@article{8689,
abstract = {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.},
author = {Chierchia, Luigi and Koudjinan, Edmond},
issn = {1560-3547},
journal = {Regular and Chaotic Dynamics},
keywords = {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.},
number = {1},
pages = {61--88},
publisher = {Springer Nature},
title = {{V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates}},
doi = {10.1134/S1560354721010044},
volume = {26},
year = {2021},
}
@unpublished{9435,
abstract = {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).},
author = {Kaloshin, Vadim and Koudjinan, Edmond},
title = {{Non co-preservation of the 1/2 and 1/(2l+1)-rational caustics along deformations of circles}},
year = {2021},
}