Shadowing Property of Hyperspace for Free Semigroup Actions

In this paper, we introduce a notion of shadowing property for a free semigroup action on a compact metric space, which is different of the notion of the shadowing property introduced by Bahabadi, called chain shadowing property. We study the relation between the shadowing property of a free semigroup action on a compact metric space X and the shadowing property of the induced free semigroup action on the hyperspace 2^X. Specially, we not only theoretically prove that \((F_{m}^{+},\mathcal {F})\curvearrowright X\) has the (chain) shadowing property if and only if \((F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}\) has the (chain) shadowing property, but also give examples to illustrate it. Finally, we compare the two notions of shadowing for free semigroup actions and obtain an interesting result that if \((F_{m}^{+},\mathcal {F})\curvearrowright X\) has the shadowing property, then it has the chain shadowing property, but not vice versa.