Large deviations for equilibrium measures and selection of subaction

2016/08/21 by Mengue, Jairo K.
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1608.05881

Abstract

Given a Lipschitz function f:\1,...,d\^ℕ → 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→∞\frac1βlog(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.

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