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The asymptotic average shadowing property and strong ergodicity. (English) Zbl 1339.37003
Summary: Let \(X\) be a compact metric space and \(f:X\to X\) 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 \(n\geqslant 1,f\times f\times\cdots\times f\) (\(n\) times) and \(f^n\) are totally strongly ergodic. Moreover, we also give some sufficient conditions for an interval map having the AASP to be Li-Yorke chaotic.

MSC:
37A25 Ergodicity, mixing, rates of mixing
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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