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.
