In this paper it is proved that a topologically stable invariant measure has no sinks or sources in its support; an expansive homeomorphism is topologically stable if it exhibits a topologically stable nonatomic Borel support measure and a continuous map has the shadowing property if there exists an invariant measure with the shadowing property such that each almost periodic point is contained in the support of the invariant measure.
"Topological stability and shadowing of dynamical systems from measure theoretical viewpoint." Topol. Methods Nonlinear Anal. 58 (1) 307 - 321, 2021. https://doi.org/10.12775/TMNA.2020.071