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The three versions of distributional chaos

Author

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  • Balibrea, F.
  • Smı́tal, J.
  • Štefánková, M.

Abstract

The notion of distributional chaos was introduced by Schweizer and Smı́tal [Trans. Amer. Math. Soc. 344 (1994) 737] for continuous maps of the interval. However, it turns out that, for continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we consider the weakest one, DC3. We show that DC3 does not imply chaos in the sense of Li and Yorke. We also show that DC3 is not invariant with respect to topological conjugacy. In other words, there are lower and upper distribution functions Φxy and Φxy* generated by a continuous map f of a compact metric space (M, ρ) such that Φxy*(t)>Φxy(t) for all t in an interval. However, f on the same space M, but with a metric ρ′ generating the same topology as ρ is no more DC3.

Suggested Citation

  • Balibrea, F. & Smı́tal, J. & Štefánková, M., 2005. "The three versions of distributional chaos," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1581-1583.
    • Handle: RePEc:eee:chsofr:v:23:y:2005:i:5:p:1581-1583
    DOI: 10.1016/j.chaos.2004.06.011
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    Cited by:

    1. Balibrea, F. & Smítal, J. & Štefánková, M., 2014. "Dynamical systems generating large sets of probability distribution functions," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 38-42.
    2. Paganoni, L. & Smítal, J., 2008. "Strange distributionally chaotic triangular maps III," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 517-524.
    3. Liao, Gongfu & Chu, Zhenyan & Fan, Qinjie, 2009. "Relations between mixing and distributional chaos," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1994-2000.
    4. Paganoni, L. & Smítal, J., 2005. "Strange distributionally chaotic triangular maps," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 581-589.
    5. Shao, Hua & Shi, Yuming & Zhu, Hao, 2018. "On distributional chaos in non-autonomous discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 234-243.
    6. Paganoni, L. & Smítal, J., 2006. "Strange distributionally chaotic triangular maps II," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1356-1365.
    7. Downarowicz, T. & Štefánková, M., 2012. "Embedding Toeplitz systems in triangular maps: The last but one problem of the Sharkovsky classification program," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1566-1572.
    8. Kim, Cholsan & Ju, Hyonhui & Chen, Minghao & Raith, Peter, 2015. "A-coupled-expanding and distributional chaos," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 291-295.

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