On the packing dimension of the Julia set and the escaping set of an entire function

Bergweiler, Walter

doi:10.1007/s11856-012-0033-0

Cited by 8 publications

(17 citation statements)

References 36 publications

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“…Remark A similar condition to (5.1) appears in a recent paper of Bergweiler [3]. The main result of [3] is that if f is a transcendental entire function with no multiply connected Fatou components and The assertion that (5.13) implies that f has no multiply connected Fatou components is justified in [3] by using results from either [25] or [6], and the assertion that (5.13) implies (5.12) is deduced from a result of Fenton [11]. We remark that this latter implication can also be deduced by using the argument from the proof of Theorem 5.…”

Section: Sufficient Conditions For Log-regularitysupporting

confidence: 63%

“…Finally we mention two other regularity conditions. First, the condition log M(2r) ≥ d log M(r), for large r, where d > 1, was mentioned in [3] in relation to a result on the packing dimension of I(f ) ∩ J(f ); see the remark after the proof of Theorem 5.1 of this paper. This regularity condition is easily seen to imply log-regularity.…”

Section: Now We Deduce Theorem 41mentioning

confidence: 99%

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Regularity and Fast Escaping Points of Entire Functions

Rippon

Stallard

2013

International Mathematics Research Notices

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Let f be a transcendental entire function. The fast escaping set A(f ), various regularity conditions on the growth of the maximum modulus of f , and also, more recently, the quite fast escaping set Q(f ) have all been used to make progress on fundamental questions concerning the iteration of f . In this paper we establish new relationships between these three concepts.We prove that a certain weak regularity condition is necessary and sufficient for Q(f ) = A(f ) and give examples of functions for which Q(f ) = A(f ).We also apply a result of Beurling that relates the size of the minimum modulus of f to the growth of its maximum modulus in order to establish that a stronger regularity condition called log-regularity holds for a large class of functions, in particular for functions in the Eremenko-Lyubich class B.

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Section: Sufficient Conditions For Log-regularitysupporting

confidence: 63%

Section: Now We Deduce Theorem 41mentioning

confidence: 99%

Regularity and Fast Escaping Points of Entire Functions

Rippon

Stallard

2013

International Mathematics Research Notices

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“…(a) If m ∈ N and ε ∈ (π/ log r n , (b n − a n )/2), where n ∈ N is sufficiently large, then Another application of Theorem 5.1(a) is given in [13,Corollary 4.1]. The next result states that, for large n ∈ N, the iterates of f behave like monomials inside C n .…”

Section: Dynamics In a Multiply Connected Wandering Domainmentioning

confidence: 98%

Multiply connected wandering domains of entire functions

Bergweiler

Rippon

Stallard

2013

Proceedings of the London Mathematical Society

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103

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Abstract. The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function f in any multiply connected wandering domain U of f . By introducing a certain positive harmonic function h in U , related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large n, the image domains U n = f n (U ) contain large annuli, C n , and that the union of these annuli acts as an absorbing set for the iterates of f in U . Moreover, f behaves like a monomial within each of these annuli and the orbits of points in U settle in the long term at particular 'levels' within the annuli, determined by the function h. We also discuss the proximity of ∂U n and ∂C n for large n, and the connectivity properties of the components of U n \ C n . These properties are deduced from new results about the behaviour of entire functions that omit certain values in an annulus.

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“…This may be possible by replacing the use of the degree two Chebyshev polynomial in the paper [6] by higher degree Chebyshev polynomials or generalized Chebyshev polynomials. Can we use such constructions to show the conditions in Bergweiler's paper [4] implying Pdim(J ) = 2 are sharp?…”

Section: Preliminarymentioning

confidence: 99%

“…In [4], a criteria on the calculation of packing dimension was given: If F (f ) has no multiply connected component, then…”

Section: Preliminarymentioning

confidence: 99%

A generalized family of transcendental functions with one dimensional Julia sets

Zhang¹

2019

Preprint

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A generalized family of non-polynomial entire functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to 1, there exist multiply connected wandering domains, and the dynamics can be completed described. For any s ∈ (0, +∞], there is a function taken from this family with the order of growth s. This is the first example, where the Julia set has packing dimension 1, and arbitrarily positive or even infinite order of growth. This gives an answer to an open problem in [6, Problem 4].

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On the packing dimension of the Julia set and the escaping set of an entire function

Cited by 8 publications

References 36 publications

Regularity and Fast Escaping Points of Entire Functions

Regularity and Fast Escaping Points of Entire Functions

Multiply connected wandering domains of entire functions

A generalized family of transcendental functions with one dimensional Julia sets

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