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Mixing properties of set-valued maps on hyperspaces via Furstenberg families

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  • Fu, Heman
  • Xing, Zhitao

Abstract

Let X be a metric space and f be a continuous self-map of X. On K(X), the set of all nonempty compact subsets of X, f induces a continuous map f¯ naturally by letting f¯(K)=f(K) for K∈K(X). In this paper Furstenberg families are heavily used to investigate the relationships between mixing properties of f and those of f¯. Consequently, several general conclusions are developed, which extent the results of Banks [Chaos for induced hyperspace maps, Chaos Solitons & Fractals 2005; 25(3):681–685.], Kwietniak and Oprocha [Topological entropy and chaos for induced hyperspace maps, Chaos Solitons & Fractals 2007; 33:76–86.].

Suggested Citation

  • Fu, Heman & Xing, Zhitao, 2012. "Mixing properties of set-valued maps on hyperspaces via Furstenberg families," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 439-443.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:4:p:439-443
    DOI: 10.1016/j.chaos.2012.01.003
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    References listed on IDEAS

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    1. Zhang, Gengrong & Zeng, Fanping & Liu, Xinhe, 2006. "Devaney’s chaotic on induced maps of hyperspace," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 471-475.
    2. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    3. Fedeli, Alessandro, 2005. "On chaotic set-valued discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1381-1384.
    4. 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.
    5. Gu, Rongbao & Guo, Wenjing, 2006. "On mixing property in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 747-754.
    6. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
    7. 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.
    8. 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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    Cited by:

    1. Félix Martínez-Giménez & Alfred Peris & Francisco Rodenas, 2021. "Chaos on Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 9(20), pages 1-11, October.

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