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Science China Mathematics

, Volume 62, Issue 12, pp 2527–2534 | Cite as

Weak mixing in switched systems

  • Yu Huang
  • Xingfu ZhongEmail author
Articles
  • 23 Downloads

Abstract

Given a switched system, we introduce weakly mixing sets of type 1, 2 and Xiong chaotic sets of type 1, 2 with respect to a given set and show that they are equivalent, respectively.

Keywords

weakly mixing chaotic set switched system 

MSC(2010)

37B05 37B55 54H20 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771459 and 11471125) and International Program for PhD Candidates, Sun Yat-sen University. The second author thanks Prof. Xingfu Zou and his department for their hospitality during his visit to University of Western Ontario. A part of this work was done when the second author was visiting University of Western Ontario. The authors are grateful to the referees for their valuable comments which have led to improvement of the paper.

References

  1. 1.
    Akin E. Lectures on Cantor and Mycielski sets for dynamical systems. Contemp Math, 2004, 356: 21–79MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bahabadi A Z. Shadowing and average shadowing properties for iterated function systems. Georgian Math J, 2015, 22: 179–184MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balibrea F, Oprocha P. Weak mixing and chaos in nonautonomous discrete systems. Appl Math Lett, 2012, 25: 1135–1141MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biś A. Entropies of a semigroup of maps. Discrete Contin Dyn Syst, 2004, 11: 639–648MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blanchard F, Huang W. Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin Dyn Syst, 2012, 20: 275–311MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bufetov A. Topological entropy of free semigroup actions and skew-product transformations. J Dyn Control Syst, 1999, 5: 137–143MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cairns G, Kolganova A, Nielsen A. Topological transitivity and mixing notions for group actions. Rocky Mountain J Math, 2007, 37: 371–397MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen Z, Li J, Lü J. On multi-transitivity with respect to a vector. Sci China Math, 2014, 57: 1639–1648MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen Z, Li J, Lü J. Point transitivity, Δ-transitivity and multi-minimality. Ergodic Theory Dynam Systems, 2015, 35: 1423–1442MathSciNetCrossRefGoogle Scholar
  10. 10.
    Costa O L V, Fragoso M D, Marques R P. Discrete-Time Markov Jump Linear Systems. Probability and Its Applications. London: Springer-Verlag, 2005CrossRefGoogle Scholar
  11. 11.
    Huang W, Li J, Ye X, et al. Positive topological entropy and Δ-weakly mixing sets. Adv Math, 2017, 306: 653–683MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huang Y, Zhong X. Carathéodory-Pesin structures associated with control systems. Systems Control Lett, 2018, 112: 36–41MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hui H, Ma D. Some dynamical properties for free semigroup actions. Stoch Dyn, 2018, 18: 1850032MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kolyada S, Snoha L. Some aspects of topological transitivity—a survey. Grazer Math Ber, 1997, 334: 3–35MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kontorovich E, Megrelishvili M. A note on sensitivity of semigroup actions. Semigroup Forum, 2008, 76: 133–141MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li J. Localization of mixing property via Furstenberg families. Discrete Contin Dyn Syst, 2014, 35: 725–740MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li J, Oprocha P, Zhang G. On recurrence over subsets and weak mixing. Pacific J Math, 2015, 277: 399–424MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li T Y, Yorke J A. Period three implies chaos. Amer Math Monthly, 1975, 82: 985–992MathSciNetCrossRefGoogle Scholar
  19. 19.
    Moothathu T S. Diagonal points having dense orbit. Colloq Math, 2010, 120: 127–138MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nadler S B. Hyperspaces of Sets: A Text with Research Questions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 49. New York-Basel: Marcel Dekker, 1978Google Scholar
  21. 21.
    Oprocha P. Coherent lists and chaotic sets. Discrete Contin Dyn Syst, 2013, 31: 797–825MathSciNetCrossRefGoogle Scholar
  22. 22.
    Oprocha P, Zhang G. On local aspects of topological weak mixing in dimension one and beyond. Studia Math, 2011, 202: 261–288MathSciNetCrossRefGoogle Scholar
  23. 23.
    Oprocha P, Zhang G. On sets with recurrence properties, their topological structure and entropy. Topology Appl, 2012, 159: 1767–1777MathSciNetCrossRefGoogle Scholar
  24. 24.
    Oprocha P, Zhang G. On weak product recurrence and synchronization of return times. Adv Math, 2013, 244: 395–412MathSciNetCrossRefGoogle Scholar
  25. 25.
    Polo F. Sensitive dependence on initial conditions and chaotic group actions. Proc Amer Math Soc, 2010, 138: 2815–2826MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rodrigues F B, Varandas P. Specification and thermodynamical properties of semigroup actions. J Math Phys, 2016, 57: 263–267MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sun Z, Ge S S. Switched Linear Systems: Control and Design. New York: Springer-Verlag, 2006Google Scholar
  28. 28.
    Wang H, Chen Z, Fu H. M-systems and scattering systems of semigroup actions. Semigroup Forum, 2015, 91: 699–717MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wang J. Almost strong mixing group actions in topological dynamics. ArXiv:1405.5971, 2014Google Scholar
  30. 30.
    Wang L, Liang J, Chu Z. Weakly mixing property and chaos. Arch Math (Basel), 2017, 109: 83–89MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wang Y, Ma D. On the topological entropy of a semigroup of continuous maps. J Math Anal Appl, 2015, 427: 1084–1100MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang Y, Ma D, Lin X. On the topological entropy of free semigroup actions. J Math Anal Appl, 2016, 435: 1573–1590MathSciNetCrossRefGoogle Scholar
  33. 33.
    Xiong J, Yang Z. Chaos caused by a topologically mixing map. In: Proceedings of the International Conference, Dynamical Systems and Related Topics. Singapore: World Scientific, 1990, 550–572Google Scholar
  34. 34.
    Zeng T. Multi-transitivity and Δ-transitivity for semigroup actions. Topology Appl, 2017, 226: 1–15MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-sen UniversityGuangzhouChina

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