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 fbof 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∞), fb) is transitive, (c) ((F(X), d0), fb) is transitive and (d) ((F(X), dS)), fb) is transitive, (2) if f : (X, d) → (X, d) is a continuous function, then the following hold: (a) if ((F(X), dS), fb) is transitive, then ((F(X), dE), fb) is transitive, (b) if ((F(X), dS), fb) 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.