Failure of the Gibbs inequality for continuous potentials

TL;DR

This paper investigates the failure of the Gibbs inequality for continuous potentials and explores the conditions under which a weaker form of the inequality may still hold in this broader context.

Contribution

It provides a detailed analysis of when the Gibbs inequality fails for continuous potentials and examines the validity of a weaker inequality form.

Findings

Gibbs inequality can fail for continuous potentials.

A weaker form of the Gibbs inequality may still be valid in some cases.

Conditions affecting the validity of the weaker inequality are identified.

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 H\"older 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.