Chain transitive induced interval maps on continua

Abstract

Let $f:I\rightarrow I$ be a continuous map of a compact interval $I$ and $C(I)$ be the space of all compact subintervals of $I$ with the Hausdorff metric. We investigate chain transitivity of induced maps on subcontinua of $C(I)$. In particular, we prove the following theorem: Let $\mathcal{M}$ be a subcontinuum of $C(I)$ having at most countably many partitioning points. Then, the induced map $\mathcal{F}:C(I)\to C(I)$ $($i.e. $\mathcal{F}(A):=\{f(x):x\in A\}$ for each $A \in C(I)$$)$ is chain transitive on $\mathcal{M}$ iff $\mathcal{F}^{2}\vert_{\mathcal{M}}=Id$.