$k$-type entropy of $\mathbb{Z}^d$ actions

We introduce the concept of $k$-type entropy for dynamical systems generated by $\mathbb{Z}^d$-actions on compact metric spaces. We investigate its fundamental properties and establish connections with classical entropy and other $k$-type dynamical notions. The $k$-type entropy of some $\mathbb{Z}^2$-actions on a two dimensional torus is also calculated.