Extender sets and measures of maximal entropy for subshifts

Abstract

For countable amenable finitely generated torsion-free G, we prove inequalities relating μ ( v ) and μ ( w ) for any measure of maximal entropy μ on a G-subshift and any words v , w where the extender set of v is contained in the extender set of w. Our main results are two generalizations of a theorem of Meyerovitch (Ergodic Theory Dynam. Systems 33 (2013) 934–953): the first applies to all such v , w when G = Z , and the second to v , w with the same shape for any G. As a consequence of our results we give new and simpler proofs of several facts about synchronized subshifts (including a result from Thomsen, Ergodic Theory Dynam. Systems 26 (2006) 1235–1256) and we answer a question of Climenhaga.