Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map ff on a compact metric space XX, the chain recurrent set, R(f)R(f) has more than one chain component, then ff does not satisfy the asymptotic average shadowing property. We also show that if a continuous map ff on a compact metric space XX has the asymptotic average shadowing property and if AA is an attractor for ff, then AA is the single attractor for ff and we have A=R(f)A=R(f). We also study diffeomorphisms with asymptotic average shadowing property and prove that if MM is a compact manifold which is not finite with dimM=2dimM=2, then the C1C1 interior of the set of all C1C1 diffeomorphisms with the asymptotic average shadowing property is characterized by the set of ΩΩ-stable diffeomorphisms.