Measure-Theoretically Mixing Subshifts with Low Complexity

TL;DR

This paper constructs a new class of measure-theoretically mixing subshifts with very low complexity, improving the known bounds and resolving a conjecture about the minimal complexity for such systems.

Contribution

It introduces extremely elevated staircase transformations that are mixing and have complexity bounds lower than previously known, confirming a conjecture of Ferenczi.

Findings

Existence of mixing subshifts with complexity $p(n) = o(f(n))$ for certain functions $f$.

Construction of extremely elevated staircase transformations.

Resolution of Ferenczi's conjecture on minimal complexity for mixing subshifts.

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.