Shadowable Points of Free Semigroup Actions

The shadowable points of dynamical systems have been well-studied by Morales in his recent paper (2016). This paper aims to generalize the main results obtained by Morales to free semigroup actions. To this end, we introduce the notion of shadowable points of a free semigroup action. Let G be a free semigroup generated by finite continuous self-maps acting on compact metric space. We will prove the following results for G on compact metric spaces. The set of shadowable points of G is a Borel set. G has the pseudo-orbit tracing property (POTP) if and only if every point is shadowable point of G. The chain recurrent and non-wandering sets of G coincide when every chain recurrent point is shadowable point of G. The space X is totally disconnected at every shadowable point of G under certain condition.