# Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

## TL;DR

This paper studies invariant functions derived from multiplier matrices of periodic points in the moduli space of endomorphisms of projective space, exploring their properties, relations, and how well they determine conjugacy classes in higher dimensions.

## Contribution

It introduces new invariant functions on the moduli space of endomorphisms of projective space and analyzes their effectiveness in classifying conjugacy classes, extending McMullen's theorem to higher dimensions.

## Key findings

- Invariant functions are in the ring of regular functions of the moduli space.
- Methods for computing these invariants are developed.
- The invariants partially determine conjugacy classes, with specific families exhibiting isospectral properties.

## Abstract

There is a natural conjugation action on the set of endomorphism of $\P^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $\P^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant functions determine the conjugacy class in the moduli space. Several different types of isospectral families are constructed and a generalization of McMullen's theorem on the multiplier mapping of dimension 1 is proposed. Finally, this generalization is shown to hold when restricted to several specific families in $\mathcal{M}_d^N$.