Abstract (EN)

It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus T0, invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian Diophantine quasi-periodic invariant tori. While the conjecture is still open, we will prove the following weaker statement: there exists an open set of positive measure (in fact, the relative measure of the complement is exponentially small) around T0 such that the motion of all initial conditions in this set is "effectively" quasi-periodic in the sense that they are close to being quasi-periodic for an interval of time which is doubly exponentially long with respect to the inverse of the distance to T0. This open set can be thought as a neighborhood of a hypothetical invariant set of Lagrangian Diophantine quasi-periodic tori, which may or may not exist.