On Recurrence and Entropy in Hyperspace of Continua in Dimension One

The uploaded paper proves h_top(f̃)=h_top(f) for graph maps by fully characterizing the recurrent continua of the hyperspace system and showing that every nondegenerate recurrent contribution has zero entropy, with the lower bound coming from the singleton subsystem. This is explicit in Theorem 1.1 and the proof that uses Theorem 4.6 and Lemma 5.1 together with the variational principle. By contrast, the model’s upper-bound argument hinges on an unsubstantiated step: that an almost conjugacy π: M→N lifts to a hyperspace factor Π: C(M)→C(N) with bounded fibers so that entropy is preserved or does not decrease across Π. The paper does not claim this, and in general the hyperspace fiber cardinalities are not uniformly bounded, so Bowen-type invariance does not transfer to Π. The paper avoids this pitfall and is correct; the model’s proof outline has a critical gap.