一维格上的相互作用、规范、DLR概率和Ruelle算子
Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice
摘要：
本文在有限字母表（或状态空间）的背景下，给出了格$\mathbb{N}$上Gibbs测度概念的几种不同含义.我们比较和分析这些“原则上”不同的概念：DLR-Gibbs措施，热力学极限和特征概率的对偶的Ruelle操作员（也称为共形措施）。除其他事项外，我们扩展了经典的概念的吉布斯规格$\mathbb{N}$在这样的方式，许多结果的相似性，在统计力学和动力系统变得明显。我们的一个主要结果是证明了利用Ruelle算子构造动力系统中Walters势的共形测度可以用Specification来表示。我们还描述了伊辛模型，与$1/r^{2+\varepsilon}$相互作用能，在热力学形式主义设置，并证明其相关的潜力是在沃尔特斯空间-我们提出了一个明确的表达。我们还提供了另一种方法来获得DLR-Gibbs措施的唯一性。
英文摘要：
In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these "in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures). Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space - we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.
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