Flatness Is a Criterion for Selection of Maximizing Measures

Abstract

For the one-dimensional classical spin system, each spin being able to get Np+1 values, and for a non-positive potential, locally proportional to the distance to one of N disjoint configurations set {(j−1)p+1,…,jp}ℤ, we prove that the equilibrium state converges as the temperature goes to 0.The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest.In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes.As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction.