# Rigidity of Newton dynamics

Abstract : We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set. In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable in the same way''. Our main tool is the concept of complex box mappings due to Kozlovski, Shen, van Strien; we also extend a dynamical rigidity result for such mappings so as to include irrationally indifferent or renormalizable situations.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03285952
Contributor : Kostiantyn Drach <>
Submitted on : Tuesday, July 13, 2021 - 4:55:37 PM
Last modification on : Wednesday, July 14, 2021 - 3:35:37 AM

### Identifiers

• HAL Id : hal-03285952, version 1
• ARXIV : 1812.11919

### Citation

Kostiantyn Drach, Dierk Schleicher. Rigidity of Newton dynamics. 2021. ⟨hal-03285952⟩

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