Persistence and expansivity through pointwise dynamics

Using the notion of topologically stable points, it is proved that every equicontinuous pointwise topologically stable homeomorphism of a compact metric space is persistent. Also, using the notion of strong topologically stable points of a Borel probability measure, it is shown that every pointwise strong topologically stable Borel probability measure with respect to an equicontinuous homeomorphism of a compact metric space is strong persistent. Further, it is established that any homeomorphism of as well as that of (0,1) does not admit any uniformly expansive point. Finally, these results are used to show that the unit circle does not admit any expansive homeomorphism.