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On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property. (English) Zbl 1225.37031
Summary: Let $$X$$ be a compact metric space and $$f: X \rightarrow X$$ be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (AASP) and other notions known from topological dynamics. We prove that if $$f$$ has the AASP and the minimal points of $$f$$ are dense in $$X$$, then for any $$n \geqslant 1$$, $$f \times f \times \cdots \times f$$ ($$n$$ times) is totally strongly ergodic. As a corollary, it is shown that if $$f$$ is surjective and equicontinuous, then $$f$$ does not have the AASP. Moreover, we prove that if $$f$$ is point distal, then $$f$$ does not have the AASP. For $$f: [0, 1] \rightarrow [0, 1]$$ being surjective and continuous, it is obtained that if $$f$$ has two periodic points and the AASP, then $$f$$ is Li-Yorke chaotic.

##### MSC:
 37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 37A25 Ergodicity, mixing, rates of mixing
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##### References:
 [1] Bowen, R., Equilibrium states and the ergodic theory of axiom a diffeomorphisms, (1975), Springer NewYork, pp. 68-87 [2] Walters, P., On the pseudo-orbit tracing property its relationship to stability, Lecture notes in mathematics, (1978), Springer Berlin, pp. 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] Aoki, N.; Hiraide, K., Topological theory of dynamical systems, Recent advances, North-holland mathematical library, 52, (1994), North-Holland Publishing Co. Amsterdam · Zbl 0798.54047 [5] Gu, R., The asymptotic average-shadowing property and transitivity, Nonlinear anal, 67, 6, 1680-1689, (2007) · Zbl 1121.37011 [6] Gu, R., On ergodicity of systems with the asymptotic average shadowing property, Comput math appl, 55, 1137-1141, (2008) · Zbl 1163.37004 [7] Honary, B.; Zamani Bahabadi, A., Asymptotic average shadowing property on compact metric spaces, Nonlinear anal, 69, 2857-2863, (2008) · Zbl 1152.54027 [8] Kulczycki, M.; Oprocha, P., Exploring the asymptotic average shadowing property, J differ equat appl, 16, 10, 1131-1140, (2010) · Zbl 1205.54034 [9] Niu, Y.X., Dynamical systems with the asymptotic average shadowing property, Appl math J Chinese univ ser A, 22, 4, 462-468, (2007), in Chinese · Zbl 1150.54037 [10] Bowen, R., Periodic points and measures for axiom A diffeomorphisms, Trans amer math soc, 154, 377-397, (1971) · Zbl 0212.29103 [11] Denker, M.; Grillenberger, C.; Sigmund, K., Ergodic theory on compact spaces, Lecture notes in mathematics, (1975), Springer-Verlag Berlin [12] Sigmund, K., On dynamical systems with the specification property, Trans amer math soc, 190, 285-299, (1974) · Zbl 0286.28010 [13] Blanchard, F.; Glasner, E.; Koyada, S.; Maass, A., On li – yorke pairs, J reine angew math, 547, 3, 51-68, (2002) · Zbl 1059.37006 [14] Huang, W.; Ye, X.D., Devaney’s chaos or 2-scattering implies li – yorke’s chaos, Topology appl, 117, 259-272, (2002) · Zbl 0997.54061 [15] Kuchta, M.; Smital, J., Two points scrambled set implies chaos, (), pp. 427-430 [16] Niu, Y.X., The large deviations theorem and sensitivity, Chaos solitons fractals, 42, 1, 609-614, (2009) · Zbl 1198.37013
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