Large Deviations for Equilibrium Measures and Selection of Subaction

  • Jairo K. Mengue


Given a Lipschitz function \(f:\{1,\ldots ,d\}^\mathbb {N} \rightarrow \mathbb {R}\), 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 \(V= \lim _{\beta \rightarrow +\infty }\frac{1}{\beta }\log (h_{\beta })\). 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.


Equilibrium measure Maximizing measure Large deviation principle 


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© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.Universidade Federal do Rio Grande do SulPorto AlegreBrazil

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