Book excerpt: "In this thesis we study the Artin-Mazur zeta function for piecewise monotone functions acting on a compact interval of real numbers. In the case of unimodal maps, we prove that the zeta function is either rational or transcendental as a formal power series, and rationality or transcendence is characterized by the asymptotic behavior of the turning point. We use the theory of finite states automata for the proof. The rationality of the zeta function in this context was stated by Milnor and Thurston [Milnor and Thurston, 1988] without proof. Next, we show that for multimodal maps, the previous characterization does not hold."--Page ii.