A Local Characterization of Unique Equilibrium States

Abstract

We study when a prescribed ergodic measure is the unique equilibrium state of a continuous potential. For continuous actions of countably infinite discrete amenable groups on compact metric spaces with finite topological entropy, we characterize this property by a local condition on entropy differences. Fixing an ergodic measure $\mu$, we extend $h(\nu )-h(\mu )$ to the cone generated by the differences $\nu -\mu$, scaling the entropy difference by the same factor. We prove that $\mu$ is uniquely attainable if and only if this entropy difference function is weak\nobreakdash-$\ast$ upper semicontinuous at the zero measure. Upper semicontinuity of entropy at $\mu$ alone does not suffice. In fact, we construct a shift space over a compact alphabet for which entropy is upper semicontinuous at every ergodic measure and continuous at a measure supported on a fixed point, but this measure is not an equilibrium state of any continuous potential.

arXiv commentThis version replaces "Which Phases Are Thermodynamically Realizable? A Local Entropy Criterion". The earlier claim that upper semicontinuity of entropy at an ergodic measure guarantees attainability is false. We construct a shift space where entropy is upper semicontinuous at every ergodic measure and continuous at a fixed-point measure that no continuous potential realizes