Chaos of trajectories

Abstract

Let f be a continuous self-map of a compact metric space X. Let $O\left(f\right)$ be the set of all trajectories under f, and let T denote its closure in the hyperspace $K\left(X\right)$ of all nonempty closed subsets of X equipped with the Hausdorff metric. The map f induces in a natural way a self-map $\overline{f}$ on the space T. In this paper, we study transitivity, sensitivity, and chaos for the induced system $(T,\overline{f})$. While we prove that this latter system can never be Devaney chaotic, we provide an example of a system $(X,f)$ for which the associated induced system $(T,\overline{f})$ is sensitive. We study which properties are inherited between the base system $(X,f)$ and its induced system $(T,\overline{f})$. We prove that several known results on the hyperspaces fail to be true on the spaces of trajectories.