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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:
 [1] Bowen, R., Equilibrium states and the ergodic theory of axiom A diffeomorphisms, (1975), Springer-Verlag New York, pp. 68-87 [2] Walters, P., (), 224-231 [3] Yang, R.S., The pseudo-orbit tracing property and chaos, Acta math. sinica, 39, 382-386, (1996), (in Chinese) · Zbl 0872.54032 [4] Yang, R.S., Pseudo-orbit tracing property and completely positive entropy, Acta math. sinica, 42, 99-104, (1999), (in Chinese) · Zbl 1014.54025 [5] Blank, M.L., Small perturbatious of chaotic dynamical systems, Russian math. surveys, 44, 1-33, (1989) · Zbl 0702.58063 [6] Sakai, K., Diffeomorphisms with the average-shadowing property on two dimensional closed manifold, Rocky mountain J. math., 3, 1-9, (2000) [7] Yang, R.S., Topologically ergodic map, Acta math. sinica, 44, 1063-1068, (2001), (in Chinese) · Zbl 1012.37007 [8] Gu, R.B.; Guo, W.J., The average-shadowing property and topological ergodicity for flows, Chaos solitons fractals, 25, 387-392, (2005) · Zbl 1080.37014 [9] Gu, R.B., The asymptotic average shadowing property and transitivity, Nonlinear anal. TMA, 67, 1680-1689, (2007) · Zbl 1121.37011 [10] Xiong, J.C., Chaos in a topologically transitive system, Sci. China ser. A, 48, 929-939, (2005) · Zbl 1096.37018
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