Topologically stable and persistent points of group actions

In this paper, we introduce topologically stable points, persistent points, persistent property, persistent measures and almost persistent measures for first countable Hausdorff group actions of compact metric spaces. We prove that the set of all persistent points is measurable and it is closed if the action is equicontinuous. We also prove that the set of all persistent measures is a convex set and every almost persistent measure is a persistent measure. Finally, we prove that every equicontinuous pointwise topologically stable first countable Hausdorff group action of a compact metric space is persistent. In particular, every equicontinuous pointwise topologically stable flow is persistent.