---
_id: '10706'
abstract:
- lang: eng
  text: This is a collection of problems composed by some participants of the workshop
    “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM
    on October 4–8, 2021.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Misha
  full_name: Bialy, Misha
  last_name: Bialy
- first_name: Corentin
  full_name: Fiorebe, Corentin
  id: 06619f18-9070-11eb-847d-d1ee780bd88b
  last_name: Fiorebe
- first_name: Alexey
  full_name: Glutsyuk, Alexey
  last_name: Glutsyuk
- first_name: Mark
  full_name: Levi, Mark
  last_name: Levi
- first_name: Alexander
  full_name: Plakhov, Alexander
  last_name: Plakhov
- first_name: Serge
  full_name: Tabachnikov, Serge
  last_name: Tabachnikov
citation:
  ama: Bialy M, Fiorebe C, Glutsyuk A, Levi M, Plakhov A, Tabachnikov S. Open problems
    on billiards and geometric optics. <i>Arnold Mathematical Journal</i>. 2022;8:411-422.
    doi:<a href="https://doi.org/10.1007/s40598-022-00198-y">10.1007/s40598-022-00198-y</a>
  apa: 'Bialy, M., Fiorebe, C., Glutsyuk, A., Levi, M., Plakhov, A., &#38; Tabachnikov,
    S. (2022). Open problems on billiards and geometric optics. <i>Arnold Mathematical
    Journal</i>. Hybrid: Springer Nature. <a href="https://doi.org/10.1007/s40598-022-00198-y">https://doi.org/10.1007/s40598-022-00198-y</a>'
  chicago: Bialy, Misha, Corentin Fiorebe, Alexey Glutsyuk, Mark Levi, Alexander Plakhov,
    and Serge Tabachnikov. “Open Problems on Billiards and Geometric Optics.” <i>Arnold
    Mathematical Journal</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s40598-022-00198-y">https://doi.org/10.1007/s40598-022-00198-y</a>.
  ieee: M. Bialy, C. Fiorebe, A. Glutsyuk, M. Levi, A. Plakhov, and S. Tabachnikov,
    “Open problems on billiards and geometric optics,” <i>Arnold Mathematical Journal</i>,
    vol. 8. Springer Nature, pp. 411–422, 2022.
  ista: Bialy M, Fiorebe C, Glutsyuk A, Levi M, Plakhov A, Tabachnikov S. 2022. Open
    problems on billiards and geometric optics. Arnold Mathematical Journal. 8, 411–422.
  mla: Bialy, Misha, et al. “Open Problems on Billiards and Geometric Optics.” <i>Arnold
    Mathematical Journal</i>, vol. 8, Springer Nature, 2022, pp. 411–22, doi:<a href="https://doi.org/10.1007/s40598-022-00198-y">10.1007/s40598-022-00198-y</a>.
  short: M. Bialy, C. Fiorebe, A. Glutsyuk, M. Levi, A. Plakhov, S. Tabachnikov, Arnold
    Mathematical Journal 8 (2022) 411–422.
conference:
  end_date: 2021-10-08
  location: Hybrid
  name: 'CIRM: Centre International de Rencontres Mathématiques'
  start_date: 2021-10-04
date_created: 2022-01-30T23:01:34Z
date_published: 2022-10-01T00:00:00Z
date_updated: 2023-02-27T07:34:08Z
day: '01'
department:
- _id: VaKa
doi: 10.1007/s40598-022-00198-y
external_id:
  arxiv:
  - '2110.10750'
intvolume: '         8'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2110.10750
month: '10'
oa: 1
oa_version: Preprint
page: 411-422
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: earlier_version
    url: https://conferences.cirm-math.fr/2383.html
scopus_import: '1'
status: public
title: Open problems on billiards and geometric optics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2022'
...
---
_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:
- access_level: open_access
  checksum: 16e7c659dee9073c6c8aeb87316ef201
  content_type: application/pdf
  creator: kschuh
  date_created: 2022-07-12T10:04:55Z
  date_updated: 2022-07-12T10:04:55Z
  file_id: '11559'
  file_name: 2022_ArnoldMathematicalJournal_Clark.pdf
  file_size: 2509915
  relation: main_file
  success: 1
file_date_updated: 2022-07-12T10:04:55Z
has_accepted_license: '1'
intvolume: '         8'
issue: '2'
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
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2022'
...