Specification and Shadowing Properties of Free Semigroup Actions

In this paper, we introduce shadowing-like properties and specification-like properties of free semigroup actions, such as periodic shadowing property, strong periodic shadowing property, special periodic shadowing property, local specification property, and local weak specification property. Furthermore, some fundamental properties of these notions are given. Finally, under some conditions, we obtain that these notions of shadowing-like properties and specification-like properties of free semigroup actions are equivalent, and the same conclusion is also obtained in the finite product systems of free semigroup actions.