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 $${\mathbb {Z}}^d$$ Z d -action. The $${\mathbb {Z}}^d$$ Z d -action on a space X has been defined in a very general manner, and therefore we introduce a $${\mathbb {Z}}^d$$ Z d -action on X which is induced by a continuous map, $$f:{\mathbb {Z}}\times X \rightarrow X$$ f : Z × X → X and denotes it as $$T_f:{\mathbb {Z}}^d \times X \rightarrow X$$ T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time $$(X,T_f)$$ ( 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 $$(X,T_f)$$ ( X , T f ) under which the chaotic behavior of $$(X,T_f)$$ ( 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.

Measurable Version of Spectral Decomposition Theorem for a Z2-Action

“…A Z d -action on a topological space is known as a multidimensional discrete dynamical system, which has been described by the two properties of its mapping. There is also abundant research about Z d -action in many different kinds of topics (See [28,[31][32][33][34][35]). One of the most interesting results in the study of Z d -action is about the spectral decomposition theorem.…”