  In this work 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, Milnor and Thurston gave a characterization for the rationality of the
Artin-Mazur zeta function in terms of the orbit of the unique turning point. We
show that for multimodal maps, the previous characterization does not hold.
