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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)
##### Keywords:
Fatou set; Julia set; Lebesque area; fast escaping sets
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