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Finite group extensions of shifts of finite type: -theory, Parry and Livšic
- Source :
- Ergodic Theory and Dynamical Systems. 37:1026-1059
- Publication Year :
- 2016
- Publisher :
- Cambridge University Press (CUP), 2016.
-
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].<br />Comment: The purpose of this repost is the addition of the Appendix D: Corrections
- Subjects :
- Finite group
Pure mathematics
Group (mathematics)
Applied Mathematics
General Mathematics
010102 general mathematics
K-theory
01 natural sciences
Invariant theory
Riemann zeta function
010101 applied mathematics
symbols.namesake
Mathematics - K-Theory and Homology
37B10, 19M05
symbols
Mathematics - Dynamical Systems
0101 mathematics
Abelian group
Topological conjugacy
Finite set
Mathematics
Subjects
Details
- ISSN :
- 14694417 and 01433857
- Volume :
- 37
- Database :
- OpenAIRE
- Journal :
- Ergodic Theory and Dynamical Systems
- Accession number :
- edsair.doi.dedup.....ed469cad77771ccb8b0f7e21085b3d07