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Iteration of a class of hyperbolic meromorphic functions. (English) Zbl 0931.30017

Let \(f\) be meromorphic in the plane, not rational of degree less than two, and denote by \(f^n\) the \(n\)-th iterate of \(f\). Let \(S_n(f)\) be the set of singularities of the inverse function of \(f^n\). Let \(B_n\) be the class where \(S_n(f)\) is bounded. The first result (Theorem A) of this paper says that if \(f\in B_n\), then there is no component of the set of normality of \(f\) in which \(f^{mn}\rightarrow \infty \) as \(m\rightarrow \infty\). In particular, there is no cycle of Baker domains of period \(n\). This latter result had previously been stated by the reviewer [Bull. Am. Math. Soc., New Ser. 29, 151-188 (1993; Zbl 0791.30018)]. As the authors correctly point out, however, his proof was based on an incorrect lemma. The key idea of the present proof is a logarithmic change of variable introduced by A. Eremenko and M. Lyubich [Ann. Inst. Fourier 42, No. 4, 989-1020 (1992; Zbl 0735.58031)] into the subject, combined with a lemma which says that if \(f\in B_n\) and \(S_n(f)\subset \{z:|z|<R\}\), then each component of \(f^{-n}(\{z:|z|>R\}\cup \{\infty\})\) is simply connected.
The other results of this paper are connected to a certain notion of hyperbolicity. Let \(P(f)=\bigcup^{\infty}_{n=1}S_n(f)\) and let \(\widehat{B}\) be the class of all functions in \(B_1\) for which the closure (with respect to the plane) of \(P(f)\) does not intersect the Julia set of \(f\). It is shown that if \(f\in \widehat{B}\), then \(P(f)\) is bounded. Moreover, it is shown that if \(f\in \widehat{B}\), then there exist \(k>1\) and \(c>0\) such that \(|(f^n)'(z)|>c k^n (|f^n(z)|+1)/(|z|+1)\) for all \(n\in {\mathbb N}\) and all \(z\) in the Julia set for which \(f^n(z)\) is defined and finite. The latter result has been used by the second author to estimate the Hausdorff dimension of Julia sets.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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