Topological stability from a measurable viewpoint

We introduce the {\em $\mu$-topological stability}. This is a type of stability depending on the measure $\mu$ 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 $\mu$-topologically stable map has the $\mu$-shadowing property for finitely supported measures $\mu$. Moreover the $\mu$-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 $\mu$ for which the map is $\mu$-topologically stable is convex. We analyze the relationship between $\mu$-topological stability for absolutely continuous measures. In the nonatomic case we show that the $\mu$-topological stability implies the set-valued stability approach in \cite{lm}. Finally, we show that every expansive map with the weak $\mu$-shadowing property (c.f. \cite{lr}) is $\mu$-topologically stable.