Equilibrium states which are not Gibbs measure on hereditary subshifts

Abstract

In this paper, we consider which kind of invariant measure on hereditary subshifts is not Gibbs measure. For the hereditary closure of a subshift $(X,S)$, we prove that in some situation, the invariant measure $\nu*B_{p,1-p}$ can not be a Gibbs measure where $\nu$ is an invariant measure on $(X,S)$. As an application, we show that for some $\B$-free subshifts, the unique equilibrium state $\nu_\eta*B_{p,1-p}$ is not Gibbs measure.