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Exploring the asymptotic average shadowing property. (English) Zbl 1205.54034
Let \(f: X\to X\) be a continuous selfmap of a compact metric space \(X\). A sequence \(\{x_i\}^\infty_{i=0}\subseteq X\) is called an asymptotic average pseudo-orbit of \(f\) if
\[ \lim_{n\to\infty} {1\over n+1} \sum^n_{i=0} d(f(x_i), x_{i+1})= 0, \]
and is said to be asymptotic shadowed in average by the point \(z\in X\) if
\[ \lim_{n\to\infty} {1\over n+1} \sum^n_{i=0} d(x_i, f^i(z))= 0. \]
A map \(f\) is said to have the asymptotic average shadowing property (abbreviated AASP) if every asymptotic average pseudo-orbit of \(f\) is asymptotically shawdowed in average by some point in \(X\). The authors prove the following statement: Assume that there is a point \(x\in X\) that has the closure of its orbit contained in some open \(f\)-invariant subset \(U\subseteq X\). Assume that \(y\in X\) is a point whose orbit is metrically separated from \(U\). Then the map \(f\) does not have the property AASP.

MSC:
54H20 Topological dynamics (MSC2010)
34D05 Asymptotic properties of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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