On the Zeta Function of a Periodic-Finite-Type Shift

Periodic-finite-type shifts (PFT’s) are sofic shifts which forbid the appearance of finitely many pre-specified words in a periodic manner. The class of PFT’s strictly includes the class of shifts of finite type (SFT’s). The zeta function of a PFT is a generating function for the number of periodic sequences in the shift. For a general sofic shift, there exists a formula, attributed to Manning and Bowen, which computes the zeta function of the shift from certain auxiliary graphs constructed from a presentation of the shift. In this paper, we derive an interesting alternative formula computable from certain “wordbased graphs” constructed from the periodically-forbidden word description of the PFT. The advantages of our formula over the Manning-Bowen formula are discussed. key words: periodic-finite-type shift, zeta function, word-based graph, Möbius inversion formula