On the preservation of Gibbsianness under symbol amalgamation

Abstract

Starting from the full--shift on a finite alphabet $A$, mingling some symbols of $A$, we obtain a new full shift on a smaller alphabet $B$. This amalgamation defines a factor map from $(A^{\mathbb N},T_A)$ to $(B^{\mathbb N},T_B)$, where $T_A$ and $T_B$ are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') $\psi:A^{\mathbb N}\to{\mathbb R}$, we can associate a unique Gibbs measure $\mu_\psi$. In this article, we prove that, for a large class of potentials, the pushforward measure $\mu_\psi\circ\pi^{-1}$ is still Gibbsian for a potential $\phi:B^{\mathbb N}\to{\mathbb R}$ having a `bit less' regularity than $\psi$. In the special case where $\psi$ is a `2--symbol' potential, the Gibbs measure $\mu_\psi$ is nothing but a Markov measure and the amalgamation $\pi$ defines a hidden Markov chain. In this particular case, our theorem can be recast by saying that a hidden Markov chain is a Gibbs measure (for a H\"older potential).