Finite group extensions of shifts of finite type: -theory, Parry and Livšic
MIKE BOYLE, SCOTT SCHMIEDING · Ergodic Theory and Dynamical Systems · 2016
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 over $\mathbb{Z}_{+}G$ , and classified the extensions up to topological conjugacy by the strong shift equivalence class of $A$ over $\mathbb{Z}_{+}G$ . Parry asked, in this case, if the dynamical zeta function $\det (I-tA)^{-1}$ (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 $\text{K}$ -theory group $\text{NK}_{1}(\mathbb{Z}G)$ is non-trivial (e.g. for $G=\mathbb{Z}/n$ with $n$ not square-free) 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 $\text{NK}_{1}(\mathbb{Z}G)$ , 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 $\mathbb{Z}G[t]$ .