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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)

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics
Full Text: DOI
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