# Rigidity of Newton dynamics

## TL;DR

This paper investigates the rigidity properties of Newton maps for polynomial root finding, establishing conditions under which their Julia sets are locally connected and showing that combinatorially equivalent maps are quasiconformally conjugate.

## Contribution

It introduces a dynamical rigidity framework for Newton maps, extending existing results to include irrationally indifferent and renormalizable cases, and proves local connectivity of Julia sets in many cases.

## Key findings

- Julia sets of many Newton maps are locally connected
- Combinatorially equivalent Newton maps are quasiconformally conjugate under certain conditions
- Extended rigidity results to include irrationally indifferent and renormalizable situations

## 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.