The Lind Zeta functions of reversal systems of finite order

TL;DR

This paper derives a decomposition theorem for Lind zeta functions of reversal systems of finite order, expressing them explicitly for shifts of finite type and sofic shifts using matrix representations.

Contribution

It introduces a decomposition theorem for Lind zeta functions of reversal systems and provides explicit formulas for shifts of finite type and sofic shifts.

Findings

Decomposition theorem for Lind zeta functions of reversal systems

Explicit matrix-based formulas for shifts of finite type and sofic shifts

Connection between group actions and Lind zeta functions

Abstract

A decomposition theorem for the Lind zeta function of a reversal system $(X, T, R)$ of finite order is established. A reversal system can be regarded as an action of a certain group $G$ on $X$. To establish an explicit formula for the Lind zeta function of $(X, T, R)$, we need to consider finite index subgroups $H$ of $G$ with induced actions given by automorphisms or by flips. When the underlying dynamical system $(X, T)$ is either a shift of finite type or a sofic shift, we express the Lind zeta function of $(X, T, R)$ in terms of matrices.