---
_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: '9435'
abstract:
- lang: eng
  text: For any given positive integer l, we prove that every plane deformation of
    a circlewhich preserves the 1/2and 1/ (2l + 1) -rational caustics is trivial i.e.
    the deformationconsists only of similarities (rescalings and isometries).
article_processing_charge: No
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Edmond
  full_name: Koudjinan, Edmond
  id: 52DF3E68-AEFA-11EA-95A4-124A3DDC885E
  last_name: Koudjinan
  orcid: 0000-0003-2640-4049
citation:
  ama: Kaloshin V, Koudjinan E. Non co-preservation of the 1/2 and  1/(2l+1)-rational
    caustics along deformations of circles. 2021.
  apa: Kaloshin, V., &#38; Koudjinan, E. (2021). Non co-preservation of the 1/2 and 
    1/(2l+1)-rational caustics along deformations of circles.
  chicago: Kaloshin, Vadim, and Edmond Koudjinan. “Non Co-Preservation of the 1/2
    and  1/(2l+1)-Rational Caustics along Deformations of Circles,” 2021.
  ieee: V. Kaloshin and E. Koudjinan, “Non co-preservation of the 1/2 and  1/(2l+1)-rational
    caustics along deformations of circles.” 2021.
  ista: Kaloshin V, Koudjinan E. 2021. Non co-preservation of the 1/2 and  1/(2l+1)-rational
    caustics along deformations of circles.
  mla: Kaloshin, Vadim, and Edmond Koudjinan. <i>Non Co-Preservation of the 1/2 and 
    1/(2l+1)-Rational Caustics along Deformations of Circles</i>. 2021.
  short: V. Kaloshin, E. Koudjinan, (2021).
date_created: 2021-05-30T13:58:13Z
date_published: 2021-01-01T00:00:00Z
date_updated: 2021-06-01T09:10:22Z
ddc:
- '500'
department:
- _id: VaKa
file:
- access_level: open_access
  checksum: b281b5c2e3e90de0646c3eafcb2c6c25
  content_type: application/pdf
  creator: ekoudjin
  date_created: 2021-05-30T13:57:37Z
  date_updated: 2021-05-30T13:57:37Z
  file_id: '9436'
  file_name: CoExistence 2&3 caustics 3_17_6_2_3.pdf
  file_size: 353431
  relation: main_file
file_date_updated: 2021-05-30T13:57:37Z
has_accepted_license: '1'
language:
- iso: eng
oa: 1
oa_version: Submitted Version
status: public
title: Non co-preservation of the 1/2 and  1/(2l+1)-rational caustics along deformations
  of circles
type: preprint
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '8691'
abstract:
- lang: eng
  text: Given l>2ν>2d≥4, we prove the persistence of a Cantor--family of KAM tori
    of measure O(ε1/2−ν/l) for any non--degenerate nearly integrable Hamiltonian system
    of class Cl(D×Td), where D⊂Rd is a bounded domain, provided that the size ε of
    the perturbation is sufficiently small. This extends a result by D. Salamon in
    \cite{salamon2004kolmogorov} according to which we do have the persistence of
    a single KAM torus in the same framework. Moreover, it is well--known that, for
    the persistence of a single torus, the regularity assumption can not be improved.
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
citation:
  ama: Koudjinan E. A KAM theorem for finitely differentiable Hamiltonian systems.
    <i>Journal of Differential Equations</i>. 2020;269(6):4720-4750. doi:<a href="https://doi.org/10.1016/j.jde.2020.03.044">10.1016/j.jde.2020.03.044</a>
  apa: Koudjinan, E. (2020). A KAM theorem for finitely differentiable Hamiltonian
    systems. <i>Journal of Differential Equations</i>. Elsevier. <a href="https://doi.org/10.1016/j.jde.2020.03.044">https://doi.org/10.1016/j.jde.2020.03.044</a>
  chicago: Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian
    Systems.” <i>Journal of Differential Equations</i>. Elsevier, 2020. <a href="https://doi.org/10.1016/j.jde.2020.03.044">https://doi.org/10.1016/j.jde.2020.03.044</a>.
  ieee: E. Koudjinan, “A KAM theorem for finitely differentiable Hamiltonian systems,”
    <i>Journal of Differential Equations</i>, vol. 269, no. 6. Elsevier, pp. 4720–4750,
    2020.
  ista: Koudjinan E. 2020. A KAM theorem for finitely differentiable Hamiltonian systems.
