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.

Low complexity subshifts have discrete spectrum

“…He there gave an example of X with a nontrivial weakly mixing measure and 𝐶 = 5/3 and again asked whether this was minimal. This was shown not to be the case in [Cre23], where examples were given of C arbitrarily close to (but above) 3/2.…”

“…He also gave an example of X with a strongly mixing measure and quadratic and asked whether this complexity was the lowest possible. This was proved not to be the case in [Cre22] and [CPR23], which provided examples first on the order of and then below any possible superlinear growth rate, establishing linear complexity as the ‘threshold’ for existence of such a measure. In a different work, Ferenczi ([Fer95]) examined the same question for weakly mixing measures, where it is known that linear complexity can occur via the well-known Chacon subshift.…”

“…There exists an infinite transitive subshift X which is uniquely ergodic, has unique measure which is weak mixing and for which lim sup 𝑝 (𝑞) 𝑞 = 3 2 . In [Cre23], it was also suggested that perhaps a subshift X having a nontrivial weakly mixing measure forces lim sup 𝑝 (𝑞) 𝑞 > 3 2 ; Theorem 5.1 answers this negatively. In fact, the examples from Theorem 5.1 satisfy lim 𝑝(𝑞) − 1.5𝑞 = −∞, in contrast to Theorem C from [Cre23], which showed that for rank-one subshifts, even total ergodicity implies lim sup 𝑝(𝑞) − 1.5𝑞 = ∞.…”

“…In [Cre23], it was also suggested that perhaps a subshift X having a nontrivial weakly mixing measure forces lim sup 𝑝 (𝑞) 𝑞 > 3 2 ; Theorem 5.1 answers this negatively. In fact, the examples from Theorem 5.1 satisfy lim 𝑝(𝑞) − 1.5𝑞 = −∞, in contrast to Theorem C from [Cre23], which showed that for rank-one subshifts, even total ergodicity implies lim sup 𝑝(𝑞) − 1.5𝑞 = ∞. The examples also satisfy lim inf 𝑝 (𝑞) 𝑞 = 1 and for any 𝑓 (𝑞) → ∞, there exist examples such that 𝑝(𝑞) < 𝑞 + 𝑓 (𝑞) infinitely often.…”

Measure-Theoretically Mixing Subshifts of Minimal Word Complexity

“…Ferenczi [Fer95] initially conjectured that mixing transformations' word complexity should be superpolynomial but quickly refuted this himself [Fer96] showing that the staircase transformation, proven mixing by Adams [Ada98], has quadratic word complexity. Recent joint work of the author and R. Pavlov and S. Rodock [CPR22] exhibited subshifts admitting mixing measures with word complexity functions which are subquadratic but superlinear by more than a logarithm. We exhibit subshifts admitting mixing measures with complexity arbitrarily close to linear: Theorem A.…”