Uniqueness and statistical properties of the Gibbs state on general one-dimensional lattice systems with markovian structure

Let $M$ be a compact metric space and $X=M^{\mathbb{N}}$, we consider a set of admissible sequences $X_{A,I}\subset X$ determined by a continuous admissibility function $A:M\times M\to \mathbb{R}$ and a compact set $I\subset \mathbb{R}$. Given a Lipschitz continuous potential $\phi :X_{A,I}\to \mathbb{R}$, we prove uniqueness of the Gibbs state ${\mu }_{\phi }$ and we show that it is a Gibbs-Bowen measure and satisfies a central limit theorem.