Extender Sets and Measures of Maximal Entropy for Subshifts
Publication Date
6-25-2019
Document Type
Article
Organizational Units
Mathematics
Keywords
37B10, 37B40, 37A35 (primary), 37A60 (secondary), Maximal entropy
Abstract
For countable amenable finitely generated torsion‐free G" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝔾G, we prove inequalities relating μ(v)" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝜇(𝑣)μ(v) and μ(w)" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝜇(𝑤)μ(w) for any measure of maximal entropy μ" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝜇μ on a G" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝐺G‐subshift and any words v,w" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝑣,𝑤v,w where the extender set of v" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝑣v is contained in the extender set of w" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝑤w. 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" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝑣,𝑤v,w when G=Z" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝔾=ℤG=Z, and the second to v,w" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝑣,𝑤v,w with the same shape for any G" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; font-size: 16px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px 2px 0px 0px; margin: 0px; color: rgb(28, 29, 30); font-family: "Open Sans", sans-serif; position: relative;">𝔾G. 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.
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Recommended Citation
García‐Ramos, Felipe, and Pavlov, Ronnie. “Extender Sets and Measures of Maximal Entropy for Subshifts.” Journal of the London Mathematical Society, vol. 100, no. 3, 2019, pp. 1013–1033. doi: 10.1112/jlms.12252.