On the stability of the penalty function for the $\mathbb{Z}^2$-hard square shift
We investigate the stability of maximizing measures for a penalty function of a two-dimensional subshift of finite type, building on the work of Gonschorowski et al. \cite{GQS}. In the one-dimensional case, such measures remain stable under Lipschitz perturbations for any subshift of finite type. However, instability arises for a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no periodic points and zero entropy. This raises the question of whether stability persists in two-dimensional subshifts of finite type with positive topological entropy. In this paper, we address this question by studying a nearest-neighbor subshift of finite type satisfying the single-site fillability property. Our main theorem establishes that, in contrast to previous results, a penalty function of such a subshift of finite type remains stable under Lipschitz perturbations.