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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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