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

Abstract

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.