{\mathbb{C}}^n$, with $s<L^{-1}$. For simplicity, we shall also say that a function is $(\alpha,\beta)$-Gevrey (respectively $\alpha$-Gevrey) if it belongs to $G^{\alpha,\beta}_{L_1,L_2}(\T^n \times U)$ (respectively to $G^{\alpha}_{L}(\T^n \times U)$).

Let us introduce the following notations: given some $\rho>0$, some domain $U \subseteq {\mathbb{R}}^n$ and some subset $S \subset \T^n \times U$, we denote by $$V_\rho U:=\{I \in {\mathbb{R}}^n \; | \; ||I-U||<\rho  \}, \quad V_\rho S:=\{(\theta,I) \in \T^n \times {\mathbb{R}}^n \; | \; ||(\theta,I)-S||<\rho  \}$$ the open $\rho$-neighborhood of $U$ in ${\mathbb{R}}^n$ and of $S$ in $\T^n \times {\mathbb{R}}^n$.

An invariant quasi-periodic Lagrangian torus $\mathcal{T}$ embedded in $\T^n \times U$ is [*doubly exponentially stable with exponent $u>0$*]{} if there exist positive constants $r_*$ and $C$ such that for any $r\leq r_*$ and any solution $(\theta(t),I(t))$ of the Hamiltonian system associated to $H$ with $(\theta(0),I(0)) \in V_r\mathcal{T}$, we have $$(\theta(0),I(0)) \in V_{2r}\mathcal{T}, \quad \forall |t| \leq \exp\left(\exp \left(C r^{-u}\right)\right).$$

Now let us come back to our first setting, that is we consider a Hamiltonian as in , with $\omega$ satisfying . In this setting, the torus $\mathcal{T}_\omega$ is doubly exponentially stable with exponent $u>0$ if there exist positive constants $r_*$ and $C$ such that for any $r\leq r_*$ and any solution $(\theta(t),I(t))$ 