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A note on stronger forms of sensitivity for dynamical systems

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  • Li, Risong

Abstract

Let (X,d) be a compact metric space and (κ(X),dH) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric dH. The dynamical system (X,f) induces another dynamical system (κ(X),f¯), where f:X→X is a continuous map and f¯:κ(X)→κ(X) is defined by f¯(A)={f(a):a∈A} for any A∈κ(X). In this paper, we introduce the notion of ergodic sensitivity which is a stronger form of sensitivity, and present some sufficient conditions for a dynamical system (X,f) to be ergodically sensitive. Also, it is shown that f¯ is syndetically sensitive (resp. multi-sensitive) if and only if f is syndetically sensitive (resp. multi-sensitive). As applications of our results, several examples are given. In particular, it is shown that if a continuous map of a compact metric space is chaotic in the sense of Devaney, then it is ergodically sensitive. Our results improve and extend some existing ones.

Suggested Citation

  • Li, Risong, 2012. "A note on stronger forms of sensitivity for dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 753-758.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:6:p:753-758
    DOI: 10.1016/j.chaos.2012.02.003
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    References listed on IDEAS

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    1. Wu, Chen & Xu, Zhengjie & Lin, Wei & Ruan, Jiong, 2005. "Stochastic properties in Devaney’s chaos," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1195-1199.
    2. Lardjane, Salim, 2006. "On some stochastic properties in Devaney’s chaos," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 668-672.
    3. 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.
    4. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
    5. Gu, Rongbao, 2007. "The large deviations theorem and ergodicity," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1387-1392.
    6. Liao, Gongfu & Ma, Xianfeng & Wang, Lidong, 2007. "Individual chaos implies collective chaos for weakly mixing discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 604-608.
    7. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
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