Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts

Abstract

We exhibit subshifts admitting weakly mixing (probability) measures, for arbitrary ǫ > 0, with word complexity p satisfying lim sup p(q) /q < 1.5 + ǫ. For arbitrary f (q) → ∞, said subshifts can be made to satisfy p(q) < q + f (q) infinitely often.We establish that every subshift associated to a totally ergodic rank-one transformation (on a probability space) satisfies lim sup p(q) − 1.5q = ∞ and that this is optimal for rank-ones.