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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Entropy and the uniform mean ergodic theorem for a family of sets
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by Terrence M. Adams and Andrew B. Nobel PDF
Trans. Amer. Math. Soc. 369 (2017), 605-622 Request permission

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
  • MR Author ID: 338702
  • 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
  • MR Author ID: 326596
  • Email: nobel@email.unc.edu
  • 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
  • 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
  • MathSciNet review: 3557787