Measure-Theoretically Mixing Subshifts of Minimal Word Complexity
Darren Creutz
TL;DR
This work resolves the link between measure-theoretic complexity and symbolic word complexity by establishing a sharp dividing line: strong mixing of all orders can occur in subshifts whose word complexity is superlinear, while non-superlinear complexity forces partial rigidity. The authors construct quasi-staircase rank-one transformations with word complexity $p(q)$ that grows sublinearly relative to any given superlinear function $f$ (i.e., $p(q)/f(q)\to 0$), and prove these systems admit strongly mixing measures and mixing of all orders. Conversely, they show that if $\liminf p(q)/q < \infty$, then every ergodic measure is partially rigid, precluding strong mixing; this connects to $S$-adic structure and known rigidity phenomena. The results collectively reveal a sharp boundary in measure-theoretic behavior at superlinear word complexity, highlighting how symbolic complexity governs possible dynamical phenomena in zero-entropy settings.
Abstract
We resolve a long-standing open question on the relationship between measure-theoretic dynamical complexity and symbolic complexity by establishing the exact word complexity at which measure-theoretic strong mixing manifests: For every superlinear $f : \mathbb{N} \to \mathbb{N}$, i.e. $f(q)/q \to \infty$, there exists a subshift admitting a (strongly) mixing of all orders probability measure with word complexity $p$ such that $p(q)/f(q) \to 0$. For a subshift with word complexity $p$ which is non-superlinear, i.e. $\liminf p(q)/q < \infty$, every ergodic probability measure is partially rigid.