The Artin–Mazur zeta function for interval maps

Abstract

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 [On iterated maps of the interval, in Dynamical systems (College Park, MD, 1986–87), vol. 1342 of Lecture Notes in Math., pp. 465–563. Springer, Berlin, 1988] gave a characterization for the rationality of the Artin–Mazur zeta function in terms of the orbit of the unique turning point under certain smoothness assumptions. We give a characterization for unimodal maps that does not depend on the smoothness of the map, and implies the previous result. We also show that for multimodal maps, the previous characterization does not hold. In the space of real polynomials of a given degree which is bigger than two, with all critical points being real, and having fixed multiplicities (that is known to be a smooth real manifold), there are real-analytic subvariety of codimention 1 such that every map of this subvariety has the same Artin–Mazur zeta function, which is a rational function. Moreover, all but one critical points of this family undergo independent bifurcations.