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Let $f$ be a map with bounded set of singular values for which periodicdynamic rays exist and land. We prove that each non-repelling cycle isassociated to a singular orbit which cannot accumulate on any othernon-repelling cycle. When $f$ has finitely many singular values this implies arefinement of the Fatou-Shishikura inequality. Our approach is combinatorial inthe spirit of the approach used by [Ki00], [BCL+16] for polynomials.

Singular values and non-repelling cycles for entire transcendental maps