Extender sets and measures of maximal entropy for subshifts

TL;DR

This paper establishes inequalities relating measures of maximal entropy for patterns based on their extender sets, generalizing previous results and providing new insights into synchronizing subshifts.

Contribution

It generalizes Meyrovitch's theorem to broader classes of groups and patterns, offering new proofs and answering open questions about synchronizing subshifts.

Findings

Inequalities relating measures of maximal entropy based on extender sets.

Generalizations of Meyrovitch's theorem to all countable amenable groups.

Simplified proofs of properties of synchronizing subshifts.

Abstract

We prove inequalities relating the measures of maximal entropy of two patterns u,v where the extender set of u is contained in the extender set of v. Our main results are two generalizations of a Theorem of Meyerovitch; the first applies to all such v,w when G=Z, and the second to v,w with the same shape and any countable amenable finitely generated torsion-free G. As a consequence of our results we give new and simpler proofs of several facts about synchronizing subshifts and we answer a question of Climenhaga.