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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.

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