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) $X_{\mathrm{\Omega }}^p$ on $\mathbb{N}$ (or $X_{\mathrm{\Omega }}^{\mathbf{p}}$ on ${\mathbb{N}}^d,d\ge 2$), every point in $[h(X_{\mathrm{\Omega }}^p),\log r]$ 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 $d\ge 1$. 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 $d\ge 1$ and $k\ge 2$.