On the Preservation of Gibbsianness under Symbol Amalgamation

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, TA) to (B , TB), where TA and TB are the respective shift maps. According to the thermodynamic formalism, to each regular function (‘potential’) ψ : A → R, we can associate a unique Gibbs measure μψ . In this article, we prove that, for a large class of potentials, the pushforward measure μψ◦π −1 is still Gibbsian for a potential φ : B → R having a ‘bit less’ regularity than ψ. In the special case where ψ is a ‘2–symbol’ potential, the Gibbs measure μψ is nothing but a Markov measure and the amalgamation π 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).