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

TL;DR

This paper introduces and studies the properties of $eta$-persistent points and measures in group actions on compact metric spaces, establishing their measurability, convexity, and stability under equicontinuity conditions.

Contribution

It defines $eta$-persistent concepts for group actions and proves their key properties, including measurability, convexity, and stability in equicontinuous actions.

Findings

The set of $eta$-persistent points is measurable and closed under equicontinuity.

The set of $eta$-persistent measures is convex.

Every almost $eta$-persistent measure is $eta$-persistent.

Abstract

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.