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