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

An infinite-dimensional torus ${\mathbb{T}}^{\infty }={\mathcal{l}}_p/2\pi {\mathbb{Z}}^{\infty }$, where ${\mathcal{l}}_p$, p ≥ 1 – space of sequences, ${\mathbb{Z}}^{\infty }$ – natural integer lattice in ${\mathcal{l}}_p$, is considered. We study the classical question in the theory of dynamical systems about the behavior of trajectories of a shift mapping on the specified torus. More precisely, some sufficient conditions are proposed that guarantee the emptiness of the ω-limit and α-limit sets of any of the shift mapping onto ${\mathbb{T}}^{\infty }$.