# Realizing ergodic properties in zero entropy subshifts

## TL;DR

This paper explores the relationship between block complexity and ergodic measures in subshifts, demonstrating new constructions that challenge previous assumptions about their correlation, especially near linear complexity.

## Contribution

It introduces novel minimal subshift constructions with near-linear complexity and uncountably many ergodic measures, and shows the existence of subshifts with slow entropy growth, revealing obstructions in applications.

## Key findings

- Subshifts with near-linear complexity can have uncountably many ergodic measures.
- Superlinear complexity subshifts cannot have uncountably many ergodic measures.
- Existence of minimal subshifts with ergodic measures exhibiting slow entropy growth.

## Abstract

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.