A Nekhoroshev Theorem for some Infinite Dimensional Systems

We study the persistence for long times of the solutions of some infinite--dimensional discrete hamiltonian systems with {\it formal hamiltonian} $\sum_{i=1}^\infty h(A_i) + V(\vp),$ $(A,\vp)\in {\Bbb R}^{\Bbb N}\times {\Bbb T}^{\Bbb N}.$ $V(\vp)$ is not needed small and the problem is perturbative being the kinetic energy unbounded. All the initial data $(A_i(0), \vp_i(0)),$ $i\in {\Bbb N}$ in the phase--space ${\Bbb R}^{\Bbb N} \times {\Bbb T}^{\Bbb N},$ give rise to solutions with $\mod A_i(t) - A_i(0).$ close to zero for exponentially--long times provided that $A_i(0)$ is large enough for $\mod i.$ large. We need $\o \partial h,\partial A_i,{\scriptstyle (A_i(0))}$ unbounded for $i\to+\infty$ making $\vp_i$ a {\it fast variable}; the greater is $i,$ the faster is the angle $\vp_i$ (avoiding the resonances). The estimates are obtained in the spirit of the averaging theory reminding the analytic part of Nekhoroshev--theorem.