---
_id: '11553'
abstract:
- lang: eng
  text: "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:\r\n• 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.\r\n• 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.\r\n• Many results in holomorphic dynamics rely on
    an interplay between combinatorial and analytic techniques. In this setting, some
    of these tools are:\r\n  • 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;\r\n  • the Covering Lemma (which controls the moduli of pullbacks
    of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);\r\n   • the
    QC-Criterion and the Spreading Principle from KSS.\r\nThe 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.\r\n• 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:\r\n   • puzzle pieces
    shrink to points,\r\n   • (under some assumptions) topologically conjugate non-renormalizable
    polynomials and box mappings are quasiconformally conjugate.\r\n• 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\r\nare the analogues
    of Lattès maps in this setting.\r\n• 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."
acknowledgement: We would also like to thank Dzmitry Dudko and Dierk Schleicher for
  many stimulating discussions and encouragement during our work on this project,
  and Weixiao Shen, Mikhail Hlushchanka and the referee for helpful comments. We are
  grateful to Leon Staresinic who carefully read the revised version of the manuscript
  and provided many helpful suggestions.
article_processing_charge: No
article_type: original
author:
- first_name: Trevor
  full_name: Clark, Trevor
  last_name: Clark
- first_name: Kostiantyn
  full_name: Drach, Kostiantyn
  id: fe8209e2-906f-11eb-847d-950f8fc09115
  last_name: Drach
  orcid: 0000-0002-9156-8616
- first_name: Oleg
  full_name: Kozlovski, Oleg
  last_name: Kozlovski
- first_name: Sebastian Van
  full_name: Strien, Sebastian Van
  last_name: Strien
citation:
  ama: Clark T, Drach K, Kozlovski O, Strien SV. The dynamics of complex box mappings.
    <i>Arnold Mathematical Journal</i>. 2022;8(2):319-410. doi:<a href="https://doi.org/10.1007/s40598-022-00200-7">10.1007/s40598-022-00200-7</a>
  apa: Clark, T., Drach, K., Kozlovski, O., &#38; Strien, S. V. (2022). The dynamics
    of complex box mappings. <i>Arnold Mathematical Journal</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s40598-022-00200-7">https://doi.org/10.1007/s40598-022-00200-7</a>
  chicago: Clark, Trevor, Kostiantyn Drach, Oleg Kozlovski, and Sebastian Van Strien.
    “The Dynamics of Complex Box Mappings.” <i>Arnold Mathematical Journal</i>. Springer
    Nature, 2022. <a href="https://doi.org/10.1007/s40598-022-00200-7">https://doi.org/10.1007/s40598-022-00200-7</a>.
  ieee: T. Clark, K. Drach, O. Kozlovski, and S. V. Strien, “The dynamics of complex
    box mappings,” <i>Arnold Mathematical Journal</i>, vol. 8, no. 2. Springer Nature,
    pp. 319–410, 2022.
  ista: Clark T, Drach K, Kozlovski O, Strien SV. 2022. The dynamics of complex box
    mappings. Arnold Mathematical Journal. 8(2), 319–410.
  mla: Clark, Trevor, et al. “The Dynamics of Complex Box Mappings.” <i>Arnold Mathematical
    Journal</i>, vol. 8, no. 2, Springer Nature, 2022, pp. 319–410, doi:<a href="https://doi.org/10.1007/s40598-022-00200-7">10.1007/s40598-022-00200-7</a>.
  short: T. Clark, K. Drach, O. Kozlovski, S.V. Strien, Arnold Mathematical Journal
    8 (2022) 319–410.
date_created: 2022-07-10T22:01:53Z
date_published: 2022-06-01T00:00:00Z
date_updated: 2023-02-16T10:02:12Z
day: '01'
ddc:
- '500'
department:
- _id: VaKa
doi: 10.1007/s40598-022-00200-7
ec_funded: 1
file:
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language:
- iso: eng
month: '06'
oa: 1
oa_version: None
page: 319-410
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
  call_identifier: H2020
  grant_number: '885707'
  name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Arnold Mathematical Journal
publication_identifier:
  eissn:
  - 2199-6806
  issn:
  - 2199-6792
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
  link:
  - relation: erratum
    url: https://doi.org/10.1007/s40598-022-00209-y
  - relation: erratum
    url: https://doi.org/10.1007/s40598-022-00218-x
scopus_import: '1'
status: public
title: The dynamics of complex box mappings
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type: journal_article
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...
---
_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).
date_created: 2022-08-01T17:08:16Z
date_published: 2022-10-29T00:00:00Z
date_updated: 2023-08-03T12:36:07Z
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
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  date_created: 2023-02-02T07:39:09Z
  date_updated: 2023-02-02T07:39:09Z
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  file_size: 2164036
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  success: 1
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'
...