Paolo Perfetti
A Nekhoroshev theorem for some 
infinite--dimensional systems
(439K, ps)

ABSTRACT.  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.