Selection of measure and a Large Deviation Principle for the general XY model

We consider $(M,d)$ a connected and compact manifold and we denote by $X$ the Bernoulli space $M^{\mathbb{N}}$. The shift acting on $X$ is denoted by $\sigma$. We analyze the general XY model, as presented in a recent paper by A. T. Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza. Denote the Gibbs measure by $\mu_{c}:=h_{c}\nu_{c}$, where $h_{c}$ is the eigenfunction, and, $\nu_{c}$ is the eigenmeasure of the Ruelle operator associated to $cf$. We are going to prove that any measure selected by $\mu_{c}$, as $c\to +\infty$, is a maximizing measure for $f$. We also show, when the maximizing probability measure is unique, that it is true a Large Deviation Principle, with the deviation function $R_{+}^{\infty}=\sum_{j=0}^\infty R_{+} (\sigma^f)$, where $R_{+}:= \beta(f) + V\circ\sigma - V - f$, and, $V$ is any calibrated subaction.