Surface entropies of \(\mathbb{Z}^2\) subshifts of finite type

Subshifts of finite type (SFTs) are sets of colorings of the plane that avoid a finite family of forbidden patterns. In this article, we are interested in the behavior of the growth of the number of valid patterns in SFTs. While entropy \(h\) corresponds to growths that are squared exponential \(2^{hn²}\), surface entropy (introduced in Pace’s thesis in 2018) corresponds to the eventual linear term in exponential growths. We give here a characterization of the possible surface entropies of SFTs as the \(\Pi_3^0\) real numbers of \([0,+∞]\).