V.I. Arnold's "Global" KAM Theorem and geometric measure estimates

TL;DR

This paper presents a detailed, explicit version of Arnold's global KAM theorem, providing measure estimates and conjugacy results for nearly-integrable Hamiltonian systems on specific phase space subsets.

Contribution

It offers a comprehensive, explicit proof of a global Arnold's KAM theorem with detailed measure estimates and constants for various phase space configurations.

Findings

Whitney conjugacy of Hamiltonian systems on measure-positive sets

Explicit measure estimates for Kolmogorov's set

Constants explicitly computed for different phase space domains

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 neighborhood 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.