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arXiv · 2511.01034

Freezing phase transition for the Thue-Morse subshift

Nicolas Bédaride·Julien Cassaigne·Pascal Hubert·Renaud Leplaideur
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Abstract

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 $\beta$, the pressure function is $P(\beta):=\sup\left\{h_{\mu}+\beta\int V\,d\mu\right\},$ where the supremum is taken over all shift invariant probabilities $\mu$ on the full shift and $h_{\mu}$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $\beta$ 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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Nicolas Bédaride, Julien Cassaigne, Pascal Hubert, Renaud Leplaideur. 2025-11-02. Freezing phase transition for the Thue-Morse subshift. https://arxiv.org/abs/2511.01034

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