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)→(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 d0, the sendograph metric dS and the endograph metric dE. 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),d0),f^) is transitive and (d) ((F(X),dS)),f^) is transitive, (2) if f:(X,d)→(X,d) is a continuous function, then the following hold: (a) if ((F(X),dS),f^) is transitive, then ((F(X),dE),f^) is transitive, (b) if ((F(X),dS),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),d0) is point-transitive.

Chaos on Fuzzy Dynamical Systems

“…The main results in this section are the equivalence between the Devaney chaos of f in K(X) and of f in F (X) and, as a consequence, the equivalence of Devaney chaos for a continuous linear operator T on a metrizable and complete locally convex space X, for its Zadeh extension T defined on the space of normal fuzzy sets F (X) and for the induced hyperspace map T on K(X). This extends previous results of D. Jardón, I. Sánchez, and M. Sanchis about the transitivity in fuzzy metric spaces [4] (see also [20]) and another result of N. Bernardes, A. Peris, and F. Rodenas [2] about the linear Devaney chaos of locally convex spaces.…”

“…We recall that Banks [21], Liao, Wang, and Zhang [22], and Peris [23] independently characterized the topological transitivity for (K(X), f ) in terms of the weak mixing property for (X, f ). Concerning the space of fuzzy sets, in [4], the authors showed (Theorem 3) the equivalences of the weak mixing property of f on X with the transitivity of f on F ∞ (X) or on F 0 (X). They also considered the fuzzy space F (X) endowed with the sendograph metric and the endograph metric.…”

“…The map f : F (X) → F (X) is called the Zadeh extension (or fuzzification) of f . In this context, Jardón et al studied in [4] the relationship between some dynamical properties (mainly transitivity) of the systems (X, f ) and (F (X), f ). In this same context, we consider in this note several notions of chaos, such as the ones given by Devaney [5] and Li and Yorke [6].…”

“…Some dynamical properties of f on the metric spaces F ∞ (X) and F 0 (X) were studied by Jardón et al in [4] in connection with the dynamics of f on the space X, and it is our aim to extend this study to some notions of chaos.…”

“…In the next section, we use the following properties of fuzzy sets on the spaces F ∞ (X) and F 0 (X) (see [4,18,19] for the details). Proposition 1.…”