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We consider entire transcendental maps with bounded set of singular valuessuch that periodic rays exist and land. For such maps, we prove a refinedversion of the Fatou-Shishikura inequality which takes into account rationallyinvisible periodic orbits, that is, repelling cycles which are not landingpoints of any periodic ray. More precisely, if there are $q<\infty$ singularorbits, then the sum of the number of attracting, parabolic, Siegel, Cremer orrationally invisible orbits is bounded above by $q$. In particular, there areat most $q$ rationally invisible repelling periodic orbits. The techniquespresented here also apply to the more general setting in which the function isallowed to have infinitely many singular values.

A bound on the number of rationally invisible repelling orbits