American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

T. Y. Li and J. A. Yorke, Period three implies chaos, The American Mathematical Monthly, 82 (1975), 985-992. doi: 10.2307/2318254.  Google Scholar

[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.  Google Scholar

[5]

M. Misiurewicz, Chaos almost everywhere, in Iteration Theory and Its Functional Equations, Springer Berlin Heidelberg, 1985. doi: 10.1007/BFb0076425.  Google Scholar

[6]

P. Oprocha, Distributional chaos revisited, Transactions of the American Mathematical Society, 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7.  Google Scholar

[7]

P. Oprocha, Invariant scrambled sets and distributional chaos, Dynamical Systems, 24 (2009), 31-43. doi: 10.1080/14689360802415114.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T. Y. Li and J. A. Yorke, Period three implies chaos, The American Mathematical Monthly, 82 (1975), 985-992. doi: 10.2307/2318254.  Google Scholar

[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.  Google Scholar

[5]

M. Misiurewicz, Chaos almost everywhere, in Iteration Theory and Its Functional Equations, Springer Berlin Heidelberg, 1985. doi: 10.1007/BFb0076425.  Google Scholar

[6]

P. Oprocha, Distributional chaos revisited, Transactions of the American Mathematical Society, 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7.  Google Scholar

[7]

P. Oprocha, Invariant scrambled sets and distributional chaos, Dynamical Systems, 24 (2009), 31-43. doi: 10.1080/14689360802415114.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Piotr Oprocha. Specification properties and dense distributional chaos. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 821-833. doi: 10.3934/dcds.2007.17.821
[2]

Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 347-356. doi: 10.3934/dcdsb.2007.8.347
[3]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[4]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[5]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[6]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[7]

Marc Briane, Vincenzo Nesi. Distributional convergence of null Lagrangians under very mild conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 493-510. doi: 10.3934/dcdsb.2007.8.493
[8]

Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191
[9]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[10]

Arsen R. Dzhanoev, Alexander Loskutov, Hongjun Cao, Miguel A.F. Sanjuán. A new mechanism of the chaos suppression. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 275-284. doi: 10.3934/dcdsb.2007.7.275
[11]

Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161
[12]

Y. Charles Li. Chaos phenotypes discovered in fluids. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1383-1398. doi: 10.3934/dcds.2010.26.1383
[13]

Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917
[14]

Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
[15]

J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653
[16]

Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic & Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
[17]

Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[18]

François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[19]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933
[20]

Jaroslav Smítal, Marta Štefánková. Omega-chaos almost everywhere. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1323-1327. doi: 10.3934/dcds.2003.9.1323

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences