Counting finite orbits for the flip systems of shifts of finite type

For a discrete system \begin{document}$ (X,T) $\end{document} , the flip system \begin{document}$ (X,T,F) $\end{document} can be regarded as the action of infinite dihedral group \begin{document}$ D_\infty $\end{document} on the space \begin{document}$ X $\end{document} . Under this action, \begin{document}$ X $\end{document} is partitioned into a set of orbits. We are interested in counting the finite orbits in this partition via the prime orbit counting function. In this paper, we prove the asymptotic behaviour of this counting function for the flip systems of shifts of finite type. The proof relies mostly on combinatorial calculations instead of the usual approach via zeta function. Here, we are able to obtain more precise asymptotic result for this \begin{document}$ D_\infty $\end{document} -action on shifts of finite type as compared to other group actions on systems available in the literature.