What is Topological about Topological Dynamics?

Christopher Good, Sergio Macias

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)
166 Downloads (Pure)


We consider various notions from the theory of dynamical systems from a topological point of view.
Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric in compact metric spaces.

We show for example that in a Tychonoff space, transitivity and dense periodic points
imply (uniform) sensitivity to initial conditions. We generalise Bryant's result that
a compact Hausdorff space admitting a $c$-expansive homeomorphism in the obvious uniform sense is metrizable.
We study versions of shadowing, generalising a number of well-known results to the topological setting, and internal chain transitivity, showing for example that $\omega$-limit sets are (uniform) internally chain transitive and
weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.
Original languageEnglish
Pages (from-to)1007-1031
Number of pages25
JournalDiscrete and Continuous Dynamical Systems
Issue number3
Early online dateMar 2018
Publication statusE-pub ahead of print - Mar 2018


  • uniformity
  • c-expansive map
  • dynamical systems
  • Compact Hausdorff space
  • compactum
  • uniform chain recurrent point
  • uniform pseudo-orbit
  • uniform internal chain transitivity
  • uniform shadowing


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