Finite group extensions of shifts of finite type

K-theory, Parry and Livšic

MIKE BOYLE, SCOTT SCHMIEDING

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 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].

Original languageEnglish (US)
Pages (from-to)1026-1059
Number of pages34
JournalErgodic Theory and Dynamical Systems
Volume37
Issue number4
DOIs
StatePublished - Jun 1 2017

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Shift of Finite Type
Group Extension
K-theory
Finite Group
Topological Conjugacy
Riemann zeta function
Equivalence classes
Group theory
Conjugacy class
Invariant
Algebraic K-theory
Finite Abelian Groups
Square matrix
Equivalence class
Fixed point
Classify
Equivalence

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

BOYLE, MIKE ; SCHMIEDING, SCOTT. / Finite group extensions of shifts of finite type : K-theory, Parry and Livšic. In: Ergodic Theory and Dynamical Systems. 2017 ; Vol. 37, No. 4. pp. 1026-1059.
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Finite group extensions of shifts of finite type : K-theory, Parry and Livšic. / BOYLE, MIKE; SCHMIEDING, SCOTT.

In: Ergodic Theory and Dynamical Systems, Vol. 37, No. 4, 01.06.2017, p. 1026-1059.

Research output: Contribution to journalArticle

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