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

Abstract

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