## On fuzzifications of discrete dynamical systems.(English)Zbl 1229.93107

Summary: Let $$X$$ denote a locally compact metric space and $$\varphi:X \rightarrow X$$ be a continuous map. In the 1970s, Zadeh presented an extension principle helping us to fuzzify the dynamical system $$(X,\varphi)$$, i.e., to obtain a map $$\Phi$$ for the space of fuzzy sets on $$X$$. We extend an idea mentioned in [P. Diamond and A. Pokrovskii, Fuzzy Sets Syst. 61, No. 3, 277–283 (1994; Zbl 0827.58037)] to generalize Zadeh’s original extension principle.
In this paper, we study basic properties of so-called $$g$$-fuzzifications, such as their continuity properties. We also show that, for any $$g$$-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on $$X$$ induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems.
Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, the endograph and levelwise topologies.

### MSC:

 93C42 Fuzzy control/observation systems 93C25 Control/observation systems in abstract spaces

Zbl 0827.58037
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