Abstract

A natural class of expansive endomorphisms \(G\in C^1\) of the infinite-dimensional torus \(\mathbb{T}^{\infty}\) (the Cartesian product of countably many circles with the product topology) is considered. The endomorphisms in this class can be represented in the form of the sum of a linear expansion and a periodic addition. The following standard facts of hyperbolic theory are proved: the topological conjugacy of any expansive endomorphism \(G\) from the class under consideration to a linear endomorphism of the torus, the structural stability of \(G\), and the topological mixing property of \(G\) on \(\mathbb{T}^{\infty}\).

中文翻译：

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无限维环面上的膨胀自同构

摘要

无限维环面\(\mathbb{T}^{\infty}\)（具有乘积拓扑的可数多个圆的笛卡尔积）的自然类膨胀自同态\(G\in C^1\ ) 是经过考虑的。此类中的自同态可以用线性展开和周期加法之和的形式表示。双曲线理论的下列标准事实证明：任何膨胀自同态的拓扑共轭\（G \）从类所考虑到环面的线性自同态，的结构稳定性\（G \） ，和拓扑混合属性在\(\mathbb{T}^{\infty}\)上的\(G \)。