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 in [15]. 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 1 n log k for k, n ∈ N. Our proof also completely characterizes the weak limits (as d → ∞) 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.