steep in a neighborhood of this point. By unicity of the Birkhoff polynomials, we then get that $H_*$ is steep in a neighborhood of the origin. So on a small $r$-neighborhood of $I=0$, $H \circ \Phi$ is an exponentially small (in $r$) perturbation of a steep integrable Hamiltonian: the double exponential stability follows by applying a version of Nekhoroshev’s theorem in Gevrey classes.

Now in the case of a family of KAM tori, under the Kolmogorov non-degeneracy condition , Popov constructed in [@Pop04] a simultaneous Birkhoff normal form for invariant tori with frequencies $\omega_* \in \Omega_{\gamma,\tau}'$: for $H_\varepsilon=h+\varepsilon f$, assuming that both $h$ and $f$ are $\alpha$-Gevrey, there exists a symplectic transformation $\Psi$ which is $(\alpha,\beta)$-Gevrey, such that $H_\varepsilon \circ \Psi(\theta,I)=h_*(I)+f_*(\theta,I)$, where $h_*$ is $\beta$-Gevrey, $f_*$ is $(\alpha,\beta)$-Gevrey, and $f_*$ is flat at every point $I \in (\nabla h_*)^{-1}(\Omega_{\gamma,\tau}')$. This immediately implies the persistence of invariant tori with frequencies in $\Omega_{\gamma,\tau}'$. Moreover, the derivatives of $h_*$ are close (with respect to $\varepsilon$) to the derivatives of $h$, hence the condition  also holds true with $h$ replaced by $h_*$, provided $\varepsilon$ is sufficiently small. This implies that $h_*$ is steep, while $f_*$ is exponentially small on a neighborhood of $\T^n \times 