×

Topological dynamics of Zadeh’s extension on upper semi-continuous fuzzy sets. (English) Zbl 1380.37024

Summary: In this paper, some characterizations are obtained on the transitivity, mildly mixing property, a-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh’s extensions restricted on some invariant closed subsets of all upper semi-continuous fuzzy sets in the level-wise metric. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and a-transitive, equicontinuous, uniformly rigid) if and only if the corresponding Zadeh’s extension is transitive (resp., mildly mixing, a-transitive, equicontinuous, uniformly rigid).

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
03E72 Theory of fuzzy sets, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akin, E. [1997] Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, (Plenum Press, NY). · Zbl 0919.54033
[2] Banks, J., Brooks, J., Cairns, G., Davis, G. & Stacey, P. [1992] “ On Devaney’s definition of chaos,” Amer. Math. Monthly99, 332-334. · Zbl 0758.58019
[3] Banks, J. [2005] “ Chaos for induced hyperspace maps,” Chaos Solit. Fract.25, 681-685. · Zbl 1071.37012
[4] Bauer, W. & Sigmund, K. [1975] “ Topological dynamics of transformations induced on the space of probability measures,” Monatshefte für Mathematik79, 81-92. · Zbl 0314.54042
[5] Bernardes, N. C. Jr. & Vermersch, R. M. [2016] “ On the dynamics of induced maps on the space of probability measures,” Trans. Amer. Math. Soc.368, 7703-7725. · Zbl 1366.37021
[6] Chen, Z., Li, J. & Lü, J. [2014] “ On multi-transitivity with respect to a vector,” Sci. China Math.57, 1639-1648. · Zbl 1311.54030
[7] Devaney, R. L. [1989] An Introduction to Chaotic Dynamical Systems, 2nd edition, (Addison-Wesley, CA). · Zbl 0695.58002
[8] Furstenberg, H. [1981] Recurrence in Ergodic Theory and Combinatorial Number Theory, (Princeton University Press, Princeton, NJ). · Zbl 0459.28023
[9] Glasner, S. & Maon, D. [1989] “ Rigidity in topological dynamics,” Ergod. Th. Dyn. Syst.9, 309-320. · Zbl 0661.58027
[10] Gu, R. [2007] “ Kato’s chaos in set-valued discrete systems,” Chaos Solit. Fract.31, 765-771. · Zbl 1140.37305
[11] Hindman, N. [1974] “ Finite sums from sequences within cells of a partition of \(<mml:math display=''inline`` overflow=''scroll``>\),” J. Combinat. Th. Ser. A17, 1-11. · Zbl 0285.05012
[12] Huang, W. & Ye, X. [2002] “ Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos,” Topol. Appl.117, 259-272. · Zbl 0997.54061
[13] Huang, W. & Ye, X. [2004] “ Topological complexity, return times and weak disjointness,” Ergod. Th. Dyn. Syst.24, 825-846. · Zbl 1052.37005
[14] Illanes, A. & Nadler, S. B. [1999] Hyperspaces, , Vol. 216 (Marcel Dekker Inc., NY). · Zbl 0933.54009
[15] Kupka, J. [2011a] “ On Devaney chaotic induced fuzzy and set-valued dynamical systems,” Fuzzy Sets Syst.177, 34-44. · Zbl 1242.37014
[16] Kupka, J. [2011b] “ On fuzzifications of discrete dynamical systems,” Inform. Sci.181, 2858-2872. · Zbl 1229.93107
[17] Kupka, J. [2014] “ Some chaotic and mixing properties of fuzzified dynamical systems,” Inform. Sci.279, 642-653. · Zbl 1360.37027
[18] Kwietniak, D. & Oprocha, P. [2012] “ On weak mixing, minimality and weak disjointness of all iterates,” Ergod. Th. Dyn. Syst.32, 1661-1672. · Zbl 1283.37026
[19] Lan, Y., Li, Q., Mu, C. & Huang, H. [2012] “ Some chaotic properties of discrete fuzzy dynamical systems,” Abstr. Appl. Anal.2012, 875381-1-9. · Zbl 1263.37031
