Abstract
We study a correspondence associating to each subshift of 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 that is, in the set of pseudowords whose finite factors are blocks of 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 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 with a pseudovariety of groups.
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