On a Paradoxical Property of the Shift Mapping on an Infinite-Dimensional Tori

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.