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.

在这项工作中，我们通过刻画\（{\ mathbb {Z}} ^ d \）作用上的混沌概念，研究了经典离散动力系统到多维离散时间动力系统的扩展。在空间X上的\（{\ mathbb {Z}} ^ d \） -动作已以非常一般的方式定义，因此我们在...上引入了\（{\ mathbb {Z}} ^ d \） -action由连续映射\（f：{\ mathbb {Z}} \ times X \ rightarrow X \）诱导的X，并表示为\（T_f：{\ mathbb {Z}} ^ d \ times X \ rightarrow X \）。基本上，我们希望将原始离散动力系统的行为（X，  f）及其诱导的多维离散时间\（（X，T_f）\）。我们强调的混沌行为是传递性和密集周期性。类似于这些混沌概念，我们考虑了多维离散时间动力系统中的k型传递性和k型密集周期性。在这个过程中，我们得到在某些条件\（（X，T_f）\）在其下的混沌行为\（（X，T_f）\）是从原始动力系统（继承X，  ˚F）。每当f打开，完全传递或混合时，条件都会变化。给出了一些例子来说明这些情况。