Summary: We consider (\(M, d\)) a connected and compact manifold and we denote by \(\mathcal B_i\) the Bernoulli space \(M^{\mathbb Z}\). The analogous problem on the half-line \(\mathbb N\) is also considered. Let \(A:B_i\rightarrow \mathbb R\) be an observable. Given a temperature \(T\), we analyze the main properties of the Gibbs state \(\hat\mu_{\frac {1}{T}A}\).
In order to do our analysis, we consider the Ruelle operator associated to \(\frac {1}{T}A\), and we get in this procedure the main eigenfunction \(\psi_{\frac{1}{T}A}\). Later, we analyze selection problems when the temperature goes to zero: (a) existence, or not, of the limit \(V:=\lim_{T\rightarrow 0}T\log(\psi_{\frac {1}{T}A})\), a question about selection of subactions, and, (b) existence, or not, of the limit \(\tilde\mu:=\lim_{T\rightarrow 0 \hat\mu_{\frac {1}{T}A}}\), a question about selection of measures.
The existence of subactions and other properties of ergodic optimization are also considered.
The case where the potential depends just on the coordinates \((x_{0}, x_{1})\) is carefully analyzed. We show, in this case, and under suitable hypotheses, a Large Deviation Principle, when \(T \rightarrow 0\), graph properties, etc. Finally, we present in detail a result due to A. C. D. van Enter and W. M. Ruszel [J. Stat. Phys. 127, No. 3, 567–573 (2007; Zbl 1147.82324)], where the authors show, for a particular example of potential \(A\), that the selection of measure \(\hat\mu_{\frac {1}{T}A}\) in this case, does not happen.