On the stability of elliptic equilibria

We consider stability of elliptic equilibria in Hamiltonian systems in the frame of Nekhoroshev's theory, recovering the steepness assumption, in the form of convexity, from an appropriate treatment of the higher orders. The singularity of the action-angle coordinates is overcome by using Cartesian coordinates. We introduce an essential refinement of the perturbative technique used in a previous work on the subject, and obtain significant improvements of results, namely better values of the exponents controlling the stability time and the confinement around equilibrium, in case the equilibrium frequencies satisfy stronger nonresonance conditions. Within the same non-resonance assumptions the new method provides instead independent informations, namely one gets a better confinement on a reduced time scale.