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The asymptotic average shadowing property and transitivity. (English) Zbl 1121.37011
Summary: We introduce the notion of the asymptotic average shadowing property (AASP) and investigate the relation between the AASP and transitivity. It is shown that a continuous map on \(X\) with the AASP is chain transitive and an \(\mathcal L\)-hyperbolic homeomorphism on \(X\) with the AASP is topologically transitive, where \(X\) is a compact metric space.

MSC:
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
39A10 Additive difference equations
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
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[1] Devaney, R.L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley Redwood City · Zbl 0695.58002
[2] Ruelle, D.; Takens, F., On the natural of turbulence, Comm. math. phys., 20, 167-192, (1971) · Zbl 0223.76041
[3] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On the devaney’s definition of chaos, Amer. math. monthly, 99, 332-334, (1992) · Zbl 0758.58019
[4] Auslander, J.; Yorke, J., Interval maps, factors of maps, and chaos, Tohoku math. J., 32, 177-188, (1980) · Zbl 0448.54040
[5] Glasner, E.; Weiss, B., Sensitive dependence on initial conditions, Nonlinearity, 6, 1067-1075, (1993) · Zbl 0790.58025
[6] Vellekoop, M.; Berglund, R., On intervals, transitivity = chaos, Amer. math. monthly, 4, 353-355, (1994) · Zbl 0886.58033
[7] Akin, E.; Auslander, J.; Berg, K., When is a transitive map chaotic?, (), 25-40 · Zbl 0861.54034
[8] S. Kolyada, L. Snoha, Some aspects of topological transitivity — a survey, in: Iteration theory, ECIT 94, Opava, 3-35, Grazer Math Ber. vol. 334, Karl-Franzens-Univ Graz. Graz, 1997, pp. 3-35 · Zbl 0907.54036
[9] 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
[10] He, L.F.; Gao, Y.H.; Yang, F.H., Some dynamical properties of continuous semi-flows having topological transitivity, Chaos solitons fractals, 14, 1159-1167, (2002) · Zbl 1098.37503
[11] Xiong, J.C., Chaos caused by topologically transitive systems, Sci. China ser. A, 35, 302-311, (2005)
[12] Walters, P., (), 224-231
[13] Blank, M.L., Small perturbatious of chaotic dynamical systems, Russian math. surv., 44, 1-33, (1989) · Zbl 0702.58063
[14] Sakai, K., Diffeomorphisms with the average-shadowing property on two dimensional closed manifold, Rocky mountain J. math., 3, 1-9, (2000)
[15] Pilyugin, S.Y., ()
[16] Zhang, Y., On the average-shadowing property, Acta sci. natur. univ. pekinensis, 37, 648-651, (2001) · Zbl 1022.37017
[17] Sakai, K., Shadowing properties of \(\mathcal{L}\)-hyperbolic homeomorphisms, Topology appl., 112, 229-243, (2001) · Zbl 0983.37024
[18] Gu, R.B., Recurrence and the asymptotic pseudo-orbit tracing property, Nonlinear anal., (2006)
[19] Gu, R.B.; Sheng, T.Q.; Xia, Z.J., The average-shadowing property and transitivity for flows, Chaos solitons fractals, 23, 989-995, (2005) · Zbl 1135.37300
[20] Walters, P., An introduction to ergodic theorem, (1982), Springer-Verlag New York
[21] Shimomara, T., Chain recurrence and POTP, (), 224-241
[22] Aoki, N., Topological dynamics, (), 625-740
[23] Zhao, J.L., Equicontinuity and POTP, Appl. math. J. Chinese univ. ser. A, 17, 179-181, (2002), (in Chinese) · Zbl 0999.54025
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