ncts

In this talk, 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 $\mathcal{T}_d$ . We prove that the entropies of isotropic and anisotropic axial products of SFTs on $\mathbb{N}^d$ are dense in $[0,\infty)$ , and the same result is also valid 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 Johnson, Kass and Madden, and Schraudner, we establish the entropy formula and structures for full axial extension shifts on $\mathbb{N}^d$ and $\mathcal{T}_d$ . Combining above mentioned results with the findings on the surface entropy for multiplicative integer systems on $\mathbb{N}^d$ enable 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.