Hereditary subshifts whose measure of maximal entropy has no Gibbs property

TL;DR

This paper investigates the conditions under which the measure of maximal entropy for hereditary closures of certain subshifts exhibits the Gibbs property, linking it to the atomicity of the Mirsky measure and exploring tautness.

Contribution

It establishes a precise criterion connecting the Gibbs property of the measure of maximal entropy to the atomicity of the Mirsky measure, answering an open question.

Findings

The measure of maximal entropy has the Gibbs property if and only if the Mirsky measure is purely atomic.

The hereditary closure's measure of maximal entropy is non-Gibbs when the Mirsky measure is non-atomic.

The paper proves that tautness of $$ is equivalent to the Mirsky measure having full support.

Abstract

We show that the measure of maximal entropy for the hereditary closure of a $\mathscr{B}$-free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, we show that $\mathscr{B}$ is taut whenever the corresponding Mirsky measure $\nu_\eta$ has full support. This is the converse theorem to a recent result of Keller.