The Artin-Mazur Zeta Function of a Dynamically Affine Rational Map in Positive Characteristic

TL;DR

This paper investigates whether the Artin-Mazur zeta function of certain algebraic dynamical systems, specifically dynamically affine maps on the projective line over fields of positive characteristic, is rational or transcendental.

Contribution

It provides a classification of the rationality or transcendence of the Artin-Mazur zeta function for dynamically affine maps in positive characteristic.

Findings

Determines conditions for rationality of the zeta function

Identifies cases where the zeta function is transcendental

Extends understanding of dynamical zeta functions in algebraic dynamics

Abstract

A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affine self-map of $\mathbb{P}^1(k)$ for $k$ an algebraically closed field of positive characteristic.