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

We show that if G is a topological graph, and f : G -> G is a continuous map, then the induced map f defined on the hyperspace C(G) of all connected subsets of G by the natural formula 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.