Periodic points, induced hyperspace maps and Li-Yorke chaos

Abstract

Let $X$ be a continuum, and let $f: X \rightarrow X$ be a homeomorphism. Let $2^X$ be the hyperspace of all nonempty compact subsets of $X$ endowed with the Hausdorff metric. Let $C(X) \subset 2^X$ be the collection of all subcontinua of $X$. Let $2^f: 2^X \rightarrow 2^X$ be the induced map defined by $2^f(A)=f(A)$, and let $C(f)$ be the restriction of $2^f$ to $C(X)$. Let $\operatorname{Per}(f)$ and $R(f)$ be the set of all periodic points of $f$ and the set of all recurrent points of $f$ respectively. In this note we prove the following: If $f: X \rightarrow X$ is a dendrite homeomorphism with $R(f)=X$, then the induced homeomorphism $2^f$ has no Li-Yorke pairs. In the second part we produce a dendroid homeomorphism $f: X \rightarrow X$ such that $\operatorname{Per}(f)=X$, and the induced map $C(f)$ is chaotic in the sense of Li-Yorke, that is, there exists an uncountable subset $S$ of $C(X)$, such that for each pair $A, B \in S$, with $A \neq B,(A, B)$ is a Li-Yorke pair of $C(f)$.