Explicit formulae and topological descriptions of action-minimizing sets for 2-locally potentials of the XY model

Abstract: We consider ergodic optimization of a symbolic dynamics with uncountable alphabet $[0,1]$ for 2-locally constant potentials with the twist condition, and give explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a remarkable difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete characterization of an analogical object of the Aubry set, called the Ma~{n}\'e set, and show that it is much larger than the Aubry set so that it contains cubes of any finite dimension.