On the preservation of Gibbsianness under symbol amalgamation
Jean-René Chazottes, Edgardo Ugalde · Cambridge University Press eBooks · 2011
. 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.