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

Consider $T(x)=dx$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to $\mathbb{R}$$,$0<\lambda<1$andtheuniquefunction$b_\lambda:S^1 \to $\mathbb{R}$$satisfying$ b_\lambda(x) = \max_{T(y)=x} { \lambda , b_\lambda(y) + A(y)}.$Wewillshowthat,when$\lambda \to 1$,thefunction$b_\lambda- \frac{m(A)}{1-\lambda}$convergesuniformlytothecalibratedsubaction$V(x) = \max_{\mu \in $\mathcal{M}$} \int S(y,x) , d \mu(y)$,where$S$ is the Ma~ne potential, $\mathcal{M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)={\sup }_{\mu \in \mathcal{M}}\int Ad\mu$. For $\beta >0$ and $\lambda \in (0,1)$, there exists a unique fixed point $u_{\lambda ,\beta }:S^1\to \mathbb{R}$ for the equation $e^{u_{\lambda,\beta}(x)} = \sum_{T(y)=x}e^{\beta A(y) +\lambda u_{\lambda,\beta}(y)}$. It is known that as $\lambda \to 1$ the family $e^{[u_{\lambda,\beta}- \sup u_{\lambda,\beta}]}$convergesuniformlytothemaineigenfuntion$\phi_\beta $fortheRuelleoperatorassociatedto$\beta A$. We consider $\lambda =\lambda (\beta )$, $\beta (1-\lambda (\beta ))\to +\mathrm{\infty }$ and $\lambda (\beta )\to 1$, as $\beta \to \mathrm{\infty }$. Under these hypothesis we will show that $\frac{1}{\beta }(u_{\lambda ,\beta }-\frac{P(\beta A)}{1-\lambda })$ converges uniformly to the above $V$, as $\beta \to \mathrm{\infty }$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \mathrm{\infty }$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \mathrm{\infty }$.