Stability Theorems in Pointwise Dynamics

TL;DR

This paper introduces new stability concepts in pointwise dynamics for homeomorphisms and measures, proving their relationships and stability properties in compact metric spaces.

Contribution

It defines minimally expansive and GH-stable points, and extends stability notions to measure-theoretic settings, establishing their implications.

Findings

Minimally expansive shadowable points are topologically stable and GH-stable.

μ-uniformly expansive μ-shadowable points are strong μ-topologically stable.

Results apply to homeomorphisms on compact metric spaces.

Abstract

We introduce minimally expansive and GH-stable points for homeomorphisms on metric spaces and $\mu$-uniformly expansive, $\mu$-shadowable and strong $\mu$-topologically stable points for Borel measures (with respect to a homeomorphism on a metric space). We prove that: (i) minimally expansive shadowable point of a homeomorphism on a compact metric space is topologically stable and GH-stable. (ii) $\mu$-uniformly expansive $\mu$-shadowable point for a Borel measure $\mu$ (with respect to a homeomorphism on a compact metric space) is strong $\mu$-topologically stable.