Boundary complexity and surface entropy of 2-multiplicative integer systems on $\mathbb{N}^d$

In this article, we introduce the concept of the boundary complexity and prove that for a 2-multiplicative integer system (2-MIS) $X^{p}_{\Omega}$ on $\mathbb{N}$ (or $X^{\bf p}_{\Omega}$ on $\mathbb{N}^d,d\geq 2$), every point in $[h(X^p_\Omega), \log r]$ can be realized as a boundary complexity of a 2-MIS with a specific speed, where r stands for the number of the alphabets. The result is new and quite different from $\mathbb{N}^d$ subshifts of finite type (SFT) for $d\geq 1$. Furthermore, the rigorous formula of surface entropy for a $\mathbb{N}^d$ 2-MIS is also presented. This provides an efficient method to calculate the topological entropy for $\mathbb{N}^d$ 2-MIS and also provides an intrinsic differences between $\mathbb{N}^d$ $k$-MIS and SFTs for $d\geq 1$ and $k\geq 2$.