Fixed points of endomorphisms of complex tori

TL;DR

This paper classifies the long-term behavior of fixed point counts of endomorphisms on complex tori, showing they grow exponentially, are periodic, or combine both behaviors.

Contribution

It provides a complete classification of the asymptotic behavior of fixed point counts for endomorphisms of complex tori.

Findings

Fixed point counts grow exponentially, are periodic, or combine both behaviors.

The classification covers all possible asymptotic behaviors.

The results deepen understanding of endomorphism dynamics on complex tori.

Abstract

We study the asymptotic behavior of the cardinality of the fixed point set of iterates of an endomorphism of a complex torus. We show that there are precisely three types of behavior of this function: it is either an exponentially growing function, a periodic function, or a product of both.