Abstract
The question of the total measure of invariant tori in analytic, nearly integrable Hamiltonian systems is considered. In 1985, Arnol’d, Kozlov and Neishtadt, in the Encyclopaedia of Mathematical Sciences, [4] and in subsequent editions, conjectured that in $n = 2$ degrees of freedom the measure of the non-torus set of general analytic nearly integrable systems away from critical points is exponentially small with the size $\varepsilon$ of the perturbation, and that for $n\ge 3$ the measure is, in general, of order $\varepsilon$ (rather than $\sqrt{\varepsilon}$ as predicted by classical KAM Theory).
In the case of generic analytic mechanical Hamiltonian systems, we prove lower bounds on the measure of primary and secondary invariant tori, which are in agreement, up to a logarithmic correction, with the above conjectures.
