Special $α$-limit sets in the hyperspace of continua: closedness and entropy for interval maps

This paper investigates the backward dynamics of hyperspace systems induced by continuous interval maps. Focusing on the hyperspace of continua, we provide a structural characterization of the $\alpha$-limit sets of backward branches for interval subcontinua. A central result of this work is the proof that the special $\alpha$-limit set of any nondegenerate subinterval is always closed. This reveals a striking topological contrast with classical single-point dynamics, where special $\alpha$-limit sets need not be closed. Finally, we prove that if a special $\alpha$-limit set contains two nondegenerate periodic continua with disjoint orbits, then the base map has a horseshoe and, consequently, positive topological entropy.