The Karoubi envelope of the mirage of a subshift. Issue 11 (3rd November 2021)

• Record Type: Journal Article
• Authors: Costa, Alfredo; Steinberg, Benjamin
• Is Part Of: Communications in algebra. Volume 49:Issue 11(2021)
• Page Start: 4820
• Page End: 4856
• Publication Date: 2021-11-03
• Subjects: Free profinite semigroup -- Karoubi envelope -- pseudovariety -- subshift -- symbolic dynamics -- zeta function; 20M07 -- 37B10; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.1931263
• Languages: English
• ISSNs: 0092-7872

Abstract

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