Justifications of spatial entropies of multi-dimensional symbolic dynamical systems

The commonly used spatial entropy $h_{r}(\mathcal{U})$ of the multi-dimensional shift space $\mathcal{U}$ is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space $\mathbb{Z}^{d}$, $d\geq 2$. This work studies spatial entropy $h_Ω(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding system $Ω=\{Ω(n)\}_{n=1}^{\infty}$ where $Ω(n)$ is increasing finite sublattices and expands to $\mathbb{Z}^{d}$. $Ω$ is called genuinely $d$-dimensional if $Ω(n)$ contains no lower-dimensional part whose size is comparable to that of its $d$-dimensional part. We show that $h_{r}(\mathcal{U})$ is the supremum of $h_Ω(\mathcal{U})$ for all genuinely two-dimensional $Ω$. Furthermore, when $Ω$ is genuinely $d$-dimensional and satisfies certain conditions, then $h_Ω(\mathcal{U})=h_{r}(\mathcal{U})$. On the contrary, when $Ω(n)$ contains a lower-dimensional part, then $h_{r}(\mathcal{U})<h_Ω(\mathcal{U})$ for some $\mathcal{U}$. Therefore, $h_{r}(\mathcal{U})$ is appropriate to be the $d$-dimensional spatial entropy.

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