# American Institute of Mathematical Sciences

March  2018, 38(3): 1007-1031. doi: 10.3934/dcds.2018043

## What is topological about topological dynamics?

 1 School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK 2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D. F., C. P. 04510. México

* Corresponding author

Received  September 2016 Revised  September 2017 Published  December 2017

Fund Project: The first author gratefully acknowledge support from the European Union through the H2020-MSCA-IF-2014 project ShadOmIC (SEP-210195797). The second named author was partially supported by DGAPA, UNAM. This author thanks The University of Birmingham, UK, for the support given during this research.

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 $ω$-limit sets are (uniform) internally chain transitive and weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.

Citation: Chris Good, Sergio Macías. What is topological about topological dynamics?. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1007-1031. doi: 10.3934/dcds.2018043
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