Large deviations for equilibrium measures and selection of subaction

TL;DR

This paper investigates large deviations for equilibrium measures associated with Lipschitz functions, establishing the existence of a limit for scaled eigenfunctions, and explicitly characterizing the deviation function, especially for potentials with multiple maximizing measures.

Contribution

It proves the existence of a uniform limit for scaled eigenfunctions and explicitly characterizes the deviation function in cases with multiple maximizing measures.

Findings

Existence of the limit $V= rac{1}{eta}\log(h_{eta})$ as $eta o \infty$

Explicit form of the deviation function for potentials with two ergodic maximizing measures

Verification that a large deviation principle holds in the studied class

Abstract

Given a Lipschitz function $f:\{1,...,d\}^\mathbb{N} \to \mathbb{R}$, 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.