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M-systems and scattering systems of semigroup actions

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Abstract

We prove that: (1) an action of a semigroup S on a compact metric space X is an M-system if and only if N(x, U) is a piecewise syndetic set for every transitive point x in X and every neighborhood U of x; (2) an action of a monoid S on a compact metric space X for which every \(s \in S\) is a surjective map from X onto itself is scattering if and only if N(U, V) is a set of topological recurrence for every pair of non-empty open subsets U, V in X. As applications, we show that: (1) if an action of a commutative semigroup S on a compact metric space X is an M-system then the system is finitely sensitive; (2) an action of a commutative semigroup S on a compact metric space X is a scattering system if and only if it is disjoint with any M-system.

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Acknowledgments

The authors thank the referee for the careful reading and many valuable comments. Research of Wang was supported by National Nature Science Funds of China (11471125).

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Authors and Affiliations

  1. Department of Mathematics, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    Huoyun Wang

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Zhijing Chen

  3. School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, 526061, People’s Republic of China

    Human Fu

Authors
  1. Huoyun Wang
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  2. Zhijing Chen
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  3. Human Fu
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Corresponding author

Correspondence to Zhijing Chen.

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Communicated by Jimmie D. Lawson.

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Wang, H., Chen, Z. & Fu, H. M-systems and scattering systems of semigroup actions. Semigroup Forum 91, 699–717 (2015). https://doi.org/10.1007/s00233-015-9736-y

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  • DOI: https://doi.org/10.1007/s00233-015-9736-y

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