广义可数马尔可夫位移的热力学形式

T. Raszeja
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引用次数: 3

摘要

对于0-1无限矩阵$A$,用$\Sigma_A$表示的可数马尔可夫移位是符号动力学和遍历理论中的中心对象。R. Exel和M. Laca引入了相应的算子代数,无限可数字母的Cuntz-Krieger代数的一种推广,以及集合$X_A$,一种与$\Sigma_A$在局部紧化情况下重合的广义马尔可夫移位(GMS)。集合$\Sigma_A$在$X_A$中是密集的,它的补集(一个有限允许单词的集合)在$X_A$中是非空时是密集的。我们发展了$X_A$的热力学形式,在其中引入了保角测度的概念,并探讨了它与$\Sigma_A$的通常形式的联系。新现象出现,如不同类型的相变和新的保形措施无法检测到的经典热力学形式$A$非行有限。给定温度的势$F$和逆$\beta$,我们研究了与$\beta F$相关的保形测度$\mu_{\beta}$的存在性。我们给出了存在一个临界$\beta_c$ st的例子,对于每一个$\beta > \beta_c$,我们都有满足$\mu_{\beta}(\Sigma_A)=0$的共形概率,并且在弱$^*$拓扑上,对于极限$\beta\to\beta_c$, $\beta >\beta_c$的共形概率集合坍缩为标准共形概率$\mu_{\beta_c}$, $\mu_{\beta_c}(\Sigma_A)=1$。详细研究了广义更新位移及其修正。我们强调了这类更新型位移的无穷发射体和极值共形概率之间的双射。证明了在足够低的温度下,对于广义更新位移上的特定势,Ruelle变换的特征测度概率的存在唯一性;由于低温,$\beta F$是暂时的,因此在标准更新换挡中不会检测到这些措施。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Thermodynamic formalism for generalized countable Markov shifts
Countable Markov shifts, denoted by $\Sigma_A$ 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 $\Sigma_A$ in the locally compact case. The set $\Sigma_A$ 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 $\Sigma_A$. 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 $\beta$, we study the existence of conformal measures $\mu_{\beta}$ associated to $\beta F$. We present examples where there exists a critical $\beta_c$ s. t. we have existence of conformal probabilities satisfying $\mu_{\beta}(\Sigma_A)=0$ for every $\beta > \beta_c$ and, on the weak$^*$ topology, the set of conformal probabilities for $\beta >\beta_c$ collapses to the standard conformal probability $\mu_{\beta_c}$, $\mu_{\beta_c}(\Sigma_A)=1$, for the limit $\beta\to\beta_c$. 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, $\beta F$ is transient.
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