V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates

article V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates published yes Luigi Chierchia author Edmond Koudjinan author 52DF3E68-AEFA-11EA-95A4-124A3DDC885E0000-0003-2640-4049 VaKa department 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. Springer Nature2021 eng Nearly{integrable Hamiltonian systemsperturbation theoryKAM TheoryArnold's schemeKolmogorov's setprimary invariant toriLagrangian torimeasure estimatessmall divisorsintegrability on nowhere dense setsDiophantine frequencies. Regular and Chaotic Dynamics 1560-3547 2010.13243 00061445470000410.1134/S1560354721010044 26161-88 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:<a href="https://doi.org/10.1134/S1560354721010044">10.1134/S1560354721010044</a>. L. Chierchia, E. Koudjinan, Regular and Chaotic Dynamics 26 (2021) 61–88. 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:<a href="https://doi.org/10.1134/S1560354721010044">10.1134/S1560354721010044</a> Chierchia L, Koudjinan E. 2021. V.I. Arnold’s ‘“Global”’ KAM theorem and geometric measure estimates. Regular and Chaotic Dynamics. 26(1), 61–88. Chierchia, L., & Koudjinan, E. (2021). V.I. Arnold’s “‘Global’” KAM theorem and geometric measure estimates. Regular and Chaotic Dynamics. Springer Nature. <a href="https://doi.org/10.1134/S1560354721010044">https://doi.org/10.1134/S1560354721010044</a> Chierchia, Luigi, and Edmond Koudjinan. “V.I. Arnold’s ‘“Global”’ KAM Theorem and Geometric Measure Estimates.” Regular and Chaotic Dynamics. Springer Nature, 2021. <a href="https://doi.org/10.1134/S1560354721010044">https://doi.org/10.1134/S1560354721010044</a>. 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. 86892020-10-21T14:56:47Z2023-08-07T13:37:27Z