Topologically stable and $\beta$-persistent points of group actions

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