2009/07/03 by Jean-René Chazottes, Jean-Rene Chazottes, Chazottes, Jean-Rene +2 · 1 citation
Mathematics · #37A50 #37A60 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #math.DS #math.PR #msc:37A50 #msc:37A60
paper · pdf · doi:10.48550/arxiv.0907.0528
20 pages, to appear in the proceedings of the 2007 BIRS Workshop on Entropy of Hidden Markov Processes and Connections to Dynamical Systems
openalex publication_date 2009/07/03 · arxiv created 2009/10/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,TA) to (B\mathbb N,TB), where TA and TB are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') ψ:A\mathbb N→\mathbb 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\mathbb N→\mathbb 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).