Second Order Hamiltonian Equations on $\T^\io$cand Almost--Periodic Solutions

Motivated by problems arising in nonlinear PDE's with a Hamiltonian structure and in high dimensional dynamical systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equations of the type $\ddot x=\dpr_x V$, [$x\in\TN$, $\dpr_x\=(\dpr_{x_1},..., \dpr_{x_N})$]. Extending methods from quantitative perturbation theory (Kolmogorov--Arnold--Moser theory, Nash--Moser implicit function theorem, \etc) we construct uncountably many almost--periodic solutions for the infinite dimensional system $\ddot x_i=f_i(x)$, $i\in \Zd$, $x\in \T^\Zd$ (endowed with the compact topology); the Hamiltonian structure is reflected by $f$ being a ``generalized gradient". Such result is derived under (suitable) analyticity assumptions on $f_i$ but without requiring any ``smallness conditions".