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

Abstract

Consider $T(x)=dx$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1\to \mathbb{R}$, $0<\lambda <1$ and the unique function $b_{\lambda }:S^1\to \mathbb{R}$ satisfying $b_{\lambda }(x)={\max }_{T(y)=x}\{\lambda b_{\lambda }(y)+A(y)\}.$ We will show that, when $\lambda \to 1$, the function $b_{\lambda }-\frac{m(A)}{1-\lambda }$ converges uniformly to the calibrated subaction $V(x)={\max }_{\mu \in \mathcal{M}}\int S(y,x)d\mu (y)$, where $S$ is the Mañe 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 $e^{[u_{\lambda ,\beta }-\sup u_{\lambda ,\beta }]}$ converges uniformly to the main eigenfuntion ${\varphi }_{\beta }$ for the Ruelle operator associated to $\beta A$. We consider $\lambda =\lambda (\beta )$, $\beta (1-\lambda (\beta ))\to +\infty$ and $\lambda (\beta )\to 1$, as $\beta \to \infty$. 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 $\beta \to \infty$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \infty$.