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The average-shadowing property and strong ergodicity. (English) Zbl 1210.37016
Let \(f\) and \(g\) be discrete dynamical systems on compact metric spaces with the average shadowing property (ASP). The paper proves the following facts:
1. Density of minimal points of \(f\) implies its total strong ergodicity (i.e. strong ergodicity of \(f^k\) for every \(k\in\mathbb{N}_+\)).
2. The product \(f\times g\) is topologically transitive and has the ASP.
3. If \(f\) is nontrivial and distal then it can’t have the ASP.
4. The full shift map over finite alphabet has the ASP.

37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
Full Text: DOI
[1] Bowen, R., Equilibrium states and the ergodic theory of axiom A diffeomorphisms, (1975), Springer New York, pp. 68-87
[2] Walters, P., On the pseudo-orbit tracing property and its relationship to stability, (), 224-231
[3] Yang, R.S., The pseudo-orbit tracing property and chaos, Acta math. sin., 39, 382-386, (1996), (in Chinese) · Zbl 0872.54032
[4] Yang, R.S., Pseudo-orbit tracing property and completely positive entropy, Acta math. sin., 42, 99-104, (1999), (in Chinese) · Zbl 1014.54025
[5] Aoki, N.; Hiraide, K., Topological theory of dynamical systems, recent advances, North-holland math. library, vol. 52, (1994), North-Holland Amsterdam · Zbl 0798.54047
[6] Blank, M.L., Small perturbations of chaotic dynamical systems, Russian math. surveys, 44, 1-33, (1989) · Zbl 0702.58063
[7] Sakai, K., Diffeomorphisms with the average-shadowing property on two dimensional closed manifold, Rocky mountain J. math., 3, 1-9, (2000)
[8] Gu, Rongbao, The average-shadowing property and topological ergodicity, Comput. math. appl., 206, 796-800, (2007) · Zbl 1115.37005
[9] Akin, E.; Auslander, J.; Berg, K., When is a transitive map chaotic?, (), 25-40 · Zbl 0861.54034
[10] Akin, E.; Auslander, J.; Berg, K., Almost equicontinuity and the enveloping semigroup, (), 75-81 · Zbl 0929.54028
[11] Aoki, N., Topological dynamics, (), 625-740
[12] Zuoling, Zhou, Symbolic dynamics, advanced series in nonlinear science, (1997), Shanghai Scientific and Technological Education Publishing House Shanghai, (in Chinese)
[13] Sakai, K., Various shadowing properties for positively expansive maps, Topology appl., 131, 15-31, (2003) · Zbl 1024.37016
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