Nielsen zeta functions for maps on infra-nilmanifolds are rational

TL;DR

This paper proves that Nielsen dynamical zeta functions for maps on infra-nilmanifolds are always rational functions, by analyzing Nielsen numbers and Lefschetz numbers and their relationships across coverings.

Contribution

It establishes the rationality of Nielsen zeta functions for infra-nilmanifold maps, extending understanding of their dynamical properties.

Findings

Nielsen number equals Lefschetz number or a related expression.

Nielsen zeta function is always rational.

Relationship holds for all powers of the map.

Abstract

In this paper we will show that for any map $f$ on an infra-nilmanifold, the Nielsen number $N(f)$ of this map is either equal to $|L(f)|$, where $L(f)$ is the Lefschetz number of that map, or equal to the expression $|L(f)-L(f_+)|$, where $f_+$ is a lift of $f$ to a 2-fold covering of that infra-nilmanifold. By exploiting the exact nature of this relationship for all powers of $f$, we prove that the Nielsen dynamical zeta function for a map on an infra-nilmanifold is always a rational function.