The Lind Zeta functions of reversal systems of finite order

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.