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

Mixing invariant extremal distributional chaos

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

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