On persistence of invariant tori and a theorem by Nekhoroshev (Q2781078)

The authors provide a proof of a theorem by \textit{N. N. Nekhoroshev} [Funct. Anal. Appl. 28, 128--129 (1994; Zbl 0847.58035)] which states that if a Hamiltonian system of \(n\) degrees of freedom possesses \(s\), \(1 \leq s \leq n\), independent integrals in involution whose level set is compact and connected and certain nondegeneracy conditions hold, there exists a \(2s\)-dimensional symplectic submanifold \(N\) fibered by \(s\)-dimensional invariant tori and action-angle coordinates can be defined on \(N\). Moreover, these invariant tori persist under small perturbations of the Hamiltonian and the integrals of motion. The proof is based on a reformulation of the nondegeneracy condition and is expressed in terms of the multiplicity of the Floquet multiplier 1 of the periodic orbits of some suitable Hamiltonian vector fields.