--- _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' ...