On semiconjugation of entire functions

Bergweiler, Walter; Hinkkanen, A.

doi:10.1017/s0305004198003387

Cited by 116 publications

(164 citation statements)

References 13 publications

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“…(Note that the set B(f ) is independent of the choice of D. The set B(f ) was first introduced in [15] where we showed that it is equal to the set A(f ) introduced by Bergweiler and Hinkkanen in [6].) We show in [16] that if a transcendental entire function f satisfies the hypotheses of Lemma 2.1, then…”

Section: Introductionmentioning

confidence: 91%

Functions of small growth with no unbounded Fatou components

Rippon

Stallard

2009

J Anal Math

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Abstract. We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components, also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.

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Section: Introductionmentioning

confidence: 91%

Functions of small growth with no unbounded Fatou components

Rippon

Stallard

2009

J Anal Math

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“…(Such functions never belong to the Eremenko-Lyubich class B.) In fact, they showed that, for any transcendental entire function, the subset A(f ) ⊂ I(f ) of "points escaping at the fastest possible rate", as introduced by Bergweiler and Hinkkanen [BH99], has only unbounded components. Also, recently [Rem07] the weak form of Eremenko's conjecture was established for functions f ∈ B whose postsingular set is bounded (which applies, in particular, to the hyperbolic counterexample constructed in Theorem 1.1).…”

Section: Corollary (Meromorphic Functions With Logarithmic Singularitmentioning

confidence: 99%

Dynamic rays of bounded-type entire functions

Rottenfußer¹,

et al. 2011

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We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative.On the other hand, we show that for many functions in B, in particular those of finite order, every escaping point can be connected to ∞ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

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“…Using arguments as in [6,36] one can show that the set A(f, D) does not depend on the choice of ρ, as long as ρ > ρ 0 . This justifies the notation where ρ is suppressed.…”

Section: Meromorphic Functions With a Direct Tractmentioning

confidence: 97%

Dynamics of meromorphic functions with direct or logarithmic singularities

Bergweiler

Rippon

Stallard

2008

Proceedings of the London Mathematical Society

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144

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Abstract. Let f be a transcendental meromorphic function and denote by J(f ) the Julia set and by I(f ) the escaping set. We show that if f has a direct singularity over infinity, then I(f ) has an unbounded component and I(f ) ∩ J(f ) contains continua. Moreover, under this hypothesis I(f ) ∩ J(f ) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f ) ∩ J(f ) is 2 and the Hausdorff dimension of J(f ) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. This method is also applied to complex differential equations.

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On semiconjugation of entire functions

Cited by 116 publications

References 13 publications

Functions of small growth with no unbounded Fatou components

Functions of small growth with no unbounded Fatou components

Dynamic rays of bounded-type entire functions

Dynamics of meromorphic functions with direct or logarithmic singularities

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