论符号融合下的吉布斯性保存
On the preservation of Gibbsianness under symbol amalgamation
摘要：
从有限字母表$A$上的全移位出发，将$A$的某些符号混合，得到了一个更小字母表$B$上的新的全移位.该合并定义了从$(A^{\mathbb N},T_A)$到$(B^{\mathbb N},T_B)$的因子映射，其中$T_A$和$T_B$是各自的移位映射。根据热力学形式主义，对于每个正则函数（“势函数”）$ψ:A^{\mathbb N}\to{\mathbb R}$，我们可以关联一个唯一的吉布斯测度$μ_ψ$。在这篇文章中，我们证明了，对于一大类潜在的，推进措施$μ_ψ\circπ^{-1}$仍然是吉布斯的潜在$ϕ:B^{\mathbb N}\to{\mathbb R}$有一个“少一点”的规律性比$ψ$。在$ψ$是“2-符号”势的特殊情况下，吉布斯测度$μ_ψ$只不过是马尔可夫测度，合并$π$定义了隐马尔可夫链。在这种特殊情况下，我们的定理可以改写为隐马尔可夫链是吉布斯测度（对于Hölder势）。
英文摘要：
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') $ψ:A^{\mathbb N}\to{\mathbb R}$, we can associate a unique Gibbs measure $μ_ψ$. In this article, we prove that, for a large class of potentials, the pushforward measure $μ_ψ\circπ^{-1}$ is still Gibbsian for a potential $ϕ:B^{\mathbb N}\to{\mathbb R}$ having a `bit less' regularity than $ψ$. In the special case where $ψ$ is a `2--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).
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