Li-Yorke chaos and topological distributional chaos in a sequence

Source abstract

We study here the topological notion of Li-Yorke chaos defined for uniformly continuous self-maps defined on uniform Hausdorff spaces, which are not necessarily compact metrizable. We prove that a weakly mixing uniformly continuous self-map defined on a second countable Baire uniform Hausdorff space without isolated points is Li-Yorke chaotic. Further, we define and study the notion of topological distributional chaos in a sequence for uniformly continuous self-maps defined on uniform Hausdorff spaces. We prove that Li-Yorke chaos is equivalent to topological distributional chaos in a sequence for uniformly continuous self-maps defined on second countable Baire uniform Hausdorff space without isolated points. As a consequence, we obtain that Devaney chaos implies topological distributional chaos in a sequence.