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Transitivity in Fuzzy Hyperspaces

Author

Listed:
  • Daniel Jardón

    (Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Calz. Ermita Iztapalapa S/N, Col. Lomas de Zaragoza 09620, México D.F., Mexico)

  • Iván Sánchez

    (Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, Mexico City C.P. 09340, Mexico)

  • Manuel Sanchis

    (Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Av. Vicent Sos Baynat s/n, C.P. 12071 Castelló de la Plana, Spain)

Abstract

Given a metric space ( X , d ) , we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f : ( X , d ) → ( X , d ) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension f ^ of f to F ( X ) , the family of all normal fuzzy sets on X , i.e., the hyperspace F ( X ) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F ( X ) with different metrics: the supremum metric d ∞ , the Skorokhod metric d 0 , the sendograph metric d S and the endograph metric d E . Among other things, the following results are presented: (1) If ( X , d ) is a metric space, then the following conditions are equivalent: (a) ( X , f ) is weakly mixing, (b) ( ( F ( X ) , d ∞ ) , f ^ ) is transitive, (c) ( ( F ( X ) , d 0 ) , f ^ ) is transitive and (d) ( ( F ( X ) , d S ) ) , f ^ ) is transitive, (2) if f : ( X , d ) → ( X , d ) is a continuous function, then the following hold: (a) if ( ( F ( X ) , d S ) , f ^ ) is transitive, then ( ( F ( X ) , d E ) , f ^ ) is transitive, (b) if ( ( F ( X ) , d S ) , f ^ ) is transitive, then ( X , f ) is transitive; and (3) if ( X , d ) be a complete metric space, then the following conditions are equivalent: (a) ( X × X , f × f ) is point-transitive and (b) ( ( F ( X ) , d 0 ) is point-transitive.

Suggested Citation

  • Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1862-:d:434026
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    References listed on IDEAS

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    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. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    3. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
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    Cited by:

    1. Jiří Kupka & Nicole Škorupová, 2021. "On PSO-Based Simulations of Fuzzy Dynamical Systems Induced by One-Dimensional Ones," Mathematics, MDPI, vol. 9(21), pages 1-26, October.
    2. 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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