Topologically stable and $β$-persistent points of group actions

Abdul Gaffar Khan, Tarun Das

Published 2020-08-13Version 1

In this paper, we introduce topologically stable points and $\beta$-persistent points for finitely generated group actions on compact metric spaces. We prove that every shadowable point of an expansive action on a compact metric space is a topologically stable point. We justify that the expansivity of an action is not a necessary condition for the topological stability of a shadowable point of that action and the existence of a dense set of topologically stable points need not imply the topological stability of that action. Finally, we prove that every equicontinuous pointwise topologically stable action on a compact metric space is $\beta$-persistent.