Justifications of spatial entropies of multi-dimensional symbolic dynamical systems

TL;DR

This paper investigates the justification of spatial entropies in multi-dimensional symbolic dynamical systems, showing conditions under which the commonly used spatial entropy accurately reflects the system's complexity.

Contribution

It establishes when the traditional spatial entropy equals the entropy on general expanding systems, clarifying the role of genuinely d-dimensional systems.

Findings

$h_{r}(U)$ is the supremum of $h_{Omega}(U)$ over genuinely two-dimensional systems.

When $Omega$ is genuinely d-dimensional and meets certain conditions, $h_{Omega}(U)=h_{r}(U)$.

If $Omega(n)$ contains lower-dimensional parts, then $h_{r}(U)<h_{Omega}(U)$ for some $U$.

Abstract

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 $h_{\Omega}(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding system $\Omega=\{\Omega(n)\}_{n=1}^{\infty}$ where $\Omega(n)$ is increasing finite sublattices and expands to $\mathbb{Z}^{d}$. $\Omega$ is called genuinely $d$-dimensional if $\Omega(n)$ 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 $h_{\Omega}(\mathcal{U})$ for all genuinely two-dimensional $\Omega$. Furthermore, when $\Omega$ is genuinely $d$-dimensional and satisfies certain conditions, then $h_{\Omega}(\mathcal{U})=h_{r}(\mathcal{U})$. On the contrary, when $\Omega(n)$ contains a lower-dimensional part, then $h_{r}(\mathcal{U})<h_{\Omega}(\mathcal{U})$ for some $\mathcal{U}$. Therefore, $h_{r}(\mathcal{U})$ is appropriate to be the $d$-dimensional spatial entropy.