Periodic Shadowing for Set-Valued Maps

We study shadowing-type properties for set-valued dynamical systems. In particular, we investigate the periodic shadowing property and its relationship with expansivity and chain transitivity. We establish that for positively expansive set-valued maps on compact metric spaces, the shadowing property implies the periodic shadowing property. Furthermore, we show that for chain transitive maps, periodic shadowing implies both shadowing and topological transitivity. We also present a general construction of set-valued maps with shadowing arising from single-valued systems admitting an isometric involution. Several examples, including systems from symbolic dynamics, are provided to illustrate the theory.