Induced dynamics of non-autonomous dynamical systems

2022/02/27 by Shao, Hua
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.13345

Abstract

Let f_0,∞=\f_n\_n=0^∞ be a sequence of continuous self-maps on a compact metric space X. The non-autonomous dynamical system (X,f_0,∞) 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,f_0,∞) implies infinite entropy of (K(X),f_0,∞) and (F(X),f_0,∞), respectively; and zero entropy of (S¹,f_0,∞) 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 non-autonomous system (I, f_0,∞) 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-\mathscrF-sensitivity) among (X,f_0,∞), (K(X),f_0,∞) and (F¹(X),f_0,∞) are equivalent, where (F¹(X),f_0,∞) is the induced normal fuzzification.

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