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.References
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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