Induced dynamics of nonautonomous dynamical systems

Abstract

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