An elementary proof of Brémont's theorem on the convergence of equilibrium measures at temperature zero

Technical Summary: An Elementary Proof of Brémont's Theorem on the Convergence of Equilibrium Measures at Temperature Zero

Problem Statement
The paper addresses the convergence of equilibrium measures $m_{tf}$ for a locally constant function $f$ on a topologically mixing subshift of finite type $(X, \sigma)$ as the parameter $t \to \infty$. In thermodynamic formalism, $t$ represents the inverse temperature, and the limit $t \to \infty$ corresponds to the zero-temperature limit. While it is known that accumulation points of $m_{tf}$ are $f$-extremal measures (measures maximizing $\int f dm$ with maximal entropy among such measures), the limit $m_{tf}$ does not necessarily exist for general continuous functions. Previous work by Chazottes & Hochman and Coronel & Rivera-Letelier demonstrated that for Lipschitz functions, the limit may fail to exist or depend sensitively on $f$. However, Brémont proved that if $f$ is locally constant, the limit exists. The challenge lies in the case where multiple $f$-extremal measures exist; the limit is then a specific convex combination of these ergodic measures, a combination not determined solely by the set of maximizing measures.

Methodology
The authors provide a new, elementary proof of Brémont's theorem, avoiding the heavy machinery of subanalytic geometry and the preparation theorem used in the original proof. The methodology proceeds as follows:

1. Reduction to Higher Block Presentation: The authors assume $f$ and a test function $g$ are constant on cylinders of length $r$. They transform the system into a higher block presentation, reducing the problem to a subshift defined by a primitive $N \times N$ matrix $B$ where $f$ and $g$ depend only on the first two coordinates.
2. Dynamical Zeta Function and Implicit Differentiation: They define a transfer matrix $B_{t,u}$ with entries $e^{tf(i,j) + ug(i,j)}$ and consider the dynamical zeta function related to $\Delta(z, t, u) = \det(I - zB_{t,u})$. Using implicit differentiation of the reciprocal of the zeta function, they derive a formula (Lemma 2.1) expressing the measure $m_{tf}(g)$ as a rational function involving $z(t) = e^{-P(tf)}$ and the partial derivatives of $\Delta$.
3. Algebraic Structure of Coefficients: The coefficients of the polynomial $\Delta(\cdot, t, 0)$ are shown to be finite linear combinations of exponentials of the form $e^{\gamma t}$ (elements of a ring $E$). Consequently, $z(t)$ is a zero of a polynomial with coefficients in $E$.
4. Elimination and Algebraic Dependence: The core technical step involves eliminating $z(t)$ from the system of equations. By treating the space of real analytic functions as an integral domain and utilizing field theory, the authors demonstrate that $m_{tf}(g)$ (denoted $y(t)$) satisfies a polynomial equation with coefficients in the field of fractions of $E$. Specifically, they show that $y(t)$ is algebraic over the field generated by the functions $t \mapsto e^{\gamma t}$.
5. Asymptotic Analysis via Elementary Lemma: The authors introduce Lemma 3.1, an elementary result stating that a bounded continuous function $y(t)$ satisfying a polynomial equation with coefficients in $E$ (not all identically zero) must have a limit as $t \to \infty$. This lemma replaces the need for the preparation theorem. The proof of the lemma relies on analyzing the dominant exponential terms as $t \to \infty$ to show that the set of accumulation points of $y(t)$ is finite and contained in the roots of a limiting polynomial, forcing convergence.

Key Contributions and Results

• Elementary Proof: The paper provides a short, self-contained proof of Brémont's theorem using standard tools from dynamical systems, linear algebra, and field theory, bypassing the complex theory of subanalytic functions.
• Convergence: It rigorously establishes that for any locally constant $f$, the equilibrium measures $m_{tf}$ converge in the weak-$*$ topology as $t \to \infty$.
• Algebraicity of the Limit: A significant result is that the limit measure assigns an algebraic number to every cylinder set. Even if $f$ takes transcendental values, the limiting weights are algebraic numbers.
• Degree Bound: The paper provides an explicit bound on the degree of these algebraic numbers. If $f$ is constant on cylinders of length $r$, the degree of the limit measure for a cylinder of length $\le r$ is at most the number of cylinders of length $r-1$ (denoted $N$ in the proof).
• Rationality of Coefficients: If $f$ takes rational values, the coefficients of the polynomial governing the limit are rational, implying the limit is an algebraic number of degree at most $N$.

Significance and Claims
The authors claim that their approach offers a more accessible alternative to Brémont's original proof, which relied on deep results regarding subanalytic functions. By framing the problem in terms of the algebraic properties of the equilibrium measure with respect to exponential functions, they isolate the convergence mechanism to a simple asymptotic argument (Lemma 3.1).

The paper notes that while the limit is a convex combination of ergodic $f$-extremal measures, the specific coefficients in this combination are not necessarily rational, even if $f$ is rational. The significance lies in the structural insight: the limit measure is not arbitrary but is constrained to be algebraic over the field generated by the exponentials of the potential values. The authors also mention that their result aligns with and provides an alternative perspective to other proofs by Leplaideur (dynamical), Chazottes et al. (algorithmic/renormalization), and Garibaldi & Thieullen (Puiseux series), particularly in the non-unique maximizing measure case.

The paper concludes by noting that the limit measure is a convex combination of ergodic $f$-extremal measures, and while the existence of the limit is guaranteed, the specific weights depend on the fine structure of the potential $f$ and are not determined solely by the set of maximizing measures.