## Abstract

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.

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 language | English |
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Pages (from-to) | 1007-1031 |

Number of pages | 25 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 38 |

Issue number | 3 |

Early online date | Mar 2018 |

DOIs | |

Publication status | E-pub ahead of print - Mar 2018 |

## Keywords

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