@article{11717,
  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 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.},
  author       = {Drach, Kostiantyn and Schleicher, Dierk},
  issn         = {0001-8708},
  journal      = {Advances in Mathematics},
  keywords     = {General Mathematics},
  number       = {Part A},
  publisher    = {Elsevier},
  title        = {{Rigidity of Newton dynamics}},
  doi          = {10.1016/j.aim.2022.108591},
  volume       = {408},
  year         = {2022},
}