Equivalent conditions of Devaney chaos on the hyperspace

Jian Li

doi:10.3969/j.issn.0253-2778.2014.02.002

paper

Equivalent conditions of Devaney chaos on the hyperspace

arXiv:1304.2447 · doi:10.3969/j.issn.0253-2778.2014.02.002

Abstract

Let $T$ be a continuous self-map of a compact metric space $X$ . The transformation $T$ induces natural a continuous self-map $T_{K}$ on the hyperspace $K (X)$ of all non-empty closed subsets of $X$ . In this paper, we show that the system $(K (X), T_{K})$ on the hyperspace is Devaney chaotic if and only if $(K (X), T_{K})$ is an HY-system if and only if $(X, T)$ is an HY-system, where a system $(Y, S)$ is called an HY-system if it is totally transitive and has dense small periodic sets.

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