# Transitivity in Fuzzy Hyperspaces

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

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

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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

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Author to whom correspondence should be addressed.

Received: 18 September 2020 / Revised: 5 October 2020 / Accepted: 9 October 2020 / Published: 24 October 2020

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures)

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 $f:(X,d)\to (X,d)$ and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension $\widehat{f}$ of f to $\mathcal{F}\left(X\right)$ , the family of all normal fuzzy sets on X, i.e., the hyperspace $\mathcal{F}\left(X\right)$ of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow $\mathcal{F}\left(X\right)$ with different metrics: the supremum metric ${d}_{\infty}$ , 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) $((\mathcal{F}\left(X\right),{d}_{\infty}),\widehat{f})$ is transitive, (c) $((\mathcal{F}\left(X\right),{d}_{0}),\widehat{f})$ is transitive and (d) $\left((\mathcal{F}\left(X\right),{d}_{S})\right),\widehat{f})$ is transitive, (2) if $f:(X,d)\to (X,d)$ is a continuous function, then the following hold: (a) if $((\mathcal{F}\left(X\right),{d}_{S}),\widehat{f})$ is transitive, then $((\mathcal{F}\left(X\right),{d}_{E}),\widehat{f})$ is transitive, (b) if $((\mathcal{F}\left(X\right),{d}_{S}),\widehat{f})$ is transitive, then $(X,f)$ is transitive; and (3) if $(X,d)$ be a complete metric space, then the following conditions are equivalent: (a) $(X\times X,f\times f)$ is point-transitive and (b) $((\mathcal{F}\left(X\right),{d}_{0})$ is point-transitive.