2 - On the preservation of Gibbsianness under symbol amalgamation

Summary

Abstract. Starting from the full shift on a finite alphabet A, by mingling some symbols of A, we obtain a new full shift on a smaller alphabet B. This amalgamation defines a factor map from (A^ℕ,T_A) to (B^ℕ, T_B), where T_A and T_B are the respective shift maps. According to the thermodynamic formalism, to each regular function (“potential”) ψ:A^ℕ → ℝ, we can associate a unique Gibbs measure µ_ψ. In this article, we prove that, for a large class of potentials, the pushforward measure µ_ψ ∘ π⁻¹ is still Gibbsian for a potential φ:B^ℕ→ℝ having a “bit less” regularity than ψ. In the special case where ψ is a “two-symbol” potential, the Gibbs measure µ_ψ 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).

Introduction

From different viewpoints and under different names, the so-called hidden Markov measures have received a lot of attention in the last fifty years [3]. One considers a (stationary) Markov chain (X_n)_n∈ℕ with finite state space A and looks at its “instantaneous” image Y_n ≔ π(X_n), where the map π is an amalgamation of the elements of A yielding a smaller state space, say B. It is well known that in general the resulting chain, (Y_n)_n∈ℕ, has infinite memory.