Profinite approach to S-adic shift spaces I: Saturating directive sequences
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Almeida, Jorge Costa, Alfredo Goulet-Ouellet, Herman |
| author_facet | Almeida, Jorge Costa, Alfredo Goulet-Ouellet, Herman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Profinite approach to S-adic shift spaces I: Saturating directive sequences Almeida, Jorge Costa, Alfredo Goulet-Ouellet, Herman Dynamical Systems Group Theory 37B10, 20M05, 20M07 (Primary) 20E08 (Secondary) 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. |
| title | Profinite approach to S-adic shift spaces I: Saturating directive sequences |
| topic | Dynamical Systems Group Theory 37B10, 20M05, 20M07 (Primary) 20E08 (Secondary) |
| url | https://arxiv.org/abs/2509.00991 |