Chaos and Turing machines on bidimensional models at zero temperature

Gregorio Luis Dalle Vedove Nosaki
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Abstract

In equilibrium statistical mechanics or thermodynamics formalism one of the main objectives is to describe the behavior of families of equilibrium measures for a potential parametrized by the inverse temperature $\beta$. Here we consider equilibrium measures as the shift invariant measures that maximizes the pressure. Other constructions already prove the chaotic behavior of these measures when the system freezes, that is, when $\beta\rightarrow+\infty$. One of the most important examples was given by Chazottes and Hochman where they prove the non-convergence of the equilibrium measures for a locally constant potential when the dimension is bigger then 3. In this work we present a construction of a bidimensional example described by a finite alphabet and a locally constant potential in which there exists a subsequence $(\beta_k)_{k\geq 0}$ where the non-convergence occurs for any sequence of equilibrium measures at inverse of temperature $\beta_k$ when $\beta_k\rightarrow+\infty$. In order to describe such an example, we use the construction described by Aubrun and Sablik which improves the result of Hochman used in the construction of Chazottes and Hochman.
零温度下二维模型上的混沌和图灵机
在平衡统计力学或热力学形式中,主要目标之一是描述由逆温度参数化的势的平衡测量族的行为$\beta$。这里,我们将平衡措施视为使压力最大化的位移不变措施。其他结构已经证明了当系统冻结时,即$\beta\rightarrow+\infty$时,这些措施的混沌行为。其中一个最重要的例子是由Chazottes和Hochman给出的,他们证明了当维数大于3时,局部常数势的平衡测度的不收敛性。在这项工作中,我们提出了一个由有限字母和局部恒定势描述的二维例子的构造,其中存在子序列$(\beta_k)_{k\geq 0}$,其中在$\beta_k\rightarrow+\infty$时,温度逆$\beta_k$处的任何平衡措施序列都发生不收敛。为了描述这样一个例子,我们使用了Aubrun和Sablik描述的结构,它改进了Hochman在Chazottes和Hochman结构中使用的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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