Topological stability from a measurable viewpoint
Keonhee Lee, Seunghee Lee, C. A. Morales
TL;DR
The paper defines $\mu$-topological stability, a measurable analogue of Walters' topological stability, and shows it aligns with pointwise stability when the measure is a Dirac measure $m_p$. It proves that $f$ is $m_p$-topologically stable iff $p$ is a topologically stable point, and develops μ-shadowing for finitely supported measures on closed manifolds of dimension at least two, along with invariance under conjugacy and full-measure restriction. It further shows that for expansive maps the set $\mathcal{M}_{top}(f)$ is convex, relates μ-topological stability to absolutely continuous measures, and, in the nonatomic case, implies the set-valued stability framework; it also connects weak μ-shadowing to μ-topological stability for expansive maps.
Abstract
We introduce the {\em $μ$-topological stability}. This is a type of stability depending on the measure $μ$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ is a topologically stable point ($m_p$ is the Dirac measure supported on $p$). On closed manifolds of dimension $\geq2$ we prove that every $μ$-topologically stable map has the $μ$-shadowing property for finitely supported measures $μ$. Moreover the $μ$-topological stability is invariant under topological conjugacy or restriction to compact invariant sets of full measure. We also prove for expansive maps that the set of measures $μ$ for which the map is $μ$-topologically stable is convex. We analyze the relationship between $μ$-topological stability for absolutely continuous measures. In the nonatomic case we show that the $μ$-topological stability implies the set-valued stability approach in \cite{lm}. Finally, we show that every expansive map with the weak $μ$-shadowing property (c.f. \cite{lr}) is $μ$-topologically stable.