Thue-Morse 子移位的冻结相变
Freezing phase transition for the Thue-Morse subshift
AI总结：
本文研究了两个符号全移位上与 Thue-Morse 替换相关的势函数，证明了当参数足够大时存在压强为零的冻结相变，并修正了前人证明中的重大漏洞。
摘要：
在两个符号的全移位上，我们考虑由 $V(x) = \frac{1}{n}$ 定义的势函数，其中 $n$ 表示无穷词 $x$ 与 Thue-Morse 替换关联的子移位元素之间的最长公共前缀。给定非负实数 $β$，压强函数为 $P(β):=\sup\left\{h_μ+β\int V\\,dμ\right\}$，其中上确界取遍全移位上所有移位不变概率 $μ$，$h_μ$ 为 Kolmogorov 熵。我们证明势函数 $V$ 存在冻结相变：当 $β$ 足够大时，压强 $P(β)$ 等于零。Bruin 和 Leplaideur 在 \cite{BL2}、\cite{Bruin-Leplaid-13} 中曾发表过类似结果，但其证明存在重大漏洞，需要实质性澄清。
英文摘要：
On the full shift on two symbols, we consider the potential defined by $V(x) = \frac{1}{n}$ where $n$ denotes the longest common prefix between the infinite word $x$ and an element of the subshift associated to the Thue-Morse substitution. Given a non negative real number $β$, the pressure function is $P(β):=\sup\left\{h_μ+β\int V\,dμ\right\},$ where the supremum is taken over all shift invariant probabilities $μ$ on the full shift and $h_μ$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $β$ large enough, the pressure $P(\be)$ is equal to zero. Similar results were previously published by Bruin and Leplaideur in \cite{BL2}, \cite{Bruin-Leplaid-13} but their proofs contained significant gaps and required substantial clarification.
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