---
_id: '11717'
abstract:
- lang: eng
  text: "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.\r\nIn
    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”.\r\nOur 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."
acknowledgement: '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).'
article_number: '108591'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Kostiantyn
  full_name: Drach, Kostiantyn
  id: fe8209e2-906f-11eb-847d-950f8fc09115
  last_name: Drach
  orcid: 0000-0002-9156-8616
- first_name: Dierk
  full_name: Schleicher, Dierk
  last_name: Schleicher
citation:
  ama: 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>
  apa: Drach, K., &#38; 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>
  chicago: 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>.
  ieee: K. Drach and D. Schleicher, “Rigidity of Newton dynamics,” <i>Advances in
    Mathematics</i>, vol. 408, no. Part A. Elsevier, 2022.
  ista: Drach K, Schleicher D. 2022. Rigidity of Newton dynamics. Advances in Mathematics.
    408(Part A), 108591.
  mla: Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances
    in Mathematics</i>, vol. 408, no. Part A, 108591, Elsevier, 2022, doi:<a href="https://doi.org/10.1016/j.aim.2022.108591">10.1016/j.aim.2022.108591</a>.
  short: K. Drach, D. Schleicher, Advances in Mathematics 408 (2022).
corr_author: '1'
date_created: 2022-08-01T17:08:16Z
date_published: 2022-10-29T00:00:00Z
date_updated: 2025-04-14T07:53:45Z
day: '29'
ddc:
- '510'
department:
- _id: VaKa
doi: 10.1016/j.aim.2022.108591
ec_funded: 1
external_id:
  isi:
  - '000860924200005'
file:
- access_level: open_access
  checksum: 2710e6f5820f8c20a676ddcbb30f0e8d
  content_type: application/pdf
  creator: dernst
  date_created: 2023-02-02T07:39:09Z
  date_updated: 2023-02-02T07:39:09Z
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  file_name: 2022_AdvancesMathematics_Drach.pdf
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file_date_updated: 2023-02-02T07:39:09Z
has_accepted_license: '1'
intvolume: '       408'
isi: 1
issue: Part A
keyword:
- General Mathematics
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
  call_identifier: H2020
  grant_number: '885707'
  name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Advances in Mathematics
publication_identifier:
  issn:
  - 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Rigidity of Newton dynamics
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 408
year: '2022'
...