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.

The dynamics of a multidimensional dynamical system may sometimes be inherited from the dynamics of its classical dynamical system. In a multidimensional case, we introduce a new map called a d 1, 2,..., 2 d k ∈ whenever either the shift map is topologically transitive or satisfies the sufficient conditions. This study helps to develop the study of k-chaotic behaviours of d -action on the multidimensional dynamical system, contributions, and its application towards symbolic dynamics.

It is well known that locally everywhere onto, totally transitive, and topologically mixing are equivalent on shift of finite type. It turns out that this relation does not hold true on shift of infinite type. We introduce the increasing gap shift and determine its chaotic properties. The increasing gap shift and the sigma star shift serve as counterexamples to show the relation between the three chaos notions on shift of infinite type.

In this article, we present a measurable version of the spectral decomposition theorem for a Z2-action on a compact metric space. In the process, we obtain some relationships for a Z2-action with shadowing property and k-type weak extending property. Then, we introduce a definition of measure expanding for a Z2-action by using some properties of a Borel measure. We also prove one property that occurs whenever a Z2-action is invariantly measure expanding. All of the supporting results are necessary to prove the spectral decomposition theorem, which is the main result of this paper. More precisely, we prove that if a Z2-action is invariantly measure expanding, has shadowing property and has k-type weak extending property, then it has spectral decomposition.