Article Contents
Article Contents

Mixing invariant extremal distributional chaos

• In this paper we give a complicated distributional chaos, that is, a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set.
Mathematics Subject Classification: Primary: 37B10; Secondary: 54H20.

 Citation:

•  [1] T. Gedeon, There are no chaotic mappings with residual scrambled sets, Bulletin of the Australian Mathematical Society, 36 (1987), 411-416.doi: 10.1017/S0004972700003695. [2] W. Huang and X. D. Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergodic Theory and Dynamical Systems, 21 (2001), 77-91.doi: 10.1017/S0143385701001079. [3] T. Y. Li and J. A. Yorke, Period three implies chaos, The American Mathematical Monthly, 82 (1975), 985-992.doi: 10.2307/2318254. [4] G .F. Liao, L. D. Wang and X. D. Duan, A chaotic function with a distributively scrambled set of full Lebesgue measure, Nonlinear Analysis: Theory, Methods & Applications, 66 (2007), 2274-2280.doi: 10.1016/j.na.2006.03.018. [5] M. Misiurewicz, Chaos almost everywhere, in Iteration Theory and Its Functional Equations, Springer Berlin Heidelberg, 1985.doi: 10.1007/BFb0076425. [6] P. Oprocha, Distributional chaos revisited, Transactions of the American Mathematical Society, 361 (2009), 4901-4925.doi: 10.1090/S0002-9947-09-04810-7. [7] P. Oprocha, Invariant scrambled sets and distributional chaos, Dynamical Systems, 24 (2009), 31-43.doi: 10.1080/14689360802415114. [8] B. Schweitzer and J. Smítal, Measures of chaos and spectral decomposition of dynamical systems of the interval, Transactions of the American Mathematical Society, 344 (1994), 737-754.doi: 10.1090/S0002-9947-1994-1227094-X. [9] H. Wang, G. F. Liao and Q. J. Fan, A note on the map with the whole space being a scrambled set, Nonlinear Analysis: Theory, Methods & Applications, 70 (2009), 2400-2402.doi: 10.1016/j.na.2008.03.024. [10] H. Wang, F. C. Lei and L. D. Wang, Dense invariant open distributionally scrambled sets and closed distributionally scrambled sets, Topology and its Applications, 165 (2014), 110-120.doi: 10.1016/j.topol.2014.01.019. [11] X. Wu and P. Zhu, Invariant scrambled set and maximal distributional chaos, Ann. Polon. Math., 109 (2013), 271-278.doi: 10.4064/ap109-3-3. [12] X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972. arXiv:1406.5822doi: 10.1088/0951-7715/29/7/1942.