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Failure of the Gibbs inequality for continuous potentials

Arantha Ranu

TL;DR

This work analyzes Gibbs-type bounds for continuous potentials on shift spaces, showing that the classical Gibbs property for Hölder potentials need not extend to continuity alone. It introduces a very weak Gibbs inequality and constructs a continuous potential on a full shift whose unique equilibrium state $\mu$ is supported on a carefully built subshift $Y$, demonstrating that the standard Gibbs ratio bounds can fail at certain points (e.g., the all-$\beta'$ point $o$). Despite this, the authors prove a meaningful a.e. Gibbs-type bound for $\mu$ with respect to a generating partition, highlighting a sharp distinction between pointwise and almost-everywhere behavior. The results illuminate the limitations of Gibbs-type inequalities in non-Hölder settings and use an odometer-based coding to realize the equilibrium state on $Y$, providing a concrete mechanism for understanding equilibrium states in symbolic dynamics.

Abstract

It is well known that the Gibbs inequality, which says that the Gibbs ratio is bounded above and below by positive constants, holds for the unique equilibrium states of Hölder continuous potentials on shift spaces, but it can fail for continuous potentials. In this article, we study the validity of a weaker form of the Gibbs inequality in this broader setting.

Failure of the Gibbs inequality for continuous potentials

TL;DR

This work analyzes Gibbs-type bounds for continuous potentials on shift spaces, showing that the classical Gibbs property for Hölder potentials need not extend to continuity alone. It introduces a very weak Gibbs inequality and constructs a continuous potential on a full shift whose unique equilibrium state is supported on a carefully built subshift , demonstrating that the standard Gibbs ratio bounds can fail at certain points (e.g., the all- point ). Despite this, the authors prove a meaningful a.e. Gibbs-type bound for with respect to a generating partition, highlighting a sharp distinction between pointwise and almost-everywhere behavior. The results illuminate the limitations of Gibbs-type inequalities in non-Hölder settings and use an odometer-based coding to realize the equilibrium state on , providing a concrete mechanism for understanding equilibrium states in symbolic dynamics.

Abstract

It is well known that the Gibbs inequality, which says that the Gibbs ratio is bounded above and below by positive constants, holds for the unique equilibrium states of Hölder continuous potentials on shift spaces, but it can fail for continuous potentials. In this article, we study the validity of a weaker form of the Gibbs inequality in this broader setting.
Paper Structure (6 sections, 20 theorems, 135 equations)

This paper contains 6 sections, 20 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction and properties of the subshift $(Y,\sigma)$
  4. Ergodic measures on $(Y,\sigma)$
  5. Construction of the potential
  6. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

For a continuous potential $\psi$ on a shift space $(Z,\sigma)$ that has an ergodic equilibrium state $\mu$. The "very weak Gibbs inequality" is satisfied for $\mu$ a.e. $z$.

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 42 more