Equilibrium states which are not Gibbs measure on hereditary subshifts

TL;DR

This paper investigates conditions under which certain invariant measures on hereditary subshifts are not Gibbs measures, highlighting specific cases like hereditary closures and B-free subshifts where this occurs.

Contribution

It demonstrates that in some hereditary subshifts, the invariant measure formed by convolving with a Bernoulli measure is not a Gibbs measure, providing new insights into measure properties.

Findings

Invariant measure $ u*B_{p,1-p}$ can fail to be Gibbs measure.

For some hereditary subshifts, the unique equilibrium state is not Gibbs.

Identifies specific hereditary subshifts where Gibbs property does not hold.

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.