Large Deviations for Equilibrium Measures and Selection of Subaction

Given a Lipschitz function , for each we denote by the equilibrium measure of and by the main eigenfunction of the Ruelle Operator . Assuming that satisfy a large deviation principle, we prove the existence of the uniform limit . Furthermore, the expression of the deviation function is determined by its values at the points of the union of the supports of maximizing measures. We study a class of potentials having two ergodic maximizing measures and prove that a L.D.P. is satisfied. The deviation function is explicitly exhibited and does not coincide with the one that appears in the paper by Baraviera-Lopes-Thieullen which considers the case of potentials having a unique maximizing measure.