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Singular values and non-repelling cycles for entire transcendental maps. (English) Zbl 07293619
Summary: Let \(f\) be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated with a singular orbit which cannot accumulate on any other non-repelling cycle. When \(f\) has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality.

MSC:
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
30D30 Meromorphic functions of one complex variable (general theory)
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