Induced dynamics of nonautonomous dynamical systems

Let f0,∞={fn}n=0∞ be a sequence of continuous self-maps on compact metric space X. The nonautonomous dynamical system (X,f0,∞) induces the set-valued (K(X),f¯0,∞) and fuzzified (F(X),f˜0,∞). We prove that under some natural conditions, positive topological entropy implies infinite (F(X),f˜0,∞), respectively; zero (S1,f0,∞) invariant subsystems (K(S1),f¯0,∞) (F(S1),f˜0,∞), respectively. confirm (K(I),f¯) (F(I),f˜) have for any transitive interval map f. In contrast, we construct (I,f0,∞) such both (K(I),f¯0,∞) (F(I),f˜0,∞) entropy. obtain is chain weakly mixing all orders if only (F1(X),f˜0,∞) so, (resp. h-shadowing multi-F-sensitivity) among (X,f0,∞), are equivalent, where induced normal fuzzification.