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On shadowing: ordinary and ergodic. (English) Zbl 1293.37014
Summary: The aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property. In particular, we deduce the chaotic behavior of a map with ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property.

37B99 Topological dynamics
37A25 Ergodicity, mixing, rates of mixing
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
Full Text: DOI
[1] Akin, E., The general topology of dynamical systems, Grad. stud. math., vol. 1, (1993), Amer. Math. Soc. Providence, RI · Zbl 0781.54025
[2] Aoki, N.; Hiraide, K., Topological theory of dynamical systems, North-holland math. library, vol. 52, (1994), North-Holland Amsterdam · Zbl 0798.54047
[3] Akin, E.; Hurley, M.; Kennedy, J., Dynamics of topologically generic homeomorphisms, Mem. amer. math. soc., vol. 783, (2003), Amer. Math. Soc. Providence, RI · Zbl 1022.37010
[4] Arai, T.; Chinen, N., P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy, Topology appl., 154, 1254-1262, (2007) · Zbl 1121.54059
[5] Auslander, J.; Yorke, J., Interval maps, factor of maps and chaos, Tohoku math. J., 32, 177-188, (1980) · Zbl 0448.54040
[6] Bowen, R., Entropy for group endomorphisms and homogeneous spaces, Trans. amer. math. soc., 153, 401-414, (1971), MR 0274707 (43:469) · Zbl 0212.29201
[7] Devaney, R.L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley Redwood City · Zbl 0695.58002
[8] Eirola, T.; Nevanlinna, O.; Pilyugin, S.Yu., Limit shadowing property, Numer. funct. anal. optim., 18, 1-2, 75-92, (1997) · Zbl 0881.58049
[9] Glasner, E.; Weiss, B., Sensitive dependence to initial conditions, Nonlinearity, 6, 1067-1075, (1993) · Zbl 0790.58025
[10] Gu, R., The average-shadowing property and topological ergodicity, J. comput. appl. math, 206, 796-800, (2007) · Zbl 1115.37005
[11] Huang, W.; Ye, X., Devaney’s chaos or 2-scattering implies li – yorke’s chaos, Topology appl., 117, 3, 259-272, (2002) · Zbl 0997.54061
[12] Lee, K.; Sakai, K., Various shadowing properties and their equivalence, Discrete contin. dyn. syst., 13, 2, 533-540, (2005) · Zbl 1078.37015
[13] Li, T.Y.; Yorke, J.A., Period three implies chaos, Amer. math. monthly, 82, 985-992, (1975) · Zbl 0351.92021
[14] Pilyugin, S.Yu.; Rodinova, A.; Sakai, K., Orbital and weak shadowing in dynamical systems, Discrete contin. dyn. syst., 9, 2, 287-308, (2003) · Zbl 1015.37020
[15] Richeson, D.; Wiseman, J., Chain recurrence rates and topological entropy, Topology appl., 156, 251-261, (2008) · Zbl 1151.37304
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