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

Yingxuan Niu and Shoubao Su

Chaos, Solitons & Fractals, 2011, vol. 44, issue 6, 429-432

Abstract: 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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:44:y:2011:i:6:p:429-432

DOI: 10.1016/j.chaos.2011.03.008

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