Gibbs Properties of Equilibrium States

TL;DR

The paper identifies extensibility as the key necessary and sufficient condition for the equivalence between Gibbs states and equilibrium states for continuous potentials on the full shift. It develops a constructive bridge between extensible potentials and Gibbsian specifications: from extensible φ one obtains a two-sided specification γ^φ for which equilibrium states coincide with Gibbs states (weak Bowen-Gibbs), and conversely every translation-invariant DLR Gibbs measure corresponds to an extensible one-sided potential φ_γ. The results establish ES(φ) ⊆ 𝒢_S(γ^φ) and 𝒢_S(γ) ⊆ ES(φ_γ), and prove a closed diagram φ → γ^φ → φ_{γ^φ} = γ, thereby unifying Gibbsian and variational viewpoints under extensibility. The framework extends beyond uniform absolute convergence, applies to higher-dimensional lattices, and encompasses classical classes (Hölder, summable variations, Walters) as well as the Dyson potential, offering a broad, constructive route to Gibbsianity of equilibrium states.

Abstract

We consider the problem of equivalence of Gibbs states and equilibrium states for continuous potentials on full shift spaces $E^{\mathbb{Z}}$. Sinai, Bowen, Ruelle and others established equivalence under various assumptions on the potential $φ$. At the same time, it is known that every ergodic measure is an equilibrium state for some continuous potential. This means that the equivalence can occur only under some appropriate conditions on the potential function. In this paper, we identify the necessary and sufficient conditions for the equivalence.

Figures (1)

• Figure :

Theorems & Definitions (18)

• Theorem A
• Theorem B
• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 3.1
• Remark 3.2
• Proposition 4.1
• proof
• Theorem 4.2