Analogue of Jacobi's theorem for infinite-dimensional tori

Dynamic properties of an infinite system of harmonic oscillators are studied. As in the classical Jacobi theorem on the trajectories of a finite system of oscillators on a torus, the conditions of periodicity, non-wandering and transitivity on an invariant torus of trajectories of an infinite system of oscillators are obtained. The ergodicity of the measure and the ergodicity of the space of functions on an invariant manifold with respect to the flow of a system of oscillators are studied. The talk is based on joint work with I.V. Volovich.