On Sufficient Conditions for Chaotic Behavior of Multidimensional Discrete Time Dynamical System

Abstract

In this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on {mathbb {Z}}^d-action. The {mathbb {Z}}^d-action on a space X has been defined in a very general manner, and therefore we introduce a {mathbb {Z}}^d-action on X which is induced by a continuous map, f:{mathbb {Z}}times X rightarrow X and denotes it as T_f:{mathbb {Z}}^d times X rightarrow X. Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time (X,T_f). The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on (X,T_f) under which the chaotic behavior of (X,T_f) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions.