We establish a principle that we call the Fatou-Shishikura injection for rational Newton maps: there is a dynamically natural injection from the set of non-repelling periodic orbits of any rational Newton map to the set of its critical orbits. This injection obviously implies the classical Fatou-Shishikura inequality, but it is stronger in the sense that every non-repelling periodic orbit has its own critical orbit. Moreover, for every rational Newton map we associate a forward invariant graph (a puzzle) which provides a dynamically defined partition of the Riemann sphere into closed topological disks (puzzle pieces). The puzzle construction is, for the purposes of this paper, just a tool for establishing the Fatou-Shishikura injection; it also provided the foundation for future work on rigidity of Newton maps, in the spirit of the celebrated work by Yoccoz and others.

中文翻译：

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理性牛顿映射的谜题和 Fatou-Shishikura 注入

我们建立了一个原则，我们称之为有理牛顿映射的 Fatou-Shishikura 注入：从任何有理牛顿映射的非排斥周期轨道集到其临界轨道集存在动态自然注入。这种注入显然暗示了经典的 Fatou-Shishikura 不等式，但在每个非排斥周期轨道都有自己的临界轨道的意义上它更强。此外，对于每个有理牛顿映射，我们关联一个前向不变图（拼图），该图将黎曼球体动态定义为封闭的拓扑圆盘（拼图）。就本文而言，拼图构造只是建立 Fatou-Shishikura 注入的工具；它还为未来关于牛顿地图刚性的工作奠定了基础，