On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems

Abstract

Consider an $n$-degrees-of-freedom real-analytic mechanical system with potential $ϵf=ϵf(x)$, $x$ being a $n$-dimensional angle variable. Then, for ‘‘general’’ potentials $f$’s and $ϵ$ small enough, the Liouville measure of the complementary of invariant tori is smaller than $ϵ|\ln ϵ{|}^a$ (for a suitable $a>0$).