---
_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
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.
Regular and Chaotic Dynamics. 2022;27(6):525-537. doi:10.1134/S1560354722050021
apa: Koudjinan, E., & Kaloshin, V. (2022). On some invariants of Birkhoff billiards
under conjugacy. Regular and Chaotic Dynamics. Springer Nature. https://doi.org/10.1134/S1560354722050021
chicago: Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff
Billiards under Conjugacy.” Regular and Chaotic Dynamics. Springer Nature,
2022. https://doi.org/10.1134/S1560354722050021.
ieee: E. Koudjinan and V. Kaloshin, “On some invariants of Birkhoff billiards under
conjugacy,” Regular and Chaotic Dynamics, 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.” Regular and Chaotic Dynamics, vol. 27, no. 6, Springer
Nature, 2022, pp. 525–37, doi:10.1134/S1560354722050021.
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: '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., & 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. Non Co-Preservation of the 1/2 and
1/(2l+1)-Rational Caustics along Deformations of Circles. 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: '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
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. Regular and Chaotic Dynamics. 2021;26(1):61-88. doi:10.1134/S1560354721010044
apa: Chierchia, L., & Koudjinan, E. (2021). V.I. Arnold’s “‘Global’” KAM theorem
and geometric measure estimates. Regular and Chaotic Dynamics. Springer
Nature. https://doi.org/10.1134/S1560354721010044
chicago: Chierchia, Luigi, and Edmond Koudjinan. “V.I. Arnold’s ‘“Global”’ KAM Theorem
and Geometric Measure Estimates.” Regular and Chaotic Dynamics. Springer
Nature, 2021. https://doi.org/10.1134/S1560354721010044.
ieee: L. Chierchia and E. Koudjinan, “V.I. Arnold’s ‘“Global”’ KAM theorem and geometric
measure estimates,” Regular and Chaotic Dynamics, 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.” Regular and Chaotic Dynamics, vol. 26,
no. 1, Springer Nature, 2021, pp. 61–88, doi:10.1134/S1560354721010044.
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: '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
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. Chaos. 2020;30(7). doi:10.1063/5.0012822
apa: Golmakani, A., Koudjinan, E., Luzzatto, S., & Pilarczyk, P. (2020). Rigorous
numerics for critical orbits in the quadratic family. Chaos. AIP. https://doi.org/10.1063/5.0012822
chicago: Golmakani, Ali, Edmond Koudjinan, Stefano Luzzatto, and Pawel Pilarczyk.
“Rigorous Numerics for Critical Orbits in the Quadratic Family.” Chaos.
AIP, 2020. https://doi.org/10.1063/5.0012822.
ieee: A. Golmakani, E. Koudjinan, S. Luzzatto, and P. Pilarczyk, “Rigorous numerics
for critical orbits in the quadratic family,” Chaos, 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.” Chaos, vol. 30, no. 7, 073143, AIP, 2020, doi:10.1063/5.0012822.
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: '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
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.
Journal of Differential Equations. 2020;269(6):4720-4750. doi:10.1016/j.jde.2020.03.044
apa: Koudjinan, E. (2020). A KAM theorem for finitely differentiable Hamiltonian
systems. Journal of Differential Equations. Elsevier. https://doi.org/10.1016/j.jde.2020.03.044
chicago: Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian
Systems.” Journal of Differential Equations. Elsevier, 2020. https://doi.org/10.1016/j.jde.2020.03.044.
ieee: E. Koudjinan, “A KAM theorem for finitely differentiable Hamiltonian systems,”
Journal of Differential Equations, 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.”
Journal of Differential Equations, vol. 269, no. 6, Elsevier, 2020, pp.
4720–50, doi:10.1016/j.jde.2020.03.044.
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: '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
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. Regular
and Chaotic Dynamics. 2019;24:583–606. doi:10.1134/S1560354719060017
apa: Chierchia, L., & Koudjinan, E. (2019). V. I. Arnold’s “pointwise” KAM theorem.
Regular and Chaotic Dynamics. Springer. https://doi.org/10.1134/S1560354719060017
chicago: Chierchia, Luigi, and Edmond Koudjinan. “V. I. Arnold’s ‘Pointwise’ KAM
Theorem.” Regular and Chaotic Dynamics. Springer, 2019. https://doi.org/10.1134/S1560354719060017.
ieee: L. Chierchia and E. Koudjinan, “V. I. Arnold’s ‘pointwise’ KAM theorem,” Regular
and Chaotic Dynamics, 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.”
Regular and Chaotic Dynamics, vol. 24, Springer, 2019, pp. 583–606, doi:10.1134/S1560354719060017.
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'
...