real-analytic or Gevrey smooth systems, under a further generic condition on $h$ and a further smallness assumption on $\varepsilon$, a sub-family $\mathcal{K}'$ of this family $\mathcal{K}$ of KAM tori (with a complementary still having measure of order at most $\gamma$) is doubly exponentially stable. We refer to Theorem \[Th2\] below and its corollary for precise statements. This result extends a previous result obtained in [@MG95b], [@GM97] in the real-analytic case, under the stronger (and non-generic) condition that $h$ is quasi-convex.

Results
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Let us now state more precisely our results. Given some bounded domain $U \subseteq {\mathbb{R}}^n$, and real numbers $\alpha \geq 1$, $\beta\geq 1$, $L_1>0$ and $L_2>0$, we define $G^{\alpha,\beta}_{L_1,L_2}(\T^n \times U)$ to be the space of smooth functions $H$ on $\T^n \times U$ such that $$||H||_{\alpha,\beta,L_1,L_2,U}:=\sup_{(k,l)\in {\mathbb{N}}^{2n}}\sup_{(\theta,I)\in \T^n \times U}\left(|\partial_\theta^k\partial_I^lH(\theta,I)|L_1^{-|k|}L_2^{-|l|}k!^{-\alpha}l!^{-\beta}\right)<\infty.$$ We shall always assume that $L_1 \geq 1$ and $L_2 \geq 1$. This space $G^{\alpha,\beta}_{L_1,L_2}(\T^n \times U)$, equipped with the above norm, is a Banach space. When $\alpha=\beta$ and $L_1=L_2=L$, we simply write $G^{\alpha}_{L}(\T^n \times U)$ and when $\alpha=1$, then $G^{1}_{L}(\T^n \times U)$ is the space of real-analytic functions which can be extended as a holomorphic function on a complex $s$-neighborhood of $\T^n \times U$ in ${\mathbb{C}}^n/{\mathbb{Z}}^n \times 