a semi-algebraic subset of $P_2(n,m_0)$ of positive codimension). Using this, we proved in [@BFN15] that the Birkhoff normal form of order $m_0$ of a Hamiltonian system at a non resonant elliptic fixed point is in general stably steep ([@BFN15], Theorem C). In a similar fashion, we have here that [*for any $H \in \mathcal{H}^\alpha_L(\omega)$ with $\omega$ satisfying , for an open set of full Lebesgue measure of $Q \in P_2(n,m_0)$, the condition holds true for the modified Hamiltonian $H_Q \in \mathcal{H}^\alpha_L(\omega)$ defined by $H_Q(\theta,I):=H(\theta,I)+Q(I)$.*]{} Indeed, the Diophantine condition on ${\omega}$ is necessary just to be able to perform the Birkhoff normal form reduction up to order $m_0$ since we are dealing here with an invariant torus while in [@BFN15] we were interested with fixed points. Apart from this, the proof that the condition holds for $H_Q$ for an open set of full Lebesgue measure of $Q \in P_2(n,m_0)$ follows exactly the same steps as in the fixed point case (see [@BFN15] Section §2.2). Hence we get the following corollary of Theorem \[Th1\].

\[coro1\] In the space of Hamiltonians $\mathcal{H}^\alpha_L(\omega)$ with $\omega$ satisfying , the set of Hamiltonians for which ${\mathcal T}_{{\omega}}$ is doubly exponentially stable with exponent $\frac{1}{\alpha(1+\tau)}$ contains an 