2025/08/31 by Almeida, Jorge, Costa, Alfredo, Goulet-Ouellet, Herman
#20M05 #20M07 (Primary) 20E08 (Secondary) #37B10 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2509.00991
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 profinite image in the free profinite semigroup over the alphabet of the induced minimal shift space. When this profinite image contains a J-maximal maximal subgroup of the free profinite semigroup (which, up to isomorphism, is called the Schützenberger group of the shift space), we say that \boldsymbolσ is 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.