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Set-valued discrete chaos

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  • Peris, Alfredo

Abstract

Given a continuous map f:X→X on a metric space (X,d), we characterize topological transitivity for the (set-valued) map f¯:K(X)→K(X) induced by f on the space K(X) of compact subsets of X, endowed with the Hausdorff distance. More precisely, f¯ is transitive if and only if f is weakly mixing. Some consequences are also derived for the dynamics on fractals and for (continuous and) linear maps on infinite-dimensional spaces.

Suggested Citation

  • Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    • Handle: RePEc:eee:chsofr:v:26:y:2005:i:1:p:19-23
    DOI: 10.1016/j.chaos.2004.12.039
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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.
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    Cited by:

    1. Sánchez, Iván & Sanchis, Manuel & Villanueva, Hugo, 2017. "Chaos in hyperspaces of nonautonomous discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 94(C), pages 68-74.
    2. Cánovas Peña, Jose S. & López, Gabriel Soler, 2006. "Topological entropy for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 979-982.
    3. Andres, Jan, 2020. "Chaos for multivalued maps and induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    4. Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.
    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. Román-Flores, H. & Chalco-Cano, Y., 2008. "Some chaotic properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 452-459.
    7. 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.
    8. Félix Martínez-Giménez & Alfred Peris & Francisco Rodenas, 2021. "Chaos on Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 9(20), pages 1-11, October.
    9. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    10. 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.
    11. 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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