Wang, Lidong; Ou, Xiaoping; Gao, Yuelin A weakly mixing dynamical system with the whole space being a transitive extremal distributionally scrambled set. (English) Zbl 1351.37061 Chaos Solitons Fractals 70, 130-133 (2015). Summary: It is known that the whole space can be a Li-Yorke scrambled set in a compact dynamical system, but this does not hold for distributional chaos. In this paper we construct a noncompact weekly mixing dynamical system, and prove that the whole space is a transitive extremal distributionally scrambled set in this system. Cited in 1 Document MSC: 37B20 Notions of recurrence and recurrent behavior in topological dynamical systems 37A25 Ergodicity, mixing, rates of mixing Keywords:distributionally scrambled set; noncompact weakly mixing dynamical system PDFBibTeX XMLCite \textit{L. Wang} et al., Chaos Solitons Fractals 70, 130--133 (2015; Zbl 1351.37061) Full Text: DOI References: [1] Li, T. Y.; Yorke, J. A., Period three implies chaos, Am Math Mon, 82, 985-992 (1975) · Zbl 0351.92021 [2] Schweitzer, B.; Smítal, J., Measures of chaos and spectral decomposition of dynamical systems of the interval, Trans Am Math Soc, 344, 737-754 (1994) · Zbl 0812.58062 [3] Liao, G. F.; Wang, L. D.; Duan, X. D., A chaotic function with a distributively scrambled set of full Lebesgue measure, Nonlinear Anal, 66, 2274-2280 (2007) · Zbl 1118.28010 [4] Gedeon, T., There are no chaotic mappings with residual scrambled sets, Bull Aust Math Soc, 36, 411-416 (1987) · Zbl 0646.26008 [5] Huang, W.; Ye, X. D., Homeomorphisms with the whole compact being scrambled sets, Ergodic Theory Dyn Syst, 21, 01, 77-91 (2001) · Zbl 0978.37003 [6] Wang, H.; Liao, G. F.; Fan, Q. J., A note on the map with the whole space being a scrambled set, Nonlinear Anal, 70, 2400-2402 (2009) · Zbl 1170.37010 [7] Oprocha, P., Distributional chaos revisited, Trans Am Math Soc, 361, 4901-4925 (2009) · Zbl 1179.37017 [8] Oprocha, P., Invariant scrambled sets and distributional chaos, Dyn Syst, 24, 31-43 (2009) · Zbl 1171.37022 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.