Positive Measure of Effective Quasi-Periodic Motion Near a Diophantine Torus

It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus \({\mathcal {T}}_0\), 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 \({\mathcal {T}}_0\) 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 \({\mathcal {T}}_0\). This open set can be thought of as a neighborhood of a hypothetical invariant set of Lagrangian Diophantine quasi-periodic tori, which may or may not exist.