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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.

Extender sets and measures of maximal entropy for subshifts