which it is known that connected components of the boundary are Jordan curves can be found in [15] and in [4].…”

Section: Wandering Domains Whose Boundaries Are Jordan Curvesmentioning

confidence: 99%

“…For example, (oscillating) wandering domains have been constructed for functions in the Eremenko-Lyubich class B (those maps with a bounded sets of singular values) [14,27,34], a landmark result because escaping wandering domains have been shown not to exist for maps in this class [22]. We also mention the recent construction by Bishop [15] of an entire function with Julia set of Hausdorff dimension 1, solving a long standing problem in transcendental dynamics. This function has multiply connected wandering domains and also exhibits other remarkable properties; for example, all the boundary components of its wandering domains are Jordan curves.…”

Section: Introductionmentioning

confidence: 99%

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Classifying simply connected wandering domains

et al. 2021

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While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

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“…Until recently it was an open question as to whether there existed an entire function f for which the Hausdorff dimension of the Julia set of f was equal to 1. However, Bishop [10] has constructed an entire function f with dim J(f ) = 1. This function has multiply connected wandering domains, and so its direct tract, defined in Section 3, does not have an unbounded boundary component.…”

mentioning

confidence: 99%