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Profinite approach to S-adic shift spaces I: Saturating directive sequences
Jorge Almeida, Alfredo Costa, Herman Goulet-Ouellet
https://arxiv.org/abs/2509.00991 https://

Profinite approach to S-adic shift spaces I: Saturating directive sequences
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 directive sequence ${\boldsymbolσ}$ 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 $\t…