Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

Abstract

Letting $(X,d)$ be a metric space, $f:X\to X$ 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 $\widehat{f}:ℱ(X)\to ℱ(X):u\mapsto \widehat{f}u$. 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 $\widehat{f}$. 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.