The Lanford–Ruelle theorem for actions of sofic groups

Abstract:

Let $\Gamma$ be a sofic group, $\Sigma$ be a sofic approximation sequence of $\Gamma$ and $X$ be a $\Gamma$-subshift with non-negative sofic topological entropy with respect to $\Sigma$. Further assume that $X$ is a shift of finite type, or more generally, that $X$ satisfies the topological Markov property. We show that for any sufficiently regular potential $f \colon X \to \mathbb {R}$, any translation-invariant Borel probability measure on $X$ which maximizes the measure-theoretic sofic pressure of $f$ with respect to $\Sigma$ is a Gibbs state with respect to $f$. This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic.

As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense.

On the expository side, we present a short proof of Chung’s variational principle for sofic topological pressure.