# Periodic points of post-critically algebraic endomorphisms

## TL;DR

This paper investigates the eigenvalues of the differential of post-critically algebraic endomorphisms on complex projective spaces, showing they are either superattracting or repelling in certain cases, extending known results from dimension one.

## Contribution

It extends the classification of eigenvalues for these endomorphisms from dimension one to dimension two and higher, removing hyperbolicity assumptions in some cases.

## Key findings

- Eigenvalues are superattracting or repelling in dimension two.
- Eigenvalues outside the post-critical set are always repelling.
- Improves previous results by removing hyperbolicity constraints.

## Abstract

A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$, a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that when $n=2$ the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one which was already obtained by Fornaess and Sibony under a hyperbolicity assumption on the complement of the post-critical set.