The Aubry set for the XY model and typicality of periodic optimization for 2-locally constant potentials

We consider the Aubry set for the XY model, symbolic dynamics $([0,1{]}^{{\mathbb{N}}_0},\sigma )$ with the uncountable symbol $[0,1]$, and study its action-optimizing properties. Moreover, for a potential function that depends on the first two coordinates we obtain an explicit expression of the set of optimal periodic measures and a detailed description of the Aubry set. We also show the typicality of periodic optimization for 2-locally constant potentials with the twist condition. Our approach combines the weak KAM method for symbolic dynamics and variational techniques for twist maps.