Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials

For the subshift of finite type Σ = {0, 1, 2}N we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non-locally constant Hölder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two different fixed points and the potential is flatter at one of these two fixed points. We prove that there always is convergence but not necessarily to the Dirac measure at the point where the potential is the flattest. This is contrary to what was expected in the light of the analogous problem in Aubry-Mather theory [1]. This is also contrary to the finite range case where the equilibrium state converges to the equi-barycentre of the two Dirac measures. Moreover we emphasize the strange behavior of the Gibbs measure: the eigenmeasure selects one Dirac measure ( at the point where the potential is the flattest) and the eigenfunction selects the other one (at the point where the potential is the sharpest).