# 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.

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