[{"author":[{"orcid":"0000-0003-2640-4049","full_name":"Koudjinan, Edmond","first_name":"Edmond","last_name":"Koudjinan","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E"},{"last_name":"Kaloshin","first_name":"Vadim","full_name":"Kaloshin, Vadim","orcid":"0000-0002-6051-2628","id":"FE553552-CDE8-11E9-B324-C0EBE5697425"}],"issue":"6","_id":"12145","scopus_import":"1","title":"On some invariants of Birkhoff billiards under conjugacy","intvolume":"        27","publication_status":"published","article_processing_charge":"No","date_created":"2023-01-12T12:06:49Z","department":[{"_id":"VaKa"}],"page":"525-537","ec_funded":1,"quality_controlled":"1","article_type":"original","publisher":"Springer Nature","isi":1,"external_id":{"isi":["000865267300002"],"arxiv":["2105.14640"]},"date_updated":"2023-08-04T08:59:14Z","year":"2022","citation":{"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.","ista":"Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under conjugacy. Regular and Chaotic Dynamics. 27(6), 525–537.","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>","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.","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>."},"abstract":[{"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.","lang":"eng"}],"doi":"10.1134/S1560354722050021","arxiv":1,"day":"03","volume":27,"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).","publication":"Regular and Chaotic Dynamics","month":"10","oa_version":"Preprint","project":[{"call_identifier":"H2020","_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","grant_number":"885707","name":"Spectral rigidity and integrability for billiards and geodesic flows"}],"language":[{"iso":"eng"}],"keyword":["Mechanical Engineering","Applied Mathematics","Mathematical Physics","Modeling and Simulation","Statistical and Nonlinear Physics","Mathematics (miscellaneous)"],"date_published":"2022-10-03T00:00:00Z","type":"journal_article","oa":1,"publication_identifier":{"eissn":["1468-4845"],"issn":["1560-3547"]},"related_material":{"link":[{"relation":"erratum","url":"https://doi.org/10.1134/s1560354722060107"}]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2105.14640","open_access":"1"}]},{"issue":"1","author":[{"last_name":"Chierchia","first_name":"Luigi","full_name":"Chierchia, Luigi"},{"first_name":"Edmond","last_name":"Koudjinan","orcid":"0000-0003-2640-4049","full_name":"Koudjinan, Edmond","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E"}],"scopus_import":"1","_id":"8689","intvolume":"        26","title":"V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates","department":[{"_id":"VaKa"}],"article_processing_charge":"No","date_created":"2020-10-21T14:56:47Z","publication_status":"published","quality_controlled":"1","page":"61-88","article_type":"original","publisher":"Springer Nature","external_id":{"isi":["000614454700004"],"arxiv":["2010.13243"]},"isi":1,"citation":{"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>","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>","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.","short":"L. Chierchia, E. Koudjinan, Regular and Chaotic Dynamics 26 (2021) 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>.","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."},"year":"2021","date_updated":"2023-08-07T13:37:27Z","abstract":[{"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.","lang":"eng"}],"day":"03","arxiv":1,"doi":"10.1134/S1560354721010044","ddc":["515"],"volume":26,"publication":"Regular and Chaotic Dynamics","month":"02","oa_version":"Preprint","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"}],"type":"journal_article","date_published":"2021-02-03T00:00:00Z","oa":1,"publication_identifier":{"issn":["1560-3547"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2010.13243"}]},{"_id":"8426","author":[{"full_name":"Buhovsky, Lev","last_name":"Buhovsky","first_name":"Lev"},{"full_name":"Kaloshin, Vadim","orcid":"0000-0002-6051-2628","last_name":"Kaloshin","first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425"}],"article_processing_charge":"No","date_created":"2020-09-17T10:43:21Z","publication_status":"published","intvolume":"        23","title":"Nonisometric domains with the same Marvizi-Melrose invariants","quality_controlled":"1","page":"54-59","publisher":"Springer Nature","article_type":"original","citation":{"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.","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>","ista":"Buhovsky L, Kaloshin V. 2018. Nonisometric domains with the same Marvizi-Melrose invariants. Regular and Chaotic Dynamics. 23, 54–59.","short":"L. Buhovsky, V. Kaloshin, Regular and Chaotic Dynamics 23 (2018) 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>."},"year":"2018","date_updated":"2021-01-12T08:19:11Z","external_id":{"arxiv":["1801.00952"]},"day":"05","arxiv":1,"doi":"10.1134/s1560354718010057","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."}],"volume":23,"extern":"1","publication":"Regular and Chaotic Dynamics","oa_version":"Preprint","month":"02","language":[{"iso":"eng"}],"type":"journal_article","date_published":"2018-02-05T00:00:00Z","publication_identifier":{"issn":["1560-3547","1468-4845"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1801.00952","open_access":"1"}],"status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}]