Frequency shifts: non-soficity, absence of specification and intrinsic ergodicity

Abstract

For $p_c\in (0,1)$, the frequency shift $\Sigma (p_c)\subset \{0,1{\}}^{\mathbb{N}}$, introduced by Becker and Baraviera, is the closure of the shift orbits of the set of sequences in which a symbol $1$ is followed by $1$ exactly when the density of ones up to that position does not exceed $p_c$. We show that $\Sigma (p_c)$ is never sofic and never has the (weak) specification property. For $p_c=\frac{k}{k+1}$, with $k\ge 1$, we describe $\Sigma (p_c)$ completely: $\Sigma (\frac{1}{2})$ is the coded system generated by $\{0^m{1}^m{\}}_{m\ge 1}$, while for $k\ge 2$ the shift $\Sigma (\frac{k}{k+1})$ is not transitive, but every invariant probability measure is supported on the coded system generated by $\{0^m{1}^{km}{\}}_{m\ge 1}$. For every $k\ge 1$, the topological entropy of $\Sigma (\frac{k}{k+1})$ is $\frac{\log 2}{k+1}$, and, using the Climenhaga--Thompson criterion, we prove that $\Sigma (\frac{k}{k+1})$ is intrinsically ergodic.

arXiv comment15 pages