(\nabla h_*)^{-1}(\Omega_{\gamma,\tau}')$: as before, the double exponential stability follows from Nekhoroshev theorem.

The plan of the article is as follows. In §\[s2\], we recall the results concerning the Birkhoff normal form for a Hamiltonian as in  and the simultaneous Birkhoff normal form for a Hamiltonian as in , following [@MP09] and [@Pop04]. In §\[s3\], we recall some results concerning stably steep polynomials, following [@Nek73] and [@BFN15]. The proof of Theorem \[Th1\] and Theorem \[Th2\] will be given in §\[s4\], using the results of §\[s2\] and §\[s3\], and a version of Nekhoroshev estimates for perturbation of steep integrable Hamiltonians in Gevrey classes, that we state in Appendix \[app\].

Birkhoff normal forms {#s2}
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In the sequel, given some $\rho>0$ and some point $I_* \in {\mathbb{R}}^n$, we define $B_\rho(I_*)=V_\rho \{I_*\}$ the open ball of radius $\rho$ around $I_*$ in ${\mathbb{R}}^n$, and for $I_*=0\in {\mathbb{R}}^n$, we let $B_\rho=B_\rho(0)$.

\[thmMP\] Let $H \in \mathcal{H}^\alpha_L(\omega)$ with $\omega$ satisfying , and define $\beta:=\alpha(1+\tau)+1$. Then there exist positive constants $\bar{r}$, $L_1$, $L_2$ and $A$, which depend only on $n$, $\gamma$, $\tau$, $\alpha$, $L$ and $||H||_{\alpha,L,B}$ and a symplectic transformation $$\Phi : \T^n \times B_{\bar{r}} \rightarrow \T^n \times B$$ whose components belong to $G^{\alpha,\beta}_{L_1,L_2}(\T^n \times B_{\bar{r}})$, such that 