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Extender sets and measures of maximal entropy for subshifts. (English) Zbl 1450.37011

This paper considers the relationship between the extender sets of two words found in the subshift and the measures of these words (i.e., their associated cylinder sets) given by a measure of maximal entropy. The first portion of the paper studies one-dimensional subshifts. Denoting the extender set of a word \(w\) from subshift \(X\) by \(E_X(w)\), the authors show that if \(E_X(v)\subseteq E_X(w)\) then \(\mu(v)\le \mu(w)e^{h_{\text{top}}(X)(|w|-|v|)}\). The authors then make use of this result to provide alternative proofs to some known results. For instance, they show that if \(X\) is a synchronized entropy minimal subshift, then it has a unique measure of maximal entropy, a result that was previously shown in [K. Thomsen, Ergodic Theory Dyn. Syst. 26, No. 4, 1235–1256 (2006; Zbl 1107.37006)].
The last part of this paper looks at \(G\)-subshifts where \(G\) is a countable amenable finitely generated torsion-free group. It is shown that if \(v\) and \(w\) have the same finite shape and \(E_X(v)\subseteq E_X(w)\), then \(\mu(v)\le \mu(w)\) where \(\mu\) is a measure of maximal entropy. This generalizes a special case of a result found in [T. Meyerovitch, Ergodic Theory Dyn. Syst. 33, No. 3, 934–953 (2013; Zbl 1309.37027)]. The paper concludes by applying this last result to hereditary subshifts.

MSC:

37B10 Symbolic dynamics
37B40 Topological entropy
37B51 Multidimensional shifts of finite type
37A05 Dynamical aspects of measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems
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