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.

中文翻译：

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合理不可见的排斥轨道数量的界限

我们考虑具有有界奇异值集的整个超越图，使得周期性射线存在并着陆。对于这样的地图，我们证明了 Fatou-Shishikura 不等式的改进版本，它考虑了合理不可见的周期轨道，即排斥周期，这些周期不是任何周期射线的着陆点。更准确地说，如果存在 $q<\infty$ 奇异轨道，则吸引轨道、抛物线轨道、西格尔轨道、克雷默轨道或合理不可见轨道的数量之和以 $q$ 为界。特别是，至多存在 $q$ 合理不可见的排斥周期轨道。此处介绍的技术也适用于更一般的设置，其中允许函数具有无限多个奇异值。