On fuzzifications of discrete dynamical systems

Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we generalize Zadeh’s original extension principle. In this paper we study basic properties, such as the continuity of so-called g-fuzzifications. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly convergent maps on X induces a uniformly convegent sequence of continuous maps on the space of fuzzy sets, and (ii) a conjugacy (a semi-conjugacy, resp.) between two discrete dynamical systems can be extended to a conjugacy (a semi-conjugacy, resp.) between fuzzified dynamical systems. Moreover, at the end of this paper we show that there are connections between g-fuzzifications and crisp dynamical systems via set-valued dynamical systems and skew-product (triangular) maps. Throughout this paper we consider different topological structures in the space of fuzzy sets; namely, the sendograph, endograph and levelwise topologies.