Large deviations for equilibrium measures and selection of subaction

Abstract

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