Let f(0,infinity) = {f(n)}(infinity)(n=0) be a sequence of continuous self-maps on a compact metric space X. The nonautonomous dynamical system (X, f(0,infinity)) induces the set-valued system (K(X), f(0,infinity))and the fuzzified system (F(X), f(0,infinity)). We prove that under system (K(X), some natural conditions, positive topological entropy of (X, f(0,infinity)) implies infinite entropy of (K(X), sic)(0,infinity)), respectively; and zero entropy of (S-1, f(0,infinity)) implies zero entropy of some invariant subsystems of (K(S-1), (sic)(0,infinity)) and (F(S-1,f(0,infinity),) respectivly. We confirm that (K(I), (sic)) and (F(I), (sic)) have infinite entropy for any transitive interval map f. In contrast, we construct a transitive nonautonomous system (I, f(0,infinity)) such that both (K(I), zero entropy. We obtain that (K(X), f(0,infinity)) and the fuzzified system (F(X), f(0,infinity)) and (F(X), f(0,infinity)) and (F(I), (sic)(0,infinity)) have f(0,infinity)) is chain weakly mixing of all orders if and only if (F-1(X), (sic)(0,infinity)) is so, and chain mixing (resp. h -shadowing and multi -F -sensitivity) among (X, f(0,infinity)), (K(X), (sic)(0,infinity)) and (F-1(X), (sic)(0,infinity)) are equivalent, where (F-1(X), (sic)(0,infinity)) is the induced normal fuzzification. (c) 2023 Elsevier B.V. All rights reserved.
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