%0 Journal Article
%T Entropy and the Uniform Mean Ergodic Theorem for a Family of Sets
%A Terrence M. Adams
%A Andrew B. Nobel
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1403.2457v1