Zadeh's extension, discrete dynamical system, topological entropy, mixing, chaos

Let $X$ be a compact metric space, let $\varphi$ be a continuous self-map on $X$, and let $\mathbb{F}(X)$ denote the space of fuzzy sets on $X$ equipped with the levelwise topology. In this paper we study relations between various dynamical properties of a given (crisp) dynamical system $(X,\varphi)$ and its Zadeh's extension $\Phi$ on $\mathbb{F} (X)$. Among other things we study various (weak, strong, mild etc.) mixing properties and also several kinds of chaotic behaviors (Li-Yorke chaos, $\omega$-chaos, distributional chaos, topological chaos etc.).