Given a Lipschitz function f : {1, . . . , d}N → R, for eachβ > 0 we denote by μβ the equilibrium measure of β f and by hβ the main eigenfunction of the Ruelle Operator Lβ f . Assuming that {μβ}β>0 satisfy a large deviation principle, we prove the existence of the uniform limit V = limβ→+∞ 1 β log(hβ). 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
