On N-Distal Homeomorphisms

• E. Rego ^[2] ; J. C. Salcedo ^[1]
  1. [1] Universidade Federal do Rio de Janeiro

     Universidade Federal do Rio de Janeiro

     Brasil

     
  2. [2] Southern University of Science and Technology (China)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
• Idioma: inglés
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• Resumen

  • 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).