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Inducing sensitivity of fuzzified dynamical systems. (English) Zbl 1401.37021

Summary: Let \((X,f)\) be a dynamical system, \((\mathcal K(X),\overline f)\) be the system on hyperspace and \((\mathbb F(X),\widetilde{f}_g)\) be its \(g\)-fuzzification. We study the relations among the various forms of sensitivity of the systems \((X,f)\), \((\mathcal K(X),\overline{f})\) and \((\mathbb F^1(X),\widetilde{f}_g)\). Firstly, we give a simpler proof of the theorem in [X. Wu and G. Chen, “Sensitivity and transitivity of fuzzified dynamical systems”, Inf. Sci. 396, 14–23 (2017; doi:10.1016/j.ins.2017.02.042)] which states that \((\mathcal K(X),\overline{f})\) is sensitive if and only if \((\mathbb F^1(X),\widetilde{f}_g)\) is sensitive. Then, we show that cm
(1)
all forms of sensitivity of \((\mathcal K(X),\overline{f})\) are the same for \((\mathbb F(X),\widetilde{f}_g)\),
(2)
all forms of sensitivity of \((\mathcal K(X),\overline{f})\) partly imply the same for \((X,f)\), and the converse holds in some cases.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
03E72 Theory of fuzzy sets, etc.
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