Nekhoroshev stability criterion for elliptic critical manifolds

Let ${\mathfrak C}_i\subset{\mathfrak C}_{gen}$ be a connectivity component of the critical manifold, and let $a\in{\mathfrak C}_i$. Suppose the nonzero spectrum of the vertical linearization is

$${\rm spec}\{DX_H^V(a)\}\setminus\{0\}=\{i\omega_1,\dots,i\omega_{2k}\},$$

where $\omega_i\in{\mathbb R}\setminus\{0\}$ for $i=1,\dots,2k$. Assume that the frequencies $\omega_r$ are rationally independent and that there exists a local degenerate, almost Lyapunov function with respect to ${\mathfrak C}_i\cap U(a)$, in the sense of the stated definition. Nekhoroshev stability criterion. Then $a$ is stable in the sense of Nekhoroshev. In particular, the second condition is always satisfied by the Hamiltonian $H$ if the index of ${\mathfrak C}_i$ is $\mu({\mathfrak C}_i)=0$. The criterion is motivated by averaging, KAM theory, and Nekhoroshev theory; the source does not provide evidence that it has been proved or disproved.