where $\,\cdot\,$ denotes the Euclidean scalar product, $||\,\cdot\,||$ is the associated norm, $n \geq 1$ is an integer, $\T^n:={\mathbb{R}}^n / {\mathbb{Z}}^n$, $B$ is some open bounded neighborhood of the origin in ${\mathbb{R}}^n$ and $\omega\in {\mathbb{R}}^n$ is a vector which is assumed to be Diophantine: there exist constants $0<\gamma \leq 1$ and $\tau \geq n-1$ such that for any $k=(k_1,\dots,k_n) \in {\mathbb{Z}}^n\setminus \{0\}$, $$\label{dio}\tag{Dio$_{\gamma,\tau}$}
|k\cdot\omega|\geq \gamma|k|^{-\tau}, \quad |k|:=|k_1|+\cdots+|k_n|.$$ The torus $\mathcal{T}_\omega:=\T^n \times \{0\}$ is then invariant by the Hamiltonian flow of $H$, and the flow restricted on $\mathcal{T}_\omega$ is quasi-periodic with frequency $\omega$. There are several questions one can ask about the stability or instability properties of such an invariant torus, for instance:

-   Is $\mathcal{T}_\omega$ accumulated by a large set of Lagrangian invariant tori?

-   Is $\mathcal{T}_\omega$ topologically unstable? Recall that the invariant torus $\mathcal{T}_\omega$ is said to be topologically stable if it admits a basis of neighborhoods which are positively invariant by the Hamiltonian flow, and that it is topologically unstable if it is not topologically stable.

-   Given an arbitrary $r>0$ small enough and an arbitrary solution starting in the $r$-neighborhood of $\mathcal{T}_\omega$, how large is the “stability" time $T(r)$ during which the solution remains in the $2r$-neighborhood 