---
_id: '12145'
abstract:
- lang: eng
  text: In the class of strictly convex smooth boundaries each of which has no strip
    around its boundary foliated by invariant curves, we prove that the Taylor coefficients
    of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In
    contrast, we prove that any two elliptic billiard maps are C0-conjugate near their
    respective boundaries, and C∞-conjugate, near the boundary and away from a line
    passing through the center of the underlying ellipse. We also prove that, if the
    billiard maps corresponding to two ellipses are topologically conjugate, then
    the two ellipses are similar.
acknowledgement: "We are grateful to the anonymous referees for their careful reading
  and valuable remarks and\r\ncomments which helped to improve the paper significantly.
  We gratefully acknowledge support from the European Research Council (ERC) through
  the Advanced Grant “SPERIG” (#885707)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Edmond
  full_name: Koudjinan, Edmond
  id: 52DF3E68-AEFA-11EA-95A4-124A3DDC885E
  last_name: Koudjinan
  orcid: 0000-0003-2640-4049
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: Koudjinan E, Kaloshin V. On some invariants of Birkhoff billiards under conjugacy.
    <i>Regular and Chaotic Dynamics</i>. 2022;27(6):525-537. doi:<a href="https://doi.org/10.1134/S1560354722050021">10.1134/S1560354722050021</a>
  apa: Koudjinan, E., &#38; Kaloshin, V. (2022). On some invariants of Birkhoff billiards
    under conjugacy. <i>Regular and Chaotic Dynamics</i>. Springer Nature. <a href="https://doi.org/10.1134/S1560354722050021">https://doi.org/10.1134/S1560354722050021</a>
  chicago: Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff
    Billiards under Conjugacy.” <i>Regular and Chaotic Dynamics</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1134/S1560354722050021">https://doi.org/10.1134/S1560354722050021</a>.
  ieee: E. Koudjinan and V. Kaloshin, “On some invariants of Birkhoff billiards under
    conjugacy,” <i>Regular and Chaotic Dynamics</i>, vol. 27, no. 6. Springer Nature,
    pp. 525–537, 2022.
  ista: Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under
    conjugacy. Regular and Chaotic Dynamics. 27(6), 525–537.
  mla: Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff Billiards
    under Conjugacy.” <i>Regular and Chaotic Dynamics</i>, vol. 27, no. 6, Springer
    Nature, 2022, pp. 525–37, doi:<a href="https://doi.org/10.1134/S1560354722050021">10.1134/S1560354722050021</a>.
  short: E. Koudjinan, V. Kaloshin, Regular and Chaotic Dynamics 27 (2022) 525–537.
date_created: 2023-01-12T12:06:49Z
date_published: 2022-10-03T00:00:00Z
date_updated: 2023-08-04T08:59:14Z
day: '03'
department:
- _id: VaKa
doi: 10.1134/S1560354722050021
ec_funded: 1
external_id:
  arxiv:
  - '2105.14640'
  isi:
  - '000865267300002'
intvolume: '        27'
isi: 1
issue: '6'
keyword:
- Mechanical Engineering
- Applied Mathematics
- Mathematical Physics
- Modeling and Simulation
- Statistical and Nonlinear Physics
- Mathematics (miscellaneous)
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2105.14640
month: '10'
oa: 1
oa_version: Preprint
page: 525-537
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
  call_identifier: H2020
  grant_number: '885707'
  name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Regular and Chaotic Dynamics
publication_identifier:
  eissn:
  - 1468-4845
  issn:
  - 1560-3547
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
  link:
  - relation: erratum
    url: https://doi.org/10.1134/s1560354722060107
scopus_import: '1'
status: public
title: On some invariants of Birkhoff billiards under conjugacy
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 27
year: '2022'
...
---
_id: '8689'
abstract:
- lang: eng
  text: 'This paper continues the discussion started in [CK19] concerning Arnold''s
    legacy on classical KAM theory and (some of) its modern developments. We prove
    a detailed and explicit `global'' Arnold''s KAM Theorem, which yields, in particular,
    the Whitney conjugacy of a non{degenerate, real{analytic, nearly-integrable Hamiltonian
    system to an integrable system on a closed, nowhere dense, positive measure subset
    of the phase space. Detailed measure estimates on the Kolmogorov''s set are provided
    in the case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded)
    set times the d-torus and (B) a domain with C2 boundary times the d-torus. All
    constants are explicitly given.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Luigi
  full_name: Chierchia, Luigi
  last_name: Chierchia
- first_name: Edmond
  full_name: Koudjinan, Edmond
  id: 52DF3E68-AEFA-11EA-95A4-124A3DDC885E
  last_name: Koudjinan
  orcid: 0000-0003-2640-4049
citation:
  ama: Chierchia L, Koudjinan E. V.I. Arnold’s “‘Global’” KAM theorem and geometric
    measure estimates. <i>Regular and Chaotic Dynamics</i>. 2021;26(1):61-88. doi:<a
    href="https://doi.org/10.1134/S1560354721010044">10.1134/S1560354721010044</a>
  apa: Chierchia, L., &#38; Koudjinan, E. (2021). V.I. Arnold’s “‘Global’” KAM theorem
    and geometric measure estimates. <i>Regular and Chaotic Dynamics</i>. Springer
    Nature. <a href="https://doi.org/10.1134/S1560354721010044">https://doi.org/10.1134/S1560354721010044</a>
  chicago: Chierchia, Luigi, and Edmond Koudjinan. “V.I. Arnold’s ‘“Global”’ KAM Theorem
    and Geometric Measure Estimates.” <i>Regular and Chaotic Dynamics</i>. Springer
    Nature, 2021. <a href="https://doi.org/10.1134/S1560354721010044">https://doi.org/10.1134/S1560354721010044</a>.
