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Description

The problem of persistence of invariant tori in infinite dimension is a challenging problem in the study of PDEs. There is a rather well established literature on the persistence of n-dimensional invariant tori carrying a quasi-periodic Diophantine flow (for one-dimensional system) but very few on the persistence of infinite-dimensional ones. Inspired by the classical "twisted conjugacy theorem" of M. Herman for perturbations of degenerate Hamiltonians possessing a Diophantine invariant torus, we intend to present a compact and unified frame in which recover the results of Bourgain and Pà ¶schel on the existence of almost-periodic solutions for the Nonlinear Schrà ¶dinger equation. We shall discuss the main advantages of our approach as well as new perspectives. This is a joint work with L. Biasco and M. Procesi.