Selection of calibrated subaction when temperature goes to zero in the discounted problem

Consider \begin{document}$T(x) = d \, x$\end{document} (mod 1) acting on \begin{document}$S^1$\end{document} , a Lipschitz potential \begin{document}$A:S^1 \to \mathbb{R}$\end{document} , \begin{document}zhongwenzy and the unique function \begin{document}$b_\lambda:S^1 \to \mathbb{R}$\end{document} satisfying \begin{document}$ b_\lambda(x) = \max_{T(y) = x} \{ \lambda \, b_\lambda(y) + A(y)\}. $\end{document} We will show that, when \begin{document}$\lambda \to 1$\end{document} , the function \begin{document}$b_\lambda- \frac{m(A)}{1-\lambda}$\end{document} converges uniformly to the calibrated subaction \begin{document}$V(x) = \max_{\mu \in \mathcal{ M}} \int S(y, x) \, d \mu(y)$\end{document} , where \begin{document}$S$\end{document} is the Mane potential, \begin{document}$\mathcal{ M}$\end{document} is the set of invariant probabilities with support on the Aubry set and \begin{document}$m(A) = \sup_{\mu \in \mathcal{M}} \int A\, d\mu$\end{document} . For \begin{document}$\beta>0$\end{document} and \begin{document}$\lambda \in (0, 1)$\end{document} , there exists a unique fixed point \begin{document}$u_{\lambda, \beta} :S^1\to \mathbb{R}$\end{document} for the equation \begin{document}$e^{u_{\lambda, \beta}(x)} = \sum_{T(y) = x}e^{\beta A(y) +\lambda u_{\lambda, \beta}(y)}$\end{document} . It is known that as \begin{document}$\lambda \to 1$\end{document} the family \begin{document}$e^{[u_{\lambda, \beta}- \sup u_{\lambda, \beta}]}$\end{document} converges uniformly to the main eigenfuntion \begin{document}$\phi_\beta $\end{document} for the Ruelle operator associated to \begin{document}$\beta A$\end{document} . We consider \begin{document}$\lambda = \lambda(\beta)$\end{document} , \begin{document}$\beta(1-\lambda(\beta))\to+\infty$\end{document} and \begin{document}$\lambda(\beta) \to 1$\end{document} , as \begin{document}$\beta \to\infty$\end{document} . Under these hypotheses we will show that \begin{document}$\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$\end{document} converges uniformly to the above \begin{document}$V$\end{document} , as \begin{document}$\beta\to \infty$\end{document} . The parameter \begin{document}$\beta$\end{document} represents the inverse of temperature in Statistical Mechanics and \begin{document}$\beta \to \infty$\end{document} means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when \begin{document}$\beta \to \infty$\end{document} .