Extender sets and measures of maximal entropy for subshifts

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 %26#x01d53e; = ℤ, and the second to v,w with the same shape for any %26#x01d53e;. 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. © 2019 London Mathematical Society

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