无限多状态下广义马尔可夫位移的热力学形式
Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States
摘要：
给定一个0-1无限矩阵$A$及其可列马尔可夫位移$Σ_A$，本文作者之一和M. Laca等人引入了一类{ \it广义可数马尔可夫移位} $X_A=Σ_A \cup Y_A$，其中$Y_A$是一个特殊的有限容许字集.对于一些研究最多的可数马尔可夫移位$Σ_A$，$X_A$是$Σ_A$的紧化，并且总是至少局部紧的。我们开发了空间$X_A$的热力学形式，探索与$Σ_A$的标准结果的连接。新的现象出现，如新的共形措施和{ \it长度型相变}：本征措施生活在$Σ_A$在高温和生活在$Y_A$在低温。利用M. Denker和M. Yuri关于迭代函数系统的定义，我们证明了在广义背景下，Gurevich压力是压力函数的自然定义。对于规范作用量，古列维奇熵是$Y_A$上新的共形测度（KMS态）存在的临界温度。我们展示了无限（甚至不可数）许多新的极值共形措施，在通常的形式主义检测的例子。证明了当势足够矫顽时，保形测度在低温下总是存在的。为了研究$X_A$上测度的弱$^*$收敛性，我们刻画了$X_A$拓扑的一个基，并给出了$Y_A$上的共形测度收敛于$Σ_A$上的共形测度的一些情形.证明了局部紧Hausdorff第二可数空间的几个保形性概念之间的等价性，包括广义Hausdorff-Deaconu群胚的拟不变测度。
英文摘要：
Given a 0-1 infinite matrix $A$ and its countable Markov shift $Σ_A$, one of the authors and M. Laca have introduced a kind of {\it generalized countable Markov shift} $X_A=Σ_A \cup Y_A$, where $Y_A$ is a special set of finite admissible words. For some of the most studied countable Markov shifts $Σ_A$, $X_A$ is a compactification of $Σ_A$, and always it is at least locally compact. We developed the thermodynamic formalism on the space $X_A$, exploring the connections with standard results on $Σ_A$. New phenomena appear, such as new conformal measures and a {\it length-type phase transition}: the eigenmeasure lives on $Σ_A$ at high temperature and lives on $Y_A$ at low temperature. Using a pressure-point definition proposed by M. Denker and M. Yuri for iterated function systems, we proved that the Gurevich pressure is a natural definition for the pressure function in the generalized setting. For the gauge action, the Gurevich entropy is a critical temperature for the existence of new conformal measures (KMS states) living on $Y_A$. We exhibit examples with infinitely (even uncountable) many new extremal conformal measures, undetectable in the usual formalism. We prove that conformal measures always exist at low temperatures when the potential is coercive enough. We characterized a basis of the topology of $X_A$ to study the weak$^*$ convergence of measures on $X_A$, and we show some cases where the conformal measure living on $Y_A$ converges to a conformal one living on $Σ_A$. We prove the equivalence among several notions of conformality for locally compact Hausdorff second countable spaces, including quasi-invariant measures for generalized Renault-Deaconu groupoids.
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