of $\mathcal{T}_\omega$?

These questions are related to, respectively, KAM theory, Arnold diffusion and Nekhoroshev theory. Concerning $(1)$, it is well-known that if $H$ is sufficiently smooth, under a generic condition, $\mathcal{T}_\omega$ is accumulated by a set of Lagrangian invariant tori of positive Lebesgue measure, which has Lebesgue density one at $\mathcal{T}_\omega$. Assuming $H$ to be real-analytic, without further assumption it is conjectured (see [@Her98]) that $\mathcal{T}_\omega$ is accumulated by a set of Lagrangian invariant tori of positive Lebesgue measure (see [@EFK15] for some partial results). Concerning $(2)$, almost nothing is known in the real-analytic category : for instance, it is still not known if there exists a real-analytic torus $\mathcal{T}_\omega$ which is topologically unstable. In the smooth category, Douady ([@Dou88]) gave examples of topologically unstable invariant tori with any given Birkhoff normal form at the invariant torus. Degenerate examples having almost all orbits oscillating between the neighborhood of the invariant Diophantine torus and infinity were constructed in [@EFK15]. As for the generic behavior, even in the smooth category this is still an open problem (see [@GK14] for some partial result in the case $n=3$).

Finally, concerning $(3)$, it is well-known that for real-analytic systems, $T(r)$ is always at least exponentially 