Puzzles and the Fatou–Shishikura injection for rational Newton maps

Drach, Kostiantyn; Lodge, Russell; Schleicher, Dierk; Sowinski, Maik

doi:10.1090/tran/8273

Puzzles and the Fatou–Shishikura injection for rational Newton maps
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by Kostiantyn Drach, Russell Lodge, Dierk Schleicher and Maik Sowinski PDF

Trans. Amer. Math. Soc. 374 (2021), 2753-2784 Request permission

Abstract:

We establish a principle that we call the Fatou–Shishikura injection for Newton maps of polynomials: there is a dynamically natural injection from the set of non-repelling periodic orbits of any 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 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). This puzzle construction is for rational Newton maps what Yoccoz puzzles are for polynomials: it provides the foundation for all kinds of rigidity results of Newton maps beyond our Fatou–Shishikura injection. Moreover, it gives necessary structure for a classification of the postcritically finite maps in the spirit of Thurston theory.

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