Abstract
Let X be a compact metric countable space, let \(f:X\rightarrow X\) be a homeomorphism and let E(X, f) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (1) (X, f) is equicontinuous, (2) (X, f) is distal and (3) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group.
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Acknowledgements
We are thankful to the referee for his (her) comments that improved the presentation of the paper. The authors thank La Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander for the financial support for this work, which is part of the VIE Project # 2422.
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Communicated by Anthony To-Ming Lau.
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Quintero, A., Uzcátegui, C. On the Ellis semigroup of a cascade on a compact metric countable space. Semigroup Forum 101, 435–451 (2020). https://doi.org/10.1007/s00233-020-10095-5
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DOI: https://doi.org/10.1007/s00233-020-10095-5