Quantitative KAM normal forms and sharp measure estimates

TL;DR

This paper provides explicit, sharp measure estimates for the persistence of KAM tori under small perturbations, improves Arnold's scheme by removing logarithmic corrections, and compares three quantitative KAM normal forms.

Contribution

It offers a detailed, self-contained proof of measure estimates for KAM tori, removing logarithmic corrections, and explicitly computes all KAM constants with applications to a simple mechanical system.

Findings

Explicit measure estimate for persistent KAM tori.

Removal of logarithmic correction in Arnold's scheme.

Comparison of three quantitative KAM normal forms.

Abstract

It is widespread since the beginning of KAM Theory that, under "sufficiently small" perturbation, of size $\epsilon$, apart a set of measure $O(\sqrt{\epsilon})$, all the KAM Tori of a non-degenerate integrable Hamiltonian system persist up to a small deformation. However, no explicit, self-contained proof of this fact exists so far. In the present Thesis, we give a detailed proof of how to get rid of a logarithmic correction (due to a Fourier cut-off) in Arnold's scheme and then use it to prove an explicit and "sharp" Theorem of integrability on Cantor-type set. In particular, we give an explicit proof of the above-mentioned measure estimate on the measure of persistent primary KAM tori. We also prove three quantitative KAM normal forms following closely the original ideas of the pioneers Kolmogorov, Arnold and Moser, computing explicitly all the KAM constants involved and fix some "physical dimension" issues by means of appropriate rescalings. Finally, we compare those three quantitative KAM normal forms on a simple mechanical system.

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