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On sensitive sets and regionally proximal sets of group actions

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Abstract

We introduce the concepts of S-sets and Q-sets for a flow (a group action on a compact metric space) and prove that a transitive flow is sensitive if and only if there exists an S-set with cardinality more than 2 and that each S-set of a transitive flow is a Q-set and the converse holds for minimal flows. Then according to cardinalities of S-sets, transitive flows are divided into several classes and some characterizations and relationships of different classes are given. This is a generalization of the \(\mathbb {Z}\)-action of Ye and Zhang (Nonlinearity 21:1601–1620, 2008. https://doi.org/10.1088/0951-7715/21/7/012).

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Authors and Affiliations

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

    Zhumin Ding, Xiaoxiao Nie & Jiandong Yin

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  1. Zhumin Ding
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  2. Xiaoxiao Nie
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  3. Jiandong Yin
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Correspondence to Jiandong Yin.

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Communicated by Jimmie D. Lawson.

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

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Ding, Z., Nie, X. & Yin, J. On sensitive sets and regionally proximal sets of group actions. Semigroup Forum 102, 408–421 (2021). https://doi.org/10.1007/s00233-021-10172-3

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