article
Rigidity of Newton dynamics
epub_ahead
yes
Kostiantyn
Drach
author fe8209e2-906f-11eb-847d-950f8fc091150000-0002-9156-8616
Dierk
Schleicher
author
VaKa
department
Spectral rigidity and integrability for billiards and geodesic flows
project
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 a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations.
Elsevier2022
eng
General Mathematics
Advances in Mathematics
0001-870810.1016/j.aim.2022.108591
408
Drach, K., & Schleicher, D. (2022). Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/j.aim.2022.108591">https://doi.org/10.1016/j.aim.2022.108591</a>
Drach K, Schleicher D. Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. 2022;408. doi:<a href="https://doi.org/10.1016/j.aim.2022.108591">10.1016/j.aim.2022.108591</a>
K. Drach and D. Schleicher, “Rigidity of Newton dynamics,” <i>Advances in Mathematics</i>, vol. 408. Elsevier, 2022.
Drach K, Schleicher D. 2022. Rigidity of Newton dynamics. Advances in Mathematics. 408, 108591.
Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances in Mathematics</i>, vol. 408, 108591, Elsevier, 2022, doi:<a href="https://doi.org/10.1016/j.aim.2022.108591">10.1016/j.aim.2022.108591</a>.
Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances in Mathematics</i>. Elsevier, 2022. <a href="https://doi.org/10.1016/j.aim.2022.108591">https://doi.org/10.1016/j.aim.2022.108591</a>.
K. Drach, D. Schleicher, Advances in Mathematics 408 (2022).
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