    Journal of Differential Equations. 269(6), 4720–4750.
  mla: Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian Systems.”
    <i>Journal of Differential Equations</i>, vol. 269, no. 6, Elsevier, 2020, pp.
    4720–50, doi:<a href="https://doi.org/10.1016/j.jde.2020.03.044">10.1016/j.jde.2020.03.044</a>.
  short: E. Koudjinan, Journal of Differential Equations 269 (2020) 4720–4750.
date_created: 2020-10-21T15:03:05Z
date_published: 2020-09-05T00:00:00Z
date_updated: 2021-01-12T08:20:33Z
day: '05'
doi: 10.1016/j.jde.2020.03.044
extern: '1'
external_id:
  arxiv:
  - '1909.04099'
intvolume: '       269'
issue: '6'
keyword:
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1909.04099
month: '09'
oa: 1
oa_version: Preprint
page: 4720-4750
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: A KAM theorem for finitely differentiable Hamiltonian systems
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 269
year: '2020'
...
---
_id: '8694'
abstract:
- lang: eng
  text: "We develop algorithms and techniques to compute rigorous bounds for finite
    pieces of orbits of the critical points, for intervals of parameter values, in
    the quadratic family of one-dimensional maps fa(x)=a−x2. We illustrate the effectiveness
    of our approach by constructing a dynamically defined partition \U0001D4AB of
    the parameter interval Ω=[1.4,2] into almost 4×106 subintervals, for each of which
    we compute to high precision the orbits of the critical points up to some time
    N and other dynamically relevant quantities, several of which can vary greatly,
    possibly spanning several orders of magnitude. We also subdivide \U0001D4AB into
    a family \U0001D4AB+ of intervals, which we call stochastic intervals, and a family
    \U0001D4AB− of intervals, which we call regular intervals. We numerically prove
    that each interval ω∈\U0001D4AB+ has an escape time, which roughly means that
    some iterate of the critical point taken over all the parameters in ω has considerable
    width in the phase space. This suggests, in turn, that most parameters belonging
    to the intervals in \U0001D4AB+ are stochastic and most parameters belonging to
    the intervals in \U0001D4AB− are regular, thus the names. We prove that the intervals
    in \U0001D4AB+ occupy almost 90% of the total measure of Ω. The software and the
    data are freely available at http://www.pawelpilarczyk.com/quadr/, and a web page
    is provided for carrying out the calculations. The ideas and procedures can be
    easily generalized to apply to other parameterized families of dynamical systems."
article_number: '073143'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Ali
  full_name: Golmakani, Ali
  last_name: Golmakani
- first_name: Edmond
  full_name: Koudjinan, Edmond
  id: 52DF3E68-AEFA-11EA-95A4-124A3DDC885E
  last_name: Koudjinan
  orcid: 0000-0003-2640-4049
- first_name: Stefano
  full_name: Luzzatto, Stefano
  last_name: Luzzatto
- first_name: Pawel
  full_name: Pilarczyk, Pawel
  last_name: Pilarczyk
citation:
  ama: Golmakani A, Koudjinan E, Luzzatto S, Pilarczyk P. Rigorous numerics for critical
    orbits in the quadratic family. <i>Chaos</i>. 2020;30(7). doi:<a href="https://doi.org/10.1063/5.0012822">10.1063/5.0012822</a>
  apa: Golmakani, A., Koudjinan, E., Luzzatto, S., &#38; Pilarczyk, P. (2020). Rigorous
    numerics for critical orbits in the quadratic family. <i>Chaos</i>. AIP. <a href="https://doi.org/10.1063/5.0012822">https://doi.org/10.1063/5.0012822</a>
  chicago: Golmakani, Ali, Edmond Koudjinan, Stefano Luzzatto, and Pawel Pilarczyk.
    “Rigorous Numerics for Critical Orbits in the Quadratic Family.” <i>Chaos</i>.
