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On N-Distal Homeomorphisms

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Abstract

In this work we study N-distal properties for homeomorphisms on compact metric spaces. For instance, we define N-equicontinuity and prove that every N-equicontinuous system is N-distal. We introduce the notions of N-distal extensions and N-distal factors. We also prove that M-distal extensions of N-distal homeomorphisms are MN-distal and present a non-trivial N-distal factor for N-distal homeomorphism having Ellis semigroup with a unique minimal ideal. It is also shown that transitive N-distal homeomorphisms have at most \(N-1\) minimal proper subsystems. Finally, we prove that topological entropy vanishes for countably-distal systems on compact metric spaces. These results generalize previous ones for distal systems (Fürstenberg in Am J Math 85:477–515, 1963; Parry in Zero entropy of distal and related transformations, Benjamin, New York, pp 383–389, 1967).

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Acknowledgements

The authors would like to thank C.A. Morales for his great help and his lectures on topological dynamics during the second semester of 2017 at UFRJ which inspired this work. We also would like to thank the referees for their valuable advice that helped us to improve the presentation of this work.

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Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    E. Rego

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, Rio de Janeiro, 21945-970, Brazil

    J. C. Salcedo

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ER and JCS wrote the main manuscript text. All authors reviewed the manuscript.

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J. C. Salcedo was partially supported by CNPq and CAPES from Brazil.

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Rego, E., Salcedo, J.C. On N-Distal Homeomorphisms. Qual. Theory Dyn. Syst. 22, 138 (2023). https://doi.org/10.1007/s12346-023-00839-3

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