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On Mean Sensitive Tuples of Discrete Amenable Group Actions

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Abstract

Let (X, G) be a G-system which means that X is a perfect compact metric space and G is a countable discrete infinite amenable group continuously acting on X. In this paper, for an invariant measure \(\mu \) of (X, G) and an integer n larger than 2, we introduce firstly the notions of \(\mu \)-mean n-sensitive tuple with respect to a Følner sequence of G and \(\mu \)-n-sensitive in the mean tuple with respect to a Følner sequence of G and we show that if \(\mu \) is ergodic, then every measure-theoretic n-entropy tuple for \(\mu \) is a \(\mu \)-mean n-sensitive tuple with respect to each tempered Følner sequence of G. Then we introduce the concepts of mean n-sensitive tuple with respect to a Følner sequence of G and n-sensitive in the mean tuple with respect to a Følner sequence of G and we prove that each n-entropy tuple is a mean n-sensitive tuple with respect to each tempered Følner sequence of G for minimal G-systems. Finally, we introduce the notion of weakly n-sensitive in the mean tuple with respect to a Følner sequence of G and we obtain that the maximal mean equicontinuous factor with respect to a Følner sequence of G can be induced by the smallest invariant closed equivalence relation containing all weakly sensitive in the mean pairs with respect to the same Følner sequence of G.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (12061043, 11661054).

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  1. Department of Mathematics, Nanchang University, Nanchang, 330031, People’s Republic of China

    Xiusheng Liu & Jiandong Yin

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  1. Xiusheng Liu
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The first draft of the manuscript was written by Xiusheng Liu and all authors commented on the previous version of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Jiandong Yin.

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Liu, X., Yin, J. On Mean Sensitive Tuples of Discrete Amenable Group Actions. Qual. Theory Dyn. Syst. 22, 4 (2023). https://doi.org/10.1007/s12346-022-00701-y

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