Zadeh's extension; fuzzy dynamical system; inverse limit space; topology

Given a compact and connected metric space (continuum) $X$, we study topological and dynamical properties of the hyperspace of normal fuzzy sets $\mathbb{F}^1(X)$ equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph metric the hyperspace $\mathbb{F}^1(X)$ is a continuum, and then for a topological graph $X$ we show how, using the inverse limit approach of Barge and Martin, the inverse limit of a fuzzy dynamical system on $X$ can be realized as an attractor of a fuzzy dynamical system on a manifold.