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.
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