V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates

    2021, Volume 26, Number 1, pp.  61-88

    Author(s): Chierchia L., Koudjinan C. E.

    This paper continues the discussion started in [10] 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 set are provided in case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the $d$-torus and (B) a domain with $C^2$ boundary times the $d$-torus. All constants are explicitly given.
    Keywords: nearly-integrable Hamiltonian systems, perturbation theory, KAM theory, Arnold’s scheme, Kolmogorov set, primary invariant tori, Lagrangian tori, measure estimates, small divisors, integrability on nowhere dense sets, Diophantine frequencies
    Citation: Chierchia L., Koudjinan C. E., V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates, Regular and Chaotic Dynamics, 2021, Volume 26, Number 1, pp. 61-88

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