It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

中文翻译：

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Ellis 半群的完全正则性

证明了 a 的 Ellis 半群$\mathbb Z$- 当且仅当前向邻近性与前向渐近性一致并且后向性与后向渐近性一致时，对一个紧凑的完全断开空间的作用是完全规则的。此外，a 的 Ellis 半群$\mathbb Z$- 要么$\mathbb R$- 前向邻近和后向邻近是传递关系的动作被证明具有至多两个左极小理想。最后，介绍了 Ellis 半群的近似简单的概念，并与上述内容相关。