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Non-existence of a universal zero entropy system for non-periodic amenable group actions

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Abstract

Let G be a non-periodic amenable group. We prove that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic measure-theoretic G-systems of entropy zero. Previously J. Serafin, answering a question by B. Weiss, proved the same for G = ℤ.

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Acknowledgements

The author is sincerely grateful to his advisor Pavel Zatitskiy for many helpful discussions. The author is also grateful to Valery Ryzhikov for drawing the author’s attention to this question.

The author thanks the anonymous referee for many remarks that significantly improved the readability of the paper.

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Authors and Affiliations

  1. Leonhard Euler International Mathematical Institute in St. Petersburg, 14th Line 29B, Vasilyevsky Island, St. Petersburg, 199178, Russia

    Georgii Veprev

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  1. Georgii Veprev
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Correspondence to Georgii Veprev.

Additional information

The work is supported by the Ministry of Science and Higher Education of the Russian Federation, agreement No. 075-15-2019-1619. The work is also supported by the V. A. Rokhlin scholarship for young mathematicians.

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Veprev, G. Non-existence of a universal zero entropy system for non-periodic amenable group actions. Isr. J. Math. 253, 715–743 (2023). https://doi.org/10.1007/s11856-022-2374-7

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  • DOI: https://doi.org/10.1007/s11856-022-2374-7

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