AI 中文总结
研究顺从剩余有限群上全转移的熵密度问题，通过科尔特斯 - 佩蒂特福勒纳镶嵌和块替换构造，将\(\mathbb{Z}\)上唯一遍历测度熵稠密结果推广到此类群，尤其适用于有限生成阿贝尔群。
AI 中文摘要
我们研究顺从剩余有限群上全转移的熵密度。熵密度意味着每个不变测度及其熵在弱*拓扑中都可以由一个特殊族中的测度逼近。在符号动力学中，根据魏斯的结果，对于\(\mathbb{Z}\)的转移作用，唯一遍历测度在遍历测度中是熵稠密的。我们将此结果推广到顺从剩余有限群上的全转移：支撑在唯一遍历子系统上的不变测度在遍历不变测度集合中是熵稠密的。这尤其适用于有限生成阿贝尔群上的全转移。证明使用了科尔特斯 - 佩蒂特福勒纳镶嵌和适用于有限指数子群的块替换构造。
英文摘要
We study entropy density for full shifts over amenable residually finite groups. Entropy density means that every invariant measure can be approximated in the weak$^*$ topology, together with its entropy, by measures from a distinguished family. In symbolic dynamics it is known, by a result of Weiss, that uniquely ergodic measures are entropy dense among ergodic measures for the shift action of $\mathbb Z$. We extend this result to full shifts over amenable residually finite groups: invariant measures supported on uniquely ergodic subsystems are entropy dense in the collection of ergodic invariant measures. This applies in particular to full shifts over finitely generated abelian groups. The proof uses Cortez--Petite Følner tilings and a block-replacement construction adapted to finite-index subgroups.
