Induced Dynamics of Non-Autonomous Discrete Dynamical Systems

Abstract: In this paper, we investigate the dynamics on the hyperspace induced by a non-autonomous dynamical system $(X,\mathbb{F})$, where the non-autonomous system is generated by a sequence $(f_n)$ of continuous self maps on $X$. We relate the dynamical behavior of the induced system on the hyperspace with the dynamical behavior of the original system $(X,\mathbb{F})$. We derive conditions under which the dynamical behavior of the non-autonomous system extends to its induced counterpart(and vice-versa). In the process, we discuss properties like transitivity, weak mixing, topological mixing, topological entropy and various forms of sensitivities. We also discuss properties like equicontinuity, dense periodicity and Li-Yorke chaoticity for the two systems. We also give examples when a dynamical notion of a system cannot be extended to its induced counterpart (and vice-versa).