any $I\in \bar{D}$ and any integer $m\geq 2$, we define the Taylor polynomial $T_I^m h \in P_2(n,m)$ (starting at order 2) by $$T_I^m h(X)=\sum_{l=2}^{m}(l!)^{-1}\nabla^l h(I).(X)^l, \quad X \in {\mathbb{R}}^n$$ and recall the following definition.

\[defK\] Let us define by $N(n,2)$ the subset of $P_2(n,2)$ consisting of non-degenerate quadratic forms. A smooth function $h : \bar{D} \rightarrow {\mathbb{R}}$ is said to be Kolmogorov non-degenerate if for all $I \in \bar{D}$, $T_I^2h \in N(n,2)$. Upon restricting $\bar{D}$ if necessary, we may always assume that its gradient $\nabla h: \bar{D} \rightarrow \bar{\Omega}$ is a diffeomorphism onto its image.

Our second main result is then as follows.

\[Th2\] Let $h \in G^\alpha_L(\bar{D})$ such that for all $I \in \bar{D}$, $$\label{condK}\tag{K}
T_I^{2} h \in N(n,2)$$ and $$\label{condS}\tag{S}
T_I^{m_0} h \in SS(n,m_0).$$ Then $h$ is KAM doubly exponentially stable with exponent $u$, for any $u<\frac{1}{{\alpha}n}$.

As mentioned in the introduction, under the Kolmogorov non-degeneracy condition on $h$, invariant tori with frequency in $\Omega_{\gamma,\tau}$ (see ) are preserved, being only slightly deformed, by an arbitrary $\varepsilon$-perturbation, provided $\varepsilon$ is sufficiently small. For Gevrey systems, this was proved by Popov in [@Pop04]. It is only the second part of the statement (double exponential stability of the set of 