[Paper Review] Thermodynamic formalism and Substitutions

Nicolas Bédaride, Pascal Hubert+1 more|arXiv (Cornell University)|Nov 10, 2015

Mathematical Dynamics and Fractals19 references3 citations

TL;DR

This paper establishes the convergence of the renormalization operator on potentials in symbolic dynamics for a class of marked substitutions—generalizing the Thue-Morse case—by proving the existence of a unique non-zero continuous fixed point. It shows that under suitable initial conditions on the potential, the iterated renormalization $ R^n(φ) $ converges to this fixed point, with explicit asymptotic limits depending on the local structure of the substitution.

ABSTRACT

This paper studies properties of a Renormalization Operator for potentials in symbolic dynamics. These operators first appeared in \cite{BLL} and the link with substitutions was done in \cite{BL1}. Their fixed points are natural candidates to have pathologic behavior such as phase transitions. If $R$ is such an operator, we study the convergence of $R^{n}(φ)$ to the non-nul fixed point. We define the family of marked substitutions, which contains the Thue-Morse substitution, and show that the associated renormalization operators on potentials admits a unique non-nul continuous fixed point. Then, we show that $R^{n}(φ)$ converges to the fixed point as soon as $φ$ has the right germ close to $\mathbb K$.

Motivation & Objective

• To study the convergence of the renormalization operator $ R^n(φ) $ on potentials in symbolic dynamics.
• To identify conditions under which $ R^n(φ) $ converges to a non-zero continuous fixed point.
• To generalize results from the Thue-Morse substitution to a broader class of substitutions via the notion of marked substitutions.
• To link renormalization dynamics with phase transitions and pathological behaviors in thermodynamic formalism.
• To show that marked substitutions ensure convergence by localizing and controlling chaotic perturbations (accidents) in the language.

Proposed method

• Define the renormalization operator $ R $ acting on potentials $ \varphi $ in symbolic dynamics.
• Introduce the class of marked substitutions, where left and right markings allow reconstruction of letters from finite prefixes and suffixes.
• Use the constant-length property of substitutions to analyze the language $ \mathcal{L} $ of infinite words and define the function $ \delta(x) $, measuring the minimal length of a word containing a given suffix.
• Analyze the shift dynamics $ \sigma^k(H^n(x)) $ to determine $ \delta_k^n $, the minimal length of a word containing the suffix starting at position $ k $.
• Apply Lemma 4.2 to derive contradictions when assuming longer language extensions, thereby proving exact values of $ \delta_k^n $.
• Compute the renormalized potential $ R^n(\varphi)(x) $ as a sum over inverse lengths, and take the limit as $ n \to \infty $, yielding explicit asymptotic expressions.

Experimental results

Research questions

• RQ1 Under what conditions does the renormalization operator $ R^n(\varphi) $ converge to a non-zero continuous fixed point?
• RQ2 How do the structural properties of substitutions—specifically markedness—affect the convergence of the renormalization operator?
• RQ3 What is the asymptotic behavior of $ R^n(\varphi)(x) $ when $ \varphi $ has a specific germ near the set $ \mathbb{K} $?
• RQ4 Can the renormalization operator for potentials be linked to phase transitions in quasi-periodic systems like substitutions?
• RQ5 How do 'accidents'—localized disruptions in language structure—affect the convergence of $ R^n(\varphi) $, and can they be controlled?

Key findings

• For any marked substitution, the renormalization operator $ \mathcal{R} $ admits a unique non-zero continuous fixed point.
• If $ \varphi(x) = \frac{1}{p} + o(1/p) $ with $ d(x, \mathbb{K}) = 2^{-p} $, then $ R^n(\varphi)(x) \to \ln\left(\frac{p}{p-1}\right) $ for $ p \geq 3 $.
• For $ p = 2 $, the limit is $ \frac{1}{2} \ln\left(\frac{4}{3}\right) $, corresponding to the case of $ \delta(x) = 2 $.
• The convergence of $ R^n(\varphi) $ is guaranteed when $ \varphi $ has the correct germ near $ \mathbb{K} $, ensuring the fixed point is attained.
• The proof shows that 'accidents' in the language (i.e., unexpected extensions of suffixes) are localized and do not disrupt convergence due to the markedness condition.
• The asymptotic limit is derived via Riemann sum approximation of the sum $ \sum_{k=0}^{2^n-1} \frac{1}{p \cdot 2^n - k} $, converging to $ \ln\left(\frac{p}{p-1}\right) $.