Sofic homological invariants and the Weak Pinsker Property
Sofic homological invariants and the Weak Pinsker Property
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DOI：
10.1353/ajm.2022.0003
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发表时间：
2018-07
影响因子：
1.7
通讯作者：
L. Bowen
中科院分区：
文献类型：
--
作者：
L. Bowen
翻译后摘要：一个概率测度保持变换的弱平斯克性质（WPP），如果对于每$\displaystyle>0$，它是可测量共轭的熵$<\displaystyle $和伯努利位移的变换的直积。在最近的一项突破中，蒂姆·奥斯汀证明了每个遍历变换都满足这个性质。此外，顺从的群体行动的自然类比也是正确的。作为对比，本文给出了一个反例，其中群$\Gamma$是一个非交换自由群，熵的概念是sofic熵。反例是随机正则图上硬核模型的一个极限。为了证明它不具有WPP，本文基于作用的模型空间的同调增长，引入了新的测度共轭不变量。主要结果表明，任何行动的WPP有次指数的同源增长的维度$0$，而反例有指数的同源增长的维度$0$。
Abstract:A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $\epsilon>0$ it is measurably conjugate to the direct product of a transformation with entropy $<\epsilon$ and a Bernoulli shift. In a recent breakthrough, Tim Austin proved that every ergodic transformation satisfies this property. Moreover, the natural analog for amenable group actions is also true. By contrast, this paper provides a counterexample in which the group $\Gamma$ is a non-abelian free group and the notion of entropy is sofic entropy. The counterexample is a limit of hardcore models on random regular graphs. In order to prove that it does not have the WPP, this paper introduces new measure conjugacy invariants based on the growth of homology of the model spaces of the action. The main result is obtained by showing that any action with the WPP has subexponential homology growth in dimension $0$, while the counterexample has exponential homology growth in dimension $0$.