The entropy structures of axial products on ${\mathbb{N}}^d$ and Trees

In this paper, we first concentrate on the possible values and dense property of entropies for isotropic and anisotropic axial products of subshifts of finite type (SFTs) on ${\mathbb{N}}^d$ and $d$-tree ${\mathcal{T}}_d$. We prove that the entropies of isotropic and anisotropic axial products of SFTs on ${\mathbb{N}}^d$ are dense in $[0,\mathrm{\infty })$, and the same result also holds for anisotropic axial products of SFTs on ${\mathcal{T}}_d$. However, the result is no longer true for isotropic axial products of SFTs on ${\mathcal{T}}_d$. Next, motivated by the work of Johnson, Kass and Madden [16], and Schraudner [28], we establish the entropy formula and structures for full axial extension shifts on ${\mathbb{N}}^d$ and ${\mathcal{T}}_d$. Combining the aforementioned results with the findings on the surface entropy for multiplicative integer systems [8] on ${\mathbb{N}}^d$ enables us to estimate the surface entropy for the full axial extension shifts on ${\mathcal{T}}_d$. Finally, we extend the results of full axial extension shifts on ${\mathcal{T}}_d$ to general trees.