Transitivity in Fuzzy Hyperspaces

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 $f:(X,d)\to (X,d)$ 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.