in the setting of zero-entropy transformations, the class of subshifts--closed shift-invariant subsets $x$ of $\mathcal{a}^{\mathbb{z}}$ for a finite alphabet $\mathcal{a}$--possesses a quantitative measure of complexity: the number of distinct `words' of a given length $p(q) = |\{ w \in \mathcal{a}^{q} : \exists x \in x \text{ s.t. w is a substring of x}\}|$.

i will discuss my work, some joint with r. pavlov, pinning down the relationship between this quantitative notion of complexity with the qualitative dynamical complexity properties of probability-preserving systems known as strong and weak mixing.

specifically, i will present results that strong mixing can occur with word complexity arbitrarily close to linear but cannot occur when $\liminf p(q)/q < \infty$ and that weak mixing can occur when $\limsup p(q)/q = 1.5$ but cannot occur when $\limsup p(q)/q < 1/5$.

the condition that $\limsup p(q)/q < 1.5$ is a (much) stronger version of zero entropy. a corollary of our work is that the celebrated sarnak conjecture holds for all such systems.