of the Hamiltonian system associated to $H$ with $||I(0)||\leq r$, we have $$||I(t)||\leq 2r, \quad \forall |t| \leq \exp\left(\exp \left(C r^{-u}\right)\right).$$

If $H$ is smooth, given any integer $m \geq 2$, there exists a symplectic transformation $\Phi_m$ defined on a neighborhood of $\T^n \times \{0\}$ which is close to the identity and such that $$\label{BNFe}\tag{BNF} H \circ \Phi_m(\theta,I)=\omega\cdot I+H_m(I)+O\left(||I||^{m+1}\right)$$ where $H_m$ is a polynomial in $n$ variables of degree $m$ without constant or linear terms (see, for instance, [@Dou88]). One also have a formal symplectic transformation $\Phi_\infty$ and a formal series $H_\infty$ such that $H \circ \Phi_\infty(\theta,I)=H_\infty(I)$. The polynomials $H_m$ (respectively the formal series $H_\infty$) are uniquely defined and they are called the Birkhoff polynomials (respectively Birkhoff formal series).

Next we need to recall the following definition from [@BFN15], which was inspired by the work of Nekhoroshev ([@Nek73]). Let $P(n,m)$ be the space of polynomials with real coefficients of degree $m$ in $n$ variables, and $P_2(n,m)\subset P(n,m)$ the subspace of polynomials without constant or linear terms.

\[stabsteep\] Given positive constants $\rho$, $C$ and $\delta$, a polynomial $P_0 \in P_2(n,m)$ is called $(\rho,C,\delta)$-stably steep if for any integer $l\in [1,n-1]$, any $P \in P_2(n,m)$ such that $||P-P_0||<\rho$ and any vector subspace 