The Karoubi envelope of the mirage of a subshift

Abstract

We study a correspondence associating to each subshift $\mathcal{X}$ of $A^{\mathbb{Z}}$ a subcategory of the Karoubi envelope of the free profinite semigroup generated by A. The objects of this category are the idempotents in the mirage of $\mathcal{X},$ that is, in the set of pseudowords whose finite factors are blocks of $\mathcal{X}.$ The natural equivalence class of the category is shown to be invariant under flow equivalence. As a corollary of our proof, we deduce the flow invariance of the profinite group that Almeida associated to each irreducible subshift. We also show, in a functorial manner, that the isomorphism class of the category is invariant under conjugacy. Finally, we see that the zeta function of $\mathcal{X}$ is naturally encoded in the category. These results hold, with obvious translations, for relatively free profinite semigroups over many pseudovarieties, including all of the form $\overline{\mathsf{H}},$ with $\mathsf{H}$ a pseudovariety of groups.

Keywords:

• Free profinite semigroup
• Karoubi envelope
• pseudovariety
• subshift
• symbolic dynamics
• zeta function

2020 Mathematics Subject Classification:

• 20M07
• 37B10

Notes

1 For every Green relation $\mathcal{K}$ and every $\phi ,\psi \in C(c,d),$ one has $F(\phi )\mathcal{K}F(\psi )\Rightarrow \phi \mathcal{K}\psi$ if $F:C\to D$ is an equivalence. Here we are applying this property in the special case $\mathcal{K}=\mathcal{H}.$

2 We give [24, Section 2.3] as a reference for this property of $\mathcal{L}\mathsf{I}$ for the sake of better readability, but the property was known before: in the language of pseudowords, it is implicit in [1, Section 3.7], and in fact it amounts to the fact that $\mathcal{L}\mathsf{I}$ is the join of $\mathsf{D}$ and its dual $\mathsf{K},$ a fact already appearing in [25].

3 The arguments used in the proof of this implication are basically the same that were used in the proof of [11, Lemma 2.2], but there one finds the assumption that $\mathsf{V}$ contains $\mathsf{Sl}$ to guarantee that $\mathsf{V}$ does contain nontrivial monoids and therefore is according to the statement in [1, Theorem 10.6.12]. As seen in our recapitulation of those arguments, such assumption is unnecessary.

Additional information

Funding

The work of A. Costa was carried out in part at City College of New York, CUNY, whose hospitality is gratefully acknowledged, with the support of the FCT sabbatical scholarship SFRH/BSAB/150401/2019, and it was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES Fundação para a Ciência e a Tecnologia; Centro de Matemática, Universidade de Coimbra.