×
Pirfalak, Farzaneh; Ahmadi, Seyyed Alireza; Wu, Xinxing; Kouhestani, Nader

Topological average shadowing property on uniform spaces. (English) Zbl 1470.37032

Qual. Theory Dyn. Syst. 20, No. 2, Paper No. 31, 15 p. (2021).
Summary: We introduce topological definition of average shadowing property. We prove that topological average shadowing property implies topological chain transitivity. In particular it is proved that for a dynamical system with dense minimal points, the topological average shadowing property implies topological strong ergodicity.

Cited in 6 Documents

MSC:

37B65 Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B02 Dynamics in general topological spaces
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)

Keywords:

topological average shadowing; topologically chain transitive; topologically ergodic; topologically strongly ergodic; uniform space
Cite Review PDF
Full Text: DOI

References:

[1] Smaoui, N.; Kanso, A., Cryptography with chaos and shadowing, Chaos Solitons Fractals, 42, 4, 2312-2321 (2009) · Zbl 1198.94126 · doi:10.1016/j.chaos.2009.03.128
[2] Shao, H.; Shi, Y.; Zhu, H., On distributional chaos in non-autonomous discrete systems, Chaos Solitons Fractals, 107, 234-243 (2018) · Zbl 1380.37038 · doi:10.1016/j.chaos.2018.01.005
[3] Li, N., Wang, L.: Sensitivity and chaoticity on nonautonomous dynamical systems. Internet J. Bifur. Chaos Appl. Sci. Engrg. 30(10), 2050146 (2020). doi:10.1142/S0218127420501461 · Zbl 1458.37027
[4] Liu, G.; Lu, T.; Yang, X.; Waseem, A., Further discussion about transitivity and mixing of continuous maps on compact metric spaces, J. Math. Phys., 61, 11, 112701 (2020) · Zbl 1460.37007 · doi:10.1063/5.0023553
[5] Li, R.; Lu, T.; Chen, G.; Liu, G., Some stronger forms of topological transitivity and sensitivity for a sequence of uniformly convergent continuous maps, J. Math. Anal. Appl., 494, 1, 124443 (2021) · Zbl 1464.37015 · doi:10.1016/j.jmaa.2020.124443
[6] Das, T.; Lee, K.; Richeson, D.; Wiseman, J., Spectral decomposition for topologically anosov homeomorphisms on noncompact and non-metrizable spaces, Topol. Its Appl., 160, 1, 149-158 (2013) · Zbl 1293.37011 · doi:10.1016/j.topol.2012.10.010
[7] Shah, S., Das, T., Das, R.: Distributional chaos on uniform spaces, Qual. Theory Dyn. Syst. 19(1) (2020) Paper No. 4, 13. doi:10.1007/s12346-020-00344-x · Zbl 1432.37035
[8] Ahmadi, SA, Shadowing, ergodic shadowing and uniform spaces, Filomat, 31, 5117-5124 (2017) · Zbl 1499.37046 · doi:10.2298/FIL1716117A
[9] Wu, X.; Luo, Y.; Ma, X.; Lu, T., Rigidity and sensitivity on uniform spaces, Topol. Its Appl., 252, 145-157 (2019) · Zbl 1407.54024 · doi:10.1016/j.topol.2018.11.014
[10] Wu, X.; Liang, S.; Ma, X.; Lu, T.; Ahmadi, SA, The mean sensitivity and mean equicontinuity in uniform spaces, Int. J. Bifur. Chaos (2020) · Zbl 1448.37012 · doi:10.1142/S0218127420501187
[11] Ahmadi, SA; Wu, X.; Feng, Z.; Ma, X.; Lu, T., On the entropy points and shadowing in uniform spaces, Int. J. Bifur Chaos, 28, 1850155 (2018) · Zbl 1403.37011 · doi:10.1142/S0218127418501559
