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

Abstract

We exhibit subshifts admitting weakly mixing (probability) measures, for arbitrary $\epsilon >0$, with word complexity $p$ satisfying $\mathrm{limsup}\frac{p(q)}{q}<1.5+\epsilon$. 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 $\mathrm{limsup}p(q)-1.5q=\infty$ and that this is optimal for rank-ones.

References in corpus (4)

• The automorphism group of a shift of slow growth is amenable
• Actions of finite rank: weak rational ergodicity and partial rigidity
• Topological Mixing Properties of Rank-One Subshifts
• Subsystems of transitive subshifts with linear complexity