KAM Theory for secondary tori

Abstract

In [3] (Rend. Lincei Mat. Appl. 26 (2015), 1-10; see also arXiv:1503.08145 [math.DS]) the following result has been announced: Theorem. Consider a real-analytic nearly-integrable mechanical system with potential $f$, namely, a Hamiltonian system with real-analytic Hamiltonian $$H(y,x)=\frac{1}{2}\sum _{i=1}^n{y}_i^2+\epsilon f(x),$$ $(y,x)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$ being standard action--angle variables. For "general non-degenerate" potentials $f$'s there exists ${\epsilon }_0,a>0$ such that, if $0<\epsilon <{\epsilon }_0$, then the Liouville measure of the complementary of $H$-invariant tori is smaller than $\epsilon |\log \epsilon {|}^a$. In this paper we provide a proof of such result.

References in corpus (1)

• Explicit estimates on the measure of primary KAM tori