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The set-valued mapping induced by a non-minimal transitive system is Li–Yorke chaotic

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  • Liu, Heng
  • Liao, Gongfu
  • Hou, Bingzhe

Abstract

Let X be a metric space, (X,f) a discrete dynamical system, where f: X→X is a continuous function. Let f¯ denote the natural extension of f to the space of all non-empty compact subsets of X endowed with a Hausdorff metric. In this paper, we prove that if f is transitive and non-minimal, then f¯ is Li–Yorke’s chaos. Furthermore, if f is non-minimal M-system, then f¯ has a s-scrambled set.

Suggested Citation

  • Liu, Heng & Liao, Gongfu & Hou, Bingzhe, 2009. "The set-valued mapping induced by a non-minimal transitive system is Li–Yorke chaotic," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 826-830.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:2:p:826-830
    DOI: 10.1016/j.chaos.2007.08.030
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    References listed on IDEAS

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    1. Fedeli, Alessandro, 2005. "On chaotic set-valued discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1381-1384.
    2. Román-Flores, Heriberto & Chalco-Cano, Y., 2005. "Robinson’s chaos in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 33-42.
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    4. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    5. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
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