Measure-Theoretically Mixing Subshifts of Minimal Word Complexity

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. $\mathrm{liminf}p(q)/q<\infty$, every ergodic probability measure is partially rigid.

arXiv commentUpdated per referee report