ENTROPY AND THE UNIFORM MEAN ERGODIC THEOREM FOR A FAMILY OF SETS

摘要

We define the entropy of an infinite family 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 C under every ergodic transformation if and only if C has zero entropy. When the entropy of 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 C if and only if the entropy of C is zero, and we obtain a corresponding result for weak mixing transformations.