Measurable Versions of Some Topological Properties for Diffeomorphisms with Continuous Foliations

Abstract

In this paper, the concepts of plaque expansivity, topological quasi‐stability, and quasi‐shadowing property for Borel measures are considered. It is proved that every plaque expansive measure with the quasi‐shadowing property is topologically quasi‐stable with respect to its continuous foliation. At the same time, some other properties of the topological quasi‐stable measures, the plaque expansive measures, and measures with the quasi‐shadowing on a compact Riemannian manifold are investigated.