A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

Abstract

We prove that if a continuous, Lyapunov stable map f from a compact metric space X into itself is topologically transitive and has the asymptotic average shadowing property, then X is consisting of one point. As an application, we prove that the identity map i X : X → X does not have the asymptotic average shadowing property, where X is a compact metric space with at least two points.