V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates - Chierchia, Luigi; Koudjinan, Edmond (Regular and Chaotic Dynamics, 2021)
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.
published
yes
Keywords: article; 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