A dynamical proof for the convergence of Gibbs measures at temperature zero

Abstract

We give a dynamical proof of a result due to J. Bremont in (4). It con- cerns the problem of maximizing measures for some given observable : for a subshift of finite type, and when only depends on a finite number of co- ordinates, it was proved in (4) that the unique Equilibrium State associated to converges to some measure when goes to +1. This measure has maximal entropy among the maximizing measures for . We give here a dy- namical proof of this result and we improve it. We prove that for any Holder continuous function (not necessarily locally constant), f, the unique Equilib- rium State associated to f + converges to some measure with maximal f-pressure among the maximizing measures. Moreover we also identify the limit measure.