A note on the dynamical zeta function of general toral endomorphisms

TL;DR

This paper provides an elementary proof that the Artin-Mazur zeta function for general toral endomorphisms is rational and offers a closed-form formula based solely on integer arithmetic, extending known results for automorphisms.

Contribution

It extends the rationality result of the Artin-Mazur zeta function to all toral endomorphisms and provides a simple, computable formula.

Findings

The zeta function is rational for general toral endomorphisms.

A closed formula for the zeta function is derived using only integer arithmetic.

The paper discusses the functional equation and links between Artin-Mazur and Lefschetz zeta functions.

Abstract

It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions.