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Freezing phase transition for the Thue-Morse subshift

Nicolas Bédaride, Julien Cassaigne, Pascal Hubert, Renaud Leplaideur

TL;DR

This work analyzes the pressure for a non-Hölder prefix-based potential defined via the Thue-Morse subshift on the full two-symbol shift and proves a freezing phase transition: for sufficiently large β, the pressure collapses to 0 and the unique equilibrium state is the TM-invariant measure with zero entropy. The authors deploy an inducing scheme on a cylinder outside TM and a transfer operator L_{z,β,V}, reducing the problem to bounding sums over return words and accident words encoded by a carefully constructed infinite matrix. They establish finiteness and convergence of these sums for β>4 and provide explicit, though non-optimal, bounds (β0<16.6 for a canonical V0) showing the phase transition, with a method to extend to a broad class of potentials Ξ. The results clarify freezing phenomena for substitution subshifts, address gaps in prior proofs, and offer an algorithmic approach for analyzing other substitutions, contributing to understanding quasicrystal-like symbolic systems within thermodynamic formalism.

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 $β$, 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.

Freezing phase transition for the Thue-Morse subshift

TL;DR

This work analyzes the pressure for a non-Hölder prefix-based potential defined via the Thue-Morse subshift on the full two-symbol shift and proves a freezing phase transition: for sufficiently large β, the pressure collapses to 0 and the unique equilibrium state is the TM-invariant measure with zero entropy. The authors deploy an inducing scheme on a cylinder outside TM and a transfer operator L_{z,β,V}, reducing the problem to bounding sums over return words and accident words encoded by a carefully constructed infinite matrix. They establish finiteness and convergence of these sums for β>4 and provide explicit, though non-optimal, bounds (β0<16.6 for a canonical V0) showing the phase transition, with a method to extend to a broad class of potentials Ξ. The results clarify freezing phenomena for substitution subshifts, address gaps in prior proofs, and offer an algorithmic approach for analyzing other substitutions, contributing to understanding quasicrystal-like symbolic systems within thermodynamic formalism.

Abstract

On the full shift on two symbols, we consider the potential defined by where denotes the longest common prefix between the infinite word and an element of the subshift associated to the Thue-Morse substitution. Given a non negative real number , the pressure function is where the supremum is taken over all shift invariant probabilities on the full shift and is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential : For large enough, the pressure 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.

Paper Structure

This paper contains 23 sections, 29 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. More definitions and tools
  5. Words, languages and special words
  6. Background on Thue-Morse substitution
  7. Accidents
  8. Main tool of the proof of Theorem \ref{['thm-tm']}
  9. Scheme of the proof
  10. Return words and minimal forbidden words
  11. Sum
  12. Technical lemmas
  13. Infinite matrices
  14. Proof of Proposition \ref{['prop:final']}
  15. Proof of Theorem \ref{['thm-tm']}
  16. ...and 8 more sections

Key Result

Theorem 1

Let $X$ be the full shift on a two-letter alphabet. Let $\mathrm{TM}$ be the subshift associated to the Thue-Morse substitution and $V(x) = \frac{1}{n}$ where $n$ denotes the longest common prefix between $x\in X$ and an element of $\mathrm{TM}$. There exists $\beta_0$ such that, for $\beta \geq \be

Figures (2)

  • Figure 1: Accidents-Dashed lines indicate infinite words in $\mathbb K$.
  • Figure 2: Global picture of the accidents in a word $u$: the first bispecial word $v_1$, then the second accident with extension $a_1v_1b_1$, and so on.

Theorems & Definitions (69)

  • Theorem
  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 59 more