## 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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## Additional Information

**Kostiantyn Drach**- Affiliation: Aix–Marseille Université, Institut de Mathématiques de Marseille, 163 Avenue de Luminy, 13009 Marseille, France
- MR Author ID: 1050262
- Email: kostya.drach@gmail.com
**Russell Lodge**- Affiliation: Department of Mathematics and Computer Science, Indiana State University, Terre Haute, Indiana 47809
- MR Author ID: 1022713
- Email: russell.lodge@indstate.edu
**Dierk Schleicher**- Affiliation: Aix–Marseille Université, Institut de Mathématiques de Marseille, 163 Avenue de Luminy, 13009 Marseille, France
- MR Author ID: 359328
- Email: dierk.SCHLEICHER@univ-amu.fr
**Maik Sowinski**- Affiliation: Universität Bielefeld, Universitätsstrasse 25, 33615 Bielefeld, Germany
- ORCID: 0000-0002-2168-4785
- Email: maik.sowinski@gmx.de
- Received by editor(s): August 17, 2018
- Received by editor(s) in revised form: November 5, 2019, and July 26, 2020
- Published electronically: January 12, 2021
- Additional Notes: This research was partially supported by the advanced grant 695 621 “HOLOGRAM” of the European Research Council (ERC), which is gratefully acknowledged.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 2753-2784 - MSC (2020): Primary 37F10, 37F25, 37C25
- DOI: https://doi.org/10.1090/tran/8273
- MathSciNet review: 4223032