The Lanford-Ruelle theorem for actions of sofic groups

TL;DR

This paper extends the Lanford-Ruelle theorem to sofic groups, showing that regular potentials have Gibbs states as equilibrium measures, and explores conditions for uniqueness and invariance of these measures.

Contribution

It generalizes the classical Lanford-Ruelle theorem to sofic groups and provides new criteria for the uniqueness and invariance of equilibrium measures.

Findings

Gibbs states are equilibrium measures for regular potentials on sofic group shifts.

Haar measure is unique of maximal sofic entropy when the homoclinic group is dense.

Criteria established for the independence of equilibrium states from sofic approximation sequences.

Abstract

Let $\Gamma$ be a sofic group, $\Sigma$ be a sofic approximation sequence of $\Gamma$ and $X$ be a $\Gamma$-subshift with nonnegative 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-theoretical 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.