I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$.
Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator
$$
\mathcal{L}_{A}(\psi)(x)=\int_{\mathbb{S}^1} e^{A(ax)} \psi(ax)\ da
$$
where $x\in(\mathbb{S}^1)^{\mathbb{N}}$ and $ax=(a,x_1,x_2,\ldots)\in (\mathbb{S}^1)^{\mathbb{N}}$ and $da$ is the normalized Lebesgue measure on $\mathbb{S}^1$. Let $\mu$ the Gibbs measure constructed from $\mathcal{L}_A$ in the standard way.
Let $\Lambda_n=[0,\ldots,n]\cap\mathbb{Z}$ and $\mathscr{T}_{\Lambda_n}$ the external $\sigma$-algebra as defined in the Georgii's book.
Question. Is there any Gibbs specification
$\gamma=(\gamma_{\Lambda_n})_{n\in\mathbb{N}}$ such that
$$
\mu(A|\mathscr{T}_{\Lambda_n})(x)=\gamma_{\Lambda_n}(A|x) \quad \mu-\text{a.s.} ?
$$
The answer is simple if $A$ is a potential depending only on finite number of coordinates or other words, if $A$ is a short range potential.
Since I am considering $A$ is lipschtiz, it seems reasonable to fix some configuration $\omega_0\in\mathbb{S}^{\mathbb{N}}$ and consider a sequence of truncated potentials as follows
$$
\Phi_n=(\Phi^n_{\Gamma})_{\Gamma\subset \mathbb{N}},
$$
where
$$
\Phi^n_{\Gamma}(x)=
\left\{
\begin{array}{rl}
-A(x_1,\ldots,x_{n},\omega_{n+1},\ldots),& \text{if}\ \Gamma=\{1,\ldots,n\};
\\
0,&\text{otherwise}.
\end{array}
\right.
$$
Now we consider the respective specifications given by these potentials and ask if the unique $\mu_n\in\mathcal{G}(\Phi_n)$, converges to the measure $\mu$ for any choice of $\omega_0$.
The main motivation to post this question is to know if there is a standard procedure to obtain the measure $\mu$ from the specification point of view. I also appreciate any comments about the aproximatoin scheme I described above. Any help or reference is welcome. Thanks.
