Transitivity in Fuzzy Hyperspaces

Daniel Jardón, Iván Sánchez and Manuel Sanchis
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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

Mathematics, 2020, vol. 8, issue 11, 1-9

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.

Keywords: fuzzy set; skorokhod metric; endograph metric; sendograph metric; Zadeh’s extension; Transitivity; weakly mixing; point transitivity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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