A theorem of Brudno says that the Kolmogorov–Sinai entropy of an ergodic subshift over $\mathbb {N}$ equals the asymptotic Kolmogorov complexity of almost every word in the subshift. The purpose of this paper is to extend this result to subshifts over computable groups that admit computable regular symmetric Følner monotilings, which we introduce in this work. For every $d \in \mathbb {N}$ , the groups $\mathbb {Z}^d$ and $\mathsf{UT}_{d+1}(\mathbb {Z})$ admit computable regular symmetric Følner monotilings for which the required computing algorithms are provided.

中文翻译：

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可计算的 Følner monotilings 和 Brudno 定理

Brudno 的一个定理说，遍历子移位的 Kolmogorov-Sinai 熵$\mathbb {N}$等于 subshift 中几乎每个单词的渐近 Kolmogorov 复杂度。本文的目的是将这一结果扩展到允许可计算规则对称 Følner 单块化的可计算组上的子移位，我们在这项工作中进行了介绍。对于每一个$d \in \mathbb {N}$, 组$\mathbb {Z}^d$和$\mathsf{UT}_{d+1}(\mathbb {Z})$承认提供了所需计算算法的可计算规则对称 Følner monotilings。