On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property

Let X be a compact metric space and f: X → X be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (Abbrev. AASP) and other notions known from topological dynamics. We prove that if f has the AASP and the minimal points of f are dense in X, then for any n ⩾ 1, f × f × ⋯ × f(n times) is totally strongly ergodic. As a corollary, it is shown that if f is surjective and equicontinuous, then f does not have the AASP. Moreover we prove that if f is point distal, then f does not have the AASP. For f: [0, 1] → [0, 1] being surjective continuous, it is obtained that if f has two periodic points and the AASP, then f is Li–Yorke chaotic.

► We study the relations between the AASP and other notions from topological dynamics.
► This work improves on existing results.
► The relation between the AASP and point distality is discussed.
► We explore Li–Yorke chaos for the map on the interval with the AASP.