Entropy and the uniform mean ergodic theorem for a family of sets

Abstract:

We define the entropy of an infinite family $\mathcal {C}$ of measurable sets in a probability space, and show that a family has zero entropy if and only if it is totally bounded under the symmetric difference semi-metric. Our principal result is 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 we obtain a corresponding result for weak mixing transformations.