Large deviations for equilibrium measures and selection of subaction

Given a Lipschitz function $f:\{1,\mathrm{.}\mathrm{.}\mathrm{.},d{\}}^{\mathbb{N}}\to \mathbb{R}$, for each $\beta >0$ we denote by ${\mu }_{\beta }$ the equilibrium measure of $\beta f$ and by $h_{\beta }$ the main eigenfunction of the Ruelle Operator $L_{\beta f}$. Assuming that $\{{\mu }_{\beta }{\}}_{\beta >0}$ satisfy a large deviation principle, we prove the existence of the uniform limit $V= \lim_{\beta\to\infty}\frac{1}{\beta}\log(h_{\beta})$. 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.