The asymptotic average shadowing property and strong ergodicity

Let X be a compact metric space and f : X → X be a continuous map. In this paper, we prove that if f has the asymptotic average shadowing property (Abbrev. AASP) and an invariant Borel probability measure with full support or the positive upper Banach density recurrent points of f are dense in X, then for all n ⩾ 1, f × f × ⋯ × f(n times) and fn are totally strongly ergodic. Moreover, we also give some sufficient conditions for an interval map having the AASP to be Li-Yorke chaotic.