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Transitivity, mixing and chaos for a class of set-valued mappings. (English) Zbl 1193.37023
Summary: Consider the continuous map \(f: X\to X\) and the continuous map \(\overline(f)\) of \(K(X)\) into itself induced by \(f\), where \(X\) is a metric space and \(K(X)\) the space of all non-empty compact subsets of \(X\) endowed with the Hausdorff metric. According to the questions whether the chaoticity of \(f\) implies the chaoticity of \(\overline(f)\) posed by Román-Flores and when the chaoticity of \(f\) implies the chaoticity of \(\overline(f)\) posed by Fedeli, we investigate the relations between \(f\) and \(\overline(f)\) in the related dynamical properties such as transitivity, weakly mixing and mixing, etc. And by using the obtained results, we give the satisfied answers to Román-Flores’s question and Fedeli’s question.

MSC:
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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[1] Ashwin, P., Attractors of a randomly forced electronic oscillator, Physica D, 1999, 125: 302–310. · Zbl 1065.94567 · doi:10.1016/S0167-2789(98)00259-0
[2] Kaczynski, T., Mrozek, M., Conley index for discrete multivalued systems, Topology and Its Applications, 1995, 65: 83–96. · Zbl 0843.54042 · doi:10.1016/0166-8641(94)00088-K
[3] Devaney, R. L., An Introduction to Chaotic Dynamical Systems, Redwood City: Addison-Wesley, 1989. · Zbl 0695.58002
[4] Banks, J., Brooks, J., Cairns, G., Stacey, P., On the definition of chaos. Amer Math Monthly, 1992, 99: 332–334. · Zbl 0758.58019 · doi:10.2307/2324899
[5] Silverman, S., On maps with dense orbits and the definition of chaos, Rocky Mountain J. Math., 1992, 22: 353–375. · Zbl 0758.58024 · doi:10.1216/rmjm/1181072815
[6] Zhou Zuoling, Symbolic Dynamics (in Chinese), Shanghai: Shanhai Sci. and Thech. Edu. Publishing House, 1997.
[7] Huang Wen, Ye Xiangdong, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology and Its Applications, 2002, 117: 259–272. · Zbl 0997.54061 · doi:10.1016/S0166-8641(01)00025-6
[8] Mai Jiehua, Devaney’s chaos implies existence of s-scrambled sets, Proc. Amer. Math. Soc., 2004, 132: 2761–2767. · Zbl 1055.54019 · doi:10.1090/S0002-9939-04-07514-8
[9] Xiong Jincheng, Chaos in topologically transitive systems, Science in China, Ser. A, 2005, 48(7): 929–939. · Zbl 1096.37018 · doi:10.1360/04ys0120
[10] Román-Flores, H., A note on transitivity in set-valued discrete systems, Chaos, Solitons and Fractals, 2003, 17: 99–104. · Zbl 1098.37008 · doi:10.1016/S0960-0779(02)00406-X
[11] Fedeli, A., On chaotic set-valued discrete dynamical systems. Chaos, Solitons and Fractals, 2005, 23: 1381–1384. · Zbl 1079.37021
[12] Klein, E., Thompson, A. C., Theory of Correspondences, New York: Wiley-Interscience, 1984. · Zbl 0556.28012
[13] Xiong Jincheng, Yang Zhongguo, Chaos caused by a topologically mixing maps, in Dynamical Systems and Related Topics, Singapore: World Scientific Press, 1992, 550–572.
[14] Xiong Jincheng, Chen Ercai, Chaos caused by a strong-mixing measure-preserving transformation, Science in China, Ser. A, 1997, 40: 253–260. · Zbl 0914.54036 · doi:10.1007/BF02874517
[15] Barge, M., Martin, J., Chaos, periodicity and snakelike continua, Trans. Amer. Math. Soc., 1985, 289: 355–365. · Zbl 0559.58014 · doi:10.1090/S0002-9947-1985-0779069-7
[16] Vellekoop, M., Berglund, R., On intervals, Transitivity = chaos, Amer. Math. Monthly, 1994, 101: 353–355. · Zbl 0886.58033 · doi:10.2307/2975629
[17] Block, L. S., Coppel, W. A., Dynamics in one dimension, Lecture Notes in Math., 1513; Berlin: Springer, 1992. · Zbl 0746.58007
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