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

2014/03/11 by Terrence Adams, Terrence M. Adams, Andrew B. Nobel +2
Mathematics · #37A35 #37A50 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Primary 37A25 #Secondary 60F05 #math.DS #msc:37A25 #msc:37A35 #msc:37A50 #msc:60F05

paper · pdf · doi:10.48550/arxiv.1403.2457

arxiv created 2014/03/11 · openalex publication_date 2014/03/11 · arxiv updated 2014/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a notion of entropy for an infinite family C of measurable sets in a probability space. We show 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 obtain a corresponding result for weak mixing transformations.

Citations

• The Gap Dimension and Uniform Laws of Large Numbers for Ergodic\n Processes
• Restricted orbit changes of ergodic ℤd-actions to achieve mixing and completely positive entropy

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