[20] Li, T. Y. & Yorke, J. A. [1975] “ Period three implies chaos,” Amer. Math. Monthly82, 985-992. · Zbl 0351.92021
[21] Li, J. [2014] “ Equivalent conditions of Devaney chaos on the hyperspace,” J. Univ. Sci. Technol. China44, 93-95.
[22] Li, J., Oprocha, P., Ye, X. & Zhang, R. [2017] “ When are all closed subsets recurrent?” Ergod. Th. Dyn. Syst., doi :http://dx.doi.org/10.1017/etds.2016.5, 1-32. · Zbl 1380.37023
[23] Liao, G., Wang, L. & Zhang, Y. [2006] “ Transitivity, mixing and chaos for a class of set valued mappings,” Sci. China Ser. A Math.49, 1-8. · Zbl 1193.37023
[24] Liu, H., Shi, E. & Liao, G. [2009] “ Sensitivity of set-valued discrete systems,” Nonlin. Anal.82, 985-992.
[25] Moothathu, T. K. S. [2010] “ Diagonal points having dense orbit,” Colloq. Math.120, 127-138. · Zbl 1200.54020
[26] Román-Flores, H. & Chalco-Cano, Y. [2008] “ Some chaotic properties of Zadeh’s extension,” Chaos Solit. Fract.35, 452-459. · Zbl 1142.37308
[27] Sharkovsky, A. N. [1964] “ Coexistence of cycles of a continuous map of a line into itself,” Ukrain. Mat. Zh.16, 61-71 (Russian).
[28] Sharman, P. & Nagar, A. [2010] “ Inducing sensitivity on hyperspaces,” Topol. Appl.157, 2052-2058. · Zbl 1200.54005
[29] Wang, Y., Wei, G. & Campbell, W. H. [2009] “ Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems,” Topol. Appl.156, 803-811. · Zbl 1172.37006
[30] Wu, X., Wang, J. & Chen, G. [2015] “ \(<mml:math display=''inline`` overflow=''scroll``><mml:mi mathvariant=''script``>ℱ\)-sensitivity and multi-sensitivity of hyperspatial dynamical systems,” J. Math. Anal. Appl.429, 16-26. · Zbl 1370.37022
[31] Wu, X. [2016] “ Chaos of transformations induced onto the space of probability measures,” Int. J. Bifurcation and Chaos26, 1650227-1-12. · Zbl 1354.37016
[32] Wu, X. & Chen, G. [2016] “ On the large deviations theorem and ergodicity,” Commun. Nonlin. Sci. Numer. Simulat.30, 243-247. · Zbl 1489.37007
[33] Wu, X., Oprocha, P. & Chen, G. [2016] “ On various definitions of shadowing with average error in tracing,” Nonlinearity29, 1942-1972. · Zbl 1364.37023
[34] Wu, X. & Wang, X. [2016] “ On the iteration properties of large deviations theorem,” Int. J. Bifurcation and Chaos26, 1650054-1-6. · Zbl 1336.37008
[35] Wu, X. [2017] “ A remark on topological sequence entropy,” Int. J. Bifurcation and Chaos27, 1750107-1-7. · Zbl 1370.37017
[36] Wu, X. & Chen, G. [2017] “ Sensitive dependence and transitivity of fuzzified dynamical systems,” Inform. Sci.396, 14-23. · Zbl 1431.37008
[37] Wu, X., Wang, L. & Chen, G. [2017a] “ Weighted backward shift operators with invariant distributionally scrambled subsets,” Ann. Funct. Anal.8, 199-210. · Zbl 1373.47027
[38] Wu, X., Wang, L. & Liang, J. [2017b] “ The chain properties and Li-Yorke sensitivity of Zadeh’s extension on the space of upper semi-continuous fuzzy sets,” Iran. J. Fuzzy Syst., accepted for publication. · Zbl 1417.54019
[39] Wu, X., Wang, L. & Liang, J. [2017c] “ The chain properties and average shadowing property of iterated function systems,” Qual. Th. Dyn. Syst., doi :10.1007/s12346-016-0220-1.
[40] Wu, X., Wang, X. & Chen, G. [2017d] “ On the large deviations of weaker types,” Int. J. Bifurcation and Chaos27, 1750127-1-12. · Zbl 1377.60044
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.