Entropy of Axial Products on Nd 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 Nd and d-tree Td. We prove that the entropies of isotropic and anisotropic axial products of SFTs on Nd are dense in [0,∞), and the same result also holds for anisotropic axial products of SFTs on Td. However, the result is no longer true for isotropic axial products of SFTs on Td. 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 Nd and Td. 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 Nd enables us to estimate the surface entropy for the full axial extension shifts on Td. Finally, we extend the results of full axial extension shifts on Td to general trees.