On a paradoxical property of the shift mapping on an infinite-dimensional tori

An infinite-dimensional torus $\mathbb{T}^\infty=l_p /2\pi\mathbb{Z}^\infty$, where $l_p$, $p\ge1$, is a space of sequences and $\mathbb{Z}^\infty$ is a natural integer lattice in $l_p$ 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.