Pointwise persistence and shadowing (Q2173232)

The authors introduce and study the notion of persistent shadowing property for homeomorphisms on compact metric spaces. More precisely, a homeomorphism \(f\) has the persistent shadowing property if for any \(\varepsilon >0\), there is \(\delta >0\) such that any \(\delta\)-pseudo orbit of a homeomorphism \(g\) with \(d_{C^0}(f,g)\leq \delta\) can be \((g,\varepsilon)\)-shadowed. They characterize the notion of persistent shadowing property from a pointwise viewpoint. It is proved that every pointwise persistent homeomorphism with shadowing property is persistent. Moreover, they show that any expansive homeomorphism with the persistent shadowing property is \(s\)-topologically stable.