A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but has uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure whose slow entropy grows slower than any given rate tending to infinitely but faster than any other rate majorizing this one yet still growing subexponentially. These constructions lead to obstructions in using subshifts in applications to properties of the prime numbers and in finding a measurable version of the complexity gap that arises for shifts of sublinear complexity.

中文翻译：

---

在零熵子移中实现遍历性质

具有线性块复杂度的子移位至多具有可数的多个遍历测度，我们继续研究这种复杂度与不变测度之间的关系。通过构造块复杂度任意接近线性但具有无数遍历测度的最小子位移，我们表明一旦块复杂度超线性，这种行为就会失败。使用不同的构造，我们表明存在具有遍历测度的最小子位移，其慢熵增长比任何给定速率都慢，趋向于无限但比任何其他速率快，但仍以次指数增长。