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Measure concentration and the weak Pinsker property
Publications mathématiques de l'IHÉS ( IF 6.2 ) Pub Date : 2018-02-15 , DOI: 10.1007/s10240-018-0098-3
Tim Austin

Let \((X,\mu)\) be a standard probability space. An automorphism \(T\) of \((X,\mu)\) has the weak Pinsker property if for every \(\varepsilon > 0\) it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than \(\varepsilon \). This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms.

This paper proves that it does. The proof actually gives a more general result. Firstly, it gives a relative version: any factor map from one ergodic automorphism to another can be enlarged by arbitrarily little entropy to become relatively Bernoulli. Secondly, using some facts about relative orbit equivalence, the analogous result holds for all free and ergodic measure-preserving actions of a countable amenable group.

The key to this work is a new result about measure concentration. Suppose now that \(\mu\) is a probability measure on a finite product space \(A^{n}\), and endow this space with its Hamming metric. We prove that \(\mu\) may be represented as a mixture of other measures in which (i) most of the weight in the mixture is on measures that exhibit a strong kind of concentration, and (ii) the number of summands is bounded in terms of the difference between the Shannon entropy of \(\mu\) and the combined Shannon entropies of its marginals.



中文翻译:

测量浓度和弱的平斯克特性

\((X,\ mu)\)为标准概率空间。一个自同构\(T \)\((X,\亩)\)具有弱平斯克属性,若对所有\(\ varepsilon> 0 \)它有一个分裂成一个伯努利移的直接产物和一个自同构的熵小于\(\ varepsilon \)。此属性由Thouvenot引入,他问它是否对所有遍历自同构都成立。

本文证明了。证明实际上给出了更一般的结果。首先,它给出了一个相对的形式:从一个遍历自同构到另一个遍历的任何因果图都可以通过任意小的熵放大而变得相对伯努利。其次,利用有关相对轨道当量的一些事实,类似的结果适用于可数可归类群体的所有自由和遍历测度保持动作。

这项工作的关键是有关测量浓度的新结果。现在假设\(\ mu \)是有限乘积空间\(A ^ {n} \)上的概率测度,并赋予该空间其汉明度量。我们证明\(\ mu \)可以表示为其他度量的混合物,其中(i)混合物中的大部分权重都表现出较强的浓度,而(ii)求和数为根据\(\ mu \)的Shannon熵与其边际组合的Shannon熵之差来界定。

更新日期:2020-04-22
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