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A probability $\hat{\nu}$ on $[0,1]\times \Sigma$ is called holonomic for $\hat{\sigma}$, if, $ \int\, g \circ \hat{\sigma}\, d\hat{\nu}= \int \,g \,d\hat{\nu}, \, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by $\mathcal H$.

For a holonomic probability $\hat{\nu}$ on $[0,1]\times \Sigma$ we define the entropy $h(\hat{\nu})$=inf$_f \in \mathbb{B}^{+} \int \ln(\frac{P_{\psi}f}{\psi f}) d\hat{\nu}\geq 0$, where, $\psi \in \mathbb{B}^{+}$ is a fixed (any one) positive potential.

Finally, we analyze the problem: given $\phi \in \mathbb{B}^{+}$, find solutions of the maximization problem $p(\phi)$= sup$_\hat{\nu} \in \mathcal{H} \{ \,h(\hat{\nu}) + \int \ln(\phi) d\hat{\nu} \,\}.$ We show an example where a holonomic not-$\hat{\sigma}$-invariant probability attains the supremum value. In the last section we consider maximizing probabilities, sub-actions and duality for potentials $A(x,w)$, $(x,w)\in [0,1]\times \Sigma$, for IFS.

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these ''in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures).

Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space -we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.

Consider $T(x) = d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $zhongwenzy<\lambda<1$ 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 $m(A) = \sup_{\mu \in \mathcal{M}} \int A\, d\mu$.

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 $\phi_\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 hypotheses 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 $\beta \to \infty$.

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