Failure of the Gibbs inequality for continuous potentials

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.

💡 Research Summary

The paper investigates the extent to which the classical Gibbs inequality—stating that the Gibbs ratio is bounded above and below by positive constants—holds for continuous potentials on symbolic shift spaces. While it is well‑known that for Hölder continuous potentials the unique equilibrium state satisfies a strong Gibbs property (there exists a constant (c>0) such that (\frac{1}{c}<\mu_\psi(