Li-Yorke Sensitivity of Set-Valued Discrete Systems

Abstract

Consider the surjective, continuous map $f:X\to X$ and the continuous map $\overline{f}$ of $𝒦(X)$ induced by $f$, where $X$ is a compact metric space and $𝒦(X)$ is the space of all nonempty compact subsets of $X$ endowed with the Hausdorff metric. In this paper, we give a short proof that if $\overline{f}$ is Li-Yoke sensitive, then $f$ is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of $f$ does not imply Li-Yorke sensitivity of $\overline{f}$.