Bounded domains of the Fatou set of an entire function. (English) Zbl 1054.37028

I. N. Baker raised the question whether the Fatou components of an entire function of order less than \(1/2\) must be bounded. He had shown that this is true for functions of much smaller growth, and he had given an example of a function of order \(1/2\) with an unbounded Fatou component. Results in this direction have been obtained by G. M. Stallard, J. M. Andersonn and A. Hinkkanen, and X. H. Hua and C. C. Yang. In the present paper, an affirmative answer to Baker’s question is given under the additional hypothesis that the function has positive lower order. As in previous papers on this question, the main idea is to use suitable minimum modulus estimates.


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