$\Lambda \subseteq {\mathbb{R}}^n$ of dimension $l$, letting $P_\Lambda$ be the restriction of $P$ to $\Lambda$, the inequality $$\max_{0 \leq \eta \leq \xi}\;\min_{||x||=\eta, \; x \in \Lambda}||\nabla P_\Lambda(x)||>C\xi^{m-1}$$ holds true for all $0 < \xi \leq\delta$. A polynomial $P_0 \in P_2(n,m)$ is called stably steep if there exist positive constants $\rho$, $C$ and $\delta$ such that $P_0$ is $(\rho,C,\delta)$-stably steep.

The set of stably steep polynomials in $P_2(n,m)$ will be denoted by $SS(n,m)$.

For a fixed $\omega \in {\mathbb{R}}^n$, we let $\mathcal{H}^\alpha_L(\omega)$ be the space of Hamiltonians $H \in G^\alpha_L(\T^n \times B)$ as in . We can now state our first main result.

\[Th1\] Let $H \in \mathcal{H}^\alpha_L(\omega)$ with $\omega$ satisfying . Assume that for $m_0:=[n^2/2+2]$, we have $$\label{condG}\tag{G}
H_{m_0} \in SS(n,m_0).$$ Then ${\mathcal T}_{{\omega}}$ is doubly exponentially stable with exponent $\frac{1}{\alpha(1+\tau)}$.

The positive constants $r_*$ and $C$ of the double exponential stability depend only on $n$, $\gamma$ $\tau$, $\alpha$, $L$, $||H||_{\alpha,L,B}$ and on some constants characterizing the condition .

It remains to explain in which sense  is generic. As we proved in [@BFN15] (this will be recalled in §\[s3\], Proposition \[propNekho2\]), $SS(n,m_0)$ is an open set of full Lebesgue measure (and hence dense) in $P_2(n,m_0)$ (more precisely, its complement is 