Iterations of rational functions and the distribution of the values of the Poincar� functions

Erëmenko, Alexandre; Sodin, Mikhail

doi:10.1007/bf01109688

Cited by 24 publications

(19 citation statements)

References 6 publications

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“…There are a variety of methods which can be used to study the value distribution of meromorphic solutions of the Schröder equation (1.1). Eremenko and Sodin [9] used methods from complex dynamics to show that the Valiron and Nevanlinna deficient values of meromorphic solutions of the autonomous Schröder equation (1.1) always coincide with the exceptional values of R(z). Ishizaki and Yanagihara [19] constructed an example showing that this is not true in general for the non-autonomous Schröder equation.…”

Section: Introductionmentioning

confidence: 99%

Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations

Barnett

Halburd

Morgan

et al. 2007

Proceedings of the Royal Society of Edinburgh: Section A Mathem

114

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It is shown that, if f is a meromorphic function of order zero andfor all r on a set of logarithmic density 1. The remainder of the paper consist of applications of identity ( ‡) to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of q-difference equations. The results obtained include q-shift analogues of the Second Main Theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.

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Section: Introductionmentioning

confidence: 99%

Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations

Barnett

Halburd

Morgan

et al. 2007

Proceedings of the Royal Society of Edinburgh: Section A Mathem

114

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“…In the case of iterations, Theorem 2 was proved by Brolin [2] (for polynomials), Lyubich [16] and Freire-Lopes-Mañé [9]. For the other proofs, see also Tortrat [27], Erëmenko-Sodin [6], Hubbard-Papadopol [12], and Fornaess and Sibony [8]. Theorem 2 can be also proven by Fornaess and Sibony's argument in the proof of [8], Theorem 6.1, where they used a crucial contradiction.…”

Section: Example 119mentioning

confidence: 70%

Nevanlinna theoretical exceptional sets of rational towers and semigroups

Okuyama

2006

Conform. Geom. Dyn.

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Abstract. For a rational tower, i.e., a composition sequence of rational maps, in addition to the algebraic and dynamical exceptional sets, various Nevanlinna theoretical exceptional sets are defined, and as we showed previously in the case of iterations, all of them are the same. In this paper, we extend this result to the cases of a rational tower with summable distortions and a finitely generated rational semigroup. We show that all the exceptional sets of a finitely generated rational semigroup are countable, and all of them are empty if and only if the algebraic one is as well (this being the smallest among them). The countability of exceptional sets is fundamental in the Nevanlinna theory, and their emptiness is important in the complex dynamics.

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“…Valiron has shown that the non-autonomous Schröder q-difference equation f (qz) = R z, f (z) , (1.1) where R(z, f (z)) is rational in both arguments, admits meromorphic solutions if q ∈ C is suitably chosen [13]. The property of meromorphic solutions of (1.1) is deeply investigated during the last decades, see for instance [3,7,10,11]. In 1998, Bergweiler et al [2] pointed out that transcendental meromorphic solutions f (z) of the functional equation n j=0 a j (z) f c j z = Q (z), (1.2) where 0 < |c| < 1 is a complex number, a j (z), j = 0, 1, 2, .…”

Section: Introduction and Main Resultsmentioning

confidence: 98%