Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates

In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with \begin{document}$ n $\end{document} degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situations) and other classical results on formal stability of equilibria. In case of Lie stable systems we bound the solutions near the equilibrium over exponentially long times. Some examples are provided to illustrate our main contributions.