Orbit counting for sofic shift-flip systems

A sofic shift is a discrete dynamical system which consists of bi-infinite sequences of labels corresponding to paths in a labeled graph. If it is subjected to a certain automorphism called a flip, then it forms a sofic shift-flip system. The flip system is regarded as an action of infinite dihedral group on the sofic shift. The distribution of finite orbits under this action may indicate the complexity of the flip system. For this purpose, the prime orbit counting function is used to describe the growth of the finite orbits. In the literature, the asymptotic behavior of the counting function has been obtained for shift-flip systems of finite type (SFT-flip systems), which are a subclass of the sofic shift-flip systems. In this paper, we will prove a similar asymptotic result for a sofic shift-flip system. The proof relies on the construction of an underlying SFT-flip system to serve as a presentation of the sofic shift-flip system. The number of finite orbits in the said system is then estimated from the SFT-flip system via combinatorial calculations. Our finding here is complete since it is applicable to both irreducible and reducible sofic shifts.