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The large time behavior of solutions to Hamiltonian partial differential equations is an important problem in mathematical physics. In the case of finite dimensional Hamiltonian systems, many features of the large time behavior of trajectories are described using the topology of the phase space. For a given infinite dimensional systems, several natural phase spaces, with different topologies, can be chosen, and the large time properties may strongly depend on the choice of such topologies. For instance, it is known that the cubic defocusing Schrodinger equation i∂tu+ ∆u = |u|u posed on a Riemannian manifold M of dimension d = 1, 2, 3 with sufficiently uniform properties at infinity, defines a global flow on the Sobolev spaces H(M) for every s ≥ 1 (see e.g. [?]). In this case, a typical large time behavior of interest is the boundedness of trajectories. On the energy space H(M), the conservation of energy trivially implies that all the trajectories are bounded. On the other hand, the existence of unbounded trajectories in H(M) for s > 1 is a long standing question [?], naturally connected to weak turbulence. In [?], [?], it was proved that dispersion properties imply a polynomial bound in time on H norms as time goes to infinity, but so far the optimality of this kind of bound has not been established. Let us mention however that, for some special solutions of cubic NLS on R×T, Hani– Pausader–Tzvetkov–Visciglia [?] succeeded in proving that H norms may not be bounded for s big enough, with a minoration of the type

On the growth of Sobolev norms for the cubic Szegő equation