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

TL;DR

This paper constructs weakly mixing subshifts with low word complexity and demonstrates that rank-one transformations have inherently high complexity growth, establishing bounds and optimality.

Contribution

It introduces subshifts with weakly mixing measures exhibiting low complexity and proves complexity bounds for rank-one transformations, showing optimality.

Findings

Existence of weakly mixing subshifts with complexity ratio below 1.5+ε

Construction of subshifts with complexity less than q + f(q) infinitely often

Rank-one transformations (not odometers) have unbounded complexity growth, optimal bounds established

Abstract

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