广义可数马尔可夫位移的热力学形式
Thermodynamic Formalism for Generalized Countable Markov Shifts
摘要：
可列马尔可夫移位，对于0 - 1无限矩阵$A$，记为$Σ_A$，是符号动力学和遍历理论的中心对象。R. Exel和M. Laca引入了相应的算子代数，Cuntz-Krieger代数在无限可数字母表上的推广，以及集合$X_A$，一种在局部紧情况下与$Σ_A$一致的广义马尔可夫移位（GMS）。集合$Σ_A$在$X_A$中是稠密的，而它的补集，一个有限允许字的集合，当非空时在$X_A$中是稠密的。我们发展了$X_A$的热力学形式，在其中引入了共形测度的概念，并探讨了它与$Σ_A$通常形式的联系。新的现象出现，作为不同类型的相变和新的共形措施未检测到的经典热力学形式主义$A$行有限。给定一个势$F$和温度的逆$β$，我们研究了与$βF$相关联的共形测度$μ_β$的存在性.我们提出的例子中存在一个关键的$β_c$。t.对于每个$β> β_c$，我们存在满足$μ_β(Σ_A)=0$的共形概率，并且在弱$^*$拓扑上，对于极限$β\toβ_c$，$β>β_c$的共形概率集合坍缩为标准共形概率$μ_{β_c}$，$μ_{β_c}(Σ_A)=1$。本文详细研究了广义更新移位及其修正，重点讨论了这类更新移位的无穷发射子与极值共形概率之间的双射关系。我们证明了存在性和唯一性的本征测度概率的Ruelle变换在足够低的温度下为一个特定的潜在的广义更新移位，这样的措施是没有检测到的标准更新移位，因为低温下，$βF$是瞬态的。
英文摘要：
Countable Markov shifts, denoted by $Σ_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Σ_A$ in the locally compact case. The set $Σ_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal measure in it, and exploring its connections with the usual formalism for $Σ_A$. New phenomena appear, as different types of phase transitions and new conformal measures undetected by the classical thermodynamic formalism for $A$ not row-finite. Given a potential $F$ and inverse of temperature $β$, we study the existence of conformal measures $μ_β$ associated to $βF$. We present examples where there exists a critical $β_c$ s. t. we have existence of conformal probabilities satisfying $μ_β(Σ_A)=0$ for every $β> β_c$ and, on the weak$^*$ topology, the set of conformal probabilities for $β>β_c$ collapses to the standard conformal probability $μ_{β_c}$, $μ_{β_c}(Σ_A)=1$, for the limit $β\toβ_c$. We study in detail the generalized renewal shift and modifications of it. We highlight the bijection between infinite emitters of the alphabet and extremal conformal probabilities for this class of renewal type shifts. We prove the existence and uniqueness of the eigenmeasure probability of the Ruelle's transformation at low enough temperature for a particular potential on the generalized renewal shift; such measures are not detected on the standard renewal shift since for low temperatures, $βF$ is transient.
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