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On ergodicity of systems with the asymptotic average shadowing property. (English) Zbl 1163.37004
The author studies the asymptotic average shadowing property of a continuous map \(f\) of the compact metric space \((X,d)\) into itself, which he introduced in [Nonlinear Anal., Theory Methods Appl. 67, No. 6 (A), 1680–1689 (2007; Zbl 1121.37011)].
A sequence \(\{x_i\}_{i=0}^{\infty}\) of points in \(X\) is called an asymptotic average pseudo-orbit of \(f\) if
\[ \lim_{n \to \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f(x_i),x_{i+1}) = 0. \] A map \(f\) is said to have the asymptotic average shadowing property, if every asymptotic average pseudo-orbit \(\{x_i\}_{i=0}^{\infty}\) of \(f\) can be asymptotically shadowed in average by some point \(z\) in \(X\), i.e., if \(\lim_{n \to \infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f^{i}(z),x_{i}) = 0.\)
The main result of the article is following: if the Lyapunov stable map \(f\) has the asymptotic average shadowing property then \(f\) is topologically ergodic.

MSC:
37A25 Ergodicity, mixing, rates of mixing
37B25 Stability of topological dynamical systems
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
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References:
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