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arXiv · 1403.2457

Entropy and the Uniform Mean Ergodic Theorem for a Family of Sets

Terrence M. Adams·Andrew B. Nobel
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Abstract

We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if $\mathcal{C}$ has zero entropy. When the entropy of $\mathcal{C}$ is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly strong mixing on $\mathcal{C}$ if and only if the entropy of $\mathcal{C}$ is zero, and obtain a corresponding result for weak mixing transformations.

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Terrence M. Adams, Andrew B. Nobel. 2014-03-11. Entropy and the Uniform Mean Ergodic Theorem for a Family of Sets. https://arxiv.org/abs/1403.2457

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