Mixing invariant extremal distributional chaos

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November 2016, 36(11): 6533-6538. doi: 10.3934/dcds.2016082

Mixing invariant extremal distributional chaos

Lidong Wang ^1, , Xiang Wang ^1, , Fengchun Lei ^1, and Heng Liu ^2,

1.	School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China, China
2.	School of Sciences, Dalian Nationalities University, Dalian 116600, China

Received July 2015 Revised June 2016 Published August 2016

Abstract
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In this paper we give a complicated distributional chaos, that is, a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set.

Keywords: Distributional chaos, scrambled set..

Mathematics Subject Classification: Primary: 37B10; Secondary: 54H2.

Citation: Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082

References:

[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:1406.5822 doi: 10.1088/0951-7715/29/7/1942.

show all references

References:

[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:1406.5822 doi: 10.1088/0951-7715/29/7/1942.

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