Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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POSITIVE EXPANSIVITY, CHAIN TRANSITIVITY, RIGIDITY, AND SPECIFICATION ON GENERAL TOPOLOGICAL SPACES

  • Received : 2021.01.16
  • Accepted : 2021.12.31
  • Published : 2022.03.31
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Abstract

We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform h-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff h-shadowing.

Keywords

Acknowledgement

We are thankful to the anonymous referees for their valuable comments and suggestions.

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