Entropy 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 $$[0,\infty )$$ , 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 et al. (Complex Syst 17(3):243, 2007), and Schraudner (Discrete Contin Dyn Syst 26(1):333, 2010), we establish the formulae and structures for entropies of 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 (Ban et al. in J Math Phys 64:16, 2023) 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.