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On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property. (English) Zbl 1225.37031
Summary: Let \(X\) be a compact metric space and \(f: X \rightarrow X\) be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (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 \(n \geqslant 1\), \(f \times f \times \cdots \times f\) (\(n\) 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 \(f: [0, 1] \rightarrow [0, 1]\) being surjective and continuous, it is obtained that if \(f\) has two periodic points and the AASP, then \(f\) is Li-Yorke chaotic.

MSC:
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A25 Ergodicity, mixing, rates of mixing
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