POLISH MODELS AND SOFIC ENTROPY
POLISH MODELS AND SOFIC ENTROPY
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波兰模型和 SOFIC 熵
DOI：
10.1017/s1474748015000468
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发表时间：
2014
影响因子：
0.9
通讯作者：
Ben Hayes
中科院分区：
文献类型：
--
作者：
Ben Hayes
我们推导出可数离散sofic群的正熵概率测度保持作用的Koopman表示的性质。我们的主要结果可被视为Sinawei因子定理的“表示论”版本。我们证明了无穷sofic群的具有完全正熵的概率保测度作用必是混合的，并且如果群是不可调和的，则它具有谱隙。这意味着，如果$\unicode[STIX]{x1 D 6 E4}$是一个不服从群，而$\unicode[STIX]{x1 D 6 E4}\curvearrowright（X，\unicode[STIX]{x1 D 707}）$是一个不强遍历的概率测度保持作用，则没有等价于$\unicode[STIX]{x1 D 6 E4}\curvearrowright（X，\unicode[STIX]{x1 D 707}）$的作用轨道具有完全正熵。这些结果的关键是一个公式的熵在波兰的存在下，但先验非紧，模型。
We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if $\unicode[STIX]{x1D6E4}$ is a nonamenable group and $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.