On the number of fixed points of sofic flip systems

In the case when $X$ is a sofic shift and $\varphi :X\to X$ is a homeomorphism such that ${\varphi }^2={\text{id}}_X$ and $\varphi {\sigma }_X={\sigma }_X^{-1}\varphi$, the number of points in $X$ that are fixed by ${\sigma }_X^m$ and ${\sigma }_X^n\varphi$, $m=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, $n\in \mathbb{Z}$, is expressed in terms of a finite number of square matrices: The matrices are obtained from Krieger’s joint state chain of a sofic shift which is conjugate to $X$.