Average shadowing and persistence in pointwise dynamics

Abstract

We introduce average shadowable measures and almost average shadowable measures for continuous maps of compact metric spaces and weakly topologically stable points and average persistent property for homeomorphisms of compact metric spaces. We prove that the set of all average shadowable measures is dense in the space of all Borel probability measures if and only if the set of all average shadowable points is dense in the phase space and every almost average shadowable measure can be weak⁎ approximated by measures having support equal to the closure of the set of all average shadowable points. Moreover, we prove that every minimally expansive point which is either persistent or α-persistent is weakly topologically stable and every mean equicontinuous pointwise weakly topologically stable homeomorphism is average persistent.