The Artin-Mazur zeta function for interval maps

TL;DR

This paper investigates the properties of the Artin-Mazur zeta function for piecewise monotone interval maps, revealing that known characterizations for unimodal maps do not extend to multimodal maps.

Contribution

It demonstrates that the rationality criteria for the Artin-Mazur zeta function in unimodal maps do not apply to multimodal maps, providing new insights into their dynamical complexity.

Findings

Rationality characterization fails for multimodal maps

Unimodal and multimodal maps exhibit different zeta function behaviors

Extends understanding of dynamical zeta functions in 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 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.