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Transactions of the American Mathematical Society

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


Authors: Terrence M. Adams and Andrew B. Nobel
Journal: Trans. Amer. Math. Soc. 369 (2017), 605-622
MSC (2010): Primary 37A25; Secondary 60F05, 37A35, 37A50
DOI: https://doi.org/10.1090/tran/6675
Published electronically: March 21, 2016
MathSciNet review: 3557787
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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.


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Additional Information

Terrence M. Adams
Affiliation: U. S. Government, 9800 Savage Road, Ft. Meade, Maryland 20755
Email: tmadam2@tycho.ncsc.mil

Andrew B. Nobel
Affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3260
Email: nobel@email.unc.edu

DOI: https://doi.org/10.1090/tran/6675
Keywords: Uniform ergodicity, uniform mixing, entropy, total boundedness
Received by editor(s): March 31, 2014
Received by editor(s) in revised form: January 13, 2015
Published electronically: March 21, 2016
Additional Notes: The second author was supported by NSF Grants DMS-0907177 and DMS-1310002