Residual Julia Sets of Meromorphic Functions

Ng, Tuen Wai; Zheng, Jian; Choi, Yan Yu

doi:10.1017/s0305004106009388

Cited by 10 publications

(4 citation statements)

References 32 publications

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“…The following result on singleton components of the Julia set is proved by Domínguez [9]. Ng et al [13] proved a generalization as follows.…”

Section: Resultsmentioning

confidence: 99%

Omitted values and dynamics of meromorphic functions

Nayak¹,

Zheng²

2010

Journal of the London Mathematical Society

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Let M be the class of all transcendental meromorphic functions f : C → C ∪ {∞} with at least two poles or one pole that is not an omitted value, and M o = {f ∈ M : f has at least one omitted value}. Some dynamical issues of the functions in M o are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker wandering domains and invariant Herman rings are proved. Eventual connectivity of each wandering domain is proved to exist. For functions with exactly one pole, we show that Herman rings of period 2 also do not exist. A necessary and sufficient condition for the existence of a dense subset of singleton buried components in the Julia set is established for functions with two omitted values. The conjecture that a meromorphic function has at most two completely invariant Fatou components is confirmed for all f ∈ M o except in the case when f has a single omitted value, no critical value and is of infinite order. Some relevant examples are discussed.

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“…The following result on singleton components of the Julia set is proved by Domínguez [9]. Ng et al [13] proved a generalization as follows.…”

Section: Resultsmentioning

confidence: 99%

Omitted values and dynamics of meromorphic functions

Nayak¹,

Zheng²

2010

Journal of the London Mathematical Society

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“…Indeed, it was proved by McMullen [14] that the Julia set of R(z) is a Cantor set of circles for sufficiently small λ > 0 and ∞ attracts all critical points of R(z) and so R(z) is hyperbolic in H( C). If Julia set of a meromorphic function f (z) which is not of the form α+ (z − α) −k e g(z) for a natural number k, a complex number α and an entire function g(z), is disconnected on C, then it has uncountably infinitely many Julia components and it was proved in Ng, Zheng and Choi [17] that it has uncountably infinitely many buried components if F f has no completely invariant components. Since J f (∞) has at most countably many points, J f has only countably components which contain points of J f (∞) and if it is disconnected on C, then J f has uncountably infinitely many components which do not contain any points in…”

Section: Expanding and Topological Dynamics Of Hyperbolic Functionsmentioning

confidence: 99%

Dynamics of Hyperbolic Meromorphic Functions

Zheng¹

2012

Preprint

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A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on the Julia set. We prove the important expanding properties for hyperbolic functions on the complex plane and with respect to the euclidean metric. We establish the Bowen formula for hyperbolic functions on the complex plane, that is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia set and furthermore, we prove that all the results in the Walters' theory hold for hyperbolic functions on the Riemann sphere.

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“…Domínguez and Fagella [13] survey the current state of affairs in this direction. Also see Ng et al [27], who prove Makienko's conjecture for locally connected Julia sets of certain meromorphic functions. The techniques of proof of Theorem 1.3 fail dramatically for trancendental functions.…”

Section: Extensionsmentioning

confidence: 99%