Gibbs Properties 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 $\phi$. 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.