A dynamical Shafarevich theorem for endomorphisms of $\mathbb{P}^N$

TL;DR

This paper establishes a dynamical version of the Shafarevich conjecture for endomorphisms of projective space over number fields, extending prior results from the case of one-dimensional projective space to higher dimensions.

Contribution

It introduces a dynamical analogue of the Shafarevich conjecture for morphisms on higher-dimensional projective spaces, generalizing previous work limited to the case N=1.

Findings

Proves a finiteness result for morphisms of degree d≥2 on projective N-space over number fields.

Extends Silverman's work from N=1 to higher dimensions.

Provides new insights into the arithmetic dynamics of endomorphisms.

Abstract

We prove a dynamical analogue of the Shafarevich conjecture for morphisms $f:\mathbb{P}_K^N\to\mathbb{P}_K^N$ of degree $d\ge2$, defined over a number field $K$. This extends previous work of Silverman and others in the case $N=1$.