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 $\varphi :B^{\mathbb{N}}\to \mathbb{R}$ 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 Hölder potential).

20 pages, to appear in the proceedings of the 2007 BIRS Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems

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