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Transactions of the American Mathematical Society

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Puzzles and the Fatou–Shishikura injection for rational Newton maps


Authors: Kostiantyn Drach, Russell Lodge, Dierk Schleicher and Maik Sowinski
Journal: Trans. Amer. Math. Soc. 374 (2021), 2753-2784
MSC (2020): Primary 37F10, 37F25, 37C25
DOI: https://doi.org/10.1090/tran/8273
Published electronically: January 12, 2021
MathSciNet review: 4223032
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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

Keywords: Rational map, Newton map, Newton graph, puzzles, Markov property, Fatou inequality, Fatou–Shishikura injection, non-repelling cycle, renormalization
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.
Article copyright: © Copyright 2021 American Mathematical Society