Rigidity of Newton dynamics Drach, Kostiantyn ; https://orcid.org/0000-0002-9156-8616 Schleicher, Dierk General Mathematics ddc:510 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. Elsevier 2022 info:eu-repo/semantics/article doc-type:Article Article http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/11717 https://research-explorer.ista.ac.at/download/11717/12474 Drach K, Schleicher D. Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. 2022;408(Part A). doi:<a href="https://doi.org/10.1016/j.aim.2022.108591">10.1016/j.aim.2022.108591</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2022.108591 info:eu-repo/semantics/altIdentifier/issn/0001-8708 info:eu-repo/semantics/altIdentifier/wos/000860924200005 info:eu-repo/semantics/openAccess