In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.

中文翻译：

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局部有限群上的有限类型移位

在这项工作中，我们证明了局部有限群上的有限类型（SFT）、sofic 移位和强不可约移位的每一个移位都具有很强的动力学性质。这些属性包括每个 Sofic 位移都是 SFT、每个 SFT 都是强不可约位移、每个强不可约位移都是 SFT、每个 SFT 都是最小熵、每个 SFT 都有最大熵的唯一度量等。此外，我们还表明，如果群上的每个 SFT 都是强不可约的，或者如果每个 sofic 移位都是 SFT，那么该群必须是局部有限的，这扩展到我们探索的所有属性。这些结果集中在两个主要定理中，这些定理通过纯动力学性质来表征群的局部有限性。为了追求这些结果，我们提出了一个正式的构建免费延期在一组上轮班G，对子组进行移位H的G，并自然地将其扩展到所有的转变G。