Entropy, Shannon orbit equivalence, and sparse connectivity
Entropy, Shannon orbit equivalence, and sparse connectivity
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DOI：
10.1007/s00208-021-02190-x
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发表时间：
2019-12
影响因子：
1.4
通讯作者：
David Kerr;Hanfeng Li
中科院分区：
文献类型：
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作者：
David Kerr;Hanfeng Li
我们说可数群的两个自由p. m. p.作用是Shannon轨道等价的，如果它们之间存在一个轨道等价，其相关的上循环划分具有有限的Shannon熵。我们表明，如果作用群是sofic和每个有一个w-正常的顺从子群，既不是局部有限的，也不是几乎循环然后香农轨道等价意味着行动有相同的最大sofic熵。这扩展了奥斯汀的结果超出了生成的服从设置，并具有这样的后果，即具有所讨论的性质的群的两个伯努利作用是香农轨道等价的当且仅当它们是测度共轭的。我们的论点适用于更普遍的行动，满足稀疏连接条件，我们称之为属性SC，并产生熵不等式的假设下，其中一个行动具有此属性。
We say that two free p.m.p. actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the acting groups are sofic and each has a w-normal amenable subgroup which is neither locally finite nor virtually cyclic then Shannon orbit equivalence implies that the actions have the same maximum sofic entropy. This extends a result of Austin beyond the finitely generated amenable setting and has the consequence that two Bernoulli actions of a group with the properties in question are Shannon orbit equivalent if and only if they are measure conjugate. Our arguments apply more generally to actions satisfying a sparse connectivity condition which we call property SC, and yield an entropy inequality under the assumption that one of the actions has this property.