### 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 SA from a square matrix over ℤ_{+}G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over ZCG. Parry asked, in this case, if the dynamical zeta function det(I - t A)^{-1} (which captures the 'periodic data' of the extension) would classify the extensions by G of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic K-theory group NK_{1}(ℤG) is non-trivial (e.g. for G = ℤ/n with n not squarefree) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of NK_{1}(ℤG), for every non-trivial abelian G we show that there exists a shift of finite type with an infinite family of mixing non-conjugate 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 non-abelian case. There is other work on basic invariants. The constructions require the 'positive K-theory' setting for positive equivalence of matrices over ℤG[t].

Original language | English (US) |
---|---|

Pages (from-to) | 1026-1059 |

Number of pages | 34 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - Jun 1 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*37*(4), 1026-1059. https://doi.org/10.1017/etds.2015.87

}

*Ergodic Theory and Dynamical Systems*, vol. 37, no. 4, pp. 1026-1059. https://doi.org/10.1017/etds.2015.87

**Finite group extensions of shifts of finite type : K-theory, Parry and Livšic.** / BOYLE, MIKE; SCHMIEDING, SCOTT.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Finite group extensions of shifts of finite type

T2 - K-theory, Parry and Livšic

AU - BOYLE, MIKE

AU - SCHMIEDING, SCOTT

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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 SA from a square matrix over ℤ+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over ZCG. Parry asked, in this case, if the dynamical zeta function det(I - t A)-1 (which captures the 'periodic data' of the extension) would classify the extensions by G of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic K-theory group NK1(ℤG) is non-trivial (e.g. for G = ℤ/n with n not squarefree) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of NK1(ℤG), for every non-trivial abelian G we show that there exists a shift of finite type with an infinite family of mixing non-conjugate 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 non-abelian case. There is other work on basic invariants. The constructions require the 'positive K-theory' setting for positive equivalence of matrices over ℤG[t].

AB - 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 SA from a square matrix over ℤ+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over ZCG. Parry asked, in this case, if the dynamical zeta function det(I - t A)-1 (which captures the 'periodic data' of the extension) would classify the extensions by G of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic K-theory group NK1(ℤG) is non-trivial (e.g. for G = ℤ/n with n not squarefree) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of NK1(ℤG), for every non-trivial abelian G we show that there exists a shift of finite type with an infinite family of mixing non-conjugate 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 non-abelian case. There is other work on basic invariants. The constructions require the 'positive K-theory' setting for positive equivalence of matrices over ℤG[t].

UR - http://www.scopus.com/inward/record.url?scp=84957795749&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957795749&partnerID=8YFLogxK

U2 - 10.1017/etds.2015.87

DO - 10.1017/etds.2015.87

M3 - Article

VL - 37

SP - 1026

EP - 1059

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -