On recurrence and entropy in the hyperspace of continua in dimension one

We show that if $G$ is a topological graph, and $f\colon G\to G$ is a continuous map, then the induced map $\tilde {f}$ defined on the hyperspace $C(G)$ of all connected subsets of $G$ by the natural formula $\tilde {f}(C)=f(C)$ carries the same entropy as $f$. It is well known that this does not hold on the larger hyperspace of all compact subsets. Also negative examples were given for the hyperspace $C(X)$ on some continua $X$, including dendrites.

Our work extends previous positive results obtained first for the much simpler case of a compact interval by completely different tools.