KAM Theory for secondary tori

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 $$H(y,x)=\frac{1}{2}\sum _{i=1}^n{y}_i^2+ϵf(x),$$ $(y,x)\in{\mathbb R}^n\times{\mathbb T}^n$ being standard action–angle variables. For “general non-degenerate” potentials $f$’s there exists ${ϵ}_0,a>0$ such that, if $0<ϵ<{ϵ}_0$, then the Liouville measure of the complementary of $H$-invariant tori is smaller than $ϵ\mathrm{\mid }\log ϵ{\mathrm{\mid }}^a$. In this paper we provide a proof of such result.