Profinite approach to S-adic shift spaces I: Saturating directive sequences

TL;DR

The paper develops a profinite framework to study S-adic minimal shift spaces by introducing profinite images of directive sequences and the notion of saturating sequences, linking recognizability to algebraic saturation within free pro-$ extsf{V}$ semigroups. It shows that recognizability implies saturating and gives concrete sufficient conditions for recognizability, including pure encodings and recurrent encodings; it also ties finite alphabet rank to finite rank for Schützenberger groups and relates the group rank to the shift’s topological rank. The methodology blends symbolic dynamics with profinite semigroup theory, using profinite categories and kernel endomorphisms to control inverse limits and interactions of tails, with implications for Bratteli–Vershik representations and flow invariants. The results pave the way for a deeper, computable understanding of Schützenberger groups of minimal shift spaces and set the stage for the subsequent papers in the series.

Abstract

This paper is the first in a series of three, about (relatively)free profinite semigroups and S-adic representations of minimal shift spaces. We associate to each primitive S-adic directivesequence ${\boldsymbolσ}$ a $\textit{profinite image}$ in the free profinite semigroup over the alphabet of the induced minimal shift space. When this profinite image contains a $\mathcal{J}$-maximal maximal subgroup of the free profinite semigroup (which, up to isomorphism, is called the $\textit{Schützenberger group}$ of the shift space), we say that ${\boldsymbolσ}$ is $\textit{saturating}$. We show that if ${\boldsymbolσ}$ is recognizable, then it is saturating. Conversely, we use the notion of saturating sequence to obtain several sufficient conditions for ${\boldsymbolσ}$ to be recognizable: ${\boldsymbolσ}$ consists of pure encodings; or ${\boldsymbolσ}$ is eventually recognizable, saturating and consists of encodings; or ${\boldsymbolσ}$ is eventually recognizable, recurrent, bounded and consists of encodings. For the most part, we do not assume that ${\boldsymbolσ}$ has finite alphabet rank although we establish that this combinatorial property has important algebraic consequences, namely that the rank of the Schützenberger group is also finite, whose maximum possible value we also determine. We also show that for every minimal shift space of finite topological rank, the rank of its Schützenberger group is a lower bound of the topological rank.

Figures (1)

• Figure 1: Path $u$ of length $n$, seen as a composition of edges $u[i]$ from $q_{i+1}$ to $q_{i}$, for $0\leq i<n$

Theorems & Definitions (49)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Remark 2.7
• Remark 2.8
• Lemma 2.9