Word complexity of (measure-theoretically) weakly mixing rank-one subshifts

Abstract

We exhibit, for arbitrary $\epsilon> 0$ , subshifts admitting weakly mixing (probability) measures with word complexity p satisfying $\limsup p(q) / q < 1.5 + \epsilon $ . For arbitrary $f(q) \to \infty $ , said subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often. We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q = \infty $ and that this is optimal for rank-ones.