What is topological about topological dynamics?

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

Shifts of finite type as fundamental objects in the theory of shadowing

“…However, shadowing can be viewed as a strictly topological property, defined in terms of finite open covers, provided that we restrict our attention to compact metric spaces. Similar observations have been made in [8,14].…”

“…This observation allows the decoupling of shadowing from the metric, and we can then take the following definition of shadowing, which is valid for systems with compact Hausdorff (but not necessarily metric) domain, an application that has recently seen increased interest [7,8,14]. Definition 7 Let X be a (nonempty) compact Hausdorff topological space.…”

Preservation of shadowing in discrete dynamical systems

“…We start with the usual metric definitions, before giving the uniform definitions which coincide with the metric ones when the underlying space is compact. Finally we follow the example of Good and Macías [22] by providing definitions in terms of open covers which coincide with the uniform definitions when the space is compact Hausdorff. We then devote a section to the preservation of each of the aforementioned types of shadowing.…”

Devaney's chaos and eventual sensitivity

“…If it is, then in any uniform structure U on X, (T, X) would be sensitive (by the uniform space version of Theorem 3.2) and that would show that Devaney's chaos is a uniform and metric space notion and that sensitivity (defined for uniform spaces) is its essential ingredient. If it is not, then one can consider the following alternative definition of sensitivity that works for topological spaces, introduced in 2018 in [3] by C. Good and C. Macías. Definition 4.2.…”