    AIP, 2020. <a href="https://doi.org/10.1063/5.0012822">https://doi.org/10.1063/5.0012822</a>.
  ieee: A. Golmakani, E. Koudjinan, S. Luzzatto, and P. Pilarczyk, “Rigorous numerics
    for critical orbits in the quadratic family,” <i>Chaos</i>, vol. 30, no. 7. AIP,
    2020.
  ista: Golmakani A, Koudjinan E, Luzzatto S, Pilarczyk P. 2020. Rigorous numerics
    for critical orbits in the quadratic family. Chaos. 30(7), 073143.
  mla: Golmakani, Ali, et al. “Rigorous Numerics for Critical Orbits in the Quadratic
    Family.” <i>Chaos</i>, vol. 30, no. 7, 073143, AIP, 2020, doi:<a href="https://doi.org/10.1063/5.0012822">10.1063/5.0012822</a>.
  short: A. Golmakani, E. Koudjinan, S. Luzzatto, P. Pilarczyk, Chaos 30 (2020).
date_created: 2020-10-21T15:43:05Z
date_published: 2020-07-31T00:00:00Z
date_updated: 2021-01-12T08:20:34Z
day: '31'
doi: 10.1063/5.0012822
extern: '1'
external_id:
  arxiv:
  - '2004.13444'
intvolume: '        30'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2004.13444
month: '07'
oa: 1
oa_version: Preprint
publication: Chaos
publication_status: published
publisher: AIP
quality_controlled: '1'
status: public
title: Rigorous numerics for critical orbits in the quadratic family
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2020'
...
---
_id: '8693'
abstract:
- lang: eng
  text: We review V. I. Arnold’s 1963 celebrated paper [1] Proof of A. N. Kolmogorov’s
    Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation
    in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp”
    asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation).
    All constants involved are explicitly computed.
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 “pointwise” KAM theorem. <i>Regular
    and Chaotic Dynamics</i>. 2019;24:583–606. doi:<a href="https://doi.org/10.1134/S1560354719060017">10.1134/S1560354719060017</a>
  apa: Chierchia, L., &#38; Koudjinan, E. (2019). V. I. Arnold’s “pointwise” KAM theorem.
    <i>Regular and Chaotic Dynamics</i>. Springer. <a href="https://doi.org/10.1134/S1560354719060017">https://doi.org/10.1134/S1560354719060017</a>
  chicago: Chierchia, Luigi, and Edmond Koudjinan. “V. I. Arnold’s ‘Pointwise’ KAM
    Theorem.” <i>Regular and Chaotic Dynamics</i>. Springer, 2019. <a href="https://doi.org/10.1134/S1560354719060017">https://doi.org/10.1134/S1560354719060017</a>.
  ieee: L. Chierchia and E. Koudjinan, “V. I. Arnold’s ‘pointwise’ KAM theorem,” <i>Regular
    and Chaotic Dynamics</i>, vol. 24. Springer, pp. 583–606, 2019.
  ista: Chierchia L, Koudjinan E. 2019. V. I. Arnold’s “pointwise” KAM theorem. Regular
    and Chaotic Dynamics. 24, 583–606.
  mla: Chierchia, Luigi, and Edmond Koudjinan. “V. I. Arnold’s ‘Pointwise’ KAM Theorem.”
    <i>Regular and Chaotic Dynamics</i>, vol. 24, Springer, 2019, pp. 583–606, doi:<a
    href="https://doi.org/10.1134/S1560354719060017">10.1134/S1560354719060017</a>.
  short: L. Chierchia, E. Koudjinan, Regular and Chaotic Dynamics 24 (2019) 583–606.
date_created: 2020-10-21T15:25:45Z
date_published: 2019-12-10T00:00:00Z
date_updated: 2021-01-12T08:20:34Z
day: '10'
doi: 10.1134/S1560354719060017
extern: '1'
external_id:
  arxiv:
  - '1908.02523'
intvolume: '        24'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1908.02523
month: '12'
oa: 1
oa_version: Preprint
page: 583–606
publication: Regular and Chaotic Dynamics
publication_status: published
publisher: Springer
quality_controlled: '1'
status: public
title: V. I. Arnold’s “pointwise” KAM theorem
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2019'
...