Quantitative KAM normal forms and sharp measure estimates

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.