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Invariant domains and singularities. (English) Zbl 0836.30016
In the dynamics of an entire function \(f\) an important role is played by the set \(\text{sing} (f^{-1})\) of all singularities of the inverse function and by \(P(f)\) which is the closure of \[ \bigcup^\infty_{n = 0} f^n \bigl( \text{sing} (f^{- 1}) \bigr). \] A component of the Fatou set of \(f\) such that \(f(U) \subset U\) and \(f^n \to \infty\) in \(U\) is called a Baker domain. Examples are known of entire \(f\) which have a Baker domain \(U\) such that \(U \cap \text{sing} (f^{-1}) = \emptyset\). The present paper improves on this by showing that \(f(z) = 2 - \log 2 + 2z - e^z\) has a Baker domain \(U\) such that \(d(P(f), U) > 0\), where \(d\) denotes Euclidean distance. It is interesting that \(\partial U\) in this example is a Jordan curve on the sphere, giving a more elementary demonstration of a phenomenon first exhibited in I. N. Baker and J. Weinreich [Rev. Roum. Math. Pures Appl. 36, No. 7/8, 413-420 (1991; Zbl 0756.30021)]. On the other hand it is proved that if \(g\) is transcendental entire with an invariant Baker domain \(V\) and if \(V \cap \text{sing} (f^{-1}) = \emptyset\), then there exists a sequence \(p_n\) such that \(p_n \in P(f)\), \(|p_n |\to \infty\), \(|p_n/p_{n + 1} |\to 1\) and \(d(p_n,U) = o(|p_n |)\) as \(n \to \infty\).
Reviewer: I.N.Baker (London)

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics
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References:
[1] Baker, Ann. Acad. Sci. Fenn 12 pp 191– (1987) · Zbl 0606.30029
[2] Baker, Ergodic Theory Dynamical Systems 8 pp 503– (1988) · Zbl 0642.30021
[3] Baker, Ann. Acad. Sci. Fenn 1 pp 277– (1975) · Zbl 0329.30019
[4] Baker, Ann. Acad. Sci. Fenn (1970)
[5] Steinmetz., Rational iteration 16 (1993)
[6] Stallard., Math. Proc. Cambridge Philos. Soc 108 pp 551– (1990)
[7] Pommerenke., Univalent functions (1975)
[8] DOI: 10.1007/BF01215911 · Zbl 0587.30040
[9] Eremenko, Ann. Inst. Fourier (Grenoble) 42 pp 989– (1992) · Zbl 0735.58031
[10] Eremenko, Leningrad Math. J 1 pp 563– (1990)
[11] DOI: 10.1112/jlms/s2-36.3.458 · Zbl 0601.30033
[12] Carleson, Complex dynamics (1993)
[13] Douady, Ann. Sci. ?cole Norm. Sup 18 pp 287– (1985)
[14] DOI: 10.1090/S0273-0979-1993-00432-4 · Zbl 0791.30018
[15] Beardon., Iteration of rational functions (1991)
[16] Baker, Rev. Roumaine Math. Pures Appl 36 pp 413– (1991)
[17] Baker, Lectures on complex analysis pp 1– (1987)
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