[12] Good, C.; Macías, S., What is topological about topological dynamics?, Discrete Contin. Dyn. Syst., 38, 1007-1031 (2018) · Zbl 1406.54021 · doi:10.3934/dcds.2018043
[13] Ahmadi, SA, On the topology of the chain recurrent set of a dynamical system, Appl. Gen. Topol., 15, 2, 167-174 (2014) · Zbl 1346.37010 · doi:10.4995/agt.2014.3050
[14] Akin, E.: The General Topology of Dynamical Systems. Graduate Studies in Mathematics, vol. 1. American Mathematical Society, Providence (1993) · Zbl 0781.54025
[15] Akin, E., Wiseman, J.: Chain recurrence and strong chain recurrence on uniform spaces. In: Dynamical Systems and Random Processes, Vol. 736 of Contemporary Mathematical American Mathematical Society, Providence, RI, 2019, pp. 1-29. doi:10.1090/conm/736/14831 · Zbl 1507.37018
[16] Morales, CA; Sirvent, V., Expansivity for measures on uniform spaces, Trans. Am. Math. Soc., 368, 8, 5399-5414 (2016) · Zbl 1339.54027 · doi:10.1090/tran/6555
[17] Wu, X.; Ma, X.; Zhu, Z.; Lu, T., Topological ergodic shadowing and chaos on uniform spaces, Int. J. Bifur. Chaos, 28, 3, 1850043 (2018) · Zbl 1387.37007 · doi:10.1142/S0218127418500438
[18] Blank, ML, Metric properties of-trajectories of dynamical systems with stochastic behaviour, Ergodic Theory Dyn. Syst., 8, 3, 365378 (1988) · doi:10.1017/S01433857000451X
[19] Dong, Y.; Tian, X.; Yuan, X., Ergodic properties of systems with asymptotic average shadowing property, J. Math. Anal. Appl., 432, 1, 53-73 (2015) · Zbl 1353.37016 · doi:10.1016/j.jmaa.2015.06.046
[20] Gu, R., The asymptotic average shadowing property and transitivity, Nonlinear Anal. Theory Methods Appl., 67, 6, 1680-1689 (2007) · Zbl 1121.37011 · doi:10.1016/j.na.2006.07.040
[21] Niu, Y., The average-shadowing property and strong ergodicity, J. Math. Anal. Appl., 376, 2, 528-534 (2011) · Zbl 1210.37016 · doi:10.1016/j.jmaa.2010.11.024
[22] Niu, Y.; Su, S., On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property, Chaos Solitons Fractals, 44, 6, 429-432 (2011) · Zbl 1225.37031 · doi:10.1016/j.chaos.2011.03.008
[23] Niu, Y.; Wang, Y.; Su, S., The asymptotic average shadowing property and strong ergodicity, Chaos Solitons Fractals, 53, 34-38 (2013) · Zbl 1339.37003 · doi:10.1016/j.chaos.2013.04.009
[24] Wu, X.; Wang, L.; Liang, J., The chain properties and average shadowing property of iterated function systems, Qual. Theory Dyn. Syst., 17, 219-227 (2018) · Zbl 1392.37009 · doi:10.1007/s12346-016-0220-1
[25] Ahmadi, SA; Wu, X.; Chen, G., Topological chain and shadowing properties of dynamical systems on uniform spaces, Topol. Appl., 275, 107153 (2020) · Zbl 1439.37020 · doi:10.1016/j.topol.2020.107153
[26] Das, P.; Das, T., Various types of shadowing and specification on uniform spaces, J. Dyn. Control Syst., 24, 253-267 (2018) · Zbl 1385.37013 · doi:10.1007/s10883-017-9388-1
[27] Hart, KP; Nagata, J-I; Vaughan, JE, Encyclopedia of General Topology (2004), Amsterdam: Elsevier Science Publishers, Amsterdam · Zbl 1059.54001
[28] Walters, P., An Introduction to Ergodic Theory. Graduate Texts in Mathematics. (2000), New York: Springer, New York · Zbl 0958.28011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.