Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

Letting (X,d) be a metric space, f:X→X a continuous map, and (ℱ(X),D) the space of nonempty fuzzy compact subsets of X with the Hausdorff metric, one may study the dynamical properties of the Zadeh's extension f̂:ℱ(X)→ℱ(X):u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations between f and f̂. More specifically, we study the transitivity, weakly mixing, periodic density in system (X,f), and its connections with the same ones in its fuzzified system.