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.

中文翻译：

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无限维环面平移映射的一个悖论性质

摘要

无限维环面\({{\mathbb{T}}^{\infty }} = {{\ell }_{p}}{\text{/}}2\pi {{\mathbb{Z}} ^{\infty }},\)其中\({{\ell }_{p}},\) \(p \geqslant 1\)是序列空间，而\({{\mathbb{Z}} ^{\infty }}\)是\({{\ell }_{p}},\)中的自然整数格。我们研究了动力系统理论中的一个经典问题，涉及\({{\mathbb{T}}^{\infty }}.\)上平移映射的轨迹行为。更准确地说，提出了充分条件，在该条件下\( {{\mathbb{T}}^{\infty }}\)上的移位映射的任何轨迹的\(\omega \) -limit 和\(\alpha \) -limit 集为空。