We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^mathbb{N}to mathbb{R}$ which depends o
nly on the two first coordinates. We are interested in finding stationary Markov probabilities $mu_infty$ on $ [0,1]^mathbb{N}$ that maximize the value $ int A d mu,$ among all stationary Markov probabilities $mu$ on $[0,1]^mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $mu_beta$ which weakly converges to $mu_infty$. The probabilities $mu_beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $frac{partial^2 A}{partial_x partial_y} (x,y) eq 0$, for all $(x,y) in [0,1]^2$, we show the graph property.
We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $mu$ is unique and the support of $mu$ is the Aubry set). Consider, for each value of $epsilon $ and h,
the entropy penalized Mather problem $min {int_{tntimesrn} L(x,v)dmu(x,v)+epsilon S[mu]},$ where the entropy S is given by $S[mu]=int_{tntimesrn}mu(x,v)lnfrac{mu(x,v)}{int_{rn}mu(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $mu(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure $mu_{epsilon, h}$ which converges to the Mather measure $mu$. We show a LDP $lim_{epsilon,hto0} epsilon ln mu_{epsilon,h}(A),$ where $Asubset mathbb{T}^Ntimesmathbb{R}^N$. The deviation function I is given by $I(x,v)= L(x,v)+ ablaphi_0(x)(v)-bar{H}_{0},$ where $phi_0$ is the unique viscosity solution for L.