  ieee: L. Chierchia and E. Koudjinan, “V.I. Arnold’s ‘“Global”’ KAM theorem and geometric
    measure estimates,” <i>Regular and Chaotic Dynamics</i>, vol. 26, no. 1. Springer
    Nature, pp. 61–88, 2021.
  ista: Chierchia L, Koudjinan E. 2021. V.I. Arnold’s ‘“Global”’ KAM theorem and geometric
    measure estimates. Regular and Chaotic Dynamics. 26(1), 61–88.
  mla: Chierchia, Luigi, and Edmond Koudjinan. “V.I. Arnold’s ‘“Global”’ KAM Theorem
    and Geometric Measure Estimates.” <i>Regular and Chaotic Dynamics</i>, vol. 26,
    no. 1, Springer Nature, 2021, pp. 61–88, doi:<a href="https://doi.org/10.1134/S1560354721010044">10.1134/S1560354721010044</a>.
  short: L. Chierchia, E. Koudjinan, Regular and Chaotic Dynamics 26 (2021) 61–88.
date_created: 2020-10-21T14:56:47Z
date_published: 2021-02-03T00:00:00Z
date_updated: 2023-08-07T13:37:27Z
day: '03'
ddc:
- '515'
department:
- _id: VaKa
doi: 10.1134/S1560354721010044
external_id:
  arxiv:
  - '2010.13243'
  isi:
  - '000614454700004'
intvolume: '        26'
isi: 1
issue: '1'
keyword:
- Nearly{integrable Hamiltonian systems
- perturbation theory
- KAM Theory
- Arnold's scheme
- Kolmogorov's set
- primary invariant tori
- Lagrangian tori
- measure estimates
- small divisors
- integrability on nowhere dense sets
- Diophantine frequencies.
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2010.13243
month: '02'
oa: 1
oa_version: Preprint
page: 61-88
publication: Regular and Chaotic Dynamics
publication_identifier:
  issn:
  - 1560-3547
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 26
year: '2021'
...
---
_id: '8426'
abstract:
- lang: eng
  text: For any strictly convex planar domain Ω ⊂ R2 with a C∞ boundary one can associate
    an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5].
    These invariants can generically be determined using the spectrum of the Dirichlet
    problem of the Laplace operator. A natural question asks if this collection is
    sufficient to determine Ω up to isometry. In this paper we give a counterexample,
    namely, we present two nonisometric domains Ω and Ω¯ with the same collection
    of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic
    orbits {Sn}n≥1 (resp. {S¯n}n⩾1) of period going to infinity such that Sn and S¯n
    have the same period and perimeter for each n.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Lev
  full_name: Buhovsky, Lev
  last_name: Buhovsky
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: Buhovsky L, Kaloshin V. Nonisometric domains with the same Marvizi-Melrose
    invariants. <i>Regular and Chaotic Dynamics</i>. 2018;23:54-59. doi:<a href="https://doi.org/10.1134/s1560354718010057">10.1134/s1560354718010057</a>
  apa: Buhovsky, L., &#38; Kaloshin, V. (2018). Nonisometric domains with the same
    Marvizi-Melrose invariants. <i>Regular and Chaotic Dynamics</i>. Springer Nature.
    <a href="https://doi.org/10.1134/s1560354718010057">https://doi.org/10.1134/s1560354718010057</a>
  chicago: Buhovsky, Lev, and Vadim Kaloshin. “Nonisometric Domains with the Same
    Marvizi-Melrose Invariants.” <i>Regular and Chaotic Dynamics</i>. Springer Nature,
    2018. <a href="https://doi.org/10.1134/s1560354718010057">https://doi.org/10.1134/s1560354718010057</a>.
  ieee: L. Buhovsky and V. Kaloshin, “Nonisometric domains with the same Marvizi-Melrose
    invariants,” <i>Regular and Chaotic Dynamics</i>, vol. 23. Springer Nature, pp.
    54–59, 2018.
  ista: Buhovsky L, Kaloshin V. 2018. Nonisometric domains with the same Marvizi-Melrose
    invariants. Regular and Chaotic Dynamics. 23, 54–59.
  mla: Buhovsky, Lev, and Vadim Kaloshin. “Nonisometric Domains with the Same Marvizi-Melrose
    Invariants.” <i>Regular and Chaotic Dynamics</i>, vol. 23, Springer Nature, 2018,
    pp. 54–59, doi:<a href="https://doi.org/10.1134/s1560354718010057">10.1134/s1560354718010057</a>.
  short: L. Buhovsky, V. Kaloshin, Regular and Chaotic Dynamics 23 (2018) 54–59.
date_created: 2020-09-17T10:43:21Z
date_published: 2018-02-05T00:00:00Z
date_updated: 2021-01-12T08:19:11Z
day: '05'
doi: 10.1134/s1560354718010057
extern: '1'
external_id:
  arxiv:
  - '1801.00952'
intvolume: '        23'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1801.00952
month: '02'
oa: 1
oa_version: Preprint
page: 54-59
publication: Regular and Chaotic Dynamics
publication_identifier:
  issn:
  - 1560-3547
  - 1468-4845
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Nonisometric domains with the same Marvizi-Melrose invariants
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 23
year: '2018'
...