Explicit estimates on the measure of primary KAM tori

TL;DR

This paper provides explicit estimates on the measure of phase points not on primary KAM tori in nearly-integrable Hamiltonian systems, detailing how these estimates depend on system parameters and phase-space domain.

Contribution

It offers explicit bounds on the measure of non-torus points, clarifying the dependence on the unperturbed system and phase-space domain.

Findings

Measure of non-torus points is O(√ε)

Explicit dependence of the constant on system parameters

Analysis of phase-space domain influence

Abstract

From KAM Theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, "primary" tori in a nearly--integrable, real--analytic Hamiltonian system is $O(\sqrt{\varepsilon})$, if $\varepsilon$ is the size of the perturbation. In this paper we discuss how the constant in front of $\sqrt{\varepsilon}$ depends on the unperturbed system and in particular on the phase--space domain.