  We consider the dynamics in a neighborhood of an elliptic equilibrium point
with a Diophantine frequency of a symplectic real analytic vector field and we
prove the following result of effective stability. Generically, both in a
topological and measure-theoretical sense, any solution starting sufficiently
close to the equilibrium point remains close to it for an interval of time
which is doubly exponentially large with respect to the inverse of the distance
to the equilibrium point. We actually prove a more general statement assuming
the frequency is only non-resonant. This improves previous results where much
stronger non-generic assumptions were required.
