Nielsen periodic point theory on infra-nilmanifolds

TL;DR

This paper extends Nielsen periodic point theory to infra-nilmanifolds, providing methods to compute Nielsen-Jiang numbers and characterizing maps where these numbers match fixed point counts.

Contribution

It introduces structural properties of infra-nilmanifolds and develops a method to compute Nielsen-Jiang numbers, advancing the understanding of periodic points in these spaces.

Findings

Infra-nilmanifolds are essentially reducible to GCD and toral cases.

A method to compute the full Nielsen-Jiang number $NF_n(f)$ is developed.

Conditions are identified under which $NF_n(f)=N(f^n)$ for all $n$.

Abstract

In this paper, we expand certain aspects of Nielsen periodic point theory from tori and nilmanifolds to infra-nilmanifolds. We show that infra-nilmanifolds are essentially reducible to the GCD and essentially toral. With these structural properties in mind, we develop a method to compute the full Nielsen-Jiang number $NF_n(f)$. We also determine for which maps it holds that $NF_n(f)=N(f^n)$, for all $n$.