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Topological entropy and chaos for maps induced on hyperspaces

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  • Kwietniak, Dominik
  • Oprocha, Piotr

Abstract

If f is a continuous selfmap of a compact metric space X then by the induced map we mean the map f¯ defined on the space of all nonempty closed subsets of X by f¯(K)=f(K).

Suggested Citation

  • Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:1:p:76-86
    DOI: 10.1016/j.chaos.2005.12.033
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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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Zhu, Wanting & Sun, Kehui & He, Shaobo & Wang, Huihai & Liu, Wenhao, 2023. "A class of m-dimension grid multi-cavity hyperchaotic maps and its application," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    2. Fedeli, Alessandro, 2009. "On two notions of topological entropy for noncompact spaces," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 432-435.
    3. Andres, Jan, 2020. "Chaos for multivalued maps and induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    4. Li, Risong, 2012. "A note on stronger forms of sensitivity for dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 753-758.
    5. 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.
    6. Daghar, Aymen & Naghmouchi, Issam, 2022. "Entropy of induced maps of regular curves homeomorphisms," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    7. Román-Flores, H. & Chalco-Cano, Y. & Silva, G.N. & Kupka, Jiří, 2011. "On turbulent, erratic and other dynamical properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 990-994.
    8. Wang, Yangeng & Wei, Guo & Campbell, William H. & Bourquin, Steven, 2009. "A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1708-1717.
    9. Andres, Jan & Ludvík, Pavel, 2022. "Topological entropy of multivalued maps in topological spaces and hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    10. Liu, Lei & Wang, Yangeng & Wei, Guo, 2009. "Topological entropy of continuous functions on topological spaces," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 417-427.
    11. 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.
    12. Hou, Bingzhe & Liao, Gongfu & Liu, Heng, 2008. "Sensitivity for set-valued maps induced by M-systems," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1075-1080.

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