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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
##### Keywords:
Fatou set; singularities; Baker domain
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##### References:
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