# Fixed points and entropy of endomorphisms on simple abelian varieties

## TL;DR

This paper studies fixed points and entropy of endomorphisms on simple abelian varieties, providing criteria to classify their growth behavior and characterizing the algebraic nature of their entropy.

## Contribution

It introduces criteria based on endomorphism algebra types for fixed point growth and entropy classification on simple abelian varieties.

## Key findings

- Fixed points grow exponentially or periodically depending on eigenvalues.
- Criteria established for zero or positive entropy of endomorphisms.
- Characterization of the algebraic structure of entropy values.

## Abstract

In this paper we investigate fixed-point numbers and entropies of endomorphisms on abelian varieties. It was shown quite recently that the number of fixed-points of an iterated endomorphism on a simple complex torus is either periodic or grows exponentially. Criteria to decide whether a given endomorphism is of the one type or the other are still missing. Our first result provides such criteria for simple abelian varieties in terms of the possible types of endomorphism algebras. The number of fixed-points depends on the eigenvalues and we exactly show which analytic eigenvalues occur. This insight is also the starting point to ask for the entropy of an endomorphism. Our second result offers criteria for an endomorphism to be of zero or positive entropy. The entropy is computed as the logarithm of a real number and our third result characterizes the algebraic structure of this number.