Entropy and measures of maximal entropy for axial powers of Subshifts

The notions of "limiting entropy" and "independence entropy" for one-dimensional subshifts were introduced by Louidor, Marcus, and the second author. It was also implicitly conjectured there that these two quantities are always equal. We verify this conjecture, which implies, among other things, that the limiting entropy of any one-dimensional SFT is of the form $\frac{1}{n} \log k$ for $k,n \in \mathbb{N}$. Our proof also completely characterizes the weak limits (as $d \rightarrow \infty$) of isotropic measures of maximal entropy; any such measure is a Bernoulli extension over some zero entropy factor from an explicitly defined set of measures. We also discuss connections of our results to various models and results arising in statistical mechanics.

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