TY - JOUR
AB - 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.
AU - Chierchia, Luigi
AU - Koudjinan, Edmond
ID - 8689
IS - 1
JF - Regular and Chaotic Dynamics
KW - Nearly{integrable Hamiltonian systems
KW - perturbation theory
KW - KAM Theory
KW - Arnold's scheme
KW - Kolmogorov's set
KW - primary invariant tori
KW - Lagrangian tori
KW - measure estimates
KW - small divisors
KW - integrability on nowhere dense sets
KW - Diophantine frequencies.
SN - 1560-3547
TI - V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates
VL - 26
ER -
TY - GEN
AB - 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).
AU - Kaloshin, Vadim
AU - Koudjinan, Edmond
ID - 9435
TI - Non co-preservation of the 1/2 and 1/(2l+1)-rational caustics along deformations of circles
ER -