Shadowable points of free semigroup actions

Abstract

The shadowable points of dynamical systems has been well-studied by Morales \cite{MR3535492}. 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 a shadowable point of $G$. The chain recurrent and non-wandering sets of $G$ coincide when every chain recurrent point is a shadowable point of $G$. The space $X$ is totally disconnected at every shadowable point of $G$ under certain condition.