Drach, KostiantynISTA ; Schleicher, Dierk
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.
Advances in Mathematics
We are grateful to a number of colleagues for helpful and inspiring discussions during the time when we worked on this project, in particular Dima Dudko, Misha Hlushchanka, John Hubbard, Misha Lyubich, Oleg Kozlovski, and Sebastian van Strien. Finally, we would like to thank our dynamics research group for numerous helpful and enjoyable discussions: Konstantin Bogdanov, Roman Chernov, Russell Lodge, Steffen Maaß, David Pfrang, Bernhard Reinke, Sergey Shemyakov, and Maik Sowinski. We gratefully acknowledge support by the Advanced Grant “HOLOGRAM” (#695 621) of the European Research Council (ERC), as well as hospitality of Cornell University in the spring of 2018 while much of this work was prepared. The first-named author also acknowledges the support of the ERC Advanced Grant “SPERIG” (#885 707).
Drach K, Schleicher D. Rigidity of Newton dynamics. Advances in Mathematics. 2022;408. doi:10.1016/j.aim.2022.108591
Drach, K., & Schleicher, D. (2022). Rigidity of Newton dynamics. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2022.108591
Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” Advances in Mathematics. Elsevier, 2022. https://doi.org/10.1016/j.aim.2022.108591.
K. Drach and D. Schleicher, “Rigidity of Newton dynamics,” Advances in Mathematics, 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.” Advances in Mathematics, vol. 408, 108591, Elsevier, 2022, doi:10.1016/j.aim.2022.108591.
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