@article{11553,
  abstract     = {In holomorphic dynamics, complex box mappings arise as first return maps to wellchosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:
• To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.
• To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.
• Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:
  • the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (AnnMath 165:749–841, 2007), referred to below as KSS;
  • the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);
   • the QC-Criterion and the Spreading Principle from KSS.
The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings.
• To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:
   • puzzle pieces shrink to points,
   • (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.
• We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings
are the analogues of Lattès maps in this setting.
• We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.},
  author       = {Clark, Trevor and Drach, Kostiantyn and Kozlovski, Oleg and Strien, Sebastian Van},
  issn         = {2199-6806},
  journal      = {Arnold Mathematical Journal},
  number       = {2},
  pages        = {319--410},
  publisher    = {Springer Nature},
  title        = {{The dynamics of complex box mappings}},
  doi          = {10.1007/s40598-022-00200-7},
  volume       = {8},
  year         = {2022},
}

@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},
}