Measure-Theoretically Mixing Subshifts with Low Complexity

Abstract

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f:\mathbb{N}\to \mathbb{N}$ with $f(n)/n$ increasing and $\sum 1/f(n)<\infty$, that there exists an extremely elevated staircase with word complexity $p(n)=o(f(n))$. This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

Revised based on helpful referee feedback