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On the iteration of entire functions. (English) Zbl 0692.30021

Dynamical systems and ergodic theory, 28th Sem. St. Banach Int. Math. Cent., Warsaw/Pol. 1986, Banach Cent. Publ. 23, 339-345 (1989).
[For the entire collection see Zbl 0686.00015.]
Denote by \(f^ n\) the \(n\)-th iterate of an entire function \(f\). The properties of the set \(I(f)=\{z\in {\mathbb{C}}:\) \(f^ nz\to \infty \}\) are studied. The main result is that \(I(f)\) is not empty for every nonlinear entire function \(f\). The proof is based on the Wiman-Valiron theory.
Reviewer: A.E.Eremenko

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
60E05 Probability distributions: general theory
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets

Citations:

Zbl 0686.00015