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- W3105819957 abstract "This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension S_A from a square matrix A over Z_+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over Z_+G. Parry asked in this case if the det(I-tA) (which captures the periodic data of the extension) would classify up to finitely many topological conjugacy classes the extensions by G of a fixed mixing shift of finite type. When the algebraic K-theory group NK_1(ZG) is nontrivial (e.g., for G=Z/4), we show the dynamical zeta function for any such extension is consistent with infinitely many topological conjugacy classes. Independent of NK_1(ZG): for every nontrivial abelian G we show there exists a shift of finite type with an infinite family of mixing nonconjugate G extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for G not necessarily abelian, and extend all the above results to the nonabelian case. There is other work on basic invariants. The constructions require the positive K-theory setting for positive equivalence of matrices over ZG[t]." @default.
- W3105819957 created "2020-11-23" @default.
- W3105819957 creator A5015520687 @default.
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- W3105819957 date "2016-02-11" @default.
- W3105819957 modified "2023-10-18" @default.
- W3105819957 title "Finite group extensions of shifts of finite type: -theory, Parry and Livšic" @default.
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- W3105819957 doi "https://doi.org/10.1017/etds.2015.87" @default.
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