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Area of the complement of the fast escaping sets of a family of entire functions. (English) Zbl 1461.37046
Summary: Let \(f\) be an entire function with the form \(f(z)=P(e^z)/e^z\), where \(P\) is a polynomial with \(\deg(P)\geq2\) and \(P(0)\neq0\). We prove that the area of the complement of the fast escaping set (hence the Fatou set) of \(f\) in a horizontal strip of width \(2\pi\) is finite. In particular, the corresponding result can be applied to the sine family \(\alpha\sin(z+\beta)\), where \(\alpha\neq0\) and \(\beta\in\mathbf{C}\).

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30D20 Entire functions of one complex variable (general theory)
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