The zeta functions of renewal systems

Abstract

A renewal system is a shift space consisting of all bi-infinite free concatenations of some finitely many words. It is known that the zeta function of a renewal system generated by a circular set coincides with the generating function of the circular set. We extend this result to the systems generated by almost circular sets. Also we establish the formulas for the zeta functions of the systems generated by codes and pure codes, respectively. Using the formulas, we show that pure codes generating shifts of finite type are circular.