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On multi-transitivity with respect to a vector

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Abstract

A topological dynamical system (X, f) is said to be multi-transitive if for every n ∈ ℕ the system (X n, f × f 2 × … × f n) is transitive. We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets, answering a question proposed by Kwietniak and Oprocha (2012). We also show that multi-transitive systems are Li-Yorke chaotic.

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References

  1. Akin E. Recurrence in Topological Dynamical Systems: Families and Ellis Actions. The University Series in Mathematics. New York: Plenum Press, 1997

    Book  Google Scholar 

  2. Akin E, Kolyada S. Li-Yorke sensitivity. Nonlinearity, 2003, 16: 1421–1433

    Article  MATH  MathSciNet  Google Scholar 

  3. Blanchard F, Host B, Maass A. Topological complexity. Ergod Theor Dyn Syst, 2000, 20: 641–662

    Article  MATH  MathSciNet  Google Scholar 

  4. Furstenberg H. Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation. Math Syst Theory, 1967, 1: 1–49

    Article  MATH  MathSciNet  Google Scholar 

  5. Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, NJ: Princeton University Press, 1981

    MATH  Google Scholar 

  6. Glasner E. Classifying dynamical systems by their recurrence properties. Topol Methods Nonlinear Anal, 2004, 24: 21–40

    MATH  MathSciNet  Google Scholar 

  7. Gottschalk W H, Hedlund G A. Topological Dynamics. Providence, RI: Amer Math Soc, 1955

    MATH  Google Scholar 

  8. Huang W, Ye X D. Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology Appl, 2002, 117: 259–272

    Article  MATH  MathSciNet  Google Scholar 

  9. Huang W, Ye X D. Topological complexity, return times and weak disjointness. Ergod Theor Dyn Syst, 2004, 24: 825–846

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang W, Ye X D. Dynamical systems disjoint from any minimal system. Trans Amer Math Soc, 2005, 357: 669–694

    Article  MATH  MathSciNet  Google Scholar 

  11. Kwietniak D, Oprocha P. On weak mixing, minimality and weak disjointness of all iterates. Ergod Theor Dyn Syst, 2012, 32: 1661–1672

    Article  MATH  MathSciNet  Google Scholar 

  12. Li J. Transitive points via Furstenberg family. Topology Appl, 2011, 158: 2221–2231

    Article  MATH  MathSciNet  Google Scholar 

  13. Li T Y, Yorke J. Periodic three implies chaos. Amer Math Monthly, 1975, 82: 985–992

    Article  MATH  MathSciNet  Google Scholar 

  14. Moothathu T S. Diagonal points having dense orbit. Colloq Math, 2010, 120: 127–138

    Article  MATH  MathSciNet  Google Scholar 

  15. Mycielski J. Independent sets in topological algebras. Fund Math, 1964, 55: 139–147

    MATH  MathSciNet  Google Scholar 

  16. Opracha P. Weak mixing and product recurrence. Ann Inst Fourier (Grenoble), 2010, 60: 1233–1257

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China

    ZhiJing Chen

  2. Department of Mathematics, Shantou University, Shantou, 515063, China

    Jian Li

  3. School of Mathematics, South China Normal University, Guangzhou, 510631, China

    Jie Lü

Authors
  1. ZhiJing Chen
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  2. Jian Li
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  3. Jie Lü
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Correspondence to Jian Li.

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Chen, Z., Li, J. & Lü, J. On multi-transitivity with respect to a vector. Sci. China Math. 57, 1639–1648 (2014). https://doi.org/10.1007/s11425-014-4797-z

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  • DOI: https://doi.org/10.1007/s11425-014-4797-z

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