Sofic同调不变量与弱Pinsker性质
Sofic homological invariants and the Weak Pinsker Property
摘要：
一个概率测度保持变换具有弱平斯克性质（Weak Pinsker Property，WPP），如果对于每个$ε>0$，它可测地共轭于一个变换与熵$<ε$和一个Bernoulli位移的直积。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$Γ$是非交换自由群，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。主要结果是通过证明WPP的任何动作在维度$0$上具有次指数同源增长而得到的，而反例在维度$0$上具有指数同源增长。
英文摘要：
A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $ε>0$ it is measurably conjugate to the direct product of a transformation with entropy $<ε$ 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 $Γ$ 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 $0$.
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