Topological chain and shadowing properties of dynamical systems on uniform spaces. (English) Zbl 1439.37020
Summary: This paper discusses topological shadowing property, chain transitivity, total chain transitivity, and chain mixing property for dynamical systems on uniform spaces and characterizes some topological chain properties for dynamical systems on compact uniform spaces. In particular, it is proved that a compact dynamical system is topologically chain mixing if and only if it is totally topologically chain transitive. Moreover, some basic properties of topological shadowing and non-wandering points on uniform spaces are obtained.
MSC:
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for 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 chain recurrence; topological chain transitivity; topological shadowing; topologically chain mixing; uniform spaceReferences:
| [1] | Ahmadi, S. A., On the topology of the chain recurrent set of a dynamical system, Appl. Gen. Topol., 15, 2, 167-174 (2014) · Zbl 1346.37010 |
| [2] | Ahmadi, S. A., Shadowing, ergodic shadowing and uniform spaces, Filomat, 31, 16, 5117-5124 (2017) · Zbl 1499.37046 |
| [3] | Ahmadi, S. A., A recurrent set for one-dimensional dynamical systems, Hacet. J. Math. Stat., 47, 1, 1-7 (2018) · Zbl 1397.54044 |
| [4] | Ahmadi, S. A.; Wu, X.; Feng, Z.; Ma, X.; Lu, T., On the entropy points and shadowing in uniform spaces, Int. J. Bifurc. Chaos, 28, 12, Article 1850155 pp. (2018) · Zbl 1403.37011 |
| [5] | Akin, E.; Glasner, E., Residual properties and almost equicontinuity, J. Anal. Math., 84, 1, 243-286 (2001) · Zbl 1182.37009 |
| [6] | Barwell, A. D.; Good, C.; Oprocha, P., Shadowing and expansivity in subspaces, Fundam. Math., 219, 3, 223-243 (2012) · Zbl 1350.37028 |
| [7] | Chu, C.-K.; Koo, K.-S., Recurrence and the shadowing property, Topol. Appl., 71, 3, 217-225 (1996) · Zbl 0861.54036 |
| [8] | Das, T.; Lee, K.; Richeson, D.; Wiseman, J., Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topol. Appl., 160, 1, 149-158 (2013) · Zbl 1293.37011 |
| [9] | Fernández, L.; Good, C.; Puljiz, M.; Ramírez, Á., Chain transitivity in hyperspaces, Chaos Solitons Fractals, 81, 83-90 (2015) · Zbl 1355.37032 |
| [10] | Good, C.; Leek, R.; Mitchell, J., Equicontinuity, transitivity and sensitivity: the Auslander-Yorke dichotomy revisited, Discrete Contin. Dyn. Syst., 40, 4, 2441-2474 (2020) · Zbl 1441.37010 |
| [11] | Good, C.; Macías, S., What is topological about topological dynamics?, Discrete Contin. Dyn. Syst., 38, 3, 1007-1031 (2018) · Zbl 1406.54021 |
| [12] | Good, C.; Mitchell, J.; Thomas, J., Preservation of shadowing in discrete dynamical systems, J. Math. Anal. Appl., 485, 1, Article 123767 pp. (2020) · Zbl 1436.37024 |
| [13] | James, I. M., Topologies and Uniformities (1999), Springer-Verlag: Springer-Verlag London · Zbl 0917.54001 |
| [14] | Kawaguchi, N., Entropy points of continuous maps with the sensitivity and the shadowing property, Topol. Appl., 210, 8-15 (2016) · Zbl 1366.37035 |
| [15] | Moothathu, T. K.S., Implications of pseudo-orbit tracing property for continuous maps on compacta, Topol. Appl., 158, 16, 2232-2239 (2011) · Zbl 1235.54018 |
| [16] | Oprocha, P.; Wu, X., On averaged tracing of periodic average pseudo orbits, Discrete Contin. Dyn. Syst., 37, 9, 4943-4957 (2017) · Zbl 1369.37031 |
| [17] | Ramírez Alfonsín, J. L., The Diophantine Frobenius Problem (2005), Oxford University Press: Oxford University Press Oxford · Zbl 1134.11012 |
| [18] | Rhodes, F., A generalization of isometries to uniform spaces, Math. Proc. Camb. Philos. Soc., 52, 3, 399-405 (1956) · Zbl 0072.40404 |
| [19] | Richeson, D.; Wiseman, J., Chain recurrence rates and topological entropy, Topol. Appl., 156, 2, 251-261 (2008) · Zbl 1151.37304 |
| [20] | Shah, S.; Das, R.; Das, T., Specification property for topological spaces, J. Dyn. Control Syst., 22, 4, 615-622 (2016) · Zbl 1376.37025 |
| [21] | Wu, X.; Liang, S.; Luo, Y.; Xin, M.; Zhang, X., A remark on the limit shadowing property for iterated function systems, UPB Sci. Bull., Ser. A, 81, 3, 107-114 (2019) · Zbl 1513.37018 |
| [22] | Wu, X.; Liang, S.; Ma, X.; Lu, T.; Ahmadi, S. A., The mean sensitivity and mean equicontinuity on uniform spaces, Int. J. Bifurc. Chaos (2020), in press |
| [23] | Wu, X.; Luo, Y.; Ma, X.; Lu, T., Rigidity and sensitivity on uniform spaces, Topol. Appl., 252, 145-157 (2019) · Zbl 1407.54024 |
| [24] | Wu, X.; Ma, X.; Chen, G.; Lu, T., A note on the sensitivity of semiflows, Topol. Appl., 271, Article 107046 pp. (2020) · Zbl 1440.37020 |
| [25] | Wu, X.; Ma, X.; Zhu, Z.; Lu, T., Topological ergodic shadowing and chaos on uniform spaces, Int. J. Bifurc. Chaos, 28, 3, Article 1850043 pp. (2018) · Zbl 1387.37007 |
| [26] | Wu, X.; Oprocha, P.; Chen, G., On various definitions of shadowing with average error in tracing, Nonlinearity, 29, 7, 1942-1972 (2016) · Zbl 1364.37023 |
| [27] | Wu, X.; Wang, L.; Liang, J., The chain properties and average shadowing property of iterated function systems, Qual. Theory Dyn. Syst., 17, 1, 219-227 (2018) · Zbl 1392.37009 |
| [28] | Wu, X.; Wang, L.; Liang, J., The chain properties and Li-Yorke sensitivity of Zadeh’s extension on the space of upper semi-continuous fuzzy sets, Iran. J. Fuzzy Syst., 15, 6, 83-95 (2018) · Zbl 1417.54019 |
| [29] | Wu, X.; Zhang, X.; Ma, X., Various shadowing in linear dynamical systems, Int. J. Bifurc. Chaos, 29, 3, Article 1950042 pp. (2019) · Zbl 1411.37025 |
| [30] | Zhang, X.; Wu, X., Diffeomorphisms with the \(\mathcal{M}_0\)-shadowing property, Acta Math. Sin. Engl. Ser., 35, 11, 1760-1770 (2019) · Zbl 1436.37034 |
| [31] | Zhang, X.; Wu, X.; Luo, Y.; Ma, X., A remark on limit shadowing for hyperbolic iterated function systems, UPB Sci. Bull., Ser. A, 81, 3, 139-146 (2019) · Zbl 1513.37019 |
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