Abstract
An infinite-dimensional torus \({{\mathbb{T}}^{\infty }} = {{\ell }_{p}}{\text{/}}2\pi {{\mathbb{Z}}^{\infty }},\) where \({{\ell }_{p}},\) \(p \geqslant 1\) , is a space of sequences and \({{\mathbb{Z}}^{\infty }}\) is a natural integer lattice in \({{\ell }_{p}},\) is considered. We study a classical question in the theory of dynamical systems concerning the behavior of trajectories of a shift mapping on \({{\mathbb{T}}^{\infty }}.\) More precisely, sufficient conditions are proposed under which the \(\omega \) -limit and \(\alpha \) -limit sets of any trajectory of the shift mapping on \({{\mathbb{T}}^{\infty }}\) are empty.
