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