Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Abstract:

Consider a real analytical Hamiltonian system of KAM type $H(p,q)$ $=N(p)+P(p,q)$ that has $n$ degrees of freedom ($n>2$) and is positive definite in $p$. Let $\Omega =\{\omega \in \mathbb R^n |\langle \bar k,\omega \rangle =0, \ \bar k\in \mathbb Z^n\}$. In this paper we show that for most rotation vectors in $\Omega$, in the sense of ($n-1$)-dimensional Lebesgue measure, there is at least one ($n-1$)-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.