Rigidity of Newton dynamics

Kostiantyn Drach; Dierk Schleicher

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arXiv · 1812.11919

Rigidity of Newton dynamics

Kostiantyn Drach·Dierk Schleicher

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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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Kostiantyn Drach, Dierk Schleicher. 2018-12-31. Rigidity of Newton dynamics. https://arxiv.org/abs/1812.11919

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